Electromagnetic Depth Soundings

ELECTROMAGNETIC DEPTH SOUNDINGS

ELECTROMAGNETIC DEPTH SOUNDINGS

L. L. Vanyan Moscow Geological Exploration Institute

and

L. Z. Bobrovnikov V. L. Loshenitzina V. M. Davidov G. M. Morozova

A. N. Kuznetsov A. I. Shtimmer E. I. Terekhin

Selected and Translated from Russian by George V. Keller

Colorado School of Mines Golden, Colorado

With an Introduction by the Translator

® CONSULTANTS BUREAU· NEW YORK· 1967

The material translated in this collection was taken from the folloWing sources:

Electrical Prospecting with the Transient Magnetic Field Method-Gosgeoltekhizdat, Moscow, 1963

Fundamentals of Electromagnetic Sounding-Nedra Press, Moscow, 1965

Concerning Some Causes for the Distortion of Transient Sounding CurvesPrikladnaya Geofizika No. 41, 1965

Concerning the F actors Distorting Frequency Sounding Curves-Razvedochnaya Geofizika No.7, 1965

Four-Layer Master Curves for Frequency EI ectromagneti c Sounding-Institute of Geology and Geophysics, Siberian Department of the Academy of the Sciences of the USSR and the All-Union Petroleum Geophysics Research Institute of the National Geological Committee of the USSR, Moscow, 1964

Library of Congress Catalog Card Number 67-19390

ISBN-13: 978-1-4684-0672-6 e-ISBN-13: 978-1-4684-0670-2 DOl: 10.1007/978-1-4684-0670-2

© 1967 Consultants Bureau Softcover reprint of the hardcover 1st edition 1967

A Division of Plenum Publishing Corporation 227 West 17 Street, New York, N. Y.100n

All rights reserved

No part of this publication may be reproduced in any form without written permission from the publisher

CONTENTS

Electromagnetic Sounding Methods - Introduction and History George V. Keller. . . . . . . . . . . . . . . . . . . . ............ .

Introduction . . . . . . . . . . . . . . . . . . . . . History of Electromagnetic Methods .....

Electrical Prospecting with the Transient Magnetic Field Method L. L. Vanyan and L. Z. Bobrovnikov ..................... .

1

1 8

19

Introduction . . . . . . . . . . • . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . 19

Part I. Physical and Mathematical Foundation for the Transient Magnetic Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . • 20

1. Time-Domain and Frequency-Domain Methods in the Theory for Transient Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2. Primary Magnetic Field of a Dipole. . . . . . . . . . . . . . . . . . . . . . . . . . 22 3. Use of the Principle of Reciprocity for Determining Transient Magnetic

Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Part n. Calculation and Analysis of Theoretical Curves . . . . . . . . . . . . . . . . 27 1. Numerical Evaluation of Transient Magnetic Fields . . . . . . . . . . . . . . • 27 2. Asymptotic Behavior of the Vertical Component of the Transient Magnetic

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3. Apparent Resistivity in the Transient Magnetic Field Method. . • . • • • . . 31 4. Computation of Wave-Limit Curves for Transient Magnetic Fields. . .. . • 33 5. Computation of Theoretical Curves for the Late Stage of the Transient

Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6. Master Curves for Transient Magnetic Fields. . . . . . . . . . . . . . . . . . . 45 7. Construction of Curves for Transient Magnetic Fields in the Far Zone. . . 46 8. Analysis of Theoretical Curves for Transient Magnetic Fields. . . . . . . . 47 9. Maximum Resolution for Transient Magnetic Fields . . . . . . . . . . . . . . . 50

10. Equivalent Curves for Transient Magnetic Fields. . . . . . . . . . . . . . . . . 51

Part m. Equipment.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1. Recording Transient Magnetic Fields. . . . . . . . . . . . . . . . . . . . . . . . . 52 2. Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3. Block Diagram for Equipment Used in the Magnetic Transient Method. . . 54 4. Generating Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . 54 5. Receiving Equipment. . . . . . • . . . . . • . . . . . . . • . . . . . . • . . . . . . . 55 6: Particular Methods for Conversion . . . . . . . . . . • . . . . . . . . . . . . . . . 64

v

vi CONTENTS

7. Alignment of an Amplifier ............................... ; 67 8. Equipment for Generating Synchronous Time Marks. . . . . . . . . . • . . . . 68 9. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Part IV. Field Methods and Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . • 69 1. Field Methods . . . . . • . . . . . . . . . . . . . . . . . . . . . . . • . . . . • . . . . . 69 2. Construction of Apparent Resistivity Curves. . . . . . . . . . . . . . . . . . . . 70 3. Topographic and Survey Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4. Reduction and Presentation of Survey Results . . . . . . . . . • . . . . . . . . . 71 5. Logistic Considerations. . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . 72 6. Basic Interpretation of the Final Stage of the Magnetic Transient • . . . . . 72 7. Use of Electric Log Data. . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . 75 8. Distortions of Magnetic Transient Curves. . . . . . . . . . . . . . . . . . . . . . 75 9. Possible Uses for the Magnetic Transient Method in Studying Structural

Geology. . . . . . . . . . . . . . . . • . . . . . . . • . . . . . . . . . . . . . . . . . . . . 75

Appendix 1. Table of Hyperbolic and Inverse Hyperbolic Functions of Complex Arguments for Computing Wave Curves for a Transient Field . . . . . . . . . . . . . 80

Appendix 2. Typical Three-Layer and Four-Layer Wave Curves for Transient Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . • . . . . . . . . . . . . . . . . 113

Appendix 3. Nomogram for Determining the Correction for Finite Length of Source Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Appendix 4. Forms for a Field Log ............• 120

Fundamentals of Electromagnetic Sounding L. L. Vanyan ......•............... 125

Part I. Geological Basis for Electromagnetic Sounding . . . . . . . . . . . . . . 125 Field Sources and Models of the Geoelectric Section . . . . . . . . . . . . . . . • . 127 Maxwell's Equations and the Vector Potential. . . . • . . . • . . . . • . . . • . . . . 128 Vector Potential in a Homogeneous Medium. . . • . . . . . . . . . . . . . . . . . . . 132 The Electromagnetic Field in a Layered Anisotropic Medium . . . .. . . . . . . 133 Electromagnetic Fields at the Surface of a Uniform Anisotropic Half-Space . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . • 142 Calculation of the Quasistatic Electromagnetic Field at the Surfaces of a Layered Anisotropic Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Part TI. Principles of Quasistatic Electromagnetic Sounding. . . . . . . . . . . . . 162 Determining the Resistivity of a Homogeneous Anisotropic Half-Space . . . . 162 Two PrinCiples of Electromagnetic Sounding . . . . . . . . . . . . . . . . . . . .. 167 Induction Sounding. . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . 169 Geometric Soundings • . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Equivalence for Thin Layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Electromagnetic Sounding in the Presence of an Insulating Screening Layer. . 190 Two Forms of Anisotropy and Their Effect on Electromagnetic Sounding. . . . 193

Conclusions . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . 198

Concerning Some Causes for the Distortion of Transient Sounding Curves L. L. Vanyan, V. M. Davidov, and E. 1. Terekhin . . . . . . . . . . . . . . . . . . . . 201

Measuring Array Located on a Sloping Surface. . . . . . . . . . • . . . . . . . . . . 201 Location of Source and Receiver at Different Heights . . . . . . . . . . . . • . . . 204

CONTENTS

Insulating Hemisphere near the Observation Point ...... .

Concerning the Factors Distorting Frequency Sounding Curves A. N. Kuznetsov. . . . . . . . . . . . . . . . . .............. .

Four-Layer Master Curves for Frequency Electromagnetic Sounding L. L. Vanyan, G. M. Morozova, V. L. Loshenitzina, E. I. Terekhin,

and A. I. Shtimmer .......•.....•...............•..

vii

205

209

217

ELECTROMAGNETIC SOUNDING METHODS-INTRODUCTION AND HISTORY

George V. Keller

INTRODUCTION

In exploration geophysics, a distinction is usually made between electrical methods in which direct current is used and those in which alternating currents are used. In the first case, the theory is developed entirely on the basis of the conservation of current, which leads ultimately to the solution of Laplace's equation in computing the resistivity of the earth from field data. In the AC methods for measuring earth conductivity, a time-varying magnetic field is generated by driving an alternating current through a loop of wire, or !through a straight length of wire grounded at both ends. If conductive material is present within the magnetic field so generated, induced or !!eddy" currents will flow in closed loops along paths normal to the direction of the magnetic field, in accordance with Ampere's law. In analysis, these conditions reduce to Maxwell's equations rather than to Laplace's equation, and the use of Maxwell's equation in computing earth conductivity constitutes the electromagnetic method of geophysical prospecting.

As with many other geophysical methods, the electromagnetic methods may be divided into two groups, one containing techniques for studying variations of conductivity with depth (depth-sounding methods) and the other containing techniques for studying lateral changes in conductivity (horizontal profiling methods). We will consider only the various depth-sounding methods in this collection of translations.

In measuring earth conductivity, one must first generate an electromagnetic field and then measure or detect the distortion in this field caused by the presence of a conductive earth. This may be done in many ways, and the variety of ways of using an electromagnetic field in studying earth conductivity has actually been a disadvantage in the application of the methods.

The three common controlled sources for electromagnetic fields used in geophysical exploration are loops of wire, short grounded lengths of wire, and long grounded lengths of wire. A current flowing in a small loop of wire generates a magnetic field which cannot be distinguished from that caused by a dipole magnet, if it is observed at moderate distances (a moderate distance being greater than about five times the diameter of the loop). The magnetic field generated by such a current-carrying loop has a strength equivalent to a dipole magnet with a moment equal to the product of the number of turns of wire in the loop, the area of the loop, and the current flowing in the wire. If the current is oscillatory, such a source is called

1

2 ELECTROMAGNETIC SOUNDING METHODS

a "harmonic," or "oscillating magnetic dipole" source. A steady current provides a magnetic field which is constant in time. An abrupt termination or initiation of current flow in such a loop leads to a transient magnetic field.

A loop may be oriented arbitrarily with respect to the surface of the earth, but generally, the plane of the loop is placed either parallel to the surface of the earth, in which case it is called a vertical magnetic dipole, or perpendicular to the surface of the earth, in which case it is called a horizontal magnetic dipole. The axis of the equivalent magnetic dipole coincides with the loop axis.

The magnetic field from a loop source, if it is time -varying, will induce currents in any conductor it cuts, and these currents in turn cause electric fields according to Faraday's law.

For a vertical magnetic dipole source located at the earth's surface, there are only three electromagnetic field components which may be observed at the surface of a uniform earth: a vertical component of the magnetic field, Hz, a radial component of the magnetic field, Hr , and a tangential component of the electric field, Ecp. The magnitudes of the three components for a homogeneous earth are given by fairly complicated expressions, as follows (Wait, 1951, 1955):

a. Vertical magnetic field

Hz = 2:2,-< {9 - [9 + 9yr + 4(yr)2 + (yr)3] e-yr } ;

b. Radial magnetic field

c . Tangential electric field

E¢ = ~ { 3 - [3 + yr + (yr)2] e-yr } . 21TOr

(1)

(2)

(3)

In these expressions, M is the moment of the dipole source, given by the product nAI (n being the number of turns in the source loop, A the area and I the current), r is the distance from the source at which the field component is being observed, and 'Y = (iO"J..Low)V2 is the propagation constant (in terms of radian frequency, w, magnetic permeability, Jl. 0, and conductivity, 0"). The symbols 10,110 K o, and K1 indicate modified Bessel functions of the argument 'Y r/2.

It is apparent that none of these three equations may be solved for conductivity in terms of the other quantities, which are all measurable. This leads to a greater difficulty in determining conductivity from electromagnetic data than is met in determining resistivity from direct-current data. One might determine conductivity by a trial and error solution of these expressions, or graphically from curves for these three equations, such as are shown in Fig. 1. The ordinate of a point on one of these curves may be found from measured values for r, H, and M, and then the abscissa for that point can be used to find the conductivity of the earth. This procedure does not always provide a unique answer, inasmuch as on some of the curves, the same ordinate may provide two different values for the abscissa. Because of this ambiguity, the method is not used in determining conductivity.

A method which 1S used to some extent in practice is curve matching. A series of values for one of the field components is measured at a variety of frequencies, and a curve is plotted on .bilogarithmic graph paper to the same scale as the theoretical curves. For a uniform

INTRODUCTION

10.------------------------, 10.------------------------.

N ::IE

~ w ~

'" "'0.1

\~

0 Mz

SIDE VIEW PLAN VIEW 0.01 LI --'---'---LLL-'---LLLlI ~-L---"---LLL..ll..L.LJ'-':';;'

100 0.01 1 10 100 fr

Fig. 1. The behavior of the magnetic and electric fields from a vertical magnetic dipole source at the surface of the earth (after Wait [1951,1955]).

3

earth, the experimental curve so plotted should have the same shape as a portion of the appropriate theoretical curve, but with the ordinates and abscissas shifted by an amount dependent on the resistivity. The conductivity can be determined from the amount of shift between the field data and the theoretical curve. A detailed description of the curve matching procedure is found in a text by Keller and Frischknecht (1966).

The curve matching procedure, though used, has several disadvantages. Measurements must be made over a diagnostic portion of the theoretical curve, one in which there is some curvature, so that the amount of shift required to make a match can be determined uniquely. This means that the approximate conductivity of the earth must be known when measurements are made. Secondly, determination of a single value of conductivity requires measurements made over a wide range of frequencies. This would appear to be wasteful of data, inasmuch as the equations indicate that a single measurement at a single frequency should be enough to determine conductivity.

A third approach to the determination of conductivity is based on the amplification of equations (1-3) which is found for large values of the product 'Y r (large values being those greater than approximately 5). In this case, the equations reduce to forms which can be solved for conductivity:

a. Vertical magnetic field

(4) b. Radial magnetic field

a = 172r8H/116iwJLoM2; (5)

c. Tangential electric field

a = 3M/217r4E¢. (6)


The assumption that the product yr is large is equivalent to saying that the distance from the source, r, is larger than a wavelength in the earth. The significance of this assumption is discussed in detail in an accompanying paper by Vanyan (1965). If this assumption can be made, a value for conductivity can be computed from a single measurement, whether or not the earth is uniform. If the earth is not uniform, this computed vallIe for conductivity is termed an apparent conductivity, as is done in the case of direct-current resistivity measurements.

With a horizontal dipole source, five of the six orthogonal field components may be observed over a uniform earth. Only the vertical component of the electric field is not observed. According to Wait and Campbell (1953b), the five observed components are:

a. Radial magnetic field (along the axis of the source dipole)

H = __ _ e_[12+12yr+5(yr)2+(yrP]+2- p )· ; M{-yr 12} r 217r3 (yr)2 yr (7)

b. Transverse magnetic field (at right angles to the axis of the source dipole)

H¢ = -- -- [3 + 3yr + (yr)2] + 1- -- ; M {e-yr 3 } 217r3 (yr)2 (yr)2 (8)

c. Tangential electric field for large yr

(9)

d. Radial electric field for large yr

(10)

e. Vertical magnetic field

(11)

Note that the expression for the vertical magnetic field from a horizontal loop is the same as the expression for the radial magnetic field from a vertical magnetic dipole, as follows from reciprocity. The behavior of all five magnetic and electric field components from a horizontal magnetic dipole source is shown in Fig. 2. The expressions for the two electric field components cannot be given in simple form for small values of yr.

As was the case with a vertical magnetic dipole source, the expressions cannot be solved in general for the conductivity of a homogeneous earth. However, for large values of yr, three of the five expressions reduce to forms which provide explicit solutions for earth conductivity in terms of measurable, quantities, and these expressions can be used to define apparent conductivity for a given source orientation and measured field component (the radial and transverse magnetic field components become independent of earth conductivity for large values of yr):

a. Tangential electric field

u - ",w~ :J -",wM'N~g.; (12)

0.01 1

?~ ~

MH PLAN VIEW

10

'fr

Hr cos e

H sin e

INTRODUCTION

10.-----------------------~

100

" "'0.1

0.01 1 10

rr

E cos 9

Er sin 9

100

Fig.2. The behavior of the magnetic and electric field components from a horizontal magnetic dipole source at the earth's surface (after Wait and Campbell [1953a] and Wait [1961]).

b. Radial electric field

c. Vertical magnetic field

5

(13)

(14)

A grounded wire may serve as the source of an electromagnetic field as well as a currentcarrying loop. In this case, if the length of the grounded wire is short compared to the distance at which the field is observed, the source may be termed a current dipole or an electric dipole. If observations are made at distances greater than about ten times the wire length, it is found that terms which contain the amount of current to the ground or the wire length separately become negiligible, and only terms containing the product of wire length and current are significant. This product, Ids, is called the dipole moment.

With a horizontal current dipole, all six components of the electromagnetic field may be observed at the surface of a homogeneous earth. According to Wait (1961):

a. Radial electric field

Ids -yr Er =--[1 + (1 + yr)e ] cos 0; 21TUT' (15)

b. Tangential electric field

E Ids [ ) -yT] . ¢ =----, 2 - (1 + yT e smO;

21TUT (16)



d. Vertical electric field for large yr

e. Radial magnetic field for large yr

H Ids. r",-sm e;

1T}'r 3

f. Tangential magnetic field for large yr

Ids H¢ '" --- cos e.

21T}'r3

(17)

(18)

(19)

(20)

As in the case of magnetic dipole sources, the general expressions for the field components cannot be solved explicitly for conductivity. Approximations valid for large values of yr must be used in order to obtain expressions for conductivity:

a. Radial electric field a =qds cos e)/(zTTr'Er) (21)

b. Tangential electric field

a =(Ids sin e)!(TTr'E¢); (22)


a =(3Ids sin !y(2TTir4w/LoHe); (23)

d. Vertical electric field

a =VwlJ.oIds cos e)/(2TTr3Ez); (24)

e. Radial magnetic field

a =EIds)2 sin2e]/[TT2iwfLc"'H~); (25).

f. Tangential magnetic field

a =EIds)2 cos2~/[4TT2iwfLcr6H¢J. (26)

The expressions for all components depend on the azimuth angle, e , which is the angle between the axis of the dipole source and the radius vector to the observation point. Curves showing the behavior of the fields about a horizontal current dipole are shown in Fig. 3.

A third idealized type of source for an electromagnetic field which is used in geophysical exploration is a long grounded wire. Field components are measured close enough to the wire so that it may be considered to be infinitely long. Only two components of the electromagnetic field from a long wire may be observed at the surface of a uniform earth - the parallel component of the electric field and the vertical component of the magnetic field. The equations for each are (Kraev, 1965):

a. Parallel electric field (27)

10.,.-----------r------,

Er COS"

0

EZ 'k ~ / .!:' ;; • NO.1

IhdS

PLAN VIEW

0.01 0.1 10 100

INTRODUCTION

b. Vertical magnetic field

H = 2IyK,<yr}, (28)

where Ko and K1 are Hankel functions of the second kind of order 0 and 1, and I is the amplitude of the current flowing in the wire at a frequency w. These expressions cannot be solved for conductivity even for large values of yr individually, but the conductivity may bEil determined from the ratio of magnetic and electric field components measured at large yr:

a=/low(H\2 4i \Ej (29)

7

Ifr

Fig. 3. The behavior of the electric field components from a horizontal electric dipole source at the earth's surface (after Wait [1961]).

In recent years, a fourth source of energy has come into use for making electromagnetic depth soundings - the natural electromagnetic energy contained in rapid variations of the earth's magnetic field. When such energy is considered to be a plane wave traveling downward into the earth, the conductivity of the earth, if it is homogeneous, can be computed from the ratio

of magnetic field strength to electric field strength (Cagniard, 1953):

A number of cases for fields with more complicated geometry than plane-wave fields have been treated in the geophysical literature, and these have been reviewed in detail in a recent text by Rikitake (1966).

(30)

In measuring the electric field in the earth, normally a short grounded wire is used. Magnetic field components may be measured with a magnetometer, though this is rarely done except in the case of the magnetotelluric method. More commonly, the magnetic field components are measured with induction coils, which detect the time-rate of change of the magnetic induction:

dB EMF = - nA-.

dt (31)

If the source is harmonic - that is, if the current to the source is a sinusoid at a specific frequency, w - in the steady state, the derivative may be replaced by a multiplying term, iw:

(32)

Thus, a voltage is measured, rather than a magnetic field component. In all cases, this voltage is proportional to the strength of the source, or the current in the source, and must be normalized for this strength. This had led to the use of the ratio of received voltage to source current, known as the mutual impedance"Z, between the source and the receiver. In many cases, an electromagnetic source will provide a measurable field at the receiver in the absence of a conductive earth, as well as in its presence. For example, a vertical component of magnetic field intensity will be observed in the plane of a vertical magnetic dipole source, even in the absence of an earth, and this is termed the primary field. The difference between such a primary field and the observed field when a conductive earth is present is called the sec-0ndary field. The mutual impedance which would be computed for the primary field is called the free-space mutual impedance, Zo, and quite commonly, field data and theoretical curves are


given in terms of the mutual impedance ratio, Z/Zo, rather than in terms of the apparent conducti vity, a a> defined in the equations above.

It is apparent that there are a great variety of techniques which might be used in electromagnetic sounding. Four types of source have been considered, and with various source-recei ver component combinations, 16 different field techniques could be used in electromagnetic depth sounding.

Depth soundings may be made .either by varying the spacing between the source and the recei ver, or by varying the frequency content of the source current. The first is termed a geometric sounding, and the second a parametric sounding. There are operating advantages to both approaches, and both are used in practice. However, control of frequency is used more commonly than variation of source -receiver separation.

With a fixed separation, measurements may be made either in the frequency domain (one frequency at a time, through a range of frequencies) or in the time domain (use of transients containing a wide spectrum of frequencies). Although it is readily shown that time-domain measurements and frequency-domain measurements are uni"quely related through the Fourier transform, the operating procedures and interpretation involved in the two approaches are quite different.

As a result, there are 45 variants which might be used in the controlled source methods, plus the magnetotelluric method, making a total of 46. Each of the 46 methods requires somewhat different instrumentation and quite different interpretation procedures and theoretical curves. Commonly, in the literature, a single method is considered at a time so that comparison between methods is difficult. The variety of methods has led to confusion in understanding the basic principles of depth sounding, and so, the two translations which accompany this introduction (Vanyan and Bobrovnikov, 1963; Vanyan, 1965) are most valuable in that they present a unified approach to electromagnetic depth soundings which has been absent in the past.

HISTORY OF ELECTROMAGNETIC METHODS

Electromagnetic methods have been used in geophysical exploration nearly as long as the direct-current methods, even though the theory has been less well understood. Until recently, the principal application has been in mining geophysics, in the search for conductive ore bodies. Only within the past decade has the theory been advanced to the point where interpretations of layered earth structures such as are of interest in petroleum exploration and engineering geology can be made.

The earliest description of a practical electromagnetic prospecting method appears to have been a patent disclosure (German patent 322,040, issued in 1913) by K. Schilowsky, who described a loop transmitter operating at 1 to 50 Kc and induction coil receivers. In 1917, H. Conklin obtained a patent (U.S. 1,211 ,197) on an inductive method in which a large loop was laid out, energized by an alternating current and the resultant electromagnetic field within the loop investigated. At about the same time, electromagnetic methods were first being used in Sweden by Lundberg, Nathorst, and Bergholm. A number of ore bodies were discovered within a few years, leading to wide application of simple electromagnetic prospecting methods outside of Sweden, as well as in Sweden, during the 1920's. The application of the method was pragmatic, in that variations in behavior of an electromagnetic field associated with conductive ore bodies were sought with no attempt being made to determine the actual conductivity of the ground.

mSTORY OF ELECTROMAGNETIC METHODS

In 1926, Ambronn (1926) in an early text on geophysics was able to cite some 400 references on electrical and electromagnetic prospecting methods. During the following decade Rust (1938) in a review on the application of electrical prospecting reported that an average of 100 papers a year on electromagnetic methods were being published.

9

The first approach to application of electromagnetic methods for studying a layered earth appears to have been the "Eltran method," based on a patent by L. W. Blau (U.S. patent 1,911,137, issued in 1933). The Eltran method consisted in the generation of an electromagnetic field with a current dipole excited with a current pulse, and detected with an electric dipole situated in line with the source dipole. It was hoped that energy reflected from boundaries between layers with different conductivities would be detected on the recorded transient at the receiver in much the same way that acoustic reflections were detected in the seismic reflection techniques. The method aroused considerable interest among oil companies for about 10 years, with a series of papers appearing which described results of field trials (see Karcher and McDermott, 1935; Statham, 1936; West, 1938; Hawley, 1938; White, 1939; Klipsch, 1939; Rust, 1940; and Evjen, 1948). With all this experimentation, there was remarkably little theoretical consideration of the method reported in the literature. A careful theoretical evaluation of the Eltran method was not reported until the work of the Socony Mobil laboratory was published (Yost, 1952; Yost and others, 1952; Orsinger and Van Nostrand, 1954). It was then apparent that for the conductive rocks normally found in sedimentary basins, the transient response to impulse excitation contained such low frequencies that it would be difficult to obtain the resolution needed to identify individual reflected events. This early disillusionment with the Eltran method has resulted in the electrical methods of exploration being used to a far lesser degree in petroleum exploration in the United States than in other countries.

The avoidance of theory for electromagnetic sounding methods among exploration geophysicists is puzzling. Excellent work on the theory of induction fields about current-carrying wires was being done at the Bell Telephone Laboratory and at other industrial laboratories even during the early days of geophysical application of electromagnetic methods. This work has been summarized in a book by Sunde (1949). A single paper describing the theoretical development of Maxwell's equations for the induction field (Peters and Bardeen, 1932) appeared in the early geophysical literature, but it apparently had little impact on field applications. Other papers began to appear later, such as those by West (1943), Wolf (1946), Haycock et al. (1949), and Belluigi (1949, 1950). The detailed development of theory for various types of sources over a uniform earth and in a simple layered earth followed shortly thereafter in a large number of papers by Wait (1951a, 1952, 1953b, c, d, 1954b, 1955, 1956a, 1958, 1961, 1962b, c, d), by Wait and Campbell (1953a,b), and by Bhattacharyya (1955,1963). These theoretical developments, though scattered widely in the literature, provide the basis for the current application of electromagnetic sounding methods. Recent papers on the theory of electromagnetic field behavior have been those by Negi (1961), Loeb (1959), Bodvarsson (1966), Bannister (1966), and Atzinger et al. (1966). Wait's development is used in several recent texts which describe electromagnetic sounding theory (Grant and West, 1965; Keller and Frischknecht, 1966). Extensive numerical tables for use in compiling theoretical curves for various types of electromagnetic soundings have recently been published (Frischknecht, 1967). These tables are compiled for sources on or above (as in airborne electromagnetic surveying) an earth consisting of one or two layers. They are complementary to the curves by Vanyan et al. which are included in this collection, inasmuch as Frischknecht's computations apply mainly to source -receiver separations which are not large compared to layer thickness, and those by Vanyan et al. apply to source -recei ver separations which are large compared to layer thickness. The ranges of parameters used in the two collections are given below.


Vanyan et al. (loop-wire geometry)

First-layer thickness 0 0 0 Separation : O. 1 to . 2

Second-layer resistivity. 1/8 1/2 2 00 First-layer resistivity . , "

Third-layer resistivity. 1/16 1/4 1 4 First-layer resistivity' , "

Fourth layer is infinitely resistant

Second-layer thickness. 1/2 2 8 First-layer thickness' "

Third-layer thickness. 1/2 2 8 First-layer thickness' "

Frischknecht (loop-loop and loop-wire)

Height . Separation'

1/32,1/16,3/32,1/8,3/16,1/4, 1/2,1,1.25,1.50,1.75,2,2.25, 2.5,3

First-layer thickness. / / / / -----=~--~------. 1 32,1 16,1 8,3 16,

Separation 1/4,3/8,1/2,3/4,1, 1.5,2,00

Second-layer resistivity. 0 0 03 0 1 0 3 3 10 First-layer resistivity' 30 ~OO' 3~~ :" , , , , ,

Utilization of electrical methods in the Soviet Union appears to have developed largely independently of the work outlined above. Early work appears to have been carried out largely with radiated fields at high frequencies. In 1923, the Institute of Applied Geophysics in Leningrad undertook a program of studies using radiation fields under the leadership of the late A. Petrowsky. In many bibliographic compilations, the name fo-r such radiation field measurements is translated as "ondometric," a term not ordinarily used in English.

These methods, while still in wide use in the Soviet Union, did not lead to the development of electromagnetic sounding methods using the induction field. Rather, such development appears to be based on early theoretical work by Fok (1926), Bursian (1936), and Kraev (1941). Following this early work, there have been a large number of papers on induction field behavior over a uniform or stratified earth published in the Russian literature. These include papers by Tikhonov and Shakhsuvarov (1956), Gillfand (1955a, b, 1965), Molochnov (1955), Pavinskii and Kozulin (1956), Kozulin (1956, 1960), Gasanenko (1959a, b, 1965), Gasanenko and Molochnov (1958), Gasanenko and Federov (1964), She inman and Frantov (1958), Velikin, Frantov, and Sheinman (1961), Jogolev, Trifonov, and Shakhsuvarov (1962), Chetaev (1962a, b, 1963, 1966a, b), Dmitriev (1965), Kuznetsov (1965), and Shakhsuvarov and Evereva (1966). The fields about a long wire have been studied by Frantov (1963, 1966). Vanyan (1957,1958,1960,1962, 1963a; Vanyan, Kaufman, and Terekhin, 1961) has published numerous papers which reduce the rather complex theory for electromagnetic field behavior to forms useable in exploration.

HISTORY OF ELECTROMAGNETIC METHODS 11

All of this theoretical development, both in this country and in the Soviet Union, is for a harmonic source. The assumption of a harmonic source allows a simple reduction of time derivatives in Maxwell's equations, but limits the solution to steady state conditions. Transient electromagnetic fields may be studied either by solving the original differential equations for special solutions or by applying Fourier transform theory to the results of a harmonic solution. Special solutions to Maxwell's equations have been used in the few papers which have appeared in the domestic geophysical literature (Bhattacharyya, 1957a, b, 1959; Wait, 1951a, 1956b,1960). The Russian literature on transient behavior has been more extensive, starting with papers by Kraev (1937), Tikhonov (1946, 1950), Tikhonov and Mukhina (1950), and Skugarevskaya (1951a, b). This early work on transient coupling between a source and receiver was considerably expanded during the late 1950's and early 1960's, as reported in papers by Tikhonov and Skugarevskaya (1957, 1958, 1959), Chetaev (1956), Tikhonov, Skugarevskaya, and Frolov (1965), Fol'd (1963), Frolov (1963, 1965), Kovtun and Novoselova (1960), Koroleva, Nikitina, and Skugarevskaya (1965), and Koroleva and Skugarevskaya (1962,1965). Tables of values for transient field behavior in soundings have been published by Tikhonov, Skugarevskaya, and Frolov (1963). The application of transient electromagnetic field behavior in soundings has been based largely on recent work by Vanyan (Vanyan, 1960; 1963b, c, 1964; Vanyan and Morozova, 1962; and Vanyan, Terekhin, and Shtimmer, 1965), in which the Fourier transform of frequency computations is used rather than transient solutions to the differential equations •

The bulk of the literature on electromagnetic sounding methods has been devoted to the theory, indicating the complexity of the problem. Relatively few papers have appeared describing the application of the methods or interpretation. Two papers on the interpretation of loop source -loop receiver measurements have appeared in English (Keller and Frischknecht, 1960; Frischknecht and Ekren, 1961). Half a dozen papers have appeared in the Russian literature on the interpretation of data obtained with a current dipole -loop receiver or with a current dipole -electric field receiver system (Enenshtein, 1957, 1962; Shakhs uvarov , 1956; Korol'kov, 1965; Kuznetsov, 1965; Davidov and Butkovskaya, 1965; and Ivanov, Nikitina, and Skugarevskaya, 1966).

A great number of papers has been published on the use of natural electromagnetic fields (the magnetotelluric method) for making soundings. The method appears to have been reported in early papers by Tikhonov (1950) and Cagniard (1953). Literature up to about 1960 has been summarized in an earlier translation (Berdichevskii, 1965). The theory for magnetotelluric methods based on plane-wave electromagnetic field behavior is of value in considering other types of controlled-source electromagnetic methods inasmuch as the plane-wave impedance can be converted to mutual coupling by a Fourier - Bessel transform (Wait, 1962b). The theory for plane-wave impedance is discussed in many papers, including those by Chetaev (1960), Bossy and Devuyst (1960), Tikhonov and Shakhsuvarov (1956), Wait (1954a), Kolmakov and Vladimirov (1961), and Sheinman (1958). Tables of plane-wave impedances (or magnetotelluric resistivities) have been published by several investigators (Jackson, Wait, and Walters, 1962; and Yungul, 1961).

The assumption of strictly plane-wave behavior for natural electromagnetic fields has been questioned many times. Price (1949, 1962) has considered the theory for fields other than plane wave, and Rikitake (1966) has published a text which includes an extensive bibliography.

In recent years, interest in VLF communications had led to the study of induction fields such as those used in geophysical exploration by many nongeophysicists. A group of papers


summarizing such work has been published (Wait, 1963) and several of the papers in this group are closely related to the problem of electromagnetic sounding in a layered earth (Burrows, J 963; Hansen, 1963; Sivaprasad and King, 1963; Maley, 1963; and Word and Patrick, 1964).

In summary, it appears that the theory for electromagnetic sounding has reached maturity, and it is quite reasonable to expect such methods to be widely utilized in the future. With the present state of development, the method is effective in studying layering in sedimentary rocks from depths of a few tens of meters, as are of interest in engineering problems, up to depths of five or ten kilometers, as are of interest in petroleum exploration. It is well within reason that the methods can be extended to studies of crustal and upper mantle structure at depths of tens of kilometers, and that in the future, such methods may be invaluable in studying the internal composition of the moon and planets.

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13

Chetaev, D. N., 1963a, "On a Dipole over a Gyrotropic Medium," Zh. Tekhn. Fiz., Vol. 33, No.6. Chetaev, D. N., 1963b, "Method for Solving the Axially Symmetric Problem of Electrodynamics in a Gyrotropic Medium," Radiotekhnika i Elektronika, Vol. 8, No. 1.

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Fol'd, I. V., 1963, "Discussion of Application of the Methods Using Transient Electric and Magnetic Fields," Geofiz. Razvedka, No. 11.

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14 ELECTROMAGNETIC SOUNDING- METHODS

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15

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Tikhonov, A. N., O. A. Skugarevskaya, and P. P. Frolov,1963, "Tables of Values for the Transient Electromagnetic Field in a Layered Space," Izv. Akad. Nauk SSSR, Ser. Geofiz., No.8.

Tikhonov, A. N., O. A. Skugarevskaya, and P. P. Frolov, 1965, "Concerning the Matter of ResolutiOn! in the Transient Magnetic Field Method," Fizika Zemli, 1965(5):42.

Vanyan, L. L., 1957, "Concerning the Theory of Dipole Electromagnetic Sounding," Prikl. Geofiz., No. 16.

Vanyan, L. L., 1959, "Some Questions about the Theory of Frequency Sounding in a Horizontally Stratified Medium," Prikl. Geofiz., No. 23.

Vanyan, L. L., 1960, "Elementary Theory for Transient Electromagnetic Fields," Prikl. Geofiz., No. 25.

Vanyan, L. L., 1960, "Magnetic Field in the Far Zone of a Dipole," Geol. i Geofiz., No.5. Vanyan, L. L., 1962, "New Method of Determining the Electromagnetic Field of a Dipole Grounded

on the Surface of a Many-Layered Isotropic Medium," Geol. i Geofiz., No. 12. Vanyan, L. L., 1963a, "Electromagnetic Field of a Harmonic Dipole Grounded on the Surface of

a Many-Layered Anisotropic Medium," Izv. Akad. Nauk. SSSR, Ser. Geofiz., 1963(8):1222. Vanyan, L. L., 1963b, "On the Parameters Characterizililg a Transient Field," Izv. Akad. Nauk SSSR,1963(11):169l.

Vanyan, L. L., 1963c, "A Transient Electromagnetic Field in an Anisotropic Layer ," Izv. Akad. Nauk. SSSR, Ser. Geofiz., 1963(10):1532.

Vanyan, L. L., 1964, "Effect of a Poorly Conducting Basement on a Transient Magnetic Field," Izv. Akad. Nauk SSSR, Ser. Geofiz., 1964(4):562.

REFERENCES

Vanyan, L. L., and V. M. Davidov, 1965, "Distortion of the Late Stage of a Transient Magnetic Field by a Nonconducting Inclusion," Fizika Zernli, 1965(6):23.

Vanyan, L. L., A. A. Kaufman, and E. 1. Terekhin, 1961, "Computation of Phase Curves for Frequency Sounding by Transform Means," Prikl. Geofiz., No. 30.

17

Vanyan, L. L., and G. M. Morozova, 1962, "Master Curves for Interpreting Transient Magnetic Fields," Dokl.Akad. NaukSSSR, Vol. 147, No.6.

Vanyan, L. L., E. 1. Terekhin, and A. 1. Shtimmer, 1961, "A Method for Computing Wave Curves for Frequency Sounding," Prikl. Geofiz., No. 30.

Vanyan, L. L., E. 1. Terekhin, and A. 1. Shtimmer, 1965, "A Method for Computing Theoretical Curves for a Transient Electromagnetic Field," Prikl. Geofiz., 1965(46):90-100.

Velikin, A. B., and G. S. Frantov, 1962, Electromagnetic Fields Applied to Induction Methods of Electrical Prospecting. Gostoptekhizdat, Leningrad.

Velikin, A. B., G. S. Frantov, and S. M. Sheinman, 1961, "Concerning Questions in the Interpretation of Multiple-Frequency Induction Electrical Prospecting," Prikl. Geofiz., No. 31.

Wait, J. R., 1951a, "Transient Electromagnetic Propagation in a Conducting Medium," Geophysics, 16(2):213-221.

Wait, J. R., 1951b, "The Magnetic Dipole over the Horizontally Stratified Earth," Can. J. Phys., 29:577-592.

Wait, J. R., 1952, "The Magnetic Dipole Antenna Immersed in a Conducting Medium," Proc. IRE, 40:1244.

Wait, J. R., 1953a, "The Fields of a Line Source of Current over a Stratified Conductor ," Appl. Sci. Res., Sect. B., 1953(4-5):279.

Wait, J. R., 1953b, "Induction by a Horizontal Oscillating Magnetic Dipole over a Conducting Homogeneous Earth," Trans. Am. Geophys. Un., 34(2):185.

Wait, J. R., 1953c, "Induction in a Conducting Sheet by a Small Current-Carrying Loop," Appl. Sci. Res., Sect. B., 3(3):230.

Wait, J. R., 1953d, "Radiation Resistance of a Small Circular Loop in the Presence of a Conducting Ground," J. Appl. Phys., 24(5):246.

Wait, J. R., 1954a, "On the Relation Between Telluric Currents and the Earth's Magnetic Field," Geophysics, 19:281-289.

Wait, J. R., 1954b, "Mutual Coupling of Loops Lying on the Ground," Geophysics, 19(2):290-296. Wait, J. R., 1955, "Mutual Electromagnetic Coupling of Loops over a Homogeneous Ground,"

Geophysics, 20(3):630-637. Wait, J. R., 1956a, "Mutual Electromagnetic Coupling of Loops over a Homogeneous Ground

an Additional Note," Geophysics, 21(2):479-484. Wait, J. R., 1956b, "Shielding of a Transient Electromagnetic Dipole Field by a Conductive Sheet," Can.J. Phys., 35:693.

Wait, J. R., 1958, "Induction by an Oscillating Dipole over a Two-Layer Ground," Appl. Sci. Res., Sect. B., 7:73-80.

Wait, J. R., 1960, "Propagation of Electromagnetic Pulses in a Homogeneous Conducting Earth," Appl. Sci. Res., Sect. B, 9:213-253.

Wait, J. R., 1961, "The Electromagnetic Fields of a Horizontal Dipole in the Presence of a Conducting Half-Space," Can. J. Phys., 39:1017-1028.

Wait, J. R., 1962a, "Theory of Magnetotelluric Fields," J. Res. Natl. Bur. Std., D. 66(5):509-541.

Wait, J. R., 1962b, "Electromagnetic Waves in Stratified Media," Macmillan, New York. Wait, J. R., 1962c, "A Note on the Electromagnetic Response of a Stratified Earth," Geophysics,

27(3):382-385. Wait, J. R., 1962d, "The Propagation of Electromagnetic Waves Along the Earth's Surface," in:

Proc. Symposium, Electromagnetic Waves, ed. byR.E. Langer, Univ. ofWisc. Press, pp. 243-290.

Wait, J. R., (Editor), 1963, "Special Issue on Eleotromagnetic Waves in the Earth," IEEE Trans. on Ant. and Prop., Vol. APll, No.3.


Wait, J. R., and L. L. Campbell, 1953a, "The Fields of an Electric Dipole in a Semi-Infinite Conducting Medium," J. Geophys. Res., 58(1):21-28.

Wait, J. R., and L. L. Campbell, 1953b, "The Fields of an Oscillating Magnetic Dipole Immersed in a Semi-Infinite Conducting Medium," J. Geophys. Res., 58(2):167-177.

West, S. S., 1938, "Electrical Prospecting with Nonsinusoidal Alternating Currents," Geophysics, 3(4):306-314.

West, S. S., 1943, "The Mutual Impedance of Collinear Grounded Wires," Geophysics, 8(2):157-164.

White, G. E., 1939, "A Note on the Relations of Suddenly Applied DC Earth Transients to Pulse Response Transients ," Geophysics, 4(4):279-282.

Williams, R. H., and C. J. Benning, 1963, "Conductivity Measurements of the Earth at ELF ," IEEE Trans. on Ant. and Prop., AP11(3):364-365.

Wolf, Alfred, 1946, "Electric Field on an Oscillating Dipole on the Surface of a Two-Layer Earth," Geophysics, 11(4):518-537.

Word, D. R., and F. W. Patrick, 1964, "A Communication System and Theory for Investigation of EM Wave Propagation in the Earth at Frequencies from 0 to 3000 cps ," Tech. Rpt. 6-60, Elec. Eng. Res. Lab., Univ. of Texas, Austin, Texas.

Yost, W. J., 1952, "The Interpretation of Electromagnetic Reflection Data in Geophysical Exploration - Part I, General Theory," Geophysics, 17(1):89-108.

Yost, W. J., R. L. Caldwell, C. L. Bear, C. D. McClure, and E. N. Skomal, 1952, "The Interpretation of Electromagnetic Reflection Data on Geophysical Exploration, Part II - Metallic Model Experiments," Geophysics, 17(4):806-826.

Yungul, S. H., 1961, "Magnetotelluric Sounding Three-Layer Interpretation Curves," Geophysics, 26:465-473.

ELECTRICAL PROSPECTING WITH THE

TRANSIENT MAGNETIC FIELD METHOD·

L. L. Vanyan and 1. Z. Bobrovnikov

INTRODUC TION

In sedimentary basins in a number of regions of the country (as for example, the VolgaUral platform), which are potentially productive of oil reserves, there are broad areas where evaporite beds and carbonates with high electrical resisti vity occur within the section.

The presence of these beds in a section prevents direct current from penetrating, and so, electrical sounding methods based on the use of a direct current source cannot be used in studying the deeper horizons.

Only in recent years have electrical prospecting methods based on the use of a timevarying electromagnetic field proved practical and economically feasible for application to deep probing through geological sequences of this type. One of these new methods for electrical exploration is the transient coupling method in which the transient electromagnetic field at the earth's surface generated by a step wave fed toa source dipole is measured.

In this method, the electromagnetic transient may be recorded with the magnetic and electric fields observed simultaneously or separately.

Both theory and practical experience have shown that it is preferable to study the transient electric field if measurements are being made over a sequence which does not include a highly-resistant, or screening, layer; it is preferable to study the transient magnetic field if measurements are made over sections which do include such screening layers.

This paper will cover questions concerning the theory and practice of field work based on the recording of transient magnetic fields. Inasmuch as the object of such a survey is the determination of the electrical properties of the earth to some depth, commonly the basement surface, such surveys are called "electrical sounding with transient fields," or, for short, ZSP, method. Also, if the electric or magnetic field is studied separately, the method is deSignated as the ZSE (electric transient method) or the ZSM (magnetic transient method) respectively.

The use of transient field surveys in electrical exploration has followed early theoretical work by Tikhonov (18), Sheinman (23), Skugarevskaya (19), and Chetaev (22), as well as much

* Published originally by Gosgeoltekhizdat (1963).

19

20 ELECTRICAL PROSPECTING WITH THE TRANSIENT MAGNETIC FIELD METHOD

basic experimental work in recording methods for transient coupling carried out by Enenshtein (9), and by Vladimirov et al. (24). Since 1957, work with the transient magnetic field method has been carried on by the Electrical Exploration Laboratory of the All-Union Institute for Geophysical Research. The theory for transient fields in the far zone was developed in this institute, and Vanyan, Terekhin, and Shtimmer (7) computed a series of theoretical curves with this theory. Considering practical problems, the question of whether the whole transient signal or only the late stages of the transient should be recorded has been considered. Appropriate field equipment was developed and put into mass production under the supervision of L. Z. Bobrovnikov, after the method was deemed ready for field application.

The results of this early work proved that the ZSP method is very effective and economical, permitting large areas to be explored rapidly. Since 1961, the method has been used widely with good results in a variety of geological investigations.

PART 1. PHYSICAL AND MATHEMATICAL FOUNDATION FOR THE TRANSIENT MAGNETIC FIELD METHOD

As was described in the introduction, the transient magnetic field method consists in recording the magnetic transient at a point on the earth's surface generated by a steady-current applied to a source dipole. Therefore, the fundamental questions about theory to be considered are:

1. Solution of the mathematical problem; that is, calculation of the magnetic field developed at the earth's surface for a dipole source located over a stratified earth;

2. Determination of practical techniques for field surveys, including such things as the choice of the field characteristic which provides the most diagnostic information about the electrical properties of the earth.

In solving the mathematical problem for the ZSP method, we make use of a Cartesian coordinate system, XYZ with the Z-axis directed downward. The XY plane coincides with the surface of the earth. We will examine cases with anisotropic layers, each characterized by a thickness hp (the index p designates the sequence of layers downward), a depth to the top of a layer Hp ' a resistivity in the longitudinal direction Pl,P' and in the vertical direction Pn,p. The index p runs from 1 to N, in considering an N -layered sequence of beds.

§l. Time-Domain and Frequency-Domain Methods in the Theory for

Transient Magnetic Fields

There are two fundamentally-different methods of analysis used in describing transient processes. It is possible to write an equation for the components of the field and obtain a nonstationary solution to such a differential equation in terms of four variables (the coordinates XYZ and time t). This method is termed the ntime-domain" solution. The time-domain method has been used in solving a variety of problems in papers by A. N. Tikhonov, O. A. Skugarevskaya, and D. N. Chetaev.

The other method for studying transient fields is the frequency-domain method. In this method, the step-wave current provided to the source dipole is resolved into a Fourier series of frequency components, each characterized by an amplitude and relative phase.

This approach is based on the use of a Fourier series to represent the signal transmitted by the source dipole. The Fourier series for a magnetic induction field, B(t) may be written as:

PHYSICAL AND MATHEMATICAL FOUNDATION OF METHOD 21

~ I 21't ) B(t)= ~IBnl cos(n-T-+~n , n-1

where I Bn I is the amplitude of a frequency component, CPn is the phase of the n-th component, and T is the period of the fundamental frequency.

Commonly, a symbolic approach is used in which each harmonic is expressed as the real part of a complex quantity:

00 00

B(t) = Re~[I Btl I cos (- w t +~) + i I Btl I sin (- UJ t + cp)] =Re~) Bn I el;:e-"·f,

where w is the angular frequency.

The product B(wn) == I Bn I eicp is called the complex amplitude of a given harmonic. Using this type of representation, the Fourier series may be written as:

-00

As the fundamental period is made longer and longer, the discrete values of angular frequency, wn become closer together, approaching a continuous function. The Fourier series merges into a Fourier integral:

+00

B(t)= 211' S B(w)u-""tdw. (1) -00

In a conductive medium, each harmonic of B(w) propagates independently, with corresponding changes in amplitude and phase. At the point where the field is measured, the sum of the various harmonics with modified phases and amplitudes is detected.

The reason that the frequency-domain method is preferred is that solution of the differential equation in the frequency domain is simple. Solving this very simple equation for each harmonic component, we obtain a relationship between amplitude, phase and frequency - the spectrum of the transient field. Summing the harmonic components, we find the transient field and the transfer function of the medium.

It is quite difficult in the frequency-domain approach to obtain the function B(t) corresponding to a function B(w). The integral in equation (1) is tabulated only for a comparatively few simple functions for B(w).

On further discussion, we will make use of the value of the integral for the function exp [- a(-iw)l!2J/-iw, which is

I +Soo - ,,-''''' 10 for t< 0 "21t_ ooe- llV

-lu

,. -i", dOl= l-cIl(a~) for t>O,

z /'

where Ij> (a y' ~ ) = ~ S e - T dt, the exponential integral. o

Taking a == 0, we have:

t<o t>O.

(2)

(3)


Integrating both forms of equation (3) with respect to t, we have

_ _e _ dw = _ '. d to = 0 for t < 0 t I +eo -i',,1 I +';" p-iwt 1

J2T: 1-i(o) 2", leo (-1<0>" . t for t> O. (4)

Usually, numerical methods are used in evaluating the integral (1). This question will be discussed in more detail below.

In calculating B(w), we assume a source dipole for the electromagnetic field to be located at the origin and directed along the :x;..axis. We will assume that the moment of this dipole is the product of the current intensity and the dipole length, AB, energized at time t == 0:

I(t)= {O for t< 0 10 for t> O.

This means that the spectrum of the step current given by equation (2) is

I(w) == Io/-iw.

§ 2. Primary Magnetic Field of a Dipole

The primary vertical magnetic field of a dipole is usually considered to be the field that would exist in the absence of any conductive layers; that is, in free space. This field is derived from the Biot-Savart law, and is given by the formula:

BO(w)=- I(w}f'<lx z 4 1t (,-3+%2)"" •

Using the Weber-Lipschitz integral, this may be expressed as:

~

Bo( SI(w)f'1l' - I I • w)= ~.sm 8me m Z J1 (mr)dm. o

For z ?: 0, this is

~

B; (w)= 5 1(:~f'1l . sin 8 me-mz J1 (tnr)dm, o

08+ ~ z (w) 5[(W) 1-'0 . a 2 -mzJ ( )d

0% =- ~·smum e 1 mr m, o

and for z :5 0, this is

x

B; (w)= S 1(:~f'1l sin 8 memaJdmr)dm, o ~

08;(01) 51(W) 1-'" • a 2 m~J ( )d iJ z = ---;r;c- sm u mel mr m.

o

It is obvious that at z == 0, the primary field is continuous [Bi(w) == BZ'(w)] , and its vertical derivative has a discontinuity equal to


+ ~ oB. (co) oB:;(O) SI(O)",O. 2

OZ - dz =- ~sme2mJl(mr)dm. (l

The discontinuity of the vertical derivative of the primary field at z = 0 represents the presence of the source field at the earth's surface. At the other boundaries, the vertical derivative of B(w) is continuous.

§3. Use of the Principle of Reciprocity for Determining Transient

Magnetic Fields

In choosing the fundamerltal equations, we make use of the principle of reciprocity, which permits us to equate the problem of determining the tangential component of electric field, ~(w). about a vertical-axis magnetic dipole located at the origin with the problem of determiriing the vertical induction field, Bzp(w), about a horizontal electric dipole source located at the origin. The field of a vertical-axis magnetic source (such as a coil of wire with small area lylng in the horizontal plane) possesses cylindrical symmetry. It is described b~ a tangential electric field, ~p, and radial and vertical components of magnetic induction, Brp and B~.

The relation between the components of a harmonic electromagnetic field and the frequency in layers not having sources is given by Maxwell's equations, which have the following form for a vertical magnetic dipole:

oB~(O) _ oB:p(co) =~EM ((I» o z d r PIp 'PP •

oE:p(O) . M

- dz =t(l)Brp«(I».

+. :r [rE:p ((I»] = i (I) B:p «(I».

(5)

(6)

(7)

Maxwell.'s first equation expresses the Biot-Savart law in differential form; that is, it relates the magnetic field to current density, jepp(w) = Eepp(w) (1/ Pl,p)' provided by the magnetic field source. The second and third equations express the law of electromagnetic induction. These indicate that a va~ing magnetic field induces a related electric field.

Equations (5), (6), and (7) are written for the quasistationary case; that is, with displacement currents being neglected.

As is well known, displacement currents in air lead to a finite velocity of propagation for electromagnetic waves (the velocity of light, c = 3 X 105 km/sec). Neglecting displacement currents in the air is equivalent to neglecting propagation times for the electromagnetic field. However, at distances no greater than 30 km, the propagation time for electromagnetic waves is of the order of 10-4 seconds, which is a hundred times smaller than the smallest time measured in the transient field method. Theory indicates that displacement currents in the earth may be neglected with even less trouble.

Substituting the expressions for B~(W) and B~(W) in terms of E;)Jp(w) in the first equation, we obtain the fundamental equation:

(8)

where k = (-iw{J.o/p)l/2, the wave number for the layer with subscript p.


It is well known that the wave number is related to the wavelength. A p by:

kp= i1t (I-i), p

or, expressing the metric wavelength in terms of wave period T.

Ap=V 107p1P.T .

Equation (8) is usually solved using the method of separation of variables. We assume that EWp'(W) may be written as the product of two functions, F 2, which depends only on z, and F 1, which depends only on r:

" F; (1 2) F1 +-,.-- 7-m F1 =O,

• (2 2) F2= m + kp F2 ,

where m is the separation parameter, which may assume any value.

Introduction of a new variable u = mr in the first equation transforms it to Bessel's equation:

The solution to this equation is a Bessel's function of the first kind of order 1 for the argument mr, J 1 (mr). The solution to the second equation is an exponential e :l:npz , ·where n = (m2 + kb)lh. The general solution for E~p(W) may be written as a linear sum of these particular solutions. Inasmuch as the primary magnetic field is expressed as an integral, we should write the solution for E~p(W) as:

where ap and bp are not functions of z and r, but depend on source strength, frequency, and the parameters describing the sequence of layers.

In evaluating ap and bp ' we make use of boundary conditions requiring continuity of the horizontal components of the electric and magnetic fields at the boundaries between layers. However, equation (6) indicates that BWp(W) is proportional to BEWp(W)/B z. Therefore, the boundary conditions may be said to require continuity of EWp(W) and BE~p(W) /Bz. Using the principle of reciprocity, we may interchange E~(w) and Bzp(w):

B2P (w) = E~l' (00) = I (ape"p" + bpe -np.) 11 (mr)dm,

and similarly. in evaluating ap and bp ' we require continuity of Bzp(w) and BBzp(w)/Bz at all boundaries except the earth's surface where the source is located. Here, the magnetic field is continuous, but its vertical derivative has a discontinuity,

~

SI(IJJ)fJil . ~sme. 2m2J) (mr)dm.

o


Obviously we must take bo = 0 in the upper half-space (z::; 0) as well as no = m. With these conditions, the boundary conditions at the earth's surface may be written in the form:

and

Jnt (at - bt)Jt(mr)dm - jmaoJ1 (mr)dm = - j~~ sin 9 2m2Jt (mr) dm. . 0 0

Combining integrands and factoring Jdmr) we have

at + b1 =ao,

n1(a1-b1)-maO=- l~"'~!'o sin92m2•

At the top of the layer with index p, we have

In order to determine Bz(w) at the earth's surface, we use the equation:

~

Bz(w) = S aOJl (mr) dm. o

with the value for ao being determined from equations (9) and (10). After a little algebraic manipulation we have

1(",) !'o 4.. sin e 2m2

n.

(9)

(10)

(11)

(12)

(13)

(14)

This last expression is particularly satisfying, inasmuch as the electrical parameters describing the various layers enter only in the term (hi + a1)/(h1 - a1), which we can designate with the symbol RN. Therefore.

I (:~!'o sin e 2m2

n. m+ R;'

In order to find R~, we divide equation (11) by equation (12):

(15)


Substituting the right-hand side of the second ipreceding, equation in this last one for the term:

we have

It is not difficult to see that for p = 1 the left part of this last equation is

If we use the designation R~ _p + 1 for the quantity:

we have

R;" = cth ( n l hi + areth :: R~_l) ,

~-1 = eth (n2h2+ areth :: R~_2)' etc.

Substituting the values for R~_l' R~_2 and so on in the expression for RN. which is given in terms of the properties of the section. the frequency and the separation parameter. m. we obtain:

(16)

Equation (16) is a valid expression for the function RN for any value of N. In our further discussions we will consider only sections in which the lowermost layer is an insulator. with kN = 0 and nN = m. Then,

For a two-layer sequence:

For a three-layer sequence:

R; = eth [nih, + areth ~~ eth (lt2h2 + areth ~ )] .

For a four-layer sequence:

R:=cth {nih, + aretg ~~. eth [n2h2+ areth ~~ eth (naha+ areth ~ )]} .

PHYSICAL AND MATHEMATICAL FOUNDATION OF METHOD

It is of some importance to note that only the longitudinal value of resistivity for an anisotropic medium enters in any of the wave numbers ki' k2' or k3 • The vertical component of a transient magnetic field is a function of only the longitudinal values for resistivity.

Using equation (13), we obtain a final integral expression for the frequency-domain representation for the transient magnetic field at the earth's surface

Considering that I(w) = I/-iw,and using the Fourier transform, the time-domain representation may also be written:

27

(17)

PART II. CALCULATION AND ANALYSIS OF THEORETICAL CURVES

§ 1. Numerical Evaluation of Transient Magnetic Fields

The expression for Bz(t) derived in the preceding section is not used in practice because of the double integral contained in it. Neither form of integral in equation (17) is tabulated (with the exception of one for frequency for a homogeneous half-space, a problem which we will not consider). Therefore, in practice numerical methods of integration are used in evaluating Bz (t).

In this respect, in order to minimize the work involved in integration, we will examine evaluation of the integral:

~

H,I.' ~ ~':; ,In H S {)

2m2], (TIlr) drn. n, m+-R~

Because it contains the factor J i (mr) , the integrand is oscillatory with a period of about 271". Values for the dummy variable m, in the integral are varied through the range 0 to 10-20, inasmuch as for the remaining range of integration, asymptotic expressions can be used. If the spectrum B(w) is computed for a spacing r = 30 (measured in units hi), the argument of the Bessel function must be allowed to cover the range from 0 to 600, which would cover about 100 periods of the oscillating function. In order to obtain reasonably accurate results in integrating an oscillatory function, it should be evaluated at 10 points in each period. Thus, in order to evaluate the integral with precision, the integrand must be evaluated several thousand times at each frequency.

Inasmuch as the spectrum must be computed for at least several hundred frequencies, it is necessary to evaluate the integrand tens of thousands of times. Rather than compute a great number of such theoretical curves, up to the present time, emphasis has been placed on studying the asymptotic behavior of the integral expression (17) for the nonstationary magnetic field.


§2. Asymptotic Behavior of the Vertical Component of the Transient

Magnetic Field

For a given geoelectric sequence, the transient process for a magnetic field depends in a complicated manner on two independent parameters: a) the separation between the source and the receiver, and b) the time, measured from the instant at which the transient process started. In view of the difficulties described in the preceding section involved in evaluating equation (17), values of rand t for which simpler evaluation methods may be used have been studied. Let us consider the behavior of Bz(t) as r --- 00, t --- 0, and as r --- 00, t --- 00. In practical terms, these conditions apply for Bz(t) at relatively large separations for very long or very short times. Trial evaluations have shown that the values of Bz (t) determined under these limiting conditions are valid for spacings which are several times as large as the depth of basement.

Fundamental studies of the asymptotic behavior of Bz(t) at large spacings have been described by Tikhonov (20) with the result:

+.. ~

I, B(t)-~5~" e51' 2m2Jl(mr)e-,wtd d 1m z - 21t 41t Sill 1m ( ) m CD,

r-_ 111-0 m + ;~ (-iw)

_~ 0 N

Thus, the magnetic field observed at large spacings depends on the .value of the function 2m 2

11,

m + RI n

as m ...... O.

(18)

An analogous theorem may also be shown using Fourier integral theory. These theorems relate the early stage of the transient process with high frequencies and the late stage of the transient process with low frequencies. Mathematically, this concept may be expressed in the following form:

+oo -

and

11m BAt) = i--5 ~!"o ,sin e5lim 2m2 J. (mr) e- I wi dm d lit 1-0 It 1t K._N (m+ ;; )<-llO) ,

-oo 0 N (19)

+Ik. OQ

. 1 51flO . 5 2m7 J,(mr)e- 1wt ItmBz (t)=2" 47t'Stne lim ( ") . dmdw, I-- K.-O m + R~ (-'''')

-oo 0

Inasmuch as wave numbers are inversely proportional to frequency, these last equations may be written as:

+~ -

lim Bz(t) = 21" 5 ~~) ,sin e r lim 1_0 JW1-+00

and ')

(19a) +oo -

I' B (t) 1 5/fio i e ~I' 2m2J, (IIIr) e- 1wt d d ,!n1 z =2'n -;rn'S n In! ( II,), m w, 1-_ t Wt-+OO III + R;' (-'''')

-~ 0

CALCULATION AND ANALYSIS OF THEORETICAL CURVES 29

At this point, it is worthwhile to define some new terms. We shall call spacings which are considerably larger than the depth to basement as being in the "far zone." Thus, the far zone may be characterized by the condition r/HN- 00. The transient process observed in the far zone may be divided into two stages.

We will designate as the "wave stage" those parts of the transient process during which the waves with wavelengths considerably shorter than the spacing are dominant. We will designate as the "late stage" those parts of the transient during which waves with lengths considerably greater than the spacing are dominant, that is, r/HN- 00, l\./HN - 0, or k1HN-O. Obviously, classification in the far zone is more general than classification in the wave stage or late stage. Therefore, we first derive an expression for Bz(t) in the far zone, and then modify it further for use with the wave stage or the late stage.

As indicated by equation (18), in order to find the expression we want, we need to find the limit:

2m2 lim m~O m+~

R~

A limiting value of zero is of no use to us. In order to analyze the behavior of the function

as m- 0, we must expand n1/RN in a Maclaurin series and find limits for the two terms:

In this expression:

R:'v lm=o =cth {IC\h\ + arcth :~ cth l IC2h2 + . _ . + arcth KN,;;-I I}

=cth {f(\h\ + arcth V*cth [f(2h2 + ___ + arcth K:-1 ]} . (20)

We will designate the value for R'N as m - 0 as RN. In order to use this expression in the preceding equation, we note that:

As m - 0, this is

iJ II,

om --R* = ;\'

(j nl /(1 dR';, I iJm-R* =-/i2am .

N N m~O


Using the recursion expressions from (16) we may express the derivatives aR~/am in terms of aRN -ti am:

For m = 0, we have

Similarly.

The final deri vati ve is

oR';. (Jii1=

R~-l

(m; "N ~ iJ ( Ill\' I) ~=(R?-l) .---.- arcth --- . um 2 /(N-I om m

Performing the differentiation indicated in the last term on the right, we find aR;/am for m = 0:

oR; ,) "N-2 1 "- = (Ri-l ) . -- . -- . om ""\,-1 "N-I

Working backwards through the recursive chain, we find:

m-O

R~_I-I

b... R2 , -1 ?" .\ - 2

R~-I "I /(') ""-2 I ---=--- X (R~-l)-.~ .. "-'-'--. PN-2R~_1 /(~ /(R /(11'_1 "N-I PX-I •

Substituting this result in equation (19a) or (19), we have

III "I "N

R* ::::::;-R + m R2 ' N N N

where R~_I-I

ltN = --'----2 PI

RN_t-p;

R7v_2 -I R~-1


Using this result, the equation for the transient magnetic field in the far zone may be rewritten in the following way:

(21)

This expression is valid both as riA 1 - 00 and as A 1/HN - 00. In further investigation of the asymptotic behavior of Bz(t) , we must distinguish between wave-stage conditions and latestage conditions. For a final value r, the wave stage is characterized by the condition that k1- 00. Therefore, the function

2m2

may be expanded into a geometric series,

3 R'i., + TtN -2m l + ....

/(1

Substituting the first two terms of this series in equation (21), we have

Inasmuch as

and

~

Sm2J 1 (mr) dm=O a

the wave stage may be expressed as:

+~

B (t) __ I S 3/1'-0 . n P (co) z -2 --Slnu--·

" 2" r4 /(~

where the frequency function p(w) is given by

p(w)=R~+ 1tN .

dw, (22)

§ 3. Apparent Resistivity in the Transient Magnetic Field Method

Before considering calculations for the wave stage for the transient magnetic field, we will consider some of the phenomena which occur at very short times.


Considering the limiting process in the Fourier integral:

Using equation (16), it is not difficult to show that:

Iimp(Ol) = 1.

Therefore, as w - 00, we have

p(w) 1 PI! -k2 ::::::::-=-k2 =-, -,

! I - IUI!,-U

Substituting this in the expression for Bz (t), we have

From equation (4):

and therefore,

+~

1 S 3lPl -lwl ,1, e

Bz(t) =-2 -2' sme ')2 dOl. (-.0 It Itr (-10>

+~

I S e-i<u1 ---:'7" d w = t.

~ (-I w)l

dw,

(23)

Thus, in the initial stage of the transtent magnetic field, the field increases in direct proportion to time. Also, the derivative 8Bz (t)/8t does not depend on time, but depends only on the longitudinal resistivity of the first layer, the system geometry, and the source strength. Therefore, the apparent resistivity is measured with the transient magnetic field strength but in terms of the time-derivative of the field strength.

We will use the following definition for apparent resistivity, Pr :

2 TOr' ()Bz(t) p .. = at·sin 8 . -c)-t-

From a consideration of equation (23), Pr - Pl,t as t -- O. Thus, apparent resistivity defined in this manner has the desirable characteristic of approaching the resistivity of the first layer as the parameter defining the depth of investigation is made small.

Considering equations (22) and (23), the ratio of apparent resistivity to the longitudinal resistivity of the first layer has the following form for the wave stage:

+-

-2'=2- P (ill) e. dw. P I J' -1 .. 1 p 1t -lot I,

(24)

Equation (24) suggests that computation of theoretical curves for the transient field is a difficult problem, inasmuch as the transient magnetic field is a complicated function of time and spacing. Only if the wave stage alone is considered does the expression for Pr simplify

CALCULATION AND ANALYSIS OF THEORETICAL CURVES

so that a simple geometric factor for apparent resistivity can be developed. Inasmuch as the limit for the wave stage is based on the condition r» Ai (A i being the maximum wavelength in the wave stage), the limiting value Ai will be increased if the spacing is increased. As r - 00, the entire transient process may be described with the wave-stage expression. The corresponding PT curves will be termed the "wave-limit" curves.

33

§ 4. Computation of Wave-Limit Curves for Transient Magnetic Fields

Computation of wave curves for the transient magnetic field is done in two steps; the first step is computation of the complex frequency function for the transient field, and the second is the numerical integration.

Let us examine the method for computing the frequency function P (w) given by equation (20) and (21). For example, for a four-layer medium, we must compute the functions:

R4 =cth (IC1h1 + arcth J,0R:J) , Pl.

R3 =cth (IC/l) + arcth ~R2)' PI,

R2 =cth ICshs,

1tt = (R~ - I ) . R~-l I?~-l

R2 PI, R2_ PI, 3--- 2 -PI, Pia

For a three-layer medium, we must compute the functions:

R, = cth ICt h2 ,

Ri-l 1t~= (R~-l) ---''----

R~-~ PI.

(25)

(26)

From equations (25) and (26), it follows that in determining values of R4 and 7T4' we have to compute a function R2 corresponding to a two-layer medium with pii = P'?a' followed by R3 with pi' = Pl2 and pi = Pl3 , and finally, followed by R4 and 7T4. The computations involved in going frJm R2 to Rg aJd from Rg to R4 are of the same type, as is indicated in comparing equation (26) with (25).

1. Let us examine the computation, R2 = cthkghg [7]. The significant quantities which must be evaluated are

h 2'11: • 2'11: .-IC8 S =-1.- - t -~- = Yl -lX].

_3 _~

h. It,

In computing, the density of points must be taken such that the values for the frequency function p(w) may be integrated accurately between two neighboring values of A/h. Examination of the functions involved indicates that P (w) varies rapidly for small values of A/h and slowly for


large values~ Therefore, it is reasonable to select values of i\ /h in a geometric progression. Based on computations carried out in the Electrical Exploration Laboratory of the All-Union GeophYSical Research Institute, for ~ost geoelectric sections, values should be taken in a geometric progression with each value being larger than the preceding one by the factor 4,[2. A typical series of values might start with 0.5000 and run to 1024.0.

Hyperbolic cotangents for complex arguments are given in Appendix 1 from the 4-place tables compiled by Gavelka [11]. In these tables, the coefficient for the imaginary part of the argument is expressed in units of right angles:

and so:

and

2 Xl=--;rX,

R2=cth (Yl-ixl ),

For each pair of values x and y, the tables list values for the modulus t and the argument T expressed in right angles.

The values for y included in the tables range from 0.00 to 1.00 in steps of 0.02. When I y I > 1, values for the inverse function l/y are given for the ranges from 1.00 to 0.20 (y:s 5.00) in steps of 0.02. For y :s 0.02, we use linear interpolation. For y > 5.00, we take t = 1.00 and T = 1.00.

In determining I cth (y ± iXi) I and arg cth (y ± iXi) for xi in the range 0 to 0.50, we find values t and T i for y and xi and then use the relations:

I cth (y±ix) 1= -h arg cth (y±ix) = ± (1----t).

If x falls in the range 0.50 to 1.00, then we determine the quantity xi = I-x, find t and T and then use the relations:

I cth (y ± ix) I . t, arg cth (y±ix) = ± (1- ---t).

If x> 1.00, it is necessary to subtract from x a whole number (2n) of right angles until the remainder is less than 1.00 in absolute value. This procedure does not change the value of cth (y ± ix):

cth [y±i(x±2n)] =cth (y±ix).

Linear interpolation is used in finding t and T for values of x and y between the values listed in the tables.

The coefficients for linear interpolation, axt, ayt, aXT and aYT are given between the appropriate values for tT and TT' expressed in ten-thousandths per 0.01 difference in the value for x or y. Finally,

t = tT + 10-2 (axt& X + aYI~y). -:t = -:tT + 10-2 (aXt .1 X + ayt Ay).

The signs of the interpolation coefficients are not given in the tables. The sign is to be taken as positive if the value of the function is increasing with increaSing argument, and as negative if the value of the function is decreaSing with increasing argument.


Let us consider an example of the use of these tables in evaluating the hyperbolic cotangent of a complex argument. Consider that we need to determine the value cth (1.8332 -i ·3.0769).

1. Subtract four right angles from x: cth (1.8332 -i • 3.0769) = cth (1.8332 + i ·0.9231).

2. Inasmuch as I x I > 0.5, we take the difference 1 - 0.9231 == 0.0769.

3. Inasmuch as y > 1, we take l/y = 1/1.8332 = 0.5451.

35

4. The closest tabulated values are xT = 0.08 and I/YT = 0.54. The corresponding values for tT and TT are 0.9534 and 0.9922, respectively.

The difference between the given value of x and the closest tabulated value is x = x - XT = -0.0031. The difference between the given value of l/y and the closest tabulated value is .6.(I/y) ;= l/y -1/YT = 0.0051.

6. We note the interpolation coefficients:

ax/ = + 3,

a y / ... -32,

aXt = -10,

ay< = - 6,

7. We find t and T:

t - 0.9534 + 10-2 [3 (- 0.0031) + (- 32).0.0051] = 0.9517,

T= 0.9922 + 10-2 [(- 10)·(-0.0031) + (- 6)·0.0051] = 0.9922.

8. Finally we have

I cth tl.8332 - i·3.0769) 1= 0.9517,

arg cth (1.8332 - i.3.0769)=arg cth (1.8332 + i·O,9231)=- (1 - 0.9922)= - 0.0078,

The argument is expressed as a number of right angles; to obtain the argument in degrees, this value has to be multiplied by 90.

II. In order to 11:0 from R2 to Rs. we must evaluate the inverse hyperbolic function for a complex argument: arcth ..; P l /p l R2 (Appendix 1). Values for this function also being given in Gavelka's tables [11] and ApJen&x 1. The tables list values for x and y for given values of t and'T.

Inverse hyperbolic functions are presented in the form:

Re (areth te±I-;) = y, 1m (areth te±I~) = =+= (1- x).

The values for t in the table vary from 0.02 to 1.00 in steps of 0.02. For t> 1, we use the expression:

R ( 1 +/-;) e areth T e- = y, 1m areth t . e±h = =+= x. ( 1 -) -

Values for T are given in fractions of right angles from 0.00 to 1.00 in steps of 0.02. Interpolation for intermediate values of t and T is done, as described in the preceding section.

Consider an example, determining the value for arcthteiT. Let t = 0.1507 and T=0.2225.

1. The closest tabulated values are t = 0.1600 and T = 0.2200, with the corresponding values for xT and YT being 0.0353 and 0.1512.


2. The differences between the given and tabulated values are

A t = - 0.0093, A 't = 0.0025.

3. The interpolation coefficients are

4. We find y and x

alx = 23,

0ty =95,

a,x= 15,

a,y== -9.

Y = Yr + 10-2 (atyAt+a,yA 't) = 0.1512 + 10-1 [95 X (- 0.0093) + (- 9).0.0025) = 0.1421,

x = xr = 10-2 (a/xAt+alxA-t) =0.0353+ 10-2 [23 X (-0.0093) + 15.0.0025) =0.0336.

5. Using the appropriate formula, we find -(1 - x) = -0.9664, with x expressed as a fraction of a right angle.

In proceeding with the computation of R3, the values of y and x which have been found are assigned indices, Y2 and x2:

Following this, the tables are used to evaluate the hyperbolic function for a complex argument:

R. = cth {(y + Ys) + i (x + x 2 )].

Each of the values found for Ra is related to a wavelength:

Using the same procedures, we go from R3 to R4• With the complex values for R2, R3, and R4 , it is not difficult to find values for 71'3 and 71'4' using equations (25) and (26).

If the values for lI. llhl are sufficiently large or small, we may obtain asymptotic expressions for the frequency function. If lI. llhl - 0, which is the same as I klhll - 00, then cthklhl -1.

Therefore, p(w) ~ 1. In the short wave region, the frequency function tends to the real part of the value (arg p(w) = 0).

For a sufficiently small value of I klhll (lI. tlhl - 00), we have

where a2 = b2 = 1.

Similarly,

CALCULA TION AND ANALYSIS OF THEORETICAL CURVES

Substituting the approximate value for R2 in the expression for R3 , and substituting this result in the expression for R4, we find values for a3' b3 , a,p and b4:

37

As / k1h1/- 0, R~ ~ aV(k1h1)3 + 2(a2b2)/3. Thus, the real part of the frequency function does not depend on frequency, and its imaginary part increases without limit as the frequency is diminished.

Similarly,

Substituting these approximate values for R 2, R3, and R4 in equations (25) and (26), and separating the real and imaginary parts, we have:

For a two-layer medium with A. 1/h1 - 00 or w - 0:

2a2

limp (Ill) = (' hN)2 +limRep(Ul). "J J • (27)

where limRe p(w) = 1/3.

For a three-layer medium:

(28)

For a four-layer medium:

(29)

Let us now consider evaluation of the integral:

+~

1 S e- iwi 2=- p(lIl)-.-dm. 27t -I", P, _<'<'

First of all, we note that / p (w) / - 00 as w - o. So that we may obtain a useful result, we subtract from p(w) its limiting value

a~ .( S, ),,)2. lklhd2= l 2nS'1l;"" ,


+~ +-

P, 1 f[ .( S, ).1)'] e-1wt 1 J'( S, A 2e-'wt pZ = 2,. J P ( m) - l , 21t S • 11; -=-r;; d m + '2""; l 2 ,. S • -;/;) _ i '" d w.

According to equation (4):

+00 +oc. J...S ,(~,~)2 e- 1wt d Ul =(_S_,_)2107

• P" .J...S27Ce-, .. t dm= (~)2107P,2,.t for t>O. 2 1t l 2,. S h, - i '" 2 1t S h2 2 1t ( - I '" F 2 ,. S h2

1 _~ 1

We introduce a new parameter, T1. which is the analog of wavelengths in harmonic processes

with the result +~

P'_lj'[() ,(S, A,)2Je- iwtd +(S, 't,)1 --- P m -l --.- -- m --.-

P 21t 2 It S h, - i w 2 It S hI • '. -00

The parameter 71' which has the dimen!3ions of length, will be called the ntransient parameter. n The transient parameter is analogous to spacing in direct-current methods; it defines the depth of investigation for the transient magnetic field method. We remember that

e-iw' = cos wt - sin UJt,

and we designate

() . (S, ).,)2 R . P It)-l 2,.s'1i; = ep (w) +tlmp (w).

Writing the Fourier integral in a form which is more convenient for computation,

+M +_

P, (S 1 '< )2 I 5 [R () sin c,' t 1 () cos (o,l t] d + i S [I () sin", t R . cos w t ] - - -2 S ',- =-2 ep m -- - mp m --- m 2- mp m --- - eplm)--- dm P 7t III 1t (0 OJ n co \ w • 'I I _~ __

As is known, the real part is an even function of frequency while the imaginary part is an odd function. This follows from the relations:

Rep(m) = Ip(w)l.cos,¥w;

Imp(m) = Ip(w)l·sin Cfl w '

where I p(w)1 and 'Pw are the modulus and argument of the complex frequency function, respectively.

From these relations it follows that the sum

I () sin", t + R () COS", t mp W -",- ep U) -",-

is odd and the second integral in the expression for PT / P1 is zero.

Therefore, .. P. (S, ,<,)2 15[R () sinwt I ()COswf]d -- --.- =- ep m --- mp m -- Ul " 2 ,. S h, 2 ,. '" III·

• 0

(30)


Here we note that for t < 0, for the conditions specified, we have PT / Pi == O. Therefore,

'"' I S[R () sin(-",t) I ()COS(-"'t)]d 1 S'"'[ R ()Sill"'l Imn(l)COSOlt} dw =0, 7" ep w (0 - mp m --h'- w =7 - ep 10 -0)-- t It -(O-

Il 0

Subtracting this last expression from equation (30), we finally have

P, _ 2 SR () sin (1Il/) d + (SI tl )2 --- ep w -- w --,-PI 7t '~, 2 1t S hI ' (31)

I (I

The principal difficulty in evaluating the Fourier integral in the form of equation (1) lies in the oscillating character of the sine function. For numerical evaluation of the Fourier integral, Filon (16), Nikolaev (15),and Solodovnikov (17) have given techniques which are based on the idea of using relatively few values of the frequency functions, with intermediate values being obtained by interpolation.

Inasmuch as the frequency function varies relatively slowly, linear interpolation may be used. In this respect, we connect adjacent values for the function pew) with a series of straightline segments.

Let us compute the real part of the frequency function for n values of frequency from Wi

to Wn. We designate the corresponding function values as Rer, Re;, ... ,Re~. We select the frequencies so that we may say:

Re;::::::limRe'p(w), w_o

Re~::::::lim'Re'p(O))= 1. ..... An approximation may be made within a reasonable error (us ually about 1%). Using the

high-frequency limiting value p(w), we may write:

2_t~,~)2- 2S~rR' () l/Sillo,t d + 2 r-SillOltd 2 w<' h -- epw- -- Ql - -- '0) PI .- .. , J 7t ,'" 'It tAr •

I '0 0

The second integral has a value of one. Therefore,

... .!::. = I + .2..SIRe/p(w) -11 sin", I do), P 1t 0' I, 0

Inasmuch as the factor contained within the parentheses has a value of zero for w > wN, the upper limit of integration may be reduced to wN:

Wn

P, t" S, ,,)~ 1 + 2 SrR ' () 'II sin", t d ---.,--'-= - epw---w p ~ 7t S hI 1t 1 Ill'

~ 0

The values for the ordinates of the function Re' pew) -1 at the points Wi' w2, •.• ,wN are designated as Rei' Re2, ..• ,ReN' and in addition, Ren = 0, Ren_i == ~n-i' Ren _2 == ~n-l + ~n-2' ... Re2 == ~2 + ~3 + ... ~n-t' Ret == ~1 + ~2 + ~3 + ... ~n-t' where ~p is the difference Rep - Rep+ 1

with p = 1,2, ... , n, and ~n == O.


Then,

We assign to the segment between wp and Wp+1Re p (w) -1 the approximate value:

Oln • wI. 1l-1wP+l n

irRe p(oo)-lI SlII",W' doo~SRel Sl~:.t dOJ+ ~ J ( ~ ~K+It1 ",-wD+')Sin"'t du •. 0' ~ ~ PWp -(I)P+1 II'J

U P-lwp N-p+l n

Inasmuch as Rei> :Et1k and b.p do not depend on frequency, the integration may be prek-p+1

sented in the following form:

fO n 0>1 cUs

r IRep(Ol) -1] sin W t dOO=(~l + t12+' .. +t1/1-d X S~dOl + (~2 + ... + ~n_I)SSin w( doo+ J III (~ lJ)

o n lUI

Wli flJ ll- J rl.ln _ 1 u'n

+A f"'-W~ sin,ot d + +A S Sill .. td..L A S Ol-'"n_1 SinOlt d +A S "'-Wn sinwt d · ... 1 --._- w. u 1 -- 0) j U , 0_- 00 1 .-- '"~ • ti)t-w2. (0 •• n-.., n- wn--Z-wn-I U) n- llln-J-(\ln OJ

to, II) 11-1 Cdn- 2 lUn- 1

+(t11S""_~.Sinwt doo+t12Sw'W-W;I ,sin",! doo+

0'1 - 11.)2 (I) w2 - w3 I., 'On ) + A S W - "'" • sin", ! d -

.. i..l1J-I W -wn-l - hln W

W J 102 w ll - 1

Using the known formulas for integration and considering that wp+1 = wp -{2, the expression within the parentheses may be reduced to the following form:

cos '02 (- cos Y2,oet (Jf2 - J)wpt

The product wpt may be expressed as the square of the ratio of the transient parameter in the first layer. T 1. to the corresponding wavelength, A 1p

~ = . = Itt =00 t ( )2 (V 1072ltt PI)2 (~)

A,p V 10' TpP,. Tp p •


Therefore,

Analysis of theoretical curves for transient magnetic fields indicates that the apparent resistivity PT , in analogy to the frequency function p(w), varies rapidly for small values of the transient parameter while it varies slowly for large values of the parameter, T 1. Therefore, in computing theoretical curves, usually values for T 1/h1 are selected in a geometric progression, with each value being larger than the preceding value by the factor V2. The corresponding values for wand t form a geometric progression with a ratio q = ..{2. The product wt also forms a geometric progression with the ratio q = ..(2, and the parameter T/A.1P forms a progression with the ratio q = Y2.

Values for the auxiliary function F(T1/A.1p) are given in Table 1, from which it may be seen that for T /A.1p- 0, we have F(TdA.1p)- 0 and for T/A.1P 2: 8, we have F( T1/A.1P) ~ 1.00.

In order to compute a transient curve, values for T 1/A.1P are written along the horizontal row on a sheet of paper, while values for the function F are written along another row beneath the first.

Computations for wave curves for the transient field are done as follows:

1. A series of values for the frequency function and Re P (w) -1 are calculated for various values of A.1/h1. These values are chosen such that for the minimum A. /h1, Re P (w) I~ 1.00 and for the maximum A./h1• Rep(w) ~lim Re p (w).

w~o

2. Values for A.1/h1 are written in the first horizontal row of the computation table in increasing order of A.1 /h1 from left to right, and the corresponding values of Re p (w) - 1 are written beneath. in the second horizontal row.

TABLE 1

'Ct!Atp 0.\05 0.125 0,149 0.176 0.210 0.250

F(::J 0.0058 0.0085 O.oII5 0.0170 0.0230 0.0341

'Ct!Atp 0.290 0.354 0.420 0.500 0.595 0.707

F(:t~) 0.0460 0.0678 0.09t5 0.136 0.191 0.269

"Ct!Atp 0.841 1.00 1.19 1.41 1.68 2.00

FCt) Atp 0.376 0.521 0,708 O.92:.! 1.111 1.162

'Ct/Atp 2.38 2.83 3.36 4.00 4.76 !i.66

FUt'p) 1.013 0.935 1.057 0.977 1.001 0.997

"'t/Alp 6.73 8.00

F( 'Ct ) A,p

1.00 1.00

42 ELECTRICAL PROSPECTING WITH THE TRANSIENT MAGNETIC FIELD METHOP

3. For each pair of values Re p(w) -1, the difference .6.Rep(w) = Rep+1 -Rep is formed and written down in a third horizontal row, beneath the second. Obviously, the values for the difference near the ends of the row must be approximately zero.

4. The sheet of paper with these values written on it is then placed on top of the computation table in such a way that the value of Tt/A.1P which is unity falls under the furthest left value of A. 1 /h1 in the first horizontal row.

5. Each value for the difference .6.Re is multiplied by the value for the function F which falls under it, and the product is written down under the values .6.Re in a fourth horizontal row.

6. The sheet of paper with the function F is shifted one space to the right, and the operation is repeated.

7. The final product is a table of values for products .6.Re . F, which are summed in vertical columns. Each of these sums corresponds to the value T 1/h1 found as the first entry in a given column.

8. To each sum is added the factor 1 + [(S1/27[S) . (T1/h1)]2 with the result, which is the apparent resistivity measured with the transient method for a section having an insulating bottom layer, being entered in the bottom horizontal row.

In conclusion, it should be noted that this approximate method of computing the transient magnetic field is valid not only for r --- co but also for any spacing for which the frequency function is available. In this case, it is no longer necessary to add the term [(S1/27[S) . (T1/h1)]2 to the final result.

§ 5. Computation of Theoretical Curves for the Late Stage of the Tran

sient Magnetic Field

Having considered the computation of the wave stage of the transient field in the far zone, we now continue to the study of the late stage. Using the asymptotic properties of the Fourier integral which allows us to associate the behavior of PT as t --- co with the behavior of the frequency function as w - 0, we obtain an approximate expression for computing the late stage of the transient process.

~ +~

B (t) 11'0' oS IS 2m'); (mr) e-iwi d d z ~-stn" 2 '-.- (j) m. 47t 7t mlim p(w) +lim..!!2..... -/(1)

w-o R'fv '0-0 R/V (32)

o

From equations (27)-(29):

where c = lim Re P (w). "'~o

We note that the given equation has the same form no matter how many layers there are in the geoelectric section. The properties of the layers enter only in the coefficients a, b, and c. Substituting these limiting values in the integrand of (32), we find:


K2h [ ( 2 2C) ] 2m+ ~ 1 1- ;rb-/i mhl

By neglecting terms in the denominator containing m 2 or m3 , we obtain a simpler expression as w - 0:

43

Using this result, the integral for the late stage of the transient magnetic field may be written in the following form:

+eo '"

1 5 I J[2m2 +2m3hl (; b- :)].[1 +:a(Klhl)'lJ1(mr) e-iwt

B (t)---- ~sjne . X -.-dmdOl. z ---23 4lt K~hl -I ••

2m+-a-

_eo 0

For numerical evaluation, we interchange the order of integration:

(33)

As was pointed out earlier, the time derivative of the vertical component of magnetic induction, BBz(t)/Bt, is a better index to earth characteristics than is the function Bz(t). Therefore, we differentiate equation (33) with respect to t:

(34)

Let us examine the second integral in equation (34):


Consider the well-known definite integral:

Differentiating this integral with respect to t, we have

+'" 2mt

_1 S<-i eo) e-lw1deo _ ( 2m)-P:;:S 21': 2m - - floS e

1'0 S-'eo

Substituting the value for this definite integral in equation (34) we obtain:

- 2mt

iJB% (I) ~ 1,,"0 • e5[2 2 + 2 8h ( 2 be)] [ 1 b I'-oh~ ( 2m )] e - -;;;s at ~-4-S1f1 m m 1"""3 - - X 1+---- -- -.--1 (mr)dm 7t a 3 a PI 1'0S ,,"oS 1 •

o

Neglecting the four terms which are relatively small, we have

This integral may be evaluated with the Weber-Lipschitz identity, so that we have for a final result

~= ~ . ~ { q2 + A [ 1 _ 5q· ]} P hi S (1 + q4)'/' (I + Q4)'/' (l + Q4)'/' • I,

where

and A=~.; Iimp(w). r 1 m_O

Thus, we see that the late stage of the transient magnetic field depends on two generalized parameters for the geoelectric section: the total longitudinal frequency function p (w) as w-O. We readily see that the relative importance of these two parameters changes with increasing spacing. With larger values of r, the parameter A has a relatively smaller effect on the late stage of the transient magnetic field. As r - 00, we have

that is, for sufficiently large spacings, the late stage of the transient magnetic field depends on only one generalized parameter for the geoelectric section - the total longitudinal conductance.

Thus, when large spacings and late times are being considered, the whole sequence of conductive beds above basement may be replaced with a single conducting sheet with a surface conductance S. The problem of transient magnetic fields over a conducting sheet was first studied by Sheinman (23).

CALCULA TION AND ANALYSIS OF THEORETICAL CURVES

Fig. 1. Set of curves for the late stage of the transient magnetic field (val ues for the parameter A are indicated on the curves).

8

Fig. 2. Set of two-layer curves for the transient magnetic field (values of r/h1 are indicated with each curve).

§ 6. Master Curves for Transient Magnetic Fields

45

In the preceding sections, we have developed formulas which may be used in computing theoretical curves for transient fields. Using these formulas, the Electrical Exploration Laboratory of the All-Union Geophysical Research Institute has computed wave curves for transient fields for representative three - and four -layer sequences, as well as curves for the late stage of transient coupling (see Fig. 1). In all, 62 three-layer curves with p~ =' 00 and 11 fourlayer curves with Pl4 =' 00 were computed (see Appendix 2). In these computations, the following values for longitudinal resistivity of the middle layer, or layers were used:

a. For a three-layer sequence:

PI, 2 3 3 7 -=3' T' 4' 9' 19' 2' 3,4,9,19, PI,

b. For a four-layer sequence:

PI, 7 -=12'

PI,

PI. 39 -=9'

P"

PI, -=00,

PI,

PI, -=4'

PI,

PI, 1 -=9'

PI,

PI, 1 -=9'

PI,

In addition, using frequency functions for a single layer resting on an insulating halfspace which had been computed on a high-speed electronic computing machine under the direc-


a 2

00

Fig. 3. Construction of a transient magnetic field curve in the far zone. (a) P2/ Pl = 1/4, h2/hl = 1, Ps= 00, r/(h1+h2) =7; (b) P2/Pl=4, h2/h1 =2, Ps'=oo, r/(h1 +h2) =5; 1) Wave curve. 2) Curve for the early stage.

tion of D. N. Shakhsuvarov (at the Institute of the Physics of the Earth, Academy of Sciences of the USSR), two-layer curves were computed at the All-Union Geophysical Research Institute for transient magnetic couple for spacing to thickness ratios: r/h1 = 2.88, 3.43, 4.08, 4.86, 5.76, 6.86, and 8.16 (see Fig. 2).

The sets of wave curves for the transient field have index designators, the first two letters of which (VS) indicate IIwave transient. II The two numbers following give the resistivity and thickness of the second layer, and so on. For example, the designator VS - Y4 - 1 - 00 indicates transient coupling curves for a three-layer sequence with PIz/P I1 = 1/4, h2/h1 = 1, and P3 = 00.

The ratio T 1/h1 is plotted along the horizontal axes of these curves with a logarithmic scale, while the ratio P T/ P1 is plotted along the vertical axis, also to a logarithmic scale. The ordinate for the horizontal axis on each of these sets of curves is 1, the abscissa for the vertical axis, 8.

Within a single set of curves, the ratio Ph. / Pl is invariant, while the ratio hZ/h1 assumes a range of values. The designator for such a1family of curves would be: VS - (PZ / PI )-M - 00 (that is, the thickness is variable). z 1

The method of plotting the curves for the late stage is different, with the spacing entering into the coordinates. This is done so that the shape of the late-stage curves will not be affected if the curves are translated along the horizontal axis.

The ratio T 1 /h1 = (Sr /Slh1)V2 is plotted along the vertical axis and P T/ P1 = (r/h1) • (S1/S) along the vertical axis. Each curve has as an index designator the appropriate value for the parameter A.

§ 7. Construction of Curves for Transient Magnetic Fields in the Far

Zone

The transient magnetic field may be described using the wave curves for comparatively short times and large spacings. Therefore, by combining the wave curves and late-stage curves, we can construct a curve for the transient curve in the far zone which is valid for the entire time interval from 0 to 00. The parameters of the geoelectric section and the ratio r /h1 are used as parameters for these curves. The first step in construction of a curve is the calculation of a wave curve or interpolation of such a curve between two already-computed curves. This is then plotted on the usual type of bilogarithmic graph paper. Next, the values for


A = (r/h1) . (S/S1) . lim Re p(w) , (r/h1) . (S1/S), and [(r/h1) . (S/S1)]1/2 are determined. The ori-w~o

gin of the late -stage plot is placed at the point with coordinates

The curve for the desired value of A is then traced on the sheet of graph paper. The method for constructing a transient curve in the far zone is shown in Fig. 3. Commonly, the wave curve approaches closely to the late-stage curve. However, if the two curves do not merge, the mid-portion of the transient curve is developed by interpolation. The larger part of the transient curve is determined by the wave curve, and only the right end is determined by the late -stage curves.

47

We should observe that the character of the geoelectric section has readily predictable effects on the wave stage and late stage. Examining the wave curves, we can observe significant differences which depend on each of the parameters of the geoelectric section. On the other hand, the late-stage curves depend only on the generalized parameters S and A; the parameters describing the individual layers do not have any effect on the late-stage curves.

§ 8. Analysis of Theoretical Curves for Transient Magnetic Fields

A most important property of transient curves is the relatively small screening effect caused by an intermediate layer with high resisti vity. As an example, consider a four-layer sequence in which the lowermost layer is an insulator, the second layer has a very high resisti vity and P1 = P3. The behavior of the apparent resistivity function P T / P 1 determined from the frequency function p(w) and of the apparent resistivity curve determined with the directcurrent method (Schlumberger array) depends on the function R [2].

For the case under consideration:

Let the resistivity of the second layer increase while its thickness decreases. Then,

[ ~ ] 1 + cth mh2 - cth mha R = cth mhl + arcth ~. P2 :::::: cth [mhl + arcth (mh, :~ + cth mha)] .

p cth mh2 +..f!.. cth mha P2

The basic difference between P (w) and it is that P (w) does not depend on the resistivity of the screening layer while R does. Therefore, as h2 - 0, P (w) ~ cthk1(h1 + h3); that is, the presence of a thin sCl'eening layer has a negligible effect on p (w) or p • On the other hand, n T -n. depends on the product R2 = h2P2 and if h2 is small while P 2/ P 1 - 00, we have R ~ cth mh1;


t

Fig. 4. Comparison of resolution of four-layer Pr and Pa curves, (a) Curve for PT for P2/ Pl =39/9, h2/h1 =1, P'/Pl=I/9, h3/hl=l, P4=00; (b) curve for Pa for P2/ Pl =4, h2/hl = 1, P3/ Pl = 1/9, hslhl = 1, P4 = 00,

that is, it approximates the case for a perfect screening layer with finite thickness, the effect of the screening layer in the transient coupling method is relatively weak compared with the effect on direct-current measurements (see Fig. 4). It should be noted that a conducting layer has the same screening effect on transient measurements as on direct-current measurements [6].

The theoretical curves in Fig. 4 illustrate the increased resolution obtained with magnetic transient method in comparison with that of the direct-current method for a geoelectric section containing a resistant screening layer. The screening effect of the second layer on the apparent resistivity measured with the direct-current method causes the minimum associated with the third layer to be 1.2Pl; the minimum observed with the magnetic transient method has a low value of 0.67 Pl'

A second unique characteristic of the theoretical curves for the magnetic transient method is that they are independent of anisotropy in horizontally-stratified rocks. This property of magnetic transient data is self-evident, inasmuch as equations (25) and (26) show that the apparent resistivity, P T/ P 1, for an anisotropic horizontally -stratified medium depends only on the longitudinal conductivities of the individual layers. Inasmuch as the transverse resistivity is not contained in these equations, anisotropy has no effect on the resistivity, PT/ Pt.

We shall now consider some of the basic characteristics of the wave curves for the transient magnetic field. Let us examine the left-hand segment of these curves, using the basic equation for apparent resistivity:

where pew) is determined using equations (25) and (26).

In these equations for sufficiently high frequencies (klhl - 00), we have RN l'::l 1, with the result that:

Substituting this approximation expression for the frequency function, P (w) , in the Fourier integral, we find the apparent resistivity for short times (t - 0):


Thus, the left-hand segment of a multiple-layer transient curve depends only on the longitudinal resistivity of the first and second layers; that is, the behavior approaches that of the corresponding two-layer case. In order to compute a two-layer curve, we expand the integrand in a series in powers of exp (-2k1h1):

cth ("I hi + arcth ~)= P"

where

Hence

Using equation (2), we have,

As T 1/h1 -- 0, the wave curve for the transient field has a horizontal asymptote. The equation for this asymptote may be derived readily from equation (2) if we consider that as x-- co, cf?(x) -1. Consequently, as T1/h1- 0, PT I P1 ~ 1.

Let us now show that the apparent resistivity approaches a left-hand asymptote of one uniformly. To do this, we must show that PTI P1 increases for-Q2 < 0 and decreases for Q2 > 0, or that a(p T I P1)/a (T 1/h1) < 0 for Q2 < 0 and that a (p T I P1)/a (Tt/h1) > 0 for Q2 > o.

From equation (31) we may show that:

(35)

where y == 47rT1/h1•

For Q2 > 0, all the terms in the summation in equation (35) are positive, and therefore, a (PT I P1)/a(T1/h1) > O. Similarly, we may show that PTlp1 decreases uniformly over Q2 > O. The uniformity of the left-hand segment of the wave curve for the transient field is an important property inasmuch as this simple behavior insures curve characteristics similar to those seen with direct-current resistivity curves. To be more specific, a layer with relatively low resistivity will decrease the apparent resistivity, PT' and a layer with high resistivity will increase the apparent resistivity.


to Z.O 3,#

(0 2,0 3,0

In order to investigate the behavior of the righthand segment of the wave curve, we will examine equation (31) for large values of T1/h1' Equation (29) indicates that as T 1/h1 - 00, the wave-stage curve for apparent resistivity will be approximately:

~=(~~)2 PI 21': S h,

When plotted to bilogarithmic coordinates, the right-hand asymptote for the wave curve has the form:

p_ '<I 21tS log -" =2log --2 log -

PI hI 8 ,

This expression plots as a straight line, forming an angle of arctan 2 = 63°25' with the logT1/h1 axis. The intersection of this asymptote with the log P T / P1 = 0 axis is PT / P1 = 1 at the point where Tdh1 = 21TS/S1' Fig. 5. Curves for the abscissas and

ordinates of the minima of three-layer type-H curves (values for P2/ P1 are given with each curve).

An important parameter for a three -layer typeH curve (one in which P1 > P2 < Pa) is the position of the minimum value in apparent resistivity - its ab

scissa and ordinate. Analysis of the wave curves shows that for a large enough second-layer thickness, the abscissa of the minimum is given by

H

(~) = 3.75 h, min (36)

The ordinate of the minimum is given by

Figure 5 shows graphs for the ratios

H and

3.75 71;-

-v1 PI,

(37) PI,' PI,

as a function of h2/h1 for different values of Pl / Pl' These curves are based on a study of the wave curves for transient magnetic fields. As 2we bay see from Fig. 5, equations (36) and (37) are valid when h2/h1 :::: 1.5.

§ 9. Maximum Resolution for Transient Magnetic Fields

The resolution of the magnetic transient method is a function of the spacing. Therefore, the combinations of times and spacings which will provide the maximum resolution are of particular interest. These are found in the wave stage. To demonstrate this, we use equa-


AD r, - , -

Fig. 6. Comparison of the resolution of three-layerJlurves for PT (a) and Pa (b).

tion (17). The parameters of the geoelectric section (particularly the resistivity of the first layer) enter into this equation only through the function R~, which is also a function of the dummy variable of integration, m. The longitudinal resistivity of the layer with index p is contained in the function R~ within a radical of form:

V 2 iWfJ-o np= m ---, , Pp

It is obvious that the variations of Pp within the radical have a greater effect on the value of the radical when the values for m are small. We would have the maximum possible relative value for np for m = O. But, according to equation (18), the behavior of the integrand as m -- 0 determines the character of the transient process as r -- 00. There-

) P, liz., )P, Ih, I J P. - g' 11;-1, P.-~; 2 Po - 19' Ii;" - 2' P.-~ fore, wave curves for the transient process which are computed for r -- 00 may serve in determining the maximum resolution. For a particular finite spacing, the resolution will be no more than that of the wave curves.

§ 10. Equivalent Curves for Transient Magnetic Fields

In order to evaluate the range for the principle of equivalence for magnetic transient curves, we will make use of the appropriate results from the preceding sections for wave curves.

Let us examine an example of a three-layer curve (type H) in which the second layer is sufficiently thin so that the transient curve depends only on the longitudinal conductance of the layer. Consider the frequency function:

As h2 -- 0, we have

However, this equivalence "is encountered only for values of h2/h1 which are significantly smaller than in the case of direct-current soundings. Comparison of wave curves representing sections with similar values of total longitudinal conductance indicates that they differ from one another by a larger amount than the corresponding direct-current resistivity curves. As an example, direct-current sounding curves and transient sounding curves for two sections are compared in Fig. 6.

The parameters describing the first case are P 2/ P 1 = 1/9, h2/h1 = 1, P a/ P 1 = 00, and S/S1 = 10. The parameters describing the second case are P 2/P 1 = 1/19, h2/h1 = 1/2, Pa/P 1 =00, and S/S1 = 9.5. The direct-current sounding curves for these two cases differ by only 5%, while the transient sounding curves differ by 40%. It is obvious that the curves differ most in the area of their minima.


PART ITI. EQUIPMENT

§ 1. Recording Transient Magnetic Fields

Considering the nature of the transient magnetic field method, we see that the important differences in comparison with the various direct-current sounding are: first, a markedly smaller screening effect produced by a layer of high resistivity in the section, so that layers beneath such screening layers may be studied; and second, the lack of sensitivity to macroanisotropy resulting from horizontal stratification. A third basic characteristic of the ZSPZ (transient) method is that a depth sounding may be made using but a single spacing between the source and receiver.

Consideration of theoretical curves shows that the resolution of the transient method is sharply reduced when short spacings are used. Both theoretical considerations and field experience have shown that the optimum source-receiver separation usually is about fives times the depth to be sounded.

First of all, we need to determine the signal amplitudes which will be observed under typical field conditions, and to do this, we manipulate equation (23) into a form which may be used in describing field conditions. We may assume that the simplest and most satisfactory system for detecting the rate of change of magnetic induction will be the detection of the EMF, e(t), induced by the magnetic field in a closed loop.

A loop lying in the horizontal plane is used in measuring the component, aBz lat. If the area of the loop is designated as q, measured in square meters, we may write the following, using equation (23):

21t r" • (t) P,= 3ABq·sln e . J . (38)

Thus, the expression for PT is similar in form to the well-known expression for apparent resistivity for the direct-current methods:

AV Pk=k j .

We may use equation (38) to calculate the magnitude of e(t) for the following conditions: PT =10 ohm-m, AB = 3 km, e = 90~ and r = 15 km.

If we measure EMF in microvolts and the area of the receiving loop in square kilometers, then e(t) Rl 0.3 qJ. Thus, taking a coil area of one square kilometer and a current of 60 A, both of which are reasonable field values, we find e(t) = 18 fJ. V. This value indicates the order of magnitude for the parameters used in deep sounding with the transient magnetic method. Considering the small size for e(t) obtained in this computation, it follows that a current of at least 25 to 30 A is required, and the coil area (the actual area of the loop multiplied by the number of turns) must be no less than a square kilometer.

§ 2. Signal-to-Noise Ratio

Noise amplitudes encountered in the ZSP method are quite Significant, inasmuch as broadband recording equipment must be used in measuring transient coupling. The noise types which may be recognized on records are loop noise, industrial noise,and natural noise.

Loop noise is generated by motion of the receiving coil in the earth's magnetic field. An effecti ve means for minimizing this is covering the cable with soil.

Industrial nois~ is found close to power distribution systems (electrified railways, factories, generating plants, and so on). High-frequency industrial noise is found also within a

,Commu'tator

EQUIPMENT

Control. vOltage

re supply Radio transmitter

Fig. 7. Block diagram for transmitter.

distance of one or two km from a high-tension line. In order to reduce industrial noise problems, surveys should be located well away from industrial areas. The distance from large sources of industrial noise, such as an electrical railway, must be at least 10 to 20 km.

53

Noise in the natural electromagnetic field may be separated into two parts -low frequency and high frequency. Low-frequency variations in the vertical component of the earth's ml}.gnetic field commonly are not of serious concern in recording transients. This is related to two factors. First, for electromagnetic waves arriving at the earth's surface from the iionosphere, the vertical magnetic component is very small. Second, low-frequency variations of the earth's magnetic field commonly have periods of 20 sec and longer, so that they do not induce large voltages in the receiving coil. High-frequency natural noise occurs at frequencies above 2 to 3 cps. This noise arises in rapid variations of the earth I s magnetic field and from distant spherics. The usual amplitude of high-frequency natural noise detected in a receiving loop with an area of one square kilometer is some tens of microvolts.

As a result, filters are necessary in a recording system for transient coupling which will reject the high-frequency noise. Naturally, such filters will distort the early stage of the transient response. In fact, because of the necessity for filtering, only the late stage of the transient coupling is recorded. In view of the high noise amplitude at frequencies above 2 or 3 cps, a high cutoff at 1 cps is used, providing a bandwidth from 0 to 1 cps. The resulting transient curve is termed the terminal stage.

If the frequency response of the filter is such that frequencies below fo = 1 cps are not significantly affected, then the transient response to a step wave will not be distorted for times ty 2: 0.16/ f 0 := 0.16 sec. Therefore, the terminal stage of transient coupling is taken to be the part occurring later than 0.1 sec following the current step.

Depending on the nature of the geoelectric section, this terminal stage may include both the minimum part and the late stage of the PT curve or only the late stage. If the total conductance is large, the terminal stage may also include part of the left-hand segment of the transient.

For depths to basement of 1.5 km or more, the total longitudinal conductance is usually at least 200 mhos, and the terminal stage of the recorded transient will provide enough information to determine the two generalized parameters, PT, min and S.

From an analysis of levels for field signals and the ratio of signal level to noise level, we may specify the following requirements for equipment:

1. A current of 25 to 30 A (or 50 to 60 A measured peak to peak) must be supplied to a source dipole having a circuit resistance of 10 to 20 ohms.


Fig. 8. Block diagram of receiving equipment.

2. The rise time for the source current, once the dipole has been energized, must not exceed 0.1 sec.

3. The amplifying and recording equipment must allow detection of signals of 10 to 15 /.LV in the frequency range 0 to 1 cps.

4. The input impedance of the amplifier must be 10 to 15 times greater than the resistance of the receiver coil.

Standard generating plants such as are used in electrical prospecting, including models ERS-23-53, ERGG-16.5-57, and ERGG-60 may be used as a primary power source.

Amplification of the transient signals may be accomplished using amplifiers of the types UMS-l, US-2, US-3.and ELU-61. Amplifiers of the type ELU-61 are used in the universal earth-resistivity system, EUL-60.

§ 3. Block Diagram for Equipment Used in the Magnetic Transient

Method

A block diagram for source equipment used in the magnetic transient method is shown in Fig. 7. The primary purpose of the source equipment is the generation of a current step of specified amplitude and duration, with the required rise time.

A direct current, developed by a rotating generator. is supplied to a switching system which closes the circuit to the source dipole at a specified instant in time. At the moment the current circuit is closed, a synchronous impulse is transmitted to the receiving equipment by radio. The radio transmitting equipment must be capable of transmitting this synchronizing reference pulse with a relatively short time delay.

A variety of switching methods may be used; mechanical, electromechanical, or electronic.

A block diagram of receiving equipment, containing elements for detecting, amplifying, and recording signals developed in the magnetic transient method is shown in Fig. 8. An ungrounded loop lying in the horizontal plane is used as a detector. The magnitude of the signal voltage induced in the ungrounded loop commonly is no more than a few tens of microvolts. Therefore, the amplifiers must have a low input noise - no more than one microvolt.

§ 4. Generating Equipment

At the present time, nearly every geological organization in the USSR does deep investigations with geophysical methods using electrical prospecting generators of types ERGG-60, ERS-23,and ERGG-16.5-57 [13]. These generators have been deSignated as standard equipment for use in direct-current soundings with the Schlumberger and dipole methods.

Generating plants of type ERGG-60 and ERS-23 each have two individual generators capable of providing 25 A at 460 V, with a power rating of 11.5 kW. These generators may be connected either in series or parallel in providing current to the load. Within the generating equipment, provision is made for commutation. or reversing the polarity of the current provided to the load. The commutator consists of four standard contactors of type KP-l. During

EQUIPMENT

switching, the rise time for a current step is about 0.10 to 0.15 sec, and regulation may be obtained after a few hundredths of a second.

55

In switching, the sequence consists of breaking the circuit to the first generator while making the circuit to the second. The opening of the switch contacts requires about 0.04 to 0.05 sec, while the closing of the switch contacts requires about 0.07 to 0.10 sec. Because of this difference in contact opening and closing times, the equipment is protected from short circuits during switching which might take place if Switching at both generators took place simultaneously.

However, when maximum power output is being used, inductive surges and arcing may take place. This res ults in burned contacts and to a longer rise time for the current pulse. Because of the difference in time for connecting and disconnecting the generator, the current surge takes place over a very short time, with the output voltage being raised. The surge may be suppressed with regulators, but the number of oscillations in the surge increases. This results in a transient process within the generator -regulator system pN -100 following switching, and the current to the source dipole varies by several percent for several tenths of a second.

The rise time for the leading edge of the current pulse may be reduced to 0.05 sec. This is done by increasing the voltage across the relay coil on the contactors to 23 V during closing., increasing the stiffness of the drawback spring on the contacts, and decreasing the voltage across the relay coil to 7 -8 V during opening.

We may note that it is possible to obtain a rise time as short as 0.01 sec using contactors of the KP-1 type. In order to do this, we need to provide a dummy or ballast load for the generator. When this is done, the time required for the build-up and decay of current does not enter into the rise time. The rise time, then,for the current wave to the dipole depends only on the closure time for the relay contacts, which is no more than a few milliseconds. A transient process takes place as the generator load is switched back and forth (during small times when the relay contacts are both open, less than 0.01 sec). The fall time for the trailing edge of the current pulse is somewhat longer, because of arcing.

Generating system model ERS-16.5-57 differs from generating systems ERGG-60 and ERS-23 in that a single generator of type PN-145 with a capacity of 33 A at 460 V and a power rating of 16.5 kW is used.

In conclusion, standard generator equipment may be used in the magnetic transient method.

§ 5. Receiving Equipment

There are two methods for amplifying slowly-varying voltages: direct amplification using either vacuum-tube or transistorized amplifiers, and conversion of dc voltages to ac voltages, followed by amplification and synchronous rectification.

Direct amplification with input levels of a few milli vol ts is diffic ult inas much as the noise in such amplifiers is large, and cannot be reduced below several tens of microvolts.

There are a large number of methods for amplifying dc voltages which are based on the general principle of converting the dc voltage to an ac voltage which has an amplitude or frequency which is proportional to the input voltage. This is followed by amplification of the ac voltage and rectification and filtering of the output. The primary difference between amplification following conversion and direct amplification is in the high degree of gain stability and lower long-term noise, as low as tenths or hundredths of a microvolt, which may be obtained using conversion-type amplifiers.

Conversion may be done using vacuum-tube circuits. Conversion with vacuum-tube circuits is obtained using such schemes as balanced modulators, multi vibrators , blocking oscilla-


tors and different types of switching systems. The main difficulty with vacuum-tube modulators is the comparatively high input Signal level which is required - (measured in millivolts or even tens of millivolts) - a comparatively high input noise figure, and a limited dynamic range for the input voltage.

There are a variety of transistorized converters which may be used. Conversion circuits in most are the same as those used in vacuum-tube circuits, but there are also other circuits based on the unique properties of semiconductor devices. Of particular interest and promise are converters based on the use of the Hall effect-variation of conductivity in a semiconductor or superconductor with variation in'"ambient magnetic field strength. However, at the present time it is very difficult to build a transistorized amplifier which has both low input noise and low thermal drift.

The input must be four-terminal, and so, the input current may be used to excite one of several types of photosensitive elements; either vacuum tube or solid-state light-sensing devices or a photoresistor to detect modulation of a light beam.

The mean drawbacks to light-sensing systems are the large parasitic or thermal EMFs (in millivolts), lack of time stability, a comparatively high drift rate, and a poor coefficient for conversion.

In addition, there is a general class of galvano-photoelectric converters. The operating principle for these is that very small rotations of the mirror on a galvanometer in the measuring circuit are detected with a photosensitive device powered with an alternating current. If the light beam reflected from the mirror on the galvanometer moves, then the current at the output of the photosensitive element changes and may be amplified as needed.

The main problems with galvano-photoelectric systems are their poor vibration characteristics, difficulties in construction and use, a high drift rate, and a comparatively low input impedance.

There are also direct-current amplifiers which make use of various magnetic modulators. The main problems with magnetic modulators are the high thermal drift and low input impedance (at best, only a hundred ohms).

In amplifying direct currents of small magnitude obtained from a high-resistance source, capacitive modulators are widely used (such conversion is said to be by means of dynamic capacitors).

Conversion is accomplished by varying the capacity of a condensor periodically. The capacity may be varied mechanically by moving one of the plates of a capacitor or by one of a number of electrical methods.

The main difficulties with capacitive modulating are the relatively high zero drift rate (several tens of microvolts per hour) and a low conversion coefficient.

Contactor-type modulators are the type most widely used; these include electromechanical (vibrator) modulators, electrodynamic modulators, electropneumatic modulators, and commutators. The primary advantages of contactor-type modulators are the high zero stability (drift is measured in microvolts per hour), low inherent noise, and high input impedance. With contactor-type modulators, it is possible to build a direct-current amplifier with a threshold sensitivity of a hundredth of a microvolt (in fact, for fairly long times, ranging from seconds to tens of seconds).

The shortcomings of contactor-type modulators are the mechanical instability (the life of the contacts commonly is only a few hundred hours) and problems of adjusting contact clearances, which must be done periodically. However, in most cases the advantages of contactortype modulators outweigh the disadvantages.

EQUIPMENT 57

TABLE 2

Amplifier characteristic UMS-1 US-2 US-3 ELU-61

Bandwidth with response within ±5o/", in cps ................ 0-4 0-10 0-15 0-10

Nominal sensitivity. IlV /mm •••••••• 1.0 1.0 0.5 1.0 Inherent noise. IlV at input ......... 1.0 2.0 0.5 1.0

Input impedance. kQ . ........... 600 300 40-300 20-1000

Characteristics of galvanometer Ao. a/mm ................ 10-8 10-6 10-6 10-6

fres. cps .............. 15 130 130 130

Ri' Q ................. 500 30 30 30 Type of modulator .............. RPB-4 VP-55 VP-55 RP-4

As noted in §2 of this chapter, at the present time, amplifiers of types UMS-1, US-2, US-3, and ELU-61, developed at the Electrical Prospecting Laboratory of the All-Union Geophysical Research Institute, are used in recording the late stage of the transient magnetic field.

These amplifiers use electromechanical converters to convert low-frequency signal voltages to an alternating square wave with a frequency of about 550 cps and an amplitude proportional to the input voltage. After modulation, the ac voltage is amplified to the desired level with a standard type of ac amplifier and then rectified. The high-frequency components of the square wave are filtered with a simple resistor-capacitor filter, and only the low-frequency amplified signal voltage is recorded with a galvanometer.

Use of the amplifier for very low frequencies is limited by the drift in the amplification factor of the amplifier and in the characteristics of the electromechanical modulator. Normally, the drift is 3 to 5 J.' V /hr .

The high-frequency limit of the amplifier is determined by the frequency of modulation, the time constants of the input circuit, the output filter, and the recording galvanometer.

Inasmuch as the modulation produces a quantized signal (that is, the signal is not continuous, but varies stepwise in time), the modulation frequency must be Significantly higher than the frequency of the signal to be amplified. If the modulation frequency is f 1 and the signal frequency is 12, there will be f 1/12 points sampled during each cycle of the signal. The reliability of sampling each point is Y2i1' Thus, the greater the ratio of the frequencies, the more precise will be the measurement.

The time constant for the input circuit must be much less than the period of the highest frequency contained in the signal. The same requirement must be met in the filter.

The resonant frequency of the recording galvanometer must be significantly higher than the highest frequency in the signal (see Table 2).

~! !LJ Relay

Fig. 9. Block diagram of UMS-l.

Figure 9 shows a block diagram of an amplifier used in recording magnetic transients (type UMS-1). The amplifier operates in the following manner. Let the switch contacts 2 and 3 be closed initially. A charge flows into the input capacitor, C!,so that a voltage U3 appears across the resistor R, which is amplified and provides a charge to the output capacitor, C2• The charging period has a length of Y 2 f 1 sec onds. Then the modulator contacts are switched, with contacts 1-2 being closed.

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EQUIPMENT

The capacitor Ci discharges through the resistor R i . The polarity of this voltage is opposite to the polarity of the voltage during charging.

59

The voltage from this charge is amplified and appears at the output of the amplifier. The voltage which charges the !capacitor Ca is the sum of two parts: a voltage I from capacitor C2 during the charging cycle for the input capacitor (contact 2 closed to 3) and a voltage from the discharge of the input capacitor (con-tact 2 closed to 1). Under steady-state opera

tion, the charging and discharging voltages are equal, and thus, the output capacitor is charged to the desired voltage level. Within the amplifier circuit there are two charges built up which are stored during the first part of the commutation cycle and then drained off into a third capacitor Ca during the second part of the commutation cycle.

The use in the amplifier circuit of three charging circuits provides a frequency roll-off characteristic for the amplifier very near the modulation frequency.

Figure 10 shows the circuit for the type UMS-1 amplifier. The ac amplifier consists of a pair of dual triodes, type 6N3P (Li -L4). The band width for the ac amplifier is 400 to 700 cps and may be changed through frequency-dependent feedback. In steady-state operation, the amplifier stages drive resistances R 4 , Ra, R 14 , and R i9 which are in the cathode circuits, and so, provide phase inversion.

A polarized relay, type RPB-4, is used for a relay-modulator. Relay contacts 1-2 serve to convert low-frequency input signals to an alternating square wave, while relay contacts 2-3 are used for synchronous rectification of the output. An R-C filter (Ri , Ci , R 2) is placed at the input of the ac amplifier to decouple the input circuit from the output circuit and from the power circuit for the relay coil.

The frequency of commutation is nominally 500 cps, but may be varied through a narrow range (±50 cps) with the resistance R 28 • This control is usually necessary in adjusting the commutator relay. A calibration circuit, using "Saturn n dry cells with a capacity of 3 A -hr, is provided. This voltage may be regulated with a monitoring meter. A small switch is used to connect the calibration voltage in place of the signal input.

Plate voltage is provided by one type 100-AMGTs-2 (BAS-BO) battery. To avoid crossfeed between stages, the supply to each stage is filtered with an R-C filter (R6Ca, R11C5 , R i6C9, R i6C11 , RaiC2Q). Heaters are powered with a single storage battery of type 5NKN-60. The chassis is not grounded to the outside case of the equipment. The heater circuit is not grounded.

The type US-2 amplifier is a later version of the UMS-1 amplifier. The modification includes use of a special relay modulator, a resonant-vibrator modulator of type VP -55, the characteristics for which are much better than those for the type RPB -4 relay. In fact, use of a resonant system in the relay circuit makes initial adjustment of the relay a simple matter. Provision is not provided for regulating the relay over a wide range, but the relay is very stable and not sensitive to vibration or shock.

While a voltage-sensitive galvanometer is used in recording the output of the UMS-1 amplifier, a current-sensitive galvanometer may be used with the US-2 amplifier. This results in a recording system which is less sensitive to erroneous recording caused by mechanical shock to the galvanometer assembly. The US-2 system contains gain control so that the sensitivity may be varied from 0.5 to 5 fJ.V Imm.


A block diagram of the US-2 circuit is shown in Fig. 11. The amplifier operates in the following manner. Let the contacts 2 -3 of the modulator relay be closed initially. The input capacitor, Cp is charged; a voltage is developed across resistance R 1, and this is amplified to charge the input capacitor, C2• The charging cycle lasts for 1/1100 sec, after which contacts 1-2 of the relay are closed, discharging the input and output capacitors.

Figure 12 shows a circuit diagram for the US-2; ac amplification stages make use of pentodes of type 6 Zh5P or 6 Zh3P. The first stage of amplification makes use of a type 6 Zh3P tube which has a very low noise figure. The two stages following make use of type 6 Zh5P pentodes. The bandwidth of the amplifier at the high-frequency end is limited by capacitors C",Hs in a frequency-dependent feedback circuit. This provides a very low noise amplifier.

In a finished amplifier with the modulator relays properly adjusted, the average noise does not exceed 1 p.V. At random times, noise termed "bounce" noise is recorded. The amplitude does not exceed 2 f.1.V, The noise is generated at the relay contacts, caused by rapid changes in the area of the contactors making contact, as well as other factors.

Relay modulators must be protected from humidity. The relay system should be encapsulated in paraffin or collodion.

Calibration of the US-2 amplifier is done in the same way as in the UMS-l, except that the square-wave calibration signal in the US-2 is developed using an auxiliary relay connected to a battery. However, closure of the relay contacts may excite microphonic noise in the amplifier tubes, resulting in recording impulsive noise.

The amplifier is fabricated on a boxlike chassis contained in a steel outer case. The chassis is insulated from the outer case with masonite.

The type US-3 (Fig. 13) is a modification of the US-2 amplifier, but differs from the earlier model in that potentiometric cancellation of spontaneous polarization is provided at the input, and the modulator has frequency selection; the commutator is driven by a precision oscillator. With this approach, the input noise for the amplifier is only 0.3 to 0.5 f.1.V (rms); other characteristics are similar to those of the US-2.

After the required amplification, Signals are recorded using standard oscillographs of types EPO-4, EPO-5, and EP07b which are used in electrical prospecting.

In 1961, the Mitishchinkii instrument factory in Moscow started production of a universal electrical exploration system, deSignated EUL-60, suitable for prospecting with the various direct-current methods, the telluric-current method, and the transient-coupling method. The EUL-60 system was developed at the Electrical Prospecting Laboratory of the All- Union GeophYSical Research Institute.

For operation with dipole-sounding methods, the telluric-current method, and the transient-coupling method, the EUL-60 system provides two universal vacuum-type amplifiers of type ELU-61.

The ELU-61 amplifier (circuit shown in Fig. 14) is a direct-current amplifier with an electromechanical modulator. The amplifier circuit contains automatic compensation, type EDA-57, developed at the Automation Institute of the Ukrainian SSR under the direction of K. B. Karandeevand L. Ya. Miziuk.

The nominal sensitivity of the amplifier is one f.1. V / mm, and may be varied in steps from 1 to 16 f.1.V/mm. Gain is controlled by varying the amount of negative feedback.

The amplifier circuit provides for change of bandwidth through wide limits. This allows the use of the optimum bandwidth,for any particular application control of bandwidth is accom-

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plished by varying the time constant of the input circuit. This usually is not precise, inasmuch as change in the time constant of the input circuit causes a change in the input resistance of the amplifier and moreover, since a large amount of negative feedback is used, this may lead to phase distortion, and in extreme cases, to self-oscillation of the amplifier. Furthermore, changes in the capacitor at the input changes the phase shift in feedback from negative to a combination of positive and negative. This may not necessarily cause oscillation, but will certainly cause distortion of the impulse response. These undesirable effects may be neglected only when the impedance of the signal source is very low.

The amplifier is provided with a calibration network so that square waves with amplitudes of 20, 40, 80, 160, or 320 JJ. V may be fed to the amplifier. The dc component of the signal (that is, spontaneous polarization at the measuring electrodes) may be cancelled with a special potentiometer at the input.

The ELU amplifier uses a standard telegraph relay, type RP-4 for a modulator. The modulator is driven at 230 cps by an R -C oscillator.

The ac amplifier consists of three stages which use pentodes of type 6J3P. The second stage has a twin-T bridge, providing negative feedback to stabilize the amplifier. This reduces the input noise to a large degree.

§ 6. Particular Methods for Conversion

The basic component of an amplifier which determines its capabilities is the converter. Figure 15 shows a Simplified circuit diagram for the input circuit of a dc amplifier with a synchronous electromagnetic converter. Inasmuch as in the general case, the time at which the converter contacts close is not precisely the same as the time at which the other converter contacts open, we must define a correction factor, y, the length of time during which contacts remain open divided by the commutation period. Obviously, 0 :s y :s l.

If we consider that the closing of the circuit to an input signal E (w) and its breaking does not lead to transient processes, then the source of the input signal [E (w)] may be represented as an equivalent generator of square waves with the frequency of commutation fk = I/Tk,and with an amplitude which follows the input signal.

During a commutation period, the converter contacts go through the following sequence: (1) The capacitor C is charged by the input voltage, E(w); during this stage, the contacts remain open for a time t3 = yTk; (2) the capacitor C discharges; the contacts remain closed; this stage has a duration tp = Tk - ts = Tk (1 - Y ) .

A study of converter operation indicates that:

a) The duration of the transient in the input circuit of a converter is determined mainly by the time constant of the circuit, and does not depend on the commutation frequency or phase.

b) In conversion, in addition to the primary phase shift between the harmonics of the converted voltage and the excitation voltage, and depending on the electrical characteristics of the converter circuit, there will also be a linear phase shift cpN which is a function of the RC time constant of the input circuit. This phase shift must be compensated because synchronous rectification is used at the amplifier output (as is

EQUIPMENT

well known, a synchronous detector has a maximum transfer coefficient for inphase voltages).

65

c) The generation of harmonics by conversion leads to development of a signal with discrete frequencies which are harmonics of the converter frequency.

d) By decreasing the time constant of the input circuit, it is possible by varying the form of the converted voltage to make the amplitudes of the higher harmonics comparable to or even larger than the amplitude of the fundamental.

In amplifying the signal, it is necessary to have a bandwidth:

AF::::::::NUl".

where N is the number of harmonics of the converted voltage which have an amplitude comparable to the amplitude of the fundamental, and so, may not be neglected.

Usually wk » wc, so that the energy in the signal is concentrated over a narrow range of frequencies. In amplifiers with a relatively high input level - greater than 1 m V - spreading the bandwidth is no problem.

In high-gain amplifiers which operate with input signals of a few microvolts, the question of the bandwidth is important inasmuch as widening the pass-band of the ac amplifier increases the noise.

Amplifier noise largely originates in the converter. It has been found that the making and breaking of mechanical contacts generates small charges. The magnitude of these signals also depends mainly on differences of potential between the contacts, how clean the contact surfaces are and abruptness with which the contacts make and break. For the best type of contacts, which are made from gold and chrome, the magnitude of these charges does not exceed 10-13 to 10-14 C.

In practical conditions, using standard silver -chrome contacts with reasonable surface cleanliness (wiped on velvet wipers), the charge developed is at least 10-9 to 10-to C. The higher noise level is caused by the development of an oxide coating on the contacts. For signals with magnitudes less than 10 MV, an oxidized surface has a large time-varying resistance (3000 to 15,000 Q).

It has been observed that the contact noise has a comparatively low-frequency character -lower than the conversion frequency. Also, there is one more noise mechanism at the contacts - noise from contact bounce. Bounce is caused by resonance in the mechanical latching of the relay arm which is not damped out immediately, permitting the relay contact to rebound slightly after contact is first made, momentarily breaking the contact. Commonly, this is observed as a discontinuity at the leading edge of the converted square-wave signal. It is obvious that such bouncing is very irregular in time. Bounce noise leads to a significant increase in the noise level of amplifier using conversion.

In addition to the ones which have been conSidered, there are many other sources of noise in an amplifier, and it would be difficult to list them all. Let us mention a few: thermal noise in components in the input circuit; noise from the Johnson effect, depending on the resistance of the input circuit; shot noise in the first amplifier tube; Johnson noise in the cathode circuit of the first amplifier stage; thermoelectric EMF's developed at junctions between unlike metals; and noise from contact with insulated, polarized surfaces.

In designing an amplifier for use with the magnetic transient method, a compromise must be reached between various requirements: 1) maximum gain and low noise; 2) minimum amplitude and phase distortion.


Fig. 16. Transient response to a one -sided rectilinear impulse.

Fig. 17. Transient response at the trailing edge of a one-sided rectilinear impulse.

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Fig. 18. Transientresponsefor a rectilinear impulse of alternating polarity.

Fig. 19. Voltage atthe output of the alternator for a sinusoidal input.

The requirement for high gain and low noise corresponds to the use of wideband response, but with steep cutoff at the band limits.

Minimum amplitudes and phase distortion may be obtained only if the amplitude -frequency characteristic is flat and the phase characteristic is linear within the pass band. It may be shown that the phase shift in an electrical system depends on the steepness of the flanks of the amplitude response curve. Thus, if the amplitude -frequency curve for an amplifier has a segment with rapidly changing slope, this may lead to considerable phase distortion. In order to obtain minimum phase distortion, the amplitude characteristic must not have rapid changes in slope within the pass band. The amplifier should also have good transient response. We may specify that the response time for a stop input should not exceed 0.05 to 0.10 sec. The transient response of various types of amplifiers has been well studied.

It is of interest to consider the transient behavior of a converter, which may be done by looking at the response to a series of impulses. If the input is a one -directional impulse of sufficient duration, two transients may be distinguished; one at the leading edge and one at the trailing edge. The transient at the leading edge starts at the zero level and approaches the steady-state level (see Fig. 16). Also, it may be shown that URz + URp = EO. Because an

ac amplifier will not pass a dc level, the transient process does not approach a steady -state level, but rather, the zero level. The wave front may be determined with a precision Tk·

The transient response at the trailing edge is given by the discharge of the capacitor C. When contacts 1 and

2 are closed, the discharge is through the resistance R, and when they are open, through the signal source resistance. If Ri «R, the discharge time of the capacitor is tp ~ 3T , and there is no varying component in the converted voltage (Fig. 17).

Thus, there is no oscillatory transient developed at either the leading edge or trailing edge of the converted voltage, and the output of the converted is a "smoothed" one-directional impulse. Also, the leading edge is specified with a precision of 1/2 Tk.

For a time-varying voltage applied to the input of the converter, the transient response is determined by the same process. The transient response at the trailing edge of the pulse may be thought of as consisting of two parts: a voltage on the capacitor C which varies from Ec to zero and a voltage on the capacitor C which varies from zero to Eo.

One may calculate that the duration of the first part of the transient is 0.7 RC, while the duration of the second phase is 3 RC. Therefore, the transient has a duration tn::: 3.7 RC. As in the earlier case, the transient returns to zero level (see Fig. 18). Therefore, for the variable component of the converted voltage, transient effects are virtually nonexistent. A study of the response when a sinusoidal input is applied permits the determination of the bandwidth of a converter (Fig. 19).

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Fig. 20. Circuit for relay control.

a

EQUIPMENT

The transfer function of a converter may be written in the form:

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67

b where m = Tc/Tk, q = e -Tk/2r , p = 1 - e -Tk/2r , and Tc is the signal period .

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Fig. 21. Recordings with the model EO- 7 oscillograph.

Consideration of equation (39) leads to the realization that the transfer function for m> 50 is approximately unity, but falls comparatively slowly at lower val ues . However, equation (39) does not take into account that the frequency of the signal and the commutation frequency may not form an integer ratio. Therefore, the transfer function varies with signal frequency and amplitude as:

[ cos~ sin~ ] U =~ __ m_ __m _ __ 1_ a 'It m + 1 + In -I m+ 1

§ 7. Alignment of an Amplifier

Fundamentally, the quality of an amplifier is determined by the relay-converter. This relay must be correctly adjusted. Figure 20 shows a method for regulating relay so that the closure time for each contact will be the same.

The Simplest means for adjusting a relay uses an electronic oscillograph of type EO-7 and as a current source a 1.6 V battery. The battery and a ballast resistance of 10,000 Q are connected in parallel to the input of one of the oscillograph channels. As the relay operates, opening and closing the contacts connected in parallel wi th the oscillograph input, voltage is applied to the input part of the time , and not for the rest of the time.

Figure 21 shows oscillograms recorded with an oscillograph of the EO-7 model . The wave form for the converted voltage must be as shown by the trace in Fig. 2a. Any other wave form is undesirable because they would indicate either an increased noise from the converteror a decreased sensitivity for the amplifier. Permissable differences in closure times of the relay contacts are no more than 30%.

The relays may also be adj usted to reduce contact bounce. Bounce is controlled by the tension of the spring on the contactor arm of the relay. Bounce and distortion of the contactors are both to be avoided.

If an amplifier is operated under conditions of high humidity or if the moisture protection of the relay is damaged, the noise in the amplifier may be sharply increased. Commonly, after immersion of an amplifier and operation for 5 or 6 hr, the relay contacts become oxidized. Under particularly adverse conditions, after lengthy storage, it is necessary to overhaul an amplifier,cleaning the relay contacts with sandpaper or emery cloth first, and then polishing them with velvet. Amplifiers should be overhauled after each 15 or 20 hr of operation.


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Fig. 22. Block diagram of an automatic relay coil voltage generator and generator for synchronous time marks.

JJUUW Fig. 23. Recording of current pulse and time marks.

§ 8. Equipment for Generating Synchronous Time Marks

Figure 22 shows a basic circuit for generating time marks and reference marks at the time of relay closure. The basic element of the system is a relay contactor with 24 positions , driven by a vacuum-tube oscillator. At the instant the relay contacts close, the cathode circuit of a type 6N3P tube is broken, developing a square pulse at the anode with a duration of 0.05 sec, which is fed to a radio transmitter.

By varying the frequency of the oscillator, the length of the current pulse can be controlled (and therefore, also the frequency of time marks) through reasonably wide limits.

Figure 23 shows a record of synchronous time marks.

§9. Summary

In field surveys with the ZSP method, in addition to the source and receiving equipment, various types of auxiliary equipment are required; cables, means for storing them, laying them out,and picking them up; electrodes and insulators, surveying equipment, and so on.

Because high currents are used in the current circuit with the ZSP method (30 or 60 A), a PMO cable is normally used with one, or better, two conductors. The current electrodes are formed from steel stakes, with a hundred or more being used at each installation. This large number of stakes is required to lower the grounding resistance to 25 n or less.

EQUIPMENT 69

As was mentioned in §1, the receiving system is a coil of wire with a total effective area of the order of one square kilometer. Usually, such a coil is formed from a multiple-conductor cable with the conductors all connected in series with a special connector. We recommend the use of a 20-wire telephone cable of type TR-VKSh 10 x 2 or TR-PKSh 10 x 2, or of a 36-wire seismic cable of type KPVS-18 x 2.

For operational simplicity, the receiving coil is usually square. If cables of type TRVKSh 10 x 2 or TR-PKSh 10 x 2 are used, the length of a side of the square coil should be 200m; if cable KPVS-18 x 2 is used, 150 m.

We do not recommend the use of a high-resistance cable. Other forms of auxiliary equipment may be used, and these are well described in handbooks of operational techniques.

PART IV. FIELD METHODS AND INTERPRETATION

§ 1. Field Methods

We will consider in a general way the types of conditions which are favorable for application of the ZSP method.

1. The depth to basement in the region to be surveyed falls in the range 1 or 1.5 to 3 or 4 km. Moreover, in applying the magnetic transient method, the resistant basement should be 3 or 4 times thicker than the overlying sedimentary column. Thin resistant layers will not appear as a resistant basement, even if the reSistivity is very high.

2. The conductivity and thickness of the sedimentary column must be sufficiently large.

3. There should not be any rapid lateral variations in the section, including relief on the basement surface.

4. There should not be any marked relief at the earth's surface.

If these conditions are met, the magnetic transient method may be used to map relief of the basement surface on a scale such as 1:1,000,000 or 1:100,000. Features of the basement surface may be recognized and contoured if they have an amplitude of at least 10 to 15% of the depth of burial.

Under favorable geologic conditions, the magnetic transient method may be used to map changes in character within the sedimentary column (as, for example, facies changes), which is of considerable interest in evaluating areas with respect to probable oil and gas deposits.

A section in which the resistivity of the various layers increases progressively with depth is not favorable for the use of the magnetic transient method.

The planning, organization,and implementation of field work with the magnetic transient method are described in the Handbook of Instructions for Electrical Prospecting [13] and Manual for the Magnetic Transient Method [14].

In conducting a survey with the magnetic transient method, the transient voltage induced in a horizontal coil by a magnetic field generated by a current dipole located at a distance from the coil which is 5 to 7 times the depth to basement is recorded. The angle, e, between the direction of the current dipole and the line to the receiving coil must fall in the range 70 to 110°. Outside this range the voltage in the receiving coil is sharply reduced, because it varies as sin e. By placing the receiving coil at various angles about a single current dipole location, one may record several different transient response curves (usually 2 to 4 locations as in-


Fig. 24. Locations of recei ving coils in recording transient magnetic fields. 1) Measuring coil for recording a transient magnetic field (from source AB). 2) Reference point for the measured apparent resisti vity (Q).

P,

Fig. 25. Two-layer curves for a transient magnetic field with an inclined coil.

dicated in Fig. 24 are used). Each of these curves is referred to the midpoint of the line between the source and receiver, just as in the case of dc dipole soundings.

The line AB may conveniently be laid out along a road. A road section made up of two or three segments with different directions may be used if each segment is considered separately in computing the geometric factor for the setup (see §2). As was found in the theoretical discussion, if the length of the source dipole is no more than 1/5 of the separation between source and receiver, the source may be considered to be an ideal dipole. Usually, the length AB is less than 1/2 of the spacing. The receiver is a coil laid out in the vicinity of the observation point. Inasmuch as the diameter of this coil is small in comparison with the spacing, it can be considered to be a point detector for the magnetic induction. Therefore, the orientation of the long side of the coil is not significant. On the other hand, any slight inclination of the plane of the receiving coil from the horizontal will have a large effect on the observed resistivity, PT. This effect comes about because an inclined coil will have a voltage induced by the horizontal component of magnetic induction which may be quite large. At short times, the effect of an inclined receiving coil is intensified, with a maximum effect being observed in the early stage of the transient coupling. Also, inclination of the coil will have least effect on the observed value of S, and the strongest effect on the value, PT min.

Figure 25 shows an example of the distortion of a two-layer transient curve caused by an inclination of 10 for the receiving coil. rt has been determined that this is the greatest inclination which is allowable. The output voltage of the receiving coil is amplified and recorded. The pulse duration is selected so that the length is at least 30 to 60% of the duration of the transient response. In order to have a steady -state response, at least 15 pulses must be transmitted Each record is calibrated, using a calibration system built into the amplifiers. A typical record of a magnetic transient is shown in Fig. 26.

§2. Construction of Apparent Resistivity Curves

Theoretical curves for a transient magnetic field are the graphical presentation of the relationship of

p, 21t r< £

P;- = 3A8ql PI . sin ~

Because Pi and hi are unknown, field data are plotted on the horizontal axis as (2'/1"t)1/2 , measured in secV2 , and on the vertical axis as PT' measured in !1-m. The expression

is used iIi computing PT. In the general case of a segmented line AB, the geometric factor is computed from the formula:

FIELD METHODS 71

Fig. 26. Typical recording of a transient magnetic field.

If e is measured in microvolts and the current I in amperes, the geometric factor for each segment is of the following form:

K 2:tr'10-6

aABq sin e Tj ,

where 1] is a correction factor for the dipole source not being ideal, and which is given in Appendix 3 as a function of the angle e and the ratio r/AB; and q is the area of the receiving coil.

§ 3. Topographic and Survey Control

Methods for obtaining topographic and survey control for magnetic transient method are the same as those used in dc resistivity soundings. The object is the determination of the map coordinates and elevation of the reference point, Q, and the values for r, AB, e, and the locations of the centers of the source dipole and the receiving coil. The locations of the points A, B, and 0 (center of the receiving coil) may be obtained from topographic maps or aerial photographs. Lines are drawn on the base map connecting the ends of the current dipole with the center of the receiving coil. The distance r is computed from coordinates taken from the map, while the angle is measured on the map with an accuracy of 0.5 0

•

Table 3 lists the permisSible rms error in determining the map coordinates so that the rms error in computing the geometric factor for the array will be no more than 3.6%.

TABLE 3

Map scale rms error, r=8km r=IOkm r=12km r=14 km r=16 km

m

I: 100,000 40 ±54 ±78 ± 99 ± 118 ± 138 I, IOU,UOO 60 ±9 ±58 ± 85 ± 106 ± 129 I, 50,UOO 20 ±66 +87 ± 105 ±123 ± 141 I: 50,000 30 ±62 ±83 ± 103 ± 121 ± 139

From Table 3 we infer that in inhabited regions we may use maps on a scale of 1:100,000 for spacings of more than 8 km, and in uninhabited regions, for spacings of more than 10 km. This is based on the assumption that locations may be spotted on the map with rms errors of 0.4 or 0.6 mm.

§4. Reduction and Presentation of Survey Results

The oscillographic records must be reduced to find the magnitude of the transient voltage. At least ten pulses are worked up in order to minimize errors. The first step in data reduction consists in drawing a base line for each pulse. If low-frequency magnetic pulsations are weak, this base line may be very nearly linear. In other cases, the base line must be drawn in by eye, following the micropulsation wave forms. Each impulse is smoothed with an averaging line to remove high-frequency noise.


For each of the selected pulses, vertical lines are drawn on the record corresponding to times t == 0.20, 0.30, 0.50, 0.70, 1.0, 1.5, 2.0,3.0,5.0,7.0, 10, and 15 sec. The unequal sampling interval is based on the conservation of transient shape when plotted to logarithmic coordinates. The logarithms of these values form roughly an arithmetic progression. In measuring times, the recorded synchronous time marks are used. The distance between the base line and the smoothed transient curve is read at each of these times (see data form 8, Appendix 4). Then, the arithmetic average for all the values for the same time is taken. These values are converted to microvolts, using the calibration data.

The third parameter needed in calculating PT is the current strength. In the magnetic transient method, no record is made of the current wave form. The current strength is read from a meter, usually. In unusual cases, when surface conditions cause extensive electrode polarization and distortion of the current pulses, it is necessary to record the current wave form with an oscillograph. Otherwise, it is necessary to record the current wave form only once or twice a month.

After values for k, e, and I are determined for each time, values for PT are computed. These values are plotted on a bilogarithmic graph, with (21Tt)t/2 being plotted along the horizontal scale. The P T curve is thus constructed for times t > 0.1 sec.

§ 5. Logistic Considerations

A brief outline of the logistic requirements for the magnetic transient method is given in Tables 4 and 5, in terms of daily productivity of a party with a single set of recording equipment, and the numbers of crew required.

§ 6. Basic Interpretation of the Final Stage of the Magnetic Transient

At the present time, interpretation of PT curves can be considered only as a first approximation. One of the obstacles in interpretation is the lack of an adequate number of computed curves for moderate spacings. Most important of all, the techniques now being used permit recording only the late stage of the magnetic transient. This would be equivalent to the use of only large spacings in direct-current soundings. However, it is well known that interpretation of left-hand (large spacing) part of a direct-current sounding curve may be essential in providing information about the geoelectric section.

Two situations may arise in interpreting the terminal stage of the magnetic transient:

1. The terminal stage includes the minimum and the right extreme of the transient curve. This case is found for sections in which there is a relatively great thickness of conductive rocks over an insulating basement. The ranges of layer thicknesses and resistivities for which this condition is met depend on all the resistivity ratios through the section, and may be found for a specific situation by computing the magnetic transient curve. As a rough approximation, it may be said that the thickness of conductive layers just above basement must be at least as great as the thickness of more-resistant near-surface layers, and that the total longitudinal conductance of the section must be at least 300 mhos.

2. The terminal stage includes only the right extreme of the transient curve. This case is found in sections in which the resistivity continually increases with depth, or for a section which is relatively thin or has a relatively high resistivity. This case is most unsatisfactory because only the value for total longitudinal conductance for the section may be determined. In the first case, an additional important parameter may be determined - the apparent resistivity at the minimum of the transient curve.

Interpretations of magnetic transient curves are made in two steps; the first step is determination of the values for S and PT ,min; the second step is determination of the average longitudinal resistivity of the section, Pl' and the depth to basement H == SPl'

FIELD METHODS

TABLE 4

M Distance Area category

aximum II I III I IV paration between

Operating Conditions A V stations,

se r, km

Normal.j Diffi - I Normal· \

Diffi - I Normal· I Diffi -km cultt cultt cultt

5- 9 1.0-2.0 3.21 2.29 2.16 1.53 1.43 1.03 10-14 2.0-3.0 3.06 2.18 2.05 1.46 1.36 0.98 15-19 3.0-5.0 2.92 2.08 1.% 1.33 1.30 0.93 20-25 5.0-6.5 2.78 1.98 1.85 1.32 1.24 0.89

• The observed voltage is more than 15 to 20 J.1.V, and the noise level is less than 200/0 of the signal level.

tThe noise level is more than 200/0 of the signal level.

TABLE 5. Crew Requirements

Job description One receiver Two receiver

crew crews

Party chief 1 1 Assistant party chief 1 Geophysicists 2 2 Instrument operators 2 3 Geophysical interpreter 1 1 Surveyor 1 1 Surveyor's assistants 1 1 Electronic technician 1 1 Computers (by work volume)

up to 100 soundings/mo 1 1 over 100 soundings/mo 1 2

Foremen 2 3 Laborers 10 13

Totals Engineers 10 14 Laborers 12 16

Overall total 22 30

73

The set of curves for the late stage of the magnetic transient may be used in determining the total longitudinal conductance. This is done by selecting the theoretical curve which is closest in form to the field curve over the portion from the minimum to the right extreme. S-line, plotted on the set of master curves, is defined by the equation

P. (S, 't)2 ~= 21ts'ii;

or

The S-line intersects the PT = 1 Q-m axis at a value


Therefore, dividing the coordinate of this intercept by ..[10'/27r , we have

.,I'107 , rn.::7. , ~ S = 2'it" . y 21t t = 503 y 21t t.

The master curves may also be used for type A sections (those in which resistivity increases with depth) as well as for type H sections (those in which the resistivity-depth function has a minimum), but in the latter case, it is quite satisfactory if the two-layer curves for the magnetic transient are used.

In determining ,the total longitudinal conductance using the two-layer master curves, a field curve is superimposed on one of the master curves so that the segment of the field curve to the right of the minimum matches with the corresponding segment of one of the theoretical curves. The intersection of the S-line with the PT = 1 U-m axis is used in determining the value for the total longitudinal conductance of the section.

The values which are found for Sand P T, min are used to construct a profile or a contour map. The most important step in interpretation is the determination of a value for the average longitudinal resistivity of the section. In order to accomplish this, it is necessary to establish a relationship between PZ and P T , min and S.

An important property of such a relationship is the fact that the value for PT at the minimum of the PT curve is independent of spacing for spacings 5 or more times the depth to basement. Because the relationship between PZ ' P T, min' and S may be different in different parts of a particular survey area, the field data are first analyzed for regional patterns in the behavior of the magnetic transient. Curves may be cataloged on the basis of the presence or absence of a minimum, or the value of PT at a curve maximum or on other indicative parameters. Several situations may be recognized, depending on the characteristics of the geoelectric section:

1. There is a unique relationship between the ratio P T, min Ipz and the value for S;

2. There is a correlation between the ratio P T , min I Pz and the value for S;

3 . The ratio P T , min I PZ is essentially constant;

4. There is a relationship between the apparent reSistivity at the curve minimum and the two parameters, PZ and the resistivity of the layer just above the basement, P NO:

pi P, .=

mm ~o

In order to know which of these situations pertains, one must know the depth to basement, H, at several places in the survey area. These reference values for depth to basement are usually obtained from drilling or seismic survey. At these points, it is a simple matter to interpret the magnetic transient curves for S2 and PT, min' and to find the values P Z = HiS and P NO = pi I P T , min' Knowing P Z ' it is simple to specify which of the four types of relationships exists between Pz and P T, min' Using a graphical representation of the proper type of relationship, the value for Pz at each survey location is determined.

Results have shown that in many areas, the minimum of the apparent resistivity curve differs from Pz by no more than 20 to 30%. These data are used to construct structure maps or profiles for the depth to basement.

Inasmuch as present recording methods do not allow the recording of the complete magnetic transient curve, further interpretation is not possible. Some indication of lateral changes in the character of sedimentary layers may be deduced from variations in PT, min and S.

FIELD METHODS 75

§ 7. Use of Electric Log Data

Electric log data, particularly that obtained with the lateral sounding technique, is of great aid in interpreting the magnetic transient data. However, one of the primary results of comparisons is that the value of average longitudinal reSistivity of a carbonate layer determined by the magnetic transient method differs significantly from the value indicated by electric logs. While lateral sounding data may give a value for the reSistivity of limestones or dolomites of some hundreds of ohm-meters, the Pr, min value is 3 to 15 Q-m even when it is known that clastic rocks are virtually absent from the section. The total longitudinal conductance determined from electric logs may be ten times smaller than the value determined from magnetic transient data.

This difference is most obvious for surveys made in the eastern part of the Russian platform. For example, the value of S computed from logs from a basement well in the Kuibyshev area was 100 mhos, while the value of S obtained from the magnetic transient method was 500 mhos. In another case, near a basement well in the Orenburg area, the magnetic conductance gave 300 mhps, while electric log data gave a value of 70 mhos. These differences may be explained in a number of ways. First, it should be noted that groundwater data indicate con-nate water with very high salinity in rocks in the eastern part of the Russian platform. E. E. Belyakova has given the salinity as 270-275 g/liter. The resistivity of such a saline water is about 0.02 Q-m. Therefore, if a section of carbonates with a total thickness of one kilometer has joint porosity with an integrated thickness of 2 m, the conductance of the section will be 100 mhos and the average longitudinal resistivity will be 10 Q-m. In a study of joint porosity in carbonate rocks from the Second Baku, A. G. Mileshina [1] has found a joint porosity of 0.005, corresponding to an integrated joint width of 5 m per km of section. One may suppose that during drilling, these joints are flushed by drilling mud with comparatively high resistivity, so that the electric logs will indicate too high a resistivity for the conductive sections in the well.

§ 8. Distortions of Magnetic Transient Curves

As has been stated in earlier sections of this paper, a thin horizontal screening layer has practically no effect on a magnetic transient curve. This is not true for a dipping screening layer. If such a screening layer has sharp relief near either the source dipole or receiving loop, the magnetic transient curve will be distorted.

One of the typical forms for the distortion of an impulse is shown in Fig. 27. The strongest distortion is seen in the early stage of the magnetic transient, and decreases with increasing 1. Results have shown that the most severe distortions are observed when the rough surface of a screening layer is close to the source dipole.

Obviously, the deeper the screening layer is, the less will be the distortion caused by relief of the screening layer. The source dipole and receiving loop should be placed at locations where surface geology indicates shallow screening layers are least likely, in order to minimize such distortions.

Fig. 27. A typical form of distortion to the curve.

§ 9. Possible Uses for the Magnetic Tran

sient Method in Studying Structural Geology

An important advantage of the magnetic transient method is the possibility of studying conductive layers lying beneath a highly resistant layer. Therefore, one of the most obvious applications of this method would be in areas where the geologic section contains such resistant screening lay-


ers. There are two such regions in the Soviet Union - the Russian and west Siberian platforms.

The geologic section in the Russian platform area contains evaporites and carbonates of Permian age which serve as screening layers, while in the west Siberian platform, similar rocks are found in the upper Cambrian. The resistivity of these screening layers is so large that in many cases, even using spacings as large as 30 km with direct-current sounding methods, it is not possible to find the depth to crystalline basement.

Based on this premise, the Electrical Prospecting Laboratory of the All-Union Geophysical Research Institute conducted preliminary field work during the 1958 field sesson in the Saratovsk, Kuibyshev, and Orenberg areas. A total profile coverage of nearly 600 km was completed. As an example, let us examine the more interesting portions of the profile surveyed in the western part of the Kuibyshev area (from Shentala to Bobrovka).

In this area, several evaporite beds of Permian age serve as screening layers when directcurrent soundings are made. As a result, information from direct-current soundings is limited to the first hundred meters. In the western part of the area, one of these zones crops out.

The results of the magnetic transient survey along the profile, except for the northern end, were in excellent agreement with theory. At the northern end of the profile, the outcrop of the evaporite rocks caused marked distortions of the P T curve, so that accurate interpretation was not possible.

The relationship between the value PT ,min and the average longitudinal resistivity for the section, Pl' was established at one well (No. 402, Kokhani region). It was found that for practical purposes, PT, min = Pl. Interpretation based on this relationship led to the following description of the structure to basement along the profile. The basic structural feature is the Mukhanovsk depression, with a depth tobasementof3000m., To the north, this depression is buttressed by the Sernovodsk uplift, while to the south, it is bounded by a step, the south side of which is bounded by a step-fault in the basement.

The 1958 field work was particularly interesting in that the results could be correlated with the results of a seismic reflection survey. This field survey was carried by the VolgaUrals field office of the All-Union Geophysical Research Institute, along with their regular seismic exploration program. A comparison of the results obtained with the two methods (Fig. 28) shows a close agreement. Differences in depth to basement obtained with the two methods do not exceed 15%. It should also be noted that the cost of the electrical profile was less than a tenth of the cost of the seismic work. The 1958 surveys also showed that the average longitudinal resistivity of carbonate rocks in the west Russian platform was no more than 10 to 15 Q-m.

In view of the excellent results obtained during 1958, the method was widely applied in regional surveys during 1959 and 1960. In 1959, field surveys with the magnetic transient method were undertaken by the Kuibyshev Oil Production Research Institute, the Buguruslansk Geophysical Group, the Tatar Geophysical Trust, the West Siberian Geophysical Group, the Oil and Coal Geophysical Trust, the Special Geophysical Group, the Bashkirian Geophysical Trust, and the Volgagrad Oil Exploration Group.

Figure 29 shows the results of a magnetic transient survey conducted by party 17/60 of the West Siberian Geophysical Group in 1960 along a profile from Aban to Dolgii Bridge over the Kansko-Taseevski uplift. The survey showed a thickness of 2 km of conductive rocks lying beneath the basement as seen with direct-current sounding methods and the basement as seen with the magnetic transient method.

FIELD METHODS

S, mhos .Ptmin IZOO

10 IODO "... __ -..J ,/'1 4Q{J

___ ... "_ 8 fOO

,. ---- -------- _ -- -- _ ---------- -- ----....-- -,-----"--- " /tOO Kokhani Bobrovka Z ZOO

----------------------~--------------------------------__t0

1000

---- z 1000

'-------r------ .1000

f o " ~ lZ km ~ H, m

Fig. 28. Results of a survey with the transient magnetic field, method along the profile Shentala-Bobrovka. 1) Relief of the crystalline basement from a seismic reflection survey. 2) Relief of the basement from transient magnetic field sounding. 3) S-curve. 4) CurveforPT,min·

Aban

~xx + + + + +

Dolgii

Most

lOtIO

1(/(J()

H, m

Fig. 29. Geoelectric section along the profile Aban-Dolgii Most (from V. 1. Pospeev and 1. N. Gomashnas). 1) Basement surface from electrical sounding data (direct-current method), which is lower Cambrian evaporites and carbonates. 2) Basement surface as seen with the transient magnetic field sounding method.

77

Based on the results obtained by the All-Union Geophysical Research Institute, parties from the Kuibyshev Oil Production Research Institute and the Buguruslansk Geophysical Group conducted many magnetic transient surveys in the Kuibyshev and Orenberg areas. These surveys were used not only to determine the depth to basement, but also to map the changes in thickness of elastic sedimentary rocks. This was done by noting increases in the value of S and decreases of the val ue of PT ,min in areas where detrital rocks were thicker.

During 1960 and 1961, use of the magnetic transient method expanded rapidly. In order to evaluate the future application of the method, in the spring of 1960, at Oktyabryskii (Bashkiria ASSR) , results obtained with the magnetic transient method were reviewed at a symposium. Twenty different organizations reviewed their experience with the method at this symposium, and recommended wider application in the solution of the following problems: 1) regional surveys of the l?asement relief for depths to basement in the range from 1 to 3 km; and 2) location of zones of thickening in detrital sedimentary rocks.

The joint use of electrical sounding methods and other geophysical methods was recommended. The best combination of methods was thought to be the use of electrical sounding along with jump-correlation seismic reflection surveys.


In a summary opinion written by a Committee of Experts from the Ministry of Geology and the Oil Production Board of the USSR in 1961 concerning the use of electromagnetic methods, it was noted that in recent years, the basic method for oil prospecting has been seismic exploration and the growing complexity of the results was being reflected in higher overall costs in oil production and a diminishing success ratio in drilling new structures.

As a result, it was felt that the philosophy and economics of geophysical exploration should be re-examined to improve efficiency.

Work conducted in recent years with new high-resolution gravitometers and a variety of new electrical exploration techniques using ac fields (telluric-current method, magnetotelluric profiling and sounding, transient sounding,and frequency sounding) provided highly mobile and inexpensi ve exploration methods in comparison with seismic methods, particularly for use in the inaccessible eastern and northern parts of the country.

The Committee of Experts suggested that wide use of the new electrical exploration methods and the high-resolution gravity method in conj unction with seismic methods and exploratory drilling would lead to a much more effective program for evaluating oil and gas reserves. Moreover, the use of such a combination of methods would be expected to reduce Significantly the amount of seismic work required in exploration.

REFERENCES

1. B. A. Andreev, Geophysical Methods in Regional Structural Geology; Gosgeoltekhizdat (1960) .

2. M. N. Berdicnevskii and L. L. Vanyan, Electromagnetic Fields in a Thin-Layered Medium; Voprosi Razvedochnoi Geofiziki, No.2 (1961).

3. M. N. Berdichevskii and T. N. Zavadskaya, Concerning the Question of Transient Electric Fields in the Earth; Izv. Akad. Nauk SSSR Ser. Geofiz., No.3 (1955).

4. L. Z. Bobrovnikov, Theory of a dc Amplifier with an Electromechanical Converter; Prikl. Geofiz., No. 26 (1960).

5. L. L. Vanyan, Some Questions on the Theory of Frequency Sounding in a Horizontally Stratified Medium; Prikl. Geofiz., No. 23 (1959).

6. L. L. Vanyan, Elements of the Theory of Transient Electromagnetic Fields; Prikl. Geofiz., No. 25 (1960).

7. L. L. Vanyan, E. 1. Terekhin, and A. 1. Shtimmer, Method for Calculating the Wave Curves for Frequency Sounding; Prikl. Geofiz., No. 30 (1961).

8. L. L. Vanyan, Concerning the Resolution of Frequency Soundings; Geol. i Geofiz., No. 9 (1960) .

9. N. P. Vladimirov, et al., Concerning Experimental Studies of Transient Electromagnetic Fields in a Many-Layered Earth; Izv. Akad. Nauk SSSR, Ser. Geofiz., No.2 (1956).

10. Questions of Petroleum Reservoir Engineering; Proc. of VSEGEI, Vol. 18 (1956). 11. R. Gavelka, Four-Place Tables of Circular and Hyperbolic Functions for Complex Ar-

guments; Viewig, Braunschweig (1931). 12. A. I. Zaborovskii, Electrical Exploration. Gostoptekhizdat (1943). 13. Instructions for Electrical Prospecting. Gosgeoltekhizdat (1961). 14. Status in Electrical Prospecting of the Transient Electromagnetic Method for Solving

Problems in Structural Geology. VNII Geofizika (1960). 15. M. V. Nikolaeva, Concerning the Approximate Evaluation of Oscillatory Integrals. Tr.

Matemat. Inst. V. A. Steklova, XXVIII (1949). 16. I. N. Sneddon, Fourier Transforms. McGraw-Hill, New York (1951).

REFERENCES

17. V. V. Solodovnikov, Statistical Dynamics of a Linear System of Automatic Regulators; Fizmatgiz (1960).

79

18. A. N. Tikhonov, Concerning Transient Electric Fields in a Homogeneous Half-Space; Izv. Akad. Nauk SSSR, Ser. Geogr. i Geofiz., No.3 (1946).

19. A. N. Tikhonov and o. A. Skugarevskaya, Concerning Interpretation of Transient Electric Fields in a Layered Medium; Izv. Akad. Nauk SSSR, Ser. Geofiz., No.3 (1958).

20. A. N. Tikhonov, On the Asymptotic Behavior of Integrals Containing Bessel Functions; Dokl. Akad. Nauk SSSR, No.5 (1959).

21. A. A. Kharkevich, Spectra and Analysis. Consultants Bureau, New York (1960). 22. D. N. Chetaev, Computation of Nonstationary Electromagnetic Fields in a Nonhomoge

neous Medium; Tr. Geofiz. Inst., Akad Nauk SSSR, No. 32 (1956). 23. S. M. Sheinman, Concerning Transient Electromagnetic Fields in the Earth; Prikl. Geo

fiz., No. 3 (1947). 24. B. S. Enenshtein, Method of Studying Transient Electric Fields in the Earth; Dokl. Akad.

Nauk SSSR, Vol. 59, No.2 (1948).


APPENDIX 1. TABLE OF HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIONS OF COMPLEX ARGUMENTS FOR COMPUTING WAVE CURVES FOR A TRANSIENT FIELD

t

Iy = 0,02 I 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 I 0,20

0,00 100 0,0200100 0,0400100 0,0599100 0,0798 99 0.099799 0,1194 S8 0,139198 0,158697 0,178197 0,1974

-- --- --- --- --- --- --- --- --- --- ---02 157 0372 ~ 0509 ~ 0677 ~i 0858 ~ 1045 ~~ 1235 ~~ 1426 ~~ 1617 ~~ 1808 ~~ 1999 29

04 157 06601:~ 07451~~ 0869 ~~ 1016 ~i 1179 ~~ 13.50 ~ 1527 ~3 1707 ~~ 1889 ~g 2071 16

06 158 09661~~ 10261~ I 1191~g 12371~g 1374 ~~ 1523 ~~ 1682 ~g 1847 ~g 2016 ~ 2188 10

08 159 12791~~ J3251~~ 13981!~ 14941~i 16091~~ 17381~ 1879 ~~ 2028 ~~ 2183 ~~ 2343 8 -- --- --- --- --- --- --- --- --- --- ---

0,10 160 O,15961~~ 0,1633136 O,I6931:g O,1774 I:g O,18711~~ O,19831~~ O,21071~~ O,22411n 0,238:21~ 0,2529 6

-- --- --- --- --- --- --- --- --- --- ---12 162 19181n 19491~~ 19991~~ 20681:~ 21521:g 22501~ 23601~g 24801~~ 26081~~ 2743 5

14 164 22441~~ 22711~~ 23141~ 23731~~ 244i:~ 25331:~ 2631 J~ 27391~~ 28561~i 2979 4

16 166 257517~ 25981fg 26361~~ 26881gg 27531~~ 28301:~ 29181:~ 3016J~ 31211~~ 3234 4

18 169 29121~g 29321n 29661~3 30121~~ 30701~ 31 39Jro 32181~~ 33071:~ 34031i~ 3507 3

-- --- --- --- --- --- --- --- --- --- ---0,20 172 O,325517~ 0,32731i~ O,33031~i O,33451~ 0,339713~ O,34591~~ O,35311~3 O,36111~i (),36991;~ 0,3794

3

-- --- --- --- --- --- --- --- --- --- ---22 176 3606175 36221;~ 36491i~ 36861~! 37331~ 37901~3 38551~~ 39281~ 40081~ 4095

3" 8 24 180 3964179 39791;~ 40031n 40371~r 40801~~ 413111~ 41901~~ 42571~~ 43311~ 4411 2 7

26 184 4332184 43451~~ 43671~; 43981~ 44371~~ 44831~~ 45371~t 4598111 46661g~ 4739 2 7

28 189 471018~ 47221ro 474i~~ 47701~~ 48051~i 48471g~ 48961~ 49521~i 50141~ 5081 2

-- --- --- --- --- --- --- --- --- --- ---0,30 195 O,509919~ 0,51 IOl~ O,51281i3 O,51531~~ 0,5 I 851ig O,52241g~ O,52681g~ O,53191g~ O,53751g1 0,5437 2

-- --- --- - ~ --- --- --- --- --- ---32 201 550120~ 551l20g 55272rl 55501i~ 55791n 56141~~ 56541~~ 57001~~ 57511~~ 5807 2

34 208 591720~ 592620~ 59412n 59612~~ 59872~~ 60182~~ 60552~ 60961~g 61421~~ 6193 2

36 216 634921~ 635721~ 637021~ 63882g 64112g 643921g 6m2~~ 65092~ 65502~ 6595 1

38 225 679822~ 680522~ 681722~ 68332i~ 68532n 687821~ 690621~ 693921~ 69752~~ 7014 I

-- --- --- --- --- --- --- --- --- --- ---0,40 235 0,726723~ O,727323~ O,72832~ O,729723~ O,73142~: O,73352n O,73602i~ O,73882i~ 0,74192n 0,7452

I

-- --- --- --- --- --- --- --- --- --- ---42 246 7758243 77632-!~ 777224~ 778324~

I 779724~ 7815218 78352~~ 78582n 788.t2r~ 7912

44 258 82742~ 827825~ 828425~ 8293~ I 830425~ 831825~ 833424~ 83522n 83722n 8394

46 272 88 I 727i 882027~ 882427g 883026~ 883826~ 884826~ 885926~ 88712~ 888525~ 8901 0

48 287 939128; 939328~ 939528~ 939828~ 940228~ 940728~ 941327~ 942027: 942727! 9435 0

-- --- --- --- --- --- --- --- --- --- ---0,50 305 I,00003~ I,00003~ I,00003Og l,OOOO30~ l,00002~ l,000029g 1,000029g I,00002~ l,000028~ 1,0000

0

HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIONS OF COMPLEX ARGUMENTS 81

t

I y = 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38 I 0,40

0,00 96 0,2165 95 0,2355 94 0,254393 0,272992 0,2913 91 0,309590 0,327589 0,345288 0,362786 0,3799

-- --- --- --- --- --- --- --- --- --- ---02 12 2188 ~1 2376 ~~ 2562 ~g 2747 9i 2930 9g 3111 8g 3290 8~ 3466 8~ 3640 8~ 3812 95

04 36 2255 ~ 2437 ~l 2619 ~ 2800 ~ 2980 ~~ 3158 ~~ 3334 ~~ 3508 ~ 3680 ~g 3850 92

06 58 2362 ~~ 2537 ~g 2712 ~ 2887 :~ 3062 :} 3235 ~~ 3407 ~~ 3577 ~ 3746 ~g 3913 87

08 77 2506 ~ 2671 ~ 2838 ~ 3005 ~~ 3173 ~ 3340 ~ 3507 ~~ 3673 ~~ 3837 ~f 3999 82

-- --- --- --- --- --- --- --- --- --- ---0,10 93 0,2681 ~~ 0,2836 n 0,2993 ~g 0,3152 ~ 0,3312 ~g 0,3473 ~g 0,3633 ~3 0,3792 ~~ 0,3951 n 0,4109 76 -- --- --- --- --- --- --- --- --- --- ---

12 107 28831~~ 3028 ~ 3175 ~~ 3325 ~~ 3477 ~~ 3629 ~~ 3782 ~~ 3936 ~~ 4088 ~~ 4240 70

14 118 31081~~ 32421~~ 33801!M 3521 ?~ 3664 ~~ 3809 ?~ 3954 ~~ 4100 ~~ 4247 ~~ 4392 65

16 128 33531~~ 34781~ 36061~~ 37381~ 38721~ 40091~~ 4147 ~~ 4285 ?~ 4425 ~g 4564 60

18 136 36161~} 37311~ 38501~~ 39741~ 41001~: 42281~~ 4358;~~ 44891~ 4622 ~ 4754 55

-- --- --- --- --- --- --- --- --- --- ---0,20 144 O,38951g~ O,40011~ 0,41121~~ O,42271~ O,43441~6 0,44651~~ 0,45871~~ 0,4711 1M O,48361~~ 0,4962 51

-- --- --- --- --- --- --- --- --- --- ---22 151 41881:~ 42871~~ 43891~ 44961~ 46061~~ 47191~i 48331~~ 49501~~ 50671~~ 5186 47

24 158 44961rs 45871~ 46821~ 47811~i 48831~ 49881~ 50951~~ 52051~~ 5315124 5426 43 56

26 164 48181~~ 49011~~ 49891~~ 50811~ 5175I :g 52731~ 53731fi1 54751~~ 55781~~ 5682 39

28 171 51531: 52301~ri 53111~~ 53951~~ 54831~~ 55731~~ 56661:~ 57611:~ 58571~~ 5954 36 -- --- --- --- --- --- --- --- --- --- ---

0,30 178 O,55031~ 0,55731~~ O,56471~ O,57251~g O,58061~~ O,58891~~ O,59741~! 0,60621~~ 1,61511:~ 0,6240 33

-- --- --- --- --- --- --- --- --- --- ---32 185 58671g~ 59311~ 59991~ 60701~~ 61431~ 62201~~ 62981~~ 63781;?i 64591~t 6542 80

34 193 62471~~ 63051gr 63661~~ 64301~ 64971~~ 65661~~ 66371~~ 67091~ 67831~ 6858 27

36 201 66431~~ 66951~~ 67491~~ 680igg 68671gf 69281g~ 69921~ 70571~ 71231~~ 7191 24

38 210 70572~~ 7l022g: 71502~ 72011~~ 72531~~ 73081~J 73641~~ 74211~~ 74791~ 7539 21 -- --- --- --- --- --- --- --- --- --- ---

0,40 219 O,74892~g O,75282~f O,75692~g O,76132~ 0,76582g~ O,77051~~ O,77531~~ O,78021~1 O,78521~~ 0,7903 18 -- --- --- --- --- --- --- --- --- --- ---

42 230 79422i~ 79742i~ 80092ig 804521~ 80822:~ 81212g~ 8161 2gri 82012g<f 82431~·i 8285 15

44 241 84172~g 84432~: 84692i~ 84972i~ 85272n 85572:~ 858821~ 86192n 86522?~ 8685 12

46 253 89172sg 89352~ 8953216 8973~ 89932~g 90142n 90362i1 90572li 908021t 9103 8

48 267 944426~ 945325~ 94632~ 947325~ 948424~ 949524g 95062~ 951723~ 952922~ 9541 4

-- --- --- --- --- --- --- --- --- --- ---0,50 282 l,000027g I,000027g 1,000026& I,00002~ 1,00OO25g I,000025~ 1,OOOO24~ 1,000024& 1,000028g 1,0000 0


7

I y-O,02 I 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

0,00 - 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 I, 0000

-- --- --- --- --- --- --- --- --- --- --02 - *3612~~~ *57672M; *69371~~ *76281~~ *8075Ws *8386m *8613~ *8787~ *8924~ *90

1806 34

04 0 1967 ~~ 36201~~~ 48691~~i 5783 ~~ 6455~g 6961m 7351m 7660m 7909~ 8115 984

06 0 1339 ~l~ 2571 ~;: 3633 ~ 4513 ~~ 5228~~ 5809~~ 628~~ 66761~ 70001~ 7Zl5 670

08 0 I016~ 1983 ~~ 2868~ 36521t3CJ 4333~~ 4921~:~ 5418~~ 5846t~ 6215m 6534 508

-- --- --- --- --- --- --- --- --- --- --0,10 0 0,0819 3~~ O,16141~ 0,2363 ~~ 0,3052 m O,3675m O,4233m O,4727~1~ O,5164~~ O,5551~~ 0,5893

410

-- --- --- --- --- --- --- --- --- --- --12 0 0689 ~~ 1364 m 2011 ~ 2620 ~~ 3186~~ 3706~ 4180~r~ 4609m 4997m 5348

345

14 0 0596 2~ 1184 2~ 1753 ~~~ 2297 m 2812~~I 3294~~ 3741~~ 4156~~ 4536m 4886 298

16 0 0527 2~l 1049 2~~ 1558 2~~ 2050 ~~~ 25201~g 2967m 3388m 3783m 4151m 4495 264

18 0 0474 2~~ 0945 23i 1406 2~ 1855 2i~ 2288!~ 27041~~ 31001~ 3475m 38301~~ 4164 237

--- --- --- --- --- --- --- --- --- --0,20 0 0,0433 2i~ 0,0862 2i~ 0,1285 2~ 0,1699 iJ~ O,21011~ o,24891g~ O,2863m o,32201~ o,35601~ 0,3883

216

--- --- --- --- --- --- --- --- --- --22 0 0399 I~~ 0796 1~ 1187 1~~ 1572 I~ 19481~~ 23131~~ 2666J~ 30061~~ 3332U~ 3644

200

24 0 0372 I~ 0741 I~ 1107 l~ 1468 I~~ 18211~ 21661~ 25011~ 28271~ 31391~I 3441 186

26 0 0349 11 0697 I~ 1041 1~~ 1379 Ifs 17161~ 20431~ 23631~ 26731~I 29751~ 3267 175 174

28 0 0.330 9 0659 l~g OJ86 1~~ 1309 I~ 16271~~ 19401~ 22461~~ 25451~~ 28351~? 3118 165 165

--- --- --- --- --- --- --- --- --- --0,30 0 0,0315 15~ 0,0628 1~ 0,0940 I~ 0,1248 I~ O,15531~ O,18521~ O,21471~ O,24351~ O,27161~~ 0,2991

157 --- --- --- --- --- --- --- --- --- --

32 0 0301 I~ 0602 1~~ 0901 1~~ 1197 1~ 14901~~ 17791~~ 20631:g 23421~~ 26161~ 2882 151

34 0 0290 14~ 0580 1U 0868 I~~ 1155 1~~ 14381~ 17171~ 19931~ 22641~ 25301~ 2790 145

36 0 0281 14g 0562 14g 0841 1~ 1119 1~ 13941~~ IE661~ 193~I~g 2198i~ 24581~ 2713 141

38 0 0274 131 0547 13~ 0819 111 1089 I~ 13581~ 16231~1 18851~~ 21441~~ 23981~ 2648

137

--- --- --- --- --- --- --- --- --- --0,40 0 0,0268 13~ O,0535 1a3 0,0801 13~ O,I066 11i O,13281~ O,15881~ O,18451~ O,20991~~ O,23491~ 0,2595

131

--- --- --- --- --- --- --- --- --- --42 0 0263 13i 0525 13~ a787 I~ Ion IJ 13051~~ 15611~~ 18141~ 20641~ 23111~i 2553

131

44 0 0259 12~ 0518 12~ 0776 1J 1032 12~ 12871~ 15401~ 17901~~ 20371~ 22811~ 2521

130

46 0 0257 1~ 0513 12~ 0768 12j 1022 12~ 1275d 152512~ 17731:Z: 2018d 22601:g 2499 128

48 0 0255 12~ 0510 12~ 0764 1~ 1016 I~ 1267d 15161~ 176312~ 20071~ 2248 l1g 2486 128

--- --- --- --- --- --- --- --- --- --0,50 0 0,0255 tJ 0,0509 12~ 0,0762 1~ 0,1014 1~ 0,1265121 O,15131J O,1760tJ O,2003J~ O,224411~ 0,2481

127


T

x y =0,22 I 0,24 I 0,26 I 0,28 I 0,30 0,32 I 0,34 I 0,36 0,38 0,40

0,00- 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,00000 1,0000 0 1,0000 0 1,0000 0 1,0000

-- --- --- ~ --- --- --- --- --- --- ---02 483 *9126~ *9203~ *9268~ *9325~ *93743J~ *94182~~ *94562g *94912rs *95222~~ *9551 46

04 460 82864~ 8432~ 8~~~ 86673~ 87623~ 88472~ 89232~ 89912~? 90522~~ 9107 86

06 '20 75 10m 7712~ 78873~~ 80413~~ 81782~ 82992~ 84082~~ 85062:~ 85952~~ 8676 117

08 371 6810~~ 7055~~~ 72693~ 74592~ 76292~: 77822g~ 79192:~ 80442~} 815821~ 8262 138

-- --- --- --- --- --- --- --- --- --- ---0,10 321 O,6197fJ O,6467~~~ O,6709i~ O,69272~ O,71232~~ 0,73002:: O,74612~~ O,76092~ O,77442~~ 0,7868 152

-- --- --- --- --- --- --- --- --- --- ---12 272 5665i~g 5951i~g 621 1m 6447~~g 66632~~ 68592~g 70392g 72052~~ 73571~r 7498 158

14 231 5206m 550lm 577Ii~~ 6019i:: 6247iZ~ 64582g~ 66521~ 68321~~ 69991~~ 7154 160

16 196 48141~g 51101g~ 53841~ 56391~~ 5876:~ 6096:gJ 63011~~ 64921~g 66701~ 6836 160

18 166 4471i~ 4771J~ 5047:~~ 5305:~r 5546:~3 5772:8~ 59841~~ 61831~: 63691~g 6544 157

-- --- --- --- --- --- --- --- --- --- ---0,20 140 O,4189:~ O,4479:~~ O,4752:~~ O,5011:g O,5254:1~ O,54S3:~: O,5700:~~ O,59041~~ O,60961~~ 0,6278 158

-- --- --- --- --- --- --- --- --- --- ---22 119 3942:~~ 4226:~~ 4496m 4753:~~ 49961ij 5:230:~~ 5446:g~ 56541~~ 58501~~ 6037 149

24 102 3730:~ 4008:g)l 4273g} 4527m 4769m 5000gg 5220:~~ 5430:~ 5629'~g 5819 145

26 87 35481~~ 38191~ri 40801~~ I 43301~g 4570:~g 48001?Z 5020:g~ 5231:g? 5432 ~~ 5625 141

28 74 33921~~ 36571~A 39121~~ 41591~~ 43961n 46251~~ 48441~ 50551g~ 5257 ~~ 5451 187

-- --- --- --- --- --- --- --- --- --- ---0,30 64 O,32571~~ O,3516 j ;g 0,3767,~~ 0,40101r~ 0,4:244 j ;g 0,44711b~ O,468916~ 0,489916~ 0,5101 ~~ 0,5296 133

-- --- --- --- --- --- --- --- --- --- ---32 54 31431~~ 33961gg 36421~~ 38811~~ 41131~~ 4337,~~ 45541g~ 47641g~ 4966 ~~ 5161 130

34 46 30451~~ 32931~1 35361f~ 37711fl 4000 l fi 42221gg 44371e~ 46461gi 4848 ~~ 5043 127

36 39 29621~~ 32071~~ 344511~ 36771g 390311~ 41231g~ 43371g~ 45451~: 4746 ~} 4941 125

38 32 28941~ 31341~g 33691r~ 3598d~ 38221~~ 40401~~ 425316~ 44591~~ 4660 ~~ 4854 123

-- --- --- --- --- --- --- --- --- --- ---0,40 26 0,2837Ii~ O,30741~ 0,3306d! 0,35341~i 0.37551~ 0,39721~~ 0,41831g3 O,43881GG 0,4588 ~~ 0,4782 121

-- --- --- --- --- --- --- --- --- --- ---42 21 27921~~ 30271~~ 32571i3 34821ig 37021~~ 39171~; 41271~; 43311~~ 4530 ~~ 4724 119

44 16 27581:~ 299011~ 32191:~ 344216~ 366116~ 38751~! 40841~~ 4288 ~~ 4486 ~~ 4680 118

46 11 27341:~ 29651U 31921:~ 3414,~~ 3632,M 38451~~ 40531~~ 4257 ~~ 4458 ~~ 4648 Jl8

48 7 272011~ 295011g 317611~ 339810g 361510~ 382810~ 403.siO~ -9

4436 9~ 4629 117 4238 99

-- --- --- --- --- --- --- --- --- --- ---0,50 2 0,271511~ 0,29451l~ 0.317111~ 0,33921U~ 0,360910~ 0,3822iO~ 0,40291O~ O. 1232 9~ 0,4430 9~ 0,4623 117


t

x I y=O,42 I 0,44 I 0,46 I 0,48 I 0,50 I 0,52 I 0,54 I 0,56 I 0,58 I 0,60

0,00 85 0,396984 0,413682 0,4301 81 0,4462 79 0,4621 78 0,4777 76 0,4930715 0,508073 0,522772 0,5370

-- --- --- --- --- --- --- --- --- --- --02 6 3981 8g 4148 8~ 4312 8~ 4473 7~ 4631 7~ 4787 7~ 4939 7~ 5089 7g 5~35 7; 5379 85

04 19 4018 ~g 4183 ~r 4345 ~b 4505 ~~ 4662 ~~ 4816 ~~ 4967 ~: 5116 ~~ 5262 ~~ 5404 84

06 31 4077 ~~ 4240 ~ 4400 ¥~ 4557 ~~ 4712 ~g 4865 ~~ 5014 ~~ 5161 ~~ 5305 ~ 5446 82

08 43 4160 ~~ 4319 ~ 4476 ~ 4630 n 4783 n 4932 ~: 5079 ~; 5224 n 5365 ~ 5504 80

-- --- --- --- --- --- --- --- --- --- --0,10 55 0,4265 ~~ 0,4420 ~~ 0,4573 ;~ 0,4723 ;~ 0,4872 i~ 0,5018 g 0,5162 ~i 0,5304 ~g 0,5443 ~~ 0,5579 78

-- -- --- --- --- --- --- --- --- --- --12 66 4391 ~~ 4541 ~~ 4689 ~g 4836 ~ 4980 ~t 5123 ~5 5263 gg 5401 ~~ 5536 ~; 5670 75

14 76 4538 ~~ 4682 ;g 4825 ~~ 4966 ~g 5106 ~~ 5244 ~ 5380 ~~ 5514 ~~ 5646 ~ 5776 7:l

16 86 4703 ~~ 4841 ~ 4979 ~~ 5115 ~~ 5250 ~~ 5383 ~~ 5514 ~~ 5644 ~ 5772 ~~ 5897 70

18 95 4887 ~~ 5019 ~~ 5150 ~ 5281 ~~ 5410 ~ 5538 ~ 5664 ~~ 5789 ~~ 5912 ~ 6033 66

-- --- --- --- --- --- --- --- --- --- --0,20 104 O,50871~ 0,5213 ~ 0,5338 ~~ 0,5463 ~~ O,~586 ~~ 0,5709 ~f 0,5830 ~ 0,5950 gg 0,6067 ~g 0,6183 63

-- --- --- --- --- --- --- --- --- --- --22 112 53051~g 54241~g 55~31n2 5661 ~ij 5779~ 5895 ~ 6011 ~~ 6125 ~ 6237 ~ 6348 59 59

24 120 55381~ 56511~ 57631~ 58751~ 59861~ 60!J61~ 6206 ~ 6314 ~ 6421 ~ 6526 56

26 1211 57871~ 58931n 59981~ 61031A~ 62081M 63121~ 64151~t 65171~ 6618 : 6718 53

28 13t] 60521~~ 61501~~ 62481~~ 63471~ 644511~ 65~21~ 66191 ~~ 67351~~ 68291~~ 6923 49

--- --- --- - --- -- --- -- --- --0,30 143 O,63311:~ O,64221ll (1,65131~ O,66051~~ O,66961~~ O,67861~~ O,68161~~ O,69651~ O,70531~~ 0,7140 45

--- --- --- --- --- -- --- --- --- --32 151 66251:~ 67091:~ 67931!~ 68771~~ 69611~~ 70441r-1 7127t~~ 72091~i 72911~ 7371 42

34 1·,8 69341~~ 701O'3~ 70B71~~ 7103143 72,ol~g 73161~~ 73921~~ 7466129 7541 125 7614 38 .J8 .~7 37

36 166 71591~ 73271~~ 73961i: 7~651~! 75331:~ 76021~ 76691~ 7737f~~ 78031~j 7869 31

38 174 75991~ 705yl~~ 77201~~ 7780 lgg 78411g~ i9011~ 79611~ 80201~~ 80791~ 8137 3U

--- -- -- --- --- --- --- -- -- --0,40 182 O,79551~~ O,80071~ O,!l0591~~ O,81111~~ 0.81631~~ O,82141~~ O,82661~~ O,83171~~ O,83671~~ 0,8417 26

-- --- -- --- --- --- --- -- --- --42 196 83'28Ig~ 83701~~ 84131~~ 8-l5i~1 8-l991~~ 85421~: 85851~~ 86271~t 86681~: 8709 21

44 2()ol 8718195 87511~ 87841~~ 88181~~ 88511;~ 888-l1;~ 89171~ 89501~~ 89821t~ 9014 17 17

46 2119 9126°~; 9H91~~ 91721~~ 919jln 92181n 92711r~ 92641n 92861~ 93081~ 9330 12

48 219 9553~1~ 956520~ 95772~ 958919~ 960119~ 96131~ 96251~ 963617~ 964817g 9659 6

--- I-- -- --- --- -- --- --- -- --0,.50 2aD I,000022~ I,000021~ I,OOOO21~ I,00002og 1,0000206 I,OOOOI~ I,OOOOI~ l,oooolSg l,oooo17g 1,0000 0

I I

HYPERBOLIC AND INVERSE HYPERBOUC FUNCTIONS OF COMPLEX ARGUMENTS 85

t

JC )1=0,62 0,64 I 0,66 0,68 I 0,70 I 0,72 I 0,74 0,76 I 0,78 I 0,80

0,00 70 0,5511 GIl 0,564967 0,578466 0,591564 0,604463 0,616961 J,6291 6U 0,6411 61 0,652767 0,6640

-- --- - --- --- --- --- --- --- - --02 4 5519 J 5657 6~ 5791 s: 5923 6: 6051 ~ 6176 6~ 6298 6~ 6417 s: 6533 5~ 6646

70

04 13 5544~ 5680 ~~ 581 4 ll 5944 U 6072 M 6196 ~~ 6318 ~g 6436 ~ 6552 5~ 6664 70

06 21 5584 ~g 5719 ~g 5852 ~g 5981 ~ 6107 ~g 6231 ~ 6351 ~~ 6468 ~~ 6583 ~~ 6694 69

08 29 5641 ~~ 5774 ~~ 590 1 ~: 6032 ~~ 6156 ~~ 6278 ~~ 6397 ~~ 6513 ~~ 66:26 ~~ 6736 63

-- --- --- --- --- --- --- --- --- --- --0,10 37 0,5713 ~ 0,5843 ~~ 0,5972 ~ 0,6097 ~ 0,6219 ~ 0,6339 ~g O,6~56 ~~ 0,6570 ~~ 0,6681 g~ 0,6789 67

-- --- --- --- --- --- --- --- -- --- --12 45 5800 ~: 5928 :~ 6053 ~l 6176 ~ 6296 ~ 6413 ~~ 6tm~ 6639 ~~ 6748 ~~ 6854 65

14 53 5903 ~~ 6027 ~~ 6149 ~ 6269 ~~ 6386 ~~ 6500 ~~ 6612 ~~ 6721 ~~ 6827 ~~ 6931 64

16 61 6020 ru 6141 ~~ 6259 ~ 6375 ~~ 6489 ~~ 6600 ~ 6708 ~~ 6814 ~~ 6917 ~ 7018 62

18 68 6152 ~~ 6268 ~ 6383 ~~ 6495 ~g 6605 ~ 6712 ~ 6817 ~1 6919 ~ 7019 ~~ 7117 59 -- --- --- --- --- --- --- --- --- --- --

0,20 75 O,6298l~ 0,6410 ~ 0,6520 ~~ 0,6628 ~ 0,6733 ~ 0,6836 ~ O,69J7 ~g O,7U36 ~~ 0,7132 ~~ 0,7226 67

-- --- --- --- --- --- --- --- --- --- --22 82 6457 ~ 6564l~ 6670 ~~ 6773 ~~ 6874 ~g 6973 :: 7070 :~ 7164 ~ 7256 ~~ 7347 55

24 89 6630 ~ 6732 ~ 6832 ~~ 6930 ~ 7027 ~~ 7121 ~ 7213 ~~ 7303 ~ 7391 ~ 7477 52

26 96 6816 : 6!H2 ~ 7007 ~~ 7100 ~~ 49 7HJI ~~ i21H ~ 7368 ~~ 7453 ~~ 7536 ~~ 7618

28 102 7015~ 7105~ 7194 ~ 7282 :~ 46 7367 ~g 7451 ~f 7533 ~~ 7614 ~g 7692 ~~ 7768

-- --- --- - --- --- --- --- -- --- --0,30 109 0,72261~ O,73111~ O,73941~ 0,7475 ~ 0,7555 ~ 0,7633 ~ 0,7709 g~ 0,7784 ~~ 0,7857 g~ 0,7928

43

-- --- --- --- --- --- --- --- --- --- --32 115 74501~ 75281~ 76041g~ 76801~

40 7753 ~~ 7825 g~ 7896 g~ 7965 ~~ 8032 gg 8098

34 122 76861~g 36 77571A~ 7827111 • 34 78951~ 79631~ 80281g~ 8093 ~~ 8156 95 31 8217 g~ 8277

36 128 79341~~ 79981~ 8C611~r 812i~g 81821~ 82411g~ 82991g~ 83561~ 8411 ~~ 8464 32

38 134 81941~ 82501~g 83051~ 8J601~~ 84131~~ 84651~ 85151g~ 85651g~ 86131~1 8661 29

-- --- --- --- --- --- --- --- --- --- --0,40 140 O,84661~: O,85141~~ O,85611~g 0.86081~~ 0.86531~g O,86981~~ O,87411~~ O,87831~ O,88251gg 0,8865

25

-- --- --- --- --- --- --- --- --- --- --42 146 87491~~ 8789t~ 8828t~~ 88661~~ 89041~ 89401i~ 89761g 901111~ 9044tl~ 9077

20

44 t52 90451~g 9075t1~ 91051~ 91351~: 9164t~ 91921i~ 92191i~ 9246t lg 927211~ 9298 16

46 158 93521ft 937311g 9393tn 9414t~g 94331~g 9453tag 947112~ 949012~ 950811g 9525 t\

48 164 96701~ 968115~ 9692t4~ 970214~ 97121~ 972213~ 973213g 9741t2~ 975012l 9759 6

-- --- --- --- --- --- --- --- --- --- --0,50 171 l,OOOOtsg I,OOOOlsg l,OOOOt~ l,OOOO14g l,OOOOI~ l,OOOO13g 1,0000136 l,OOOO12g l,OOOOl~ 1,0000

0


x I I y-O,42 I 0,44 I 0,46 I 0,48 I 0,50 I 0,52 0,54 I 0,56 I 0,58 I 0,60

0,00 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1.0000 0 1,0000

-- --- --- --- --- --- --- --- --- --- --02 225 *95762g *96002~~ *96221~g *96421~ *966017g *967716A *969315~ *970814~ *972213~ *9735 13

04 222 91582~~ 92051g~ 92471~6 92871;~ 93241~~ 93581n 938g1f~ 94191:~ 94471~~ 9473 25

06 216 8750~ 88191~~ 88811~ 89391~j 89931~~ 90441~ 90901~~ 91341i~ 91751~ 9214 37

08 2m 8357/~~ 84451~r 85271~~ 86021~ 86721~~ 87381~ 87991~~ 88561~ 89101~ 8960 48

-- --- --- --- --- --- --- --- --- --- --0,10 197 O,79821~~ o 80881~~ O,81861~~ O,82771~~ O,83631~& 0,84421~~ 0,85171~1 0,85871~ O,86521~i 0,8714 57

-- --- --- --- --- --- --- --- --- --- --12 185 76291~i i7501~~ 7863161 79691~~ 80671:~ 81601~~ 82471~ 83281~ 84051~~ 8477 65 53

14 172 72981~~ 74331~~ 75591g~ 767ii~ 77871~g 78911~ 79891~~ 80811~; 81681~i 8250 72

16 159 69911~~ 71371~~ 727~1~~ 74021~i 75231~~ 763i~ 77451~~ 78461~~ 79421~~ 8033 78

18 146 67091i~ 68n31~~ i0091~~ 714i~ 72771~~ 739911~ 751511~ 762511~ 77301~ 7828 82

-- --- --- --- --- --- --- --- --- --- --0,20 1~3 O,64501~? 0,66121~~ 0,67651~ O,69101~~ O,70481~ 0,71781g O,73011~ 0,74191~ 0,75301~ 0,7636 86

-- --- --- --- --- - --- --- -- --- --22 121 62141~~ 63821~~ 65~ 11~~ 66921~i 68361~~ 69731~ 7103 ~~ 7226 ~g 73H ~ 7456 88

24 109 60001~~ 61131~~ 63371~~ 6t931~ 6642 ~i 6784 ~ 6920 ~ 7049 ~ 7172 86 7290 91 59

26 97 5808 ~~ 5986 ~~ 6152 ~~ 6312 ~~ 6466 ~~ 6612 ~~ 6752 ~j 6886 ~~ 7014l~ 7136 92

28 87 5636 ~~ 5815 ~~ 5985 ~~ 6149 ~~ 6305 ~~ 6456 ~~ 6599 ~g 6737 ~ 6869 2 6995 93

-- --- -- -- --- -- -- -- -- -- --0,30 77 0,5483 ~ 0,5663 ~~ 0,5836 ~~ 0,6002 ~8 0,6161 ~~ 0,6315 ~~ 0,6461 ~i 0,6602 ~~ 0,6738 ~ 0,6868 94

-- -- -- --- -- -- -- -- -- -- --32 68 5349 ~r 5530 ~~ 5704 ~ 5872 ~ 6033 ~~ 6189 ~~ 6339 ~~ 6482 ~g 6620 ~~ 6753 94

34 59 5231 ~i 5413 ~ 5~88 ~~ 5758 ~~ 5921 ~~ 6078 ~g 6230 ~~ 6376 ~~ 6516 ~~ 6651 94

36 51 5129 ~: 5312 ~~ 5488 ~g 5658 ~ 5823 n 5982 ~~ 6135 ~~ 6283 ii 6425 ~~ 6562 94

38 43 5043 ~~ 5225 ~~ 5402 ~~ 5573 ~1 5739 ~~ 5899 ~~ 6053 i~ 6203 ~~ 6346 ~5 6485 94

-- --- -- --- -- -- -- -- -- -- --0,40 36 0,4971 ~~ 0,5153 ~~ 0,5331 ~~ 0,5502 ~~ 0,5669 ~g 0,5829 ~ 0,5985 ~~ 0,6135 ;~ 0,6280 ~g O,6~21 94

- -- --- -- -- -- -- -- --- --- --42 29 4913 ~i 5095 ~~ 5273 ~ 5445 ~g 5612 ~~ 5773 ~~ 5929 ~~ 6081 ~; 6227 ~r 6368 94

44· 22 4868 ~i 5051 ~~ 5228 ~ 5401 ~~ 5568 ;~ 5730 ~~ 5887 ~~ 6038 ~~ 6185 ~: 6327 I 94

46 16 4836 ~~ 5019 ~g 5197 ~ 5369 ~~ 5537 ~7 5699 ~~ 5856 ~~ 6008 ~~ 6156 ~~ 6298 94

48 9 4817 ~? 5000 8~ 5178 ~ 5350 8~ 5518 8i 5680 7~ 5838 ;~ 5990 7~ 6138 7i 6281 94

- --- -- --- -- --- --- -- -- -- --0,50 :1 0,4811 9~ 0,4994 s~ 0,5172 8~ 0,5344 ~ 0,5512 8~ 0,5674 7~ 0,5832 7~ 0,5984 7: 0,6132 7~ 0,6275

94


T

x I I ),=0,62 I 0,64 I 0,66 I 0,68 I 0,70 I 0,72 0,74 0,76 0,78 I 0,80

0,00 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000

-- -- -- I- -- -- --- -- -- --- --02 132 *9748123 "975912g *977011~ *978011g *97901og *97991~ *9808~ *9816 9~ *9824S: *9832

6

04 131 94971~ 95201t~ 95421t~ 95621~ 95821~ 960010g 9617 9~ 9634 9A 96508~ 9664 12

06 1:19 92501~~ 92841t= 93161t~ 93471~~ 93761~~ 9403 ~~ 9429~ 9453~g 9477~~ 9499 18

08 127 90081~ 90531~~ 90951M 91351~ 91731~A 9209 ~~ 9243 ~~ 9276 ~~ 9307~ 9336 24

-- -- -- -- -- -- -- --- -- --- --0,10 123 O,87i21~ 0,88281~ 0,88801GG O,89291~ 0,8976 ~ 0,9020 ~~ 0,9062~~ 0,9102 ~~ 0,9140 ~~ 0,9176

29

-- -- --- -- -- -- -- -- -- -- --12 118 85451~~ 86101~ 86711g: 87291gg 8784 ~ 8837 ~~ 8886 ~: 8933 ~~ 8978~: 9021

34

14 114 83271~ 84011~ 84711~ 8537 ~ 8600 ~ 8659 ~g 8716~~ 8770~~ 8822~ 8871 39

16 108 81191~ 82011gg 8279 ~ 8352 ~ 43 8423 ~~ 8489 ~ 8553~ 8613 ~~ 8671 ~~ 8726

18 102 7922 ~ 8011 :~ 8096 :A 8177 ~ 8254 &~ 8327 ~~ 8397~ 8463~ 8527~ 8588 47

-- -- -- -- -- -- -- -- -- -- --0,20 96 0,7737 : 0,7833 : 0,7924 ~ 0,8011 ~ 0,8094 ~ 0,8173 ~ O,8249~ 0,8321 ~i O,8390~ 0,8456

50

- -- - -- -- -- -- -- -- -- --22

90 7563 ~r 7665 ~ 7762 :~ 7855 ~: 7944 ~~ 8028 ~ 8109 ~g 8186 g~ 8260~ 53 8331

24 tI3 7402 ~ 7S09 ~~ 7612 ~g 7709l~ 7803lg 7893~ 7978~ 8060~ 8139M 8214 56

26 77 7253 ~~ 7365 ~ 7472 ~ 7574 : 7673 ~~ 7767~ 7857~~ 7943 ~~ 8025 t; 8105 59

28 70 7116~ 7232 ~g 7344 ~ 7450 ~~ 7552 ~g 7650~ 77.j3~ 7834~ 7920 ~~ 8003 61

-- -- --- -- --- -- -- -- -- i--- -0,30 64 0,6392 ~ 0,7112 ~~ 0,7226 ~~ 0,7337 ~ 0,7442 ~r O,7544~ 0,7641 ~~ 0,7734 ~ 0,78:.14: 0,7910

62

-- -- -- -- -- -- -- -- -- -- --

32 57 6831 ~~ 7003 ~~ y 63

7234 ~~ 7343~ 7448~ 75481~ 7644~ 7737~ 7826 64 7121 57

34 51 6782~ 6907 ~~ 7027 ~ 7143 ~ 7254~ 7361~ 7464~ 7563 :~ 7658: 7750 65

36 45 6694 ~ 6822 ~~ 6944 ~~ 7062 ~ 7176~ 7286~ 7391 ~I 7492 ~g 7589 ~; 7683 66

38 38 6619 ~ 6748 ~~ 6873 ~g 6993 ~ 7108~ 7220~ 7327 ~~ 7430 ~~ 7529 ~g 7625 67

-- -- -- -- -- -- --- -- f-- I--- -0,40

~,

0,6556 ~ 0,6686 ~ 0,6813 ~? 0,6934 ~~ 68 0,7051 ~~ O,716t ~ 0,7273 ~~ O,7378~ 0,7479 ~~ 0,7576

-- -- -- -- -- -- -- -- f-- -- -42 26 6505 ~ 6636 ~~ 6763 ~~ 6886 ~: 7005 ~~ 7119 ~~ 7229 ~~ 7335 ~~ 7437 ~~ 7535

68

44 20 6465 ~ 6597 ~ 6725 A~ 6849~ 6968~ 7083~ 7194ll 7301 M 7405~ 7504 69

46 15 6436·~~ 6569 A: 6698 M 6822 ~g 6942 a 7058M 7170~ 7277~ 7381~ 7481 69

48 • 6419 6~ 6553~ 6682 ~ 6806 ~ 6927J 7043J 71Msl 7263J 7367 ~ 7468 89

-- I--- -- - --- - - --- ~ ---- ~ O,SO a 0,6413 t.~ 0,6547 ~ 0,6676 ~ 0,6801 J O,6922J O,7038J O,71SOJ O,7258J O,7363~ 0,7463 89


t

I y=o.s2 I 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 I 1,00

0,00 55 0,675164 0,685852 0,6963 SI 0,706449 0,716348 0,725941 O,73524S 0,7443 44 0,7531 G 0,7616

-- --- --- --- --- --- --- --- --- --- --02 3 6756~ 6864 5~ 6968 S~ 7069 4~ 7168J 7264 4~ 7357 J 7447.J 7535.J 7620 S5

04 9 6774 5~ 6880 5~ 6984 5~ 70854~ 7183J 7279 4 7371 J 7461 ~ 75484~ 7633 55

06 15 6803 ~~ 6908 At 7011 M 7l111~ 7209 !~ 7303 !~ 7395 !~ 7484 !~ 75711~ 7655 54

08 21 6843~ 6947 ~~ 7049 !~ 7148 !~ 7244 !~ 7337 !~ 7428U 7516 !~ 7602 !~ 7685 54

-- --- --- --- --- --- --- --- --- --- --0,10 27

0,6895 ~~ 0,6997 ~g 0,7097 ~~ 0,7195 ~~ 0,7289 ~~ O,731l1 ~~ 0,7470 ~1 0,7557 ~~ 0,7641 ~~ 0,7723 53 -- --- --- --- --- --- --- --- --- --- --

12 33 6958 ~~ 7058 ~~ 7156 ~g 7252 ~~ 7344 ~~ 7435 ~~ 7522 ~g 7607 ~~ 7690~ 7770 52

14 88 7032 ~~ 7130 ~g 7226 ~~ 7319 ~~ 7409 ~~ 7497 ~1 7583~ 7666 ~5 7746 ~g 7625 50

16 44 7116 !~ 7212 !i 7305 !~ 7396 ~~ 7484 ~~ 7569 ~~ 7653~ 7733 ~~ 7812~ 7888 49

18 49 7212 ~~ 7305 ~~ 7395 ~~ 74821~ 7568 1i 7650 !6 7731 ~~ 7809 ~~ 7885 ~~ 7959 48

--- --- --- --- --- --- --- --- --- --0,20 55

0,7318 ~~ 0,7407 ~1 0,7494 ~~ O,75791~ 0,7661 :6 0,7741 ~~ 0,7818 ~~ 0,7894 ~~ 0,7967 ~1 0,8038 46

--- --- --- --- --- --- --- --- --- --22 60 7434 ~g 7520 ~~ 7603 ~t 7684 g~ 7763~ 7839 ~~ 7914 j~ 7986~ 8056~ 8124 44

24 65 7561 ~t 7642 ~6 772'2 3~ 7799 ~~ 7874 ~~ 7947 g~ 8018~ ~087 ~g iH53 i~ 8218 42

,0 7637 ~~

1 7774 ~~ 7849 3~ 7922 ~~ 7993 ~g 8063 ~~ 6130 g3 8195 3~ 8258 ~~ 831Y 26 40

28 i5 7843 ~~ 7915 ~~ 7985 ~~ 8055 ~g 8121 ~~ 8186 ~~ 8249 ~~ 831q~ 8370 ~~ 8428 37

--- --- --- --- --- --- --- --- --- --0,30 80 O,799i! ~~ U,8065 ~~ 0,8131 ~~ 0,8195 ~~ 0,8258 g~ 0,8318 ~~ 0,8377 ~~ 0,8-134 ~~ 0,8489 ~ U,85-13

J5 --- --- --- --- --- --- --- --- --- --

32 85 8162 gi 8224 ~2 8285 ~b 8344 ~~ 32 8402 ~~ 8457 ~~ 8511 ~~ 8564 ~g 8615 ~~ 8664

34 89 8335 ~~ 8392 ~t 8447 ~j 8501 ~~ 8553 ~~ 860-1 ~~ 8653 ~! 8701 ~~ 8747 ~~ 8792 29

36 ~4 8517 ~~ 8568 ~~ 8617 ~~ 8665 ~3 "9 8757 ~~ 8801 ~t 8844 ~i 8886 ~6 8926 26 8712 h

38 98 8~06 ~~ 8751 ~~ 8795 ~i 8837 ~~ 8878 ~~ 8918 ~g 8956 g 8994 I~ 9030 I~ 90E5 21

--- --- --- --- --- --- --- --- --- --0,40 102 0,890-l i~ 0,894:2 i~ 0,8979 i~ 0,9015 ~~ 0,9050 ~~ 0,9084 ~~ 0,9117 ~~ 0,9149 I~ 0,9180 i~ 0,9210

2<1

--- --- --- --- --- --- --- --- --- --42 lOG 911 Ol~~ 9141 i~ 9171 i~ 9201 i~ 9229 ~~ 9257 ~3 928-l ~~ 9310 ~g 9335 ;~ 9359

10

44 110 932~1~~ 93-161n 9370 ii 9392 i~ 9414 i~ 9435n 9456 ~g 9476n 9495 88 9514 12

46 114 95-1211g 955810~ 957410~ 9590 9~ 9604 9~ 9619 9~ 9633 8~ 96-16 8~ 9660 8~ 9672 8

48 117 976811~ 97761~ 978!10~ 97921O! 9800 9~ 9807 ~ 9814 9~ 9821 8g 9828 8~ 9834 4

--- --- --- --- --- --- --- --- --- --0,50 120 I,OOOOllg I,OOOOllg l,oooolOg 1,00001O~ I,OOOOlog I,00009g I,00009g 1,000086 l,00008g 1,0000

0

HYPERBOLIC AND INVERSE HYPERBOIJC FUNCTIONS OF COMPLEX ARGUMENTS 89

t

1 +=0,98 1 0,96 0,94 0,92 0,90 0,88 0,86 0,84 0,82 1 0,80

0,00 42 0,770043 O,77Q543 0,7871 43 0,795843 0,804544 0,813244 0,821944 0,830744 0,839544 0,8483

--- --- --- --- --- ~ --- --- --- --

02 2 7704 4~ 7789 4~ 7875 4~ 7961 ~ 80~8 4~ 8135 4~ 8223J 8310 4~ 8398 4~ 8486 42

04 6 7717 4~ 7801 4g 7887 4g 7972 4g 8059 43 8146 4~ 8233 4~ 8320 4~ 8407 ~ 8494 42

06 )\ 7738 ~~ 7821 ~g 7906 4~ 7991 ~ 8077 4g 8163 4~ 8249 4~ 8335 4~ 8422 4~ 8508 42

08 15 7767 ~1 7849 !~ 7933 !~ 8017 !~ 8101 ~~ 8186~~ 8272 !1 8357 !~ 8442 !~ 8528 41

--- --- --- --- --- --- --- --- --- --0,10 19 0,7804 !i 0,7885 !~ 0,79671: 0,8050 !~ 0,8133 !g 0,8217 !~ 0,8301 !~ 0,8385 !~ 0,8469 !g 0,8553

40

-- --- --- --- --- --- --- --- --- ---2"' 7849 ~~ 7929 ~~ 8009 ~b 8090 ~~ 8172 !i 8254 !i 8336 !~ 8418 !I 8501 !r 12

., ~ 40

14 28 7902 ~~ 7980 ~~ 8059 ~g 8138 ~6 8218 ~g 8298~ 8378 ~b 8458 ~g 8539 !g 8619 39

16 32 7963 ~1 8039 ~~ 8115 ~~ 8192 ~~ 8270 ~~ 8348 ~~ 8426 ~~ 8504 ~~ 8582 ~~ 8660 38 38

18 33 8032 ~~ 8105 ~~ 8179 ~~ 8254 ~~ 8329~ 8404 ~~ 8479~ 8555 ~~ 8631 ~~ 8706 36

--- --- --- --- --- --- --- --- --- --,20 39 0,8108 ~~ O,8179~~ 0,8250 ~~ O,8322~ 0,8394 ~~ 0,8466 ~~ 0,8539 3~ 0,8612 ~~ 0,8685 ~~ 0,8757

35 o --- -- --- --- --- --- --- --- --- --

22 43 8191 42 8259 ~~ 8328 ~~ 8396 ~~ 8466~ 8535~ 8605~ 8674 31 8744~ 8813 34 34

24 47 8282 ~~ 8347 i~ 8412 j~ 8477~ 8543 ~~ 86;9 ~~ 8676 ~~ 8742 ~i 8808 3~ 8874 32

26 51 8380 ~i 8441 ji 8503 ~f 8565 ~i 8627 ji 86891~ 8752 ~~ 8815 g~ 8877 ~i 8940 30

28 54 8484 ~~ 8542 ~g 8600 ~~ 8658 ~~ 8716 ~~ 8775 ~~ 8833 ~~ 8892 ~~ 8951 ~~ 9009 28

--- --- ~ --- --- --- --- --- --- --o ,30 58 0,8596 ~~ O,R6-t9 ~i 0,8703 ~~ 0,8757 ~~ O,881q~ O,8865~ 0,8920 ~~ 0,8975 ~~ 0,9029 ~~ 0,9083

26 --- --- --- --- --- --- --- --- --- --

32 61 8713 ~~ 8762 ~~ 881q~ 8861 ~~ 891qg 8961 ~g 901q~ 9061 ~~ 9111 ~1 9161 24

34 64 8836 g~ 8881 ~g 8926 ~~ 8971 g~ 9016 ~3 9062~ 9107 ~g 9152 ~~ 9198 ~3 9243 22

36 67 8965 ~3 9005 g6 9045 gg 9086 ~b 9126~ 9166~ 9207 ~g 9248 ~g 9288 ~g 9328 20

38 70 9100 ~~ 9135 ~~ 9170 ~~ 9205 ~g 9240 f~ 9276 ~~ 9311 f~ 93461~ 9382 t~ 9417 17

-- --- --- --- --- --- --- --- --- --,40 72 0,9239 ;g 0,9269 n 0,9299 ~~ 0,9329 ~~ 0,9359 f~ 0,9389 ~~ 0,9419 n 0,9449 n 0,9478 tg 0,9508

15 o --- --- --- --- --- --- --- --- --- --

42 75 9383 i~ 9408 ~~ 9432n 9456 ~~ 9481 ~~ 9505~ 9530 fg 9554 ~~ 9578 f~ 9602 12

44 77 9532 i~ 9551 7~ 9569 6~ 9588~ 9606 6~ 9625 6g 9644 5~ 9662 5~ 9681 5~ 9699 9

46 79 9685 7g 9697 7~ 9710 7~ 9722 6~ 9735 6~ 9748 6A 9760 5~ 9773 s~ 9785 6~ 9798 6

48 81 98417g 9847 ig 9854 7i 9860 6g 9866 sg 987363 9879 sg 9886 sg 9892~ 9898 3

-- --- --- --- --- --- --- --- --- --,50 83 1,00008g 1,00007g 1,0000 7~ 0 o 1,0000 78 1,00006Z I,OOOO~ I,00006g 1,0000sZ 1,000056 1,0000


T

I y = 0,82 I 0,84 0,86 0,88 0.90 0,92 0,94 0,96 0,98 1,00

0,00 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 -- --- --- --- --- --- --- --- --- ---

02 84 *9839 8~ *9846 7~ *9852 7~ *9858 7~ *9864 6g *9870~ "9875 6~ *9880~ *9885 5~ *9890 4

04 84 9679~ 9692 7~ 9705 7~ 9717 7~ 9729 6~ 97406~ 9751 6~ 9761 sg 9771 57 9780 7 " 06 83 9520 I~ 9540 IZ 9559 7~ 9578 7g 9595 6~ 9612~ 9628 6~ 9643 5~ 9657 5~ 9671 11

08 81 9364 rg 9390 Ig 9416 I~ 9440~~ 9463~ 9485 ~~ 9506~~ 9526 ~g 9546 5g 9564 14

--- --- --- --- --- --- --- --~ ---0,10 80 0,9211 I~ 0,9244 i~ 0,9275 ;~ 0,9305~~ 0,9334 ~~ 0,9361 ~~ 0,9387 yg 0,9412 ~~ 0,9436 ~~ 0,9459 17 -- --- --- --- --- --- --- --- --- ---

12 78 9062 i~ 9101 I~ 9138 n 9174 ~ 9208 ~~ 9240~ 9271 f~ 9301 ~ 9329 7: 9356 20

14 75 8918 ~~ 8963 ~i 9005 ~Z 9046 ~~ 9085n 9122 fg 9158 ff 9192n 9225 f~ 9256 23

16 72 8779 ~~ 8829 ~~ 8877~ 8323 ~~ 8967 ~i 9009 ~b 9049 ~g 9087 f~ 9124 tg 9159 26

18 69 8616 ~~ 8701 ~i 8754 ~~ 8805 ~~ 8853 ~~ 8900 ~~ 8944 ~~ 8986 ~g 9027 i~ 9066 29

--- --- --- --- --- --- --- --- ---0,20 66 O,8b19 ~~ 0,8579~~ 0,8637 ~~ 0,8692~ 0,8745 ~i 0,8795 ~~ 0,8844 ~g 0,8890~~ 0,8935 ~~ 0,8977 32 --- --- --- --- --- --- --- --- ---

22 62 8399~ 8464~ 8526 gg 8585~ 8642 ~~ 8696~Z . 8748~ 8798 ~~ 8846 ra 8892 34

24 59 8286 ~~ 8355~ 8421 3~ 8484~ 8545~~ 8603 ~~ 8659 ~~ 8712~ 8763 ~~ 8812 36

26 55 8180M 8253~ 8323~ 8390 ~~ 8454 ~f 8515~ 8574 ~~ 8631 i~ 8685 ~~ 8737 38

28 51 8083~ 8159 ~~ 8232 ~~ 8302 ~1 8369 ~~ 8434 ~1 8496~ 8555 ~g 8613 ~~ 8667 40

-- --- --- --- --- --- --- --- --- ---0,30 47 0,7993 jg 0,8072 ~~ 0,8148 3~ 0,8221 ~ 0,8292 ~~ 0,8359 ~~ 0,8424 ~y 0,8486~ 0,8546 ~~ 0,8603

41 -- --- --- --- --- --- --- --- --- ---

32 4:! 7911 ;1 7994 ~~ 8072~ 8148 g~ 8221 ~ 8291 ~~ 8358~ 8422 3~ 8484 ~~ 8544 4a

34 3!! 7838 ~f 7923 ~f 8004 ~~ 8082~ 8157 ~~ 8229 3~ 8299~ 8365 ~~ 8429 ~~ 8491 44

:36 34 ~773 ~~ 7860 ~~ 7943~ 8024 i !BOI ~~ 8175 ~~ H246~ 8314 g~ 8380 ~~ 8444 45

38 29 7717 ~~ 7806~ 7891 ~~ 7973 ~~ 8051 ~ 8127 ~~ 8200 ~~ 8270 ~~ 8338 ~~ 8402 46

-- --- --- --- --- --- --- --- --- ---0,40 25 0,7669 ~~ 0,7759~ 0,7846 ~~ "2 0,8010 ~~ 0,8087 ~~ 0,8161 ~~ 0,8233 ~~ 0,8301 ~g 0,8367

47 0,7929 40 -- --- --- --- --- --- --- --- --- ---

42 ~o 7630 ~g 7721 ~~ 7809 ~g 7894l~ 7975 17 8054~ 8129 ~~ R202 k~ 8271 A~ R339 47 :19

H In 7600 ~~ 7692 ~~ 7781ll 786611 7949 !~ 8028 ~~ 8104 1: 8178 ~i 82481~ 8310 4~ 3,

46 11 7578 II 7671 n 7760 ~g 7847 W 7929~ 8009 3~ R086 3~ 8161 3~ 8232 3~ 8300 48

48 7 7565 4; 7658 4~ 7748 4g 7835 4~ 7918,IZ 7998 3~ 8076 3~ 8150 3g 8222 3~ 8290 48

-- --- --- --- --- --- --- --- --- ---0,50 0,7560 4~ 0,7654 .j~ 0,7744 4~ 0.7831 4~ 2 0,7995 3~ 0,8072 3~ 0,8146 3~ O,8218~ 0,8287

49 0.7914 ,10


T

1+=0,98 1 0,96 0,94 0,92 0,90 0,88 0,86 0,84 0,82 0,80

0,00 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 -- --- --- --- --- --- --- --- --- ---

02 55 *9894~ *9899 5~ *9903~ *9908~ *9912 ~ *9917 4~ *9921 8g *99263~ *9930~ *9934 2

04 55 9789~ 9798 sg 9807 4: 9816 4~ 9825~ 98344l 9843~ 9851 3~ 9860~ 9868 5

06 54 9685 5~ 9698 5~ 9712 4~ 9725 4~ 9739 4~ 9752 4~ 97653~ 9778~ 97903~ 9803 7

08 54 9582 5~ 9600 4g 9618 4~ 9636 4~ 9653 4~ 9671 4~ 9688~ 9705~ 9722~ 9738 9

-- --- --- --- --- --- --- --- ---0,10 53 0,9481 f~ O,95031~ O,9525fl 0,9547 11 0,9569 ii 0,9591 i~ 0,9612 ~ 0,9633 ig 0,9654 n 0,9675 11 -- --- --- --- --- --- --- --- --- ---

12 51 93831~ 9409 i~ 9435 i~ 9461 i~ 94871A 9513 ~~ 9538 ~~ 9564~~ 9589 i~ 9613 13

14 50 9287n 9317 i~ 9347n 9377 t§ 9407~ 9437~ 9466~ 9495~ 9524 i~ 9553 15

16 48 9194 i~ 9228 i~ 9262 i~ 92961~ 9328 i~ 9363 n 9397~ 9429~ 9462n 9494 17

18 47 9104 i~ 9142 i~ 9180 1~ 9218 ~~ 9255n 9292~ 9329~ 9366 i~ 9402~ 9438 19

-- --- --- --- --- --- --- --- --- ---0,20 45 O,9019ii o,9060il 0,9102 ~~ 0,9143 ~~ 0,9184~ 0,9225 ~~ 0,9265 ~~ 0,9305 ~A 0,9344 ~6 0,9383 "21 -- --- --- -- --- --- --- --- --- ---

22 42 8937 ~A 8983 ~g 9027 ~~ 9072~ 9116~ 9160~~ 9203 ~~ 9247 ~~ 9289 ~~ 9332 23

24 40 8860 ~~ 8908 ~~ 8956 ~~ 9004~ 9051 g~ 9098~ 9145 ~~ 9192~ 9237 ~~ 9283 24

26 38 8788~ 8839 ~~ 8890 ~~ 8940 ~~ 8991'rs 9041 ~~ 9090~~ 9140 ~~ 9188 ~~ 9237 26

28 85 8721 ~i 8775 ~~ 8828 ~~ 8881 ~~ 8934 ~~ 8987 ~~ 9039 ~~ 9091 ~~ 9143~ 9194 27

-- --- -- --- --- --- -- --- -- -- ---0,30 P2 0,8659 ~~ O,8715~ 0,8771 rs 0,8826 ~~ 0,8882 ~~ 0,8937 ~~ 0,8992 ~i O,9046~ 0,9100 ~~ 0,9154

28 -- --- -- --- --- --- -- -- --- -- ---

32 29 8602 ~~ 8660 ~~ 8719 ~g 8777 ~~ 8834 ~~ 8892 ~~ 8949 ~~ 9006 ~~ 9062 i~ 9117 29

34 27 8551 ~ 8611 ~ 8671 ~6 8731 ~ 8791 ~ 8851 ~6 8910~ 8969 ~~ 9027 ~4 9084 30

36 24 8506 ~~ 8568 ~i 8630 ~l 8691 ~~ 8753 1i 8814~ 8875~ 8936 ~~ 8996 ~g 9055 31

38 21 8466 ~g 8529l~ 8593 ~~ 8656 A~ 8719 n 8782 ~~ 8845 ~~ 8907 ~1 8968 ~6 9019 32

-- --- --- --- --- --- --- -- --- --0,40 18 0,8432 ~ 0,8497 ~~ O,85621~ 0,8626 A~ 0,8691 M O,87551~ 0,8819 A~ 0,8883 ~i 0,8945 ~i 0,9007

32 -- --- --- --- --- --- --- -- --- -- ---42 14 8405~ 8470M 8536M 8602l~ 8668~ 8733M 8798g 8862 Ag 8926 ~g 8989

33

44 11 8383ll 8450~ 8516~ 8583~ 8649 :J 8715 :J 8781 ~ 8846 3~ 8911 3~ 8975 33

46 8 8357 J 8435~ 8502~ 8569 3~ 8636~ 8703 3g 8769~ 8835~ 8901 3~ 8965 34

48 5 8358~ 8426~ 8493:J 8561 :J 8628 3~ 8695~ 8762~ 8828~ 8894~ 8959 34

-- --- --- --- --- --- --- -- --- ---0,50 2 O,8355~ O,8423~ 0,8490~ 0,8558 al O,8626 3i 0,8693 J O,8760J 0,8826J O,8892J 0,8957

34


t

1+-0,781 0,76 0,74 0,72 0,70 0,68 0,66 0,64 0,62 0,60

0,00 41 0,857044 0,865743 0,874443 0,882942 0,891442 0,899741 0,907840 0,915839 0,923638 0,9311

02

04

06

08

0,10

I I

16

18

0,20

24

26

28

0,30

.'34

.16

:18

0,40

1 44

4 43

7 1:1

10 43

13 42

~I a9

23 38

26 31i

30 33

33 31

35 29

8573 4~

!-iS81 4~

8594 _I~

8613 4i

0,86361~

8665 !i

;m81~

8781 ~~

0,8830 ~J

X8R3 ~~

8940~

9002 ~1

90E8 ~~

37 -05 27 0,9137 27

39 25 921q~

~~ I 9188 ~~

~~ I 9368 ~~

ti I 9·152 t~ l--

46

I 15

42 I 4, 12

I 0,9538 i~

9626g

-!4 14~ -16 4~

48 50

0.50 51 "

971745 9

1.0000 4~

8660 41 8667 4~

8680 4g

8697 I~

1',871911

K747 J;~

8778 16 ~9

KX 15 ~~

8356 ~~

0,3902 ~~

8951 ~~

9005 ~~

9063 ~i

9125~~

0,9191 ~~

9260~

9332 ~~

!J408 ~g

9486 f~

O,95671~ -

9650 i§

9735 4~

9822 4~

9910 4j

IIOOOO~

8746 4~

8753 4~

8765 4~

8781 4~

0.8802 ~~

R827 ~~

HS.S7 ~~

8892~

R930 ~~

0,8973 ~~

9020 ~~

9070 ~~

9125 ~6

9183 ~~

9377 ~~

9447 :;g

9fi20 f~

-0.9596 ~~

9673n

9753 4g

9834 4~

9917 4~

-1,0000 4~

8831 4~

8838 4~

8849 4~

8864 4~

O,8884,lg

89081~

8935M

8967 ~~

9003M

0,90-13 ~

q087~

9134 ~i

9185 ~g

9239 ~~

8916 4~

8922 l~

89324f

8946 46

0,8964 4~

&987 M '10l3 ~~

9042 H I 9076 ~l,

0,9113 j~

9154 ~g

9198 ~~

9245 ~~

9295 §~

I

8999 -11

9004 41 9014 .13

9027 46

0,9044 1'~

9064 ~~

9088 h~

9116 :~ci

9147 ~~

O,9181 11

q219\~

9260 3~

9303 §~

9350 5~

----- r---- -----0,9297 §~ 0,9348 ~~ 0,93':19 ~~

----- r---- ---9357 ~~

91~0 ~i

~486n

95541i

0,9624 ~~ -

9697 f~

9771 3~

9846 3~

9923 3~

9404 ~~

9463 ~~

952 .. :n 9587 f~

0,9652 f.:

9719 ~t

9788 1 9858 3~

9928 ~~

-- -1. 0000 s~ 1. 0000 3~

9451 ~~

9505 ~I

9561 i~

9620 i~

O,96~0 ~~

9741 11 9805 3~

9869 3~

9934 J~

9080 .j~'

9085 43 9094 3~

9106 3g

0,91:22 3~

9141 3~

9163 U Y1881~

9217 ~!

0,9248 1~

921<3 ~;

9a20 ~5

9361 ~~

0403 ~~

0,9449 ~1

9546 ~g

9598 i~

'1651 n O,9i06 i~

9763 i~

9821 2~

9880 2~

9940 3~

9160 3~

9165 3~

9173 3~

9184 3~

0,9198 3~

9215 3~

92:16 ~~

9259 ~~

9285 ~~

O,93141i

93-16 ,~g

9;J80 ~~

941n~

9456 ~g

9237 3~

9242 3~

9249 3i 9259 3~

0,9272 3~

9288 3~

9306 3~

9327 ~1

9351 M

0,9378 ~1

'1406 ~~

9438 ~~

9471 ~~

9507 ~~

---- r----O,9497~! 0,9544 ~~

----- I----

9541 ~~

9586 i~

9633 i~

9682 i~

9584 ~~

9625n

9668g

9712 i~

-- r---:-:- <)3 0,9733 i3 0,9/.)8 i2 -- -

9784 ig 9837 2~

9891 2Z ,

9805 i~

9852 2~

9901 2~

9950 2~

--I,00002g

9312

9316

9323

9332

0,9344

9358

9375

9394

9415

0,9439

9461'>

9493

9524

9556

0.9590

9625

9662

9701

9741

0,9782

9824

9867

9911

995.~

1,0000


t

1+ = 0,581 0,56 0,54 0,52 0,50 0,48 0,46 0,44 0,42 0,40

0,00 36 0,938435 0,945333 0,951931 0,958229 0,964027 0,969525 0,975523 0,979020 0,983018 0,9866 --- --- --- --- --- --- --- --- ---

02 1 9385 31 9454 3l 95203?. 9583 2g 9641 2~ 9695 2g 9745 2g 9790 2g 9831 1g 9866 36

04 2 9388 3~ 9457 :J 9523 3~ 9585J 9643 2~ 9697 2~ 9747~ 9792 2~ 98321~ 9867 36

06 3 9394 :J 9463~ 9528 3~ 95B92~ 9647 2~ 9700 2; 9749 2~ 9794 2~ 98331~ 9869 36

08 5 9402 a1 9470 3~ 9534 3~ 9595 2~ 9651 2~ 9704 2~ 9753 2~ 9797 2~ 98361~ 9870 35

--- --- --- --- --- --- --- --- ---0,10 6 0,9413 ail O,94793~ 0,9542 36 0,9602 2: 0,9658 2~ O,97092~ 0,9757 2~ 0,9800 I~ O,98391~ 0,9873 35

--- --- --- --- --- --- --- --- ---12 i

9426 3~ 9491 31 9552 2g 9611 2i 9665 2~ 9716 2g 9762 2~ 98051~ 98421~ 9876 34

14 8 9441 3g 9504 3~ 9564 2~ 9621 2~ 9674~ 4' 9769 2~ 9810 I~ 9847 I~ 9879 33 9723 23

16 10 9458 3i 9519 2~ 9577 2~ 9633 2~ 9684 2~ 9732 2~ 9776 26 98161~ 9851 I~ 9883 32

18 11 94771g 9536 2~ 9593 2~ 9646 2~ 9695 2~ 9742 2~ 9784 I~ 98221~ 9857 I~ 9887 31

• --- --- --- --- --- --- --- ---0,20 12 0,9499 ~~ 0,9555 ~~ 0,9609 2~ 0,9660 2~ O,9708 2g 0,9752 25 O,97931~ 0,9830 I~ 0,9863 I~ 0,9892 30 --- --- --- --- --- --- --- --- ---

22 13 9522g 9576 ~~ 9628 2~ 9676 2~ 9722 2; 9764 1g 9803 1g 98381~ 98691~ 9897 28

24 14 9547 ~~ Y5\i8~! . 96H ~~ 9693 2~ '1737 2~ 9777 I~ 9813 I~ 98461~ 98761~ 9902 27

26 IS 9574 ~~ 9622 ~~ 9668g 9712 26 9752 1g 9790 I~ 98251~ 98561~ 98841~ 9908 25

28 16 9603 ~~ 9648 ~t 9691 ~6 9731 :g 9769 1g 9804 I~ 9836 I~ 98661~ 9892 It 9914 24

--- --- --- --- --- --- --- --- ---0,30 17 0,9633 ~~ 0,9675 ~ri 0,9715 :~ 0,9752 i~ 0,9787 I~ O,98191~ O,98491~ O,g8761~ 0,9900 It 0,9921 22 --- --- --- --- --- --- --- --- ---

32 18 9665 :~ 9703 ~~ 974() g 9774 :~ 9806 1g 98351~ 9862 I~ 9887 I~ 9909 1g 9928 20

34 19 9698g 9733 ~~ 9766 :~ 9796 g 9825 :g 9852 1g 9876 1; 9898 1g 9918 g 9935 18

36 19 9733U 9764 :~ 9793 i~ 9820 g 9845 :g 9869 Ii 9890 I~ 9910 g 9928 g 9943 16

38 20 9769 :~ 97951~ 9820g 9844 n 9866 1g 9887 g 9905 ~ 9922 g 9937 ~ 9950 14

--- --- --- --- --- --- --- --- ---0,40 21 O,98051~ 0,9828 :1 o,98i91ri 0,9869 I~ 0,9887 I~ 0,9905 ~ 0,9920 ~ 0,9935 ~ 0,9947 ~ 0,9958 12 --- --- --- --- --- --- --- --- ---

42 21 98431~ 9861 I~ 9878 I~ 9894 I~ 9909 I~ 9923 ~ 9936 ~ 9947 ~ 9958 ~ 9967 9

44 22 9882 I? 9895 I~ 9908 1~ 9920 1~ 9932 11 9942 ~ 9952 ~ 9960 ~ 9968 ~ 9975 7

46 22 99212g 9930 ll 9938 I~ 9947 1~ 9954 11 9961 Ig 9968 ~ 9973 ~ 9979 ~ 9983 5

48 22 9960 2g 9965 1g 99691~ 9973 I~ 99771~ 9981 Ig 9984 ~ 9987 ; 9989 t 9992 2

--- --- --- --- --- --- --- --- ---0,50 22 I,00002g I,OOOOlg 1,00001g 1,0000 1~ 1,000015 I,OOOOlg 1.0000 g l,oeoo b 1,0000 g 1,0000 0


x 1+.,.0,78 0,76 0,14 0,72 0,10 0,68 I 0,66 I 0,64 1

0,62 0,60 I

0,00 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1.0000 0 1,0000 -- --- --- --- --- --- --- --- ---

02 33 *9938 3~ *9942 2g *9946 2J *9950 2~ *9954 2~ *9958 2~ *99611~ *99651~ *99681~ *9971 2

04 33 987631 9885 2~ 9893 2~ 9900 2~ 9908 2~ 9916 2~ 9923 1~ 9930 1g 9937 1g 9943 4

06 33 9815 3A 98272~ 98J9 2~ 9851 2~ 9863 2~ 9874 21 98851~ 98951~ 9905 1~ 9915 6

08 32 9755 ag 9711 2g 9787 2~ 98022~ 9818 2~ 983L 2~ 9847 1~ 98611~ 98741~ 9887 8

-- --- --- --- --- --- --- --- --- ---0,10 32 0,9695 ~g 0,9716 ~ 0,9735 ig 0,9754 2~ 0,9773 2~ 0,9792 2g 0,9810 1~ O,98271~ 0,9843 1~ 0,9859 10 -- --- --- --- --- --- --- --- --- ---

12 31 9637 i~ 9661 i~ 9685~ 9708 i~ 9730n 9752 i~ 9773 1g 97941~ 981 4 1g 9833 12

14 30 9581 i~ 9608n 9635~ 9662ig 9688 ~1 97131~ 97381~ 97611~ 97841~ 9806 14

16 29 9526 i~ 9557 i~ 9588 i~ 9618 i~ 9647 il 96761~ 9703g 97201g 97561~ 9761 16

18 28 9473 i~ 9507 i~ 9541 i~ 9575 iA 9607 ig 96391~ 9670U 9700 :~ 9729:: 9756 18

-- --- --- --- --- --- --- --- --- -0,20 27 0,9422 i~ 0,9460 i~ 0,9497i~ 0,9534 i~ O,95691~ 0,9604 g O,96381~ o,96711~ O,97021~ 0,9733 19 -- --- --- --- --- --- --- --- --- ---

:22 26 9373 ~1 9415 ~g 9455 ~~ 9494~ 9533 {~ 9571 :~ 9608 :~ 9643 :~ 9677 ~~ 9710 21

24 25 9327 ~~ 9372 ~~ 9415 ~~ 9451 ~y 9499 ~~ 9539 ~g 95791~ 96171~ 9654g 9689 22

26 23 9284 ~~ 9331 ~g 9377 ~~ 9422 ~~ 9466 ~~ 9510 ~~ 9551 M 9592~ 9631 g 9669 24

28 22 924~ ~~ 9293 ~~ 9342 ~~ 9390 ~~ 9436 ~g 9482 ~~ 9526 ~~ 9569 ~i 9610 ~3 9650 25

-- --- --- --- --- --- --- --- --- ---0,30 20 0,9206 ~~ 0,9258 ~~ 0,9309 ~~ 0,9359 ~~ 0,9408 ~! 0,9456 ~~ 0,9502 ~ 0,9547 g 0,9591 ~~ 0,9632

20 -- --- --- --- --- --- --- --- --- ---

32 18 9172g 9226 ~~ 9:n9~g 9331 M 9382 ~~ 9432 ~~ 9481 g 9528 ~g 9573 2~ 9616 27

34 17 9141 ~~ 9197 ~~ 9252 ~i 9306 ~~ 9359 ~~ 9411 ~~ 9461 ~~ 9510 2~ 9557 2g 9602 28

36 15 9114 ~~ 9171 ~~ 9228 ~~ 9284g 29 933'9 ~~ 9392 ~~ 9444 2~ 9494 2~ 9542 2~ 9589

38 13 90891~ 9149M 9207 ~~ 9264 ~g 30 9320 2~ 93i5 2~ 9428 2~ 9480 2~. 9530 2~ 9578

-- --- --- --- --- --- --- --- --- ---0,40 11 O,9069 1g O,YI29 ~g 0,9189 2~ 0,9248 2~ 0,9305 2g 0,9361 2~ 0,9415 2~ 0,9468 2~ 0,9519 2~ 0,9568

31 -- --- --- --- --- --- --- --- --- ---42 9 9052 3~ 9114 gg 9174~ 9234 2~ 9292 2g 9349 2~ 9405 2~ 9458 2~ 9510 2~ 9560 31

44 8 9039 3I 9101 3~ 9163 ~ 9223 3g 9282 2~ 9340 2~ 9396 2~ 9451 2~ 9503 2~ 9554 32

46 5 9029 3~ 9092 3t 9155 3( 2216 3ci 32 9275 2~ 9334 2~ 9390 2~ 9445 2~ 9498 2~ 9550

48 3 9024 3~ 9087 3~ 9150 ai 9211 ~ 9271 2~ 9330 2~ 9387 2~ 9442 2~ 94952~ 9547 32

-- --- --- --- --- --- --- --- --- ---0,50 1 0,9022 3~ 0,9085 31 0,9148 31 0,9209 s& O,9"270J 0,9328 2~ 0,9385 2~ O,94412~ O,94942~ 0,9546

32


T

x I 1+ = 0,58\ 0,56 0,54 0,52 0,50 0,48 0,46 0,44 0,42 0,40

0,00 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 0 1,0000 --- --- --- --- --- ---

02 14 *9975 1g *9978 11 *9980 2 10 *9983 i *9985 7 *9988 ~ *9990 f *9992 t *9993 r *9995 1 1

04 14 99491~ 9955 I~ 9961 3 10 9966 ~ 9971 7 99:5 ~ 9979 ~ 9983 ~ 9986 ~ 9989 3 2

06 14 99241~ 99331l 9941 5 10 9949 : 9956 ~ 9963 g 9969 3 9975 ~ 9980 ~ 9984 4

OS 14 9899 1~ 9911 1~ 9922 6 10 9932 ~ 9942 ~ 9951 ~ 9959 ~ 9966 ~ 9973 ~ 9979 5

--- --- --- --- --- ---0,10 11 0,9875 1~ O,98891~ 0,9903 8

10 0,9916 ~ 0,9928 ~ 0,9939 ~ 0,9949 ~ 0,9958 : 0,9966 ~ 0,9973 6 --- --- --- --- --- ---12 13 98511~ 9868 I~ 9884 9

9 9900 ~ 9914 j 9927 ~ 9939 ~ 9950 ~ 9960 ~ 9968 8

14 13 982716 9848 1g 9866 11 9 9884 ~ 9901 ~ 9916 ~ 9930 ~ 9942 ~ 9954 ~ 9963 9

16 13 9805U 98271~ 9849 12 9 9869 g 9888 ~ 9905 ~ 9921 ~ 9935 ~ 9948 ~ 9959 10

18 12 9783g 98081~ 9832 13 9 9854 I~ 9875 1g 9894 ~ 9912 : 9928 ~ 9942 ~ 9954

11

--- --- --- --- --- ---0,20 12 0,9762 U 0,9790 1~ 0,9816 15

8 0,9840 II O,98631~ O,9884 1g 0,9903 ~ 0,9921 ~ 0,9936 ~ 0,9950 12 --- --- --- -- --- ---22 11 97421g 97721~ 9800 16

8 9827 I~ 9851 I~ 9874 1g 9895 ~ 9914 ~ 9931 ~ 9945 13

24 11 97231g 97551~ 9785 17 7 9814 I~ 9840 I~ 9865 If 9887 1ci 9907 g 9926 ~ 9941

14

26 10 9705 I? 97391~ 9771 18 7 9802 1~ 9830 13 98561~ 9880 11 99011~ 9921 ~ 9937 15

28 9 9688 1~ 9724 1~ 9758 19 7 9790 I~ 9820 I~ 98481~ 9873 If 98961~ 9916 ~ 9934 16

--- --- --- --- --- ---0.30 9 0,9672 1~ 0,97\0 1~ 0,9746 20

6 0,9780 I~ O,98111~ 0.9840 1~ 0,9867 I~ 0,9891 d 0,9912 ~ 0,993\ 17 --- --- --- --- --- ---32 8 9658 2b 9698 1g 9735 21

6 9770 I~ 98031~ 9833 I: 9861 Ig 9886 1~ 9908 15 9928 18

34 7 9645 2~ 9686 1g 9725 22 5 9762 1~ 9796 1ci 9827 1~ 9856 1g 9882 I~ 9905 1~ 9925 18

33 6 9634 2~ 9676 2g 9716 22 5 97541~ 97891~ 9821 d 98511~ 98781~ 9902 1~ 9922 19

38 6 9624 2~ 9667 2~ 9708 23 4 9747 1~ 97831~ 98161~ 9847 1~ 98741~ 9899 11 9920 19

--- --- --- --- --- ---O,~O 5 0,9615 2~ 0,9660 21 0,9702 23

3 O,974\1~ O,97781~ O,98121~ 0,9843 1~ 0,9871 11 0,9896 11 0,9918 2U --- --- --- --- --- ---42 4 9608 2: 9653 2~ 9696 24

3 9737 I~ 97741~ 98091~ 9840 1~ 9869 11 9899 11 9917 20

44 3 96022~ 9648 2~ 9692 24 2 97331~ 97711~ 98061~ 98381~ 9867 1~ 9893 11 9916 20

46 2 9598 2~ 9645 2~ 9689 24 2 9730 1~ 97691~ 98041~ 9837 1~ 9866 11 9892 11 9915 21

48 1 9596 2~ 9643 2~ 9687 25 1 97291~ 9767 1~ 98031~ 9836 1g 9865 1g 9891 I~ 9914 21

--- --- --- --- --- ---0,50 0 9,9595 2g 0,9642 2~ 0,9686 25

0 O,9728 1g O,9767 1g O,9803 1g O,9835 1g 0.9865 1g O,9891 1g 0,9914 21


t

x I I ~ = 0,38 1

0,36 I 0,34 I 0,32 I 0,30 I 0,28 0,26 I

0,24 I 0,22 I 0,20 I I

0,00 15 0,989713 0,992311 0,9944 9 0,9961 7 0,9975 5 0,9984 3 0,9991 2 0,9995 1 0,9998 1 0,9999 --- --- --- --- --- --- --- --- ---

02 0 9897 1g 9923 1~ 9945 ~ 9962 ~ 9975 g 9984 g 9991 0 9995 ~ 9998 ~ 9999 15 2

04 0 9898 1g 99241~ 9945 ~ 9962 ~ 9975 g 9984 ~ 9991 0 9995 ~ 9998 ~ 9999 15 2

06 1 98991~ 9924 I~ 9945 g 9962 ~ 9975 g 9984 g 9991 0 9995 ~ 9998 ~ 9999 IS 2

08 1 9900 I~ 9925 I~ 9946 g 9963 g 9975 g 9985 g 9991 0 9995 ~ 9998 ~ 9999 IS 2

--- --- --- --- --- --- --- --- ---0,10 I 0,9902 I~ 0,9927 16 0.9947 ~ 0,9963 g 0,9976 g 0.9985 g 0,9991 g 0,9995 ~ 0,9998 ~ 0,9999 15 --- --- --- --- --- --- --- --- ---

12 1 9904 I~ 9928 16 9948 ~ 9964 g 9976 g 9985 g 9992 g 9996 ~ 9998 ~ 9999 14

14 2 9907 I~ 9930 1~ 9950 I 9965 g 9977 0 9986 g 9992 g 9996 ~ 9998 ~ 9999 H 8 4

16 2 9910 Ii 9932 § 9951 I 9965 ~ 9978 0 9985 g 9992 g 9996 ~ 9998 ~ 9999 14 7 4

18 2 9913 IT 9935 § Y953 I 9967 ~ 9979 ~ 9987 g 9992 g 9996 ~ 9998 ~ 9999 13 7

--- --- --- --- --- --- --- --- ---0,20 2 0,9917 Ii 0,9938 § 0,9955 ~ 0,9969 ~ 0,9979 ~ 0,9987 g 0,9993 g 0,9996 ~ 0,9998 ~ 0,9999 13 --- --- --- --- --- --- --- --- ---

22 3 99211~ 9941 I 9957 1 9970 ~ 9980 ~ 9988 g 9993 g 9996 ~ ~998 ~ 9999 12 8 7

24 3 9925 16 9944 ~ 995J I 9972 ~ 9982 1 9989 g 9993 ~ 9997 ~ 99Q8 ~ 9999 11 8 6

26 3 9929 ~ 9947 ~ 9962 ~ 9974 ~ 11 9983 ~ 9989 g 9994 ~ 9997 ~ 9998 ~ 9999

28 3 9934- ~ 9951 ~ 9965 ~ 9975 ~ 9984 ~ 9990 g 9994 ~ 9997 ~ 9999 g 9999 10

--- --- --- --- --- --- --- --- ---0,30 3 0,9939 ~ 0,Q955 ~ 0,9967 ~ 0,9977 ~ 0,9985 ~ 0,9991 g 0,9995 ~ 0,9997 ~ 0,9999 g 0,9999

9 --- --- --- --- --- --- --- --- ---32 4 9945 ~ 9959 ~ 9970'1 9979 1 8 9986 1 9992 g 9995 ~ 9997 ~ 9999 g *0000

34 4 9950 ~ 9963 ~ 9973 ~ 9981 1 7 9988 ~ 9992 g 9996 ~ 9998 ? 9999 g 0000

36 4 9956 ~ 9967 ~ 9976 ~ 9984 1 9989 ~ 9993 ~ 9996 ? 9998 ? 9999 g 0000 7

38 4 9962 ~ 9972 ~ 9980 ~ 9986 ~ 6 9991 ~ 9994 ~ 9997 ~ 9998 ~ 9999 g 0000

--- --- --- --- --- --- --- --- ---0,40 4 0,9968 ~ 0,9976 ~ 0,9983 ~ 0,9988 ~ 0,9992 ~ O,9S95 ~ 0,9997 ~ 0,9999 g 0,9999 g 1,0000

5 --- --- --- --- --- --- --- --- --- ----42 4 9974 ~ 9981 ~ 9986 ~ 9990 ~ 4 9994 ~ 9996 ~ 9998 ? 9999 g 9999 g 0000

44 4 9981 ~ 9986 ~ 9990 ~ 9993 ~ 9995 1 9997 : 9998 g 9999 g *0000 g 0000 3 1

46 4 9987 ~ 9990 i 9993 i 9995 : 9997 1 9998 ~ 9999 g 9999 g 0000 g 0000 2 I

48 4 9994 ~ 9995 i 9997 i 9998 A 1 9998 A 9999 A 9999 g *0000 g 0000 g 0000

.--- --- --- --- --- --- --- --- ---0,50 4 1,0000 ~ 1,0000 6 1,0000 6 1,0000 A 1,0000 ~ 1,0000 ~ 1,0000 g 1,0000 g 1,0000 g 1,0000

0


T

x 1 +=0,38 1

0,36 1

0,34 1

0,32 1

0,30 1

0,28 1

0,26 1

0,24 1

0,22 1

0,20

--

0,00 0 1,0000 0 I,oroo 0 1,0000 0 1,0[100 0 1,0000 0 1,0000 0 1,0000 0 0,0000 0 1,0000 0 1,0000 -- --- --- --- --- --- --- --- --- ---

02 3 *9996 i *9997 ~ ~9998 ~ *9998 ~ *9999 ~ *9J99 g OOJO g 0000 g 0000 g 0000 1

04 3 9992 i 9994 i 9996 01

9997 1 9998 ~ 9999 g *9999 g 0000 g 0000 g 0000 1 1 1

06 3 9988 ~ 9991 i 9993 1 9995 1 9997 1 9998 g 9999 g *9999 g 0000 g 0000 2 1 1 1

08 3 9984 ~ 9988 ~ 9991 1 9994 1 9996 1 9998 ~ 9999 g 9999 g 0000 g 0000 2 1 1 1

--- --- --- --- --- --- --- --- ---0,10 3 0,991:>0 ~ 0,9985 ~ 0,9989 ~ 0,9992 1 0,9995 1 0,9997 ~ 0,9998 g 0,9999 g 1,0000 g 1,0000 3 1 1 -- _0_- --- --- --- --- --- --- --- ---

12 3 9976 ~ 9982 1 9987 ~ 9991 1 9994 1 9996 ~ 9998 ~ 9999 g *9999 g 0000 4 2 1

14 2 9972 ~ 9979 1 9985 ~ 9990 1 9993 ~ 9996 ~ 9998 ~ 9999 g 9999 g 0000 4 2

16 2 9968 ~ 9976 1 9983 1 9988 ~ 9992 g 9995 ~ 9997 ~ 9998 g 9999 g 0000 5

18 2 9965 ~ 9974 1 9981 1 9987 ~ 9991 g 9995 ~ 9997 1 9998 g 99S9 g 0000 5 3

--- --- --- --- --- --- --- --- ---0,20 2 0,9961 ~ 0,9971 o~ 0,9979 1 0,9986 ~ 0,9990 g 0,9994 ~ 009997 ? 0,9998 Y 0,9999 g 1,0000 6 3 -- --- --- --- --- --- --- --- --- ---

22 2 9958 ~ 9969 ! 9977 ~ 9984 1 9990 g 9994 ~ 9996 ~ 9998 ? 9999 g 0000 6

24 2 9955 ~ 9966 ~ 9976 ~ 9983 1 9989 g 9993 ~ 9996 ~ 9998 ~ 9999 ~ 0000 7

26 2 9952 ~ 9964 1 9974 ! 9982 ~ 9988 g 9993 g 9996 ~ 9998 ~ 9999 g 0000 7

28 2 9949 ~ 9J62 1 9973 ! 9981 1 9988 ~ 99J2 g 9996 ~ 9998 ~ 9999 g 0000 8

--- --- --- --- --- --- --- --- ---0,30 2 0,9947 ~ 0,9960 ~ 0,9971 ! 0,9980 1 0,9987 ~ 0,9992 g 0,999.5 ~ 0,9998 ~ 0,9999 g 1,0000 8 --- --- --- --- --- --- --- --- ---

32 2 9944 ~ 9958 ~ 9970 1 9979 ~ 9986 ~ 9992 g 9995 ~ 9997 ~ 9999 g OOJO 8

34 1 9942 ~ 9957 1 9969 1 9978 ~ 9986 ~ 9991 0 9995 ~ 9:97 ~ 9999 g *9999 9 6 2

36 1 9940 ~ 9955 1 9968 1 9978 ~ 9985 ~ 9991 0 9995 ~ 9997 ~ 9899 g 9999 9 6 2

38 1 9939 ~ 9954 ~ 9967 g 9977 ~ 9985 g 9991 0 9995 ~ 9997 ~ 9999 g 9999 9 2

-- --- --- --- --- --- --- --- --- ---lO,40 1 0,9937 ~ 0,9953 ~ 0,9966 g 0,9977 ~ 0,9985 ~ 0,9990 g 0,9995 ~ 0,9997 ~ 0,9999 g 0,9999 9 -- --- --- --- --- --- --- --- --- ---

42 1 9936 ~ 9952 ~ 9966 g 9976 ~ 9984 ~ 9990 g 9994 ? 9997 ~ 9999 g 9999 10

44 1 9935 ~ 9952 ~ 9965 g 9976 ~ 9984 ~ 9990 g 9994 ~ 9997 ? 9999 g 9999 10

46 0 9935 g 9951 0 9965 g 9976 ~ 9984 g 9990 g 9994 ~ 9997 ? 9999 g 9999 10 7

48 0 9934 g 9951 0 9965 g 9975 ~ 9984 ~ 9990 g 9994 ~ 9997 ~ 9999 g 9999 10 7

-- --- --- --- --- --- --- --- --- ---0,50 0 0,9934 g 0,9951 ~ 0,9965 g 0,9975 ~ 0,9984 g 0,9990 g 0,9)94 ? 0,9997 ~ 0,9999 g 0,9999 10

I


x

I t = 0,02 I 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

1,00 64 0,0127 64 0,0255 64 0,0382 63 0,0508 63 0,0635 63 0,076063 0,0886 62 0,1010 62 .0,1134 61 0,1257 - --- --- --- --- --- - --- ---0760~ 98 0 0127 6~ 0254 6~ 0381 ~ 0508~ 0634 6g 0885J 1010 6g 1133 6~ 1256 64

96 0 0127 ~ 02546~ 0381 ~ 0507 ~ 0633~ 07596~ 0884 6~ 1008 6~ 1132 6t 1254 64 94 0 0127 6g 0253 6~ 0380 6g 0506~ 06326~ 0757 6~ 0882 6~ 1006 6~ 1129 6t 1252 63 92 0 0126 6g 0252~ 03796~ 0504 6~ 0630 6~ 07556~ 08796~ 1003 6~ 1126 6i 1248 63 --- --- --- --- --- --- --- ---

0,90 0 O,0126 6g O,02516~ 0,0377 ~ 0,0502 6~ 0,0627 6~ 0,0751 6~ O,08756~ O,0998 6i 0,1121 6i 0,1243 63 -- --- --- --- --- --- --- --- ---88 0 0125 6g 0250 6~ 0375 6~ 0499 a~ 01j24a~ 0747 6~ 0871 6~ 0993 6f 1115 6r 1237 62 86 0 0124 6g 0248 6~ 0372 6~ 04966~ 0620 6~ 0743 6~ 086S ar 0987 af 1109 6g 1229 62 84 0 0123 6g 0247 a~ 0370 at 0493 6i 0615 6i 0737 6r 0859 6~ 0980 66 1101 sci 1221 62 82 0 0122 61 0244 61 0366 6i 0488 ai 0610 6~ 0731 68 0852 6ci 0972 sci 1092 66 1212 61 -- --- --- --- --- --- --- --- ---

O,SO 0 0,0121 61 0,0242 61 0,0363 6~ O,0484~ O,0604~ O,0724J 0,0844 66 0,0964 sci 0,1083 5~ 0,1201 61 -- --- --- --- --- --- --- --- --- ---78 0 0120 66 0:24°ab 0359~ 0479 6g 0598~ 0717 5~ 0836 5~ 0954 5~ 1072 5~ 1189 60

76 0 0118 5~ 0237 5~ 0355 5~ om 5~ 0591 5~ 0709 5~ 0826 5~ 0943 5~ 1060J 1176 59 74 0

0117 5~ 0234 5~ 0350~ 0467 5~ 0584~ 0700 5~ 0816 5~ 0932J 1047 J 1163 58 72 0 0115 5~ 0230~ 0345~ 0461 5~ 0575 5i 0690 5~ 0805 5~ 0919 5~ 1034 5~ 1148 5~ -- --- --- --- --- --- --- --- --- ---

0,70 0 0,0113 5~ 0,0227 5~ 0,0340 5~ O,0454 5i 0,0567 si 0,0680 6~ 0,0793 5~ 0,0906 5~ 0,1019 ~ 0,\132 57 -- --- --- --- --- --- --- --- ---68 0 0112 5~ 0223~ 03$5~ 0446 S: 0558~ 0669~ 0781 5~ 0891 ~ l003~ 1114 56

66 0 0110 sA 0219 5; 0329 J 0438 5~ 0548 5~ 0658 5~ 0767 ~ 0877 5~ 0987 5g 1096 65 64 0 0108 5~ 0215 6~ 0323~ 0430 s: 0538~ 0645 5~ 0753 5~ 0861 5~ 0969 5~ 1077 54 62 0 0105 5~ 0211 ~ 0316 5~ 0421 si 0527 5g 0633 sg 0738~ 0844J 0950 sg 1056

53 -- --- --- --- --- --- --- --- --- ---0,60 0 0,0103 s~ 0,0206 5~ 0,0309 5~ 0,0412 5~ 0,0516 5~ 0,0619 5~ 0,0723 5~ 0,0827 5~ 0,0931 ~g 0,1035 52 -- --- --- --- --- --- --- --- --- ---

58 0 0101 56 0201 ~ 0302 sci 0403 5¥ 0504 5¥ 0605 51 0706 5~ 0808 5~ 0910 ~~ 1012 50

56 0 0098 4~ 01964~ 0296 4~ 0393 4~ 0492 4~ 0590~ 0689 ~ 0789 ~g 0889~ 0989 49

54 0 0096 4~ 01914~ 0287 4~ 0383 4~ 0479 4~ 0575 4~ 0672 4~ 07691g 0866l~ 0964 48 52 0 0093 4~ 0186 4~ 0279 4i 0372 4~ 0465 4~ 0559 4~ 0653 4~ 0748!g 08431~ 0939 46 -- --- --- --- --- --- --- --- --- ---

0,50 0 0,0090 4~ 0,0180 4~ 0,0270 4~ 0,0361 4~ 0,0452 4~ 0,0543 4~ o,06341~ o,07261~ 0,0819 n 0,0912 45 -~ --- --- --- --- --- --- --- ---

48 0 0087 4~ 0174,J 0262 4: 0349 ~ 0437 ~ 0526~ 0615!g 0704ll 07941~ 0884 44 46 0 0084 4~ 0169 4~ 0253 4~ 0338 4g 0423 4~ 0508 4~ 0594 1g 0681ll 0768~ 0856 42 44 0 0081 4~ 0162 41 0244 4~ 0326 4~ 0408 4~ 0490 4~ 0573g 0657 !~ 0741!g 0826 41 42 0 0078 3~ 0156J 0234~ 0313~ 0392 4~ 0472~ 0552lb 0632n 0714!1 0796

39 -- --- --- --- --- --- --- --- ---

0,40 0 0,0075 3~ 0,0150 J 0,0225 3~ 0,0300 ~ O,0376~ 0,0453 ~ 0,0529 M 0,0607 ~~ 0,0685 16 0,0765 37 -- --- --- --- --- --- --- --- --- ---38 0 0072~ 0143~ 0215~ 0287 3~ 0360 3~ 0433 ~~ 0507 M 0581 M 0656~ 0732 36 36 0 0068 3~ 0137 ~ 0205J 0274 3~ 0343~ 04131g 0483~ 0554M 0626~~ 0699

34

34 0 0065 3~ 0130~ 0l95~ 0260 3~ 0326~ 03931~ 0460~ 0527 ~1 0596~ 0665 32 32 0 0061 ai 0123 3~ 0184 3~ 0246 3I 0309 a~ 0372~ 0435 ~~ 0499M 0565~ 0631 31 -- --- --- --- --- --- , --- --- --- ---

0,30 0 O,0058J O,0116~ 0,0174 2~ O,0232J 0,0291 al: O,0350~ 0,0410 ~~ 0,0471 ~t 0,0533 ~~ 0,0595 29


x

T t=O,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

28 0 0054J 0109 2~ 0163 2~ 0218~ 02732~ 0329~ 0385~ 0442~ 0500~ 0559 27 26 0 oo512~ 0101 2~ 0152 2~ 0203~ 0255 2~ 0307 ~~ 0360~~ 0413 ~~ 0467 ~~ 0522 25 24 0 0047~ 0094 2! 0141 2~ 0188 2~ 0236 2~ 0285 ~! 0333 ~: 03831~ 0433 ~~ 0484 23 2,

22 0 0043 2~ 0086 :ci 0130 2~ 0173 2g 0217 2~ 026:!M 0307 Jg 0353 ~g 0399 ~~ 0446 22 -- --- --- --- --- --- --- --- --- ---0,20 0 O,O039~ O,O079~ O,0118 2g O,Ol58 2g 0,0198 ~e O,0239~1 0,0280 ~r 0,0322 ~~ 0,0364 ~~ 0,0408 20 -- --- --- --- --- --- --- --- --- ---

18 0 00361~ oo711~ 0107 1g 01431~ 01791g 0:!161~ 0253 :~ 02911~ 0329~ 0368 18 16 0 00321~ 00631~ 00951~ 01271~ 0160 l~ 0193 :~ 0226 :~ 0259 :~ 02941~ 0329 16 14 0 00281~ 0056 1: 00841~ 0112 1: 0140 l~ 0169:~ 01981: 0228 :~ 0258 :~ 0289 14 12 0 00241~ 0048 I~ 00721~ 00!.6 1g 01201g 0145 :~ 0170 :~ 0196 :~ 02221~ 024/l 12 -- --- --- --- --- --- --- --- --- ---

0,10 0 O,O0201~ O,0040 lri 0,0060 18 O,0080 lg 0,0101 :g 0,0121 :~ 0,0141 It O,O1631~ O,O1851~ 0,0207 10 -- --- --- --- --- --- --- --- --- ---

08 0 0016 ~ 0032 : 0048 ~ 0064 ~ 0081 1g 0097 I~ 01141~ 01311~ 01481~ 0166 8

06 0 0012 ~ 0024 ri 0036 ~ 0048 ~ 00611~ 00731~ 00861~ 00981~ Ollll~ 0125 6 04 0 0008 ~ 0016 : 0024 ~ 0032 : 0040 I~ 00491~ 0057 I: 00661~ 00741~ 0083 4 02 0 0004 ~ 0008 ~ 0012 ~ 0016 g 0020 l g 0024 I~ 0029 I~ 0033 I~ 00371~ 0042 2 -- --- --- --- --- --- --- --- ---

0,00 0 0,0000 ~ O,OOuO ~ 0,0000 g 0,0000 g O,OOOOlg 0,0000 I~ 0,0000 I~ o,oovo 18 0,000013 O,uooo 0

y

T t ... 0,02 I 0,04 0,06 0,08 0,10 0,12 I 0,14. 0,16 0,18 0,20

I !

1,00 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 -- --- --- --- --- --- --- --- --- ---

98 0 0006 ~ 0013 g 0019 g 00251~ 00311~ 0037 1g 0043 2~ 0049 2g 0055 2~ 0060 3 96 0 0013 : 0025 ~ 0038 ~ 0050 I~ 0062 I~ 0074 I~ 0086~ 0098 2~ 0110 2~ 0121 6 94 0 0019 : 0038 g 0056 g 00751~ 0093 I~ 01l1 lg 0129 2~ 0147 2~ 0164 2~ 0181 9 92 0 0025 1g 0050 I~ 00751~ olOog 0124g 01481g 0172 ~~ 0196 ~~ 0219n 0241 13 -- --- --- --- --- --- --- --- --- ---

0,90 0 O,O0311~ O,O0631~ O,0094 1g 0,0124 :~ O,Ol551g O,O1851~ 0,0215 ~~ 0,0244 ~: o,o.m ~~ 0,0301 16 -- --- --- --- --- --- --- --- ---88 0 00371~ 0075 1~ 01121~ 0149 :~ 01861~ 0222 :: 0258 ~~ 0293 ~~ OJ27n 0361 19 86 0 0044J 0087J 0130 2~ 0173 ~~ 0216 ~r 0258 ~~ 0300~A 0341 ~~ 0381 ~Z 0421 22 84 0 0050~ 0099 2~ 01492~ 0198 ~~ 0247 ~~ 0295 ~: 0342 ~~ 0389 ~: 0435 ~~ 0480 25 82 0 oo562~ 0111 2~ 0167 2~ 0222 ~~ 0277 ~~ 0331 ~~ 0384 ~~ Ot36 ~: 0488 ~~ 0539 28 --- --- --- --- --- --- --- --- ---

0,80 0 O,o062 3f O,Ol23 3r 0,0185 ai 0,024655 0,030613 O,0366~~ O,04l5 ~~ 0,0484 ~~ 0,0541 ~g 0,0597 31 -- --- --- --- --- --- --- --- ---78 0 0068S: 0135~ 0203 sg 0270~ 0336 ~~ 0402~ o 167 ~~ 0530 ~~ 0593 ~r 0655 34 76 0 00743~ 0147 s~ 02203~ 0293 ~~ 0365 ~~ 0437 ~ 0507~ 0577~ 0645~ 0713 37 74 0 0079,J 0159J 0238s~ 0316~ 0394~ 0471~ 0547~ 0623~~ 0697 ~~ 0770 40

72 0 00854 0170 4~ 0255J 0339~ 0423lt 0505!r 0587 :g 0668~ 0748 ~: 0826 48


y

I t - 0,02 I 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

0,70 0 0,0091 4~ 0,0181 4~ 0,0272 4~ 0,0362 !~ 0,0451 ~~ 0,05391~ 0,0627 ~g 0,0713~~ 0,0798 ~~ 0,0882 45 -- --- ~-- --- --- --- --- --- --- ---

68 0 0096 4~ 0193 4~ 0288 4~ 03841} 0478 !~ 0572 1; 0665 19 0757 ~~ 0848 ~;, 0937 4S 4" 66 () 0102 5~ 0203 5~ 0305 sll 0406 ~6 0506M 0605 16 0704 19 0801 ~~ 0897 ~~ 0992 51 49 49

64 0 0107 5~ 0214 5~ 0321 5~ 0427 ~~ 0533g 0637 ~~ 0741 ~i 0844 ~J 0945 ~~ 1045 54 62 0 01125~ 0225 5~ 0337 5~ 0448M 0559 ~g 0669 ~~ 0778 !S 0886 ~~ 0993 ~j 1098 56 .,4

-- --- --- --- --- --- --- --- --- ---0,60 0 0,DI18 5~ 0,0235 5~ 0,0352 5~ 0,0469 ~~ 0,0585 ~~ 0,0700 ~~ 0,0814 ~~ 0,0928 ~~ 0,1040 ~~ 0,1150 59 -- --- --- --- --- --- --- --- --- ---

58 0 01236~ 0245 61 0367 6~ 0489 ~~ 0610 ~~ 0730 ~g 0850 ~~ 0968 ~~ 1085 ga 1201 61 oS .56 0 0127 6~ 0255 6~ 0382 6~ 0508 ~g 0634 12 0760 ~~ 0884 ~~ 1008 ~g 1130 ~i 1251

64 63 54 0 0132 6~ 0264 6~ 0396 6~ 0528 10 06591~ 0789 ~~ 0918 ~~ 1047 ~~ 1174 ~~ 1300 66 6.5 6,

52 0 0137 6~ 0274 6g 0410 6~ 0546 6~ 0682 ~~ 0817 ~j 0951 ~~ 1085 ~3 1217g 1348 68 -- --- --- --- --- --- --- --- --- ---

0,50 0 0,0141 7i 0,0283 7f 0,0424 i~ 0,0564 7~ 0,0705 ~6 0,0844 ~ri 0,0983 (\~ 0,1122 ~~ 0,1259 ~~ 0,1395 71 -- --- --- ---' --- --- --- --- --- ---48 0 0146 7~ 0291 7~ 0437 7~ 0582 7~ 0727 g 0871 ~~ 1015 i~ 1157 i~ 1299 ~1 1440 73 46 0 0150 7? 0300 7~ 0450 7~ 0599 7~ 0748 ~~ 0897 ~1 1045 }~ 1192 ~~ 1339 ~g 1485 75

44 () 0154 7; 0308 7j 0462 7~ 0616 7~ 0769 ~~ 0922 i~ 1074 ~~ 1226 ~~ 1377 ~~ 1528 77

42 0 0158 i~ 0316 7~ 0474 7g 0631 7g 0789 ~~ 0946 i~ I10q~ 1258 ~~ 1414 ~~ 1569 79 -- --- --- --- --- --- --- --- --- ---

0,40 0 0,0162S~ 0,0324 st 0,0485 sY 0,0647 S~ 0,0808 ~1 0,0969 ~~ 0,1130M 0,1290 ~~ 0,1450~ 0,1609 SI -- --- -'-- --- --- --- --- --- --- ---38 0 0165 R~ 0331 Hj 0496 S~ 0661 H~ I 0826,i ' 0991 M 1156g 1320 ~~ 1484 !~ 164R 83 36 0 0169 si 0338 s~ 0507 sl 0675 8; 0844 S~ 1012 ~l 1181 ~~ 1349 ~1 1517 ~~ 1685 S4 34 0 0172 8~ 0344 S~ 0516 8~ 0689 s~ 0861 8~ 1033 ~:; 1203 ~~ 1377 ~j 1548 J~ 1720 86

32 0 o 175 R~ 0351 R~ 0526 8~ 0701 s~ 0877 8~ 1052 k~ 1227 ~~ 1403 ~~ 1578 ~~ 1754 88 -- --- --- --- --- --- --- --- --- ---0,30 0 0,0178 8~ 0,0356 ~~ 0,0535 s~ 0,0713 8~ 0,0892 8~ 0.1070 8~ 0,1249 ~~ 0,1428 M 0,1607 M 0,1786 89 -- --- ---- --- --- --- --- --- ------

28 n 0181 9: 0362 9~ 0543 9; 0724 9~ 0906 9; 1087 9i 1269 ~? 14.51 ~i 1633 ~f 1816 91 26 0 0184 9~ 0367 9~ 0551 9i 0735 9~ 0919 9~ 1103 9~ 12885i I-m~~ 1658 ~~ 1844 92 24 0 0186 95 0372 9~ 0558 9j 0745 93 0931 9~ 1118 9~ 1305 9~ 1-193 ~~ 1682 ~i 1871 93 22 II 0188 9~ 0376 9~ 0565 9~ 0754 gt 0943 9~ 1132 9.~ 1322 9~ 1512 ~~ 1703 ~~ 189.5 91

--- --- --- --- --- --- -- --- --- -

0,20 II 0,0190 9~ 0,0381 9~ 0,0571 9~ 0,0762 9j 0,0953 9j 0,11-15 93 0,1337 9~ 0,1530 9~ 0, 172.3 b~ 0,1918 9.1 -- --- --- --- --- --- --- -- --- ---

18 0 0192 9~ 0384 9~ 0577 ~ 0769 9i 0963 9~ 1155 9~ 1350 9~ 1545 9~ 1741 9~ 1938 96 16 0 0194 9~ 0388 9~ 0582 9~ 0776 9~ 0971 9~ 1167 9~ 1363 9~ 1550 9~ 175810~ 1957 97 14 n 0195 9~ 0391 9~ 0586 9~ 0782 9~ 0979 9~ 1176 9g 13H 9~ 1572103 I 7721O{ 1973

98 12 n 0197 9~ 0393 9~ 0590 9§ 0787 9~ 0985 9~ 1184106 1383103 158310f 1785,0~ 1987 98

--- --- --- --- --- --- --- --- ---0,10 " 0,0198 9~ 0,0395 9~ 0,0593 9~ 0,07921O~ 0,099110~ 0,119010~ 0,1391 1Oi 0,1 59310f 0,17951(,1 0,2000 99 -- --- --- --- --- --- --- --- --- ---

08 0 0198 ~ 0397 9~ 05961O~ 07951O~ 099510~ 3 139810~ 160010~ 180410j 2010 99 1196101

06 0 0199108 03981O~ 059810~ 0798106 0999IO! 1200lOi 14031O~ 160610~ 1811 IO~ 2017 100 04 0 0200lOg 0399106 05Q9106 0800101 1001lOJ 120310~ H061O~ 16111O~ 1816!O~ 2023 100 02 0 0200 lOg 040010~ 060010~ 0801 !OJ 1003101 12051O~ 1408!O~ 1613!O~ 181 '\ol 2026 100 --- --- --- --- --- --- --- ---

0,00 0 O,0200lOg 0,0400i~ 0,O601 1O? O,08021O~ 0,10031O~ 0,12061Og O,I4091Og O,16141Og O,1820!O~ 0,2027 Ion


x

T t = 0,22 I 0,24 I 0,26 I 0,28 I 0,30 I 0,32 I 0,34 I 0,36 I 0,38 I 0,40

1,00 61 0,137960 0,150060 0,161959 0,173859 0,185558 0,1972 57 0,208657 0,220056 0,231255 0.2422 --- --- --- --- --- --- --- --- ---

98 0 1378 Jl 1499 s8 1619 5g 1737 5g 1855J 1971 5~ 2086 5~ 2199~ 2311 5g 2422 61 96 1 1376J 1497 J 1617 5~ 1735 s~ 1853 s~ 1969 5~ 2083 5~ 2197 5~ 2309 s~ 2419 61 94 1 1373~ 1494~ 1613 5~ 1732 s~ 1849~ 1965 s~ 20?0 5~ 2193 5; 2305 s~ 2415 61 92 2 1369~ 1489~ 1609 s~ 1727 S~ 1844 s~ 1960 5~ 2074 5~ 2188 5~ 2299 s~ 2410 61 --- --- --- --- ---

0,90 3 O,1364~ 0,1484 5~ 0,1603 S~ 0,1721 S~ 0,1837 s~ 0,1953 s~ 0,2067 s~ 0,2181 s~ 0,2292 S~ 0,2403 60 --- --- --- --- --- --- --- --- ---

88 3 1357 ~ 1477 s~ 1595 5~ 1713 s~ 1829 s~ 1945 si 2059s~ 2172 s: 2284 S~ 2394 60 86 4 1349 66 1468 S~ 1587 5~ 1704 s~ 1820~ 1935 5~ 2'l49J 2162.J 2273~ 2384 60 84 4 13W sg 1459 sg 15i6 sg 1693~ 1809 s~ 1924 s~ 2038 s~ 2150 s~ 2262~ 2372 60 82 5 1330 5~ 1448 s~ 1565 s~ 6

1797 s~ 1911 s~ 2024~ 2137 s~ 2248~ 2358 59 1681 58 --- --- --- --- --- ---7 --- ---5 0,1319 5~ 0,1436 5~ 0,1552 5~ 0,1668 5~ 0,1783 5~ 0,1897 s~ 0,2122 J 0,2233 5~ 0,2343 0,80 59 0,2010 56 --- --- --- --- --- --- --- ---

78 6 1306 s~ 1423 ~ 1538 5~ 1653 5~ 1767 s~ 1881 s: 1994 5~ 2105~ 2216 5~ 2326 58

76 6 1292 5~ 1408 s4 1523 5~ 1637 s~ 1751 cg 18ti3~ 1976 5~ 2087 ~ 2197 5~ 2307 68

74 7 1277 5~ 1392 5~ 1506 5~ 1619 5~ 1732 ~ 1844 ~~ 1956 i~ 2067 ~g 2177 ~ 2286 57

72 8 1261 5~ 1375 s~ 1488 5~ 1600~ 17121~ 1824~ 1935ll 2045ll 2155g 2264 &7 --- --- --- --- ---11 --- ----

0,70 8 0,1244 ~ O,1356~ 0,1468 ~~ 0,1580 ~ O,I691ll 0,180211 O,2022~ O,2131ll 0,2240 56 0,1912 55 --- --- --- --- ---

~~ --- -

68 9 1225~ 1336~ 1447M 1557 ~~ 1668 ~~ 1778 1~ 1997 ~~ 2105 A~ 2214 56 66 9 1206 1~ 1315 ~ 1425 11 ' 1534 1; . 1643~ 1752 ~~ 1861 ~ 1970~ 2078 A: 2186 55 64 10 1l85~ 1293 A~ 1401 ~~ 1509 ~~ 1617 ~~ 1725~ 1833 ~: 1941 ~: 2048~ 2156 54 62 10 1163M 1269 ~~ 1376~ 1482~ 1589M 1696M 1803 ig 1910 ig 2017 ~g 2124

53 --- --- --- --- --- --- --- --- ---0,60 11 O,1139g 0,1244 ~~ 0.1349 ~~ 0,1454 M O,1560 1g 0,1665 ~ 0,1771 ~g 0,1877 ~~ 0,1983 g 0,2090 52 --- --- --- --- --- --- --- --- ---

58 11 1115 ~~ 1218~~ 1321 ~~ 1425l~ 1529 ~~ 1633 ~~ 1738 g 18431~ 19~8 ~g 2C53 51 56 12 1089 ~~ 1190 ~1 1292 Af 13941~ 1496 ~~ 1599M 1702 ~~ 18061~ 1 910 1~ 2015 51 54 12 1063 ~~ 1162 M 12611g 1361 ~g 14621~ 15631~ 1665 ~~ 17681~ 187q~ 1975 49 52 13 1035l~ 1132 !~ 1229l~ 1327 !~ 1426 ~~ 1526~ 1626 ~~ 1727 ~~ 1829 ;: 1932

48 --- --- --- --- --- --- --- --- ---0,50 13 O,1006l; 0,1100 !g O,U96ll O, I292 lg 0,1339 !~ 0,1486 ~g 0,1585 ~~ 0,1685 ~~ 0,1785 ~i O,I1l87

47 --- --- --- --- --- --- --- --- ---48 14 0976 !~ 1068 ~~ 1161 !~ 1255l~ 1350 ~g 1445 ~~ 1542 ~~ 1640 ~ 1739 ~6 1839

46 46 14 0945l~ 1034 ~~ 1125 !~ 1216 !~ 1309 ~~ 1403 n 1497 ~~ 1593 ~~ 1691 ~~ 1789

44 44 15 0912~ 0999 U 1087~ 1176 ~g 1266 ~~ 1358 ~~ 1451 ~~ 1545 ~~ 16~0 ~~ 1737

43 42 15 0879 ~~ 0963 ~g 1048l~ 1135ll 1222 ~~ 1311 ~~ 1402 ~~ 1494 ~~ 1587 ~~ 1682

42 --- --- --- --- --- --- --- --- ---0,40 16 O,Oll45 n 0,0926 ~i 0,1008 ~g 0,1092 ~~ 0,1177 ~~ O, I263 ll 0,1351 ~~ 0.144q; 0,1532 ~~ 0,1624

40 --- --- --- --- --- --- --- --- ---38 16 0809 18 0888~ 0967 ~5 1048 ~~ I 130 ~~ 1213 ~~ 1298 ~~ 1385 ~~ 1474 ~~ 1564 39 39 36 17 0773 18 0848 ~~ 0924 ~~ 1002~ 1081 ~ci 1162 ~~ 1244 ~~ 1328 ~~ 1414~ 1502 37 38 34 17 0736 ~~ 0808~ 0880~ 0955 ~~ 1031 ~~ 1l08~ 1188 ~~ 1269 ~~ 1352 ~~ 1437

35 32 17 0698~ 0766 ~1 0835~ 0906~ 0979 ~~ 1053 ~~ 1129 ~~ 1207 ~~ 1287 ~~ 1369

34 --- --- --- --- --- --- --- --- ---0,30 18 0,0659 ~~ O,0723~ O,0789~~ O,0857~ O,0926~ 0,0996 ~~ O,1069~ O,1144~ O,I22U ~ 0,1299 32


x

I t = 0,22 I 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38 0,40

28 18 0619 ~? 0680 ~i 0742 24 0806~ 0871 n 0938 ~~ 1007 ~A 1078 ~~ 1151 ~ 1226 30 32 26 18 0578 ~g 0635 ~~ 069t ici 0754 ~~ 0815 ~~ 0878~ 0943~ 1010 34 1080~ 1151 28 3., 24 19 0537n 0590 ~~ 0645 ~~ 0700 ~~ 0758 ~~ 0817 gJ 0878~ 0941 ~ 1006~ 1073 26 22 19 0495 ~~ 0544 ~~ 0594 ~~ 0646 ~~ 0699~ 0754 ~~ 0811 ~~ 0870 gg 0930~ 0993 24 --- --- --- --- --- --- --- --- ---

0,20 19 0,0452 ~~ 0,0497 ~~ 0,0543 ~~ 0,0591 ~~ 0,0640 rs 0,0690 ~~ 0,0743 ~~ 0,0797 ~~ 0,0853 ~~ 0,0911 22 --- --- --- --- --- --- --- --- ---18 20 0408 ~~ 0449 ~t 0491 ~~ 0535 ~g 0579~ 0615 ~~ 0673 ~~ 0722 ~~ 0773~ 0827 20 16 20 0364 ~~ 0401 ~~ 0439 ig 0477 ~~ 0517 ~l 0559~ 0601 ~~ 0646 ~g 0692 ~! 0740 18 14 20 0320 i~ 0352 ~~ 0385n 0419 i~ 0455 ~~ 0491 ~~ 0529 ~~ 0568 ~~ 0609 ~l 0652 16 12 20 0275n 0303 i~ 0331 i~ 0361 i~ 0391 ~~ 0423 ~~ 0455 r; 0489 r~ 0525 t~ 0562 14 --- --- --- --- --- --- --- --- ---

0,10 20 0,0230 i~ 0,0253 i~ 0,0277 ~~ 0,0302 r~ 0,0317 r~ O,0354 n 0.0381 rl 0,0410 tg 0,0439 t~ 0,0471 11 --- --- --- --- --- --- --- --- ---08 21 0184 2~ O~OJ ig 0222 27 0242 ~g 0262 ~i 0284 rf 0306 r~ 0329 :g 03531g 0378 9 , 10 06 21 0138 2~ 0152 2~ 0167 2~ 0182 3~ 0197 3~ 0213 3~ 0230 3~ 0247 4~ 0265 tci 0285 7 04 21 0092 2~ 0102 2~ 0111 2~ 0121 ag 0132 a~ 0142 3~ 0154 a~ 0165 4~ 0177~ 0190 s 02 21 0046 2~ 0051 2~ 0056 2g 0061 3g 0066 a~ 0071 a~ 0077 sg 0083 4~ 0089~ 0095 2 --- --- --- --- --- --- --- --- .---

0,00 21 0,0000 2~ O,OOO02g O,OOO02g 0,0000 ag o,ooooSZ o,ooooag 0,00003g 0,0000 4~ O,OOoo~ 0,0000 0

y

I t = 0,22 I 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38 0,40

1,00 a 0,0000 0 0,0000 a 0,0000 0 0,0000 a 0,0000 a 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 --- --- --- --- --- --- --- --- ---

98 30 006633 0071 3g 0077 38 0082 4~ 0086 4~ U091 4g 0096 4~ 0100 sg 0104 s~ 0108 ~ .3

96 30 0132 3~ 0143 3~ 0153 3~ 0163 4~ 0173 4~ 0182 4~ 0191 4~ 0200~ 0209 s~ 0217 6

94 30 0198 3~ 0214 3~ 0229 3~ 0245 4~ 0259 4~ 0273 4~ 0287 4~ 0300 sg 0313 5~ 0325 8 92 30 0263 if 0185 i~ 0306 fg 0326 i~ 0346 4~ 0364 4g 0383 4g 0400 5~ 0417 5~ 0433

11 --- --- --- --- --- --- --- --- ---

0,90 aD 0,0329n 0,0356~ 0,0382 rj 0,0407 t~ 0,0432 t~ 0,0455 t~ 0,0478 t~ 0,0500 f~ 0,0521 f5 0,0542 14

--- --- --- --- --- --- --- --- ---88 30 0394 ~~ 0426 f~ 0458 f~ 0488 t~ 0518 t~ 05t6 t~ 05741~ 0600 fg 0625 r~ 0650

17

86 30 0459n 0497 f~ 0533 r~ 0569 ~~ 0603~ 0637 ~~ 0669 i~ 0700 f~ 0729 ~~ 0758 19 84 30 0524 ~J 0567 ~t 0609 ~g 06~9 ~g 0689 4~ 0727 i~ 07641~ 0799 ~~ 0833 f~ 0866 22 82 29 0588 ~~ 0637 ;~ 0684 g~ 0730 ~g 0774 ~~ 0817 ~f 0859 ~6 0899 ~g 0937 f~ 0974 25

--- --- --- --- --- --- --- --- ---0,80 29 0,0652 ~~ 0,0706 ~~ 0,0758 ~~ 0,0809 ~~ 0,0859 ~~ 0,0907 ~3 0,0953 ~~ 0,0998 ~g 0,1041 ~i 0,1082 28 -- --- --- --- --- --- --- --- --- ---

78 29 0716 ~~ 0775 ~~ 0833 ~~ 0889 ~~ 0943 ~~ 0996 ~~ 1047 ~~ 1097 ~~ 1144 ~~ 1190 ao

76 29 0779 j~ 08433i 0906 ~i 0968~ 1027 ~~ 1085 ~~ 1141 ~~ 1195 ~~ 1248 ~~ 1298 33 74 29

0811 ~~ 09113! 0980 g~ 1046 ~~ 1111 ~i 1174 ~i 1235 ~~ 1294 ~~ 1351 ~~ 1405 36

72 28 0903 ~~ 09i8M 1052 ~~ 1124 ~~ 1194~ 1262~ 1328 ~~ 1392 ~~ 1453 g& 1513 38


y

1: I I t = 0,22 I 0,24 I 0,26 I 0,28 I 0,30 1

0,32 I 0,34 I 0,36 I 0,38 0,40

I I 0,70 28 0.O964~ 0,10~5 ~ 0,1124~~ 0,1201 g~ 0, 1271 ~~ 0,1350 ~ 0,1421 ;~ 01489 49 I 0,1556 g~ 0,1620 41 ' 33 -- --- --- --- --- --- --- --- --- ---

68 28 102" ~~ 1111 ~~ 1196 ~7 1278 ~~ 1358 ~~ 1437 ~~ 1513 4~ 1587 ~~ 1658 3\ 17:27 44 3,

66 27 108.5 ~~: 1176 ~~ I ~66 ~~ 1354:<8 wog 1523 ;~ 1604 1r. WS3 ;~ 1760 :;~ itm 47 ·I~ 39 64 ':!i 1144 30 12~1 ~~ 1336 ~~ 1429 ~~ 1520 !~ I f09!~ 1695 j~ 1779 ~~ 1861 01 19-!0 49 4' 39

6:2 26 1202 ~i 130 I o~ 1405 ~~ 1503 ~~ 1.599 1~ 16941~ 1785 4~ 1874 ~R 1~61 ~~ 2015 52 "tJ q., ~;J

-- --- --- --- --- --- --- --- --- ---0,60 26 0,1:259~~ 0, 13.i7 :~1 o,l-m~; 0,1576 ~; O,Ib78 ~g 0,1'77~~ 0,11)75 .;~ 0,1969i~ 0,2061 ~~ 0,2150 :;5 -- --- --- --- --- --- --- --- --- ---

58 26 1316 ~~ 1428~~ 15~0 ~~ 1649 ~; 1756 ~~ 18t.>0 ~i 1963 ~4 20631~ 2160 ~¥ :l2.55 57 "6 '").1 .,0 56 2.; 1371 ?R 1439 ~" 1605 3~ 17;20 30 1832 ~~ 1942 ~l! 2050 ~; 2156 ri~ 2259 ~p, 2359 60 .·9 .,~ S, ,1G 0·)

54 2:") 1425 ~~ 1648 3~ 1670 ~~ l'i9J ~~ 1908 3R 2023 ~~ 2137 ;~ 2248 ~~ 2356 ~~ 2462 ,:2 [,1 58

.52 24 1478 2" 1607 i~ 1733 ~~ 1859 ~~ 1982 ~; 2103 ~g 2222 ~~ 2138 4; 2~52 ~~ 2564 65 64 57 -- --- --- --- --- --- --- --- --- ---0,50 23 0,1530 i¥ n, 1663 ~~ 0, 179o n 0, 1926~! 0,2054 ~~ 0,2181 ~i O,2:l0() ~I 0,2428 ~~ 0,2548 i~ 0,2665 6, -- --- --- --- --- --- --- --- --- ---

48 23 1580 ~~ 1719 28 1856 ~~ 1992 ~~ 2126 ~~ 2258 ~~ 2388 ~.\ 2516 ~~ 26~:! ~~ 2763 70 69

46 22 1629 ~~ 1,73 ~; 1915 ~~ 2056 ~~ :1196~~ 2333 ~~ 2169 ~~ 2603 ~~ 2734 4~ 2863 72 6., 44 21 1677 ~: 1826 ~~ 1973 ~~ 2i 19 ~~ 2264 ~i 2407n 2548 ~g 2688 ~~ 2S~5 :~ 2690 75 42 21 1723 ~~ 1876 ~~ 2029 ~~ 2180 ~! 2330 ~ 2478 ~~ 2626 ~~ :!771 ~~ 2914 n 3056 77 ,0 -- --- --- --- --- --- --- --- --- ---

(1,40 20 0,1768 ~; o,19:!6 ~~ 0,2083 n 0,2239 ~~ 0,2394 ~~ 0,2')·18 ~~ 0,2701 ~ 0,2852 n 0,:j002 ;: 0,~1.j9 79 -- --- --- --- --- --- --- --- ---2774 ~~ i 38 19 22' 19i3 ijt 2135 ~~ 2296 ~g 2457 ~5 2616 ~~ 2931 ~~ . 3087 ~~ 3241 82 1811 81 I

36 19 1852 ~! 2019 ~~ 2186 ~3 2352 ~~ 2517 ~ 2682 ~ 2845 ~~ 3008 ~~ 3169 ~b 3330 84

34 18 !802 ~~ 2063 ~~ 2234 ~~ 2405 ~~ 2575 ~~ 2745 ~~ 2914 34 3082 ;~ 3250 :~ 3416 86 8·1 32 17 19~9 ~~ 2105 ~~ 2280 ~~ 245.5 ~~ 2630 ~~ 281,5 ~~ 2980 ~~ 3154 ~~ 3327 ~~ 3')00 as

-- --- --- --- --- --- --- --- --- ---0,30 JG 0, I (!65 ~~ 0,2144 ~6 O,2J24 ~~ 0,2:;04 ~~ 0,:268:3 ~~ 0,2863 ~ci 0.3043 ~~ O,3~2:2 ~0 0.3102 ;[. 0,:1580 GO -- --- --- --- --- --- --- --- --- ---

28 1 ; 1999 b~ 21ti2 !\i 2366 ~~ 25~9 f~ 2734 ~~ 2918 ~~ 3103 ~~ 3288 ~~ 3473 ~~ :1638 91

26 J.! 2031 ~l~ ~2i7 ~'; 2405 ~~ 2593 ~~ 2781 ~1 2970 ~~ 3160 ~~ 3350 ~1 3510 :\1 3731 93 911

24 13 2060 ~~ 2251 ~~ 2442 ~~ 2633 ~~ 2826 g 3019 ~] .3213 ~~ 340:1 ~~ 360.'j ~; 3801 P5

22 12 2088 ~~ 22a 1 ~; 24 7fi ~~ 2671 ~~ 2867 ~~ 3065 ~~ 321i4113 34631~i 3665[8'i 3867 fl6 -- --- --- --- --- --- --- --- --- ---

0,70 11 o .J 13 ~~ O,:!,l!U &;; 0)507 ~~ 0.27061.tTr 0,2lJI'61&i (),31071~: 0,3310,Gi 0,35141~~ 0,37 ,;od~ 0,:lY28 9H ,--- --- --- --- --- --- --- --- --- ---

IS 10 2136111~ 2335:6e ~5J6Ji 27381?~ 29411r~ 31461i~ 33331?~ 3j611~~ 37i21~~ 3lJ8~ 9" 16 9 21571b~ 23591,1~ 25621,1~ 27661!)~ 29731b~ 31811b~ 3391 1?)ri 36041z,i 381Rd~ 4036 lOll

14 8 217bH~ 23791't,~ 2.:;8Jli[~ 27921.1~ 300 11&~ 32121b~ 31~6),~ 36121g 38601i: 4081 lUI

I~ ;

2191lU~ 23V7J(1~ :2C!)51i,~ :!8141;~ 30261!'~ 3240))~ 34561b~ 36751:i 38971:~ 4L2 102 -- --- --- --- --- --- --- --- --- --- ---

0,10 r, 0,221).11,,; O,~413iO~ 0.162210~ (l,28:)310~ 0,3047 1!'~ 0,3263Ib~ 0.34821 ::! U,37031g 0,3Q2:'1I;~ 0,4156 103 -- --- --- --- --- --- --- --- --- ---

08 5 2~17J[>~ 2425iO~ 2636li:~ 28 j9!O~ 306410~ 3282118 35f31g 37271]~ 395411~ 4185 10~

06 4 2::2\0; 243510~ 264710~ 28611O~ 3078lib 329711? 351911~ 374511~ 3974li~ 4207 104

04 3 :2;:3!1O~ 2442w~ I 26551r~ 28701n~ 3087 l1g 330811~ 353111~ 375811~ 3989i1~ 4223 104

02 2 223':\1'~ "446 2\ <)"- 0 28751O~ 309311t 331411~ 3539111 376611ci 399811~ 4233 105 __ ~ __ 107 ~I(J8 -- --- --- --- --- --- --- ---O,UO 1 0.2237 1UA 0,:2 q81U~ I 0, 661 J(~ O,2tl7710~ 0,309511 ; 0,331611~ 0,3541 11! 0,376911~ 0,40UIIl~ 0,4237 105


x

1: I I t ... 0,42 I 0,44 I 0.46 I 0,48 I 0,50 I 0,52 I 0,54 0,56 I 0,58 I 0,60

1,00 55 0,2531 54 0,2639 53 0.274552 0,284951 0,2952 51 0,305350 0,315249 0,325048 0,334647 0,3440 -- --- --- --- --- --- --- --- --- ---

98 0 2531 ~ 2638 5g 2744 5g 2848 5~ 2951 5~ 3052~ 3151 4g 3249 4g 33454~ 3440 55 96 1 2528 sl 2636J 2742 5~ 2846 5~ 2949 51 3050 5~ 3149 4~ 3247 4A 33434~ 3438 55 9-4 2 2524~ 2632 5~ 2738 5~ 2842 5~ 2945 5~ 3046~ 3146 4~ 3244J 3340 4~ 343.5 55

92 8 2519 5~ 2626 5g 27325~ 2837 S~ 2940 5~ 3041 ~ 3141 4~ 3239 4~ 3335 4~ 3430 55 -- --- --- --- --- --- --- --- --- ---0,90 4 0,2512 s: 0,2619 5~ O,2i25 5~ 0,2830 5~ 0,2933 5~ O,3034~ 0.3134 4~ O,32324~ 0,3329 4~ 0,3424 S5 -- --- --- --- --- --- ---

88 4 2503 s: 2610 5~ 2717 5~ 28215~ 2924 51 3026~ 3126 4~ 3224 4~ 3321 4~ 3417 54 86 5 2493 5~ 2600 53 2706 5~ 2811 5~ 2914 5~ 3016~ 3116 4~ 3215 4~ 3312J 3408 54 84 6 2~81 5~ 2588 5g 2694 5g 2799 5~ 2903 sf 3005 sg 3105 5g 3204 4~ 3302 4~ 3398 54 82 7 2467 5~ 2574 5~ 2681 5~ 2786 5~ 2889 5I 2993 5g 3093~ 3192 4g 3290 4~ 3386 54 -- --- --- -- --- --- --- ---

0,80 8 O,2~52 5~ O,2559 5g 0,2666 5~ O,27715~ 0,2874 5~ 0,2977 5I 0,3078 5~ 0,3178 44 O,3276~ 0,3373 54 -- --- --- --- --- --- --- --- ---78 9 243-4 5~ 2542.J 2648 sg 2754 5g 2858 5~ 29615~ 3062~ 3162 4: 3261 4: 3359 54 76 9 2416 5~ 2523 5~ 2630~ 2735J 2839 5~ 2942 S~ 3044~ 3145~ 3244 4~ 3342 54 74 10 2395 ~~ 2502 ~g 2609 ~g :27l5 ~g 2819 ~g 2923 ~~ 3025 A~ 3126 Ml 3226 4~ 3324 54 72 11 2372g 2480 ~~ 2586M 2692M 2797 M 2901 M 3003M 3105~ 3205 ~g 3305 54 -- --- --- --- --- --- --- --- --- ---

0,70 12 0.2348 ~~ 0,2455 ~~ O,2562~ 0,2668 ~~ 0,2773 ~~ 0,2871 ~~ 0,2980 ~~ o,3082M 0,3183 M 0,3283 54 -- --- --- --- --- --- --- - --- ---68 13 2322 ~! 2429 13 2536 ~~ 2641 ~g 2747 ~~ 28511~ 2955 ~~ 3057 ~~ 3159 12 3260 54 5~ so 66 14 2293 ~: 2400 ~~ 2507~ 2613 ~~ 2719 ~~. 28231~ 2927~i 3030M 3133 ~~ 3234 54 64 15 2263 ~~ 2370 13 2476~ 2582 ~g 2688~g 2793 A~ 2898 ~~ 30011~ ;)104 ~i 3207 54

62 16 2231 Ag 2337 Ag 2444~ 2550 ~g 2655~ 27611~ 2866 A~ 2970l~ 3074~ 3171 53 -- --- --- --- --- --- --- --- --- ---0,60 17 0,2196 M O,2302M 0,2408 ~g O,2515~ 0,2620 Ag O,2726M 0,2831 A~ 0,2936 A~ 0,3041 A~ 0,3145 53 -- --- --- --- --- --- --- --- ---

58 18 2159~ 2265 Ag 23711~ 2477~ 2583 A~ 2689~ 2795~ 2900~ 3005 ~~ 3110 5:i 56 19 2120~ 2225~g 2331 ~g 2437~ 2543~ 2649~ 2755 ~g 2861 Ag 2967 Ag 3073 53

54 20 2079 ~~ 2183 ~~ 2289~ 2394 ~~ 2500 ~~ 2606 ~~ 2713 ~~ 2819 ~~ 2926~ 3033 fi2

52 21 2035 ~~ 2139 ~~ 2244~ 2349 ~~ 2455 ~~ 2561 ~g 2668 13 2775 ~~ 2882 ~~ 2989 52 54 -- --- --- --- --- -- --- --- --- ---0,50 23 0,1989 ~~ O,2092~~ O,2196~ 0,2301 ~ O,2406~ O,2513~ 0,2619 ~l 0,2727 ~l 0,2835 ~; 0,2943 51 -- --- --- --- --- --- --- --- --- --- ----

48 24 1940 ~~ 2042~g 2145~ 2250~ 2355 ~g 2461 ~g 2568 ~~ 2676 ~~ 2784 25 2893 51 55

46 25 1889 ~6 1990 ~~ 2092 ~~ 2196 ~~ 2300 ~~ 2406 27 2513 ~~ 2621ll 2730 ~~ 2840 :'1) .,1 53 -14 21;

1835 ~b 1935~~ 2036 ~7 2138 ~~ 2242~ 2348 ~~ 2454 ~~ 2563 ~~ 2672 ~g 2i82 49

42 27 1778 ~~ 1876 ~g 1976~ 2078 ~g 2181 ~~ 2286~ 2392 ~l 2500 ~! 2610 31 2721 48

O~ 55 -- --- --- --- --- --- --- --- --- ---0,40 29 O,17l9 ~~ 0,1815 ~~ 0,1914 ~ O,2014~ O,2116~ O,2220~ 0,2326 ~~ O,2434~ O,2543~~ 0,2654 47 -- --- --- --- --- --- --- --- --- ---

38 30 1657 ~} 1751 ~~ 1848~ 1946~ 2047 g~ 2150 gg 2255~ 2363 ~~ 2472 ~~ 2584 46

36 31 1592 f~ 1684~ 1779~ 1875~ 1975~ 2076~~ 2180 37 2287~ 2396 ~g 2507 4.5 53 3-1 33 1524~~ 1613~ 1706 ~~ 1801 ~~ 1898: 1998 ~~ 2\o1~ 2206~ 2315~ 2426 44

32 34 1454~ 1541 ~ 1630 ~~ 1722 ~~ 1817 !g 1915~: 2016 ~~ 2121 ~ 2228 ~~ 2338 42 -- --- --- --- --- --- --- --- --- ---0,30 35 0,1380 ~~ 0,1465 ~~ 0,1551 :g O,1640:~ 0,1733 :~ 0,1828 ~6 0,1927 ~f 0,2029 ~g O,2135~ O,224,i 41


x

~ I I t = 0,42 I 0,44 0,46 0,48 0,50 0,52 0,54 0,56 0,58 0,60

28 36 1304~ 1384 :g 1468 :1 1554 !~ 1644 :~ 1736~ 1833 ~b 1933 ~~ 2037 ~~ 2145 39

26 38 1225 ~g 1302 !~ 1381 :~ 1464 :3 1550 :~ 1640 !~ 1733 ~g 1830 51 1935 ~~ 2037 37 SI

24 39 1143 ~~ 1216 ~~ 1292 ~~ 1370 :I 1452 !~ 1538 ~~ 1628 ~~ 1722 ~~ 1820 ~~ 1923 35

22 40 lC59 ll 1127 ~~ 1198 ~~ 1273 ~~ 1350 ~l 1432 ~~ 1517 ~~ 1607 ~~ 1702 ~~ 1801 33 --- --- --- --- --- --- --- --- ---

0,20 41 o,om~~ O,1035~ 0,1102 ~ 0,1171 ~~ O,I244g1 O,I321 5g 0,1401 ~g 0,1487 ~~ 0,1576 ~~ 0,1672 30 --- --- --- --- --- --- --- ---

18 42 0882~ 0941 :i IOJ2 gg 1066 g~ 1134 ~ 1205 g~ 1280 ~~ 1360 ~~ 1444 ~~ 1534 28

16 43 0791 ~~ 0844 ~g 0899 ~~ 0957~ 1019~ 1084~ 1154 ~~ 1227 g~ 1305 ~g 1389 25

14 41 0697 ~~ 0744 ~ 0793 ~~ 0846 ~~ 0902 ~~ 0960 ~~ 1022 g~ 10~8 ~~ 1160~ 1236 22

12 45 0601 ~g 0642 ~~ 0685 ~~ 0731 ~~ 0779 ~~ 0331 ~i 0886 ~; 0945 ~~ 1008 ~~ 1075 20 --- --- --- --- --- --- --- --- ---

0,10 46 0,0504 i~ 0,0538 ~~ 0,0575 ~~ 0,0613 ~i 0,06j5 ~~ 0,0698 ~~ 0,0745 &g 0,0796 ~~ 0,0850 ~j 0,0908 10 --- --- --- --- --- --- --- --- ---

08 46 0405 i~ 0433 fg 0462 f~ 0494n 0527 ¥~ 0563 ~~ OEOI ~~ 0642 ~~ 0687 ~~ 0735 13

06 47 0305 f? 0326 r~ 0348 f~ 0372 ~~ 0397 ~~ 0424 ~~ 0454 n 0485 :~ 0519 ~~ 0556 10

04 47 0204 s~ 0218 s~ 0233 sg 02496~ 0266~ 0284 :g 0304 :f 0325 ~g 0348 ~~ 0373 7

02 47 0102 s~ 0109 s: 0117 s~ 0125 6~ 0133 sg 0112 7~ 0152 7~ 0163 8A 0175 8J 0187 3 --- --- ---oOo:iO 62 I 0 0000 57

--- --- ---0,00 48· 0,0000 SA o,oUG0 5g o,oouosg 0,0000 7~ 0,0000 7~ O,OOJO 85 o,ouoo 8~ 0,0000

0 • 0 ' 0

y

I t = 0,42 I 0,44 0,46 0,48 0,50 0,.52 0,54 0,56 0,58 0,60

1,00 0 (\0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0300 0 o,oeoo 0 0,0000 0 0,00(10 0 0,0000

- --- --- --- --- --- --- --- --- ---98 54 0112 5g 0116 5g 0119 sg 0123 6~ 0126 6~ 01296t 0131 6~ 0134 6; 0136 6; 0139

2

96 54 0224 E~ 0232 5~ 0239 6~ 0245 6~ 0251 6~ 0257 6j 0263 6~ 0268 5~ 0273 6~ 0277 4

94 54 0337 sJ 03+7 5~ 03.)8 6J 0363 6~ 0377 6J 0386 6: 0394 6~ 04)2 6~ 0409 fi8 0416 6 3

92 St 0449 s~ 0463 s~ 0477 6~ 0491 6~ 0503 6~ 0515 6~ 05:26 E~ 0536 6~ 0516 6~ 0555 8 -- --- --- --- --- --- --- --- --- ---

0,90 54 0,0561 sg 0,0579~ 0,0597 6~ 0,0613 6~ 0,0629 6~ 0,0644 t~ 0,0658 5~ 0,0671 6~ o 0683 6~ 0,069,5 10 -- --- --- --- --- --- --- --- --- ---

88 54 0673 ~f 0695 ~~ 071 6n 0736n 0755 6~ 07,3 6~ 0790 6~ 0806 6; 0821 6~ 0834 12

86 54 0785 f~ 0311 f~ 0836 ~~ 0859n 0382n 0903n 0922 6~ 094 J 6~ 0958 6~ 0975 14

84 54 0837 5j 0927 ~~ 0956 ~1 0983 62 IOll8 ~~ 1032 ~~ 1055 ~: 10i7 ~g '1097 6~ 1115 16 15. 1~

82 54 1010 f~ 1043 ~~ 1076n 1106 ~~ 1135 ~~ 11112 ~~ 1I88n 12 13n 1235 fi 1257 18

-- --- --- --- --- --- --- --- --- ---0,80 54 0,1122 rg 0,IJ59~~ O,I 195n 0,1230 f~ 0,1262 ~ 0,1293 ~ 0,1322 n 0,1349 ~~ 0,1375 ;g 0,1399

20 -- --- --- --- --- --- --- --- --- ---

78 51 1234 g~ 1276 ~~ 1316 ~J 1354 ~~ 1390 ~~ 1424 n H56n 1486n 151.5 ;~ 1541 22

76 54 1346 ~~ 1392 ~~ 1436 g1 1478~ 1517 ~~ 1555 ~~ 1591 n 1624~ 1656 ;g 1685 21

74 54 1458 ~~ 1508 ;~ 1556 ~g 1602 ~~ 1646 ~t 1687 ~e 1726 ~~ 1763 ~~ 1797 ;~ 1830 26

72 54 -6 1624 ~~ 1677 g~ 1727 ~~ 1774 ~~ 1819~~ 1862 ~g 1902 ;~ 1940 ;~ 1975

29 1570 27


y

I t= 0,42 I 0,44 0,46 0,48 0,50 0,52 0,54 0,56 0,58 0,60

0,70 54 0,1682 ~e 0,1741 g~ 0,1797 g~ 0,1851 ~~ 0,1903 g: 0,1952 g~ ~l 0,2042 ~~ 0,2083 ;~ 0,2122 31 0,1998 22 -- --- --- --- --- --- --- --- ---

68 :;3

1793 ~g 1857 ~~ 1918~ 1976~ 2032~ 2085g~ 2135 ~~ 2183 ~g 2227 ~i 2269 33

66 53 56 1973 ~~ 2039 ~~ 2101~ 2161 ~ 2219 g~ 2273 ~~ 2324 ~l 2373~~ 2418 36 1904 34 64 53 2016 ~~ 2089~ 2159~ 2227 ~~ 2291 ~~ 2353~ 2411 ~ 2467 ~~ 2519 ~~ 2568 38

62 53 2126 ~~ 2205~ 2280~ 2352~~ 2421 ~ 2487 ~i 2550 ~g 2610 ~~ 2666 ~i 2720 41

-- --- --- --- --- --- --- --- --- ---0,60 53 0,2237 ~g 0,2320 ~~ 0,2401 gg 0,2478 ~~ 0,2552~ O,2623~ 0,2690 ~g 0,2754 ~~ 0;2815 ~~ 0,2872 43 -- --- --- --- --- --- --- --- --- --- ---

58 52 2347 ~~ 2435 ~g 2521 ~~ 2603ro 2683 ~~ 2758 ~~ 2831 ~~ 2899 ~~ 2965 ~r 3026 46

56 52 2456 ~~ 2550~ 2641 ~ 2729 ~~ 2813 ~~ 2894~ 2972 ~~ 3045 ~~ 3116~ 3182 49

54 52 2565 ~~ 2664 ~~ 2761 ~ 2854 fs 2944 ~~ 3031 ~i 3113 ~~ 3192~ 3267 ~ 3339 51 52 51 2672~ 2778 ~i 2880~ 2979~ 3075 ~~ 3167 ~ 3255 ~~ 3340 ~ci 3421 ~~ 3497 54

-- --- --- --- --- --- --- --- --- ---0,50 51

0,2779 ~~ 0,2891 ~~ 0,2999 ~~ 0,3104 ~i 0,3206 ~~ 0,3304 ~ 0,3398 ~~ 0,3488 ~~ 0,3575 ~; 0,3657 57 --- --- --- --- --- --- --- --- --- ._--

48 50 2885 ~~ 3003 ~~ 3117 ~~ 3229~ 3336 ~~ 3440 ~~ 3541 ~; 3637 ~~ 3730 78 3818 60 ¥.

46 49 2990 ~~ 3114~ 3234 ~g 3352 ~i 3466 g~ 3577 ~~ 72

3787 ~~ 3886 :I~ 3981 6~ 3684 5~

44 49 3093 g~ 3223 ~~ 3351 ~~ 3475 ~~ 3596 ~g 3714~ 3827 ~~ 3937~ 4043 ~~ 4144 66

42 48 3195 ~~ 3332~ 3466 ~~ 3597~ 3725~ 3849 ~i 3970 ~~ 4087n 4200 ~~ 4309 70 -- '--- --- --- --- --- --- --- --- ---.

0,40 47 0,3295 ~g 0,3438 ~~ 0,3579 ~~ 0,3717 ~g 0,3852 ~ 0,3984 ~~ -1 0,4238 ~~ 0,435& ~~ 0,4475 73 0,4113 62

-- --- -.-- --- --- --- --- --- --- --38 46 3393 ~~ 3543 ~~ 3691 ;~ 3836 ~~ 3979 ~~ 41l9~~ 4255 ~~ 4388 ~~ 4517 ~~ 4641 76 ,.) 36 45

3488 ~~ 3645 ~~ 3800 ~~ 3953 59 4104 ~~ 4251 ~~ 4396 i~ 4537 ~~ 4675 79 4808 79 75 f,7 34 43

3581 ~~ 3745 ~? 3908 ~6 4068 ;r 4226 ~~ 4382 ~~ 4535 !~ 4685 ~j 4832 ~i 4975 83 I,)

32 42 3672 45 3843 ~~ 4012 ~ 4180 ~3 4347 ~~ -l.:ill ~t 4673 ~g 4833 4~ 4989 ;~ 5141 86 85 -- --- --- --- --- --- --- --- --- ---0,30 40

0,3759 ~~ 0,3937 ~~ 0,4114 ~~ 0,4290 ~ 0,4464 ~i 0,4637 ~ 0,4809 ~~ 0,4977 ~~ 0,5144 ~~ 0,5307 89 -- --- --- --- --- --- --- --- --- ---

~8 39 3843 ~~ 4028 ri~ 4212 ~g 4396 g] 4579 57 4760 ~~ 4941 ~~ 5120 ~~ 5296 ~~ 5471 9:1 91 26 :17 3923 ~ 4114 ~J 4306 ~ 4498 ~ 4689 ~~ 4880 60 5070 6~ 5259 ~~ 5446 ~~ 5632 !Hi 95 90

24 :i.~ 3999 ~ 4197 41 43961~ 45951ri~ 47951M 49951f~ 51951?'5 539-1 g~ 5593 ~~ 5790 99 99 ~:! 33 40701~~ 42751~ 44811ri1 46881~ 48961g~ 51051g~ 53141~g 55241~~ 573416~ 5944 102 --- --- --- --- --- --- --- --- ---

0,20 RI 0,413812~ 0,43481~ 0,45611ri~ 0,-H751ri~ 0,4991 1g 0,52091n 0,54281~b O,56481ti 0,58701~~ 0,6092 1(1.')

--- --- --- --- --- --- --- --- ---18 2M 419912g 44161iri 46351n 48561g 50801t~ 53071n 55351~~ 57651~~ 59981~~ 6233. 107 16 25 42551ii 44781i1 47031]1 4931 In 516211~ 53961t~ 56341~g 58741~~ 61181g~ 6365 110 14 2:1 43061i~ 45331i~ 476~1~~ 4g981r~ 02361~i 54781~~ 57241;~ 59741~i 62291~g 6488 112 12 20 43501i~ 45821i~ 48181~~ 50581~~ 53021~~ 55511~~ 58051~g 6064 4:; 63291g~ 6599 111 132 --- --- --- --- --- --- --- --- ---

0,10 17 o,438811~ O,46241~b 0,48(j51~~ 0,51091~~ O,53591~~ O,56141~f, O,58751~~ 0,6I"21~~ 0,64161:6 0,6696 116 --- --- --- --- --- --- --- --- ---08 14 44201~~ 46591g 4903 1~ 515~1~~ 54071~~ 56671~~ 59341~~ 6208d] 648911~ 6779 117 12,> 06 11 44451~j 46871~! 49341~~ 51861~~ 54441~~ 57091~~ 59811~~ 62601~~ 65481~~ 6844 119 04 8 446312~ 47061~~ ~9551g 521011i 5471 1j1 57391~ 60141g 62981!~ 65901~~ 6892 120 02 5 447312~ 4718d 4969d 522513i 5488v;~ 5757p;~ 6035111~ 6321 1,g 66161g fj92:c! 120 --- --- .--- --- --- --- --- --- --- ---.-

0,00 2 0,447712~ 0,4722d 0,497312~ O,5230J:j~ 0,5493j.'~ 0,576313~ 0,6042H~ 0,6328H~ 0,66251;~ O,69~1 120

I I I


x

~ I I t = 0,62 I 0,64 I 0,66 I 0,68 I 0,70 I 0,72 I 0,74 I 0,76 I 0,78 0,80

1,00 46 0,353346 0,3624 45 0,371 4 44 0,380243 0,388842 0,397342 0,401':641 0,413740 0,421739 0,4296 -- --- --- --- --- --- --- --- --- ---

98 0 3533 4g 3624 ~ 3713~ 3801 4g 3888 4g 3972 4~ 4055 4~ 4137 43 4217 3g 4295 46 96 1 3531 4~ 36"22 4~ 3712,J 3800 4~ 3886 4~ 39714~ 4054 4t 4136 4& 4216 3~ 4294 46 94 2

3528 4~ 3619 4~ 3709,J 3797 4A 3883 4~ 3968.J 4052 4t 4134 4& 4214 3~ 4293 47 92 2 3523 4~ 3615 4~ 3705 4~ 3793 4; 3880 4; 3965 4~ 4049 4~ 4131 4& 4211 3~ 4290 47 -- --- --- --- --- --- --- --- --- ---

0,90 3 0,3517 ~ 0,3609 4 0,3700 ~ 0,3788~ 0,3875 4; 0,3961 4~ 0,4045 4~ 0,4127 4~ 0,4208 4~ 0,4287 47 -- --- --- --- --- --- --- --- --- ---88 4 3510J 3603 4~ 3693 4~ 3782~ 3870 4g 3955 4~ 4040 4~ 41:2:2 4~ 4204 .j~ 4_83 47 86 4

3502 4~ 3595 4~ 3686 4~ 3775 4: 386J 4g ~

4035 4~ 3 4199 4~ 4279 47 3949 42 411741

84 5 3492 4~ 3585 4~ 3677 4~ 3767 ~ 3855.J 3942 4j 4027 4~ 4111 4f 4193 4e 4273 47

82 6 3481 4~ 3575 4~ 3667 4~ 3757 4~ 3846 4~ 3933 4~ 4019 4~ 4103 4; 3 4:267 47 4186 41 -- --- --- --- --- --- --- --- --- ---

0,80 7 0,3469 4~ 0,3563 4~ 0,3655 4~ o,:m64~ 0,3836 4~ 0,.3924 4~ 0,40104~ U,40~5 .~ 0,41794~ 0,4261 48 -- --- --- --- --- --- --- --- --- ---78 7 3~55 4~ 3549 4~ 3642 4~ 3734 4~ 3824 ~ 39134~ 4000 4~ 4Ull6 4~ 4170 4; 4c53 48 76 8 3439J 353~ 4~ 3628 4~ 3720 J; 3811 4~ 39014: 3939~ 4076 4~ 4161.j~ 4244 48 74 9 3422J 3518 4~ 3612 4~ 3705 4~ 3797 41 3888 4~ 3977,J 4064 4~ 4150 4~ 4235 49 72 10

3403 4~ 3499 4~ 3595 4~ 3689 4~ 3782 4~ 3873 4~ 3963~ 4051 ~ 4139j 4224 49 -- --- --- --- --- --- --- --- --- ---

0,70 11 O,3382 1g 0,3479 !~ 0,3576 4~ 0,3671 4~ 0,37"5 4~ 0,3857 4~ O,3~48 .j~ 0,-.038 4; 0,4126 4~ O,~!13 49 -- --- --- --- --- --- --- --- --- ---68 12

3359 !~ 3458 !§ 3555 !~ 3651 !~ 3746 4~ 3839J 3\132 4~ 4023 4~ 4112 4~ 4200 50

66 13 3335~ 3434 !~ 3532 !~ :3630 !~ 3726 !~ 3820 !~ 3914 4~ '*006 4 4097 4~ 4186 50

64 14 3308M 3408 ~g 3508 !§ 3606 !~ 3703 !~ 3800 l~ 3894 !~ 3988 4~ 408U 4~ 4171 51 62 15 3279 §i 3331 ~ci 3481 ig 3581 !~ 3679 !~ 3i771~ 38731~ 396SW 4062 4~ 4155 51 -- --- --- --- --- --- --- --- --- ---

0,60 16 0,3248 ~~ 0,:1.351 ~~ O,34521~ o,3553M 0,3653 ~~ 0,3752!§ 0,3350 !~ O,3947ll O,40U!~ 0,4137 52 -- --- --- --- --- --- --- --- --- ---58 17 3214 ~~ 33181~ 3421 ~~ 3523 ~~ 3625 ~ci 3725 §g 38251~ 3924 !~ 4021!~ 4117 52 56 19 3178 ~g 3283 ~g 3337 g 34911~ 35941f 36961i 3798 §ci 38981~ 3998 !§ 4096 53 54 20 3139 ~~ 3245 J~ 3351 J~ 3456 ~~ 3561 J~ 3665 J~ 3768 ~f 3871 Jt 3972 ~~ 4072 53 52 22 3097 ~1 3:104 ~~ 331q~ 3418 ~~ 3525lg 3631 g 3736 ~~ 3841 §~ 3944 ~t 4047 54 -- --- --- --- --- --- --- --- --- ---

0,50 23 0,3052 ~~ 0,3160 ~~ O,3269 U 0,3377 U 0,3486 ~~ 0,35941~ 0,3701 ~g 0,3808 ~ O,3914 1g 0,4019 54 -- --- --- ---

3333~~ --- --- --- --- ---

48 25 3003 ~~ 3113 ~~ 3223 ~~ 3443 ~§ 3553 ~g 3663 §~ 3773 ~~ 388'1 ~~ 3989 55

46 27 2950 ~~ 3061 ~~ 3173 ~~ 3285 ~~ 3397 ~~ 35\0 ~~ 31j22~ 3734 ~~ 3845 ~~ 3956 55

44 29 2894 ~~ 3006 ~~ 3119 ~~ 3233 ~~ 3347 ~~ 3462 ~~ 3577 ~~ 3691 ~} 3806 ~~ 3920 56

42 31 2833 ~~ 2946 ~~ 3061 ~~ 3177 ~~ 3293 ~~ 3410~~ 3527 ~~ 3645 ~~ 3763 ~~ 3830 56 -- --- --- --- --- --- --- --- ---

0,40 33 0,2768 ~~ 0,2882 ~~ 02998 ~~ O,3115~~ O,3234~ 0,3353 ~g O,34H~~ 0,3594 ~f 0,3715 ~t O,383.~ 57 -- --- --- ---3049~~

--- --- --- --- ---38 35

'2697 ~~ 2313 ~~ 2930 ~6 3170 ~~ 3292 ~~ 3415 ~g 3539 ~g 3663 ~~ 3788 57 36 38 26'21 ~~ 2738 ~~ 2856U 2971 ~~ 3099 ~~ 3224 ~j 335:> ~~ 3477 ~J 3605~ 3735 57 34 41 2540 ~~ 2657 ~g 2776 i:i 2898 ~2 3023 ~g 3150 ~~ 3279 ~~ 3409 ~~ 3542 ~~ 3675 57 32 44 2452 ~ 2569 ~6 2639 ~~ 2813 ~~ 293Y~ 3068 ~~ 320D ~~ 33:l5~ 3471 ~g 3609 57 -- --- --- --- --- --- --- --- ---

0,30 47 0,2358 ~~ 0,2475 ~b 0,2595 ~~ ---47

0,2843 ~~ 0,2979 ~~ 0,3114 ~~ 0,3252 t~ 0,3392 ~g 0,3535 57 0,2720 64

I I I


x

I t = 0,62 I 0,64 0,66 0,68 0,70 0,72 0,74 0,76 0,78 0,80

28 50 2257 M 2373 ~& 2493 ~J 2618 g~ 2747 ~7 2882 :g 3018 ~~ 3159 ~~ 3304 ~! 3453 56 26 54 2148 ~ 2263 ~g 2383 ~ 2507 ~~ 2637 ~~ 2772 ~ 2912 ~~ 3057 ~1 3206 j~ 3359 55 24 57 2031 ~~ 2144 ~~ 2263 ~g 2387 ~~ 2517 ~~ 2652 r? 2794 ~~ 2942 ~~ 3095 ~~ 3254 54 22 61 1906 ~~ 2016 ~~ 2133 ~~ 2255 ~~ 2384 ~~ 2520 ~~ 2663 ~~ 2813 ~~ 2970 ~~ 3134 52 -- --- --- --- --- --- --- --- --- --- ----

0,20 65 0,1772 ~~ 0,1879 ~~ 0,1992 ~g 0,2112 ~~ 0,2239 ~~ 0,2374 ~~ 0,2517 ~~ 0,2669 ~6 0,2828 i! 0,2997 50 -- --- --- --- --- --- --- --- --- ---18 69 1630 ~1 1732 ~: 1840 ~g 1956 ~~ 2080 ~~ 2213 ~l 2355 ~~ 2506 ~1 2668 ~~ 2840 48 16 73 1478 ~~ 1574 ~~ 1677 82 1788 ~6 1907 ~~ 2035 ~~ 2174 ~ 2323 ~1 2485 ~~ 2659 45 :;S 14 76 1318 ~ 1406 ~~ 1503 ~~ 1605 91 1717 ~~ 1840 ~~ 19731~g 21181~3 22771~~ 2451 41 :;6 19

12 80 1149 ~g 1228 ~j 1315 ~j 1409 ~~ 15121g~ 16261~~ 1750IA~ 18881~~ 204JI~~ 2210 37 -- --- --- --- --- --- --- --- --- ---0,10 84 0,0972 ~ 0,1041 ~~ O,11l6 ~~ O,I1991~~ O,12911~1 O,I3931g O,15061~~ O,16321~~ O,17741~g 0,1934 32 -- --- --- --- --- --- --- --- --- ---

08 1\(; 0787 ~~ 0845 ~~ 0!J081~J~ 0977IM \O551!~ 11411~g 12391~~ 13491~~ 14741;~ 1618 16 J.l

06 89 0596 ~~ 06411~~ 06901~ 07441~b 08051~~ 0873134 U951 1:! \0391~t 11421~3 1261 20 39 04 91 0400 98 04311~~ 046411~ OiiOII~l 05431~! 05911~~ 06451~~ 07071~~ 07801S~ 0866 14 IS 02 93 0201 10g 021610J 023311g 02521n 02741~~ 029811~ 03261ftl 0358Ii~ 03961~~ 0442 7 -- --- --- --- --- --- --- --- ---

0,00 94 O,OOoolO~ O,OOOOIO~ 0,0000116 O,OOOOI2~ O,OOOOI3~ O,ooool4g o,oooo16g o,oooo17g O,OOOOI9~ 0,0000 0

y

I t -= 0,62 I 0,64 0,66 0,68 0,70 0,72 0,74 0,76 0,78 0,80

I 1,00 0 0,0000 0 0,0000 tJ 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 0 0,0000 -- --- --- --- --- --- --- --- --- ---

98 69 0141 '~ 0143 71 0144 7i 01467t 014871 0149 7~ 0150 7f 0151 7~ 0152 '~ 0153 I

96 69 0282 7~ 0275 7,~ 0289 7~ 0292 7~ 02957t 0298 7~ 0301 7~ 0303 7~ 0305 7~ 0307 2

94 69 0422 7~ 0428 7~ 0434 7~ 0439 7~ 0443 7~ 0447 7~ 0451 7~ 0455 7~ 0458 76 0460 3 , I

92 70 056471 0572 7~ 0579 7~ 0585 7~ 0592 7~ 0597 7~ 0602 7~ 0607 7~ 0611 7~ 0614 4 ~- --- --- --- --- --- --- --- --- ---

0,90 iO 0,D70571 O,u715 7~ 0,0724 7~ 0,0733 7~ 0,074071 0,0747 7~ 0,0754 7~ 0,0759 7~ 0,0765 7; 0,0769 5 -- --- --- --- --- --- --- --- --- --88 70

0847 '~ 0859 7~ 0870 i~ 0880 74 0890 7~ 0898 7l 0906 7~ 0913 7~ 0919 i7 0924 6 5 ~

86 70 0990 7~ 1004 7~ 1017 7~ 1029 7~ 1040 7~ 1050 76 1059 7~ 1067 71 \074 78 1081 M " ~

84 70 1133 7~ 1149 7~ 1164 7j 1178 7g 1190 'g 1202 7~ 1212 ii 1222 '~ 1230 7~ 1238 9

82 71 1276 7~ 1295 7~ 1312 7~ 1328 7~ 1342 7~ 1355 7~ 13677i 1378 7~ 1388 7~ 1396 IU -- --- --- --- --- --- --- --- --- --- ---0,80 71 0,1421 i~ 0,1442 i~ 0,1461 7~ 0,1479 7~ 0,1495 7~ 0,1510 7~ 0,1523 7~ O,I535 7g 0,1546 7g 0,1556 11 -- --- --- --- --- --- --- --- --- ---

78 7\ 1566 ;~ 1589 it 1611 rg 1631 76 1649 7~ 1665 7~ 1680 7~ 1694 7~ 1706 8~ 1717 ):j 9 76 72

1713 i~ 1738 i~ 1762 I~ 1784 n 1804 7g 1822 7~ 1839 7~ 1854~ 1868 8~ 1880 14

i4 72 1860 r: 1888 ;g 1914 ;g 1938 Ii 1960 ;~ 1981 7~ 1999 8~ 2016 8A 2031 8~ 204·1 15

72 73 2008 it 2039 i~ 2068n 2094 i~ 2119n 2141 ~g 2161 8§ 2179 8~ 2196~ 2211 17


y

I t = 0.62 I 0,64 0,66 O,6g 0,70 0,72 0,74 0,76 0,78 0,80

0,70 73 0,2158 i~ 0,2192 I~ 0,2223 I~ 0,2252 ;g 0,2278 ~g 0,2303 ~l 0,2325 ~~ 0,2345 8~ O,2363 8J 0,23-9 18 -- --- --- --- --- --- --- --- --- ---

68 74 2309 i~ 2345g 2:180 i~ 2411 ~~ 2440 n 2467 y~ :2491 ~~ 2513 ~ri 2533 8(. 2550 20 66 74 2461 ~3 2501 i~ 2538 Ii 2.m ~~ 2604 n 2633 ~~ 26. 9 ~~ 2683 n 2705 ~~ 2724 21 64 75 2615 ~~ 2658 ;~ 2698 ~g 2735 ~~ 2770 ~~ 2801 ~~ 283" ~~ 2856 ;~ 2880 ~3 2901 23 62 7ti 2770 ~~ 2816 ~~ 2860 g,~ 2900 ~J 2938 ~~ 2972 n 3003 n 3032 ~~ 3057 ~i 3080 25 -- --- --- --- --- --- --- --- --- ---

0,60 76 O,2~26 ~~ 0,2977 ~~ 0,3024 g~ 0,3068 ~(~ 0,3108 ~~ 0,:3146 ~~ O,31hO ~~ 0,:1210 ~~ 0,3238 i~ 0.3263 27 --- --- --- --- --- --- --- --- ---

58 77 3085 ~~ 3139 ~~ 3190~~ 3238 g~ 3281 g:, 33:22 ~~ 3359 ~i 3392 i~ 3423 ~~ 3450 29 56 78 3245 g~ 3303 ~~ 3359 ~~ 3410 ~~ 3457 ~g 3501 ~g 3541 i~ 3578 i~ 3611 i~ 3640 31 54 79 3406 ~1 3470 ~g 35:29 ~~ 3585 g~ 3636 g~ 3684 ~~ 3727 ~~ 3767 i~ 3803 i~ 3835 3. 52 79 3570 ~~ 3638 ~~ 3702 ~~ 3762 ~~ 3818 ~~ 3870 ~ 3917 ~~ 3;60 k~ 399J ~~ 4033 36

-- --- --- --- --- --- --- --- --- ---0,50 80 0,3735 ~~ 0,3809 ~~ 0,38',8 ~~ 0,3943 g~ 0,4003 ~~ 0,40,',9 ~~ o,ml ~l 0,4158 ~i r,42Col~~ 0,4238

3~ --- .--- --- --- --- --- --- --- ---

48 81 3902 ~6 3931 ~~ 4056 8~ 4126 ~~ 4192 ~~ 4:253 ;~ 43('8 ~~ 43601~j 441 61~~ 4448 42 30

46 81 4071 ~~ 4156 ~3 4237 ~g 4313 ~~ 4384 1~ 4450 ~~ 45111~~ 45671g: 46171<6 4663 45 23

44 82 4241 ~~ 4333 ~~ 4421 ~i 4503 ~~ 4580 ~~ 46511g~ 47181~t 477!Jl~ 48341~~ 4885 48 42 82 4413 gg 4512W 4607 ~~ 4595 ~~ 47791~ 485~1~~ 4930;~~ 49971~~ 50571~~ 5112

52 --- --- --- --- --- --- --- --- ---0,40 83 O,4S!l7 ~~ 1',4694 ~l O,-l7!J5 ~~ 0,4892 ~~ O,.j9831~~ O,50581~g O,'j1471~~ o,52201g O,5~871j,; U,5348

56 --- --- --- --- --- --- --- --- ---38 83 4761 ~Z 4877 ~~ .j987 96 50911~g 51901~~ 5283108 53701~~ 54501j~ 5524118 5591

fiO -'2 43 3~

;36 .os;; ~9Ji ~1 SOtil ~~ 5181 ~~ 52941~j 54021~~ ,j50:11~~ 55981l! 56861~t 57ti81~: 5842 65 ."

34 83 5113 ~~ 5247 ~~ 5376 ~~ 5500103 56171~~ 57281g 5832117 59301~~ 60201~~ 6102 6) 59 32 83 5290 ~g 5 :34 i6 fi574 ~7 570Rl~1 58361~i 595g11~ E0731~~ 6I801~~ 62~ ott 6371

74 --- --- --- --- --- --- --- --- ---0,30 83 0,5465 ~~ 0,5622 ~ O,:m3 ~~ O,59191~g O,60581M O,til921g O,63191~~ O,64381~~ 0,1 5481~t O,b651

80 --- --- --- --- --- --- --- ---28 82 5542 ~~ 5809 ~~ 59721~ 61 j) 1~~ 62841~~ 643\1~3 657t11~~ 67021~~ 68261~j 6941

86 26 81 5815 ~~ 5995 ~~ 61721~g 63Hl~ 651i~t 66731~~ 682812 , 69751~~ 71131~: 7242

\·2 n 24 79 5986 15 6180 ~~ 6371 ~~ 65581~i 674!1~~ 69181~~ 70901~1 71531~~ 74091~~ 7555

98 97 22 77 83 6361 16:\ 65671~g 6770;~ 69~010~ 71651~~ 735513:1 75381:~ 77I 31~~ 7879

105 6153J(J~ ~2

--- --- --- --- --- --- --- --- ---0,20 74 O,63141~: O,6S371~~ O,67591~ O,6979:~~ O,71Y8:6~ 1"4 O,76!4:J~ O,78281~~ O,tlO261~~ 0,8214

111 O,74131U5 --- --- --- --- --- --- --- --- ---

18 71 64691;~ 67071n 69451i~ 7184l?~ 7422m 7658l~~ 7892134 81 21 ltg 8344:~~ 8559 118 115

16 66 66161~~ 68681~~ 89 73811~~ 7640:~ri 7899g~ 8158g~ 8414g~ 86'i6:~j 8912 125 712412"

14 61 675113~ 70191~~ 72921~~ 756g1~~ 7848:~~ 813lm 8416m 8703::~ 89S8:~) 9270 132

12 55 687.'il~i 71581~': 74461~~ 774 l~i 80431~l 8351 l1~ 8664l~i 8982:j~ 93031~~ 9625 138 -- --- --- --- --- --- --- --- --- ---

0,10 49 O,69841~~ O,72801~~ O,75851~~ -8 lH O,7898 ltil O,82201~~ o,855:!I~l 0,8893m o,92441~~ 1'0 0,9603183 0,9970

--- --- --- --- --- --- --- --- ---08 41 46 73851~~ 77031~~ 80331~i 83741~~ 87291~~ 9097 102 9480:~~ !J87~g~ *0:291

141 7077 154 191

06 ~3 7151 :i7 74691~~ 77991i~ 81431~~ 85011~; 88751~~ 92682~~ 9680~~ *0114~2S, 0572 153 159

O-l 24 7206,~~ 75311~6 35 8!241:A 85951~ 89852~~ 93972~~ 98332~~ 02972~~ 0794 157 7870177

02 15 72391i~ 75691~~ 7913j~3 82741~ 86532~g 90532i~ 94772;~ 99292!~ 04142~~ 0937 159 --- --- --- --- --- --- --- --- ---

0,00 5 O,725016~ O,7582d O,7928j8~ O,829119~ O,86732~g O,90762U O,95052~~ O,99622l~ 1,(;4542~~ 1,0966 159

110 E1.ECTRICAL PROSPECTING WITH THE TRANSIENT MAGNETIC FIELD METHOD

x

~ I t - 0,82 I 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,00

1,00 38 0,437238 0,444837 0,452236 0,459436 0,466535 0,473534 0,480333 0,487033 0,493632 0,5000 -- --- --- -- --- --- --- --- --- ---

98 0 4372~ 4448 3~ 4521 ~ 4594~ 4665~ 4735~ 4803~ 4870 3g 4936 3g 5000 38 96 1 4371 ~ 4447 ;: 4521 ~ 4593~ 4665~ 4734~ 48033~ 4870~ 4936~ 5000 39 94 1 4370 al 4445 3~ 4520~ 459'.l3~ 4664~ 4734~ 4802~ 4870~ 4935~ 5000 39

92 1 4368J 4444 3~ 4518J 4591 J 4663 3~ 4733~ 4802~ 4869~ 4935 3g 5000 39 -- --- --- --- --- --- --- --- --- ---0,90 2 0,4365 3~ 0,4441 3~ 0,4516 3~ 0,4589 at; 0,4661 ~ 0,4732 3~ O,48013~ 0,4868 ~ O,4935~ 0,5000 39 -- --- --- --- --- --- --- --- --- ---

88 2 4361 ~ 4438~ 4513 3~ 4587 J 4659~ 4730~ 4800 ai 4868~ 4935~ 5000 39

86 2 4357 3~ 4434~ 4510J 4584 3~ 4657 ~ 4728 3~ 4798 ai 4867 ~ 4934~ 5000 39

84 3 , 4353 3~ 4430~ 4506J 45813~ 4654~ 4726 3~ 4797 ~ 4866ai 4934~ 5000 40

82 3 4347 J 4425~ 4502~ 4578 3~ 4652 sA 4724~ 4795J 4865ai 4933~ 5000 40 -- --- --- --- --- --- --- --- --- ---0,80 3 0,4341 ~ 0,4420 :J 0,4498 3~ 0,4574 3~ 0,4648 3~ 0,4721 sA 0,4793 3~ O,4863~ O,4932~ 0,5000 4U -- --- --- --- --- --- --- --- --- ---

78 4 4334~ 4414 :J 4492 3~ 4569~ 4644 s~ 4718~ 4791 ~ 4862~ 4932~ 5000 41 76 4 4326~ 4407 4~ 44863~ 4564~ 46403~ 4715 3~ 4788~ 4860~ 4931 ~ 5000 41 74 5 4318 41 4399~ 4480 :J 4558 :J 4635~ 4711 J 4786J 4858J 4930 sg 5000 42 72 5 4J084~ 4391 41 4472 4~ 4552 :J 4630~ 4707 ~ 4783 3~ 4856J 4929J 5000 42 -- --- --- --- --- --- --- --- --- ---

0,70 6 O,4298 4g O,43824~ O,4464~ O,4545 4g O,4625J O,4703~ 0,4779 3~ O,48543~ O,49283~ 0,5000 43 -- --- -'-- --- --- --- --- --- --- ---68 r. ,t!87 4~ 4372 l~ 4455 It 4538~ 4618~ 4698 3~ 4775 3~ 4852 :I~ 4927 :l~ 5000 1:\

66 7 4274~ 4361 4g 4446 4~ 4529 41 4612~ 4692~ 477ls~ 4849~ 4925 3~ 5000 44

64 8 42614~ 4349~ 4435J 4520 4~ 4604 41 4686 4g 4767 ~ 48463~ 4924J 5000 45 62 8 4246,J; 43354~ 4424 4g 4511 4g 4596J 4680 4~ 4762~ 4843~ 4922 3~ 5000 46 -- --- --- --- --- --- --- ---

0,60 9 O,4229J 0.43214~ 0,4411 ~ O,4500,J O,46874~ 0,4673J 0,4757 4~ 0,4839 4~ 0,4921 ~ 0,6000 46 -- --- --- --- --- --- --- --- --- ---

58 10 4212 4~ 4305 4~ 4397 4~ 44884~ 4577 4~ 4665 4~ 4751 4~ 4836 4~ 4919 4~ .5000 47 56 11 4193~ 42884~ 4382J 44754~ 4567 J 4656 4: 4745 4~ 4831 4~ 4917 4~ 5000 49

54 12 4172~ 4269J 4366J 4461 4i 4555 4~ 4647 4~ 4738 4~ 4827 J 4914 4~ 5000 50

52 12 4149 ~~ 4249 Ml ' 9 4446 4g 4642 4~ 4637 4~ 4730 J 48224~ 4912J 5000 51 4348 49 -- --- --- --- --- --- --- --- --- ---

0,50 14 O,4124M O,4227M 0,4329 Ml 0,4429 J 0,4528 4~ O,4626J O,47224~ 0,4816 4~ 0,4909 4~ 0,5000 52 -- --- --- --- --- --- --- --- --- ---48 15 4096 ~~ 4202~~ 4307 ~~ 44115i 4513J 4614~ 4713 4~ 4810 4~ 4906 4~ 5000 54 46 17 4066 ~~ 4175~ 4284~ 4391 ~ 4496~ 4600 5i 4703~ 4804~ 4903~ 5000 55 44 18 4033 ~~ 4146~ 4258~~ 4368~~ 4478~ 4585 5~ 4692 5g 4796 51 4899~ 5000 57

42 20 3997 ~g 4114 ~g 4229M 4344 ~~ 4457~ 4569~ 4680 5~ 4788 5~ 4895 5~ 5000 59 -- --- --- -- --- --- --- --- --- ---0,40 22 O,3957~ O.4078~ 0,4198 ~~ 0,4317 ~~ O,4434M O,4551 5g 0,4666 5~ 0,4779 5~ 0,4891 ~ 0,5000 so -- --- --- --- --- --- --- --- --- ---

38 24 3913 ~i 4038~ 4163~ 4286 ~~ 4409A~ 4531 ~g 4651 5~ 4769 5~ 4886 5~ 5000 63

36 27 3864 ~~ 3994~ 4123~ 4253ll 4381 ~! 4508M 4634 6~ 4758 6~ 4880 G~ 5000 65 34 30 3810 ~~ 3945~ 4080 ra 4215 A~ 4349~ 4483 ~~ 4615 Ag 6 4874 6~ 5000 67 4745 64

32 33 3749~~ 3890~T 4031 ~~ 4172~1 4313 ~~ 4454 ~g 4593 ~~ 4731 6~ 4867 6* 5000 70 -- --- --- --- --- --- --- --- ---0,30 'ir1 0,3681 ~ O,3827~! O,3975~ O,4124~ 0,4273 ~~ 0,4421 ~~ O,4569~~ 0,4714 7~ 0,4858 71 0,5000 73


x

I t = 0,82 I 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 I,on

28 41 3604 ~~ 3757 ~~ 3913 ~~ 4069 ~~ 4226 ~~ 4384 ~~ 4540 ~~ 4696 7~ 4849 7~ 5000 76 26 47 3517 ~t 3678 ~~ 3841 ~~ 4006 32 4173 ~l 4341 ~ 4508 A:; 4674 ~~ 4837 8~ 5000 79 83 24 53 3417 ~~ 358(; ~ 3758 ~ 3933 ~g 41 \1 ~1) 4290 25 4469 19 4648 ~~ 4825 8~ 5000 82 90 9J 22 60 3304 ~ 3480 ~~ 366:! ~~ 3848 4~ 4038 ~ 4231 ~ 4425 ~~ 4618 ~~ 48\0 9~ 5000 85 9,

-- --- --- --- --- --- --- --- --- ---0,20 69 O,3173~~ 0,3358 ~~ 0,3550 ~~ O,37481g~ O,39521~l 0,4!601~~ O,437115~ O,~532Il~ O,47921O~ 0,5000 88 -- --- --- --- --- --- --- --- --- ---

18 79 3022 ~~ 32151~i 34171~~ 3629 60 38491~~ 40751t~ 43051~~ 45371i~ 47"01~~ 5000 91 110

16 9J 28461~ 30461~ 32591I~ 34851i~ 37:221~~ 39691~~ 42:231~~ 44821~ 47421M 5000 93 14 104 264°12~ 28461?8 30691~g 33091~~ 35651~~ 38371~~ 41:2°1~~ 44111~~ 47061l~ 5000 95 12 120

239816! 2607m 28J7m 3090:~~ 33671~~ 8-

398tl~J 43171~i 46581~t 5000 94 3666159 -- --- --- --- --- --- --- --- --- ---

0,10 138 O,2115:~~ O,2320m 0,2552:~i 0,2815:~~ 0,:3\1 tl~~ O,3-HO:~~ 0,380019" O,418723~ 0,45" 12~l 0,.5000 90 -- --- --- --- --- --- --- --- --- ---

08 158 17841g~ 1976)g 220tl~g 2-464:~'; 27i2:~J 3130!3~ 353ll~~~ 39972~~ 4491 ~o 5uOO 83 2.,4 06 178 14011~ I 569igt mom 2016i~t 2318m 2690~~~ 3148~~~ 3697~;~ 43273~~ 5000 70 04 1<7 09692~~ 10952~~ 12512~~ 1450i~~ 1708~~J 2053~19 2522~J? 3165~~~ 4009l~~ 5000 52 235 02 212 04962~~ 05652~~ 06513~ C7653i~ 0921m I I 46ii1 H94;~~ 2087~~~ 3181~lci 5000 27 -- --- --- --- --- --- --- --- --- ---

0,00 221 0,0000~4g 0,0000285 O,ooo032g 0,0(00386 O,OO,,04sg O,ooo057g O,OOOOHZ O,oooolOci4 0,00JOI5J(' -0

y

I t = 0,82 I 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,00

1,00 0 O,oroo 0 0,0000 0 O,ooao 0 0,0000 0 0,0000 0 0,0000 0 0,1)000 0 0,0000 0 0,0000 0 0,0000 -- --- --- --- --- --- --- --- --- --- ---

98 77 1054 7~ 0155 7~ 0155 7g 0156 7g 0\.56 ig 0157 7g 0157 7g 0157 7~ O1577~ 0157 79 0 96 77 03£187f 0310 7; 0311 7? 0312 7~ 0313 7g 0313 7~ 031475 031475 0314 ;5 0314 79 I 94 77 0463 7; 0465 7? 0467 7~ 0468 i~ 0-:69 7~ 0470 ;~ 0471 7Z 0472 7~ 047275 0472 79 I 92 77 0618 7; 0620 7~ 0623 7~ 0625 7~ 0626 ii 062875 062975 062975 0630 ;~ 0630 79 2 -- --- --- --- --- --- --- --- --- --- ---

0,90 77 0,0773 7~ 0,0777 7~ 0,0780 7~ O,07827i 0,07847i 0,07867i 0,078775 0,078875 0,0/8875 0,0789 79 2 -- --- --- --- --- --- --- --- --- --- ---88 78 0929 7g 0934 7~ 0937 7~ 0940 7i 0943 7i 0945 7i 0946 8~ 0947 Be 0948 8g 0948 80 2

86 78 1086 i~ 1091 7~ 1096 7~ 1039 sg I \02 8~ 11058~ 1106 H~ 1108 8~ 110888 1\09 80 3 84 78 124t 7~ 1250 7g 1255 8~ 1259 sg 1263 8~ 1265 8~ 126B 8: 1:269 8t 127086 1270 al 3 82 79 1404 sg 1410 8g 1416 sg 1421 8J 1425 8~ 1428 81 1430 8~ 1432 s~ 1433 Bb 1433 81 4 --- --- --- --- --- --- --- --- --- ---

0,80 80 0,1564 f~ O,15728~ O,1578 8! O,1.18J 8~ 0,151i8 8~ 0,1.':91 8i O,1594 8f 0,1596 8~ 0,153786 O,159i 82 4 --- --- --- --- --- --- --- --- --- ---

78 81 1726 8~ 1735 8~ 17-!2 8~ m88~ 1753 8~ 1757~ 17U0 8f I i62 8f 1763 s~ li63 83 5 76 81 1890 8~ 1899 8~ 1907 Hg 1914~ 19198~ 1924 8~ 1927 8t 1929 Sf 1931 8ci 1931 84 5 74 82 2056 8~ 20668.l 2075 8: 2082 8~ 2088 8~ 2093~ 2097 8~ 2099~ 2101 8g 2101 85 6 72 83 22238~ 2235 8: 2244 8~ 2253 8~ 2259 sg 2265 8~ 2269 8~ 2271 ~ 227388 :i274 85 6


y

': I I t = 0,82 I 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,00

0,70 84 0,2394 8~ 0,2406 8~ 0,2417 s: 0,2425 s: 0,2433 8~ 0,2439 8~ 0,2443 8~ O,2446 8I 0,2448 8~ 0,2449 8• 7 -- --- --- --- --- --- --- ---68 86 2566 ~ 2579 8~ 2591 8~ 2601 ~ 2609S: 2615 s: 2620 ~ 2624 8i 2626 8g 2626 89 8 66 87 2741 ~ 2756 ~ 2769 8g 2779 8: 2788 ~ 2795 ~ 2801 9g 2804 ~ 2806 9g 2807 90 9 64 88 2919 ~ 2935 ~ 2949 "2 2961 9~ 2971 9l 2978 ~ 2984 ~ 2988 9~ 2991 ~ 2991 92 9 62 90 3100 9~ 3118 9~ 3133 9~ 3146 9~ 3157 9~ 3165 ~ 3172 ~ 317691 3179 9~ 3179 94 10 - --- --- ~-- --- --- --- --- --- --- ---

0,60 92 0,3285 i~ 0,3304 ~ 0,3321 9~ 0,3335 9~ O,3347 9g 0,3356 9~ 0,3363 ~ 0,3368 9~ 0,3371 sg 0,3371 96 11 -- --- --- --- --- --- --- --- --- --- ---

58 93 3414 ~ 3495 9~ 3513 9~ 3528 9~ 3541 9~ 3551 9: 3558 9g 3564 9~ 3567 9~ 3568 98 12 56 95 3666 ~ 3689 i~ 3709 ~ 3726 ~ 3740 ~ 31511a<.l 375910g 376510~ 3768101 3769101 13 54 97 3864 ig 38891W 391010~ 39281°l 39431~ 3955102 396410~ 39711~ 39751~ 3976103 14 5 52 100 40651~l 40931~~ 41161~ 413610: 41531~ 41661~ 41761~ 418310~ 41871~ 4188106 16

-- --- --- --- --- --- --- --- ---0,50 102 O,42731~~ O,43021~g O,43281~ O,435010~ O,43681~ O,43831~ 0,439310: O,44011og O,44051~ 0,4407109

17 -- --- --- --- --- --- --- --- --- --- ---48 105 44851~ 45181~ 45461~ 457011g 459011l 460611~ 461811~ 462611~ 463111~ 4633113 19 46 108 47041~~ 4140l U 4771 IIi 479il1 481911~ 483611~ 485011g 48591lg 486411I 4866117 20

44 III 49291M 496911~ 500311~ 50321g 50561:8 50751l~ 509012g 510012l 5106121 5108121 22

42 114 51621~~ 520511~ 52431~g 52751~ 53021~~ 532312: 533912~ 535112g 535712~ 5359126 25 -- --- --- --- --- --- --- --- --- --- ---0,40 118 O,54021~~ O,54501~~ O,54921~: O,55271~~ O,55571~~ O,558112~ O,55991~ O,56111~ 0,5618131 0,5621131 27 -- --- --- --- --- --- --- --- --- --- ---

38 122 56511~j 57041~~ 57501~g 57901~~ 58231~g 58491~ 58691~ 588313~ 589113~ 5894137 30 36 126 5908129 59671~~ 60191~~ 60631~~ 61001~ 613011~ 6152t4~ 616814: 6117t4~ 6180 141

3~ 30 34 130 61761~ 62421;~ 63001~ 63491~~ 639111~ 642411~ 644914~ 64671'ig 64'lSt5g 6481150 37 32 135 64541~~ 652SIg 65931~ 66491ra 66961l~ 67331t~ 676:l1~g 67S215~ 679415~ 6798 159

41 -- --- --- --- --- --- --- --- --- --- ---0,30 110 O,67431:~ O,68271~~ O,69001g~ O,69641~~ O,70161~~ O,70601~~ O,70931~~ O,711616f O,113016~ 0,1134 1118

46 -- --- ---' --- --- --- --- --- --- --- ---

28 145 70451~~ 71401~~ 72231~~ 1296166 13571~g 14C61r~ 74441;3 747017~ 748617g 7491 179 52 30 26 lSI 13601~~ 74681~: 15631~g 7647175 17171gg 77741~~ 78181n 784918~ 786719~ 1873 191 59 35 24 156 16891~~ 78121;~ 79221~ 80191~~ 81011~ 81671~~ 82192~r 82552n 827720~ 8284 20S

67 22 162 80331n 81751~ 83021g~ 84151:~ 85112~~ 85902M 86512~~ 86942~8 872022! 8728 222

77 -- --- --- --- --- --- --- --- --- --- ---0,20 168 O,83911~~ O,85551~g O,87052~~ O,883821} O,89522~} O,90462~~ O,91202~~ O,9173~~ O,920424~ 0,9214 243

89 -- --- --- --- --- --- --- --- --- --- ---18 173 87641~~ 8956~ 91322~: 92902~~ 94282~~ 95432!~ 96332g~ 96972~~ 973626g 9148 267 1112 16 177 9150l~~ 9375ng 95852~~ 97762~~ 99442~~ *00862~~ *01992~ *02802~1 ·03282sg *0344 298 119 14 179 9546l~ 9812m *0064~~~ *0297i~~ *05062~g 0685~~ 08293~~ 09343~~ 09973ici 1018 3.~7 138 12 178 9946i~~ *0262~~~ 0567m 0855m 11 19m 13503~~ 15403~g 168OS~~ 1766~ 1194 366 160 -- --- --- --- --- --- --- --- ---

0,10 172 1,0342l~~ I,071 6m 1,1087i:i8 1,144'ji~g 1,1186~~~ 1,2093n~ 1,23514~ 1.25484~f 1,26704~5 1,2710 458 186 -- --- --- --- --- --- --- --- --- ---08 161 0719~~g 1159~~~ 1609~gl 2061~I 359 292t;~! 3289~~ 35805~g 31665~g 3830 560 214 2504,09 06 141 1056~ 1567~6~ 2106~~i 2669~~6 3249~~ 3829~j: 4377~~ 4842~~~ 51596~ 5271 721 242 04 III 1328~~ 1905~~~ 2532~~ 3215~~~ 3960~ 4764:~~ 5607~~ 642q~~ 7055i~g 7301101~ 267 02 71 15063~~ 213111~ 2827~~ 3611~~~ 4509m 5558~~~ 6807~ 8296~~ 99021ill *07681734 285 -- --- --- --- --- --- --- --- --- --- ---

0,00 25 1,1568~ 1,2211:: 1,29334~~ 1,3758~~ 1,4722~l 1,5890m 1,7380~~ 1,94591~: 2,29761~7 00 ~

291

THREE-LAYER AND FOUR~LAYER WAVE CURVES

APPENDIX 2. TYPICAL THREE-LAYER AND FOUR-LAYER WAVE CURVES

FOR TRANSIENT FIELDS

First-Thick-

Second- Third-layer

Set No. layer re- layer re-Resis. Thickn.

sistivity ness

sistivity

1 1 1 2/3 co co

2 1 1 3/7 co co

3 1 1 1/4 co co

4 1 1 1/9 co co

5 1 1 3/2 co co

6 1 1 7/3 co co

7 1 1 4 co co

8 1 1 9 00 co

9 1 1 19 co co

10 1 1 7/12 1/4 1

11 1 1 co 1/9 5

12 1 1 39/9 1/9 1

Note: On the sets of curves, the index with each curve is the relative thickness of the second layer, h2/h1 •

For the four-layer cases (10-12), the fourth layer has an infinite resistivity.

113


e

0®®80000 N2

e

THREE -LAYER AND FOUR-LAYER WAVE CURVES 115

Pt 1 CD ®® 0) 0) 0 0 0 N3 f1

e

11

---___ 8

11 ---.. h.

.'It -_. hI


e

e __ 0 11

h~

THREE-LAYER AND FOUR-LAYER WAVE CURVES 117

N7

N8


M9

Nl0

THREE -LAYER AND FOUR-LAYER WAVE CURVES 119

Ntl

cr -----~ h,

N12


Date Point

APPENDIX 3

Nomogram for determining the correction for finite length of source dipole.

APPENDIX 4. FORMS FOR A FIELD LOG

Listing of Array Geometric Factors

I Equipment r, m AB, e Coil area, number 1\ m

m2 serial no. of turns

Operator:

Form 1

Correction for depar-ture from Comments true dipole condition

FORMS FOR A FIELD LOG 121

Form 2 Record for Generator Assembly No. __

Date S~+illog<'Ph T~e~ Pulse No. Rshunt (on !source Iaverage Com-AB, monitoring !(without (without ments

no. no. recordi duration AB m meter) ohms doubling) doubling)

Operator:

Form 3

Record for Recording Unit No. __

Date Sounding Oscillo- Time of Pulse I Area of re- MN,m Sensi- L1 V gr' Filter Com-

graph no. recording duratfon ceiving coil, tivity cutoff ments no. IlV

q, m2

Operator:


FOnD 4 Record of Standardization

Amplifier assembly no.,-~ ___ _

OSCillograph no.-----

AVgr• Amplitude. mm True ampli- True ampli-

Record Sensiti-AVgr • tude AVgr tude AVgr Filter amplifier oscillo- For cali- For stan- Com-For cali- For stan- ( calibration (calibration no. vity cutoff JlV graph. brated ments

brated dardized dardized of am plified. of oscillo-mV amplifier system oscillo- JlV graph), mV

graph system

I l I I I Operator:

Recwder:

Form 5 List of Records

Index Index Inventory Mag- Elec- number number

Date netic tric Numberof of records of records Comments sound- sound- records for re- for gen-

Rejected I Accepted lng no. ing no. corder erator

Designation I

A!

Bf

AB

0

Q,

r,

0

Q%

r,

Interpreter:

Computer:

FORMS FOR A FIELD LOG

Form 6

Record of Location

Petrovka AB = 6km. A -at bridge across Bystritsa River 2 km SW of the village of Petrovka. Laid out from the river bank toward the village of Sidorovka. reaching a bend in the road. From the bend in the road. the line was taken cross country (two tie marks were established on this segment). It then followed road to a ford across the river. B-ford at river 2 km north of the village of Ivanovka.

Form 7

Computations from Tables

Operation I X Y I Comments

5913460 9595000

2 5908730 959530

1-2 4730 230

4735 I

1-2 -2- 2365 115

5911 095 9595115

2 .;915140 9607550

4045 12435

13076

5911095 9595115

2 5907450 9608845

1-2 3645 13730

14205 I

123


Sec. Y2ti

I I 2 I 0,2 1,12 10,8 11,0 0,3 1,37 12,0 12,0 0,5 1,77 14,0 13,2 0,7 2,10 17,2 16,2 1,0 2,51 21,4 20,5 1,5 3,08 23,8 :23,6 2 3,55 22,5 21,8 3 4,34 In,5 16,9 4 5,00 11,8 11,3 5 5,60 8,5 tj,4 6 6,15 5,2 fi,2 7 6,.~6 4,0 5,3 8 7,10 2,6 4,8 9 7,50 1,0 3,6

10 7,92 0,7 2,2

Interpreter:

Computer:

Example of the Reduction of a record of Transient Magnetic Field for AB = 8038 m, q = 826,600 m2

Amplitude of deflection

Pulse number Aav Agr .6.Vgr , .6. V ,

fa IlV Il V

3 \

4 I 5 I 6 I 7 I 8 I 9 I 10

I I 10,8 )(,,5 11,2 10,5 9,6 10,7 10,2 10,8 10,6 35,3 40 - 59,6 11,3 12,0 12,0 11,2 9,5 11,8 11,2 11,8 11,5 - - - -12,6 14,5 14,0 14,0 11,2 13,5 13,0 13,2 13,3 - - - -15,8 17,0 16,0 16,5 14,0 16,0 15,0 15,9 16,0 19.1 21,0 20,3 20,0 17,2 19,6 ·18,5 19,0 19,7 22,7 24,5 24,4 24,0 22,0 22,0 22,8 21,5 23,1 21,0 24,6 23,2 22,2 22,0 2D,6 23,2 20;2 22,2 14,2 21,2 16,2 16,0 16,0 15,0 17," 11,2 16,4 9,4 16,0 10,2 11,0 10,2 10,4 11,2 9,0 11,0 6,!! 12,4 6,8 7,5 6,9 6,8 7,5 5,0 7,7 5,U 8,4 4,0 5,0 3,8 4,5 5,0 2,8 5,0 4,0 6,0 2,8 3,11 :.1,2 3,2 4,0 1,3 3,6 2,6 5,2 1,4 2,2 1,4 2,4 3,U 0,5 2,6 1,2 4,6 0,8 1,8 0,8 2,U 2,0 0 1,8 0 3,0 0 1,5 0,5 1,2 1,4 0 1,05

Form 8

Pt Comments

16,25 17,6

15-sec 20,4 24,6 pulses 30,2 35,4 4Vgr·Aa 34,0 4V= 25,2 Agr 16,9 11,7 7,66 5,5 4,0 'l,73 1,61

v

FUNDAMENTALS OF ELECTROMAGNETIC SOUNDING*

1. 1. Vanyan

PART I. GEOLOGICAL BASIS FOR ELECTROMAGNETIC SOUNDING

The application of electromagnetic sounding methods is based on differences in the electrical properties of various rocks, such as the electrical resistivity, p, and the dielectric constant, e.

The resistivity of a rock (usually in the range from a few Q-m to several hundred Q-m) is determined by the porosity of the rock and the resistivity, Po,of the water contained in that pore space:

where P v , the formation factor, is a parameter which depends on porosity. An empirical relationship between P v and porosity, kp,is given in Fig. 1.

The resistivity of the water in the pore structure is determined by the concentration, C, of salts in it, as shown in Fig. 2.

Consideration of the relationship between resistivity, porosity, and water salinity indicates that the resistivity of a particular rock may vary over a wide range. For example, a limestone with 10% porosity saturated with water carrying 50 gm/liter of salt in solution has a resistivity of about 15 Q-m, according to the curves in Figs. 1 and 2. If the salinity of the water is increased to 200 gm/liter, the resistivity of the rock is decreased to 4 Q-m.

However, some rocks have very high resistivities, as high as 103_104 Q-m, and though rarely, even higher. These rocks are evaporites or crystalline igneous rocks which have virtually no porosity. Beds of halite, gypsum, and anhydrite which occur in sequences of terrestrial sedimentary rocks commonly act as insulating screening layers, but insofar as electromagnetic sounding methods are concerned, the surface of the crystalline basement acts as the electrical basement in sedimentary basins.

A sedimentary column frequently is made up of a great number of thin layers, each with its own resistivity. If such a collection of fine layers, as apparent on an electric log, is grouped together and considered as a single homogeneous layer, it must be considered to be anisotropic; that is, the resistivity to current flowing along the bedding planes, Pt, is different than the resisti vity to current flowing across the bedding planes, P n .

* Published originally by Nedra (1965).

125

126 FUNDAMENTALS OF ELECTROMAGNETIC SOUNDING

PII

200

? 100

60

JO 20

10

6

" J

Z

J IJ 7 910 20 "0 kp' r.

Fig. 1. Relationship between P v and porosity for various types of rocks (from V. N. Dakhnov). 1) Sands. 2) Sandstones. 3) Limestones.

Fig. 2. Variation of Po for various concentrations of NaCl (from V. N. Dakhnov).

For a sequence of horizontal layers, the resistivity Pt is termed the horizontal or longitudinal resistivity, while Pn is termed the vertical or transverse resistivity. The coefficie,.t of ani30tropy, A = (p nl Pt)V2, defines the difference between the two values. Also, the homogeneous composite bed is assigned the geometric average resistivity value,

Qm = -V QnQ! = QtA = Q.,JA.

Values for the coefficient of anisotropy for some types of sedimentary rocks are listed in Table 1 (Dakhnov [7]).

As may be seen from the values listed in Table 1, the coefficient of anisotropy for sedimentary rocks is usually no more than 1.5 -2.0.

The increase in temperature with depth in the earth causes the rock resistivity to decrease. This is caused by a decrease in the resistivity of the pore water, which is given by the formula [7]

Q = Q18 [1 +0.025(t _180 C))-l,

where P 18 is the resistivity of the pore water at 18°C.

This equation indicates that the 100° to 200° temperature rise which may take place through a sedimentary column can lead to a 3- or 4-fold decrease in resistivity.

It has also been shown that the resistivity of crystalline rocks decreases with increasing temperature. Some experiments have shown that at 1000°C, rocks which are normally nonconductors will have resistivities as low as a few tens of ohm-meters.

According to Sharkov [8] the theoretical relationship between resistivity and temperature for olivine is given by:

l~OOIl (1 0 10-6 ) ') 10-6 10l7C +~. p Q=-' . ,

where TO is the absolute temperature and p is the pressure in atm.

This last equation indicates that the resistivity in the earth's crust decreases markedly with increasing temperature. The increase in pressure only slightly modifies this decrease. Table 2 lists values of resistivity for olivine at high pressures and temperatures, appropriate to the given depths within the earth.

A typical geoelectric section for the upper part of the earth's crust includes sedimentary rocks which may be several kilometers thick, resting on a crystalline basement with very high resistivity; however, at greater depths, of the order of 100 to 200 km, the resistivity decreases to values comparable with those in the sedimentary column.

The ratio of the dielectric constant in rocks to the dielectric constant of free space which is (1/3611") • 10-9 ~ 10-11 in the MKS system of units, varies from about 4-5 to about 12-14. The presence of moisture in sedimentary rocks increases the dielectric constant, E.. Layering in

GEOLOGICAL BASIS FOR ELECTROMAGNETIC SOUNDING 127

TABLE 1. Values for the Coefficient of Anisotropy TABLE 2

Rock type A Il. I TOK' I031 atrri:l04 1

Q. km Q-m

Poorly bedded shales 1.02-1.05 50 1.1 2 4.105

Shales with sand lenses 1.05-1.15 100 1.5 4 1000

Sandstone layers 1.10-1.59 130 1.i 5 100 200 1.9 ti 20

Endurated shales 1.10-1.59 :!50 2.1 8 8

Shaly slate 1.41-2.25

rocks must necessarily lead to a difference in the values of dielectric constant measured perpendicular to and parallel to the layering.

The magnetic permeability, IJ., of rocks usually does not differ Significantly from the magnetic permeability of free space, lJ.o' We will assume that IJ. = lJ.o' In the MKS system, lJ.o = 47T X 10-7 •

Field Sources and Models of the Geoelectric Section

The purpose of electromagnetic sounding is to determine the electrical character and thickness of beds in the geoelectric section. Two conditions must be satisfied before such a determination can be made. First, the layers in the geoelectric section must differ in electrical resistivity or dielectric constant (we will find that variation in resistivity is the basic requirement). Secondly, there must be developed an electromagnetic field in the earth with a magnitude which depends on the properties of the geoelectric section.

As an idealized model of the geoelectric section, we will use a medium consisting of a series of horizontal layers separated by horizontal planes (Fig. 3). The layer designated with the index p has a longitudinal resistivity Ptp' a transverse resistivity Pnp' a thickness hp, and a depth Hp. The layers are numbered consecutively from the surface down, with the upper halfspace having the index O. The last layer is infinitely thick, and has the index N.

The electromagnetic field incident on the layers will in general be time -varying. For a rate of variation approaching zero, we would have the special case of a static electromagnetic field.

Excitation may be provided using a generator or using, a natural source. In the first case, the immediate source of excitation commonly is a wire grounded at both ends or a horizontal ungrounded loop. The diameter of the loop must be small in comparison with the spacing (the distance at which the field is detected), so that it may be treated as a vertical magnetic dipole with moment M, equal to the product of the current, J, and the area of the loop, Q. With a grounded wire, two cases must be considered: 1) If the length of the wire is small in comparison with the spacing, it may be treated as an electric dipole with moment I, equal to the product of current and the wire length, deSignated as AB; 2) if the grounded wire is long, then we have to consider the field as being the superposition of contributions from short elements of the wire.

Natural fields have their origin from time-varying currents in the ionosphere, at an altitude of 80-100 km. Over limited areas, these currents may be considered to be linear with infinite extent, so that they may be represented by a series of dipoles placed above the earth's surface.

Thus, in the most general case for the source of an electromagnetic field, we need to consider electric and magnetic dipoles placed at a height ho above the surface of the earth. Taking ho as 0, we would have the field for a dipole located on the earth's surface, from which we could find the field for a long wire by integration.


The current supply to the source may have several types of time dependence. The simplest form of current variation would be a continuous sinusoid (harmonic) which is expressed mathematically in the following form:

J(t) = Je-i(OJ 1-(0.

Here, J is the current magnitude, w is the radiation frequency w = 27T/T. T being the period, and cp is the phase. Usually the phase of the source current is taken as zero.

Fig. 3. Model for a geoelectric section. Another type of excitation which has come into wide usage because of its practical simplicity is the abrupt start of a constant current:

{ 0 for t<O, J(t) = J for t>O.

With such a source, a transient electromagnetic field is generated in a conductive medium, which gradually approaches the static field behavior [10-13].

Using harmonic spectral analysis, the transient field may be expressed in terms of its harmonic components. The Fourier transform integral gives the relationship between the transient time function, f(t), and its frequency domain representation, F'(w):

+00

j(t)= 2~ J F'(w)e-iOJtdw. -co

It should be noted that a single value of f (t) for a single instant of time reflects the result of integrating the harmonic spectrum over the entire frequency range.

The Fourier integral shows that the transient and harmonic electromagnetic field expressions contain precisely the same information about the geoelectric section. However, there are Significant differences between the two approaches. In the frequency domain, variations in the amplitude and phase of a sinusoidal signal are measured as a function of frequency. In the time domain, the time variation of one or another field component is measured following the instant the field is excited.

Since the magnitude of the frequency components in a step source current is inversely proportional to -iw, F'(w) = F(w)/-iw and the Fourier transform integral can be written as:

+co 1 J e-i OJ t

j(t)= 2n F(w) -im dw. -co

(1)

In the following paragraphs, we will use expressions for the frequency domain obtained by Fourier transform of transient measurements made in the time domain.

Maxwell's Equations and the Vector Potential

The behavior of an electromagnetic field from any type of source in a homogeneous medium is described by Maxwell's equations. The first of these relates the magnetic field inten

!.ity, H(t),tolhe conduction current density, Jpr(t) = if (t)/p, and the displacement current density, jsm(t) = e8E(t)/8t:


- - oE (t) curl H (t) = E (t)/Q + 8 -O-t -

and

- - flo OE (t) curl B (t) = E (t) Q + 8110 -O-t -,

where 13 = JloH is the magnetic induction.

Maxwell's second equation relates the induced electric field to the rate of change of magnetic induction:

curl if (t) = _ O~;t) . Two other equations express the continuity of magnetic and electric lines of force in the

absence of free charges:

div H(t) = 0,

div E' (t) = 0.

We must assure continuity of the tangential components of the electric and magnetic fields at the boundaries between layers. Also, at very large distances from the source, we may require that the electromagnetic field approach zero.

For a harmonic variation with angular frequency, W, Maxwell's equations assume a simpler form:

curl jJ = E (It; - i wello) ,

curl if = iwB, div B = 0,

div E = 0,

(2)

where 13 andE are the complex amplitudes of the magnetic induction and the electric field, respectively.

Usually, solutions are obtained to Maxwell's equations in terms of a vector potential and a scalar potential, which are subject to two auxiliary conditions.

1. On the basis of the conditions that div 13 = 0, we may always express a vector potential, A, which satisfies the following condition:

B =curl::·L (3)

The poloidal character of the magnetic field indicates that it originates from an electric dipole. Substituting curl A in place of 13 in Maxwell's second equation, we have

curl E = i wcurlA or

curl (E - i w A) = o.

If the curl of a vector is zero, that vector can be represented as the gradient of some function U, called the scalar potential. Thus,

E=iwA -gradU. (4)

The choice of the vector potential is arbitrary to some extent. As we will see below, a solution may be obtained to Maxwell's equations using a vector potential A which does not have a horizontal component, perpendicular to the direction of the dipole moment (Ay =

0) [9].


Since we are considering anisotropic layers in which the electrical properties in the vertical and horizontal directions are different, we have"

Writing Maxwell's first equation in terms of the rectilinear coordinates, and expressing the magnetic and electric fields in terms of the vector and scalar potentials, we obtain the following (Chetaev [15]):

a ( aAx) a (aAx aAz ') (110' )(. au) Ty - au - Tz az - a;- = Qt - l we tllo l w Ax - Ox '

a (OAz) a ( aAx) (110' )( au) Tz au - ax - au = Qt" - l WBtllo - av ' a (OAx aAz) a (OAz) (110 . )(0 au) dx az - a;- - Ty ffY = ~ - l WBnllo l W Az -Tz .

(5)

With the second equation, the relationship between vector and scalar potentials may be written as

Using the designation

we obtain:

aAx + aAz (. 110) U ax az = l WBtllo - Qt .

aAx + aAz = div A iJx az '

; WI1o' 2 --- - W Btllo = kt.

Qt

iw . -U = -. dIVA.

k t

(6)

(7)

The relationship between U and A allows us to obtain the following expression from the first of equation (5):

_o2Ax_o2Ax O'Az=_~iwA_~ o2Ax a2A z , 2 [

iJy2 iJe' + iJxoz iw x k~ ( iJz' + axaz)]

or

a2Ax I a"Ax + a"Ax _ k"A = 0 ax' T ay' az' t x •

(8)

Introducing the Laplacian operator, defined as

We obtain an equation for the horizontal component of the vector potential:

(8')

Similarly, from the third equation in (5), we have

__ ij2Az _ d2 Az o2.4 x _ ( i Wfto (J)2e t ) A. _.i..!'!.. (£.r!.. _ . )( a2Az -t- a2Ax) V.I' dy' + V.c d; - Q" + nl 0 • k' Qn l Wenllo de' I dx de '

t


where kt -..-- = .\. hit

<V

Thus, we see that the parameter A, representing anisotropy in resistivity and anisotropy in dielectric constant as a function of frequency, is a generalized form of the coefficient of anisotropy, A. At sufficiently low frequencies,

Using the designations from [15], we have

or

ii2 Az + OZAl -I- _1_ iJzA l _,.." A = (1 __ 1_) (i2A x • (IX" oy"' ,\2 d:2 n z .\.. dxdz (9)

Thus, the electromagnetic field of an electric dipole may be written in terms of the components of the vector potential, Ax and Az , and the scalar potential, U:

Bx = OAl , dy

E . A ilL" x = I W x- dx'

B = oA.< _ ,lAz E _ aU Ii iJz dX' Y - - -uy ,

B - - ,JA x E . A au Z - dy' ' z = I W z - --;;z- .

Continuity of tangential components of the magnetic and electrical fields implies continuity in Ax, aAx /az, Az , and U. Consequently, there is continuity of the tangential derivatives aAz/ax, aAz/ay, au/ax, au/ay, and therefore, in Bx , By, Ex,and Ey .

Thus, there is continuity both in the horizontal components of the vector potential A and in its vertical derivatives at the boundaries between layers. This continuity of the vertical derivatives does not mean that the vertical components are continuous, inasmuch as the vertical deri vati ve of the scalar potential is discontinuous:

u = .!:..!!!... ( oAx + OAz) • k~ ox OZ

II. USing the condition div E = 0, it is possible to find a vector potential A* which satisfies the following equation:

1;* = i wcurlA*. (10)

In this case, the source of the poloidal electric field is a magnetic dipole. If the dipole moment is directed in the vertical direction, a horizontal (tangential) electric field will be excited in the layers. Substituting this curl expression in Maxwell's first equation, we obtain the following result for a vertical magnetic dipole:

- * (i cu/-to ) - ;-curlB = -Q-t - + w2et!!o curlA* = -kt[curlA*. (11)

It follows from equation (11) that the magnetic induction may be expressed in terms of a vector potential, A * , and a scalar potential, U* :

B* = -k~A* - grad U*'. (12)


Substituting equation (10) for the electric field in Maxwell's second equation, we have

curl curl A * = 13* or using the identity for \7x\7x,

\72)* - grad div A* = -B*. (13)

Combining equations (12) and (13), we have

\72A* = k~A*, U* = -div A*, and so

B* = -k~A* +graddiv A*. (14)

The symmetry of the field about a vertical magnetic dipole allows us to write the vector potential with only a vertical component, A~ , from which we may write:

the azimuthal electric field,

(15)

the vertical magnetic field,

(16)

and the radial magnetic field,

ii"A' II" = __ z • r rir rio (17)

Continuity of the tangential components of the electric and magnetic fields requires continuity in the sole component of the vector potential,Ai,and its vertical derivative, aAilaz.

We should note that the electromagnetic field of a vertical magnetic dipole does not depend on the parameters P nand E. n' This is a consequence of the absence of the vertical component of the electric field.

Moreover, the equations and boundary conditions for Ax and Ai coincide. This permits us to use the horizontal component of the vector potential about a grounded dipole in studying the electromagnetic field about a vertical magnetic dipole.

If the dipole is raised above the earth's surface, then the vector potential very close to the source will tend towards the value for a homogeneous medium having the wave number ko. The expression for vector potential in this case is developed in the next section.

Vector Potential in a Homogeneous Medium

The electromagnetic field of an electric dipole in a homogeneous medium may be written in terms of a vector potential having a single component, A~. In view of the symmetry in the problem, we will rewrite equation (8) in a spherical coordinate system. With spherical symmetry, the vector potential does not depend on e or cp, so that

where r = (r2 + z2)1/2.

The solution of the equation ii' (r A~)

---'.,-;;- =k~A~ r rir'


or

which tends to zero at large distances is the function A~ == (C/r) exp (-ktr). In determining the value for the constant C , we note that for w == 0, A~ == C /r, and B~ == 8A~/ 8y == (C sin e) /r2.

It follows from the Biot-Savart law that Bz == (IJ.!osin e)/47rr2, and consequently,C == IJ.!o/47r, so:

o I fLo e-kt r Ax=-----.

4n r (18)

Similarly, for a vertical magnetic dipole:

(19)

The Electromagnetic Field in a Layered Anisotropic Medium

Let us consider the solution to the equations for the components of the vector potential about a horizontal electric dipole. We will seek an expression for Ax in the form of functions with cylindrical symmetry, that is, depending only on the vertical coordinate and the distance from the source.

In cylindrical coordinates, equations (8) and (9) for Ax and Az have the forms:

82Ax +.!. 8Ax + 82Ax _ k2A = 0 arZ r Dr fiZ2 t x , (20)

82Az +.!. 8A z + --L 82Az _ k 2 A = (1 __ 1_) 82A.,. 8r2 r 8r J...2 8z2 n z J...2 ox 0: ' (21)

Since the scalar potential must be continuous at the boundaries between layers, we have

u = !:..!:!... ( 8Ax + eAz) k~ 8x 8z'

The vertical component,Az,may be written as a derivative of a function W which has cylindrical symmetry with respect to x. Then"

u-!:!E...~ (A 8W) - e ax x+ 8z ' t

and for W we obtain the following equation in cylindrical coordinates:

(22)

Equations (20) and (22) may be solved by separation of variables. In so doing, particular solutions ax and ware each given as products of two functions, one of which depends on z, w, p, and e, while the other depends only on r:

ax = X 'IjJ (r), w =c Z 'IjJ (r).

Substituting these expressions for ax and w in equations (20) and (22), we have

'IjJ"/'IjJ + 1/r.'Ij;'/$: = k~ - X"/X,

1jJ"/1jJ+lIr.1jJ'hjJ=k~,- ~~' +(1-~$X'/Zt A-Z A-


where the primes indicate differentiation with respect to z in the case of X and Z, and with respect to r in the case of l/J.

Since the left-hand and right-hand sides of these equations are functions of different variables, the equality can hold only if each side equals some constant, which we shall write as -m2• Then, we obtain ordinary differential equations for l/J, as well as for X and Z:

and

and

where

'¢" + 1/r~l' + m2'¢ = 0,

X" - (m' + k~) X = 0

X" -n2X = 0,

z" - A.2 (m2 + k~l) Z = (7\2 - 1) X'

(23)

(24)

(25)

The solution to equation (23) is a Bessel function of the first kind of order zero, Jo(mr).

Let us now consider solutions to equations (24) and (25). The general solution to the inhomogeneous equation for Z is made up of the general solution V for the corresponding homogeneous differential equation

Z" - (An)2Z = 0

and the particular solution to the inhomogeneous equation. By direct substitution in the lefthand side of equation (25) we can readily show that the particular solution to the inhomogeneous equation may be expressed by the function -1/m2.x'. Thus,

But we have from equation (24)

and so

As a result, the general solution to the inhomogeneous equation has the form:

Z=V-X'/m2. (26)

The vector potential in a homogeneous medium (18) may be written as Sommerfeld's integral:

co

A~ = /4110 J..!!!:.. e- no I z+ho I Jo (mr) dm, it no

o

and so, Ax and Az assume the following forms:

co

Ax = ~J XJo (mr) dm, 4it

o

(27)

(28)


and

00

A I Ito oW I Ito 0 J ZJ ( d z = 4it 7k" = 411: ax 0 mr) m. (29)

o

Since the particular solutions for the components of the vector potential have the form of the product of a Bessel function which is independent of the properties of the medium, and of functions X or Z, the continuity of the solution is the same as the continuity of the functions X, XI, Z, or the expression (1/kl)(X + Z I).

Using boundary conditions for the functions X and Z, we find conditions for the function V. Continuity of V follows from the continuity of Z and XI. The same is true of the boundary condition for the vertical deri vative, VI, which may be obtained readily using the property of continuity of the scalar potential, from which the continuity of the quality (X + Z I)/kt follows.

Substituting the sum of the general and particular solutions for Z I, we have

1/k~(X + Z') = 1/k~(X + 1/' - 1/m2X") = 1/k~ (X + V' _ m2/~k~ X) = 1/k~V' -1/m2X.

Since the function X is continuous at the boundaries between layers, there will be continuity in 1/kiVI.

The similarity of the equations and the boundary conditions for X and V should be stressed. The similarity permits us to make use of a more general function, Y, which satisfies the following equation in the layer with index p

(30)

as well as boundary conditions requiring continuity in Y and bpY I and the conditions of attenuation of the electromagnetic field with distance: Y -- 0 as z -- ±oo. It is easy to see that for a = nand b = 1, the function Y is the function X, while for a = An and b = 1/kl, the function Y is the function V.

The solution to equation (30) in the layer with index p has the form:

(31)

Let us now consider, as was done by N. V. Lipskaya, the ratio of the function Y to its vertical derivative, designatingYp!y1 as -Rp/ap .

In the first layer

and for z = 0,

for z = h,

d l e- alz + C1 e aIZ

d1 e- alz _ C\ e aIZ


In the second layer

and for z = h,

It may be easily shown that we have the identity

From the continuity of Y and bY' at z = hi' it follows that

al e- alhl + C1 ealhl = d2 e- a2hl + C2 ea2ht,

a b Cd e- Glhl - C ealhl ) - a b (a e- a2hl C ea2hl ) '1 1 1 1 - 2.2 2 - 2 •

Dividing the first of these equations by the second, we have

d, e- ailil -:- CI e a1h1 _ a,b, d2 e- a2hl + C2 ea2hl

dl e- allll_ CI ealhl - a2"2 d2 e a2hl_ C, ea2h1

(32)

(33)

Using identity (32) and equation (33), we may express the value for the function Hi at the earth's surface in terms of its value at the surface of the second layer:

(34)

By mathematical induction, it is a simple matter to extend equation (34) to any number of layers:

Since the condition that the field decreases with distance from the source requires use of a function YN of the form YN = dNexp (-aNz) , RN(HN) = 1, and, as a consequence,

(35)

Making the substitutions a = nand b = 1 in this last equation, we have an expression for X at z == 0:


Similarly, for V we set a = An and b = l/ki:

(37)

The relationships given by equations (36) and (37) are not sufficient to determine X and V'. We must also consider a boundary condition at the earth's surface which takes into account the fact that the source is located at a height 110 above the surface.

The electromagnetic field in the upper half-space is usually given in the form of a sum of a primary electromagnetic field, which increases without limit close to the source, and an induction field:

where A ~) 0 is the vector potential in a homogeneous medi um, while A~l) 0 is the vector potential , , for the induction field:

00

A(O) = I/Lo J ~e-no I z+ho I J (mr) dm x 0 4n no 0,

o

00

A ( 1) I /Lo J C nOl J ( ) d x 0 = ~ 0 e 0 mr m, o

from which 00

A _1/Lo J' ( m e- l1o I z + "0 I /- C 11 0Z) J ( ) d x 0 - -, - -- - 0 e 0 mr m. -'!:t 120 (38)

o

In the first layer, the vector potential has the form:

co

• l~tn (C "1" d -n1z)J ( )d ."ix 1 = -- . 1 e + 1 e 0 mr m. 4:t o

(39)

In view of the continuity of Ax and BAx /Bz at the boundary. (z = 0), we use the equations

m -noho + C C' d X no e 0= 1,- 1= l'

- m e-noho + noCo = n1 (C1 - d1) = X;.

The constant Co is found from this set of equations:

so that we may find a value for the function X = Xo = X t at the earth's surface expressed in terms of the ratio Xl/X~:


Inasmuch as the ratio x~/xi at the earth's surface was related to the geoelectric parameters for the layers and frequency through the function R* , we have

(40)

In considering the function V, we note that the condition for its continuity at the boundary holds only in the absence of a source in the layer. Thus, in the upper half- space, X(z) = X(O)elloZ

and X' (0) = noX(O). However, the presence of a dipole over the earth makes the last condition invalid, since it was indicated above that for Z = 0 we have

that is,

X' - C -110110 X - C + ~e-noho J - no 0 - me, - 0 IZO '

Thus, at the earth's surface we have a discontinuity in X' of size -2me-noho , and as a consequence the boundary condition for V = Z + X'/m 2 assumes the following form, considering the continuity of Z:

(41)

In the upper half-space, the function V, as well as the function X, must have the form: Vo = O!o enoz , so that at z = 0, V~ = noVo. Using this last relationship, we can find an equation relating Vi and V~:

With this equation and the expression for the ratio vtlv~ at the earth's surface, giving the dependence on the geoelectric parameters and frequency, we have two equations in two unknowns, so that we may solve for Vi and V~ at z = 0:

(42)

Having the functions Xi' xL Vi,and V~, it is a simple matter to write expressions for the horizontal and vertical components of vector potential, the vertical derivative,and the scalar potential at the earth's surface:

(43)

]J,(mr)dm, (44)


00

vAx = _ 1 ILo f n1/R* VZ 2lt

m el-TlOhO

+ /R* 10 (mr) dm, "0 "1

(45)

o

00- 2) 00' 2 ) = IILoiCJJ~j' (X1+V~- \X1 10(mr)dm= IILoi~ -dV /' (V:- k/~ Xl 10(mr)dm

4lt k~ v,r m 4:t k t x, \ m o 0

00

= __ I_IL_o_jO)_ cose f"[ 1 __ k_'~_{_]e_nOhOll(mr)dlll' 2ltk~ _/;: Jj' "I 2 + _ _ "0+ ll*

• k/ {"o .\1"1 o

(46)

The electromagnetic field about a current filament at a height ho above the earth's surface may be found by integrating the fields for electric dipoles along the x-axis over an infinite range. Since the vertical component of the vector potential Az=aW/ax=[x/r]a/arrW(r)] is indirectly a function of x, we have:

+00 J Az dx = O.

-00

Thus, the electromagnetic field of a current filament is represented by a vector potential with only one component, parallel to the current.

We find the magnitude of this component by evaluating the integral

+00 00 +00 f Ill /' me- noho ,. Ax dx = ---.,--Il- + '11* J 10 (mr) dx dm .

... Jl 110 IlII

-00 ~ ~~

With some simple manipulation, we have the identity

+00 00 00

j~ I ( )d -,)J'1 ( )d1f ~_ ~_') {' rlo(mr)dr o mr x - _ 0 mr "r y - _ .~ .j rr--y-

-co u 0

= co~ my m

As a result, the sole component of the vector potential for an infinity-long current-carrying wire parallel to the x-axis at a height ho above the earth's surface does not depend on x, and at z = 0 has the form

and the vertical derivative is

00

Ax = IILo J :t

o (47)

(48)


In the same case, the scalar potential is

U=~ aAx_O k: ax - •

Equations (43)-(46) simplify somewhat for a dipole source at the earth's surface (ho = 0):

00

A = lllo J x 231: (49) o

00

f[ n,

A _ lllo ~ 7i* z - 2ri ax. m (no+ ~~ )

o

R*/mAn1 1 ] X J (mr)dm k2/k2 n 'R*/An . 0 , o t1 OT 1 (50)

oc

(lAx lllo J IR* m -az = - 231: n1 n +n JR* Jo (mr) dm, o 0 1 (51)

00

u - ! Ilo i ro e f (1 kr 1 ) - - 231: kl COS k~/k11no +R*IA11l1 - 1l0+ndR* J 1 (mr) dm.

o (52)

Making use of the symmetry between Ax and Ai. we have the following expressions for a vertical magnetic dipole:

00

* Milo [ m J ( d A Z =-2- + /R* 0 mr) m, 31: " no nl

o * 00

aA z MlloJ m az = -21i""" n/R* no+nr/R* Jo (mr) dm. o

(53)

(54)

Using equations (3), (4), (15)-(17),and (49)-(54) at z = 0, we can seek integral expressions for the components of the electromagnetic field of a dipole located at the surface of a horizontally-stratified anisotropic medium:

00

E; = - M~~i ro ! J 1l0+:dR* Jo (mr) dm, (55 ')

o 00

* Milo 1 a a [ m Bz = - -2-- -a r-a 'JR* Jo (mr) dm, 31: r r r. nOTnl

o (55 ") 00

B; = - M21l0 -a a In/R* + m /R* Jo (mr) dm, 31: r no nl

o

co

B - _ I flo Ii ( m z - 2;[ iJy oJ "0+ lid H* Jo (mr) dm,

u (56 ')

(56")


~ ro ~ E T 110 i W I m T 110 i (i) (j :r J (1 ki I ' x = --- Jo mr dm...l- ----- - J lIZr dm

:!:t • 1I0+ndR* () I '):tk~ ().c r k~I!.'Z n -l-il.* \n 1I0+nliR*) l( ) . o ~. t IOU t I 0 I • , 1 I

If we consider the dipole to be situated at the surface of a uniform anisotropio half-space, expressions for Xi> xL V l' and Vr may be obtained from equations (41) and (43), letting hl - 00

in equations (36) and (37). Then, R * = R * = I, and

., 'VI = - m :\, n, (k51k;-lno+ 1/ Allld '

The simple nature of the electromagnetic field in the uniform lower half-space allows us to determine the functions Xl' X~, V1, and vI not only at the surface but also for' z > O. In this case, c1 = 0 in egu~tion (31) so that it follows that X varies with depth as e -nlz while V varies with depth as e-nl Az.

As a consequence, in a uniform half-space atz 2: 0, we have

v __ 2m -nl' ..tl.l ----e ,

llO -r III

, 2e-171Al%

Vl = m (k~Ik;IIlO+1/;\ Tid'

(57)

With equation (57), the expressions for the components of the field in a uniform anisotropic half-space assume the form

(58')

(58")

00

1110 {) J me- nlZ Bz=---- Jo (mr) dm,

2n iJy. nO+nl (59')

o

(59")


The integrals in (55) and (56) have not been tabulated and must be evaluated numerically.

An important special case allowing the examination of the basic characteristics of electromagnetic sound methods is that of a source over a uniform anisotropic half-space.

Electromagnetic Fields at the Surface of a Uniform Anisotropic Half

Space

Taking z = 0 in equations (58), let us consider the field of a harmonic vertical magnetic dipole, located at the surface of a homogeneous anisotropic half-space:

00

E* - M 110 i (oJ a J m rp - - 2 -c) -+--Jo (mr)dm,

Jt r no III (60')

o

00

* M 110 1 a a J' m Bz = -----r- ---Jo(mr)dm. 2:1 r or or 1I0+nl

(60") o

00

B* ,1I/lo 0 J' mn, l' = - -.,- - --- J o (mr) dm,

.:.:1 'Ir 110+": o

In evaluating the expression for the azimuthal component of the electric field, we multiply the numerator and denominator of the integrand by no - n1:

00

E*- Ml1oiro ~J' mno-mn, J ( d rp - - 2:t ar .,~ 0 mr) m.

k(j-kt 1 o

co

In evaluating an integral of the form SmnJo (mr) dm we may use the well-known Sommer-u

feld relation: 00

J!!!... e- n I Z I Jo (mr) dm = /I

o

e - R. 1 rT2"+Z2

/r2+:2 .

Differentiating both sides of this identity with respect to z and setting z = 0, we have

Je;' - hr

. mnJo(mr)dm= - T (1 + kr). o


Substituting the value for this last integral in the expression for the azimuthal component of the electric field, we obtain the desired equation [5]:

E*= 3MI1oiCtl -x[e-kO"(1-Lkr+J..k~r2)_e-htlT(1--Lk r+.!..k2r2)] (61) <p >_ (" -, ) I 0 .> U I t 1 ., t I •

2:t r' ki I - ko <J <J

We can find the vertical component of the magnetic field in a similar manner:

B~ = - 59(1"0 2) X [e- kOT (1 + kor + 4/9k5r2 + 1/9k5r3),- e- kt IT (1 + kt Ir+ 4/9k'71r:2 + 119M Ir3)]. 211r ktl-kO

(62)

Let us examine the behavior of the azimuthal component of the electric field as an example of the particular behavior of the electromagnetic field of a dipole. We will examine the variation in E~ with distance from the source.

If I kQr I « 1 and I kUr I «1 (either a very small distance or a very low frequency), we have

Thus, for very low frequencies or short spacings the geoelectric properties of the lower halfspace have no effect on the electric field of a harmonic magnetic dipole. With increasing spacing, the amount of electromagnetic energy diSSipated in the earth increases. For I kt1r/ »1, the second term in the square brackets in equation (61) can be neglected. In this case

E' ""'" 3M 110 i Ctl - kOT (1 'k --L 1/3 k2 2) <p 4 ( 2 .') e T or I or.

211 r k t 1 - kii (63)

Let us consider the quantity kf1 - kJ. Inasmuch as all of the available information on the electrical character of the earth is contained in the wave number,k~1' the accuracy with which it can be determined is greater, the less kJ is in comparison with kE1'

Considering that kJ == -w2 eo fJ.o, ki1 == -W2et1/tO - iW/to / Ph, and assuming that Pt :::: 103 ,

eo == 10-9/367T ,and et1 Rl 10-10 , we can estimate the upper limit of frequencies needed in frequency sounding. With f :::: 105 cps, / kJ/k~1/ < 6 . 10-2, that is, the effect of displacement currents in the upper half-space is no more than a few percent. It is a simple matter to show that under these conditions, displacement currents in the earth may be neglected, so that

w~olQt 1» W2Bt 1~0. k~1 ""'" -i w~o/Qt I.

A=A.

This quasi static approximation in which displacement currents are neglected and the dielectric constant is considered to have no effect on the behavior of the electromagnetic field is adequate in the great majority of cases. The exception would be the use of electromagnetic sounding methods at very shallow depths, not exceeding a few tens of meters. We will not consider such applications in this paper.

Neglecting displacement currents, the electric field about a vertical magnetic dipole can be expressed in the following form [17]:

(64)


As has already been pointed out, at small spacings, the harmonic electric field, E~, does not depend on the resistivity of the lower half-space. With increasing spacings, a dependence does appear, along with loss, which has the form 1 e-kUr I.

Because

we have

The expression we have obtained is the product of two terms, one of which is an attenuation factor, e- Y';"'o/2I!tl,r ,and the other of which is aphasor, ei YW"O/2I!tl,r, Taking a wavelength A. as the distance over which the phase rolls through an angle of 271', we have

or

/" = --;::=2=;;[ =:=== -{ Ull'o/2Qt

y w/lo/2Qt /" = 2n:

2:t = y 107Qt T • y' 4;[2, 10-7/ Qt T (65)

Since the wavelength is proportional to v'Pt we will ascribe the appropriate index to it for each layer. The wave number for the layer with index p has the form

kp = 2n: ( 1 - i)//"p,

Substituting the expression for kU in equation (64), we have

{ -21t..!.....(1-i) [ 2' 2]} E' = - 3M Qt 1 1 _ e Al 1 + 2n: (1- i) ~ _ 8~ l ~ •

QJ 2nr4 1.1 3 1.2 1

(66)

(67)

Thus, the electric field for a vertical magnetic dipole at the surface of a uniform anisotropic half-space depends on the spacing, the longitudinal resistivity, and the dimensionless ratio of spacing to wavelength.

For r ~ A. 1 e-271'r/A. 1 :s 0.002, the expression for the electric field Simplifies markedly. In this range, it is proportional to the resistivity and inversely proportional to the fourth power of the spacing:

E' ilM Qt 1 cp = - 2nr4 • (68)

Equation (67) has a simple physical significance if we consider that the electromagnetic field of a dipole consists of two parts: one which propagates in the upper half-space and diffuses into the earth, and the other which propagates directly through the earth with exponential attenuation. Obviously, at distances greater than one wavelength from the source, the second part will be attenuated to practically nothing, and only the first part will remain.

In order to examine the manner of diffusion of the electromagnetic field into the lower conductinghalf-space,let us examine E* for z > 0 over the range of spacings greater than a wavelength. First of all, we can note that the electric field lines in the lower half-space are everywhere horizontal. Taking ko = 0 (the quasistatic approximation) in equation (58 '), we have for z > 0


In evaluating the integral, we make the following changes

co

E" = - MQtl..!!..-.- J (m? - mn) e-n1z J(mr)dm cp 2;1; ar 1 0

o

=,-- M~tl :r [j m2e-nlzJo(mr)dm- Jcomnle-nlZJo (mr) dm] . o 0

It is a simple matter to see that for I kttr I »1, the second term in the square brackets tends to zero,

J"" co -k Yr2+z2 -nlz a2 m n z a2 e t 1

mn1 e J 0 (mr) dm = 7fT J - e - 1 J (mr) dm= 7Ji -vrz::FT ' o Z "0 n1 , o· Z r2 +z1

that is, the integral decreases exponentially with increasing I kttr I . With respect to the first term, the integral may be represented with the well-known in

tegral of V. A. Fok:

"" I e~:lZ J o (mr) dm = 10 [k~, (Vr2 + Z2 - z)] x Ko [k~, (Vr~ + Z2+ z)] • o

where 10 and Ko are modified Bessel functions.

The Bessel functions may be replaced with asymptotic expressions for I kUr I » 1

lo(u)= Jf eU • Ko(u)=,13t e- u

211 u JI Tu

and Fok's integral assumes the simpler form:

Differentiating both sides of this equation with respect to z, we have

co

S e- n1z Jo (mr) rim = e- kt 1 z Ir. o (69)

The integral we are concerned with can be expressed in the same form as (69) if we make use of the following identity for Bessel functions

2J ( ) _ ij'Jo(mr) 1 aJo(mr) m 0 mr - - art - r or

Finally, for r » A ,

(70)

so that

(71)


Thus, at spacings greater than a wavelength in the earth, the electric field attenuates exponentially with depth in the earth, and decays along the surface in inverse proportion to the fourth power of the distance from the source.

We should note that the decay in electric field intensity with spacing is not accompanied by any phase roll. At sufficiently large distances from the source, the electric field at the surface is in quadrature to the source current. Phase roll which is constant for a given depth is found if the field is explored as a function of depth. Thus, for r > ~'"1' the electromagnetic field permeates into the conducting half-space as aplane wave with horizontal constant-phase surfaces. The only distinction between this field and a plane wave field is the fact that surfaces of equal magnitude do not coincide with surfaces of equal phase.

Returning to equation (62), we may write the quasi static approximation for the vertical component of the magnetic field about a vertical magnetic dipole:

B* 9MQtt [1 -k r(1 k "I 2 2 3 3"1 z=2ni(1lr5 -e tl +tlr+ 9ktlr+l/gktlr) (72)

As in the case of the electric field [equation (68)], the vertical component of magnetic induction for r ::: ~ 1 may be written in the form of the product of two terms; one which is related to spacing as (1/rli) , and the other which is related to the dipole strength frequency and resistivity:

B* <"=' 9M Qtl z 2niror". (73)

Let us now examine the remaining component of the electromagnetic field of a dipole -the horizontal magnetic field.

Using the same approach in calculating E*cp and neglecting displacement currents, equation (60 m) takes the following form:

Because

we have

00

J m2q+l Jo (mr) dm = 0, o

00

B * M flo {) J' 2 J d r = --2 - a m n1 0 (mr) m. 2n k t 1 r

o

Differentiating both sides of equation (70) with respect to z and setting z = 0, for I ktlr I» 1 , we have

and so

00

J m2n1J 0 (mr) dm = -kt l/fd, o

(74)

Thus, for ! kU r I »1, the horizontal component of the magnetic field is inversely proportional to the fourth power of the distance from the source.

GEOLOGICAL BASIS FOR ELECTROMAGNETIC SOUNDING

For spacings small in comparison with a wavelength, or for very low frequencies, we have the following approximation to equation (60 m):

ro

r = - -1.- ~ m 0 mr) m :; ° (z = 0). B* M!Loa{J( a 'l:t ur.

o

Consequently, under these conditions, the dependence of the horizontal magnetic dipole on the resistivity of the half-space disappears.

The properties of a quasistatic magnetic dipole source located on the earth's surface are such that at large spacings, the intensity of the horizontal magnetic field is proportional to the square root of the longitudinal resistivity. At the same time, the vertical component of magnetic induction, as well as the intensity of the horizontal electric field, is proportional to the first power of resistivity. Hence, it follows that variation in resistivity must have considerably more influence on the vertical component of the magnetic field and on the electric

147

field than on the horizontal component of the magnetic field. Thus, in electromagnetic sounding, when a vertical magnetic dipole is used as a source, it is best to measure the components E~ or B~.

Use of the Fourier transform method allows us to make use of the results of the analysis of harmonic electromagnetic fields to study the transient generated by the initiation of a current step in a dipole source. In so doing, we will consider only the components E ~ and B~ and their relationship to resistivity.

The transient electric field from a vertical magnetic dipole, e~ (t) for the quasistatic approximation may be found readily using the Fourier transform:

This expression may be evaluated using forms from a table of Fourier integral transform pairs:

z

- --dw= 1 +JCO e- i rot { ° 2n -iUl 1

-00

for t<! 0,

for t>O,

+00 _ irot 10 for t<O, in J e -ht IT e -i Ul dw = 1 _ <l> (r -. / !Lo )

-00 V 2Qtl t for t >0,

where <l> (z) = V ; J e- x2 / 2 dx is the integral of the Gaussian probability density function. o

Let us define a transient parameter

't'1 = 2n: V;::"2t'--Q-t-/-;-Il-o = -V 1072n: t Qt l'

(76)

(77)

having the dimension of length, so that the transient electromagnetic field may be expressed in terms of the dimensionless ratio r /T , just as in the harmonic case, the field is expressed in terms of the ratio of separation to wavelength. It is not difficult to find a formal relationship between A and T. An expression for the transient parameter is obtained from the formula for wavelength, substituting 21ft for the period of oscillation, T.


For simplification, we will define the symbol u = 21Tr/T1. Differentiating equation (77) with respect to r, we obtain the Fourier integral which we need in evaluating eJ (t) and b~ (t):

+00

1 I -k re- ioot y2 -u2j2 -,- kt1re t1 __ . -doo = ~ue 2lt -l W It '

-00

(78)

(79)

(80)

Carrying out the integration, we find the transient electric field for t > 0, that is, after the initiation of current [16]

(81)

From the properties of the Gaussian integral, it follows that as t - 0, cp (u) ~ 1

00

because I e-:x2j 2 dx = ~ , and that as t - 00, cp(u) ~ o. o

Consequently, the variation of e*cp with time may be represented in the following manner. Up until the moment current is initiated (t < 0), e~ = O. At the moment current is initiated, the magnitude of the electric field rises abruptly to a value 3MPt1 /21Tr4 , inasmuch as T 1 ~ 0, exp (-u2/2)/T 1 ~ 0, and exp (-u2 /2) /d ~ O.

Then, e*cp varies according to formula (81), dropping to nearly zero in view of the fact that as t - 00, all of the term within the square brackets tend to zero.

Turning now to consideration of the transient magnetic field, we note that it is preferable to work with the time rate-of-change of the field, inasmuch as the expression for this is quite similar to the expression for e~ (t). Thus, from equation (72):

+ OJ

o{/ (I) " 11 I I"~ • • 1 e- i 00 t z _ ". (if 1 r l -h f r (1 I k f- 41 k- .- + 1/ k" 3) d -U-l - _c - :!:Y f" 2;t, - C I .,- t 1r - . 9 tI' 9 t jr ~ (f). (82)

-co

Using the identities (76)-(80), we find the time rate-of-change of the transient vertical magnetic field:

dv: (t) _. flM (if 1 [<I> ( ) _ • I ~ -ue '2 (1 + 2/ ..1... 41)] ~ .- - ~:t r" U V:t CUll 3 Ill, 9 • (83)

As in the case for e~, the time derivative of the vertical magnetic field rises abruptly from zero at the instant current is initiated, reaching a value 9MPt1 / 21Tr5 , and then decreases to zero over some length of time. Physically, the time -deri vati ve of the magnetic transient is measured in the field when an induction coil is used as a field sensor. The EMF induced in such a receiving coil is proportional to the time rate-of-change in magnetic flux cutting the coil.

Considering the electromagnetic field about a grounded electric dipole in the quasistatic approximation, we note that for ko = 0, equations (59) for the field components in a uniform anisotropic half-space assume the forms:


co

B - - [flo ..!!..-J z - 2:t dy

u

OO[ _] [fl iJ2 n e- IlIZ e-1I1 ·\1 z B = __ 0 __ 1 _ J mr dm, X 2:t iJx iJy J m (m + nil II< 0 ( )

o

E = I!-to iffi Joo me-nlZ J (mr) dm- IQtl .!..... -=- r (A n e-n1 Al Z _ k~ 1 e- llIZ ) J (mr)dm x 2:n: m + n 0 2Jt OX r J 1 1 III + n 1 , (84) o 1 '0 '

co

I !-to i ffi iJ J [ "1 e- niZ e-Til Al Z ] E -.----- - J mr dm Z-· 2:n: ax 1'1 (m+n I ) m o()-

o

00

I Qt I a J (' - -Til Al Z k~ 1 e-niZ ) - -2- cos 8 -a Al n1 e - --,-- J 1 (mr) dm. Jt z mTIl, o

The vertical component of the magnetic field of a grounded electric dipole at z = 0 may be found using the relation between the vector potentials A and A* along with equations (59') and (64):

Obviously, for r > A l' we have

Evaluating the horizontal electric field at z = 0, we arrive at the following result:

00 00

E = I!-toiffif __ I~_J (mr)dm _ I Qt 1 ~--=-J [ ')" n __ k~ 1 (m-n d ] " 2Jt m·'- r 0 2~ ax ,. 1 1 2 2 X

I 'I Jl, In -n o 0 1

00 00

I !-to iffi J. m I 01 1 a x f ( A) J ( ) d xJ1 (mr)dm=--- ---Jo(mr)dm--~-·--- m-n1 +.i'l nl X 1 mr m. 2Jt . In + III 2n d.x r ~

o u

The first integral entering into the expression for Ex is obtained by the same means which was used in evaluating E~.

(85)

(86)

In evaluating the second integral, consider the following integral form, which may readily be integrated by parts:

fe-niZ Jdmr) dm = - + {e- I1IZ dJo (mr) = - + {[t'-I1JZ J o (mrll;;o - [Jo (mr) de- nI .} = J 0 o.

1 [ -k Z j~ m -ni' J ( ) d] e- kt l' z e-ki Y,2+.' - - - e t 1 + Z - e 0 mr m = --_ -- . r III r Vr2+o2

·0

Differentiating the left- and right-hand sides of this equation with respect to z and setting z = 0, we have

150

Similarly,

With ktt = 0

FUNDAMENTALS OF ELECTROMAGNETIC SOUNDING

co , -k r In] mr(dm) = kl1r+e 11

1 1 ~~r7.2----

00

"

klllr+e -kn JT

v nl] 1 (mr) dm = rO

U

00 J m]l (mr) dm = rIo '

o

After some simple manipulations, the final expression for the complex Ex at the earth's surface is

Usually, we measure the electric field along the equatorial axis of a dipole (at y = r, x = 0, e = 90°) or along the polar axis (at x = r, y = O,and e = 0,.

In the first case

and in the second case

(87)

(88)

(89)

If the frequency of oscillation is low enough that the separation is very much less than a wavelength, then along the equatorial axis of the dipole:

E ' = _ 1(11 J:\J = _ I (1m J

'" 2:1 r" 21t r3 , (90)

and along the polar axis:

(91)

Making use of the Weber-Lipschitz integral in evaluating the electric field in the earth for e = 90°, we have

C'O

E -- I\!IJ (J XJA -m.\l=J ( )d __ 1!)1lIJ [1'2+(\ _)~J-3/2 x - - :!:1 (I.r r "1 m c 1 mr m - 21t • 1 ~ • (92)

o

Thus, in contrast to all of the various components of the field which we have examined so far, the electric field for a grounded electric dipole for r« A 1 (small separations or low frequencies) depends on the reSistivity of the lower half-space. In this respect, the resistivity value which controls the electric field is the geometric average resistivity; that is, the product of longitudinal resistivity and the coefficient of anisotropy.


At large spacings or at high frequencies ([ kttr [ »1), the electric field assumes the form

E => [IJtt (3~ - 2) x 2nr3 r' • (93)

Two things should be noted in this behavior: first, in the wave zone, the coefficient of anisotropy has no effect on the quasistatic behavior of the electric field, and it is a function of the longitudinal resistivity only; secondly, both for r« ~"1' and for r» ;\1' the electric field varies in inverse proportion to the cube of the separation. In this respect, the behavior of the electric field of a grounded electric dipole differs from the behavior of the other components. For example, the vertical magnetic field about a grounded electric dipole varies as 1/r2 close to the source, but as 1/r4 further from the source.

This fact, that the vertical value for resisti vity has no effect on the electric field about a grounded electric dipole in the wave zone, leads to the conjecture that the electric field is horizontally polarized in this zone. In order to see this, let us examine the vertical component, Ez •

Simplifying equation (84) for the vertical component of the electric field, we have

A2 2 -»1 Al z A 2k 2 e-iil Al Z n (m _ n ) e- n1Z] X, (mr) dm = - 1m e - 1 n - 1 1 1

cc

Evaluating the integral J A: m2 e - iiI Al Z 'I (mr) dm by parts, we obtain an expression for o

E2 at z > 0:


(94)

This expression shows that the vertical component of electric field intensity decays exponentially with distance. This is the basic difference in the behavior of Ex and Ez , which results in the electric field being essentially horizontally polarized in the wave zone.

Let us examine how the horizontally polarized electric field, Ex, given by equation (84), is attenuated in the lower half-space for r »:>. 1. The first integral has already been considered in the analysis of the electric field of a magnetic dipole. In the wave zone

Evaluating the second integral, we can write it in the following form:

and we note that:

f(nIAle-nIAIZ - nle-n1r) Jdmr)dm = ;z ICe- nlz _e-n1A1 Z) Jdmr) dm o 0

As has been shown before,

DO -h z J nz e tl e- 1 J 1 (mr)dm=-r--

o

In the wave zone, with I kt1r I »1 and I kn1r I »1,

ze- kt1 y~

1/ r2 +z2

Since kn!A! = ktt, the last expression reduces to zero. Thus, in the wave zone the second integral takes the form:

00 n -ht 1Z

Considering that for r» 1..1 • J e -nlz J 0 (mr) dm ~ _e_r - I it is a simple matter to find the o

horizontal component of the electric field intensity about a grounded electric dipole.

E ~ I Qt? (3 x 2 _ 2) e -ht l Z

x 2nr3 r2 ' (95)


so that along the equatorial axis of the dipole

Ex.= _ I Qt1 e-Rt ,z . nr3 f

and along the polar axis

Thus, for r » }"1, the electric field of a grounded electric dipole is attenuated in the conductinghalf-space at the same rate as the corresponding component of the field of a vertical magnetic dipole. Attenuation is controlled by the term exp (-ktlz), which causes the electromagnetic field to exhibit the character of plane waves with horizontal equal-phase surfaces.

It may be shown that the horizontal magnetic field of a grounded electric dipole is a weaker function of resistivi1;y than are the components Ex and Bz •

For example, according to equation (84), we have for the component Bx at the earth's sur-face.

00

B. = - 1110 ~J'_I-J (mr) dm. x ":t OJ: ay Ill' It 0 _. T 1

o

For I ktlr I - 0 (small separation or low frequency); the horizontal magnetic field is given by

co

I~to a JI B.~= ----- -Jo (mr) dm 4:( ax ay lit ' o

and does not depend on the resistivity. In the wave zone I kt1r I - 00 (large separation or high frequency)

That is, the horizontal magnetic induction is proportional to the square root of resistivity, in contrast to Ex and Bz , which are proportional to P t1'

Let us now consider the transient electromagnetic field of an electric dipole grounded at the surface of a uniform anisotropic half-space. As was done in the case of the vertical magnetic dipole, we will examine the time rate-of-change of bz (t) for t > 0:

T::O

abz(t)=:lIQtlSiIlO....!.-rll_ -kIJT(1+k r+1/3e r2)1~d(J)= dt 2n r' :.!:t ~ . ell I I -/ W

- co

= ;ll ~t 1 :ill e [cD (u) _ -. /2 e -u2/2 u (1 + U~/3)] . _:tr V :t (96)

Thus, as in the case for e*cp' the quantity 8bz (t)/8t rises ~bruptly at the moment current is initiated to a magnitude (3Ipt1 sin (J) /271' r 4 , while as t - 00, 8bz (t) /8t - O.

Using the expressions for the axial and equatorial components of the harmonic elect-ric field, we find the corresponding transformed processes:

Along the equatorial axis of the dipole

{ 112 l

ex(t) = - 'n~3' (i)(u)- {~1t ue- T +A:.!' [1-cD( ~JJr (97)



(98)

To complete our consideration of dipole sources, we will now examine the electromagnetic field generated by a current filament in the quasistatic approximation. If the current filament is parallel to the x-axis, the electromagnetic field may be characterized in terms of the vector potential Ax (47) and its vertical derivative (48) which have the following forms for the case of a uniform earth:

co

J!to Ie-mhO Ax = -- --+-- cos my dm, It m nl

o 00 aA JII In e-mho __ x = __ ,.._0 1 cos my dm. az It m+nl

o

In evaluating the first integral, we first multiply the numerator and divide by n1 - m:

Ax- J;:o J nl;-m e-mhOcosmydm=J:o J ~1 e-mhOcosmydm-+fme-mhOcosmydm . 00 (00 00) . k t 1 k t 1 k t 1 • 000

Both of these integrals may be found in tables:

00 I_~ ,[ me - mho COS my dm = 0

o (l-h~/ '

where S1, 1 is the Lommel function, related to Struve's H-function and Neuman's N['function as follows:

Thus, the vector potential for a current filament parallel to the x-axis has the following form:

(99)

Let us now examine the behavior of the vector potential, AX' at small and large distances from the source. Equation (99) indicates that the vector potential depends on the product of the wave number and the distance from the source, kt .J h~ + y2. Therefore, we can examine the behavior of the field as I kul- 0 rather than the behavior as .Jh5 +y2 ~O. In this case

00 J!to J~ e- mho

4:0:=-- ---cosmydm, • 2it m o

co

(lAx = J !to /' e- mho COS my dm. az ~:t ~

o


Thus, with Iku .Jh~+ )121-0, the resistivity of the half-space has no effect on the electromagnetic field of a current filament.

In the case in which the modulus of the product kt1 Hi;ll + y2 is large, we can use an asymptotic expansion for Lommel's function [6] for I z I »1, S1,1(z) ~ 1 + l/z2 + ... , that is, S1,1(z) ~ 1 for I z I »4 with an accuracy of several percent.

Consequently, for I kt11"h.6 + y21 2: 4 or for l'h6 + y2 2: A /2, we have

Using this last relationship, we find that at distances greater than half a wavelength in the earth from the source, the vector potential may be approximated as

(100)

Taking the vertical derivative of the vector potential, we may write it in the following form:

ClO

aAx J /10 J' nl (n, -m) - mho d (f; = --;t k:!. e cosmy m =

o t I

co 00 co

= J /10 (f e - mhO COS my dm + J ";.2 e - mho COS my dm ~ r ~nl e - mho COS my dm) , it k t I . Ii t I

0 00

The first two integrals are found in tables:

00 J -mho d ho e cosmy m = -2--2 ' h +y o 0

00

J 2 - mho d 2ho ( 4y2 1) m e cos my m = (2 2" -2 --. - . h +y)" ho+y o 0

The third integral may be expressed in terms of the Lommel function:

00 00

J mnl - mho d a f IZ 1 - mho d -z-e cosmy m = - -h- --2-e cosmy m ,= k t 1 dO. kl I

o 0

= _ _ 13_ f S1.1 [kt I (hO+iy)] 5 1,1 [kt 1 (ho-iY)]}, ahO l 2ktJ (ho+iY) + 2k t d ho-;Y)

Using the approximation for the Lommel function which is valid at distances greater than a half wavelength, we have

00

j' mn -mh h~-l -2 _I e 0 COs my dm = 'J

kt 1 kt 1 (h~+y·)~ o


Finally,

(101)

For sufficiently large values of [kttYh6 + y2[

(102)

With equations (100) and (102), we can find the electric and magnetic field components at distances greater than "h 1 /2 from a current filament:

(103)

(104)

[ 1 (' h~-l \] 1 + -k-'- 1 + 2-2--2 ) • t 1 lO ho + y (105)

It is a simple matter to show that, as is the case for dipole sources, when the condition [ktd~ + y21 ;::: 4 is met, the field attenuates with depth as exp (-kt1z), suggesting plane-wave behavior. However, there is a basic difference between the field from a current filament source and the field from a dipole source as they propagate over the earth's surface. Examining the horizontal magnetic field from a current filament, we realize that under the assumed conditions, the field does not depend on the resistivity of the lower half-space. This follows from the fact that, if the current filament is raised to a sufficient height that I kUho I ;::: 16, ho > 2"h 1> we have

B """ Jflo _h_o_ !1 n: h2+ 2 o y

within about 5%.

It is of interest to compare this expression for By with that for the primary magnetic field. Taking k1 = 0, we find the value for By in free space:

ClO

B J flo J -milo d J ~tn 110 U = -.)- (' cosmy In = -.-) ---.,--...

_:t _:t 11- + y-O 0

Thus, for ho ;::: 2"h 1, the horizontal magnetic field is precisely twice the primary field. This phenomenon has a simple physical interpretation. For a sufficiently high earth conductivity, an incident electromagnetic wave will be almost completely reflected at the earth's surface, which results in doubling the horizontal magnetic field.

The condition ho ;::: 2"hl is satisfied for rapid variations of the natural electromagnetic field which have periods principally in the 20-30 sec range. If the longitudinal resistivity in a sedimentary column is of the order of 10 n-m, then for T = 25 sec, "h 1 =V107 PtlT = 50 km, while the height of ionospheric currents, as is well known, is greater than 90-100 km. In that case

GEOLOGICAL BASIS FOR ELECTROMAGNETIC SOUNDING

E = J Ilo i ro ho x l't k lh2 ")' t 1 O+U

J 110 2hoY B z =--- .,.J ., ;t k t 1 (lzii+y-t

If the distance from the source satisfies the condition V h~ + y2 ~ A /2, but not the condition ho ::=:; y /5; then it follows from equations (103)-(105) that:

157

These conditions are met in the mid-latitudes for geomagnetic disturbances of the "bay" type, with y ~ 1500-2000 km. The duration of a bay is no more than 2-3 hr, and for Ptt ~ 10 Q-m, the wavelength, At::=:; 1000 km. The source of magnetic bays is a narrow band of current in the polar auroral zone; that is, near 70° north latitude at altitude of several hundred km.

The long duration of bays permits diffusion of the field deep into the earth, to hundreds of km, so that investigations made with such a source permit us to study electrical conductivity in the mantle.

Thus, two types of plane waves may propagate into the earth, depending on the ratio of the height of the current filament to the wavelength in the earth. For relatively low altitude sources, if y is sufficiently large, the electromagnetic field assumes the character of plane electromagnetic waves. Such waves are usually referred to as "oblique ," inasmuch as they propagate laterally from the source and then refract nearly vertically into the earth.

If, on the other hand, the source is at a relatively high altitude at small distances y, the field also diffuses into the earth as a plane wave, with the wave vector being nearly vertical over the entire travel path from the source to the observation point. Such a wave is usually referred to as having "normal incidence." The equal amplitude and equal phase surfaces are parallel.

The analysis of electromagnetic fields over the surface of a uniform half-space allows us to draw a series of conclusions.

1. Depths of investigation greater than tens of meters permit the use of quasistatic approximations in most cases. This means that the electromagnetic field will be controlled by resistivity and will not depend on dielectric constant.

2. For spacings which are small in comparison with a wavelength in the earth and for the types of source which have been conSidered, the electromagnetic field components do not depend on the resistivity of thelowerhalf-space. The electric field for a grounded electric dipole, for which the magnitude is proportional to P mt at w = 0, is an exception.

3. With increasing distance between the source and the observation point along the earth's surface, the effect of the resistivity of the lower half-space becomes more important, reaching its maximum at distances greater than a wavelength. At these distances, which have been termed the wave zone, the components of the electromagnetic field have magnitudes proportional to the resistivity (with the exception of the horizontal component of the magnetic field), and decrease with distance from the source in inverse proportion to some power of distance in the horizontal direction, or exponentially with depth. Thus, in the wave zone, that part of the electromagnetic field which propagates in the earth with an attenuation of the form e-kr


characteristic of attenuation in a uniform conducting medium, almost disappears. The earth's surface acts as a source of waves propagating vertically into the conducting half-space.

4. The horizontal component of the magnetic field depends on the electrical properties of the lower half-space to a lesser extent than does the vertical component or the electric field.

Thus, it is best if the components Bz and Ex are used for electromagnetic sounding. The study of the horizontal magnetic fields from ionospheric currents is of special interest. The horizontal magnetic field, which is weakly dependent on the resistivity of the earth, characterizes basically the intensity of ionospheric sources.

Calculation of the Quasistatic Electromagnetic Field at the Surfaces

of a Layered Anisotropic Medium

As was shown in the preceding sections, the vertical magnetic and horizontal electric components of the quasistatic field are most important in electromagnetic sounding. The corresponding expressions in the frequency domain are found from equations (55) and (56) by taking ko = 0 and kp = -iwJ.Lo / Ptp:

ex>

* .M 110 i I!l fJ J' m Erp = 2Jt a,: m+llt / R* 10 (mr) dm, o

ex> ex>

E = 1110 i I!l J' m 1 (mr) dm _ ~.!.- =-f ( At ;;t _ k; I ) 1 (mr) dm '" 211:, m + 11,1 R* 0 211: dX r n* m + nd R* 1 ,

o 0 •

ex>

B 1110' e a J' III 1 ( d z = -2- SIn -d + IR* 0 mr) m. 11: r. m nl o

These equations express the components of the harmonic electromagnetic field in the form of an integral with a dummy variable m which serves as the variable of integration for a sum of Bessel functions Jo(mr) and J 1(mr) , with complex coefficients which are functions of frequency and of the electrical properties of the section. For large values of the argument, the Bessel functions may be written as damped cosinusoids:

10 (mr) = V 2 cos (mr - -411:) , 11: mr

")12 ( 311:) 11 (mr) = t' -;r:;;;;: cos mr - 4'" .

It is obvious from these asymptotic expressions that the decrease of the Bessel functions with spacing increases at high "frequencies" of the harmonic distance m. As a result, in calculating the electromagnetic field for r - 00, we may instead evaluate the integral for m - O.

For example, for a uniform half-space, as m - 0


Substituting this simplified expression for X in the integral for E;, we find that as r - 00

00

JmJo(mr) dm = 0 II

00

E*""", MQtl~f m2Jo(mr)dm= - 3:)1:Qr!1. <P 2;t or -"

o

This approximate expression is precisely the same as the asymptotic expression for the electric field of a vertical magnetic dipole (68), valid for r » A l'

In general, it is known that as r - 00

00 00

J g(m)Jo (mr) dm""'" J g(m)Jo(mr)dm. o 0

where y(m) -y(m) as m - O.

We can make use of this method in considering the electric field over a many-layered section for the condition r » A l' We note that the function X is a very simple expression. in the approximation as m - 0

x = 2m 2mR 2m2R" mT'-nl/R* """'-k---"-'

t 1 t"i 1

where

(106)

Using the simplified expression for X, we find E~ for spacings greater than a wavelength in any of the layers:

(107)

We may find Bi in a similar manner. as well as all of the components of a harmonic electromagnetic field of a grounded electric dipole in the wave zone,

After some simple operations, we obtain

* IIJ! Qt 1I~ fl- ""'" 1,

... :!.:t i w r S

Ex ""'" J .;' 1~2 (3 cos2 8 - 2). _:t r

(108)

(109)

(110)

In the case of an ionospheric current at an altitude 110 2: 2A 1 above the earth, it is easy to obtain the approximations:

E ""'" JfJ.o iWhnR x (" "J' ;t Ie 11- ..Ly-

t 1 o·

B ~ J fL" __ '10_ y~ .•.• ,

:t II~-I-y· (111)


while for the conditions y:::: 5ho, Y :::: A. 1 /2

H. = J flo ...'!.!!.... (1 _ _ l_l_) , .I n!l" lit Iho

(112)

It should be stressed that equations (107)-(110) describe the components of the electromagnetic field of an electric dipole source at distances greater than a wavelength in any of the layers. If the basement has a high resistivity, then the wavelength in the basement will be large. Therefore, usually this condition may be stated by saying the distance from the source is greater than a wavelength in the rocks above basement, but comparable to a wavelength in the basement. In such a case, the components of the electromagnetic field differ significantly from the values computed using the wave zone expressions, equations (107)-(110).

Only one approach is available for computing at arbitrary spacings - numerical integration of equations (55) and (56). Usually, a correction to the values for the desired component in a homogeneous medium, ~~, D.Bi, D.Ex,or D.Bz is computed:

co

A£* _ Jl[ flo i (0) J ( ... _ XO) J ( .. ) d u <p -, A. m I /Ill m, .JJt

o co

M/ = - !of flo ~.!....-r -!-J (X - Xli) J (mr) dm z .1::1 r dr lir ... n , U

00 00

6.Eo:= Ifl40iCil f(X-Xo)Jo(mr)dm- IQtl~.!..-J (Z'-ZO')JI(mr)dm, Jt • 4Jt iJx r

u 0

00

I1Bz = I :n0 sin 8 J (X - XU) mJI (mr) dm.

o

(113)

The computation of a component of the harmonic electromagnetic field consists in evaluating the differences X - x<> and Z' - Zo, for a sequence of values of m and integrating. It should be noted that at sufficiently large values for the dummy variable m, R* ~ R* ~ 1. Therefore, the infinite upper limit of the integral may be replaced with some finite value, mo, without unreasonable error.

The fundamental term in equations (113), which depends on the electrical properties of the earth as well as on frequency, i:s the complex quantity R* or Rt:. The recursive nature of equations (36) and (37) allows us to compute real and imaginary parts to these quantities for a given layered sequence, with the recursion being repeated a number of times equal to the number of layers.

After the real and imaginary parts of the functions (X -Xo) and (Z' - Zo,) have been evaluated, the integration is carried out. This is made difficult by the oscillatory character of the integrand, which resembles a slowly damped cosinusoid. Good results can be obtained using quadrature methods only in favorable cases where the range of integration is short. This is because usually the quadratic formulas are implemented using a polynomial approximation of the integrand of a given degree. For example, in using Simpson's rule for integrating, the integrand is approximated with a second-degree parabola, while in using the trapezoidal rule, a linear approximation is made.

A more economical means for integrating an oscillatory function may be used if two conditions are satisfied: 1) the integrand consists of the product of a slowly-varying amplitude co-


efficient x(m) and an oscillatory function t(mr); 2) the oscillatory function is integrable when multiplied by a weight mP,

The approach is to approximate the integrand with a low-order polynomial over short ranges. Let m-l, roo, and ml be three values for the independent variable for which there are corresponding values of x-l' xo, and xl' and the difference ml - m-l is taken sufficiently small that the function x (m) can be satisfactorily approximated with a parabola over the interval:

and

x (m) = lo + lIm + l2m2,

Using Lagrange interpolation, we have

-1Il_ 1 -1Il1

~o = (lIto-lIt_d (1II0-1Il 1) , -1I1_ t -flI 0

~I = (1II1-III_d ("1 1 -111 0) ,

1 I Y-I=(IIl_I- 1Il0)(III_I- III I)' Yo= V"o-lII-d(lIIn-lIIt)'

1

Considering the interpolation, we have

In computing the components of a harmonic electromagnetic field, the oscillatory term is a Bessel function:

where

mIo (mr) dm = -;:- II (mr) , J ill

I m2Io (mr) dm = :. [mrIo (mr) + (m2r2 - 1) Idmr) + Ii I (mr)] ,

J 11 (mr) dm = - 10 (mr),

r mIdmr)dm=- }2 [mrIo(mr)+Idmr)-lidmr)],

" J m2Idmr) dm = - ~; [mrIo (mr) + 2ldmr)J,

is the Bessel integral function [6].


The same problem in numerically evaluating an oscillatory integrand is found also in computing the transient magnetic field from its spectrum. In evaluating equation (1), we make use of the relationships

e- iCllt = cos rot - i sin rot, F (ro) = Re F (ro) + i 1m F (ro):

+00 +00

f(t)= ~lTJ [ReF(ro)Si~Wl_.ImF(ro)coswwt]dro+ ~:n: f [ImF(ro) Si:Wl + ReF(ro) coswwt]dro. -00 -00

It is well known that the real part is an even function of frequency and the imaginary part is an odd function. The odd term in the second integral is zero, so:

+00

f(t) = ~ f [ReF(ro) sinwt - ImF(ro) coswt] dro. 2:n: w w

- 00

Up until the moment current is initiated, that is, for t < 0, j (t) == 0, and so:

+00

O=2~ J [_ReF(ro)Si:wt_ImF(ro) coswwt] dro.

- 00

Subtracting this last integral fromj(t), we have

+00 00

f (t) = ~ J' Re F (ro) sin UJ t dro = 2 f Re F (ro) Sin wI dro, n UJ IT. W

-00 U

where the role of the variable m has been assumed by the angular frequency, w; then,

~ sin w I R '" = --w- , X = e ero.

In this case CIl1 Co sin (tl t j -w- dro = Si (rol t) - Si (ro_1 t),

00_1

where Si (x) is the sine integral;

J . cosw/ Sill ro t dro = - --t- ;

J . t d sin w t (tl cos (tl f ro SIn ro ro = -t-2- - t

The range of variation of the variables m or w (from zero to infinity) is divided into a series of short segments, such that the function 'X. may be approximated satisfactorily along each segment by a second-degree parabola. After evaluating the integral along each segment, the res ults are summed.

PART II. PRINCIPLES OF QUASISTATIC ELECTROMAGNETIC SOUNDING

Determining the Resistivity of a Homogeneous Anisotropic Half-Space

In the preceding section, we examined the first problem in electromagnetic sounding -the computation of the field of a given set of electrical properties for the section. These com-

PRINCIPLES OF QUASISTATIC ELECTROMAGNETIC SOUNDING 163

putations are fundamental to the inverse problem - determining the resistivity and the thickness of layers from measurements of the components of an electromagnetic field at the earth's surface.

We will consider the simplest case first - a homogeneous anisotropic half-space. We note at once that all the components of the electromagnetic field for both types of source, except the electric field of a grounded electric dipole, reflect the effect only of the longitudinal resistivity. The transverse resisti vity and the coefficient of anisotropy enter into the expression for the sole component of the electric field for a grounded electric dipole but only at relatively low frequencies. If the wavelength of the electromagnetic field or the transient field parameter is less than the distance from the source (wave-zone behavior), all of the components, without exception, are controlled by the longitudinal resistivity. In such cases Pt1 enters into the expression for the amplitudes of the fields for magnetic and electric dipoles in the form of a multiplying coefficient, which makes its evaluation simpler. Thus, in the wave zone, according to equations (68), (73), (86), and (93), or to equations (81), (83), (96), (97), and (98), we have

2nr4 * 2nr5 iJB; 2nr3 E = ~ aB, . Qt 1 = 3M Erp = 9M --at = 1 (3 cos. ~ - 2) '" 31 sin 0 dt '

(114)

In the time domain,

(115)

As a result, for determining the longitudinal resistivity for the lower half-space, we need to determine the amplitude of the horizontal electric field or the rate -of-change of the vertical magnetic induction, multiplied by a coefficient that depends on spacing and the dipole moment. Graphs showing val1J.es obtained from equations (114) and (115) are given in Figs. 4-12.

All of the curves approach the value Ph for small values of A1 or T1' As A1 or T1 is increased, the effect of the reSistivity on all of the components except Ex and ex decreases and the curves approach a descending asymptote so that in the limit, the field is determined by frequency or by delay time.

Graphs for the behavior of the product (27fr4/3I sine) . (BBz/Bt) as a function of wavelength are given in Fig. 4 for two values of Pt1> one of which is four times as large as the other. Providing the wavelength does not exceed the spacing, values are shifted on the graph so that they differ by the factor 4. If the wavelength is greater than (1.5-2)r, then the effect of resistivity is sharply reduced. This reflects the fact that the wave zone is best for determining longitudinal resistivity.

In determining the transverse resistivity or the coefficient of anisotropy, we make use of the components Ex or ex; as may be seen in Figs. 6, 8, 11, and 12, differences in the coefficient of anisotropy are not reflected in wave-zone behavior, but become more pronounced with increasing wavelength or transient parameter, and reach a maximum as w- 0 or t -- 00; that is, in the direct-current region. In this case, according to equations (9) and (91), we have

along the dipole equator

along the dipole polar axis

E __ 1Qml . " - 2nr3 '

E _ IQm, x-~.

164

0.1

FUNDAMENTALS OF ELECTROMAGNETIC SOUNDING

\ \ \ \ l

Y"

.--~.\ ·45" \

90·

\ \ , , , ,

10

.......... .......... ---

-45·

0.1

-90 0

\ \ \ \ \ \ \ ,

\ \

\"'..2 \ ,

fa

, .... ..... --

Fig. 4. Relationship of the modulus and argument of (27Tr4/ 3 I sin e)(aBz/at) = (27Tr4/3M) E; to A tiro 1) Modulus;

Fig. 5. Relationship between the modulus and argument of (27Tr5 / 9M) (a B~/ a t) and A tI r. 1) Modulus; 2) argument.

2) argument; 3) curve for the modulus with Ptl decreased by a factor of 4.

Fig. 6. Relationship between the modulus of [27Tr3Ex/I(3cos2e -2)] and Al/rfore =0°. Curve indices are the values for the coefficient of anisotropy.

PRINCIPLES OF QUASISTATIC ELECTROMAGNETIC SOUNDING

Fig. 7. Relationship of the argument of [2?Tr3Ex/I(3cos 2e - 2)] to At /r for e = 0°. The curve indices are the values for the coefficient of anisotropy.

II'wI /I'tl 3

o.s

Fig. 8. Relationship of the modulus of [2?Tr3Ex/ 1(3 cos 2e -2)] to A tJr for e = 90°, The curve indices are the values for the coefficient of anisotropy,

165

Multiplying the amplitude of the stationary electric field, Ex,bythefactor 2?Tro/I(3cos 2e-2), we obtain in place of Ptt either P mt /2 (for the equatorial array) or 2p mt (for the polar array),

In order to find the value P mt directly, one must multiply the electric field amplitude by the factor -2?Tr3/I (for e = 90°) or by ?Tr3/I (for e = 0°).

Thus, combined measurements of the electric field in the wave zone and in the directcurrent zone make possible the determination of both unknown values: the longitudinal resistivity and the coefficient of anisotropy.


~"'I JO'

-30'

Fig. 9. Relationship of the argument of [21rr3Ex/I(3cos28 -2)] to A./r for (J == 90 0 • The curve indices are the values for the coefficient of anisotropy.

1 P 10

Fig. 10. The relationship for (21rr4/3M)e~ == (21rr4/3I sin8)x (abz! at) and (21r ~ / 9M)(ab~/ at) to r/r. 1) e~ == abz/at; 2) abi lat.

P-r/Ptf fO

__ ----If __ ---2.[2

__ --2 1{2

Fig.H. Relationship of [21rr3ex/I (3cos 2(J-2)] to r/r for 8 == 00 • Curve indices are values for the coefficient of anisotropy.

The problem of determining resistivity with time-domain measurements is similar. With r 1, the effect of resistivity on e~, aez/ at, and abz / at becomes weaker, while the terminal value of the ex reflects the coefficient of anisotropy. After the transient response is finished, we have

on the dipole equator

on the dipole polar axis

IQmt • ex=- 2ar3 ,

I Qmt ex= --a-' aT

PRINCIPLES OF QUASISTA TIC ELECTROMAGNETIC SOUNDING

P-rlft! fO

--------If _--------------2~

1!P,~tl~~~~;;;;~~~~~~2 r;lr

~----.f2

Fig. 12. Relationship of [27rI'3ex/I(3 cos 2e -2)]

to T tlr for e = 90°.

e -Z1fz/A

Fig. 13. Attenuation of a harmonic electric field with depth. Curve indices are values for the wavelength in the earth.

167

In determining the resistivity of a homogeneous half-space, we have referred the measured field amplitudes to the source dipole moment. What do we do, then, when the source is an ionospheric current with an unknown intensity? Usually, the solution to this problem lies in the simultaneous measurement of two components of the electromagnetic field. The intensity of the electric field or the vertical magnetic induction, which are more sensitive functions of the resistivity, is compared to the less sensitive horizontal components of magnetic induction [14]. If the condition 110 > 2A.1 is met for short period oscillations, then it is possible to determine Ph from the ratio Ex lEy = iw/kt1' from which it follows that

n f.to I Ex 12 ,,11 = W B;" . (116)

At mid-latitudes, some types of electromagnetic disturbances allow the determination of Ph not only from this last equation, but also from the ratio of the two magnetic components:

from which it follows that

Wf1.o y2 I Bz 12 QIJ = --4- B;" . (117)

It should be noted that in using the last method, it is assumed that the distance y from the source, which is a current in the ionosphere at 70° north latitude, must be known.

The disadvantage in using electromagnetic sounding of natural origin is that the observed fields are proportional to .[Ptl, rather than to Pu as in the cases of controlled sources.

Two Principles of Electromagnetic Sounding

In the preceding section, it was indicated that it is possible to determine the longitudinal resistivity and the coefficient of anisotropy from measurements of the magnitude of the components of an electromagnetic field at the surface of a uniform half-space. However, the objective of a sounding is the determination of the structure of a nonuniform'half-space; that is, the determination of the resistivities and depths of a sequence of layers. What physical principles, then, can be used in electromagnetic soundings?


to

1-f/J (21l zjr,j

Fig. 14. Attenuation of a transient electric field with depth. Curve indices are values for the transient parameter.

Fig. 15. Decrease of a static electric field with depth. Curve indices are values for the spacing.

For the answer to this question, we recall that within the limits of the wave zone - that is, for relatively large distances between source and receiver - a harmonic electromagnetic field for any type of source decreases exponentially with depth according to the law e-27rz /A 1 (see Fig. 13).

At depths greater than 0.5;\ ,the current density is a fraction of a percent of the corresponding value at the surface. By varying the period of oscillation, it is possible to control the wavelength and so, the depth of penetration of current, or in other words, to make frequency soundings. The corresponding values for the components of the electromagnetic field at the earth's surface will be related to the resistivities of the layers penetrated by the electromagnetic waves. For example, if the depth of penetration is less than the thickness of the upper layer in a two-layer sequence, the components of the electromagnetic field are practically the same as the values appropriate to a uniform half-space with the resistivity, Pt!. If the depth of penetration is considerably greater than the thickness of the upper layer, the components will be the same as those for a uniform half-space with the resisti vity of the lower layer, Pt2'

Similar results are obtained from an analysis of transient fields in the earth, which may be described by the following expression at sufficiently large spacings:

+00 J /

' -i OJ! ? ) _')_ e- hl ' _c_. _ dw = 1 _ li) ( _:t z • ..... Jt ~ -l (tl Tl (118)

-00

Graphs giving the attenuation of a transient electric field with depth are shown in Fig. 14 for several values of the transient parameter or, what is equivalent, for several delay times. At depths greater than 0.571 the current density does not exceed a fraction of a percent of the corresponding value at the earth's surface.

The moment that current is initiated in the earth is taken as time zero. With increaSing delay time, the transient time parameter is defined as

"[I = V 211: t Qt 1 . 1U7

and, as shown in Fig. 14, computations indicate an increasing depth of penetration - that is, we make a transient-field sounding.


The "skin-effect" (the decrease in depth of penetration of electric currents in a conductor with the increase in their rate-of-change) serves as the basis for frequency sounding, magnetotelluric sounding and transient-field sounding. We will call all these forms of sounding "induction sounding." The essence of induction sounding is the variation in apparent resistivity in relation to penetration (referred to wavelength or transient parameter) for a fixed, large spacing.

However, the induction principle is not the only principle on which electromagnetic soundings may be based. If, in induction sounding we make use of the skin effect, which is most important in the wave zone - that is, in the relatively high frequency range - then, a second approach to sounding may be based on the use of a static electromagnetic field. It is adequate to say that only the electric field of a grounded electric dipole can be used for this purpose, because all other components are independent of the electrical properties of the earth at zero frequency.

The physical basis for a sounding in which the static electric field of a grounded electric dipole is used consists in observing the relationship between current density and depth by varying the spacing. Therefore,such a sounding is termed a geometric sounding. Using equations (90) and (92), it is not difficult to find the ratio of amplitudes of the static electric field at depth in a uniform isotropic half-space and at the surface. For e = 90 0 , this ratio is:

Ex (z) _ [ z· ] -3f. Ex (0) - 1 + -;2 .

As may be seen from this expression, the greater the spacing, the less rapidly will the electric field decrease with depth. Graphs giving the variation in current density over vertical planes, which characterize the variation in depth of penetration of current as a function of spacing,are shown in Fig. 15.

Thus, the basis for geometric sounding is as follows: with increasing spacing, using the static electric field, the relative effect of horizontal layers at greater depths becomes relatively more important.

As an illustration of the geometric principle of sounding, we can consider a two-layer medium, on the surface of which a grounded electric dipole has been placed. The static electric field will be measured at a fixed distance r from the source. If the thickness of the upper layer, hi' is large, then the electric field at the observation point will be practically the same as that at the surface of a uniform half-space with the geometric average resistivity, P mi' Consider now that the thickness of the layer is decreased. At the limit hi = 0, we would obtain a uniform medium with the characteristics of the lower layer, with a resistivity Pm2" Thus, for decreasing hi (or the ratio ht/r, since the distance was fixed), the magnitude of the electric field at the earth's surface varies from a value corresponding to a uniform half-space with a reSistivity, Pm1lto a value corresponding to a half-space with the resistivity Pm2" It is easy to see that the same results would be obtained if hi were held constant and the ratio hi/r were varied from infinity to zero by varying the spacing.

Thus, the controllable parameter in geometric sounding is the spacing. Theoretically, one should make use of a time-invariant electric field for geometric sounding. If this is not done, the depth of investigation at large spacings is limited by the skin effect. In practice, a boxcar current pulse is used, with a duration which is considerably longer than the transient process.

Induction Sounding

Let us now assume that we have no information about the electrical properties of the geologic section and that we use the expressions developed in the preceding sections for a uni.,. form half-space to determine longitudinal resistivities.


t' ..

o--~~~--~--------~~

o

Fig. 16. Two-layer amplitude curves for frequency sounding in the wave zone. Curve indices are values for P 12 / P t1'

Fig. 17. Two-layer phase curves for frequency sounding in the wave zone. Curve indices are values for Pt2/Pt1'

We will multiply the magnitude of the harmonic field components by the coefficients contained in equations (114). So doing, in the wave zone, according to equations (107)-(110), we will obtain the product Ptt . R2 in place of pt!. We may now examine the behavior of the function pttR2 in relation to wavelength.

For frequency sounding in a two-layer sequence

R = cth (kt 1 hI + arcLh JI Qt ~ ) = cth [2;1; ~ (1 - i) + arcth V Qt 2 ]. Qt l I., I1t 1

If Al «h1, then R ~ 1, that is, Pt!R2 ~ Pt!. If now Ai »hi ,

R2 ~ cth2 (arcth -V I1t 2 ) = Q/ 2 ,

Qtl I Q/1

that is, pttR2 ~ Pt2' As a consequence, with increasing wavelength the product,puR2,varies from P U to P t2· Therefore, we will call this quantity the effective or apparent resistivity, P w.

In developing theoretical curves, it is convenient to make use of a relative apparent resistivity value, P w / Pt!, which varies from unity to ptd Pt! in a two-layer sequence. The complex function Pw / Pt! consists of a modulus (magnitude) and an argument (phase), graphs of the


two parts being given ih Figs. 16 and 17 as function of Al/hl' As is obvious from Fig. 16 the modulus of apparent resistivity approaches the long-wave asymptote monotonically, while the curve oscillates about the asymptote [pw / Pt! [ == 1 for short waves and the phase oscillates about C{Jw == O.

Next, designating 21Thl/A 1 == x and arcthl(Ptdpt! == y, we obtain

cth (kl hI + arcth v' ~:: ) = cth [(x + y) - ix] = 1-cth (x+y) cth ix cth (x+y)-cthix

and as a consequence

1 +ctgB Z cthB (z+- y) ctgB z+eth2 (zTY) ,

i-cth (x+y) ctg x i cth (x+y)-ctg.z:'

2-2 ctg.r [ethl (x+yJ-ll 2 cosr sin x arg R - arctg cth (z+y) (ctg2z-t-1l = arctg eh (z+Ylsh(x + y)

If I ctgxl == 1 or 21Thl/Al == 1T /4 + (1T/2)n [that is, Al/hl == 8/(2n + 1)], then IpW/Ptll ==1. As a result, the two-layer amplitude curve for apparent resistivity intersects the horizontal. axis at the points Al/hl == 8,8/5,8/9, •..• The amplitude of the oscillation decreases with decreasing A l/hl'

The two-layer phase curves intersect the C{Jw == 0 axis points where sin 2x == 0, or 41Th/A 1 ==

1Tn (that is, where Al/hl == 4/n). As a result, for A/hl == 4, 4/3, 4/5, ... , the apparent resistivity is entirely real. For A l/hl > 4, the phase C{J w > 0 providing Pt2 > PU and the phase C{Jw < 0 providing Pt2 < Ptl' With increasing wavelengths, the curves pass through an extremum, and with further increase in wavelength, the phase tends asymptotically to zero.

For Pt2 »Ptl or Pt2« PU' a diagnostic behavior at long wavelengths is observed. In the first case

cth (kt I hI + arcth -V ~: ~ ) ~ cth klIhI'

and for A 1 »h1 ([ kth11 - 0),

where S1 == h1/ Pt!. Therefore, I Pw I increases in direct proportion to the period of oscillation, T , with a 90 0 phase shift, leading with respect to current at the source, and depends only on h1/Pt!. In the second case, we use the identity

cth (kll hI + arcth ~) = th (kn hI + arth ~) .

For Ph «Pt1 and I kt1h1 [ - 0,

Thus, in this case, I Pw I is independent of Pu and decreases in inverse proportion to the period of oscillation, T, with a 90 0 phase shift lagging the current at the source.

This principle may readily be extended to a many-layer sequence, with the roles of S1 and h1 being assumed by the corresponding values S == ~hp/ Ptp and H == ~hp'


Fr/.Ptf 10

100 f,ln,

1/"

0.'

IJ o

Fig. 18. Two-layer transient-field curves in the wave zone. Curve indices are values for

Pt2 I Pt!·

It should be noted that the phase shift is not an independent meas ure of the electrical properties of the geologic section. As has been shown in [12], the value of cp (wo) is related to the magnitude of the apparent resistivity by the integral relationship:

d :0' (119)

It is obvious from equation (119) that the phase shift at a particular frequency is a function of the derivative of logarithms [dIn 1 pw I] I[dln (wi wo)] over the whole frequency spectrum.

However, val ues for the deri vati ve of logarithms enter into the integral with a variable weight, given by the function

In I oo/hlO+ I I. 00/000- 1

Consideration of this last expression indicates that there is a singularity to the integrand at w == wo, and quickly recovers as w departs from the value wo' Therefore, val ues of the deri va-ti ve of logarithms close to Wo have the most

effect on the value cp (wo). Speaking crudely, the value for cp (wo) is proportional to the deri vati ve of logarithms at w == Wo •

Such is the qualitative character of the theoretical curves for frequency sounding in the wave zone. These curves are fundamental to the resolution of the inverse problem in frequency sounding of curve matching and theoretical curves, which is the best approach to interpretation.

Magnetotelluric sounding is very close to frequency sounding in that, according to equations (111), (112), (116), and (117) the apparent resistivity PT is also given by the product Pt1R2.

Making use of equation (1), we may find the resistivity in the time domain, Pr , by Fourier transformation of the expression for P w:

+00

I J~ p- i III t Q. ~~ -.,- Q",--. -dw .

... :t -l tl)

-co

At relatively high frequencies in the wave zone, the wave-stage of the transient field is developed such that:

+00

J -;",/ QII R2 -1'-. -dw.

-zoo - co


8

Fig. 19. Two-layer amplitude curves for frequency sounding with ptdptl = 1/4. Curve indices are val ues for r /h1o

Let us now examine the behavior of Pr for a two-layer sequence as a function of the transient parameter. For this purpose, we expand the quantity R 2 in a series of terms of the form e -2klhl:

where

Q= J!~ -V~ . ~+~

173

Substituting this series in the Fourier integral, we find the relative apparent resistivity, PT' for the wave-stage:

Two-layer wave curves for Pr/Ptt are shown in Fig. 18 which indicates that as the ratio Tt/h1 increases, the relative apparent resistivity varies from unity to Pt2/Ptl. Thus, for T dh1- 0, q,[47rn/(T t/h1)]l::::: 1 and PT / Ptl l::::: 1, while for T l/hl- "",

cD(~)=O Tdhl

and

In contrast to the curves for frequency sounding, the two-layer time-domain curves are monotone to the left.

From the simple example of a two-layer sequence we have seen that the curves for P w, P T, and PT have a general similarity to one another, since the basis for each of these methods is the skin effect.


\p..,\/Ptt 10

\ I \ I Ij

I I

8

lOa A,/h,

Fig. 20. Two-layer amplitude curves for frequency sounding with pt21ptt = 4. Curve indices are values for r/ht .

Basic to the validity of induction sounding methods is the requirement that measurements be made in the wave zone for a particular source, such that a harmonic electromagnetic field attenuates exponentially with depth. However, when a controlled source is used, it is difficult to assure that the spacing is large enough so that it exceeds a wavelength in all of the layers. Therefore, we must consider the properties of induction sounding for intermediate values of the ratio riA.

Let us now consider the two-layer theoretical curves for frequency sounding with the component Bz for a sequence in which the resistivity in the second layer is less than that in the upper layer. Let P t2 = V4 P tt. As may be seen from the graphs (Fig. 19), the apparent resistivities are essentially the asymptotic values,pwlptt =R2,for short period waves. With increasing wavelength, departure from the asymptotic behavior becomes more important.

The curves Pwlptt exhibit maxima, and then smoothly approach a decreasing asymptote, so that in the limit the apparent resistivity depends only on the spacing. With larger spacings, the right asymptote shifts to longer wavelengths. At the same time, the practical limit for the wave zone is shifted in the same sense. The practical limit for the wave zone may be taken as the value of Al/h for which the difference between I Pwll Ph and R2 is no more than 5%. In the limit for r - 00, the curve for P wi Ph coincides with R2 over the whole frequency range.

Theoretical curves for apparent resistivity in the time domain exhibit a similar behavior.

A basic difference from the behavior described above for theoretical curves arises in those cases in which the apparent reSistivity in the second layer is higher than that in the upper layer, so that there is a decreased attenuation of the electromagnetic field in the lower layer. As a consequence, at spacings greater than a wavelength in the upper layer, the electromagnetic field propagates with low loss in the second layer and spreads from the surface as a plane, horizontally polarized wave similar to that from the source, which is refracted back up into the surface layer. The portion of the en,ergy which propagates directly from the source through the upper layer attenuates as e-21Tr/At, so that for r 2: At, the contribution is negligible.


Two-layer curves for I Pw 1/ Pt1 using the component Bz are shown in Fig, 20 for ptdpu == 4. Let us examine the curves for r/h1 == 45. At short wavelengths, I Pwl / Pt1 ~ R2. With increasing wavelength, the modulus of apparent resistivity exhibits an interference minimum, followed by an increase which exceeds 50% I R21. This combination of a minimum and maximum results from the effect of electromagnetic waves traveling along the boundary between the first and second layers, and then refracting upwards. On decreasing the spacing to r/h1 == 8, the amplitude of the maximum is more than 100% I R21 as a result of lower attenuation in the second layer.

Examining the similarity between the electromagnetic waves propagating along the upper and lower boundaries of a conducting layer, we find an essential difference between them. It may be concluded that for any given resistivity in the second layer, we can select a spacing for which r > A z. In this case, the electromagnetic field in the lower layer is practically completely attenuated, and the approximation may be made that P w / P t1 ~ R 2, which characterizes the wave zone. However, it may not be technically feasible to obtain large enough spacings if the lower layer resisti vity is very high. For example, if the wavelength in the upper layer is 10 km and Pt2/PU == 100, then the wavelength in the lower layer is A2 ==VPtdpt1A1 == 100 km. Thus, only for fantastic separations, greater than 100 km, would we be able to use the approximation Pw ~ pUR 2 .

Since in practice such large spacings cannot be used, it is of interest to consider other asymptotic cases where the spacing is conSiderably less than a wavelength in the basement, so that attenuation can be neglected. Within the realm of this approximation, we might consider a perfect insulator, As indicated by computations, two-layer theoretical curves for I Pwl/ptt for Pt2 == 00 and for Pt2 == (100 to 150) Pt1 are essentially identical over a range of spacings satisfying the inequality r:5 6h1. For larger spacings, the difference between the results becomes larger - indicative of attenuation in the basement.

We will consider the nature of the electromagnetic field of a dipole situated on a surface layer resting on an ins ulating basement for r » Ai' that is I k1 r I -+ 00.

In this case, equation (36) assumes the form

nIIR* = n1 th(nihi + arth minI) = k t I th (k t lltl + arth mlkl1 ) =

-k thkIJhl+m/kll k hk I ' (1 h2 k ) - t 1 l' Ik th k h = 11 t tIll T m - t t Ihl ' ,m l tIt I 1

Using this expression, we find an asymptotic formula for X:

With the asymptotic expression for X, it is not difficult to find Ax and A~ for r »A1' and then for E ~, B~ , and Bz , we find

Computing the approximate value for scalar potential, we note that for Pt2 -+ 00, m -+ 0:

l ' R-*~ th(k" h~ll) 1 1 1 1m ~c tlll+arct - = = =-m-O Qt2 ( ,.1Qt;) cthkt1h1 R'

cth k t Ihl + arcth V -;;;;--Qt 1 ,

and the expression V~ - (kftlm2)X1o in the formula for scalar potential assumes the form


\P,.,I!Pr l

'00 A., /11,

8

Fig. 21. Comparison of two-layer amplitude curves for frequency sounding with ptd Pt1 »1. 1) r» A2 » A 1; 2) A 2 »r » A 1.

k2 [ 2 cth k h 2 ] __ /1_ 11 1-~(2cth2kllhl-1) m kl'l k2 I 1 .

From this approximate expression, it follows that for r »A 1, the apparent resistivity on the surface of a layer resting on an insulating basement,

(120)

is the same for all the components of the field which we have examined.

For r » A 2' that is, in the case of complete attenuation of waves propagating through the basement, the apparent resistivity for P t2 » P t1 has the form:

Q.,!QII= lim cth2 (k l lhl + arcth V~) = cth2 k/lhl . Qt2~CO \ Qtl

The expression obtained for a perfectly in-sulating basement differs from Pw I Pt1 in the wave

zone by the term cth2kt1h1 -1, which may be given a simple physical Significance. The difference cth 2 kt1h1 - 1 presents its contribution to the apparent resistivity as that part of the electromagnetic excitation which propagates in the insulating basement and refracts into the conducting layer as a horizontally polarized wave traveling upward to the surface. If the wavelength in the layer is small in comparison with its thickness (I kt1h11 »1), the electromagnetic field attenuates before it reaches the insulating basement. In this case, cth2kt1h1 ~ 1 and cth2kt1h1 -1 ~ 0; that is, there is practically no energy propagating in the basement. If, on the other hand, the wavelength is considerably larger than the layer thickness (I kuh11 «1), then cth2kuh1 ~ktth1 »1 and cth2ktth1 -1 ~ cth2ktth1. Under some conditions, for long waves, there is no difference between the electromagnetic energy propagating along the earth's s urface and along the boundary between the conducting layer and the insulating basement, so that the amplitudes of the electric and magnetic fields are doubled (Fig. 21).

The apparent resistivity for riA 1 --- 00 and riA 2 --- 0 generally is the same as in the wave zone. It is computed using the same coefficients (114) and (115), independent of the type of array or the direction of separation. However, the basic difference is that for a sufficient increase in spacing, the effect of attenuation in the basement is inevitably observed, no matter how low the conductivity may be. Attenuation with increasing spacing leads to the result that the electromagnetic field begins to depart significantly from the li¥1iting value for Pt2 == 00

and approach values appropriate to the "true" wave zone. Therefore, the range of spacings satisfying the inequality A 2 »r» A 1, may be termed the pseudo-wave zone. This last inequality is commonly well satisfied in the case of highly resistant crystalline basement rocks underlying sedimentary rocks in platform areas. Since the apparent resistivity in the time domain is related to P w through the Fourier transform, the same properties hold for PT.

Thus, for induction sounding with dipole excitation, the most satisfactory type of electrical section is one in which the resistivity of sediments decreases with depth and in which the basement is practically insulating. In this situation, the spectrum of the electromagnetic field does not change appreciably with spacing provided the spacing is greater than a wavelength in the layers above the basement.


Fig. 22. Set of master curves for Pw computed from E~ or Bz with ptd Pu =

1/4. The curve indices are values for

r/h1•

38,0

11.0

161

Fig. 23. Set of master curves for cp w computed from E ~ or Bz , with pt2 / Pt1 = 1/4. Curve indices are values for r/h1.

The situation in which the basement has a relatively high resistivity, but not large enough that attenuation may be neglected, is considerably less favorable. The effect of slow attenuation of the waves at the surface of the basement with spacing results in a pronounced dependence of the frequency characteristics of the electromagnetic field on the distance from the source.

As a consequence, induction electromagnetic sounding has been effective in studying the surface of the crystalline basement in platform areas where the basement resistivity is at least a hundred times greater than that in the overlying sedimentary rocks. Under such conditions, spacings considerably less than a wavelength in the basement are commonly used so that it is possible to ignore attenuation and to consider the idealized model of a geoelectric section with an insulating basement.

Typical three- and four-layer theoretical curves for induction sounding are shown in Figs. 22-36 for ptdpti = 1/4, h2/h1 = 4,and Pt3/ pt1 = 00

It is evident from these curves that the best differentiation between curves is found for sections in which the resistivity of the second layer is lower than that of the first or third layers. The diagnostic portion of the curves seems to be the minimum. It is interesting to note that with increasing spacing, the minimum apparent resistivity approaches the true resistivity for the second layer. However, for spacings large enough that the minimum portion of the curve falls wi thin the wave zone, further increase in the spacing does not change the value of apparent resisti vity.

We might also take note of the fact that for large enough val ues of h2, the minimum value for P w may be somewhat less than the true resistivity of the second layer. This is a result of the appearance of an interference minimum, characteristic of wave curves P w.

As may be seen from Figs. 22, 25, and 28, I P w Imin practically coincides with the limiting value for spacings greater than 5 -8 times the depth to the insulating basement. Somewhat

smaller separations are required for the polar component of the electric field and somewhat larger spacings are required for the equatorial component. Significantly larger spacings are needed for measurements of Bz and Bi. As may be seen from the curves presented, the maximum value for P T is within the wave zone at smaller spacings than the corresponding values

for I Pw Imin·


PriP"

0.1

8

Fig. 24. Master curves for PT computed from e~ or bz , with Pt2/Ptl = 1/4. Curve indices are val ues for r /h1•

00

J8.0 rr-==-______ :JZ.O )4"1 (7'==-______ :26.9 vt==-_____ --.:Z2.6 ~~rr--___ ===19.0 P': 16.0

Fig. 25. Set of master curves for I Pw I computed from Ex (8 =90°), with Pt2/Ptl = 1/4. Curve indices are values for r/h1•

The diagnostic characteristic of the theoretical curves for P w for a monotonically increasing resistivity as a function of depth is the appearance of a sharp minimum at intermediate spacings. It is a simple matter to give a physical explanation of this behavior considering that at sufficiently large spacings the electromagnetic field in the basement is related to waves traveling along the earth's surface and on the surface of the insulating basement. If the basement is covered by a layer with intermediate resistivity, then there is still another wave in this layer, directed upward and attenuating slowly with distance. As a result of interference between the three waves, a diagnostic minimum on the Pw curve is developed.

In finishing this discussion, it should be remembered that induction soundings sense the longitudinal resistivity of the rock layers. The sole exception would be a sounding in which the electric field from a grounded electric dipole is used, for which the observed resistivity


a

o~~----~--------~--__ ~~~

Fig. 26. Set of master curves for CfJw' computed from Ex (e = 90°) with Pt2/Ptt = 1/4. Curve indices are values for r/h1 •

...---------J8.0 -==-_________ J2.0 r,/h, --: ________ :26.9

,~_:--------~zz.G ~~---------------__ ~O ~ 160

s 8

Fig. 27. Set of master curves for PT computed from ex (e =

90°) with Pt2/Ptt = 1/4. Curve indices are values for r/h1 •

179

depends on the coefficient of anisotropy. The effect of anisotropy on such measurements will be examined in a later section.

Geometric Soundings

We can utilize measurements of the intensity of the electric field of a grounded electric dipole with w = 0 in order to measure the geometric mean resistivity of a uniform half-space.

Using equation (59 m), we may find Ex on the surface of a layered anisotropic half-space, taking w = 0:

(X)

E x = - 1Qm1 [R [..!..J1(mr)+'x2 -da ..!..J1(mr)] mdm, 231. r r rr~

(121) o


00

___ 18.0 ___ --]2.0

__ ---15.9 " __ ---12.5

_-===~~~ __________ ~~~~;;:;~~/~!~O ~ / 15.0 • II,

0.2 8

Fig. 28. Set of master curves for I Pwf computed from Ex (8 == 0) with Pt2/ Ptl == 1/4. Curve indices are values for r/h1-

.:!w

JIf

o........,,=.j...,.--~------1t--~:------=~~~~ tI,/h, /0 100

Fig_ 29_ Set of master curves for CfJw computed from Ex (8 == 0) with Ptdptl == 1/4. Curve indices are values for r/h1-


Pt / p{ I 5 00

~ __ JI.O ~--Jl.O

":.---15.9 ~---12.5 __ ::::::::::::j-------j~~=====~/9.p, III 15.0 I ,

6 s

Fig. 30. Set of master curves for PT computed from Ex (e = 0) with Pt2 / Pt1 = 1/4. Curve indices are values for r/h1•

JI.O no

-==:::::==~~-tit1/W-------L:Ir\-l,. 11. 9 • ./" ll.1- ",{lit 1!.fJ

15.D

Fig. 31. Set of master curves for' Pw' computed from E~ or Bz with Ptdpt1 = 4. Curve indices are values for r /h1•

181


Fig. 32. Set of master curves for CfJw computed from E~ or Bz with Pt2/ Ptl = 4. Curve indices are values for r/h1•

5 d

00

t-----JIJ.O v-----JZP 1-------l6.9

v;.. ______ ll.6 r~ 190 ;--------~O

Fig. 34. Set of master curves for I P w I computed from Ex (e = 90 0

) with PtdPtl =4. Curve indices are values for r/h1•

00

J8.Q JZ.O 21.1 Zl.i 11.0 16.0

---===-+-------+P.--~/0:70 [,I " ,

Fig. 33. Set of master curves for PT computed from e~ or bz with Pt2 / Ptl = 4. Curve indices are values for r/h1•

00

~==~--~---------~~~~AJh, 100

Fig. 35. Set of master curves for CfJw computed from Ex (e = 90 0

) with Pt2/Ptl = 4. Curve indices are values for r/h1.

where

PRINCIPLES OF QUASISTA TIC ELECTROMAGNETIC SOUNDING

00

~-::'-=:'::==:==36.0 f/'-_____ Jl,O

~=-----25,9 1/ _______ 22,5 ,V:..-_______ '9,O

f" 16.0

1-=-+-+---------'o"--o---lIjh,

s 8

Fig. 36. Set of master curves for PT computed from Ex (() = 90°) with Pt2/Pt1 = 4. Curve indices are values for r/h1'

R- I' 1 h [ \ h h Qm 2 h ( A I . h Qm (N - 1) )] = 1m -=- = ct m ~ II-/- arct -- X ct m 2 ,12 -/- ' •• -/- arct . "'-0 R* Qm 1 QmN

Along the equatorial axis of the dipole, with x = 0 and y = r:

00

Ex = - I'2~m/ J RmJ. (mr) dm. o

Along the polar axis of the dipole, with x = rand y = 0:

00

183

(122)

Rx = - I QmlS n [.i.. J 1 (mr) -/- r-aa .i..J1 (mr)] mdm = - I Q," IJH[mrJo (mr) - J 1 (mr)] mdm. (123) 2lt r r r '271 r

o u

In a manner similar to that used for induction soundings, we can introduce a relative apparent resistivity, Pk / Pm1ofor the static case as the ratio of the intensity of the electric field on the surface of a layered medium to the intensity on the surface of a uniform anisotropic half-space:

I Ex Q" Qm. =

J (1m 1 (3 cos2 8-1) '271 r3

Along the equatorial axis of the source dipole


00

el/em 1 = r2 f HmJ. (mr) dm, o

co

erlQml = r2 f R[mrJo{mr)-J.{mr)]mdm. o

(124)

(125)

(126)


As an example, let us examine the variation of Pk for a two-layer sequence as a function of spacing, considering the equatorial array . For this purpose, we expand R in a series of terms of the form e -2mA 1h1:

00

R = cth (mAt ht + arcth ~::) = 1 + 2 ~qne-2mAlhln,

where

q = Qmz-Qml , Qm2+Qml

n~l

Substituting this series in the expression for relative apparent resisti vity (125) and integrating the result, we have

00

= 1 + 2 ~qnr3 [;'2 + (At ht n) 2]-3 /2,

Fig. 37. Two-layer set of master curves for Pk' Curve indices are values for P m2 I P m1'

n'=1

At small spacings, PklPm1 ~ 1, while as r - 0()

00

/ 1 2 ~ n 1 I 2q 1 + q / QlI Qm t ~ + q = T -, - = -1-- = Qm 2 Qm I' l-q -q

n=l

Thus, with increasing spacing, the value for Pk varies from the geometric average resistivity for the upper layer to the geometric average resistivity for the lower layer (see Fig. 37).

The electrical properties of the layers enter into the expression for P kip m through the recursion expression Ii, as a function also of the frequency with harmonic constants m. As was shown earlier, for small spacings, harmonics with m - 0 are the most significant. It may be said that rand m are related inversely. This suggests the possibility of examining qualitatively the behavior of the relative apparent resistivity, using the simpler function R with the argument 11m. For example, for a two-layer sequence

Ii = clh (mh1Al + arcth Qm2) tor pm > Pm ; Qm 1 2 1

Ii = lh (mh1Al + ar;th em 2 ) for pm < Pm . em 1 2 1

If 11m - 0, then Ii ~ 1, while 11m - 0() , then R ~ cth (arcth Pm2 I Pm1) = Pm2 I Pm1'

Comparing the curves for Pk' Pmloand Ii (Fig. 38), it is easy to see the similarity,

The behavior of Pk at large spacings in the case in which there is an insulating basement is of special interest. If Pm2 - 0() , then Ii ~ cth mh1A1o while the dummy parameter of integration m - 0, showing that R ~ l/mh1A 1, and as a result, we have for an equatorial array

co

Q,/ Qm 1 = ---=-- J J 1 (mr) dm = _r_ AI h, AlII.

u


or

f..p" ,i,.

where

Thus, the apparent resistivity obtained

Fig. 3~ Comparison of Pk(r),_Pr(r) and R(I/m). 1) Pk; 2) Pr; ::3) R.

with a dc geometric sounding using relatively large spacings is a function only of a generalized parameter for the conductive layer above the basement - the longitudinal conductance, which is the ratio of the actual thickness of the layers to the longitudinal resistivity, or the ratio of the effective thickness to the geometric average resistivity. This result may be extended to a multiple-layer sequence without difficulty.

Apparent resistivity measured with a polar dipole array, Pr' behaves in a similar manner (see Figs. 38 and 39) because, as may be seen from equation (123), it differs from Pk only by a differentiation [1]:

It should be noted that the polar dipole array exhibits a higher resolution than the equatorial dipole array. In Fig. 38, the minimum value for P r is much closer to the val ue P 2 than the minimum value for Pk. However, at the same time, a spacing 1.5-2 times greater than that for the equatorial array is required with the polar array.

In addition to dipole arrays, geometric soundings are made using a long current line symmetric about the measuring electrodes. Considering that a dipole moment is I = Jdx, and that a line with length 2r is made up of a series of elementary dipoles, we may find the intensity of the electric field

+r 00 r ~

Ex= J gmtJdx~J{' RmIl(mx)dm= Jg mt 1'~J'RmIl(mx)dmdx= Jf!I11I!YimI j (II1T)rllll. 2:1 ax Jt " ox. Jt •

-r 0 0 0 II (127)

On the surface of a homogeneous anisotropic half-space ,Ex = J Pm1hrr2, so that the apparent resisti vity for a symmetrical array (the Schlumberger array) assumes the form:

"" Q,,/Qm 1 = ,.2 J Jim I} (lilT) dm, (128)

u

that is, it provides exactly the same results as an equatorial dipole array. The advantage of the symmetrical array (the Schlumberger array) is that the electric field which must be measured is relatively large, decreaSing with increasing spacing only as 1/r2. The disadvantage of this array is the amount of work involved in using it, because a sounding is made by increasirfg the length of the current line stepwise, and this length is twice the spacing. Also, the measured field represents an average of electric properties in the ground over a much larger lateral area than in the case of a dipole array.

In summary, we find that with either geometric soundings or induction soundings, the apparent reSistivity is a function of a penetration depth. If the layers in a sequence are all thick, then, during a sounding, the apparent resistivity will vary from Ptt or Pm1 to Pt2 or Pm2' and


Prj?tr.1 then to P1:3 or Pm2' and so on. However, if the layer h2 is not quite thick enough, the effect of the third layer will become apparent before the measured resistivity reaches the value P t2 or P m2. This means that in many cases, the effect of a thin layer on the apparent resistivity curve may not be definitive, and evaluation of that layer cannot be precise.

01

o

Let us now consider what electrical properties are necessary for a thin layer to be accurately determined with an electromagnetic sounding.

Equivalence for Thin Layers

We will now investigate various cases of equi valence for a three -layer sequence.

Fig. 39. Two-layer set of master curves for Pro Curve indices are values for

Pm2/Pm1·

Let us divide three -layer sequences into two classes such that for the first class the condition P1:3 > Pt2 or PIl13 > Pm2 is satisfied and for the second class the condition P 1:3 < P t2 or P m3 < P m2 is satisfied.

In defining an equivalence parameter for thin layers, we will examine the functions R * and R * , inasmuch as they are definitive functions of the resistivities and thicknesses of the layers. If

then

Similarly, if P m3 > P m2 (n3 < n2) and h2 -+ 0,

-_ n2hz Az + ~ Qm 2

nl Qm 1 n3 Qm , - h2 A2 + iiI Qm 1 - h2 + Ul Qm 1 =---- "",n1Qml-- -_-- =n1Qml-- ----0

Ilz Qm 2 1 +- h A n2 Qm 2 Qm 2 n3 Qm 3 Qt 2 na Qm 3 n22 2 0

----

nsQms

Substituting these expressions in the equations for R* and R* , we have

(129)


(130)

Thus, we see that the resistivity and thickness of a thin layer resting on a poorly conducting basement do not affect the electromagnetic field separately, but rather, a generalized parameter - the longitudinal conductivity S = h / P t = hA / Pm enters.

Equivalence for thin layers in terms of their longitudinal conductances is exhibited by all the modifications of electromagnetic soundings both induction and geometric. However, the critical thickness for which h2 and P t2 can be distinguished by their effects on an electromagnetic field depends on the type of sounding.

The physical significance of the longitudinal conductance of a layer may be seen readily by conSidering current density in the layer.

If the current density is uniform from the top to the bottom of a layer, the resistance of a rectangular prism with a height equal to the thickness of the layer, and with the other sides one meter long, is:

Thus, for equivalence with respect to S2' it is necessary that current flow be restricted to the horizontal plane and that current density be constant from the top to the bottom of the layer. In thin layers resting on a poorly conducting basement, that basement sharply limits vertical current flow and variation of the horizontal component as a consequence of skin effect is nearly absent. The lack of a vertical component to current density means that the anisotropy of the layer will not affect the longitudinal conductance.

Having considered equivalence for the first class of layer sequences we may now examine the second class, consisting of sequences containing a thin layer resting on a conducting basement.

If P t3 < P 12 (n3 > n2), then for h2 > 0

1/, I (' I 11.) II, - ct I n."t., + arct 1-- =-1/2 - - 113 112

l·L_l_~ I 1l"}.h2 na

I n2 112h2 -;- h;

II, =/12

As a result, all components of the electromagnetic field with the exception of the electric field from a grounded electric dipole, for this case are determined from an approximation to the function R * :

(131)

As may be seen from this expression, the resistivity of a thin layer resting on a conducting basement has no effect on the magnetic field. Thin layers will exhibit equivalence, providing the same effects for a range of resisti vities .

The nature of equivalence for sequences of the second type is more complicated when the function it * is considered.

For h2- 0 and P m3 < P m2' we have

-112Qmz

~l Qm 1 cth (n2 A2 h2 + arcth ~2 Qm 2 ) = ~l Qm 1 1 + 112 A2 h2 lia Qm " = 112Qm2 lIaQma IIzQm2 + ~2Qm2

112 Az hz 113 Qm a


-_ n2 A2 h2 + ~2 Qm 2 _

= ~1 Qm 1 ____ --:n~3....:::Q~m'-"s- """ ____ n-!.ol.::.Qm:::...o..l_-=-__

n 2 Qm z 1 +n2Az hz ~z Qm Z m Zh2 Qnz-i Olfto h2+na Qm 3

"s Qm3

Thus, the properties of the second layer enter the expression for R* as the product h2Pn2' as well as in terms of the thickness h2. The generalized parameter hPn = hApm is termed the transverse resistance of a layer, and is usually designated by the symbol T.

After substituting these results in the expression for 'R*, we have

(132)

It is easy to see that here we are dealing with a composite equivalence. For high frequencies,

and so

(133)

Consequently, for high frequencies or at large spacings, the electric field, which is a function of 'R* , behaves in the same manner as the magnetic field, in that only the thickness of a thin layer resting on a conducting basement can be detected. On the other hand, when w> 0, as in a geometric sounding, we have

R = lim (4-) = th (mhlAl + arcth T Q:;"'l ) = (0-,0 R* m z Qm3

(134) .

In this case, the electric field is relatively unaffected by changes in h2 and Pn2' providing their product remains constant. It may readily be seen that T2 is the resistance which would be measured with vertical current flow in a prism with a cross section of 1 m 2, with the upper surface at the top of the layer and the lower surface at the bottom of the layer. A conducting basement "attracts" the current flow lines from an electric dipole so that their direction in the thin layer is nearly vertical. At the same time the magnetic field is generated by horizontal current flow, so that the transverse resistance of a layer has essentially no effect on it. The principal contribution to the magnetic field is made by currents generated in the conducting basement. Changes in the resistivity of a poorly conducting thin layer have relatively little effect on the magnetic field. Changes in thickness which raise or lower the upper surface of the layer do change the intensity of the magnetic field.

For a qualitative evaluation of the limits of equivalence, we can consider a thin conducting layer immersed in a homogeneous half-space. The behavior of the minimum value for [ Pw [ (in the wave zone) and for Pk representing the thin layer is shown in Fig. 40 as a function of thickness and longitudinal conductance. It may be seen readily that a ten percent anomaly in [ P w [ is caused by a layer with a conductance of 20% of that of the overlying medium, while for the same contrast,S2/S1' the anomaly for Pk is 50%. Thus, a thin conducting layer affects an induction sounding less than a geometric sounding. It is evident from Fig. 40 that when the thickness of a thin conductive layer is no more than 50-20% of the thickness of the overlying


-- a

4 ~=-:=-===-=:-::::--:::-::::::- --- b - c

.r,/S,·S _____ -- -·-·- d n.z

Fig. 40. Locus of extreme values for 1 Pw 1 and Pk for a sequence of three layers. Contours: a) 82/81 for 1 Pw I; b) 82/81 for Pk; c) TdT1 for 1 Pw I; d) Td T1 for Pk. Curve indices are values of 8 2/81 for a and b and values of TdT1 for c andd.

189

medium, the value for Pw is essentially independent of the thickness and is controlled only by the longitudinal conductance; that is, there is equivalence with respect to 82. In the case of Pk' equivalence holds for h2 < (0.25-0.35)h1. It may be said that the equivalence with respect to 82 covers a somewhat broader range for geometric soundings.

The behavior of maximum ~alues for IPwl and Pk for the case of a thin resistant layer immersed in a uniform conducting half-space is also shown in Fig. 40. Each curve is characterized by a single value for the ratio T2/T1. It may be seen from Fig. 40 that a 10% anomaly in Pk is developed when T2 ~ 1/4T1 . As the transverse resistance is increased, the anomaly in Pk is increased. If the thickness of the thin layer is no more than 0.4 to 0.8 of the depth of cover, then Pk,max depends only on T2, and not on h2 or Pn2 separately.

Exactly the same behavior is shown by the curves for 1 P wI max. If the thickness of the thin resistant layer is no more than 20% of the depth of burial, then the layer causes a change of no more than 10% in 1 Pwl, and the change does not depend on the resistivity. This property of induction soundings is advantageous in cases in which a thin resistance layer interferes with the exploration of deeper conducting layers. However, this same feature becomes a disadvantage if the object of exploration is the thin resistant layer. In this case, a geometric sounding has higher resolution.

8ummarizing the contents of this section, we may state the following conclusions.

1. The electromagnetic field in the presence of a thin layer resting on a poorly conducting basement depends on the longitudinal conductance of the thin layer.

2. All components of an electromagnetic field, except the electric field from a grounded electric dipole, in the presence of a thin layer resting on a conducting basement depend only on the thickness of the thin layer.

3. The static electric field of a grounded electric dipole depends on the transverse resistance of a thin layer resting on a conducting basement.


4. The electric field of a grounded electric dipole in the wave zone, as well as the magnetic field, depends only on the thickness of a thin layer resting on a conducting basement.

Electromagnetic Sounding in the Presence of an Insulating Screening

Layer

In the preceding section it was established that a thin layer resting on a conducting basement affects the functions Rand R* differently. The first function is practically independent of the resistivity of the layer, while the function R* , and so, the electric field from a grounded electric dipole also, is a function of the transverse resistance, T, which is proportional to the resistivity of the layer. This distinction is most pronounced in the case of a thin insulating layer.

If ktthl- 0 while P2 == 00, such that T 2 == 00, then

(135)

Removal of the thin horizontal insulating layer has no effect on the function R* which in this case represents a two-layer sequence, and while the second layer serves as a screening layer, this screening layer "drops out" of the sequence of layers.

Somewhat different results are found in an analysis of the electric field from a grounded electric dipole. The expression for the function R* which is used in the equation for the electric field in this case assumes the form:

(136)

Using this expression for R* and equation (52), we find the scalar potential:

If the thickness of the first layer is small in comparison with the spacing and the wavelength, then cthfilAlhl ~l/iilAlhl' and the scalar potential takes the form:

00

u= 12~t coseS(~l k?~)Jl(mr)dm. m'R* u

Evaluating the scalar potential at large spacings so that the integral may be evaluated for m-O, we have

(137)

It is apparent from this equation that the scalar potential consists of two parts. The first part does not depend on frequency and contains information only about the depth of burial of the screening layer. The second part reflects the presence of the screening layer, but the effect


I I

I

I I

I I rl

,...,,"""t----+-f--+-""'"""",-----G-----...l.~,jhf 100

Fig. 41. Comparison of the curve for I Pw I computed from Ex (with r /hi = 11.3) with the wave curve. 1) Wave curve I Pw I .

decreases at lower frequencies, and as a result the depth of cover to the screening layer has a weak effect on the electric field of a grounded electric dipole. For w- 0 (that is, with direct current), complete screening takes place. This is illustrated in Fig. 41, where the Fheoretical curve for Ex with r /hi = 11.3 is shown for a four-layer sequence in which the second layer is an insulating screen, with Pt2/PH = 00, h2/hi = 0.1, Pt:J/ PH = 1, ha/hi = 3, and Pl4/Ptl =00.

Comparison with the wave curve indicates that the electric field of a grounded electric dipole is practically constant over the frequencies at which the wave curve indicates the effect of the insulating basement.

The physical effect of a thin resistant layer on the electric field of a grounded elec

tric dipole is that as a result the vertical component of current density is polarized by the screening layer, and in addition to the normal field, there is added an induction field for a series of induced vertical electric dipoles which do not generate a vertical magnetic field.

Thus, the removal of a thin ins ulating layer has practically no effect on the magnetic field, but at the same time, such a layer strongly screens the electric field to more deeply lying conductive layers.

If the thickness of the insulating screen is nonzero, it has some effect on the magnetic field. As examples, theoretical curves for IpwI, CPw, and PT are given in Figs. 42-44 for sequences with the characteristics:

QI 2/QI 1 = 00, h/h1 = 2, QI :/Q11 = 1/ 4 , h.jh1 =~, Qt 41QI 1 = 00.

As may be seen from Figs. 42-44, for a large enough thickness of an insulating screening layer, the values for I Pw I and PT are increased Significantly over a range of relatively small values for i\ 1 and T 1, and at the same time, the phase shift is increased.

If a screening layer has essentially no thickness but has a finite transverse resistance, the behavior of the magnetic field is little different than in the cases already considered. There is a weak dependence of the magnetic field on the resisti vity of the screening layer. With the electric field, in the case of a finite transverse resistance, it is always possible to go to a large enough spacing that the effect of the screening layer is reduced to a minimum.

Thus, as m - 0,

(138)

that is, for r - 00, the electric field of a grounded electric dipole, just as in the case of the magnetic field, is independent of the transverse resistance of the screening layer. The effect of a screening layer on induction soundings in sequences of layers with an insulating basement is particularly sharply reduced when the spacing corresponds to the S-range for a geometric sounding. With such a choice of spacing, the screening layer has a uniformly weak effect on both electric field soundings and magnetic field soundings. The theoretical curves for frequency sounding using Ex are shown in Fig. 45 for a four-layer sequence with the properties:

Q2/Q1 = 8, h21hl = 1, Q:/Q1 = lis, h2Ihl=2, Q/(h = 00.


IP", lip,.

9.2 8

00

Fig. 42. Four-layer set of master curves for I P w I computed from E ~ or Bz • Curve indices are values for r/h1.

Fig. 43. Four-layer set of master curves for CPw computed from E*cp or Bz . Curve indices are values for r/h1•

00

'f5

32

1z.6 11.3

16

0.2 8

Al hi

Fig. 44. Four-layer set of master curves for PT computed from e; or bz • Curve indices are values for r/h1•

Fig. 45. Four-layer set of master curves for I Pw I computed from Ex' Curve indices are values for r Ih1•

It follows from these curves that for r/h1 ?: 22.6, the effect of the screening layer on the electric field is much reduced. As may be seen from the curves for geometric sounding with the equatorial dipole array (Fig. 46) for r/h1 ?: 22.6, the value for Pk depends only on the total longitudinal conductance; that is, the electric field is essentially horizontal. Therefore, in the choice of such a spacing in induction sounding where the electric component is measured, reference should be made to the results of geometric dipole soundings.


.P.I .p, 3

S /

A comparison of Figs. 45 and 46 indicates that the value for IPwlmin is much closer to the value for longitudinal resistivity of the screened layer than is Pk, min.

Two Forms of Anisotropy and Their

Effect on Electromagnetic Sounding

For a horizontally-stratified microaniso-Fig. 46. Four-layer curve for P k. tropic sequence of layers, the recursion relation

R * depends only on the longitudinal (horizontal) resistivity. Therefore, the transverse (vertical) resistivity does not effect those varieties of electromagnetic sounding which depend on this function.

These varieties include all induction sounding methods excepting only the methods in which the electric field from a grounded electric dipole is measured, in which case, generally, the anisotropy has some effect. This is a consequence of the fact that the function ii* , which depends on the coefficient of anisotropy, enters into the expression for electric field.

However, the effect of anisotropy is not the same over different frequency ranges. Thus, the effect is essentially absent for frequencies in the wave zone. Furthermore, using the condition for the wave zone that I kr I »1 or I k I »m, we find

Inasmuch as

R* = cth [niAI hI + arcth ~1 Qm 1 cth (nz A2 h~ + .. 0)] = 1Z2Qm2

the function R* assumes the following form in the wave-zone range of frequencies:

that is, it depends only on the longitudinal resistivities of the layers.

(139)

As a consequence, microanisotropy has an effect only over the low-frequency range for the given type of sounding. This reflects the fact that at sufficiently high frequencies the electromagnetic field penetrates to the conducting layers essentially as a horizontally-polarized plane wave, and moreover, the vertical component of current density is essentially absent. In the case of low frequencies, the electric field becomes approximately that of a direct-current dipole, in which the vertical component of current density is not zero.

However, for one important class of layer sequences, the vertical component of current density is nearly absent even at low frequencies. This is the class of layer sequences with an insulating basement. If the spacing is more than several times the depth to basement, the direct-current field in the vicinity of the observation points is essentially horizontal, inasmuch as, under such conditions, the conductive sequence of layers is equivalent to a surface conductance.


-----'~Vi :::t:::::======= I, Vl;l

It, --o:::ot'CO"'"--lf+--+--------lt,fh,

I

Fig. 47. Two-layer amplitude curve, [ Pwl, computed from Ex (8 = 90°) with P 2 = 00.

Curve indices are the values for the coeffi-cient of anisotropy.

For a qualitative evaluation of sequences of layers in which the transverse resistance of the horizontal layers has no effect on the harmonic electric field of a grounded electric dipole, we will consider the theoretical curves for I Pw I for 8 = 90° and r /hi = 3.36, assuming for the sake of simplicity that the geoelectric section consists of a uniform anisotropic layer covering an insulating basement (Fig. 47). An analysis of the results indicates that for rlhi 2: 3.36 and A:::o; 2.0, the anisotropy of the layer resting on basement has practically no effect on I Pw [.

Considering now a geometric sounding, it follows from equation (121) that the anisotropy has an important effect on the direct-current electric field. Anisotropy leads to the concept of an effective thickness, hA, and the geometric

mean resistivity, Pm = APt. Thus, it is not possible to determine the true characteristics of the layer without additional information (we should note that the longitudinal conductance of the anisotropic layer, hAIPm, is the same as the true value,h/Pt).

However, it is not possible to determine the transverse resistivity with induction sounding in most cases. Thus, the most information about an anisotropic layer may be obtained by the combined usage of induction and geometric soundings.

A second form of anisotropy is macroanisotropy, in which a sequence of layers which may be distinguished from one another on an electric log act as a single layer with averaged properties insofar as an electromagnetic sounding is concerned.

Let us examine the conditions for which the averaging of several layers is permissible by taking the example of a simple repetitive structure. We will assume that microanisotropic layers alternate, forming an infinite sequence in which the properties of all odd-numbered layers are the same as those of the first layer and the properties of all the even-numbered layers are the same as those of the second layer.

The electromagnetic field at the surface of such a sequence of layers is determined from functions R* and R* which may be written in the following form for an infinite sequence of layers:

Inasmuch as the properties of the layers, starting with the third, are repetitive, and the number of layers is infinite, according to M. N. Berdichevskii, we find


Expanding terms, we obtain an expression for the square of the function R* :

.!2_~ R*2_R* nl n.

and on solving for R* , we have

cthnshs+.!2 cthn1h1 nl

195

(140)

The negative sign for the radical has been discarded, inasmuch as it leads in the special case of a uniform half-space to the value R* = -I, rather than the correct value,R* = 1. Similarly

- -~. 12m ._~. 12m 1

Let us now examine a repetitive structure consisting of layers with a thickness which is much less than the wavelength or the spacing. If hi - 0 and h2 - 0, then

n.

(142)

The term (hi + h2)/ (hi/ PU + h2/PU> is the average longitudinal resistivity for a pair of layers. Inasmuch as the section being considered consists of an infinite sequence of pairs of layers, the value

can be called the average longitudinal resistivity of the whole section. The symbol Pl is commonly used in the literature for the average longitudinal resistivity of a sequence of layers.

Thus, the function X, which determines the behavior of all of the components of an electromagnetic field of a grounded electric dipole, assumes the following form in the simple laminated medium:

(143)

As a consequence, for sufficiently large spacings and wavelengths, a laminated half-space is equivalent to a uniform anisotropic half-space with the longitudinal resistivity Pt.


2 ---=======""'8 ==::::-_--- 2

It, ", 100

~ _____ fh

---'fa

0.1

8

Fig. 48. Set of master curves for I P w I for a periodic structure. Curve indices are values for Pt2/Pt!.

It is easy to see that the average longitudinal resistivity of a number of layers with total thickness H and total longitudinal conductance S equals the longitudinal resistivity of a uniform layer with the same thickness and longitudinal conductance.

The wave curves for frequency sounding computed for a laminated medium are shown in Fig. 48 for various values PU/ Pt!. for h2 = hi. Using these curves. it is a simple matter to choose the minimum wavelength for which the macroanisotropic sequence is equivalent to a uniform medium with an error in the modulus of apparent resistivity of no more than 5%. For example. with Pt2 = 2pti. the minimum wavelength is 38hi . Inasmuch as it is usually required that r ?: A i in making a frequency sounding. it is necessary in this case that the spacing be more than 38 times the thickness of the first layer. Thus. for r ?:

38h1 and A1 »38h1• the geoelectric section which consists of an infinite number of layers with equal

thickness and alternating resistivities (Pt!. 2pt1) is equivalent to the homogeneous medium with Pt = 1.33pt1·

Similar results are obtained from an analysis of the electric field of a dipole. grounded at the surface of a laminated geoelectric section. Simplifying this expression for R* as h1 -00 0 and h2 -00 O. we have

(144)

The value

is the transverse resistivity averaged for a pair of layers and is called the average transverse resistivity. Pn.

The expressions for the generalized parameters of the section show a striking similarity. if we consider the average longitudinal conductivity. a to in place of the average longitudinal resisti vity.

(145)

The average transverse resistivity. Pn. is

(146)

Obviously. for a continuous variation of Pt and P n with depth. equations (145) and (146) transform as follows:


uo

__ -31

------8 _-----_4

_-------1

Fig. 49. Set of master curves for Pk for a periodic structure. Curve indices are values for ptdpu.

and the average is found by integrating over the depth range from Zo to Zo + h.

The average resistivities characterized the laminated medium in the cases in which the individual layers are isotropic. Thus, for a pair of layers

By analogy to the microanisotropic case, we may introduce the concept of a geometric average resistivity, ISm = VPnPt ' so that the formula for VI = lllPml / mR* takes the form

F' = {lm V m 2 -i wflo/QII • m

This last expression is the same as the corresponding expression for a uniform microanisotropic half-space. Thus, the electric field of a dipole grounded at the surface of a laminated medium depends not only on the average longitudinal resistivity, but also on the average geometric average resistivity for the sequence of layers.

At higher frequencies, the effect of the average longitudinal resistivity becomes more important, inasmuch as for k - 00

As a result, in the wave-zone range of frequencies for induction sounding using either magnetic or electric field measurements, results are a function only of the average longitudinal resistivity for the, laminated medium. On the other hand, with measurements of the electric field from a grounded electric dipole at low frequencies, the importance of the geometric mean resistivity increases. In the limit for w = 0

Qm2 _ Qm!

R* = ~ Qm 1 Qm 2 + 2 cthmA2h2+ Qm2 cthmAzhz

Qml

(

Qm2 Qm! )2 h Ah f-Qm\ I Ah ----- ct m 2 2- --Ctlln \ \ ...!.. Qm \ Qm 2 + Qm 2

2 cth m Az h2 + Qm 2 cth m AI hi cth m A2 h. + Qm 2 cth m Al hi Qm 1 Qm 1

(147)

This expression gives the relationship between the parameters in a geometric electrical sounding and the properties of the laminated medium. Theoretical curves of geometric soundings in a laminated medium are shown in Fig. 49.

At large spacings, the apparent reSistivity Pk is approximately Pm. Thus, for all types of electromagnetic sounding which have been considered in this section, a simple repetitively


stratified medium is equivalent to a uniform anisotropic half-space. However, this equivalence leads to essentially different results for parametric soundings (frequency and transient) and for geometric soundings.

Let us now consider a finely laminated medium of limited thickness resting on an insulating basement. If the depth to basement is much greater than the thickness of an individual layer, then the effect of the basement first becomes apparent on that portion of a sounding curve where the apparent resistivity is approximately Pt and the low-frequency part of the sounding curve assumes the form characteristic of a two-layer sequence. As a result, in this case the group of thin layers is equivalent to a single uniform anisotropic layer. This equivalence essentially reduces the resolution with which the geoelectric section can be studied. However, there are several basic properties of a multilayer sequence which may be determined with induction sounding despite a rather broad range for equivalence.

In the first place, the low-{requency part of a sounding curve provides either the total longitudinal conductance S (for an insulating basement) or the total depth H to basement, if it is a perfect conductor.

In the second place, if a conductive sequence of layers resting on an insulator has the form of a number of alternating beds, as in the case of the repetitive structure, the apparent resistivity at intermediate frequencies is approximately the average longitudinal resistivity, Pt. As a result, the depth to basement, which is H = Spt may be determined in this case.

Such is not the case with a geometric sounding. The important parameters, it seems, are the geometric average resistivity Pm and the total longitudinal conductance, S. Significant differences between Pm and Pt mean that the depth cannot be determined from the value of S without additional information about the degree of anisotropy.

As may be deduced from the material in this section, the difference between micro- and macroanisotropy basically depends on scale. Layers which appear to be uniformly microanisotropic on closer examination may turn out to consist of an alternating sequence of fine, isotropic laminae. A more exact analysis of the behavior of an electromagnetic field in a laminated medium using numerical methods would require a great deal of time.

The most important result of the analysis of the effect of anisotropy is that with induction soundings the average longitudinal resistivity is measured, while with geometric soundings, the geometric average resistivity is measured. One must recognize that in both cases, there is an incomplete determination of the available geologic information inasmuch as an anisotropic medium is characterized by two independent resistivity values.

A complete description, including both the longitudinal and transverse resistivities, can be obtained only by making combined measurements with both types of sounding. However, for the solution only of the structural problem - that is, the determination of depth of a nonconducting basement, determination of the longitudinal resistivity is by far the more important, because it can be used to convert the value of total longitudinal conductance to a total depth to basement.

Conclusions

In this paper, I have made an analysis of the properties of electromagnetic soundings which allows a comparison of the advantages and disadvantages of the various types of sounding.

An essential feature of induction soundings is that measurements are made at a fixed spacing. This leads to a significant logistic and operational advantage in comparison with geometric soundings. However, the need for complicated equipment to make measurements over a wide range of frequency largely negates this advantage.

REFERENCES 199

The characteristic feature of induction soundings is that they provide the possibility of measuring the actual longitudinal. resistivity in a horizbntally-stratified sequence of rocks. The longitudinal resistivity is extremely important because it must be known in order to compute depth to an insulating basement from observed values of total longitudinal conductance.

In the case of geometric soundings, the geometric average resistivity is measured. As a result, we can find all of the properties of a micro- or macroani~otropic medium only by making measurements both with induction sounding and geometric soundings. However, the depth to basement may be found using only induction soundings.

A thin resistant layer has a much smaller effect on an induction sounding curve than on a geometric sounding curve. This might be considered a disadvantage for geometric soundings if a thin resistant layer screens more deeply lying rocks which are the object of a survey.

A sequence of rocks in which a nonconducting basement is covered by a considerable thickness of conductive layers is the most favorable condition for the application of electromagnetic sounding methods. This has led to the widespread application of various forms of sounding in the platform areas. However, the upper part of the sequence in platform areas frequently contains evaporite and carbonate layers, so that the induction methods in which the magnetic field is measured are the most useful for studying basement relief.

REFERENCES

1. L. M. Alpin, Theory of Dipole Sounding. Gostoptekhizdat (1950) [English translation in: Dipole Methods, for Measuring Earth Conductivity, Consultants Bureau, New York (1966)].

2. M. N. Berdichevskii, Electrical. Prospecting by the Telluric Current Method. Gostoptekhizdat (1960) [English translation, Quart. Colo. School Mines, 60(1):1-216 (1965)].

3. L. L. Vanyan, Some Questions on the Theory of Frequency Sounding in Horizontal Layers, Prikl. Geofiz., No. 23 (1959).

4. L. L. Vanyan and L. Z. Bobrovnikov, Electrical Prospecting with the Transient Magnetic Field Method, this collection.

5. L. B. Gasanenko,"The Normal Field of a Vertical Harmonic Low-Frequency Magnetic Dipole," Research report, Leningrad Univ., No. 249 (1958).

6. I. M. Ryshik and I. S. Gradshtein, Tables of Series, Products, and Integrals, Plenum Press, New York (1963).

7. V. N. Dakhnov, Electrical Exploration for Oil and Gas Deposits, Gostoptekhizdat (1953). 8. V. N. Sharkov, "On the Electrical Conductivity and Temperature in the Earth's Core,"

Izv. Akad. Nauk SSSR, Ser. Geofiz., No.4 (1958). 9. A. I. Zaborovskii, Electrical Exploration. Gostoptekhizdat (1963).

10. A. P. Kraev, "Aperiodic Electromagnetic Processes in Absorptive Media," Dissertation, Leningrad State Univ. (1936).

11. O. A. Skugarevskaya, "Theoretical Study of Transient Electromagnetic Fields in Layered Media," Dissertation, Inst. Fiz. Zemli, Akad. Nauk SSSR (1959).

12. V. V. Solodovnikov, Introduction to Statistical Dynamics of an Automatic Control System. Gostekhteorizdat (1952).

13. A. N. Tikhonov, "On Transient Electrical Currents in a Uniform Conducting Half-Space," Izv. Akad. Nauk SSSR, Ser. Geofiz., No.3 (1946).

14. A. N. Tikhonov, "On Determining the Electrical Characteristics of Layers Deep in the Earth's Crust," Dokl. Akad. Nauk SSSR, Vol. 73, No.2 (1950).

15. D. N. Chetaev, "On the Field of a Low Frequency Dipole on the Surface of a Uniform Anisotropic Conductive Haif-Space, " Zh. Tekhn. Fiz., Vol. 32, No. 11 (1962).

16. S. M. Sheinman, "On Transient Electromagnetic Fields in the Earth," Prikl. Geofiz., No.3 (1947) .


17. S. S. Stefanescu, "Theoretical Studies on Electrical Pros pecting of the Subs urface ," Inst. Geol. Romane, Studii Tekhn. Econ. Geofisica, Ser. D, No.1 (1947).

CONCERNING SOME CAUSES FOR THE DISTORTION OF TRANSIENT SOUNDING CURVES·

L. L. Vanyan, V. M. Davidov, and E. I. Terekhin

The results of field studies show that occasionally the curves obtained in sounding with a transient magnetic field (the ZS method) are distorted in character. These distortions are most severe in the left-hand segments of the curves, and may substantially increase or decrease the value of PT computed for small times. In extreme cases, the value for PT computed for the left-most segment of the transient coupling curve may be negative (this part of the transient is opposite in polarity to the excitation pulse). Examples of such distorted curves are shown in Fig. 1. These distorted curves obviously cannot be matched well with theoretical ZS curves computed for sequences of horizontal layers. In a number of cases it has not been possible to recognize the presence of such distortions merely from the form of the curves (for example, curve ZS-81 in Fig. 2), but such curves differ significantly from the type curves established for a particular area (as for example, curve ZS -84 in Fig. 2), permitting distortions to be recognized.

The cause of the distortions in ZS curves is either a violation of the condition of reciprocity in locating the source and receiver arrays, or the presence of lateral inhomogeneities in the earth.

The latter possibility has been evaluated in an approximate manner, with a determination being made of distortion caused by having the receiving array on a sloping surface, by having the source dipole AB and the receiving array at different elevations as well as by having an insulating lens near the receiving array. The results are described below.

MEASURING ARRAY LOCATED ON A SLOPING SURFACE

With a coil lying strictly in the horizontal plane, the induced voltage is

AU OB. D. 1 = -qdi'

where q is the effective area of the coil and Bz is the vertical component of magnetic induction.

* This paper appeared originally in Prikladnaya Geofizika, No. 41, pp. 86-94 (1965).

201

202 CAUSES FOR DISTORTION OF TRANSIENT SOUNDING CURVES

It It It

I/J 11.9

.~

10 ' tJ"flii

a b c

Fig. 1. Examples of distorted frequency sounding curves, from results of field party No. 510 of VNIIGeofizika in the Orenberg area, 1959. [a) ZS-91, r = 17,350 m, AB = 6100 m, e = 76.5°; b) ZS-94, r = 20,400 m, AB = 7100 m, e =74°;

c) ZS-151, r = 1B,470 m, AB = 7500 m, e = BO°.]

Pc za.s

-+-------------r--,~O-~

Fig. 2. Comparison of electromagnetic sounding curves. 1) ZS-B1, r = 17,150 m, AB = 5300 m, e = 6B030'; 2) ZS-B4, r = 17,690 m, AB = 600() m, e

If the plane of the coil is inclined with respect to the horizontal plane, the voltage induced in the coil is

A aBn o.U2 = -q---ae '

where Bn is the magnitude of magnetic induction projected on the direction of the normal to the coil plane. Inasmuch as the vertical component of magnetic induction is measured near the equatorial plane of the source dipole (the angle e varies from 70° to 110°) where By »Bx , we have

"u ( fJBz (JBII • ) 0. 2 = -q ---aecos a - at sma .

Here, O! is the angle of inclination of the coil plane measured in a direction perpendicular to the axis of the source dipole. For O! = 00, D.U2 = D.U1•

For a harmonic signal, we have

6.U2 = q [Bz (w) i w cosa - By (w) i w sin uJ.

For a small angle of inclination

L\U2=qBdW)iW[1- ~2]-qBu(W)iWU 113°; 3) ZS-B2, r = 16,7BO m, or AB = 5300 m, e = 105°.

L\U2 = qBz(w)iw-qBu(w) iwa-qBz(w)iw ~2 •

In practice, the angle O! rarely exceeds 2 to 3 0, or 0.03 to 0.05 radians, and so the third term may be neglected. Hence, the voltage induced in an inclined coil with the angle of inclination O! measured along a line perpendicular to the source AB is

L\U2 = qi wBz (w) - qiwBu (w) u. (1)

In the wave zone

(2)

MEASURING ARRAY ON SLOPING SURFACE 203

and . B () I ABRq -V .

ql co II co = --n;:a- -/ coJ!olh , (3)

where Pw is the apparent resistivity measured with the frequency sounding method and R is a parameter related to the impedance, Z, seen by plane waves propagating vertically through the earth:

R = - !!!. z. too

Substituting (2) and (3) in (1), we have

!!. 3IABq IABq V' U2 = ~Qro - ---n;:a R a -/ COIlOQl •

As is well known [6], with measurements of the vertical component of magnetic field, Pw is determined from the expression

with an inclined measuring coil

or

211: r 4 AU Qro = 3qAB -1- ;

(4)

(4 ')

The second term in equations (4) and (4') characterizes the error generated by the horizontal magnetic field By in an inclined coil. The size of this error depends on the resistivity of the first layer. The second term is proportional to P~,h while Pw is proportional to the first power of P l' Thus, as Pi increases, the relative size ofthe error caused by the inclination of the coil becomes less by the factor pih.

As is well known, the transient response to a step input is given by the Fourier integral +00

t f - e-irot Q~ = 211: Qro -i 00 d co. (5)

The specific integrals which result are not usually included in standard tables of integrals. However, considering that distortions to the forms of the curve occur primarily at short times (at the high-frequency end of the spectrum), we can obtain approximate values for a homogeneous half-space. We substitute the values R = 1 and Pw = Pl in equation (4):

- 2 1/r Qro = Q1 - :3 r a V -i CO~loQ1

and equation (5) assumes the form: _ +00

~= 1-~ar-1- J' V -iOO/lo e-~rot dco, l!l 3 211: 111 -, 00

-00

The integral transform in this equation is found in a number of tables [5]. After some manipulation, we have:

Q" 2 1 j- (l r h 1 j - = 1 - -3 V 8", -, were 1'1 = Y 107 Ql 2", t , Ql 'fl

The second term specifies the error made in measuring PT with a coil inclined at an angle a with the receiving coil at a distance r from the source.

In the general case, the integral in equation (5) can be evaluated with a high-speed computing machine.


LOCA TlON OF SOURCE AND RECEIVER A T DIFFERENT HEIGHTSI

Let us consider a three-layer geoelectric section of type H (p 1 > P 2 < P3) in which the surface layer varies in thickness. In order to simplify calculations, we will assume that the surface layer has a resistivity of tens of thousands of ohm-meters, so that it may be treated as an insulator. Our idealized section can be treated as a two-layer section. Let the source dipole be located at a point where the thickness of the surface layer is zero, and the measuring coil at a height l above the first conducti.ve layer.' It is known [2, 3] that the transient vertical magnetic field may be expressed as a double integral:

+f'oo -i CiJ t I A-B- () fOO 2m e-m1 10 (mr) dm d Bz (t) = - in. e_i (0 ~ sin e X Or ---+-"-n'--l ---'-- w.

-00 0 m R (6)

Inasmuch as the general form of the integral in equation (6) is not tabulated, we will restrict our considerations to the wave stage of transient coupling and to the S-stage. Using an approximation for the wave stage, the quantity

2m

may be expanded in a Maclaurin's series in powers of m/k. Considering only the first two terms, we have'

+00 00

Bz (t) = - 2: J' .e_-i OJ t. I AB /lo sin a () J' (2mR 2m2R2 )

= " -i (0 4n X Tr ~ - ~- e-m1lo (mr) dmdw = -00 0 1

sin e 2n

Neglecting terms of the order (l/r)2, inasmuch as such terms will rarely exceed 0.001, we have

Computing apparent resistivity from the Bz (t) values:

_ +co

~ = J' R2 e-~CiJt (1 + !:!.l..)dw. Qi -tffi R (7)

-co

Equation (7) shows that the second term in the parentheses acts as a correction for the height of the observation point, increasing at higher frequencies (k1 =-V-iw/J.o / P1), and therefore, its effect on the transient process is maximum at short times (in the high-frequency region). Taking R:::::l 1, which is true at sufficiently high frequencies, we have,

_ +00

~ = 1 + _1_ f l'" f /lo e-i OJ t dw.

Ql 2n V Ql V -; (0 -co

(8)

SOURCE AND RECEIVER A T DIFFERENT HEIGHTS 205

Integrating, we have -~= 1 +V8l't~. QI t'l (9)

It follows from this last equation that the error depends on the ratio of the height, 1, to the transient parameter, That any instant in time. For example, if 1 == 300 m and p == 4 11-m, th~J~or t == 0.1 sec,V81T· 1 IT1 == 0.30; for t == 0.9 sec, V81T. 1 IT1 == 0.10; and for t == 1.6 sec,

81T • liT 1 == 0.07.

We may use an expression for the magnetic field of a dipole over a conducting surface developed by Sheinman [4] for evaluating the effect of distortion on the S-stage. It follows from this expression that

-Q,; r 8, ~=h;s

[ ( 21 )2]&/2 ' 1+ q2+--;:- (10)

where, for a two layer sequence

For h1 == 2 km, p 1 == 4 11-m, t == 3 sec, r == 8 km, and 1 == 300 m, we would have

t' r -II = 13.5, Ii; = 4, q2 = 1.2, and ~ = 0.07. r

As a result, the error in this example would be 6%. Other computations indicate that elevation of the observation point primarily distorts the early part of the transient. The error is no more than 2-3% in the late part of the transient.

Similar errors as those caused by inclination of the coils and by elevation of the observation point above the source dipole may readily be noted as the result of geologic structure. The magnitude -%ar in equation (6) corresponds to the value of 1 in equation (9).

INSULATING HEMISPHERE NEAR THE OBSERVATION POINT

Consider that a lens of high-resistivity rock is located near the receiving coil. In order to make analysis possible, the lens is assumed to be a hemisphere with radius a, centered at point A along the equatorial axis of the source dipole. The distance from the center of the hemisphere to the source is taken as rio the distance from the center of the hemisphere to the receiving coil as r2' and we assume that r1» r2'

The current dipole moment theoretically varies with time according to the expression

{o for t < 0, 1 (t) = 10 for t > O. (11)

In practice, the closure time for relay contacts is about 0.1 sec. Therefore, equation (11) may be given more exactly as follows:


The val ue for a in this expression is chosen to satisfy the condition that at t = 0.1 sec, I(t)=0.95To. This means that a ~30. Thus,I(t)lt>o~Io(1-e-30t).

Let us examine distortion about the minimum of the curve, which usually is the wave stage. The current density at point A is

j (t) = E (tl = IOQ~ (1 _ e-30t ), Ql Q1nr1

where Pe is the apparent resistivity determined from the electric field. If the radius of the hemisphere is small in comparison with the distance, r1' it may be assumed that the electric field is uniform over the surface of the hemisphere. It should be noted that the transient parameter 71 for the terminal stage of the transient process is of the order of kilometers. Thus, for t = 0.1 sec and P 1 = 4 Q-m, 7 1 ~ 5 km. Therefore, the effect of a local inhomogeneity with a dimension of tens of meters will be the same as for a direct current. If we reflect the lower half-space into the upper half-space, the inhomogeneity assumes the form of a sphere. The secondary field associated with an insulating sphere is the same as that of an electric dipole with a moment:

( -301) 9 I'(t) = 10 1-e a Qe.

2n r: Ql

The vertical magnetic field caused by the inhomogeneity at the observation point is calculated as the field for a constant-current dipole in view of the small value of the ratio r2/71:

B' . e 10 a3 Qe ( -301) z = - SIn 2 --2 --3--- 1-e . 411: r 2 211: r 1 Ql

The vertical magnetic field of the source dipole at the observation point is

The apparent resistivity is proportional to the deri vati ve aBz / at:

A comparison of these two equations shows that the ratio of the distortion field to the signal is maximum for short t, decreasing rapidly with increasing time. Moreover, aBz fat rapidly decreases with distance from the center of the sphere. As an example, let us consider the distortion under the following conditions: r1 = 15 km, P 1 = 10 Q-m, e = 90° and a = 400 m.

For t = 0.1 sec, r z = 400 m, and e 2 = 90°, we have

aB~

dt ~O.2, {jBzo

at

INSULATING HEMISPHERE NEAR OBSERVATION POINT 207

That is, the distortion amounts to 20%. With t = 0.2 sec, the distortion is decreased by a factor of 20. Thus, in this example, the distortion has the form of a sudden impulse. It is of interest to note that with e 2 = 270°, the polarity of the distortion is inverted.

Let us now examine another example, with Pe» P l' This case is found when a shallow screening layer is present in the section. If the depth to the screening layer is 600 m, then S = 600 m/10 U-m = 60 mhos, and P = 15,000/60 = 250 U-m. Under these conditions, at the instant t = 0.1 sec, the distortion is nearly five times larger than the signal.

At t = 0.2 sec, the distortion is decreased to 25% and at t = 0.40 sec, it is only a few tenths of a percent. This example indicates that the distortion of a transient curve is accentuated when the section contains a resistant screening layer.

With increasing distance from the center of the insulating hemisphere, the amount of distortion decreases in proportion to the square of the distance.

The sources of distortion which have been considered in this paper are not the only ones possible, but they are representative. It should be noted that inclination of the measuring coil, elevation of the observation point above the source dipole, and the presence of an insulating inclusion all affect the left-hand part of the transient curve, including the curve minimum, more than the right-hand part, representing the late stage of the transient. This partly explains the experimental observation that values of longitudinal conductance determined with the transient sounding method are reliable.

Making corrections to values of PT for the effects of coil inclination, elevation of one coil over the other or the presence of a conducting or insulating inhomogeneity close to dipole is not feasible at the present time. Thus, in carrying out field meas urements with the transient sounding method, the inclination of the coil in a direction perpendicular to the line AB must not exceed 1%, and the difference in elevation between the line AB and the receiving coil must be no more than 1% of the separation between them. If surface inhomogeneities in resistivity are noted in the survey area either from geologic mapping or from detail resistivity soundings, care must be taken to locate the dipoles far from such inhomogeneities.

REFERENCES

1. A. I. Zaborovskii, Electrical Prospecting. Gostoptekhizdat (1963). 2. L. L. Vanyan, "Some Questions on the Theory of Frequency Sounding over Horizontal

Layers," Prikl. Geofiz., No. 23 (1958). 3. L. L. Vanyan, "Elements of the Theory for Transient Electromagnetic Fields," Prikl.

Geofiz., No. 25 (1947). 4. s. M. Sheinman, "On Transient Electromagnetic Fields in the Earth," Prikl. Geofiz.,

No.3 (1947). 5. I. M. Ryshik and I. S. Gradshtein, Tables of Sums, Products, and Integrals, Plenum

Press, New York (1963). 6. L. Z. Bobrovnikov, L. L. Vanyan, Yu. S. Korol 'kov, A. P. Pryakhin, and E . I. Terekhin,

utilization in Electrical Exploration of the Transient Field Method for Solving Problems in Structural Geology. Files of VNII Geofiziki (1960).

CONCERNING THE FACTORS DISTORTING

FREQUENCY SOUNDING CUR VES*

A. N. Kuznetsov

In recent years, new techniques based on the study of nonstationary electromagnetic fields (frequency sounding method, ChZ , and transient method, ZS) have been widely used in applied electrical exploration.

When ZS or ChZ curves are interpreted, difficulties arise very commonly, with the curves which are to be interpreted being distorted; that is, they do not match any of the curves in albums of theoretical curves for horizontally homogeneous media.

Distorted curves are obtained in several parts of the Russian Platform where complicated geoelectric sections are found. Such distorted curves account for up to 20% of the total number of soundings made in some areas [4). The results of field surveys reported by a number of individuals (Alekseev, Davidov, and Melamed) indicate that the primary cause for distorted curves is the geologic structure in the survey area, with surface relief being a secondary source of trouble (when receiver and transmitter are not coplanar). One of the most important causes for distorted curves is the presence of nonhorizontal contacts between zones with different conductivities [1, 4). The present paper describes an experimental study of the factors leading to distortion of frequency sounding curves (ChZ curves). The study was carried out using the electrolytic model facility at VNII Geofizika [3). The first step in the study was the limited recording of ChZ curves. Data were obtained over a two-layer section with a nonhorizontal surface at the base of the first layer, over a three -layer section of type H in which the conductive second layer had the form of a disclike lens, and over four-layer sections containing thin insulating screening layers (either horizontal or dipping) with limited lateral extent.

An analysis of the results so obtained indicated three basic types of distortion:

1. In some cases, depth of the minimum of the p w curve was shallower than the corresponding minimum for curves for a horizontally layered medium at the same r/h distance (Figs. 1, 2, 3), with r being the spacing and h the depth beneath the center of the array;

2. In some cases, the depth of the minimum in the Pw curve was much greater than for the corresponding curves for a horizontally homogeneous medium (Figs. 2, 4).

3. In still other cases, the low-frequency asymptote is nonexistent, or assumes a much lower value than p 1 (Fig. 5).

* This paper appeared originally in Razvedochnaya Geofizika, No.7, pp. 16-24 (1965). 209

210 FACTORS DISTORTING FREQUENCY SOUNDING CURVES

a I~I

1.0~--~~--~------~

10.0

MN Mii A8

H

b

----;r---___ /

__ ~~~~~-------------L h,

10.0

8 1 B2 b---A3 Fig. 1. Distortion of a frequency sounding curve in a section with a thin insulating screen. a) Curve in the central part of the screen; b) curves over the edge of and outside the screen. 1) Experimental frequency sounding curve in the direction perpendicular to the edge of the screen; 2) experimental frequency sounding curve parallel to the edge of the screen; 3) theoretical curve from the two-layer set for a horizontally layered medium. [(I) r/h i == 5.6, (II) r/H == 3.0. h i ==l1cm, h 2 ==0.7cm, h3 == 9.5 cm, Pi == P3 == 0.5 Q-m.J

b

~ , h H

l "O'----'~--~--+----- IO~--~--r-------~ , n

,l 1,0 --_....,--.fII---f---------- Ii

10,0

Fig. 2a. Distortion of frequency sounding curves over a syncline. 1) Experimental curve, strongly distorted, l<r; 2) experimental curve, weakly distorted, l > r; 3) theoretical curves from the two-layer set. [(I) r /h == 5.7, (II) r /h == 3.3.J

c

10.0

, oWl I~

l.o-....-~I'---I------ ~ n

2 I ! 3 10,3

Fig. 2b. Distortion of frequency sounding curves over an anticline. a) Frequency sounding curves over the crest of the anticline, l == 0, r/h == 10; b) frequency sounding curve close to the crest of the anticline, l == 10cm, r/h ==7.0; c) frequency sounding curves with the AB dipole over the crest of the anticline, l == 0, r/h ==3.2. 1) Experimental curve ,strongly distorted, l< r; 2) experimental curve,weakly distorted, l > r; 3) theoretical curves from the two-layer set.

FACTORS DISTORTING FREQUENCY SOUNDING CURVES

:tP liAS

I~~I I~~I a b

0

..t J. HAl 11118

8,0 B,O

El' 8 2

Fig.3a. Distortion of frequency sounding curves over a monocline. a) For r/H == 4.0. Experimental curves: 1) Strong distortion, 1 == O. 77r, a == 8 0 ; 2) weak distortion, l == 5r, a == 20

;

PW,min increases with increasing a. b) For r/H == 3.0. Experimental curves, weakly distorted; (I) a == 80

, 1 == 1.3r; (II) a == 2 0 , l == 6.5 r.

Mil Mil Mil A8 At! All

~ F 1 .. '11 j" fJ •

p·'BR

; }}j)

a b 11&1 ,?, I ~----

(j)

~0>-----1'-..!--' ----VT W_.._.c---ff--'------ ~

O,05n-m

..i "O-.c---fi---;------ -;;

10.0

Fig.3b. Distortion of frequency sounding curves over a wedge. a) Comparison of experimental frequency sounding curves perpendicular over a wedge for different distances of the dipole from the outcrop of the wedge: strong distortions (I) for 1 == (2/3)r, (II) for l == r; weak distortion. for 1 == (4/3)r. b) Comparison of experimental frequency sounding curves over a wedge with theoretical curves from the two-layer set for a horizontally layered medium. (Curve indices are CD for r/h == 9.0; ® for r/h == 5.2.) 1) Experimental curve, strongly distorted; 2) experimental curve, weakly distorted; 3) theoretical curve from the two-layer set.

211


10

5 If

3

2 (5

10

1.5

60

60

MN

~hJ AS A8 MN

40 20 o 60 80 100 l. em

a

20 a 20 60 80 100 l. em b

Fig. 4 .. Graphs for the ratio Pw,max /Pw,min (1), the product~Tmin . PW,min (2), and the longitudinal conductance S (3) over an inclined step. H == 15 cm, h == 5 cm. a) Array axis parallel to strike; b) array axis perpendicular to strike. The ordinate is: Pw,max; S; .JTmin· Pmin x 105 Q-m-sec1/2.

The first two types of distortion lead to curves which are similar in appearance to curves for a horizontally homogeneous medium. However, such curves cannot be matched with any of the theoretical curves for horizontal layers. The third type of distortion can be recognized even with a superficial examination ofthe data. The greatest distortion is seen at the low frequency end of the curve.

Curves of the first type are obtained over a section containing a thin, nonconducting screen (Fig. 1). The right, or low-frequency asymptote in such cases assumes a value appropriate for the portion of the section above the resistant screen, while the minimum for the curve over the insulator is shifted to the left and up in relation to a two-layer theoretical curve.

Strong distortions to the curves are noted at the border of insulating screening regions when the array is oriented parallel to the border and the array center is within a distance l~ AB/2 (or MN/2) from the border (Fig. Ib).

Analogous distortions are noted also on curves for the perpendicular array when the measuring dipole MN is close to the border.

Distorted curves of the first type are found also in cases with inclined conducting layers. Over a number of structures (Figs. 2, 3) with one of the dipoles at a distance l < r from the surface trace of the dipping surface, we noted an irregular behavior of Pw min with Pw min , , increasing as l decreases. In co mparison, the value for PW,min for a comparable horizontally-uniform medium would increase, as a result of the increase in the ratio r /h.


A 8

M -+ -N

MN ..oj t:! lAB Mil ~t=-- l :jAB MN

~~Jt~jl;;;;;; )1 -1'~ L

a b 1f ... I.O .m I'P.,I. I). m

0.'----+-:---==----+-----: ___ -1'1; sec1/ 2

0.05 -

:[ ~

:t

'. ...... ·m

_A_ zL ----'"\. •

..... .... >, 10-) ". "

.... \lJ'

···m

e

'.5~[ --=----:-':----:':-' -_\~/--:,-' ----'-' -~. '8() -60 -40 -20 0 20 4() 60 l. em

Fig. 5. Frequency sounding curves and characteristic curves over a conducting lens. a) Array oriented perpendicular to profile; b) array oriented along profile; c) curves of characteristic points over conducting lens . 1) Experimental frequency sounding curves for various distances of the array from the center: (I) l = 55 em, (II) l = 15 cm, (III) l = 0, (IV) l = 10 cm. 2) Curves for PW,min; 3) curve forv'Tmin; 4) curve for P W ,ac for f = 150 kc.

The effect of dipping conductive beds is observed also for soundings over a monocline. The distortions observed near the outcrop and far from the outcrop are nearly the same.

213

Obviously, not only dip of conducting beds but also a marked change in thickness of such beds will affect the transient curves. In this respect consider the graph for P w ,min obtained ?ver a cupula. As one of the dipoles crosses the crest of such a cupula, the minimum Pw,min IS accentuated.

Transient curves of the second type are found near regions of rapid change in the direction of dip of the electrical basement (Figs. 3, 4). The invariant conditions for distortion of the transient curve is the location of the electrodes for one dipole (or for both dipoles) over


different limbs of the structure with a large distance of the dipole center at spacings l :::::: AB/2 (or MN/2) from the boundary with the sharply varying section. In this respect, if the array is moved along a profile across the width of the structure, no distortion is noted. On the other hand, if the axis of the array is parallel to the long dimension of the structure, then close to the crest of the anticline or the upper lip of a step, a distortion is noted to the transient curve which is characterized by the fact that the ratio P w ,max:! P w ,min is larger than the corresponding ratio for a theoretical curve for a horizontal-layered medium with r/h = r/hi , where hi is the depth to the crest of the anticline or the upper limb of a step (Fig. 4). The curve is commonly stretched out and cannot be matched with any of the theoretical curves in the catalog.

In the same case when the axis of the array is parallel to the strike, the curve is very ragged: sharp extremums are associated with the top and bottom of the step. Usually, a significant maximum (40-45%) is formed on the curve for Pw,max/Pw,min over the top of the step.

Distortions of the third type were obtained for soundings made over a model of a conducting lens of graphite. This model represented an inclusion with a resistivity markedly different than the resistivity of the surrounding medium (p 1 = 0.05 Q -m; P 2 = 10-4 Q-m).

Frequency sounding curves along two profiles centered over the lens are shown in Fig. 5. In one case, the sounding axis is directed along the profile, and in the second, the sounding axis is perpendicular to the profile but with the electrodes symmetrically placed with respect to the center of the lens. Thi s marked change in resistivity of the section with no accompanying change in thickness causes a highly diagnostic distortion. Close to the lens, the values P w,ac and Pw,min decrease sharply whileVTmin increases: commonly the curves assume a lowfrequency linear asymptote with P w,ac < Pi. The distortion is especially strong when one of the dipoles is situated close to the center of the lens. On such curves, the interference minimum on the low-frequency portion of the Pw curve is absent or weak, with a general tendency for values of Pw to be depressed at lower frequencies. The left-hand (high frequency) portions of these curves are less distorted, and fall close to one another. All of the curves close into the left-hand asymptote with the value of the resisti vity for the first layer, P w = Pi = 0.05 Q-m.

In conclusion, we must take note of the characteristics of interference minimums on twolayer frequency sounding curves over horizontally inhomogeneous media.

Curves for the longitudinal conductance S and corresponding curves for the product V T min . P w,min over an inclined step are compared on Fig. 4. It is apparent that the values for Sand "T min· P w ,min vary in the same manner, differing only by a multiplying factor of 105; the maximum difference between S andVTmin . PW,min .105 is no more than 25-30%. This relationship is of interest in the practical analysis of field data, and, obviously, in some cases may be used for simplified frequency profiling with a narrow band of frequencies around the minimum, providing as a product the total value for S from the coordinates of the minimum point on the frequency sounding curve.

Thus, the data which have been presented allow examination of the nature of distortions to frequency sounding curves for a number of simple horizontally inhomogeneous sections.

Further experimental work must be done with soundings over more complex, multiplelayer sequences in media with nonhorizontal boundaries.

REFERENCES

1. G. P. Alekseev"and V. M. Davidov, "Results of the Application of Electromagnetic Sounding Methods in the Kuibyshev Basin" in: New Methods of Electrical Prospecting for Oil and Gas. Gostoptekhizdat (1963).


2. L. L. Vanyan, "Some Questions on the Theory of Frequency Sounding over Horizontal Layers ," Prikl. Geofiz. No. 23 (1959).

215

3. A. N. Kuznetsov, "Technique for Modeling Frequency Electromagnetic Soundings ," Razvedochnaya i Promislovaya Geofizika, No. 51 (1964).

4. B. M. Melamed, "Description of Work with the Electromagnetic Sounding Method in Northwest Bashkiria," in: New Methods of Electrical Prospecting for Oil and Gas. Gostoptekhizdat (1963).

FOUR-LAYER MASTER CURVES FOR FREQUENCY ELECTROMAGNETIC SOUNDING*

L. L. Vanyan, G. M. Morozova, V. L. Loshenitzina E. I. Terekhin, and A. I. Shtimmer

Thcl theoretical curves presented here for use with the variable-frequency electromagnetic sounding method for a four-layer sequence were compiled by L. L. Vanyan and G. M. Morozova at the Computer Center of the Siberian Department of the Academy of Sciences of the USSR, and prepared for publication by E. I. Terekhin and A. I. Shtimmer at the All- Union Research Institute for Applied Petroleum Geophysics.

The theoretical curves are given as the relationship between the frequency f and the complex value for apparent resistivity, defined as

2"r' aB Pw = -3-r-L-s-in-e -a-t - ,

where IL is the current dipole moment, r is the spacing, e is the angle between the vectors IL and r, and B is the vertical component of magnetic induction.

The ratio of the wavelength in the first layer, A, to the thickness of the first layer, hi' is plotted along the horizontal scales on the curves to a logarithmic base with a modulus of 46.9 mm:

A,/h, = (la' p,)

(fh~) •

Curves for the magnitude of complex resistivity are plotted to the left on each sheet, with a vertical scale to a logarithmic base with a modulus of 46.9 mm, while curves for the phase angle are plotted to the right on each sheet to an arithmetic scale.

The vertical reference line passes through the values A l/h! = 8 on the magnitude curves, and through the values A/hi =4 on the phase curves.

A line denoting S, the total longitudinal conductances of the three top layers, is shown on the magnitude curves.

The curve indices are as follows: the first two figures denotes values for P21 PI and h2/hl' respectively; the next two figures denotes values for P31 PI and h3/hl' respectively; and

* Published originally by the Academy of Sciences of the USSR, Siberian Department, Institute of Geology and Geophysics, and the National Geological Committee, USSR, All- Union Petroleum Research Institute, Moscow, 1964.

217

218 FOUR-LAYER MASTER CURVES FOR FREQUENCY ELECTROMAGNETIC SOUNDING

the last character indicates that the ratio pi Pi is infinite. The figure with each curve in a set is the value for rP2/hiPi.

Parameters for Sets of Master Curves

Set First-layer Second-layer I Third -la yer

number p h P I h I p I h

1 1 1 (Single-layer curves for parallel electric dipoles)

2 1 1 (Single-layer curves for one electric dipole, one magnetic dipole)

(All remaining curves sets are for one electric dipole, one magnetic dipole)

3 1 1 1/8 1/2 1/16 1/2 4 1 1 1/8 1/2 1/16 2 5 1 1 1/8 1/2 1/16 8 6 1 1 1/8 1/2 1/4 1/2 7 1 1 1/8 1/2 1/4 2 8 1 1 1/8 1/2 1/4 8 9 1 1 1/8 1/2 1 1/2

10 1 1 1/8 1/2 1 2 11 1 1 1/8 1/2 1 8 12 1 1/8 1/2 4 1/2 13 1 1 1/8 1/2 4 2 14 1 1 1/8 1/2 4 8 15 1 1 1/8 2 1/16 1/2 16 1 1 1/8 2 1/16 2 17 1 1 1/8 2 1/4 1/2 18 1 1 1/8 2 1/4 2 19 1 1 1/8 2 1 1/2 20 1 1 1/8 2 1 2 21 1 1 1/8 2 4 1/2 22 1 1 1/8 2 4 2 23 1 1 1/8 8 1/16 1/2 24 1 1 1/8 8 1/4 1/2 25 1 1/8 8 1 1/2 26 1 1 1/8 8 4 1/2 27 1 1 1/2 1/2 1/16 1/2 28 1 1 1/2 1/2 1/16 2 29 1 1 1/2 1/2 1/16 8 30 1 1 1/2 1/2 1/4 1/2 31 1 1 1/2 1/2 1/4 2 32 1 1 1/2 1/2 1/4 8 33 1 1 1/2 1/2 1 1/2 34 1 1 1/2 1/2 1 2 35 1 1 1/2 1/2 1 8 36 1 1 1/2 1/2 4 1/2 37 1 1 1/2 1/2 4 2 38 1 1 1/2 1/2 4 8 39 1 1 1/2 2 1/16 1/2 40 1 1 1/2 2 1/16 2 41 1 1 1/2 2 1/4 1/2 42 1 1 1/2 2 1/4 2 43 1 1 1/2 2 1 1/2 44 1 1 1/2 2 1 2 45 1 1 1/2 2 4 1/2 46 1 1 1/2 2 4 2 47 1 1 1/2 8 1/16 1/2

FOUR-LAYER MASTER CURVES FOR FREQUENCY ELECTROMAGNETIC SOUNDING 219

Parameters for Sets of Master Curves

Set First -la yer Second -layer Third-layer

number p h P h P h

48 1 1 1/2 8 1/4 1/2 49 1 1 1/2 8 1 1/2 50 1 1 1/2 8 4 1/2 51 1 1 2 1/2 1/16 1/2 52 1 1 2 1/2 1/16 2 53 1 1 2 1/2 1/16 8 54 1 1 2 1/2 1/4 1/2 55 1 1 2 1/2 1/4 2 56 1 1 2 1/2 1/4 8 57 1 1 2 1/2 1 1/2 58 1 1 2 1/2 1 2 59 1 1 2 1/2 1 8 60 1 1 2 1/2 4 1/2 61 1 1 2 1/2 4 2 62 1 1 2 2 1/16 1/2 63 1 1 2 2 1/16 2 64 1 1 2 2 1/4 1/2 65 1 1 2 2 1/4 2 66 1 1 2 2 1 1/2 67 1 1 2 2 1 2 68 1 1 2 2 4 1/2 69 1 1 2 2 4 2 70 1 1 2 8 1/16 1/2 71 1 1 2 8 1/4 1/2 72 1 1 2 8 1 1/2 73 1 1 00 1/2 1/16 1/2 74 1 1 00 1/2 1/16 2 75 1 1 00 1/2 1/16 8 76 1 1 00 1/2 1/4 1/2 77 1 1 00 1/2 1/4 2 78 1 1 00 1/2 1/4 8 79 1 1 00 1/2 1 1/2 80 1 1 00 1/2 1 2 81 1 1 00 1/2 1 8 82 1 1 00 1/2 4 1/2 83 1 1 00 1/2 4 2 84 1 1 00 1/2 4 8 85 1 1 00 2 1/16 1/2 86 1 1 00 2 1/16 2 87 1 1 00 2 1/4 1/2 88 1 1 00 2 1/4 2 89 1 1 00 2 1 1/2 90 1 1 00 2 1 2 91 1 1 00 2 4 1/2 92 1 1 00 2 4 2 93 1 1 00 8 1/4 1/2 94 1 1 00 8 1 1/2 95 1 1 00 8 4 1/2


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Electromagnetic Depth Soundings

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