Three-dimensional TEM modelling of near-surface …...Three-dimensional TEM modelling of...

Three-dimensional TEM modelling of near-surface resistivity variations

Mads Wendelboe Toft

Department of Earth Science University of Aarhus, Denmark

November 2001

Three-dimensional TEM modelling of near-surface resistivity variations Mads Wendelboe Toft, Department of Earth Science, University of Aarhus, Denmark.

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Contact info:

Mads Wendelboe Toft

5/102 Spit Road

Mosman NSW 2088

Sydney, Australia

Tel.: +61 2 9969 2789

[email protected]

URL: http://www.madstoft.com


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Contents

1 Acknowledgements ................................................................................................5

2 Introduction .............................................................................................................7 2.1 This work and its relevance..................................................................................7 2.2 This thesis ............................................................................................................8

3 The Transient Electromagnetic Method................................................................11 3.1 Electromagnetic methods in general ....................................................................11 3.2 The TEM method..................................................................................................12 3.2.1 The principle behind the method ....................................................................................12 3.2.2 Sources of errors ............................................................................................................13

4 EM Theory................................................................................................................15 4.1 The Maxwell Equations and the constitutive relations..........................................15 4.2 The Wave Equations ............................................................................................17 4.3 Boundary conditions.............................................................................................19 4.4 Solution to the wave equations ............................................................................20

5 The TEMDDD modelling code ................................................................................25 5.1 Approaches to numerical modelling of EM fields. ................................................25 5.2 TEMDDD ..............................................................................................................26 5.3 Model discretisation..............................................................................................27 5.4 Executing the program .........................................................................................28

6 Verifying the TEMDDD modelling code................................................................31 6.1 Earlier work with the TEMDDD modelling code ...................................................31 6.2 Things to consider before starting ........................................................................32 6.3 The reference codes ............................................................................................33 6.4 Verifying the code over a homogeneous half space ............................................34 6.4.1 The models .....................................................................................................................34 6.4.2 Estimating the total depth of the model space. ..............................................................35 6.4.3 Estimating the total width of the model space. ...............................................................37 6.4.4 Estimating the required number of matrix operations.....................................................39 6.4.5 The optimised final model parameters for the homogeneous half-space ......................41 6.5 Verifying the code over a layered model ..............................................................43 6.5.1 The models .....................................................................................................................43 6.5.2 Optimising the upper vertical nodal distribution for shallow resistivity contrasts............44 6.6 Verifying the code over a 3D model .....................................................................47 6.6.1 The models .....................................................................................................................48 6.6.2 Comparison of TEMDDD responses with other multi-dimensional modelling codes.....49 6.7 Summary of the achieved results .........................................................................53

7 The Modelling implementation ..............................................................................55 7.1 TEMDDDModelCreator ........................................................................................55 7.1.1 Introduction .....................................................................................................................55 7.1.2 Defining the model in TEMDDDModelCreator ...............................................................55 7.1.3 Projecting the TEMDDDModelCreator onto the TEMDDD model space .......................56 7.1.4 Applying the correctional end-inhomogeneities..............................................................57 7.1.5 Distributing small shallow random resistivity variations..................................................59 7.2 EM1DINV interpretation and visualisation the data..............................................61 7.2.1 Reformatting the TEMDDD output-files ..........................................................................61


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7.2.2 The EM1DINV inverse modelling code .......................................................................... 61 7.2.3 The multiple-layer inversion model ................................................................................ 63

8 The steepness of geological structures............................................................... 65 8.1 The Models .......................................................................................................... 65 8.2 Sloping structure with a conductive base............................................................. 66 8.2.1 90-degree slope: The one-dimensional regions of the profile........................................ 67 8.2.2 90-degree slope: 3D effects observed before the edge of the slope............................. 67 8.2.3 90-degree slope: 3D effects observed after the edge of the slope................................ 69 8.2.4 90-degree slope: Features of the residuals ................................................................... 71 8.2.5 Decreasing the steepness of the sloping structure........................................................ 72 8.3 Sloping structure with a resistive base................................................................. 74 8.3.1 90-degree slope: 3D effects observed before the edge of the slope............................. 74 8.3.2 90-degree slope: 3D effects observed after the edge of the slope................................ 77 8.3.3 90-degree slope: Features of the residuals ................................................................... 77 8.3.4 Decreasing the steepness of the sloping structure........................................................ 78

9 The influence of a small near-surface resistivity variation on early-time data. 79 9.1 Background.......................................................................................................... 79 9.2 The Models .......................................................................................................... 80 9.3 The general behaviour of TEM fields for shallow resistivity variations................. 81 9.3.1 100 ohmm resistivity variation, central loop configuration. ............................................ 85 9.3.2 100-ohmm resistivity variation, offset configuration ....................................................... 86 9.3.3 25-ohmm resistivity variation, central loop configuration ............................................... 91 9.3.4 25-ohmm resistivity variation, offset configuration ......................................................... 92 9.3.5 Summary ........................................................................................................................ 95 9.4 Using total-field data for the offset configuration ................................................. 95 9.4.1 introduction..................................................................................................................... 95 9.4.2 EA-field inversions for two shallow resistivity variations ................................................ 96

10 The influence of a shallow, random resistivity distribution ............................... 101 10.1 The model ............................................................................................................ 101 10.2 Inversion-results of three different configurations................................................ 103 10.2.1 Central loop, z-field inversion......................................................................................... 103 10.2.2 55 meter offset, z-field inversion .................................................................................... 104 10.2.3 55 meter offset, EA-field inversion ................................................................................. 105 10.2.4 Comparison of the three inversions ............................................................................... 106

11 Conclusion .............................................................................................................. 109

12 References .............................................................................................................. 111


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1 Acknowledgements

I wish to thank my two advisers Esben Auken and Kurt Sørensen for interesting and

motivating guidance during my time as master student.

The use of the three-dimensional TEM modelling code TEMDDD was made

possible due to the kindness of the developer of the code, Knútur Árnason of the

National Energy Authority of Iceland.

Thanks to Thorkild Maack Rasmussen of the Geological Survey of Denmark and

Greenland (GEUS) for supplying Arjuna 2D reference responses.

Thanks to thank Egon Nørmark and Anders Vest Christiansen for putting their

powerful computers at my disposal during the summer holidays of July 2001. A special

thank to Jens Ensted and Jesper Heidemann Langhoff for reviewing this thesis and

contributing with constructive criticism. Also thanks to Niels B. Christensen for his

helpfulness and always open mind.

Finally, I wish to thank my lovely wife, Alisha for her mildness and valuable help

during these last months of hectic work and writing.

Mads Wendelboe Toft


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2 Introduction The following thesis is written at the end of a long period of work at the Laboratory

of Geophysics under the Department of Earth Science at the University of Aarhus.

The work began with the aim of investigating how multidimensional structures in

Danish geological settings affect one-dimensional inversion of transient

electromagnetic (TEM) soundings. The optimal test site would consist of a geological

setting with a near-vertical boundary and a relatively high resistivity contrast. The

challenge was to identify a site where it was possible to record a continuous

undisturbed profile over a distance of about 1000 meters. Thus, surface resistivity

maps for large areas in Eastern Jutland were studied to try to identify possible field

sites. The main problem in finding a near perfect field site was mainly the presence of

human made electrical conductors such as telephone wires, farm houses, fencing for

cattle and underground power cables, which are a major source of errors in TEM

investigations.

Several test profiles were recorded using different source-receiver configurations

and a spatial sampling as dense as 20 meters between the individual soundings.

Though, after some investigations it soon became evident that explaining the observed

effects would require a totally controlled environment. Furthermore, the chosen field

site still had power cables and roads intersecting the profile which made it impossible

to obtain a continuous profile. It was therefore decided to take a modelling approach to

the problem.

2.1 This work and its relevance Through many years using of the Time Domain Method, it has become evident that

there are a number of questions that needs to be answered to fully understand its

limitations and strength. Over the last ten years the method has been used

commercially in Denmark, mainly in the prospecting of groundwater aquifers. During

this period a lot of high quality data has been collected and interpreted with good

results. However, there are projects from areas in which interpretation of the collected

data is impossible when assuming a simple one-dimensional earth. Soundings in these

areas are not disturbed by man made conductors, such as fencing for cattle, and


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geological mapping in the areas present (Gravesen, 1997) suggests that 3D features

could be present.

The question of when a model essentially can be regarded as one-dimensional

instead of two- of three-dimensional is of special importance in most of environmental-,

engineering- and hydrogeophysical methods. The physical understanding of the

electromagnetic fields has been known for decades, but it is only in recent years that

computer power and memory requirements have been able to comply with the

complexity of the problem and allow direct modelling of them.

2.2 This thesis After a brief introduction to this thesis, an introduction of the transient

electromagnetic method is given in chapter two. This is followed by an outline of the

electromagnetic theory behind the method. The basic Maxwell’s equations are

presented, and wave equations and their solutions are derived and the transiente

response is discussed.

In chapter five the three-dimensional modelling code TEMDDD is introduced with a

short description of the input parameters. The one- and multi-dimensional. Chapter six

documents the resource-demanding task of verifying the TEMDDD modelling code

against the 1D inversion code EM1DINV and 2D/3D codes EM3D and ARJUNA.

Various parameters are tested for a range of resistivities and the complexity of the

models is stepwise increased. Finally, responses from simple three-dimensional

models are calculated with TEMDDD and compared to responses from two alternative

modelling codes. Chapter seven describes the modelling implementation. The main

features of the user interface are described and the simple calculations involved in

simplifying the input parameters to the TEMDDD code are described. Furthermore, the

one-dimensional inversion code EM1DINV used in the interpretation of the TEMDDD

responses is presented.

In chapter eight the first real use of the modelling code is carried out. A series of

profiles containing models with varying steepness and resistivity distribution is

calculated. The resulting responses are explained with respect to the physics behind

the method and the question of when a model essential can be regarded as one-


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dimensional is discussed. This is followed by an investigation of the influence of a

small, shallow 3D resistivity variation in chapter nine. The chapter focuses on

explaining the responses for three different transmitter-receiver configurations in detail.

In chapter ten, the complexity of the model is increased. A whole layer of cubes with a

random resistivity variation replaces the single shallow resistivity variation. Again, the

observed effects for the three transmitter-receiver configurations are described, and the

pros and cons for each configuration are discussed. Chapter eleven gives short

summary of the archived results.


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3 The Transient Electromagnetic Method In the following chapter a short introduction to the transient electromagnetic method

(TEM) is given. The basic principles behind the method are explained and sources of

errors are described.

3.1 Electromagnetic methods in general One must differ between two main principles when working with electromagnetic

methods, the frequency domain and the time domain approach. Common for both of

them is that they are controlled by the basic physics of the Maxwell Equations. In fact,

one of them is simply the Fouier transform of the other.

In the frequency domain methods, a harmonic magnetic signal containing one single

frequency (the primary magnetic field) is transmitted and the relative amplitude of the

earth response (the secondary magnetic field) is measured. Furthermore, the phase

shift is recorded. The main drawback of the method is that the measured response field

does not only contain the earth response, but also the primary transmitted field. The

primary field is of much larger amplitude than the induced secondary field, and the

separation of these two magnetic field components requires precise knowledge of the

primary field. The one-frequency approach does allow a very efficient noise reduction

by using filters.

The time domain method is different from the frequency domain methods in two

ways. Firstly, it is decaying secondary field as a function of time that is measured, and

secondly the primary field is not present during the time measured. The primary field is

thus not registered and there are no problems with separating the two field

components. Because the secondary signal contains a wide range of frequencies the

undesired noise cannot be filtered out. To obtain a high signal to noise ratio one must

repeat the measurement several times to stack the data. In this way the stochastic

electromagnetic noise level is reduced.


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3.2 The TEM method The transient electromagnetic method was originally developed in the seventies for

mining exploration with the aim of identifying conducting ore deposits. In recent years,

it has shown to be very useful in identifying major aquifers in Danish fluvial deposits

and large areas have been mapped by the method.

Since the early pioneer work with the method in Denmark was carried out in the

early nineties a lot of important discoveries and developments have been made in the

field procedures, data acquisition and the data interpretation. Among recent important

developments are the inclusion of the band limitations in the data inversion (Effersø et

al, 1999), the development of a continuous TEM method (Sørensen et al, 1995) and

most recently the joint inversion of high-moment TEM central-loop and offset data as

proposed by Krivochieva et al (2001).

3.2.1 The principle behind the method

The TEM method is generally based on the simple physical fact that when a current

in a coil is changed, a magnetic field is induced, and vice versa. Applying a current to a

large coil creates a magnetic field, which is stable after some time. This is seen in the

‘Time-on’ period in the upper plot in figure 3.1a.

Figure 3.1: a) The transmitted signal and the secondary response. b) The flow of TEM eddy current below the transmitter at early and late times. (McNeill, 1990).

After the field has stabilised the current is turned off very rapidly (thus the term

transient EM) and this induces a primary magnetic field, which is proportional to the

effective area of the loop and the current (Figure 3.1a, middle). The current can be

turned off in several ways, and for theoretical studies the turn-off ramp can even be a

a) b)


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true ‘step-off’ response. Just after the current is turned off, eddy currents are instantly

generated near the transmitting wire, maintaining the magnetic field. With the passage

of time these horizontal currents diffuse down- and outward. As a result of the finite

conductivity of the ground these currents decay with time (Figure 3.1b, lower). The

decay of the currents induces a secondary field (Figure 3.1a, bottom), which is only

dependent of the electric conductivity of the ground, and that can be measured by a

induction receiver coil (McNeill, 1990).

The general TEM system used in environmental studies consists of a transmitter-

and a receiver-unit. The transmitter is usually a large square loop made of standard

flexible cable with a side length between 20 and 100 meters. A current up to 100

ampere can be applied to the transmitter loop. For most investigations in Denmark the

current is normally between one and three ampere in a 40 by 40 meter single-turn loop

transmitter. The receiver is usually a rigid multi-turn coil that is easily handled by one

person in the field. The coil can be mounted with a small battery powered preamplifier

and it has a diameter between a half and one meter and an effective area of around 40

m2. Alternatively, the transmitter coil can also be used as receiver coil, though this is

not customary in Danish hydrogeophysical investigations.

The receiver can be placed either inside the transmitter (central-loop configuration)

or outside the transmitter (offset configuration) and all three components of the field

can be used to gain information about the conductivity of the underground. For ground

TEM systems, usually only the vertical component of the secondary field is measured

in a central loop configuration, but in recent years steps have been taken to use other

configurations. For airborne systems a offset configuration is used, and all three

components of the field is recorded.

3.2.2 Sources of errors

The origins of errors in the TEM method are very diverse and one must differ

between a couple of categories, which include instrumental-, geometrical-, geological-,

electromagnetic- and cultural effects.

Instrumental errors include drift in the in the calibration of the instrument, which

typically occur if the temperature of the electronics in the receiver unit changes.


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Another error is the inability of the amplifiers to handle the large dynamic range that is

typical for the TEM method. Further more, exact knowledge of the transmitted

waveform is important. Together with instrumental errors, geometrical errors are

serious errors, which can influence the outcome of a sounding. This includes all

divergences from the assumed geometric layout of the transmitter and receiver, i.e.

loop shape, source-receiver spacing, and orientation of the transmitter and receiver.

Especially for offset configurations these errors can make it impossible to interpret the

data.

A source of error that is of special interest in this thesis is the geological noise. This

category includes a wide range of earth characteristics, which might influence the

outcome of a TEM sounding using a 1-D interpretation model under certain

approximations. For the present thesis, an investigation of the effect of near-surface

inhomogeneities on different transmitter-receiver configurations will be presented later.

Coherent electromagnetic noise is a factor that limits the depth of investigation for

the TEM method, and by synchronous averaging in time domain this depth can be

increased. The noise can be defined as all unwanted electrical or magnetic signals that

are detected by the receiver sensor and include geomagnetic signals and power-line

radiation. The unwanted low frequency (< 1 Hz) geomagnetic signals are generated by

variations in the Earth’s magnetic field due to variations in the solar wind. The high-

frequency signals originate from atmospheric lightening, also referred to as ‘spherics’.

Man-made noise arise mainly from 50/60 Hz power lines, but also from motor pumps,

Navy VFL communication transmissions and radio/radar stations. Another man-made

source of error is cultural effects arising from induced currents in metallic conductors

such as fences, power lines and buried pipes, which produce anomalous responses

(Spies et al, 1991).


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4 EM Theory The following chapter gives a short introduction to the electromagnetic theory

behind the TEM method. Where possible, the presentation of the relevant equations is

kept to the time domain, although some arguments require equations stated in

frequency domain.

4.1 The Maxwell Equations and the constitutive relations The foundation of all electromagnetic phenomena is the Maxwell Equations, which

are all first-order linear differential equations. They are uncoupled, but can be coupled

by empirical constitutive relations, which reduce the number of functions.

The Maxwell Equations are empirical equations based on experiments of for

example Faraday and Ampere (Ward et al, 1988). All electromagnetic phenomena

obey these equations described in time domain as

0t

=∂∂

+be×∇ (4.1)

jh =∂∂

−td

×∇ (4.2)

0=⋅b∇ (4.3)

ñ=⋅d∇ (4.4)

in which e is the electric field intensity [V/m],

b is the magnetic induction [Wb/m2 or Tesla],

h is the magnetic field intensity [A/m],

j is the electric current density [A/m2],

d is the dielectric displacement current [C/m2] and

ρ is the electric charge density [C/m3].


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The first of the Maxwell’s Equations is Faraday’s Law, which states that a magnetic

field that changes in time generates an electric field that is perpendicular to the

magnetic field. The second equation is Ampere’s Law, which states that magnetic fields

occur perpendicular to the electrical current and that the total current is the sum of the

conductive current and the displacement current. The last two equations are both the

laws of Gauss describing the magnetic induction and electric displacment. The first of

the two states that the divergence of the magnetic field is zero, which ultimately means

that no magnetic mono-poles exist. The last equation states that the divergence of the

displacement current is equal to electric charge density. The Maxwell’s Equations,

stated above, are coupled through the constitutive relations in the frequency-domain.

They are

ErEεD~

⋅ω= ,...)P,T,t,,,( (4.5)

HrHµB~

⋅ω= ,...)P,T,t,,,( (4.6)

ErEσJ~

⋅ω= ,...)P,T,t,,,( (4.7)

in which ε, µ and σ are the tensors describing, respectively, the dielectric permittivity,

the magnetic permittivity, and the electric conductivity as a function of angular

frequency ω, electric field strength E, position r, time t, temperature T, and pressure P.

(Ward et al, 1988) .

To simplify analysis, a number of assumptions, which apply to most electromagnetic

earth problems, are made:

• All media are linear, isotropic, homogeneous, and possess electrical

properties, which are independent of time, temperature or pressure.

• The magnetic permeability µ is assumed that of free space.

Despite that, anisotropic media are sometimes considered to aid in interpretation of

data as well as temperature and pressure is taken into account in deep crustal surveys.

Furthermore, the time dependence of electric conductivity due to changes in soil

moisture in shallow geotechnical surveys cannot always be neglected.


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In the following, the simplification of the three equations above in time domain will

suffice. The relations are restricted to non-dispersive cases where µ, ε and σ are

independent of time.

ed ε= (4.8)

hb µ= (4.9)

ej σ= (4.10)

4.2 The Wave Equations The wave equations describe the wave properties of electric and magnetic fields as

they travel trough a conducting media. Taking the curl of equation 4.1 and 4.2 and

applying the constitutive relations in time domain (equation 4.8, 4.9 and 4.10) yields

0=∂∂

µ+the ×∇×∇×∇ (4.11)

eeh ×∇×∇×∇×∇ σ=∂∂

ε−t

(4.12)

If the vector functions h and e are piecewise continuous and have continuous first and

second derivatives the operators ×∇ and t∂∂ may be interchanged. This applied to

the equations above yields:

( ) 0=∂∂

µ+ he ×∇×∇×∇t

(4.13)

( ) eeh ×∇×∇×∇×∇ σ=∂∂

ε−t

(4.14)

Substituting the expressions for h×∇ and e×∇ given by equation 4.1 and 4.2,

respectively, using the vector identity


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a-aa 2∇∇∇×∇×∇ ⋅≡ (4.15)

and remembering that 0=⋅e∇ and 0=⋅ h∇ , for homogeneous regions, equation 4.13

and 4.14 turn out to be

02

=∂∂

µσ−∂∂

µεtt 2

ee-e2∇ (4.16)

02

=∂∂

µσ−∂∂

µεtt 2

hh-h2∇ . (4.17)

These are the wave equations for the electric and magnetic fields in time domain.

Although the wave equations are fully valid, there are a number of assumptions that

can be made to further simplify the expressions. These assumptions are generally

referred to as the quasi-stationary approximation and are best considered in the

frequency domain version of the wave equations.

Fourier Transformation of the wave equations with respect to time yields

( ) 02 =µσω−µεω+ ΕΕ∇2 ι (4.18)

( ) 02 =µσω−µεω+ ΗΗ∇2 ι (4.19)

which are also known as the Helmholtz equation. The expression in the brackets

describes the differences between the displacement currents and the conducting

currents. For frequencies less than 100 kHz the conducting currents are much larger

than the displacement currents when real earth materials are considered. This mean

that µεω2 << µσω and equation 4.18 and 4.19 may be simplified to

0=µσω− ΕΕ∇2 i (4.20)


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0=µσω− ΗΗ∇2 i (4.21)

This argument is also valid for the Helmholtz equations in time domain, equation 4.16

and 4.17, which ultimately means that the term involving the dielectric permittivity ε can

be neglected.

Applying the Fourier transform with respect to frequency to equation 4.20 and 4.21,

and thereby returning to the time domain leads to

0=∂∂

µσ−tee2∇ (4.22)

0=∂∂

µσ−thh2∇ (4.23)

These equations are diffusion equations that apply to real earth materials and lead to

the lack of resolution in EM methods. (Ward et al, 1988)

4.3 Boundary conditions The boundary conditions will not be derived but simply stated with a short

description of each of them. For a complete derivation please refer to appendix 1.2 in

Ward and Hohmann (1988).

Normal B: The normal component of the magnetic induction B is continuous over an

interface between two media with different electrical properties.

0)( =⋅− nBB 21 (4.24)

Normal D: The normal component of the dielectric displacement D discontinuous over

an interface due to the accumulation of a surface charge density ρs.

sρ)( =⋅− nDD 21 (4.25)


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Tangential E: The tangential component of the electrical field E is continuous.

0)( =−× 21 EE n (4.26)

Tangential H: The tangential component of the magnetic field H is continuous assuming

that no surface currents exist. This is true for most non-metallic media.

0)( =−× 21 HH n (4.27)

Normal J: The normal component of the current density J is continuous over an

interface.

0)( =⋅− nJJ 21 (4.28)

4.4 Solution to the wave equations The wave equations are second order linear differential equations. Ward and

Hohmann (1988) consider two sets of basic solutions. The first set describing the

sinusoidal dependence while the second set describes the impulsive, electric and

magnetic fields in a full-space.

The sinusoidal time dependence eiωt at frequency ω for plane waves is described by

t)ω(kz

0t)ω(kz

0 e +−−−+ += ii eeee (4.29)

t)ω(kz

0t)ω(kz

0+−−−+ += ii ee hhh (4.30)

where k = α - iβ. In the quasi-static approximation, conduction currents dominate over

displacement currents thus making α and β identical quantities defined by

α=β=(ωµσ/2)1/2. +0e and +

0e are the amplitudes of the electric wave at t = 0 propagating

in the positive and negative z-direction, respectively. The same indices apply for the

magnetic wave. The solution of equation (4.29) and (4.30) for the decay in the positive

z-direction may then be written as


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tωzβα0

izi eee −−+= ee (4.31)

tωzβα

0izi eee −−+= hh (4.32)

The expression e-βz represents the attenuation of the electromagnetic wave, which

becomes evident because β is real. Thus, the expression gets smaller with increased

depth, z. The electromagnetic wave is reduced in amplitude by a factor of 1/e at a

distance described by the skin depth, δ :

21

21

fσ1503

ωµσ2δ

=

= (4.33)

As seen by the expression for the skin depth the attenuation depends on both the

frequency and the conductivity of the media and relatively high conductivity and

frequency result in a faster attenuation with respect to depth.

Another solution to the wave equations is derived by Fourier transformation of the

positive part of equation (4.29) and (4.30) assuming displacement currents are

neglected. It describes the behaviour of impulsive electric and magnetic fields at the

surface of a media (z=0):

( ) 4tzµσ

0

0 2

23

21

21

etπ2

zµσ −+

+

=

he

he

(4.34)

Figure 4.1a shows the calculated electric/magnetic field in a 100 Ωm whole space due

to a 1-D impulse in the field. As a function of time the field exhibits a relatively short

peak with a long tail.


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By setting the time derivative of equation (4.34) equal to zero one finds that the

maximum occurs at

6zt

2

maxµσ

= (4.35)

Similarly, setting the derivative of equation (4.34) with respect to z equal to zero can

derive the occurrence of the maximum in figure 4.1b.

21

maxt2z

µσ

= , (4.36)

where zmax is the penetration depth. Differentiating equation (4.36) with respect to time,

one gets an expression for the travel velocity of the maximum

21)t2(1V

µσ==

dtdzmax (4.37)

As seen from equation (4.36) the penetration depth is proportional to t1/2. This is similar

to the skin depth described by equation (4.33), which is proportional to 1/ω1/2.

0 50 100 150 200 250Distance [ m ]

0

1

2

3

4

5

6

7

8

9

Ampl

itude

0.00 0.04 0.08 0.120.02 0.06 0.10 0.14Time [ ms ]

0

1

2

3

4

5

6

7

8Am

plitu

de

t =µσz2/6 z =(2t/µσ)1/2

a) b)

Figure 4.1: a) Electric/magnetic field as a function of time 100 meters from a 1-Dimpulse in a 100 Ωm whole space. b) Electric/magnetic field at 0.03 ms as a functionof distance from an impulse (Ward et al, 1988).


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For measurements at the centre of a circular loop, an analytic expression can be

derived. Ward et al (1988) evaluates the vertical component at the surface of a

homogenous earth to

[ ]ka2232z e)akka33(3

akIH ii −−+−−= (4.38)

By inverse Laplace transformation of the frequency-domain expression in equation

4.38 divided by iω the step response of the vertical magnetic dipole in time-domain

becomes

θ

θ−+

πθ= θ− )a(erf

a231e

a3

a2Ih 22

az

22

(4.39)

in which erf is the error function and θ is

t4θ 0σµ

= .

The time derivative of equation 4.39 expresses the time-domain decay of the

magnetic field intensity in the centre of a circular loop. It is obtained by differentiating

equation 4.39 with respect to time (Ward et al, 1988)

θ+θ

π−θ

σ−=

∂∂ θ− 22

21

a223

0

z e)a23(a2)a(erf3au

It

h (4.40)

Equation 4.39 and 4.40 describes the vertical magnetic field intensity and the

derivative of the vertical magnetic field intensity, respectively. An example of the

behaviour of these two expressions is shown in figure 4.2. As seen by the figure both

the magnetic field intensity and its derivative maintain a positive sign over the entire

time interval the middle of the transmitter coil. However, when the observation point is

outside the transmitter the sign changes from negative to positive as is passes under

the receiver point. For late times the field intensity decay in the ground with a t-3/2–

dependency, while the time-derivative decay with a t-5/2–dependency.


____________________________________________________________________________

24

ap

an

in

m

Fvtchoo

igure 4.2: The behaviour of theertical magnetic field intensity inhe centre of a 50-meter radiusircular loop with on a 100 Ωmalf-space. At zero time a currentf one ampere is turned abruptlyff (Ward et al, 1988).

For late times θ becomes small in equation 4.39 and 4.40 and they can be

proximated by the following two equations:

2321

2230

23

z t30

aIh −

πµσ

≈ (4.41)

2521

2230

23z t

20aI

th −

πµσ

−≈∂

∂ (4.42)

In time-domain methods an induction-coil censor usually records the decaying field

d it is therefore appropriate to speak of magnetic induction instead of magnetic field

tensity. This is easily done as the relation between magnetic field intensity and

agnetic field induction is given by b=µ0h.


____________________________________________________________________________

25

5 The TEMDDD modelling code In the following chapter the emphasis is put on the introduction of the TEMDDD

modelling code. The different input parameters are explained and an example of an

input-file is given. Though this thesis is not a study of numerical modelling methods

itself, different approaches to 3D EM modelling are mentioned.

5.1 Approaches to numerical modelling of EM fields. Numerical solutions are archived by approximating the relevant differential or

integral equation and solving a large matrix equation. The solutions can be divided into

two major categories that can be solved in either frequency- or time-domain. The two

categories are differential- and integral-solutions, which are fundamentally different in

several ways.

In the integral equation approach, the Maxwell equations are solved by replacing

the anomalous conductivity distribution by an equivalent distribution of currents. The

solution is then carried out by integrating over the volume of scattered currents. Integral

equation solutions involve relatively advanced mathematics, but once implemented

they are usually faster than the differential-equation solutions. This is because the

unknown field only needs to be found in the anomalous regions of the model. This fact

makes this solution best suitable for modelling simple structures which can be used for

survey design and multidimensional interpretation catalogues. (Ward, 1988)

Differential equation solutions are the easiest to implement, but yield large matrices

that are relatively time consuming to solve. This is because the entire model-earth is

modelled on a grid for the relevant EM method. Therefore, this approach is preferred to

model relatively complex geology. Differential equation solutions can be subdivided in

to two groups: Finite element or finite difference. The first step in finite element analysis

is to divide the configuration into a number of heterogeneous elements. The elements

can be small where geometric details exist, and much larger elsewhere. The corners of

these elements are called nodes, and the goal of the finite-element method is to

determine the field quantities at these nodes. Because the elements can take almost

any shape this approach is very useful for modelling geological settings that typically

vary smoothly and contain some kind of sloping structure. The finite-element method


____________________________________________________________________________

26

can be formulated in both the time- and frequency-domain and the results are easily

converted using Fouier-transform.

The finite difference time domain

method, also referred to as the FDTD

method, is different from the finite-

element method since it uses a direct

solution of the Maxwell time

dependent curl equations. The model-

space consists of two staggered and

interleaved grids of discrete points.

One grid contains the points, at which

the electric field is evaluated, while

the other grid contains points where

the magnetic field is evaluated. A

conceptual sketch of this is shown in

figure 5.1.

The FDTD method is a time-stepping procedure. This means that the electric field

values at time t are used to find the magnetic field values at t+∆t. The electric and

magnetic fields are alternately calculated at each time step, and the fields are

propagated throughout the grid until a steady state solution is obtained. One of the

drawbacks of the method is that the basic elements of the grid are cubes. This demand

curved surfaces or slopes to be approximated by a staircase surface (Hubing, 1991).

5.2 TEMDDD TEMDDD is a three-dimensional finite difference time-domain (FDTD) code for

calculating the transient electromagnetic response due to at step-current excitation of a

conducting half-space. The code calculates forward responses over multi-dimensional

subsurface structures, which can be as complex as the discrete three-dimensional

nodal grid allows. It is an implementation of a staggered grid discretisation solution for

the electric field in the time domain. The solution is formulated in terms of vector

differential equations in time, which are solved by Lanczos Spectral Decomposition

(SLMD). This method is a specialised way of solving the differential equations.

Figure 5.1: The staggered grid of the finitedifference method (Hubing, 1991).


____________________________________________________________________________

Solutions are found by building orthogonal polynomials that approximate the solution to

the differential equations. For further description of the numerical solution of this

problem, please refer to Árnason (1995) and Druskin et al (1988).

The conducting half-space can have any resistivity structure defined by blocks of

constant and isotropic resistivity. They must have boundaries that coincide with the

planes of the rectangular discrete grid. The source can be either a grounded dipole or a

loop, which must be defined by the nodal points. A maximum of 50 receivers can be

placed at the centre of any nodal plane at the model surface. The transient response is

calculated as function of time in the current-off regime after a step current of 1 ampere

is turned off. The time derivative of all three components of the magnetic field (dBx/dt,

dBy/dt and dBz/dt) and the two horizontal components of the electric field (Ex and Ey)

can be calculated at all receiver positions (Árnason, 1999).

5.3 Model discretisation The model space is described in a right-handed coordinate system with the z-

direction pointing downwards (see figure 5.2). N nodal points in the x-direction, M nodal

points in the y-direction and L nodal points in the z-direction define the discrete grid.

The x- and y- grid dimensions are equal as it

saves substantial computation time. This means

that N is equal to M, and that the individual nodal

points have the same spacing in the two

dimensions. The nodal points are numbered from

0 to N-1 in the x- and y-direction and from 0 to L-

1 in the z-direction with the first point at the origin

(O) of the coordinate system.

z

y

x

O(0,0)

Figure 5.2: The orientation of thecoordinate system

27

The resistivity structure in the total model consists of two levels of resistivity

definitions. The first level is the layered host, which is defined only in one dimension,

i.e. the z-direction. In the two other directions the resistivity value is constant. The one

dimensional layered host is defined by L-1 values, each giving the value between two

nodal points in the z-direction. The second resistivity level defines the three-

dimensional resistivity structures in the layered host. The modelling code is designed

so that these resistivity structures overwrite the layered host wherever they are defined.


____________________________________________________________________________

28

Whenever a multi-dimensional body is defined in a position that partly or fully overlaps

a previously defined body, the latter of the resistivity definitions in the overlapping area

is valid. The start and end nodal number for each dimension defines the bodies as

rectangular boxes. This is followed by the resistivity for each box. A maximum of 1000

individual bodies can be defined.

5.4 Executing the program The program is executed by typing ‘temddd.exe’ in a MS-Dos Prompt. Once

executed, the program will prompt the date and time of the run and ask for an input file

containing the model parameters.

Mon Feb 12 11:46:23 2001

Input file:_

The program can also be called by typing

Temddd.exe Inputfile.dat > Inputfile.log,

which executes the TEMDDD using the input file named ‘Inputfile.dat’ and writes the

screen output to the file ‘Inputfile.log’. This commando line is especially useful when

embedded in a batch file.

In figure 5.3 an example of a simple input file is given. The first line contains the

number of nodes in the x and y direction, the number of nodes in the z direction. The

last integer in that line contains the number of matrix operations to be performed. This

is an important parameter because it defines how far the code will take the iterative

solving process, i.e. when the calculations will terminate and write the output. The

second and third line contains the distances between the individual nodes in the x/y-

and z-direction. To obtain the true nodal distance in meters, the individual distances

must be multiplied by the overall grid scale stated in line twelve. This scaling factor is

adjusted to stabilise the code and ensure successful termination. For the parameters

used in this thesis, scales between 10-4 and 10-2l for both the sensitivity- and grid-scale

have proven themselves useful. In line four the scaled resistivities that apply between

each nodal layer, defined in line three, are defined. They make up the layered

background host in which the multidimensional inhomogeneities are later defined.


____________________________________________________________________________

29

In line five the number of inhomogeneities are stated and in line six the first

inhomogeneity is stated by defining nodes in three dimensions followed by the desired

resistivity in that region. The first two integers define the nodal interval in the z-

direction, while the following two sets of integers define the nodal interval in the y- and

x-direction, respectively.

Line seven contains the node numbers that define the transmitter loop position at

z=0. The loop is first defined by at set of minimum nodal numbers for the x- and y-

direction, respectively. The two maximum nodal numbers follow this. In line eight the

number of receivers are stated and in the following lines their positions at z=0 on the x-

y mesh are defined by nodal numbers. After each receiver position the number of

components of the time-derivative of the decaying field is defined. If the value is 0

(zero) only the z-component is calculated, while a value of 1 (one) yields a calculation

of all three components of the field.

Line eleven contain the time interval within the fields are calculated and line twelve

and thirteen contain the relative grid- and resistivity-scale, respectively, as explained

before. The last two lines are the filenames, which contain the results of the

calculations in dB/dt and apparent resistivity, respectively.

#1 17,10,3000 #2 128000 64000 32000 16000 8000 4000 2000 2000 2000 2000 4000 8000 . . .128000 #3 500 1000 2000 4000 8000 16000 32000 64000 128000 #4 10000 10000 10000 10000 2500 2500 2500 2500 2500 #5 1 #6 1 4 6 10 6 10 5000 #7 7 7 9 9 #8 2 #9 8 8 1 #10 8 10 1 #11 5.0e-006,1.0e-002 #12 1.0e-002 #13 1.0e-002 #14 testrun.db #15 testrun.ohmm Figure 5.3: An example of a simple input file for TEMDDD. The line numbers on the leftare not included in an original input file.


____________________________________________________________________________

30


____________________________________________________________________________

31

6 Verifying the TEMDDD modelling code In the following chapter the individual steps in verifying the code is explained. First

the parameters are optimised over a homogenous half-space and a layered 1D model.

Finally, the code is compared to known responses from other 2D/3D modelling codes.

6.1 Earlier work with the TEMDDD modelling code The TEMDDD modelling code is not an official or commercial available code.

Therefore only little work involving this code has been published.

Kreutzmann and Árnason, (2000) presented three-dimensional TEM modelling of

Icelandic geothermal systems. In the paper a comparison between the magneto-telluric

and TEM method for maximum detection of geothermal systems was performed. The

calculations of the three-dimensional TEM responses were done using TEMDDD.

The geology in Iceland is different from Denmark. The modelling parameters will

subsequently be different for solving problems that are relevant to the Danish

geological environment. Kreutzmann and Árnason uses the code for investigations at

‘shallow’ depths for of less than 1000 meters. This is by far beyond the depth of interest

for Danish investigations. Whereas the target of interests can be relatively deeply

buried in the Icelandic case, targets in Denmark are usually in the upper 200 meters of

fluvial sediments. The resistivity contrasts are also on a different scale. Because of the

basaltic rocks systems that are common in Iceland, resistivities range from as low as

one ohmm to several thousands ohmm. These extreme resistivities can be found in

Denmark, but resistivities normally range from 5 to 200 ohmm in areas of groundwater-

or environmental interest.

Recently Krivochieva et al (2001) used the code in an investigation of simultaneous

inversion of data from central loop and offset TEM measurements. Even though the

investigated model is a single rectangular box in a three-layer half-space the

resistivities and target depths are more similar to what could can be in Denmark.

The imbedded box is 100 x 500 x 20 m in x-, y- and z-directions, respectively and

has a resistivity of 10 ohmm and the layered host has a 20m/30 ohmm top layer


____________________________________________________________________________

32

followed by a 80 m/100 ohmm layer and ends in a 10 ohmm base. This could very well

be a simplified model of a Danish Quaternary aquifer. The top layer being a till

overburden followed by a saturated sand body and finally ending in Tertiary clays. The

three-dimensional box could resemble a clay wedge relocated by the ice.

The fact that successful results from the use of the code has been published is

positive but it does not decrease the need for a thorough verification of the code over a

broad selection of models before its limitations and strengths are ‘fully’ understood.

6.2 Things to consider before starting Working with a 3D modelling code for the first time, there are a number of elements

that must be considered in the pre investigation. The main objective is to acquire

knowledge of what modelling parameters are needed for what type of model. What is

needed really means; “Does the model converge with known results from other

modelling codes”. The model parameters that are needed to make them converge are

functions of both the resistivity of the overall model space and the shape and resistivity

contrast of the model. As one cannot perform this investigation for every model that is

run, a number of simpler models are made. The optimised parameters from these

models are then adapted to more complex models.

One must consider the following questions before starting with the tests of the code:

• In what time range do I want the response calculated? • In what resistivity range do I want to be able to calculate responses? • What transmitter and receiver configurations are wanted? • How long calculation times am I willing to accept per response? • How much computer capacity is available? • What error am I willing to accept on the data?

Table 6.1 offers some answers to these questions, although it may be revised later in

the process.

Beginning with the simplest possible model, a homogeneous half-space, the next

step is optimising the input parameters for a layered half-space. Getting known one

dimensional model responses for comparison with TEMDDD responses is a simple


____________________________________________________________________________

case and there are a number of codes available for this purpose. Things get more

difficult when one wants a range of two- or three-dimensional responses.

T

c

t

t

p

m

w

b

d

6

d

i

e

c

a

s

s

i

p

d

E

Time: 5*10-6 – 10-2 seconds Resistivity range 10-100 ohmm Tx-Rx configurations: 40mx40m central loop, 40mx40m w/ 55m offset. Cpu and memory capacity 5 x 650MHz P III CPU with 2.5 Gb of RAM memory. Error tolerance 5%

33

able 6.1: The range of the parameters to be considered in testing the TEMDDD code.

Because of the complexity and the large amount of data generated, the following

hapter will not show the whole verifying procedure as it actually took place. Instead

he final optimised model parameters for the 3D modelling will be used, and one by one

he different parameters will be altered. This will show that it really is the right

arameters that finally are chosen. E.g. all parameters except the total width of the

odel space is fixed and this width is gradually narrow to show what the minimum

idth requirements are for that specific conductivity model. This systematic proof will

e repeated for the following modelling parameters: Total model width, total model

epth, vertical nodal density, and number of matrix operations performed.

.3 The reference codes Three different codes are used in the verification of the TEMDDD code. The one-

imensional forward responses are carried out using the commercially available 1D

nversion code EM1DINV (Effersø et al 1999; Foged 2001). The code has been used

xtensively in Denmark in recent years and is considered well debugged. With the

ode, it is possible to calculate the time derivative of all three B-field components for

ny transmitter-receiver configuration. The 1D code is described further in chapter

even.

Two different codes are used in the verification of TEMDDD over multi-dimensional

tructures. EM3D (Newman et al, 1986; Wannamaker et al, 1984), which is a 3D

ntegral forward modelling equation. The comparison data is taken from EM3D models

ublished in Auken (1995). ARJUNA (Raiche et al, 1999) is an 2.5D finite element code

eveloped for the AMIRA project in Australia. The program calculates the time-domain

M response of a general heterogeneous 2-D structure excited by a 3-D source. The


____________________________________________________________________________

34

6.4 Verifying the code over a homogeneous half space The simplest model to calculate the electromagnetic response over is a

homogeneous half-space. Several parameters are necessary to estimate:

• The required total width of the model. • The required total depth of the model. • The required number of matrix operations.

6.4.1 The models

A suite of models has been selected to represent the whole range of resistivities.

The models are homogeneous half-spaces with resistivities of 10 ohmm, 20 ohmm, 50

ohmm and 100 ohmm.

The manual for the code (Árnason, 1999) suggests that the xy-grid dimension

defined in the centre of the mode has a constant grid spacing. Outside this area the

grid spacing should be increasing form the centre and outwards according to a power

law. In the z-direction a power law downward increasing nodal spacing follows a

relatively thin layer in the top.

Following these instructions a first model was set up with the following model

parameters. The central constant grid spacing was set at 10m and the whole area of

constant grid spacing at the centre of the model was 100 meters wide. This was done

to be able to calculate responses for a great variety of transmitter-receiver

configurations. The grid density was set to 10 nodal points per decade outside the

constant grid spacing area in the x- and y-direction. In the z-direction the nodal points

in the first 100 meters are distributed so they’ll fit the later task of projecting the model

space onto the different resistivity models that investigated. The top layer is 0.5 meters,

after which the layer thickness is either constant or downward increasing. After the 100

meters, the distribution is 10 nodal points per decade.

In the presented dB/dt plots a suite of models are presented and compared with the

relevant EM1DINV references response. To avoid overcrowded plots the relative

differences have only been plotted for the model chosen for the further investigation.


____________________________________________________________________________

35

The basic parameters for this model have been highlighted in the parameter-box

accompanying each plot.

6.4.2 Estimating the total depth of the model space.

Estimating the total depth of the model space is of particular importance for the

relatively late time data. The fields travel out from the transmitter with the velocity

determined by the resistivity of the media. To give a first estimate of the required size

of the model space, the diffusion depth is calculated for each of the resistivities. These

are stated in table 6.2. As described earlier, the diffusion depth is the depth at which

the local electric field reaches its maximum value for a given time, and it can be

calculated by using equation 4.36 in the theory chapter. It’s important to note that these

results are only valid for the maximum of the field and are therefore an expression for

the absolute minimum depth of the model. Using

these simple calculations and expecting

significant differences between this result and

the Em1dInv reference response a number of

forward responses have been calculated.

Figure 6.1 shows the results from the calculated forward responses for a central

loop (upper plots) and a 55m offset configuration (lower plots). To isolate the effect of

making the total model depth smaller all other model parameters are chosen

conservatively, e.g. the number of matrix operations is set to be 4000 even though later

investigations might show that 2500 would be enough. The parameters for each model

are stated in the info box in the plot, and the optimal model is highlighted. The green

line in the plot is the relative difference between the optimal model and the EM1DINV

1D response.

Decreasing the total model depth affects only the relatively late times. The shallower

the model, the more the response differs from the reference response. The required

Resistivity Diffusion depth [Ohmm] [meters]

10 178 20 252 50 399

100 564 Table 6.2: The diffusion depth at 10 ms.


____________________________________________________________________________

36

Figure 6.1: Estimating the total depth required for four different resistivity models of 10, 20, 50 and 100 ohmm. Blue lines are the TEMDDD responses, while the reds are the EM1DINV reference responses. The green line is the relative difference between the highlighted response and the EM1DINV reference response (red line).

10-6

10-5

10-4

10-3

10-2

10-1

0.2

0.1

0 -0.1

-0.2Relative Difference

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

0.2

0.1

0 -0.1

-0.2Relative Differenc

10-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-6

10-5

10-4

10-3

10-2

10-1

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

10-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

12 3 45

1 2 3 46

5

12 3 45

12 3 46

5

2 3 4 57 6

2 3 4 576

12 3 46

51 3 46

5

2

Res

[Ωm

]: 10

Offs

et [m

]: 7

Wid

th [m

]: 62

00N

,M

: 53

Nt

:

4000

# D

epth

[m]

L

1

809

34

2

654

32

3

355

30

4

244

28

5

174

26

6

129

24

Res

[ Ωm

]: 10

0O

ffset

[m]:

7W

idth

[m]:

6200

N,M

: 5

3N

t

: 40

00#

Dep

th[m

] L

1

3

047

4

02

1

945

3

83

1

249

3

64

809

3

45

531

3

26

355

3

0

1

Res

[ Ωm

]: 10

0O

ffset

[m]:

55W

idth

[m]:

6200

N,M

: 5

3N

t

: 40

00#

Dep

th[m

] L

1

3

047

4

02

1

945

3

83

1

249

3

64

8

09

3

45

5

31

3

26

3

55

3

0

Res

[Ωm

]: 20

Offs

et [m

]: 7

Wid

th [m

]: 62

00N

,M

: 53

Nt

:

4000

# D

epth

[m]

L

1

809

34

2

654

32

3

355

30

4

244

28

5

174

26

Res

[ Ωm

]: 50

Offs

et [m

]: 7

Wid

th [m

]: 62

00N

,M

: 53

Nt

:

4000

# D

epth

[m]

L

1

304

7

40

2

194

5

38

3

124

9

36

4

80

9

34

5

53

1

32

6

35

5

30

7

24

4

28

Res

[ Ωm

]: 10

Offs

et [m

]: 55

Wid

th [m

]: 62

00N

,M

: 53

Nt

:

4000

# D

epth

[m]

L

1

809

34

2

654

32

3

355

30

4

244

28

5

174

26

6

129

24

Res

[ Ωm

]: 20

Offs

et [m

]: 55

Wid

th [m

]: 62

00N

,M

: 53

Nt

:

4000

# D

epth

[m]

L

1

809

34

2

654

32

3

355

30

4

244

28

5

174

26

Res

[Ωm

]: 50

Offs

et [m

]: 55

Wid

th [m

]: 62

00N

,M

: 53

Nt

:

4000

# D

epth

[m]

L

1

304

7

40

2

194

5

38

3

124

9

36

4

80

9

34

5

53

1

32

6

35

5

30

7

24

4

28

1


____________________________________________________________________________

37

model depths for the four resistivity models are chosen to be the shallowest that fits the

reference response within the 5% tolerance limit that was decided on earlier.

The four model depths are stated in table 6.3. As seen in the table, the need for a

deeper model increases with the increased resistivity. One exception is the 100 ohmm

half-space, which require a smaller model space. This could be because the 100 ohmm

model is not chosen as conservatively as one could wish for. The reason for this is that

the optimum conservatively model for the 100 ohmm half-space would require too

many model parameters and matrix operations. Furthermore, models with a relatively

large number of nodal points and matrix operations tend to behave unstable at a

certain point, and unexpected sign reversals are sometimes seen at late times.

One solution to this problem could be to

reduce the number of nodal points per decade

in the model space and thereby reduce the

need for a large number of matrix operations.

The reason for not doing this is that a number

of tests have shown that a nodal density of 10

points per decade is reasonable for the

modelling purpose pursued in this thesis. Even though the number of model

parameters in the 100-ohmm model probably is too small the difference from the

reference response is still acceptable. This could be caused by two different errors that

cancels each other out; The smaller model space causes the late time response to be

underestimated while a slight underestimation of the number of matrix operations have

been seen to cause a lift of the calculated response at late times.

6.4.3 Estimating the total width of the model space.

Parallel to the approach in the investigation into the required depth, the required

width of the model space has been estimated. In this investigation the same approach

has been taken to isolate the effect of changes in width of the model space. The same

four resistivity models as in the previous example are used, and all model parameters

except the model width and number of nodal points in the x- and y-direction, are fixed.

Resistivity Model depth L [Ohmm] [meters] [ # ]

10 809 34 20 809 34 50 3047 40

100 1249 36 Table 6.3: Required model depth and the corresponding number of nodes in the z-direction.


____________________________________________________________________________

38

10-6

10-5

10-4

10-3

10-2

10-1

0.2

0.1

0 -0.1

-0.2 Relative Difference

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

0.2

0.1

0 -0.1


10-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-6

10-5

10-4

10-3

10-2

10-1

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

10-6

10-5

10-4

10-3

10-2

10-1

1 23 4 5 67

1 2 3 4 5 67

12 34

5

1 2 34

5

1 23 4 5 67

1 23 4 5 6 7

12 34

5

6

1 2 3 456

Res

[ Ωm

]: 10

Offs

et [m

]: 7

Dep

th [m

]: 30

47L

: 40

Nt

:

4000

# W

idth

[m] N

,M1

2

510

452

2

010

433

1

610

414

1

040

375

840

356

680

337

550

31

Res

[Ωm

]: 20

Offs

et [m

]: 7

Dep

th [m

]: 30

47L

: 40

Nt

:

4000

# W

idth

[m] N

,M1

2

510

452

2

010

433

1

290

394

840

355

550

31

Res

[ Ωm

]: 50

Offs

et [m

]: 7

Dep

th [m

]: 30

47L

: 40

Nt

:

4000

# W

idth

[m] N

,M1

6

200

5

32

4

940

5

13

3

940

4

94

3

140

4

75

2

010

4

36

1

290

3

97

1

040

3

7

Res

[Ωm

]: 10

0O

ffset

[m]:

7D

epth

[m]:

3047

L

:

40N

t

: 45

00#

Wid

th[m

] N,M

1

778

0

55

2

620

0

53

3

394

0

49

4

314

0

47

5

251

0

45

6

161

0

41

Res

[Ωm

]: 10

0O

ffset

[m]:

55D

epth

[m]:

3047

L

:

40N

t

: 40

00#

Wid

th[m

] N,M

1

778

0

55

2

620

0

53

3

394

0

49

4

314

0

47

5

251

0

45

6

161

0

41

Res

[Ωm

]: 20

Offs

et [m

]: 55

Dep

th [m

]: 30

47L

: 40

Nt

:

4000

# W

idth

[m] N

,M1

2

510

452

2

010

433

1

290

394

840

355

550

31

Res

[Ωm

]: 10

Offs

et [m

]: 55

Dep

th [m

]: 30

47L

: 40

Nt

:

4000

# W

idth

[m] N

,M1

2

510

452

2

010

433

1

610

414

1

040

375

840

356

680

337

550

31

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

Res

[ Ωm

]: 50

Offs

et [m

]: 55

Dep

th [m

]: 30

47L

: 40

Nt

:

4000

# W

idth

[m] N

,M1

6

200

5

32

4

940

5

13

3

940

4

94

3

140

4

75

2

010

4

36

1

290

3

97

1

040

3

7

Figure 6.2: Estimating the total width required for four different resistivity models, 10, 20, 50 and 100 ohmm. Blue lines are the TEMDDD responses, while the reds are the EM1DINV reference responses. The green line is the relative difference between the highlighted response and the EM1DINV reference response (red line).


____________________________________________________________________________

39

The resulting forward response (Figure 6.2) looks much like what was observed for

the model depth. For all resistivity models there are a relatively small difference

between the calculated response and the reference response until 2 ms. As the total

model space is narrowed the differences gets larger at late times, and earlier times are

effected.

For the 100 ohmm there is a difference between how the central-loop and the offset

configuration reacts to the narrowing of the model. The 100 ohmm offset configuration

reacts as expected but the central loop responses exhibits unexpected behaviour: A

very wide model space (response #1), consisting of relatively many model parameters,

leads to an overestimation of the response at late times. Furthermore, and a very

narrow model creates an even higher response (response #5). The very wide model

(#1) could be a result of the combined effect of too many model parameters and too

few matrix operations, as described earlier.

Like in the previous depth investigation, the differences observed as the model

space is narrowed are restricted to the relatively late times after about 1ms. It is

interesting that it actually is possible to half the

width of the model space and still produce

usable responses until this time. The

narrowest model that fits the reference

response with the fewest model parameters

within the selected error tolerance of 5% is

selected for each resistivity model. The result

from this selection can be seen in table 6.4.

6.4.4 Estimating the required number of matrix operations.

To save valuable CPU-time, the number of matrix operations (Nt) performed for

each resistivity model is optimised. The objective is to find the model that fits the

reference response within the 5% tolerance limit with the least number of matrix

operations. Again all other model parameters are kept constant, and only the Nt value

is changed. The required number of matrix operations is closely linked to the overall

number of nodal points. Therefore it is the size of each optimised resistivity model that

is used in this investigation. To observe the effect, the number of matrix operations has

Resistivity Model width M,N [Ohmm] [meters] [ # ]

10 1610 41 20 2010 43 50 3047 47

100 6200 53 Table 6.4: Required model width for four different half-space resistivities. N,M is the corresponding number of nodes in the x,y-direction.


____________________________________________________________________________

40

10-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

0.2

0.1

0 -0.1


10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

0.2

0.1

0-0.1

-0.2 Relative Difference

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-6

10-5

10-4

10-3

10-2

10-1

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-6

10-5

10-4

10-3

10-2

10-1

1 23 45

6

3 45

6

1 2 3 45

6

12

1 2

3 45

6123 4

56

1 23 45

6

1 2 3

4

56

123

4

56

7 7

Res

[Ωm

]: 10

Offs

et [m

]: 7

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 16

10N

,M

:

41

#

N

t

1

25

00

2

20

00

3

15

00

4

10

00

5

5

00

6

100

Res

[Ωm

]: 20

Offs

et [m

]: 7

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 20

10N

,M

:

43

#

N

t

1

30

00

2

25

00

3

20

00

4

15

00

5

10

00

6

500

Res

[ Ωm

]: 50

Offs

et [m

]: 7

Dep

th [m

]: 30

47L

: 40

Wid

th[m

] : 3

140

N,M

: 4

7#

Nt

1

3500

2

2500

3

2000

4

1500

5

1000

6

5

00

Res

[ Ωm

]: 10

0O

ffset

[m]:

7D

epth

[m]:

1249

L

:

36W

idth

[m]:

6200

N,M

: 5

3#

Nt

1

4500

2

4000

3

3500

4

3000

5

2500

6

15

00

7

5

00

Res

[ Ωm

]: 10

Offs

et [m

]: 55

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 16

10N

,M

:

41

#

N

t

1

25

00

2

20

00

3

15

00

4

10

00

5

5

00

6

100

Res

[ Ωm

]: 20

Offs

et [m

]: 55

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 20

10N

,M

:

43

#

N

t

1

30

00

2

25

00

3

20

00

4

15

00

5

10

00

6

500

Res

[ Ωm

]: 50

Offs

et [m

]: 55

Dep

th [m

]: 30

47L

: 40

Wid

th[m

]: 31

40N

,M

:

47

#

N

t

1

35

00

2

25

00

3

20

00

4

15

00

5

10

00

6

500

Res

[ Ωm

]: 10

0O

ffset

[m]:

55D

epth

[m]:

1249

L

:

36W

idth

[m]:

6200

N,M

: 5

3#

Nt

1

4500

2

4000

3

3500

4

3000

5

2500

6

15

00

7

5

00

Figure 6.3: Estimating the number of matrix operations required for four different resistivity models, 10, 20, 50 and 100 ohmm. Blue lines are the TEMDDD responses, while the reds are the EM1DINV reference responses. The green line is the relative difference between the highlighted model and the 1D reference response.


____________________________________________________________________________

been minimised step by step for all the resistivity models. The result of these runs can

be seen in figure 6.3.

In table 6.5 the result of the optimisation is summarised. It is seen that higher

resistivities require more matrix operations to be performed. This is linked to the

increased number of nodes in the high-resistivity model.

A slight underestimation in the number of

matrix operation can result in a slight lift of the

model curve at late times while server

underestimation results in an underestimation

of the model response. An example of this is

half

halv

the

6.4.

T

ope

the

resp

par

for

F

betw

betw

are

extr

offs

rela

Aga

reve

Tabope

Resistivity Matrix operations [Ohmm] [ # ]

10 2500 20 3000 50 3500

100 4500 le 6.5: Required number of matrix rations .

41

seen for both configurations in the 100 ohmm

-space. Compared to the optimum model the number of matrix operations can be

ed in all four resistivity models and still produce responses that are consistent with

reference response until about 1ms.

5 The optimised final model parameters for the homogeneous half-space

he effects of changes in the size of the model space and the number of matrix

rations have been investigated. All parameters have been found individually, and

final result of these optimisations can now be combined to one single optimised

onse for each half-space resistivity. This is shown in figure 6.4. The final model

ameters are summarised in the small info box for each model, and the parameters

the central loop- and offset model are the same.

or the 10 ohmm half-space the central loop model show relative differences

een -7 and +7 percent. The offset configuration shows relative differences

een -9 and +20 percent. The large differences observed around the sign-reversal

not alarming. Only two data points on each side of the sign reversal define the

eme values. Furthermore, similar differences can be observed when comparing

et responses for two one-dimensional codes. The 20 ohmm half-space shows

tive differences between –15 and +3 percent for the central loop configuration.

in the offset configuration shows relatively large differences around the sign

rsal, but moderate differences elsewhere.


____________________________________________________________________________

42

10-6

10-5

10-4

10-3

10-2

10-1

0.2

0.1

0.0

-0.1

-0.2Relative difference

TEMD

DDEM

1DIN

Vre

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

0.2

0.1

0.0

-0.1

-0.2Relative difference

TEMD

DDEM

1DIN

Vre

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

TEMD

DDEM

1DIN

Vre

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

TEMD

DDEM

1DIN

Vre

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

TEMD

DDEM

1DIN

Vre

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

TEMD

DDEM

1DIN

Vre

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DDEM

1DIN

VRe

lative

diffe

renc

e

10-6

10-5

10-4

10-3

10-2

10-1

Tim

e [ s

]

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DDEM

1DIN

VRe

lative

diffe

renc

e

Res

[ Ωm

]: 10

Offs

et [m

]: 7

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 16

10N

,M

:

41

Nt

: 2

500

Res

[Ωm

]: 20

Offs

et [m

]: 7

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 20

10N

,M

:

43

Nt

: 3

000

Res

[Ωm

]: 50

Offs

et [m

]: 7

Dep

th [m

]: 30

47L

: 40

Wid

th[m

] : 3

140

N,M

: 4

7N

t

: 35

00

Res

[Ωm

]: 10

0O

ffset

[m]:

7D

epth

[m]:

1249

L

:

36W

idth

[m]:

6200

N,M

: 5

3N

t

: 450

0

Res

[ Ωm

]: 10

Offs

et [m

]: 55

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 16

10N

,M

:

41

Nt

: 2

500

Res

[ Ωm

]: 20

Offs

et [m

]: 55

Dep

th [m

]: 80

9L

: 34

Wid

th[m

]: 20

10N

,M

:

43

Nt

: 3

000

Res

[Ωm

]: 50

Offs

et [m

]: 55

Dep

th [m

]: 30

47L

: 40

Wid

th[m

] : 3

140

N,M

: 4

7N

t

: 35

00

Res

[Ωm

]: 10

0O

ffset

[m]:

55D

epth

[m]:

1249

L

:

36W

idth

[m]:

6200

N,M

: 5

3N

t

: 450

0

Figure 6.4: The responses from the final optimised models, the EM1DINV reference response and the relative difference between the two. The input parameters for each model are stated in the info box.


____________________________________________________________________________

43

Before 1e-6 second the 50 ohmm central loop response shows a relative difference

of +5 percent, and at the latest times the differences are –15 percent. The differences

for the offset response are relatively large at early times because of the position of the

sign reversal. At later times the differences are –15 percent. For the 100 ohmm half-

space the relative differences are +5 percent at the earliest times for the central loop

configuration. At the later times the differences vary between –2 and +10 percent. The

offset configuration shows differences between –5 percent at the earliest times and –15

percent at the latest times.

The final models all show relative differences that are below 7 percent at the early

times. One exception is offset responses, which show extreme differences around the

sign-reversal. The latest times are influences by the limited size of the model and the

number of matrix operations for each model.

6.5 Verifying the code over a layered model The next step is optimising the model parameters over a layered earth. The

optimised parameters found for the homogenous half-space are used, and the vertical

nodal-distribution is investigated.

6.5.1 The models

Because the main purpose with the optimised model is to investigate the effect of

near surface inhomogeneities. Therefore, the layered resistivity models investigated in

this chapter have layer boundaries that are relatively shallow. Two opposite models

have been chosen to investigate the ability of the code to resolute a relatively thin layer

in the top of the model.

The models consist of a half-space with a 20-meter layer inserted in the 5–25 meter

depth interval. The first model consists of a 10 ohmm half-space with an inserted layer

of 100 ohmm, while the second model is a 100 ohmm half-space with an inserted layer

of 10 ohmm. For each of the two models five different vertical nodal distributions for the

upper 25 meters have been generated and the TEMDDD response has been

calculated for a central loop- and a 55 meters offset configuration.


____________________________________________________________________________

44

Figure 6.5 shows two opposite

resistivity models, and the 5

different vertical nodal

distributions. Model # 1 has a 25-

centimetre distance between the

two top nodal points, followed by 5

stepwise increasing vertical nodal

distances until the depth of five

meters. Thereafter, five nodal

increasing layers define the 20 meter inserted resistivity layer. This is followed by a 10

nodal points per decade vertical distribution in the rest of the model space. The vertical

nodal distribution is stepwise coarsened through the five models. In #5 the top

resistivity layer and the inserted resistivity layer is defined by only two nodal distances.

Below 25 meter depth there is a 10 nodal points per decade vertical distribution in the

rest of the model space. The responses for the two resistivity models, each with five

vertical nodal distributions, are calculated for both central loop- and 55 meter offset

configuration, making a total of 20 responses to be compared with the EM1DINV

reference response.

6.5.2 Optimising the upper vertical nodal distribution for shallow resistivity contrasts

The optimisation of the upper vertical nodal distribution in models with a relatively

shallow resistivity contrast is of particular interest. This is because the optimised model

parameters ultimately will be used to investigate the influence of shallow 3D

inhomogeneities for different transmitter-receiver configurations.

The result of this comparison is shown in figure 6.6. The first two graphs (a and b)

show the 10 ohmm half-space with an inserted 100-ohmm layer in the 5 – 25 meter

interval. From the graph it is evident that a choice of a 25 centimetre top nodal distance

results in a response that differs several hundred percent at early times compared to

the 1D reference response. This is observed for both the central loop- and the offset

configuration. The lack of accuracy for this nodal distribution could be caused by the

relatively high ratio between the horizontal- and vertical dimensions of each nodal block

1.00 1.50 2.50

4.00 6.00 10.0 13.0 16.0 :::

2.50 2.50

10.0 10.0

13.0 16.0 :::

5.00

20.0

25.0 31.0 :::

0.25 0.50 0.75 1.00 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 10.0 :::

0.50 1.00 1.50 2.00 3.00 4.00 6.00 7.00 10.0 13.0 :::

10/100 Ωm 5 m

10/100 Ωm

100/10 Ωm 20m

Resistivity model

Vertical nodal distribution [m]# 1 # 2 # 3 # 4 # 5

Figure 6.5: Five different vertical nodal distributionsin the shallow region for two resistivity models


____________________________________________________________________________

10-6

10-5

10-4

10-3

Tim

e [ s

]

10-7

10-6

10-5

10-4

10-3

10-2 dB/dt

1 2 3 4 5 Ref.

Diff.

10-6

10-5

10-4

10-3

Tim

e [ s

]

1 2 3 4 5 Ref.

Diff.

10-6

10-5

10-4

10-3

Tim

e [ s

]

0.20

0.10

0.00

-0.1

0

-0.2

0

Relativedifference

1 2 3 4 5 Ref.

Diff.

a)c)

d)

10-6

10-5

10-4

10-3

Tim

e [ s

]

1 2 3 4 5 Ref.

Diff.

b)

Res

[Ωm

]: 10

/100

Offs

et [m

]: 0

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

Nt

:

4000

#

t min

[m]

L

1

0.

25

32

2

0.

50

28

3

1.

00

25

4

2.

50

23

5

5.

00

18

Res

[Ωm

]: 10

/100

Offs

et [m

]: 55

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

Nt

:

4000

#

t min

[m]

L

1

0.

25

32

2

0.

50

28

3

1.

00

25

4

2.

50

23

5

5.

00

18

Res

[Ωm

]: 10

/100

Offs

et [m

]: 0

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

Nt

:

4000

#

t min

[m]

L

1

0.

25

32

2

0.

50

28

3

1.

00

25

4

2.

50

23

5

5.

00

18

Res

[ Ωm

]: 10

/100

Offs

et [m

]: 55

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

Nt

:

4000

#

t min

[m]

L

1

0.

25

32

2

0.

50

28

3

1.

00

25

4

2.

50

23

5

5.

00

18

Figure 6.6: Optimising the required numbers of nodal layers to accurate define a shallowthin layer in two opposite models. a),b): A 10 ohmm half-space with a 100 ohmm layer inthe 5-25 meter depth interval. c),d): A 100 ohmm half-space with a 10 ohmm layer in the 5-25 meter depth interval. The relative difference between the EM1DINV reference responseand the best-fitted model (highlighted model parameters) is shown

45


____________________________________________________________________________

46

in the upper nodal layer. With the usual 10 meters horizontal nodal distance in the

central part of the model, this ratio gets as high as 40 in model #1.

The following two models (#2 and #3) have a top nodal layer of 0.5 and 1 meter,

respectively. The relative difference between model #3 and the EM1DINV reference

response is also shown because model #3 shows the best fit to the reference

response.

For the central loop configuration the relative difference is about - 3 percent at 5e-6

seconds and gradually approaching +7 percent one decade later. At later times the

difference approaches zero and after 1 ms it is stable at +1 percent. For the offset

configuration the difference between model #3 and the reference response, plot 6.6b is

-2 percent at 5e-6 seconds, slowly approaching zero towards the sign reversal. The

relative differences can be quite extreme in the close proximity to the sign reversal, and

in this case it is about +400 percent. At the local maximum following the sign reversal

around 2e-4 seconds the difference reaches -37 percent. Hereafter it decreases and

reaches zero after 1 millisecond. The large differences that are seen around the sign

reversal for relatively large transmitter-receiver separations and high conductivity are

due to the extreme gradient of the fields as they pass the receiver

Further coarsening the upper nodal distribution (model #4 and #5) leads to distinct

differences between the calculated response and the EM1DINV reference response.

However, these are still not as large as seen in the very dense nodal distribution in

model #1. Model #5 shows differences for both transmitter-receiver configurations of

more than 20 percent, which is caused by the inadequate nodal discretisation in the

upper layers.

The calculated responses from the 100 ohmm half-space with a 10 ohmm layer are

shown in plot c) and d) in figure 6.6. The upper vertical nodal distribution is the same

as in plot a) and b) but the resistivity model is reversed. In contrast to the observations

in the minimum model, the relatively fine vertical discretisation of the upper layers of

model #1 does not cause large differences in the calculated response. For the central

loop configuration there is a high degree of conformity between model #1 and #2. As

the nodal distribution is coarsened in model #3, differences become significant, and

these are enforced in models #4 and #5. The relative differences for those two models


____________________________________________________________________________

47

are several hundred percent. The offset configuration shows a similar pattern, though

#1 shows some differences around the local maximum after the sign reversal.

For both configurations, the calculated response from model #2 gives the best fit

compared with the EM1DINV reference response. In the case of the central loop

configuration the relative difference at 5e-6 seconds is around +2 percent and reaches

+8 percent at 1e-4 second. After this it gradually approaches zero and stabilise at 1ms.

For the offset configuration the relative difference at 5e-6 seconds is -2 percent raising

to +2 two percent just before the sign reversal. Like the offset responses in plot b) the

relative differences are considerable in the immediate vicinity of the sign reversal.

Despite the big differences this is not considered problematic as only 3 data points

around the sign reversal exhibits relative differences greater than +/- 5 percent. After

the sign reversal the relative difference approaches zero percent and stabilise here

after 0.1ms.

The vertical nodal distribution has been optimised for 1D shallow resistivity

variations. For the 10 ohmm half-space with a 100 ohmm layer the optimum distribution

was found to be #3 in figure 6.5. For the 100 ohmm half-space with a 10 ohmm layer

the optimum nodal distribution was found to be #2 in figure 6.5. For the first resistivity

model, it is notable that relatively large differences to the EM1DINV reference response

occur if the upper nodal distance is below 1 meter.

Test runs have shown that variations in the vertical nodal distribution do not

influence the accuracy of the calculated responses significantly. Outside the equally

spaced centre of the model the value of 10 nodal points per decade is maintained

throughout the verification of the code.

6.6 Verifying the code over a 3D model The final step in the optimisation of the model parameters is comparing 3D

TEMDDD responses with responses from other 3D codes.


____________________________________________________________________________

48

6.6.1 The models

The models in the following investigation have been chosen because reliable 3D

responses were available from Auken (1995). In the paper the integral code EM3D

(Newman et al, 1986; Wannamaker et al, 1984) is used. Furthermore Thorkild Maack

Rasmussen of the Geological Survey of Denmark and Greenland (GEUS) have been

very helpful by calculating additional reference responses using the 2.5D EM modelling

code ARJUNA (Raiche et al., 1999).

Two opposite models have been

used in the following investigation.

The main layout of the two 3D-

resistivity models is shown in figure

6.7. A three-dimensional, elongated

beam is surrounded by a layered

host. The beam is 50 meters in the z-

direction, 100 meters in the y-

direction and 800 meters in the x-

direction. It ispositioned in the depth

interval from 20 to 70 meters. The geometric centre of the beam is located at (x,y,z)

equal to (0,0,45) meters. The resistivity of the beam is either 10 or 100 ohmm creating

a sharp contrast to the host in which it is embedded. The layered host consists of a 20-

meter overburden with a resistivity of 30 ohmm. This is followed by a layer, which fill

out the rest of the model space with a resistivity of either 100 or 10 ohmm. The

elongated model allows direct comparison between 3D responses and the 2.5D

ARJUNA response. Tests have shown that the relative differences between 3D

TEMDDD response and a 2D TEMDDD response for the models are less than 1

percent.

All responses are calculated along the beam’s centre in the x-direction at three

different transmitter positions as shown in figure 6.8. For each transmitter position one

receiver was placed in the centre of the transmitter and one was placed 40 meters

offset in the +y direction. TEMDDD only allows transmitter corners to be placed at

nodal points, and receivers to be placed at the centre of four adjacent nodal points.

This makes it impossible to place a receiver at a distance of 40 meters from the centre

of the transmitter with the usual 10 meters spacing in the central area of the model

10/100 ohmm

100/10 ohmm

30 ohmm100m20m

50m

Figure 6.7: The 3D reference model. Thebeam is 800m meter in the direction x-direction.


____________________________________________________________________________

49

space. Thus, one is forced to bypass this by making an alternative central grid

distribution. This new nodal distribution must both satisfy the requirement of the 40 by

40 meters transmitter, the position of the beam and the 40 meters receiver offset. The

exact horizontal nodal distribution in shown in figure 6.8

0m

50m

100m

10 10 10 10 10 10 10 1088 81010 10 10 10

10 10 10 10 10 10 10 1088 81010 10 10 10

10 10 10 10 10 10 10 10 6 10 1088 810 10

131621

10Nodal pointNodal distanceReciever TransmitterBeam

Figure 6.8: Outline of the alternative nodal distribution in the z-direction in the central area of the model. As seen the central loop configuration is not a real central-loop, but a

755 22 ≈+ meter offset configuration.

6.6.2 Comparison of TEMDDD responses with other multi-dimensional modelling codes

The responses from the three modelling codes are plotted in figure 6.9 for three

different distances from the centre of the beam and two different transmitter-receiver

configurations. The six columns represent the three points were the responses have

been calculated for the two resistivity models, respectively. The top row shows the

responses from the central loop configuration while the bottom one shows the

responses from the 40 meters offset configuration.

Each of the plots has five different curves plotted. First the 3D TEMDDD response,

which is has been calculated using the experience that was acquired earlier in this

chapter. The basic parameters involved in the calculations of these responses are

stated in the small info box in each plot. Next the 2D ARJUNA response and the 3D

EM3D response which both serve as reference responses. The 1D EM1DINV response

is also included to give an impression of the relative effect of the 3D beam. Finally, the

relative difference between the 3D TEMDDD- and the EM3D response is shown. The

difference between the 3D TEMDDD and the 2D ARJUNA response is not shown as

the ARJUNA code has obvious problems resolving the early time response. These

differences are possibly caused by inadequate nodal discretisation in the upper layers.


____________________________________________________________________________

50

Figu

re 6

.9: C

ompa

rison

bet

wee

n th

e TE

MD

DD

resp

onse

and

EM

3D/A

RJU

NA

resp

onse

s fo

r tw

o op

posi

te 3

D re

sist

ivity

mod

els.

a)

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3 dB/dt

TEMD

DD

ARJU

NA 2D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 0

mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

0W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

b)

10 o

hmm

c)TE

MDDD

ARJU

NA 2

D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 1

00m

Res

[Ωm

] : 3

0/10

/100

Offs

et [m

]: 0

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

L

:

32N

t

: 40

00t m

in[m

] :

0.2

5

10 o

hmm

d)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

10-1

1

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3 dB/dt

TEMD

DD 3D

ARJU

NA

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 0

mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

40W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

e)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

TEMD

DD 3D

ARJU

NA 2D

EM3D

EM1D

INV

1DDi

ffere

nce

Posi

stio

n.: 5

0mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

40W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

f)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]]

TEMD

DD 3D

ARJU

NA 2

DEM

3D 3D

EM1D

INV

1DDi

ffere

nce

Posi

stio

n.: 1

00m

Res

[Ωm

] : 3

0/10

/100

Offs

et [m

]: 40

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

L

:

32N

t

: 40

00t m

in[m

] :

0.2

5

10 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

TEMD

DD

ARJU

NA 2

DEM

3D

EM1D

INV

1D

Diffe

renc

e

10 o

hmm

100

ohm

m

30 o

hmm

Posi

stio

n.: 5

0mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

0W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

g)TE

MDDD

3DAR

JUNA

2D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 0

mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

0W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

h)TE

MDDD

3DAR

JUNA

2D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 5

0mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

0W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

i)

0.2

0.1

0.0

-0.1

-0.2

0.3

0.4

0.5

-0.3Relative

difference

TEMD

DD 3D

ARJU

NA 2D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 1

00m

Res

[Ωm

] : 3

0/10

/100

Offs

et [m

]: 0

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

L

:

32N

t

: 40

00t m

in[m

] :

0.2

5

10 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

j)

10-6

10-5

10-4

10-3

10-2

Tim

e [ s

]

TEMD

DD 3D

ARJU

NA 2D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 0

mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

40W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm

k)

10-6

10-5

10-4

10-3

10-2

Tim

e [ s

]

TEMD

DD 3D

ARJU

NA 2

DEM

3D

EM1D

INV

1DDi

ffere

nce

10 o

hmm

100

ohm

m

30 o

hmm

Posi

stio

n.: 5

0mR

es [Ω

m] :

30/

10/1

00O

ffset

[m]:

40W

idth

[m]:

6200

Dep

th[m

] : 2

980

N,M

: 5

3L

: 32

Nt

:

4000

t min

[m]

: 0

.25

10 o

hmm

l)

0.2

0.1

0.0

-0.1

-0.2

0.3

0.4

0.5

-0.3Relative

difference

10-6

10-5

10-4

10-3

10-2

Tim

e [ s

]

TEMD

DD 3D

ARJU

NA 2

D

EM3D

EM1D

INV

1D

Diffe

renc

e

Posi

stio

n.: 1

00m

Res

[Ωm

] : 3

0/10

/100

Offs

et [m

]: 40

Wid

th [m

]: 62

00D

epth

[m] :

298

0N

,M

: 53

L

:

32N

t

: 40

00t m

in[m

] :

0.2

5

10 o

hmm

10 o

hmm

100

ohm

m

30 o

hmm


____________________________________________________________________________

51

The data from the EM3D responses were not calculated at the same discrete times

as the TEMDDD responses. To be able to compare the two responses directly, and

calculate the relative differences for each discrete time, the EM3D data set was

resampled using the standard MATLAB routine ‘Interp1.mat’.

Generally, comparing the 3D responses with the 1D EM1DINV response it is evident

that the part of the response that is due to the beam moves to later times as the

distance to the beam is increased. The anomaly produced gets smaller with an

increased distance. This is especially pronounced for the offset configuration.

Differences at the latest times are due to a slight underestimation of the total size of the

model space or the required number of matrix operations.

The calculated central loop responses from the resistive beam in the conductive

media (plot 6.9a, b and c) show relatively small relative differences between the

TEMDDD response and the EM3D reference response. At the position directly above

the beam (plot a) there is no difference at all at the earliest times, but after 1E-05

seconds the difference rises to +/- 10 percent. Plot b) and c) show relative differences

of +/- 5 percent. The offset configuration (plot 6.9d, e and f) the relative differences are

larger than 20 percent around the sign-reversal. Relatively few data points on each

side of the reversal define these large differences. Furthermore, the observed

differences are comparable to the differences described for the half-space and layered

half-space, though they stretch to later times.

The six responses from the conductive beam in a resistive media are seen in figure

6.9g, h and i. Like in the case of the resistive beam in a conductive media, the

responses are calculated for the central loop and the 40m offset configuration.

Because of the characteristic of the resistivity model combined with the limited

frequency bandwidth involved in the calculation the EM3D reference response displays

distinct inaccuracy and instability at times later than about 2E-4 seconds. These effects

are enforces trough the MATLAB resampling and thus the calculated differences

between the EM3D reference data set and the TEMDDD data set appears to be

relatively large.

For the central loop response directly above the conductive beam (plot g) the

TEMDDD response shows differences of about +/-10 percent at times earlier than 4E-


____________________________________________________________________________

52

4. At the later times the EM3D response is very influenced by instability but the

differences are around +10 percent on average. The central loop TEMDDD response

for the position 50 meters from the centre of the beam (plot h) shows relative

differences around 0 percent at the earlier times. After 2E-4 seconds the relative

difference rises dramatically to around +50 percent. The ARJUNA response also

shows higher response than the EM3D response. Both the receiver and transmitter are

positioned directly above this high-contrast abrupt boundary for this response. This

could be the cause of these large differences observed. The responses calculated at a

100 meter distance from the centre of the beam (plot 6.9i) shows relatively low

differences at early times. After 2E-4 seconds the difference rises to around -20

percent, but approaches 0 percent at the latest times.

The three offset responses for the conductive beam in a resistive media is seen in

plot j, k and l. As in the case of the offset responses for the resistive beam in a

conductive media, large differences are observed around the sign reversal. For the

response calculated directly above the inserted beam (plot j) these large differences

diminish over the first 1½ decade and around 1E-4 seconds the relative difference

between the two responses is 0 percent. Large differences of up to +50 percent are

observed at later times. At the transmitter position 50 meters from the beam (plot k),

relatively small differences of about +/- 10 percent are seen after 1E-4 seconds. The

Large differences at the latest timed are obviously due to the instability of the EM3D

response. At the 100 meters transmitter position (plot l) even smaller differences

between the EM3D and TEMDDD codes are seen. These differences are in the range

of +/- 10 percent.

The relatively large differences for some responses in figure 6.9 could be connected

to the alternative central nodal distribution in the central part of the model. Furthermore,

the EM3D and ARJUNA responses are calculated for an exact central-loop

configuration. This configuration is not possible with the TEMDDD code because the

transmitter is defined at the nodal points, while the transmitter is defined between four

adjacent nodal points. Thus, the closest the TEMDDD model can come to a real central

loop configuration is a 7.07m offset configuration.


____________________________________________________________________________

53

6.7 Summary of the achieved results Three main steps have been taken in verifying the TEMDDD modelling code. First

the input parameters were optimised for a homogeneous half-space in the resistivity

range from 10 to 100 ohmm. This was done for a central loop configuration and the 55

meters offset configuration. Except in the close vicinity of the sign reversal, the

obtained results showed differences compared to the EM1DINV code of less than five

percent. During the investigations several important parameters were found, such as

the required size of the total model space and the required number of matrix operations

for different resistivity models.

Next, the vertical nodal distribution was optimised using layered half-space. Again

the transmitter-receiver configurations used were central loop and 55 meters offset.

The results from these showed differences compared to the EM1DINV code of less

than +/-10 percent.

Finally the code’s ability to resolve 3D structures was verified using two opposite

resistivity models. The relative differences were generally around +/- 10 percent, but

larger differences were also observed for some responses.

Although not investigated as thorough as the z-component, simple comparison

tests have shown that observed errors for the x- and y-component of the secondary

electromagnetic field are in the same order of magnitude as described for the z-

component.


____________________________________________________________________________

54


____________________________________________________________________________

55

7 The Modelling implementation

A substantial part of the work in this thesis has been the pre phase work with the

TEMDDD modelling code. This work included getting familiar with the limitations and

strengths, but also how to implement the code into various programming routines.

These routines were all written from scratch in Matlab© and included three programs:

TEMDDDModelCreator, TEMDDD2Patem and EM1DINVPlot.

7.1 TEMDDDModelCreator

7.1.1 Introduction

As the name indicates, the Matlab routine ‘TEMDDDModelCreator’ is used to

create input model files for the TEMDDD code. When a few simple models are used

over small profiling distances one can ‘hand-edit’ the input files. This is both time

consuming and unreliable because a simple mistype can lead to miscalculations and

consequently loss of valuable CPU-time. For longer and more complicated model

profiles a stable and thoroughly tested program for generating the input files is

essential.

7.1.2 Defining the model in TEMDDDModelCreator

The program TEMDDDModelCreator uses a predefined model file and a semi-

graphical interface in ASCII format. The model file contains information about the

desired model dimensions in TEMDDD, e.g. the total width and depth of the TEMDDD

model space. Furthermore, the layered background is defined together with information

about the shallow multidimensional inhomogeneities. In the semi-graphical ASCII-file

the larger multidimensional structures are defined as seen in figure 7.1.

Figure 7.1: Example of a simple semi-graphical interface ASCII file.

1 0000000000000000000000000000 2 0000000000001111000000000000 3 0000000000001111000000000000 4 0000000000001111000000000000 5 0000000000001111000000000000 6 0000000000000000000000000000 7 0000000000000000000000000000 8 0000000000000000000000000000 9 0000000000000000000000000000 10 0000000000000000000000000000


____________________________________________________________________________

56

Areas with no three-dimensional resistivity definition are assigned the integer ‘0’

(zero), while the three-dimensional structure is defined with integer between one and

nine. The resistivity values of these integers in the final model are defined in the

TEMDDDModelCreator model file as explained above. Each integer in the example

represents a certain volume or area in the final TEMDDD input file. For the example in

figure 7.1 the value is 5 meter, which makes the model a 20 meter wide box in the

depth interval from 5 to 25 meters. The last dimension, which is defined in the

TEMDDDModelCreator model file, was also set to 20 meters, making the

inhomogeneity a 20 by 20 by 20 meter cube. For more complex structures, like in figure

7.2, the three-dimensional structure is subdivided into a suite of horizontal rectangles

that are defined as individual inhomogeneities in every TEMDDD input file.

7.1.3 Projecting the TEMDDDModelCreator onto the TEMDDD model space

In order to generate long profiles, (~1000 meters) it is unavoidable to split up the

model into a suite of smaller models and calculate the responses from these

individually. This is because TEMDDD uses the same grid dimensions in both the x-

and y-direction and a very long model therefore also would be very wide. Furthermore,

a large area with constant grid spacing in the centre model would be needed, which

would require a number of nodal definitions that would exceed even the biggest

computers available. A new TEMDDD input model is generated each time the source-

receiver configuration is moved over the structure. This means that it is the same

predefined nodal grid is used for all sounding positions as the nodal grid is ‘stepped’

through the large structure. This is illustrated in figure 7.2 with a sloping structure.

This has several advantages since the source-receiver configuration always has the

same position in each TEMDDD input file, and the nodal distribution always is identical.

If no resistivity variation exists in the top nodal layer the calculations in TEMDDD can

be greatly reduced because the calculated surface matrix in TEMDDD can be re-used

in the calculation of the next sounding in the profile.

The three-dimensional structure can be regarded a piece-wise one-dimensional in

the horizontal plane when defining the structure as a group of horizontal rectangular

bodies. This is possible because the three-dimensional structure has a constant size

perpendicular to the profiling direction that coincides with a nodal position.


____________________________________________________________________________

57

Furthermore, the group of horizontal rectangular bodies each has boundaries in the z-

direction that coincides with a nodal plane. Thus, the problem of approximating the

position of the end of the individual horizontal rectangular bodies becomes a question

of defining their length instead of volume.

As seen in the example in figure 7.2 the boundaries of the 3D bodies in the

TEMDDDModelCreator model space does not always coincide with the position of the

nodes in the y-direction. The distance between the individual soundings in the profiles

is usually identical to the nodal spacing the central part of the TEMDDD model. This

means the boundary of a structure will always coincide with a nodal position in this

area. In the z-direction the TEMDDD model space is defined in such a way that each

nodal plane always coincide with a required boundary.

In cases where the boundary does not coincide with a nodal position the body

boundary is moved to the nearest node, thus making the three-dimensional body

smaller or bigger than first intended. This is shown in scenario #3 and #2 in figure 7.2,

respectively. To ensure accurate calculations the correctional end-inhomogeneities are

introduced.

7.1.4 Applying the correctional end-inhomogeneities

The concept of the correctional end-inhomogeneities is built on conductivity

preservation. The boundary of a structure is moved to the nearest nodal point and the

resistivity in this area is corrected to maintain a constant integrated conductivity. The

1010 10 10 1012151924

1010 10 10 1012151924

00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001111111111111100000000000000000000000000000000011111111111111110000000000000000000000000000000111111111111111111000000000000000000000000000001111111111111111111100000000000000000000000000011111111111111111111110000000000000000000000000111111111111111111111111000000000000000000000001111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

0-50 10 20 30-10 605040-40 -30 -20 70-60-70

10Nodal pointNodal distanceInhomogeneity

1010 10 10 1012151924#1#2#3

y

z

Figure 7.2: The concept of projecting the TEMDDDModelCreator model space (upper)onto the TEMDDD model space (lower). The three examples show three different resultsof stepping trough the model. For each scenario the TEMDDD model space is moved 10meters across the TEMDDDModelCreator model space.


____________________________________________________________________________

58

correction is made by defining the correctional end-inhomogeneities after the original

inhomogeneity definition in the TEMDDD input-file. In this way the original resistivity

definition is overwritten by the value of the correctional end-inhomogeneity.

This is shown in figure 7.3, which shows an example in which the Inhomogeneity is

made smaller. The figure is a detailed view of scenario #3 in figure 7.2. As seen from

the figure the correctional end-inhomogeneity is only applied to the end of each

individual, horizontal, rectangular body. More precisely: The correctional end-

inhomogeneity is applied between the two last nodes in the approximated body,

overwriting the original resistivity definition in this region.

By only regarding the area affected by the correction (Ideal Length), the goal is to

maintain constant conductivity before and after the boundary of the body is moved.

This is done by defining the resistivity of the correctional end-Inhomogeneity (ρcorr.).

Before the correction is made the integrated conductivity (σbefore) in the area simply is

bodybefore ρ

ILσ = (7.1)

After the correction is made the integrated conductivity (σafter) in the area consists of

contributions the background resistivity (ρbackg.) from the area and the resistivity of the

correctional end-Inhomogeneity (ρcorr)

corr.backg.after ρ

ALρ

ALILσ +

−= (7.2)

121519

Ideal Length (IL)Approx. Length (AL)

10Nodal pointNodal distanceInhomogeneity

InhomogeneityCorrectional

0 10 20 30-10 40

Figure 7.3: The principle of introducing the correctional end-inhomogeneities. Theexample shows a close-up on scenario #3 in figure 7.2.


____________________________________________________________________________

59

The absolute value of the difference between the Ideal Length (IL) and the

Approximated Length (AL) is used to comply with both situations of increasing and

decreasing the size of the rectangular bodies.

Setting equation 7.1 equal to 7.2 and solving for the resistivity yields

1

backg.bodycorr ρ

ALILρILAL

−

−−⋅=ρ , (ρbody < ρbackg. ) (7.3)

To ensure that the correctional end-inhomogeneities always are assigned a positive

value when IL > AL, the resistivity-value of the three-dimensional body must be smaller

than the background resistivity.

The described correction is made for both the beginning and end of each of the

rectangular bodies that make up the whole three-dimensional structure. For small

three-dimensional structures there are two situations in which special actions has to be

taken to avoid unwanted effects from the correction. Firstly, if two neighboring nodes

define the whole rectangular body the calculated correctional end-inhomogeneities will

overwrite each other in the TEMDDD input-file. In such a case a second category of

correctional end-inhomogeneities will be calculated, again preserving the constant

integrated conductivity in the area before and after the correction is made. Secondly, If

the same node is closest to both the start and the end of an the rectangular body the

approximated length (AL) will get the length zero. In this case one of the ends of the

body is moved to the next nodal position and the single correctional end-inhomogeneity

is calculated as described just above.

7.1.5 Distributing small shallow random resistivity variations

Besides being able to project large structures onto the TEMDDD model space, the

TEMDDDModelCreator can also be set to make a random distribution of small

rectangular inhomogeneities. This feature is used to simulate highly inhomogeneous

overburdens such as a till, and observe its effects on the three-dimensional TEMDDD

response. The parameters for this feature include individual inhomogeneity dimension,

mean and spreading factor of the resistivities, and distribution interval in the z-direction.

The inhomogeneities can be defined in any size that is a multiple of the nodal-distance

in the central part of the TEMDDD model-grid.


____________________________________________________________________________

60

To avoid a large number of inhomogeneities in the TEMDDD input-file, they are only

defined in the centre of the TEMDDD input-file where the nodes are equally distributed.

Outside this area the resistivity maintains a constant value equal to the mean of the

inhomogeneities. Typically, the central area of the model is around 150 meters in the x-

and y-direction. The dimensions of the inhomogeneities are chosen in such manner

that the individual boundaries always coincide with a nodal position.

The distribution of the resistivities is a log-distribution, which is calculated using the

following formula:

log(f) log(p)10 ⋅+= RM (7.4)

where

M= array containing the log-distributed resistivities R = array containing distributed values with mean 0 and variance 1 p = mean values of the resistivities f = spreading factor of the resistivities

Figure 7.4 show three different realisations of the formula using a mean resistivity value

of 100 ohmm and spreading factors of 1.2, 1.5 and 2.0.

For specific modelling tasks the user must adjust the value of the spreading factor,

mean and inhomogeneity size to fit the problem in interest.

Figure 7.4: Examples of the resistivity distribution for different Spreading Factors usinga mean resistivity of 100 ohmm.

0 100 200 3000

0.05

0.1

0.15

0.2

Rel

ativ

e di

strib

utio

n

Factor=1.2

0 100 200 3000

0.05

0.1

0.15

0.2

Resistivity (Ohmm)

Factor=1.5

0 100 200 3000

0.05

0.1

0.15

0.2Factor=2.0


____________________________________________________________________________

61

7.2 EM1DINV interpretation and visualisation the data After calculation of the TEMDDD responses the results are still in basic ASCII-

format and a reformatting process for interpretation and visualisation of the data is

necessary.

7.2.1 Reformatting the TEMDDD output-files

For easy 1-D interpretations and quick raw-data preview the programs

TEMDDD2Patem and EM1DINVPlot has been made. TEMDDD2Patem is a

reformatting program that reads all the dB/dt output-files after the TEMDDD execution

and transforms them into two different formats.

The first format gives the user a quick preview of the dB/dt data in profile mode as

seen in figure 8.1b. The format file is loaded directly into the program PATEM which

was originally developed for visualisation of data from the Pulled Array Transient

Electromagnetic Method (PATEM) (Sørensen et al, 1995). With the PATEM program, it

is possible to se the raw data in both profiling mode and single sounding mode. This it

makes it very suitable for a quick overview over the data before any 1-D interpretations

are made.

The second format is used as input files for the 1D inversion algorithm EM1DINV.

All information about transmitter- and receiver positions, waveform etc is included in

this file. Both formats are made for all combinations of receiver positions and field

components.

7.2.2 The EM1DINV inverse modelling code

The one-dimensional interpretations of the TEMDDD data is carried out using the

commercially available 1D inversion code EM1DINV (Effersø et al 1999; Foged 2001).

The inversion is carried out as an iterative damped least-square approach (Menke,

1989), formally writing

( ) ( )nnnnnn ddCGGCGmm −α++= −−−+ obs

1d

T11d

T1 I , (7.5)


____________________________________________________________________________

62

where m denotes the model vector, Gn is the Jacobian matrix, Cd is the data covariance

matrix, α is the damping factor, I is the identity matrix, dobs denotes the observed data

vector and dn denotes the forward data vector based on the previous model vector mn.

Three residuals are used in the inversion. Data residual, a residual connected to the

a priori information on the model parameters and a residual related to the a priori

information on the mutual model parameter combination. The data residual is given by

( ) 21N

1measured

2elmodmeasureddata

)var(N1R

−= ∑

=i i

iid

dd, (7.6)

where N is the number of data points, dimodel is the calculated model response for the ith

data point. dimeasured is the ith measured datum, and var(di

measured) is the variance of the

measured datum. The parameter residual for the apriori information on the individual

parameters is given by

( ) 21P

1apriori

2elmodapriori

parameter)var(P

1R

−= ∑

=j j

jj

p

pp, (7.7)

where P is the number of model parameters , pjapriori is the apriori value of the jth

parameter in the model. pjmodel is the jth model parameter, and var(pj

apriori) is the variance

of the apriori value. The a priori information on the mutual model parameter

combination is given by

( )( )

211L

1apriori

2elmodelmod

sintconstra)ln(var

)ln()ln(1L

1R

σ

σ−σ

−= ∑

−

= +

+

k 1kk,

1kk , (7.8)

where L is the number of layers in the model, σkmodel is conductivity of the kth layer, and

))var(ln( apriori1kk, +σ is the variance of the apriori smoothness constraints of the layers k

and k+1.


____________________________________________________________________________

63

The total residual is a weighted sum of the relevant residuals. For a multiple layer

model with fixed layer boundaries it is given by

( ) 212

sintconstra2datatotal R)1L(RN

1LN1R

⋅−+⋅

−+= (7.7)

For a more detailed description of the code please refer to Effersø et al (1999) and

Foged (2001).

7.2.3 The multiple-layer inversion model

The EM1DINV model consists of a multiple-layer model with fixed boundaries. The

basic idea behind the multiple-layer is to produce a smooth model (de Groot-Hedlin et

al, 1990; Poulsen et al, 1999). The smoothness constraints are introduced as an

uncertainty between conductivities of neighbouring layers. Throughout this thesis the

value of this the uncertainty is 0.3. This ensures stability during the inversion, and

result in responses without unwanted resistivity fluctuations. Tests have shown that the

constraint value does not restrict the variability in the investigated models. This is

mainly due to the usage of a L2-norm during the inversion, which results in relatively

smooth models.

The input model is a 10 ohmm half-space, but for some inversions other half-space

resistivities was chosen. This was due to converging problems for the EM1DINV code.

The layer distribution is arranged so that the uppermost layer thickness is one meter,

increasing in order to a power law, and ending with a 30 meter layer in the 125 to 155

meter depth interval. For all data used with EM1DINV a standard deviation of 5 percent

was used.

The multiple-layer approach produces smooth models, which reveals relatively

small resistivity variations. For 1D interpretation of 3D data this is preferred, because a

conventional inversion model makes it difficult to decide how many layers to use. This

ensures that the interpreter is fully independent and cannot influence the outcome of

the final interpretation.


____________________________________________________________________________

64


____________________________________________________________________________

65

8 The steepness of geological structures The question of when a model essentially can be regarded as one-dimensional

instead of multi-dimensional is of special interest in the use of electromagnetic

methods. The reason for this is that in the interpretation of the data 1D-asumptions are

often made to simplify the solution of the inverse problem. This chapter investigates the

effect of applying a 1D inversion to 3D data from specific sloping structures.

8.1 The Models Three-dimensional responses have been calculated over two opposite resistivity

models each with slopes of 90, 45, 22.5 and 11.25 degrees and a specific resistivity

contrast of 80-10 ohmm. These models are simple, making it easier to isolate effects

and thus achieve an understanding of the method, its limitations and strengths. After

gaining an understanding by investigating a suite of models, the complexity and

dimensional expansion can gradually be increased.

A sketch of the models can be seen in figure 8.1a. The slopes of the structure are

not as ideal as shown in the figure. Except in the case of the 90-degree slope, the

characteristics of the discrete finite difference grid makes it impossible to generate the

ideal sloping structure. Instead it is approximated by several small 10-meters steps.

Plotting each time gate over a whole profile as dB/dt versus profile coordinate (figure

Figure 8.1: a) The outlineof the two opposite modelsin investigating the effectsof a sloping structure. b)Responses calculated overthe 22.5% slope with a 10-ohmm base and 80-ohmmfill-in. The dB/dt plotshows the gates in thetime interval from 5e-6 to2.8e-3 seconds along theprofile. Results from thisdata set can also be seenin figure 8.2c.

10m

70m90o 45o 22.5o 11.25o

80 / 10 ohmm

10 / 80 ohmm

Profile coordinate [m]

a)

b)


____________________________________________________________________________

66

8.1b) gives a smooth curve for each time gate. If the approximations of the slopes were

inadequate the 10-meters step would show up in such a plot as small wriggles

corresponding to the individual steps. Thus, figure 8.1b is a good quantitative proof that

the approximation made is useful. Similar plots for other models investigated in this

chapter produces similar results.

All profiles are 600 meters and responses are calculated at discrete intervals of 20

meters. The model parameters such as number of matrix operations, total width and

depth of the model and vertical nodal distribution were chosen according to the result

achieved earlier. The

numerical values of these

parameters are shown in

table 8.1. The transmitter

is a 40 by 40 meter loop

and the receiver is

positioned 7.07 meter

from the centre of the

loop.

8.2 Sloping structure with a conductive base The 31 responses from each of the four sloping structures with a conductive base

are shown in figure 8.2. The four sub-figures (a-d) each consists of 3 plots. The

uppermost shows the true model, the one-dimensional EM1DINV interpretations are in

the middle and the relative differences between the two plots at the bottom. Just below

the EM1DINV interpretations the data-, resistivity- and total-residuals are shown for

each inversion result.

For a better understanding of the features seen in figure 8.2, a suite of single

responses for each sloping model are plotted in figure 8.3. The responses represents

positions at each side of the upper edge of the slope, and the relevant EM1DINV one-

dimensional responses are plotted alongside to make it easier to identify the 3D

effects.

Parameter Conductive base Resistive base Total width [m] 4920 6180 Total depth [m] 2433 2433 N,M 49 51 L 39 39 Nt 4000 4000 Table 8.1: The basic model parameters used in the investigation of the sloping structure. N,M is the number of nodes in the x,y-direction, while L is the number of nodes in the z-direction. Nt is the number of matrix operations performed by TEMDDD.


____________________________________________________________________________

67

8.2.1 90-degree slope: The one-dimensional regions of the profile

Figure 8.2a shows the results from the 90-degree slope. Clear differences emerge

when comparing the true model with the one-dimensional interpretations. Turning the

attention to the two ends of the profile, where the true model essential is one-

dimensional differences are still observed. The very start of the profile could be

regarded as strict one-dimensional model with a 10-meter resistive layer in top followed

by a conductive base. Thus, one would expect to see a perfect agreement between this

model and the one-dimensional EM1DINV interpretation. Likewise, at the end of the

profile the model is essential one-dimensional, consisting of an 80-meter resistive layer

followed by a conductive base. In both cases the EM1DINV interpretation is not

consistent with the true model.

These differences not are effects of the three-dimensional model, but a

consequence of the choice of the one-dimensional multi layer model. If layer

boundaries in the true model does not coincide with the fixed boundaries in the

EM1DINV model at least one extra layer is needed to make a transition from one

resistivity to another. The resulting resistivity of this extra layer would be a geometric

mean according to the position of the true model layer boundary between the two

EM1DINV layer boundaries. Furthermore, the EM1DINV modelling code uses a L2 –

norm, which consequently produces relatively smooth models. A L1–norm would be

helpful in this investigation because it produces more blocky models. This would make

the resistivity transitions in the one-dimensional part of the profile more ‘blocky’.

8.2.2 90-degree slope: 3D effects observed before the edge of the slope

The effects of the 90-degree slope before the edge of the slope in figure 8.2a have

been marked zone I in the difference plot. At the very beginning of the profile, at

coordinate 700 meters, the model is essentially one-dimensional as described above,

but moving closer to the edge the effects of the 3D start to emerge. Looking at the

difference plot is the easiest way to identify the 3D effect. At profile coordinate 800

meters there are clear differences between the true model and the EM1DINV one-

dimensional interpretation. As one approaches the edge at 990 meters three

developments take place. The amplitude of the anomaly gets bigger, the anomaly

occur closer and closer to the surface, and the depth interval in which the 3D-effect

occurs is narrowed.


____________________________________________________________________________

700 800 900 1000 1100 1200 13000

50

100

150

Dep

th [m

]

0

50

100

150

Em1DInv Model Interpretation

Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference

Residuals: Vres Total Data

TEMDDD Model Space (True Model)700 800 900 1000 1100 1200 1300

0

50

100

150

Dep

th [m

]

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference


TEMDDD Model Space (True Model)

I

II

III

a) b)

800 900 1000 1100 1200 1300 14000

50

100

150

Dep

th [m

]

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference



0

50

100

150

Dep

th [m

]

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference



c) d)

Figure 8.2: The modelling results from four different sloping structures with the same resistivitycontrast.

68


____________________________________________________________________________

69

All three described developments are effects of the three-dimensional structure and

can be separated from the effects of the inversion-model, described early in this

chapter.

At profile coordinate 940 meters in figure 8.2a it is evident that influence from the

three-dimensional edge-structure is considerable. The dB/dt-curve from this response

is seen in figure 8.3a along with the EM1DINV one-dimensional reference response.

The transmitted field propagates down- and outwards, and at early times (before 0.1

ms) there is good agreement between the TEMDDD- and the one-dimensional

reference response. This is corresponding to the upper 25-30 meters of the EM1DINV

interpretation seen at profile coordinate 940m in figure 8.2a. After 0.1ms the

transmitted field has propagated down- and outward to such extend that it has reached

the sharp edge at 990 meters. Because of the high resistivities from that point off, the

proportion of the field travelling in this region induces a much smaller secondary field.

Thus, the sum of the response at that time becomes less than one would expect from

the one-dimensional model. From plot of the relative differences it is evident that the

amplitude of the 3D effect in increased as one moves closer to the edge. This is

caused by the larger and larger proportion of the field that travels in the low conductive

region.

8.2.3 90-degree slope: 3D effects observed after the edge of the slope

Again, looking at figure 8.2a after profile coordinate 990m (zone II and III) it is

evident that the method cannot track this side of the sharp edge. Instead a zone of

intermediate resistivities ranging from 10 to 80 ohmm is present in zone II. This zone

stretches from the upper edge of the slope (x,z) = (990,10) and downward in the

resistive layer with a angle of approximately 40 degrees. At the depth of 80 m, it

coincides with the lower horizontal conductive boundary, and below this level only

small effects in zone III are observed from the three-dimensional structure.

The observed 40-degree angle of the upper part of zone II is produced because extra

response is induced in the conductive part of the vertical wall adding to the responses

after profile coordinate 990 m. The longer one moves away from the edge at 990

meters the later in the time the effect from the conductive wall occurs. The time delay,

at which the effect from the conductive wall occurs, is determined by the resistivity of


____________________________________________________________________________

70

a)

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

Slop

e: 9

0 dg

Bas

e: C

ondu

ctiv

ePr

ofile

coo

rdin

ate:

940

m1D

mod

el p

aram

eter

s:1s

t lay

er:

10m

/ 80

Ωm

2nd

laye

r:10

Ωm

b)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

c)TE

MDDD

3DEM

1DIN

V 1D

Rel D

iff.

d)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

e)TE

MDDD

3DEM

1DIN

V 1D

Rel D

iff.

f)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

TEMD

DD 3D

EM1D

INV

1DRe

l Diff

g)TE

MDDD

3DEM

1DIN

V 1D

Rel D

iff.

0.2

0.1

0 -0.1

-0.2

0.3

0.4

0.5

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

0.6

0.7

0.8

0.9

Relative difference

h)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

0.2

0.1

0 -0.1

-0.2

0.3

0.4

0.5

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

0.6

0.7

0.8

0.9

Relative Difference

Slop

e: 9

0 dg

Bas

e: C

ondu

ctiv

ePr

ofile

coo

rdin

ate:

1040

m1D

mod

el p

aram

eter

s:1s

t lay

er:

80m

/ 80

Ωm

2nd

laye

r:10

Ωm

Slop

e: 4

5 dg

Bas

e: C

ondu

ctiv

ePr

ofile

coo

rdin

ate:

1040

m1D

mod

el p

aram

eter

s:1s

t lay

er:

55m

/ 80

Ωm

2nd

laye

r:10

Ωm

Slop

e: 4

5 dg

Bas

e: C

ondu

ctiv

ePr

ofile

coo

rdin

ate:

940

m1D

mod

el p

aram

eter

s:1s

t lay

er:

10m

/ 80

Ωm

2nd

laye

r:10

Ωm

Slop

e: 2

2.5

dgB

ase:

Con

duct

ive

Prof

ile c

oord

inat

e:96

0 m

1Dm

odel

par

amet

ers:

1st l

ayer

:10

m /

80Ω

m2n

d la

yer:

10Ω

m

Slop

e: 2

2.5

dgB

ase:

Con

duct

ive

Prof

ile c

oord

inat

e:10

60 m

1Dm

odel

par

amet

ers:

1st l

ayer

:45

m /

80Ω

m2n

d la

yer:

10Ω

m

Slop

e: 1

1.25

dg

Bas

e: C

ondu

ctiv

ePr

ofile

coo

rdin

ate:

1120

m1D

mod

el p

aram

eter

s:1s

t lay

er:

40m

/ 80

Ωm

2nd

laye

r:10

Ωm

Slop

e: 1

1.25

dg

Bas

e: C

ondu

ctiv

ePr

ofile

coo

rdin

ate:

960

m1D

mod

el p

aram

eter

s:1s

t lay

er:

10m

/ 80

Ωm

2nd

laye

r:10

Ωm

Figure 8.3: Example of single soundings at each side of the upper edge of the slope. Theupper row contains responses 50 meters before the edge, while the lower row containsresponses from the middle of each slope. For the 90-degree slope the responses aretaken 50 meters at each side of the edge.


____________________________________________________________________________

71

the resistive media. This is because it is within this media that that the field travels

before reaching the conductive wall in which current is induced. The intermediate

resistivities in zone II are due to the arrival of secondary response from both the

resistive and conductive medial at the same time.

In zone III the resistivities are about 15 percent higher than one would expect from

the one-dimensional model due to the extra induces secondary fields in the upper part

of the conductive host before profile coordinate 990m. According to the difference plot

in figure 8.2a an area of well determined resistivities of 10 ohmm exists in the middle

between zone I and III. This area can be regarded as an transition area between zone

I, which contains too little response and zone III, which contains too much response

8.2.4 90-degree slope: Features of the residuals

The data-, total- and vertical resistivity residuals for each of the 1D models are

plotted below the EM1DINV interpretation in figure 8.2a.

The relative variations in the three residual values in figure 8.2a are quite uniform. In

the one-dimensional sections of the profile the values of the data residuals are around

0.25, while the vertical resistivity residuals are around 0.7. The total residuals are

around 0.40. Starting form the left side of the profile these three values stay constant

until around profile coordinate 950 m, where obvious differences start to occur.

Between coordinate 950 and 1010 all three residuals drop drastically to about half of

the initial value, where after they rise again and gain their initial value at coordinate

1110m.

It is interesting that a rise in complexity of the true model yields a lowering of the

residuals as seen between profile coordinate 950 and 1110 m in figure 8.2a. One

would expect that such a rise in complexity would make it more difficult to fit 3D

responses with a 1D model, and thus yield a rise in the residuals. If one looks closer at

the EM1DINV interpretation and remembers that the residual of the vertical resistivity

also was referred to as a measure of the ‘relative smoothness’ the explanation become

obvious. As explained above there exists an area containing intermediate resistivity

values, named ‘zone II’, and it is around this zone one observes the lowering of the

residuals.


____________________________________________________________________________

72

Since the true model only consists of two layers with a varying thickness and a

resistivity contrast of 10 – 80 ohmm, the value of the resistivity residuals must be

controlled by the transition between these two layers. By observing the ends of the

profile, it is seen that the Em1DINV interpretation challenges the applied resistivity

constraints, and yields the same values of the resistivity residual. This is consistent

with the former observation that the transition from 80 to 10 ohmm takes place over the

same number of layers in both ends of the profile, which are essentially one-

dimensional. Because the area between profile coordinate 950 and 1110 m consist of

intermediate resistivities the transition from 80 ohmm to 10 ohmm takes place over

many more layers than at the ends of the profile. This challenges the constraints of the

vertical resistivities less, and therefore yields a lower resistivity residual.

At profile coordinate 960m an example of the just opposite is present. The transition

between the 80-ohmm layer and the 10-ohmm layer still takes place over about 3

layers in the top of the model. The high-amplitude anomaly caused by the lack of

response in the 30 – 60 meter depth interval add to the complexity of the EM1DINV

inversion result. These two effects combined give rise to a decrease in the smoothness

of the model, which is equivalent to a small increase in the vertical resistivity residual

from 0.7 to 0.8.

8.2.5 Decreasing the steepness of the sloping structure

All the variations observed and described in the 90-degree slope are present to

some extend when the steepness of the slope is decreased. This is seen throughout

figure 8.2, ending in figure 8.2d with a slope of 11.25 degree. Furthermore, the

explanations given to the effects in figure 8.2a regarding the resistivity distribution and

residual variation all apply to the other three figures as well.

It is seen by figure 8.2 that the observed differences between the true model and

the EM1DINV interpretation is increasingly smeared as the slope is reduced. In zone I,

the absolute relative differences decreases from more than 50 percent in the 90-degree

slope to less than 10 percent in the 11.25 percent slope. Furthermore, the distribution

of the zone gets more widely spread as the slope is decreased. The responses in the

top row in figure 8.3 supports this observation. The explanation for the increased smear


____________________________________________________________________________

73

of zone I can be found in the fact that the lower the steepness of the slope is the longer

the longer the slope itself becomes.

Zone II is decreased as the slope is decreased, but because of the large amount of

induces response in the conductive media in the 10 – 80 meter depth interval, the

amplitude of the relative difference stays almost constant. Zone III is only moderately

influenced by the decrease in steepness. Relative differences of around 10 to 15

percent are seen at the foot of all four slopes, and the total sizes of the area remain

approximately constant.

In general the observations as the steepness of the slope is decreased can be

summarised to the following short statements:

• In a conductive media the area of reduced response is smeared (zone I). • In a conductive media the relative difference is reduced (zone I). • In a conductive media the area of increased response is constant (zone III). • In a resistive media the area of increased response is reduced (zone II).

As the steepness of the slope is decreased, the behaviour of the residuals is similar

to what was observed for the resistivities: The amplitudes are lowered and the affected

area is increased. This goes for all three residuals. This is particularly interesting when

determining at what slope steepness the model can be regarded at one-dimensional. A

constant residual over the whole profile would indicate that the same number of layers

was needed to make the transition from the 80 ohmm to 10 ohmm. This means that no

intermediate resistivity values were present in the final inversion result and that the

three zones were no longer distinct.

The 11.25 percent slope in figure 8.3d shows just these features. The division into

the three zones is only possible by looking at the plot of the differences. The absolute

values of these differences are below 10 percent. Furthermore, all three residuals

remain constant through out the whole profile. This leads to the conclusion that this

three-dimensional model can be regarded as essentially one-dimensional. Looking at

figure 8.3, it is obvious that the responses on both sides of the upper edge of the

slopes approach their one-dimensional EM1DINV reference responses as the

steepness of the slope is decreased. The two responses from the 11.25 degree slope


____________________________________________________________________________

74

in figure 8.2 shows differences compared to the EM1DINV reference response that are

around five percent, which is equal to the standard deviation assigned to the data.

8.3 Sloping structure with a resistive base The structures of the four models in this section are identical to the ones described

in the former section. The resistivity distribution is reversed so that the models now

have a resistive base and a conductive fill in. The true models, one-dimensional

interpretations and relative differences of the four models are shown figure 8.4. The

models are divided into three zones with each their characteristics.

To give a better understanding of the features seen in figure 8.4 at suite of single

responses for each sloping model have been plotted in figure 8.5. The responses

represent positions at each side of the upper edge of the slope, and the relevant

EM1DINV one-dimensional responses are plotted alongside for easy identification of

the 3D effects.

8.3.1 90-degree slope: 3D effects observed before the edge of the slope

As seen in the difference plot in figure 8.4a, the model has been divided into three

zones. The conductive fill-in after the edge has a substantial influence on the one-

dimensional interpretations. At the very start of the profile, at coordinate 700m, the

influence from the three-dimensional structure is relatively little, but as one moves

closer to the edge, differences start to occur. The differences in the resistive base are

referred to as zone I, and are characterised by intermediate resistivities of about 25

ohmm. As one move closer and closer to the edge the zone stretches closer to the

surface, but still continues down to more than 150 meters depth. The left side of zone I

dip with an approximately 30-degree angel. Figure 8.5a shows an example of a

response from this zone at profile coordinate 960m. As expected the TEMDDD curve

show a lift in the response at relatively late times compared to the EM1DINV reference

response. This is because of the induced response in the conductive fill in after profile

coordinate 1000m.


____________________________________________________________________________

75

Figure 8.4: Graphical representation of the results from four different sloping structures withthe same resistivity contrast.

700 800 900 1000 1100 1200 13000

50

100

150

Dep

th [m

]

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference



50

100

150

Dep

th [m

]

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference



I II

III

800 900 1000 1100 1200 1300 14000

50

100

150

Dep

th [m

]

Relative difference

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference



0

50

100

150

Dep

th [m

]

0

50

100

150


Dep

th [m

]

0

0.3

0.6

0

50

100

150

Difference

Dep

th [m

]

4 6 9 15 25 40 65 110 200

Resistivity [Ohmm]

−0.4 −0.25 −0.1 0 0.1 0.25 0.4

Relative difference



a) b)

c) d)


____________________________________________________________________________

76

Furt

a)

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

Slop

e: 9

0 dg

Bas

e: R

esis

tive

Prof

ile c

oord

inat

e:94

0 m

1Dm

odel

par

amet

ers:

1st l

ayer

:10

m /

10Ω

m2n

d la

yer:

80Ω

m

b)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

c)

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

d)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

e)

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

f)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

g)

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

2 1.5

1 0.5

0 -0.5

-1 -1.5

-2

Relative difference

h)

10-6

10-5

10-4

10-3

10-2

Tim

e [s

]

10-1

0

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

dB/dt

TEMD

DD 3D

EM1D

INV

1DRe

l Diff.

2 1.5

1 0.5

0 -0.5

-1 -1.5

-2

Relative difference

Slop

e: 9

0 dg

Bas

e: R

esis

tive

Prof

ile c

oord

inat

e:10

40 m

1Dm

odel

par

amet

ers:

1st l

ayer

:80

m /

80Ω

m2n

d la

yer:

10Ω

m

Slop

e: 4

5 dg

Bas

e: R

esis

tive

Prof

ile c

oord

inat

e:94

0 m

1Dm

odel

par

amet

ers:

1st l

ayer

:10

m /

10Ω

m2n

d la

yer:

80Ω

m

Slop

e: 2

2.5

dgB

ase:

Res

istiv

ePr

ofile

coo

rdin

ate:

940

m1D

mod

el p

aram

eter

s:1s

t lay

er:

10m

/ 10

Ωm

2nd

laye

r:80

Ωm

Slop

e: 1

1.25

dg

Bas

e: R

esis

tive

Prof

ile c

oord

inat

e:94

0 m

1Dm

odel

par

amet

ers:

1st l

ayer

:10

m /

10Ω

m2n

d la

yer:

80Ω

m

Slop

e: 4

5 dg

Bas

e: R

esis

tive

Prof

ile c

oord

inat

e:10

40 m

1Dm

odel

par

amet

ers:

1st l

ayer

:45

m /

80Ω

m2n

d la

yer:

10Ω

m

Slop

e: 2

2.5

dgB

ase:

Res

istiv

ePr

ofile

coo

rdin

ate:

1060

m1D

mod

el p

aram

eter

s:1s

t lay

er:

45m

/ 80

Ωm

2nd

laye

r:10

Ωm

Slop

e: 1

1.25

dg

Bas

e: R

esis

tive

Prof

ile c

oord

inat

e:11

40 m

1Dm

odel

par

amet

ers:

1st l

ayer

:50

m /

80Ω

m2n

d la

yer:

10Ω

m

igure 8.5: Example of single soundings at each side of the upper edge of the slope. Thepper row contains responses 50 meters before the edge, while the lower row containsesponses from the middle of each slope. For the 90-degree slope the responses areaken 50 meters at each side of the edge.


____________________________________________________________________________

77

8.3.2 90-degree slope: 3D effects observed after the edge of the slope

The intermediate resistivities of zone I continue approximately 75 meters after the

edge of the slope because of the lack of response. This is due to the surface near

resistive base before profile coordinate 1000m. This, combined with the description of

the effects in zone I before the edge of the slope, lead to the following conclusion:

Before the edge the intermediate resistivities are caused by extra induced response in

the conductive fill-in, while they are caused by the lack of response after the edge of

the slope because of the shallow resistive base.

After the right flank of zone I, a swift transition in the resistive media leads to zone

III, which is characterised by a lack of response compared to the true model. The zone

contains resistivities in the range between 80 and 150ohmm, with the lowest

resistivities at the bottom centre of the semicircle-like area. The high resistivities in the

centre of zone III are caused by the lack of response as the transmitted field

propagates trough the conductive 80meter-layer around profile coordinate 1150m and

enters the shallow resistive base before profile coordinate 1000m. At the far right side

of the zone it approaches the one-dimensional model.

Like zone III, zone II is characterised by higher resistivities of around 25 ohmm,

which is higher than expected compared to the true model. These relatively high

resistivities are caused by the lack of response as the primary field enters the shallow

resistive base before profile coordinate 1000m. The extension of the zone stretches

vertically from the surface down to a depth of 80 meters, and horizontally from profile

coordinate 1000m to 1175m forming a triangular shape. This lack of response is also

recognised in the response plot in figure 8.5b, which shows a single response at profile

coordinate 1040m. The plot clearly shows a lower TEMDDD response compared to the

EM1DINV reference response at times later than approximately 0.2 milliseconds.

8.3.3 90-degree slope: Features of the residuals

The three residuals seen in figure 8.5a show patterns, which are similar to the

observed in the example with the conductive base in figure 8.2. The absolute values of

the constant level residuals are also comparable; the vertical resistivity residual and the

data residual are around 0.6 and 0.2, respectively, while the total residual is around

0.45. In contrast to conductive base with resistive fill-in, the anomaly in the residuals


____________________________________________________________________________

78

not only is restricted to the narrow area around the edge of the slope; changes in the

residual are is also observed in area of relatively low response between profile

coordinate 1100 and 1250m. In this profile interval a slight rise in the residuals occurs.

This is caused by the decreased ‘smoothness’ of the model due to the low resistivities

in zone the resistive base (zone III).

8.3.4 Decreasing the steepness of the sloping structure

As the steepness of the slope is decreased from 90 degrees in figure 8.4a to 11.25

percent in figure 8.4d, the amplitude of the observed 3D effects are decreased. In

figure 8.5d, 8.4f and 8.5h the transition from the area of reduced response in zone II,

and the area of increased response in zone I is visible as a characteristic cross-over

between the TEMDDD- and the EM1DINV reference response. From the comparison

between these two responses, it is clear that the relative differences between the

curves are decreased as the steepness of the slope is decreased.

The model The TEMDDD responses never reach the same level of one-

dimensionality as was seen in the case with the conductive base. With an 11.25

percent slope there are still relatively considerable differences between the true model

and the 1D interpretation. This is seen in figure 8.4d and plot 8.5g,h.

The residuals reflect the observed changes in the resistivity distributions. As the true

model approaches the one-dimensional, the 25ohmm intermediate resistivities of zone

I and II approaches the true value of the resistive base. This leads to a decrease in the

smoothness of the model, and thus the overall fall in the residuals becomes less

profound.


____________________________________________________________________________

79

9 The influence of a small near-surface resistivity variation on early-time data

The objective of this investigation is to describe the effects of a near-surface

resistivity variation on the early time data. This is done for a central loop configuration

and a 55 meter offset configuration, using z-component data. Furthermore, an

investigation of the use of the total field-components instead of the conventional z-

component in the inversion is presented. For the central loop configuration data is

plotted at the centre of the transmitter loop. For all offset configurations the plotting

point is midways between the transmitter and receiver.

9.1 Background When using relatively high magnetic moments in a central loop configuration,

undesired effects can arise. Firstly, electronic challenges occur in constructing the

receiver equipment because the spike coming

from the turn-off of the primary field makes the

receiver amplifiers unstable. Furthermore, leak

currents in the transmitter coil after turnoff adds to

the secondary response and makes the recorded

data unreliable. Secondly, in some geological

sediment induced polarisation effects (IP) occur

(Flies et al, 1989). This effect is most profound in

the central loop configuration and moves to later

times as the receiver is moved further away from

the transmitter. These effects can be avoided by

moving the receiver coil well outside the

transmitter coil.

As the time-derivative of the z-component has

traditionally been used for the central loop

configuration, this has also been adapted to the

offset configuration. This posses two important

problems. Firstly, the existence of a sign-reversal

Figure 9.1: The influence of errorin the receiver offset. Thecalculated relative difference isbetween the 55m and the 56.6moffset.

10-5 10-4Time [ s ]10-7

10-6

10-5

10-4

dB/d

t

-0.8

-0.4

0.0

0.4

0.8

Rel

ativ

edi

ffere

nce

55m offset, dBz/dt55.5m offset, dBz/dt56.5m offset, dBz/dt58m offset, dBz/dtRel. diff. 55m - 56.5m

Res [Ωm]: 50 (½-space)Offset [m]: 55-58Trans.[m]: 40x40Compon. : dBz/dt


____________________________________________________________________________

80

in the time-derivative of the z-component makes the configuration extremely sensitive

to errors in the transmitter-receiver separation. This is illustrated in figure 9.1 where it is

evident that deviations from the assumed offset distance of just a few tens of

centimetres gives relatively large errors compared to the expected response. In the

figure the relative difference between the z-component responses for a 55m and a

56.5m offset, corresponding to an offset error of less than three percent, yields relative

differences of up to 100 percent around the sign reversal. At later times the errors from

the offset variations becomes smaller and smaller.

Secondly, the configuration has a high sensitivity to lateral resistivity variations in

the near-surface region. This often makes it impossible to perform a usable 1D

inversion and interpretation of the data-set without cutting considerable data in the

early times.

9.2 The Models The TEMDDD models consist of a 50-ohmm, homogenous half-space with a

quadratic box with a side length of 20 meters placed with its centre in a depth of 15

meters. The box is positioned between profile coordinate 550 and 520m. Two different

resistivities of 25 ohmm and 100 ohmm are assigned to the block to create a high- and

low-resistivity contrast. The model is shown in figure 9.2.

Furthermore, an equivalent sheet-model is generated using the Australian LEROI

code (Weidelt, 1983; Raiche et al, 1999). This is done to obtain an equivalent B-field,

which the TEMDDD is not capable of calculating. Using the B-field instead of the dB/dt

field makes the understanding of the offset configuration easier, and helps explaining

the observed effects of the small resistivity variation. The sheet is assigned the same

conductivity as the TEMDDD cube and is positioned as shown in figure 9.2.

True three-dimensional TEMDDD responses are calculated over a 130-meter profile

at discrete intervals of 10 meters. All input parameters used in the TEMDDD

calculations are chosen according to the experiences gained earlier in this thesis. A

TEMDDD homogeneous half-space response is used for calculating the 3D effect of

the resistivity variation at selected positions along the profile. The one-dimensional

EM1DINV interpretations are done with a 20-layer model.


____________________________________________________________________________

81

The transmitter-receiver configurations used are the conventional 40 by 40-meter

central loop and a 40 by 40-meter transmitter with a 55m offset receiver. The

transmitter passed directly over the imbedded cube in such manner that at profile

coordinate 510m the transmitter completely surrounds the cube with a ten-meter

distance from the edge of the cube to the transmitter wire. The receiver is laid out to

the left of the transmitter and passes over the cube at a distance of 5 meters from the

one edge.

For the central loop configuration, data is plotted at the centre of the transmitter

loop. For all offset configurations the plotting point is midways between the transmitter

and receiver. This terminology kept throughout the rest of this thesis.

9.3 The general behaviour of TEM fields for shallow resistivity variations

To explain the observed effects from a three-dimensional resistivity variation in a

half-space, it is important to understand the way the secondary magnetic field behaves.

Because of the complexity of the response, especially for the offset configuration, the

description is given with reference to the induced secondary field (B-field) instead of

the time-derivative of the field (dB/dt). Just after the current is turned off the secondary

field is downward positive inside the transmitter coil and negative outside the coil. As

time passes the current passes down- and outward and at some point in time the

z=0

Top view

Side view

TEMDDD

LEROI

Tx edge

Rx

cube

sheet

10m

10m

Figure 9.2: The outline of the model including the TEMDDD cubic body and theequivalent LEROI sheet.


____________________________________________________________________________

82

secondary field outside the transmitter becomes positive as it passes under the

observation point.

For a central loop configuration the receiver is positioned in the centre of the

transmitter, and the primary field is downward positive at this point at all times. For a

resistivity decrease in a region near the transmitter, the induced secondary field is also

reduced, and vices versa. All together, a traverse over a shallow 3-D body yields a

peak in the response when the transmitter and receiver are positioned directly above it.

One special feature for the central-loop configuration is that for very small, shallow

resistivity variations (e.g. a thin sheet) inside the transmitter-loop the effect of the body

is decreased, since the field travels downwards and outwards from the transmitter wire

(Spies, 1974).

For the offset configuration the situation gets more complex when dealing with 3-D

bodies at different positions in and around the transmitter-receiver system. In figure 9.3

three different scenarios are presented. The schematic model shows an offset

configuration with the transmitter (Tx) and receiver (Rx) on the ground. The imbedded

three-dimensional resistivity variation is resembled by a coil, which is excited by the

transmitter. The arrows represent the induction in the ground after transmitter turn-off,

and after the secondary magnetic field has passed under the receiver coil. It is

important to note that for small resistivity variations, the three-dimensional resistivity

variation and the host can be regarded as two separate ‘systems‘, which do not interact

after the current is turned off in the transmitter-coil (Esben Auken, personal

communication, Oct. 2001).

In figure 9.3a the ‘body’ is positioned to the left of the Tx-Rx layout. At the turn-off of

the transmitted current in Tx, a negative secondary magnetic field is induced in the

small body, causing a current to flow within it. This causes a positive field-component in

the receiver-coil (B), which consequently adds to the secondary field arising from the

induction in the host. Assuming that the secondary B-field has already passed the

offset receiver at the earliest recording-time (which for example is just the case for

receiver offsets of less than 55 m on a 50 ohmm half-space at 0.005ms), the secondary

B-field (H) from the host will also be positive. This has the consequence that no matter

if the 3-D body is more or less resistive than the host, the recorded B-field at early

times will always be higher than B-field from the host alone. This is somewhat different


____________________________________________________________________________

83

from the central-loop configuration where a higher resistivity yielded a lower induction

in the receiver coil.

In figure 9.3b the transmitter-receiver layout is moved so that the small resistivity

variation is positioned in the middle of it. The induced secondary host-response is still

positive at the receiver position (H), while the response from the body (B) is negative at

that position. Again this result is different from what one would observe in the central-

loop configuration because a conductive body between the transmitter and receiver

actually will decrease the total response from the host and body. In figure 9.3c the

small body is moved to the right of the transmitter-receiver layout, and the resulting

response arise exactly as explained for the situation in figure 9.3a.

Altogether, a densely sampled profile over some small resistivity-variation would

yield a small positive peak just as the transmitter passes over the body. This is followed

by a negative peak compared to the background responses as the area between the

transmitter and receiver passes over the resistivity-variation. Moving the system further

over the resistivity-variation a small positive peak will occur again as the receiver

passes over it. This pattern of positive-negative-positive is characteristic for the TEM

offset configuration, and examples of this will be given over the following sections.

H

H

H

B

B

B

TxRx

TxRx

TxRx

Body

Body

Body

a)

b)

c)

Figure 9.3: The schematic outline ofmeasuring the secondaryfield over a small three-dimensional body. Thefigure shows the effect oftraversing over the bodyin three steps. Thesymbol by the coilsrepresents the directionof the current (⊗ is thedirection in the paper). Band H show the directionof the response from thebody and host,respectively.


____________________________________________________________________________

84

a)

10-6

10-5

10-4

10-3

10-9

10-8

10-7

10-6

10-5

10-4

10-3

dB/dt

b)

10-6

10-5

10-4

10-3

c)

10-6

10-5

10-4

10-3

0.2

0.1

0.0

-0.1

-0.2Relative Diff.

10-6

10-5

10-4

10-3

d)e)

10-6

10-5

10-4

10-3

10-8

10-7

10-6

10-5

10-4 dB/dt

f)

10-6

10-5

10-4

10-3

g)

10-6

10-5

10-4

10-3

0.2

0.1

0.0

-0.1

-0.2Relative Diff

10-6

10-5

10-4

10-3

h)

TE

MD

DD

3D

TE

MD

DD

Hal

f-sp

ace

Rela

tive

Diff

eren

ce

Rx

Tx

Figu

re 9

.4: T

he C

entr

al-lo

op (l

eft s

ide)

-and

offs

et (r

ight

sid

e) in

terp

reta

tions

for a

100

ohm

m c

ube

in a

50

ohm

m h

alf-s

pace

.


____________________________________________________________________________

85

9.3.1 100 ohmm resistivity variation, central loop configuration.

The results from the 3D TEMDDD forward calculations for the 100 ohmm resistivity

variation is shown in figure 9.4, left side. Four time-plots at the top the models show the

TEMDDD response at specific positions along the profile compared to the TEMDDD

half-space response. The relative difference between the two is plotted alongside. The

transmitter and receiver positions of the plotted responses are shown just above the

true model plot and below this the EM1DINV interpretations, the residuals and the plot

of the relative differences.

Through plot a) to d) it is obvious that the largest effect from the resistive cube is

observed when the transmitter fully surrounds it and the receiver is placed above its

centre. This is seen in plot a), where the relative differences between the three-

dimensional TEMDDD response and the TEMDDD half-space response is -5 percent at

the earliest times. Thereafter the relative difference steadily diminishes and by 0.1

millisecond it is below –1 percent and essentially equal to the half-space response.

Moving the transmitter-receiver layout 20 meters to the right, and thereby placing

the left side of the transmitter loop directly above the centre of the cube, moves the

time of maximum relative difference to about 0.01 millisecond as seen from plot b). The

constant difference value is interesting because the receiver no longer is positioned

above the cube but 15 meters from the edge at profile coordinate 535m. The reason for

this is that a relatively large proportion of the received secondary field still originates

from the cube because the left side of the transmitter wire is placed directly above it.

The observed time shift between the time of maximum difference from plot a) to b) is

due to the increased distance between the resistive cube and the receiver.

At plot c) the transmitter-receiver layout is moved another 20 meters, so that the

nearest loop side is positioned 10 meters from the edge of the box. At the earliest time

there is no relative difference, but it rises and peaks at 0.02 milliseconds with a

difference of 2 percent. After this it steadily diminishes and after 0.1 millisecond the

difference is below one percent. The peak of maximum difference is further shifted in

time compared to plot a) and b) because of the increased distance between the

resistive cube and the receiver. The maximum is also decreased to about half of the

value compared to plot a) and b). This is caused by the increased distance between

the receiver and the resistive cube. This effect is also recognised in plot d), where the


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86

nearest loop side is 30 meters from the edge of the cube. The time of maximum

difference is delayed further to about 0.05 milliseconds, and the absolute amplitude is

decreased to less than one percent.

The one-dimensional EM1DINV interpretations clearly show the central loop

configuration’s relatively low sensitivity to shallow resistivity variations. Because of the

small differences between the TEMDDD and EM1DINV code, which were described in

the earlier chapter on verifying the code, the effect of the resistive cube is slightly

underestimated in the inversion.

The true model is not recognised in the interpretations, and only by looking at the

plot of the differences a clear pattern of the effects of the resistive cube emerges. It is

seen that the maximum effect of the cube is observed when either side of the

transmitter loop is placed above the resistive cube and when the transmitter-receiver

layout is on either side of the cube. This is observed at either side of the cube at profile

co-ordinate 455 to 485m and 535 to 555m. The cube’s effects on the one-dimensional

interpretations between profile co-ordinate 485 and 535m is relatively small, which is

due to the central loop configurations’ relatively low sensitivity inside the transmitter

loop. This is parallel to observations done by Spies (1974) using a thin sheet

approximation, which simulated a vertical dike. The residual profile plotted below the one-dimensional interpretations does shows

little variation. Both the values of the vertical resistivity, the data and consequently the

total residual are all relatively low throughout the whole profile. The values of these

three residuals are all around 0.1, which suggest a good data fit between the real data

and the inversion result.

9.3.2 100-ohmm resistivity variation, offset configuration

Figure 9.4, plot e) to h), shows four responses using a 55 meters offset

configuration over the same 100 ohmm cube described before. The symbols above the

true model explain the precise positions of the transmitter wire and the receiver.


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87

In figure 9.5a the earliest times of the B-field result from the equivalent sheet-

calculations are plotted. A blue line indicates the position of the sheet, and the vertical

arrows mark the position of the soundings in figure 9.5b.

At plot 9.4e, the right side of the loop is positioned just above the resistive box. The

55-meter offset receiver is laid out at a 75-meter distance from the edge of the box.

The observed differences are relatively low, which is due to the offset configuration’s

low sensitivity to lateral resistivity variations outside the transmitter-receiver layout.

Figure 9.4.e corresponds to a position at figure 9.5 between mark 1) and 2), which

shows a slight increase in response.

In plot 9.4f the small negative resistivity variation is now positioned inside the

transmitter loop. The observed differences compared with the half-space response are

considerably larger than in plot 9.4e because the box now is positioned between the

receiver and the transmitter. Comparing the 3-D TEMDDD response with the TEMDDD

half-space response makes it difficult to directly explain the presence of the small

resistive body because it is the time-derivative of the B-field that is plotted. The sheet

calculations of the B-fields in figure 9.5 makes the observed effect easier to explain.

The equivalent B-field response in figure 9.5a is found at profile coordinate 490m in

figure 9.4, right side. It clearly shows that the B-field response is decreasing as the

sheet gets closer and closer to the middle between the transmitter and receiver.

In plot 9.4g the transmitter-receiver layout is moved 20 meters further to the right so

that the resistive box is positioned just to the left of the transmitter, which means that

the box is just in the middle between the transmitter and the receiver. As seen from the

420 440 460 480 500 520 540 560 580Profile coordinate [ m ]

0

100

200

300

B-fi

eld

[ pT

]

0.006 ms0.0065 ms0.007 ms0.0075 ms0.008 ms0.009 ms0.01 ms0.011 ms1) 2)

3)

0.01 0.1Time [ ms ]

100

200

300

908070605040

B-fi

eld

[ pT

] 1)2)

3)

a) b)

Figure 9.5: a) A perpendicular B-field profile over a 20x20 m vertical sheet with aconductance of 0.2 S in a 50 ohmm half-space. The arrows mark the middle of the Tx-Rxlayout for the soundings plotted in figure 9.4 (right side). b) The Three individualsounding at positions marked in figure 9.5a. The grey area indicates the time interval forwhich the data is plotted in the profile.


____________________________________________________________________________

88

plot in figure 9.4g relatively large differences are observed compared to the TEMDDD

half-space response, and from the EM1DINV interpretations it is evident that it is this

area that the maximum sensitivity is present. The equivalent B-field sheet response in

figure 9.5 is positioned at mark 3). The sheet-responses in this area are almost

homogeneous at ten meters to each side of the sheet. This is probably caused by the

fact that it is a vertical sheet, and that it is not a perfect representation of the

20x20x20m cube. The B-field response from the cube would probably yield a more

peaking response at the central of the cube because of the wider spatial distribution of

the conductance.

In figure 9.4h the transmitter-receiver layout is moved further to the right and the

receiver is now positioned 5 meters to the right of the cube’s right edge. The dB/dt-

response shows a lower TEMDDD 3D-response than the TEMDDD half-space

response after the sign reversal. The equivalent LEROI B-field response is located at

profile coordinate 550m in figure 9.5 and is almost equal to 2) in the same figure.

Parallel to the explanation given for figure 9.3a, a small increase in the B-field response

is observed at this position.

When comparing the EM1DINV interpretation of the central-loop to the offset

interpretations (figure 9.4), distinct differences are observed. For the offset

interpretation, is clear that there is a high sensitivity to the specific shallow, resistive

inhomogeneity. The position of the centre of the inhomogeneity, plotted in the middle

between the transmitter and receiver, is a relatively good estimation. The overall spatial

distribution of the body is also corresponding to the true model, but the resistivity of the

inverted model is around 65 ohmm, which does not fit with the 100ohmm in the true

model. This underestimation of the resistivity is a clear symptom of the 1D

interpretation of the 3D data. If the difference-plot at the bottom of figure 9.4, (right

side) is considered it is seen that differences from the true model also is observed in

the 30-150 meter depth interval, below the resistive cube. These differences are

relatively small (below +/- 10 percent) and are also a symptom of the 1D interpretation

of the 3D data.

A notable feature of the residuals is the unexpected increase in the residuals at

profile-coordinate 555m, which is caused by the described increase in the shallow

resistivity values. Test, where the 40x40m transmitter was replaced by a 10x10m,


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89

yielded perfect symmetry for the residuals around 510m. This suggests that observed

effects are caused because does not represent a perfect dipole. This is consistent with

observations made by Spies et al (1991) pp 335, which states that transmitter-receiver

separations must be at least five times the transmitter loop diameter to yield differences

to an dipole of less than one percent.

Although the EM1DINV interpretations clearly show the presence of a resistive body

it is important to note the behaviour of the residuals plotted below. Before profile-

coordinate 480m the data-, Vres-, and total-residual assume relatively low values of

about 0.15. As the middle of the transmitter-receiver layout is positioned over the left

edge of the cube and moves over it, the residuals increase in value. These peak when

the cube is positioned in the middle between the transmitter-centre and the receiver at

510m. The values of the data-, Vres- and total-residual at this point are around 0.6, 0.2

and 0.85, respectively.

The early time of this TEMDDD response is

plotted with the result from the EM1DINV

interpretation at profile coordinate 510m in

figure 9.6. The standard five percent error-bars

are plotted with the TEMDDD-response and

the poor data-fit clearly show up just after the

sign-reversal and reveals that a 1D

interpretation in this case is inadequate. The

maximum difference is 15 percent. In a ‘real-

data’-situation the interpreter would be forced

to cut considerable in the data to achieve a

better fit. The editing of the data set would

typically consists of cutting all data-points

before the peak of the positive part of the

response (In his case until around 2e-05

second). This will lead to a better data-fit, but

will also cause early-time information about

one-multidimensional parts of the response to

be lost. This is consistent with recent

observations done by Krivochieva et al (2001).

10-5 10-4Time [ s ]

10-7

10-6

10-5

10-4

dB/d

t

TEMDDD 3D dataEM1DINV inversion result

Figure 9.6: The TEMDDD data fromthe resistive cube at profilecoordinate 510m plotted alongsidewith the EM1DINV inversion result.Error bars are +/- five percent.


____________________________________________________________________________

90

a)

10-6

10-5

10-4

10-3

10-9

10-8

10-7

10-6

10-5

10-4

10-3

dB/dtb)

10-6

10-5

10-4

10-3

c)

10-6

10-5

10-4

10-3

0.2

0.1

0.0

-0.1

-0.2Relative diff.

10-6

10-5

10-4

10-3

d)e)

10-6

10-5

10-4

10-3

10-8

10-7

10-6

10-5

10-4 dB/dt

f)

10-6

10-5

10-4

10-3

g)

10-6

10-5

10-4

10-3

0.2

0.1

0.0

-0.1

-0.2Relative diff.

10-6

10-5

10-4

10-3

h)

TE

MD

DD

3D

TE

MD

DD

Half-

space

Rela

tive

Diff

ere

nce

Rx

Tx

Figu

re 9

.7: T

he C

entr

al-lo

op (l

eft s

ide)

-and

offs

et (r

ight

sid

e) in

terp

reta

tions

for a

25

ohm

m c

ube

in a

50

ohm

m h

alf-s

pace

.


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91

As the centre of the transmitter-receiver layout is moved away from the centre of the

resistive cube, the fit to the 1D-model gets better and the residuals decrease.

9.3.3 25-ohmm resistivity variation, central loop configuration

Figure 9.7 (left side) shows the 25 ohmm 20-meter cube imbedded in a 50 ohmm

half-space and the result from the central loop calculations.

Through plot a) to d) it is obvious that the largest effect from the conductive cube is

observed when the transmitter fully surrounds it, and the receiver is placed above its

centre. This is seen in plot a) where the relative difference between the three-

dimensional TEMDDD response and the TEMDDD reference half-space response

shows that the half-space response is about ten percent higher at the earliest times.

After this, the relative difference steadily diminishes and by 0.1 millisecond it is below

one percent and essential equal to the half-space response.

Moving the transmitter-receiver layout 20 meters to the right thereby placing the left

side of the transmitter loop directly above the centre of the cube moves the time of

maximum relative difference to about 0.015 millisecond as seen in plot b). The

observed time shift between the time of maximum difference from plot a) to b) is due to

the increased distance between the resistive cube and the receiver.

At plot c) the transmitter-receiver layout is moved another 20 meters, so that the

nearest loop side is positioned 10 meters from the edge of the box. At the earliest time

there is no relative difference, but it rises and peaks at 0.025 milliseconds with a

difference of 7 percent. After this it steadily diminishes and after 0.1 millisecond the

difference is below one percent. The peak of maximum difference is shifted further in

time, compared to plot a) and b). This is because of the increased distance between

the resistive cube and the receiver. The maximum is also decreased to about half of

the value compared to the observed from plot a) and b). This is caused a consequence

of moving the receiver further away from the resistive cube. This effect is recognised in

plot d), where the nearest loop side is 30 meters from the edge of the cube. The time of

maximum difference is shifted further to about 0.05 milliseconds, and the absolute

amplitude is decreased to about 1 percent.


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92

The one-dimensional EM1DINV interpretations clearly show the central loop

configuration’s relatively low sensitivity to shallow resistivity variations. The true model

is only weakly recognised in the interpretations, and only by looking at the plot of the

difference plot, a clear pattern of the 3D effects emerges. The difference-plot show

resistivity values in the upper five meters of the 475-535m profile-interval that are ten to

fifteen percent higher than the 50-ohmm background. Between profile coordinate 460m

and 555m in the 30 to 70 meter depth the difference plot shows the effect of the

conductive cube. At an proximate angle of 45 degrees the induced response in the

cube arrive later and later in each response as one moves further away from the cube.

This effect was also thoroughly described in the earlier chapter on different sloping

structures, and is consistent with results of Auken (1995) and Farquharson (1999).

The values of the residuals relatively low throughout the whole profile. The values

of these three residuals are all around 0.1, which suggest a good data fit between the

real data and the inversion result.

9.3.4 25-ohmm resistivity variation, offset configuration

Figure 9.7, plot e) to h) show four responses using the 55 meters offset

configuration over the same 25 ohmm cube as described before. Again, the symbols

above the responses show the exact position of the transmitter and receiver for each

response.

At plot e) the transmitter wire is laid out so that the right side of the loop is

positioned just above the conductive box. The 55-meter offset receiver is positioned at

a 75-meter distance from the edge of the box. The observed differences are relatively

low, before and after the sign-reversal, which is due to low sensitivity of the offset

configuration to lateral resistivity variations outside the transmitter-receiver layout.

Figure 9.8 show the results from the equivalent B-field LEROI sheet model. The

position of the response in 9.7e is between mark 1) and 2) (profile coordinate 455m),

which show a slight increase in response. This effect further increased as the

transmitter-receiver layout is moved closer to the cube in figure 9.7f, and is consistent

with the explanations for figure 9.3c.


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93

In Plot 9.7f the conductive cube is positioned inside the transmitter loop. As

explained in figure 9.3b the response is decreased for a conductive body positioned

between the centre of the transmitter and the receiver, which is also seen in figure 9.8

(profile coordinate 495m).

Plot 9.7g shows the response from profile coordinate 515m, where the conductive

cube is positioned halfway between the receiver and the center of the transmitter.

Relatively large differences are observed between the TEMDDD 3D- and the TEMDDD

half-space response, which again reflect the high sensitivity in this region. The

equivalent B-field LEROI sheet model in figure is seen in figure 9.8 (mark 3) ). The B-

field responses between profile coordinate 490m and 530m are relatively

homogeneous. This is because all the conductivity is concentrated in a very small

interval in the profiling direction. The B-field response for the true 3D TEMDDD

response would probably exhibit a more peaky B-field response around profile

coordinate 510m.

In plot 9.7h the transmitter-receiver layout is moved further to the right, and the

receiver is now positioned five meters right to the cube’s edge. The plot show

increased values in the dB/dt-values after the sign-reversal. The equivalent B-field

LEROI sheet-response in figure 9.8a at profile coordinate 555m also show an

increased response.

If the EM1DINV interpretations are compared to the true model there are some

differences. In the upper five meters of the profile interval 485m-510m, the resistivity

values are slightly overestimated at values around 70ohmm. In the 5 – 25 meter depth

420 440 460 480 500 520 540 560 580Profile coordinate [ m ]

0

100

200

300

B-fi

eld

[ pT

]

0.006 ms0.0065 ms0.007 ms0.0075 ms0.008 ms0.009 ms0.01 ms0.011 ms

0.01 0.1Time [ ms ]

100

200

300

908070605040

B-fi

eld

[ pT

] 1)2)

3)

b)a) 1) 2)

3)

Figure 9.8: A perpendicular B-field profile over a 20x20 m vertical sheet with aconductance of 0.8 S in a 50 ohmm half-space. The arrows mark the middle of the Tx-Rxlayout for the soundings plotted in figure 9.7 (right side). b) The three individualsounding at positions marked in figure 9.8a. The grey area indicates the time interval forwhich the data is plotted in the profile.


____________________________________________________________________________

9

interval the spatial distribution of the conductive body is actually well resolved, though

the body seems a more elongated. The resistivity value at the centre of the body only

differs from the true value by a few percent. In the 25-70 meter depth interval between

profile coordinate 485m and 535m, an area or relatively high resistivity is observed.

The area is distributed in the same profile interval as the conductive body. Like in the

shallow region the resistivities in the centre of this resistive body are around 70ohmm.

Although the 1D EM1DINV interpretations clearly show the presence of ‘some’

resistivity variation, the result is ambiguous because of the presence of the high-

resistivity areas above and below the well-resolved resistive cube. This ambiguity is

further enforced by the fact that all three residuals show extreme values in the area

around the conductive body. This area is between profile coordinate 480m and 555m,

where the cube is positioned between the transmitter and receiver. The ‘smoothness’

of the 1D models in this area, represented by the Vres-residual show maximum values

of about 0.4, while the total- and data-residual show maximum values that both

exceeds 1.5.

These high residual values cannot be

lowered in any other way than cutting data

away. Practically the interpreter will remove all

early time data until the maximum after the sign-

reversal, and thus loose the early time

information from the sounding. This will result in

lower residuals, but does not solve the fact that

a 1D model does not satisfy the 3D data set.

This is one of the most important problems in

the interpretation of z-field offset TEM data sets.

Figure 9.9 show the result of the inversion from

at profile coordinate 510m. The reasons for the

high residual values are clear when comparing

the TEMDDD data-set with the five percent

error-bars to the result of the 1D EM1DINV

inversion result. At the peak after the sign-

reversal there are more than 27 percent

FtcwE

10-5 10-4Time [ s ]

10-7

10-6

10-5

10-4

dB/d

t

TEMDDD 3D dataEM1DINV inversion result

igure 9.9: The TEMDDD data from

he conductive cube at profileoordinate 510m plotted alongsideith the EM1DINV inversion result.rror bars are +/- five percent.

4

difference between the two responses.


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95

9.3.5 Summary

The investigation of the two different resistivity-models for two transmitter-receiver

configurations clearly illustrates the difficulties that arise when moving the receiver

outside the transmitter loop. Compared to the central-loop configuration even a

relatively small resistivity variation between the transmitter and receiver will act as a

source of geological noise when using the offset configuration. Equally important, the

interpreter is discouraged because of high residual values and the confidence in the

data set is reduced. When cutting in the data, tests have shown that the final result of

the interpretation is very dependent on how much data is cut away. In the best case the

early-time (shallow) information is reduced, but in the worse case the interpretation will

give misleading results for the shallow regions.

9.4 Using total-field data for the offset configuration

9.4.1 introduction

Using z-component offset data involves challenges that are very hard to come

around when measuring over a three-dimensional earth. Smith et al (1996) and

Poulsen (2000) describe similar problems for time-domain airborne electromagnetics

(AEM). In these papers the emphasis is put on the use of all three dB/dt-field

components combined to one total-field-component. This total-field-component,

referred to as Energy Amplitude (EA), is calculated very simple as the length of the

resulting vector when adding the three field-components.

2z

2y

2x BBBEA

)))++= , (9.1)

where the ‘ ^ ’ denotes the time-derivative of the B-field component.

The recent ground TEM systems have become more and more AEM-like. E.g. the

PATEM system is fully automated in respects to receiver gain, transmitter moment and

spatial sampling rate. Furthermore, the high transmitter moments have required the

receiver sensor to be positioned offset to the transmitter. In some sense all this puts

the PATEM-system in the same category at the AEM-systems. Therefore, there is no

reason why the practical improvements in the latter should not be applied to the


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96

PATEM-system if it seems sensible. In figure

9.10 all three components of the response in

figure 9.7g are plotted. The z-component used

earlier is plotted in green, while the x- and y-

component is plotted in blue and red,

respectively. Finally the calculated EA-field is

plotted in black. In the small info-box the

precise offset position of the receiver is

specified. This is important in regards to the y-

component, which is zero for one-dimensional

models with the receiver directly in the y-

direction. Because the receiver is positioned 5

meter offset in the y-direction it is not always

zero. However, changes in its value can still be

used as an indicator of lateral resistivity

variations that is not symmetric about the line of

traversing Smith et al (1996).

9.4.2 EA-field inversions for two shallow resistivity variations

In the following, the EA-fields from the same two simple models as described earlier

in this chapter are investigated. The EA-fields are calculated by using equation 9.1 on

the derivative of the three B-field components from TEMDDD. In figure 9.11 the result

from these calculations are shown. To the left is the resistive 100 ohmm block in a 50

ohmm half-space, and to the right is the conductive 25 ohmm block in a 50 ohmm half-

space. The data from the EM1DINV interpretation is plotted midways between the

centre of the transmitter and the receiver.

In plot 9.11a the right side of the transmitter-loop is positioned on the left edge of

the resistive cube. Relative differences compared to the half-space of minus two

percent are observed. Moving the transmitter-receiver system further to the right (figure

9.11b) move the negative peak in difference to later times but maintains the maximum

difference at around two percent. Furthermore, the plot shows a higher response at

the earliest times. Plot c in figure 9.11 show the response from the resistive cube

positioned between the transmitter and receiver. Compared to the half-space response,

10-5 10-4Time [ s ]

10-7

10-6

10-5

10-4

dB/d

t

dBx/dtdBy/dtdbz/dtdBtotal/dt

Res [ohmm]: 50/25Tx[m] : 40x40x-offset [m] : 55y-offset [m] : 5Position : 540m

Figure 9.10: The three componentsof the time-derivative of the B-fieldand the total B-field. The responsecorresponds to figure 9.7g.


____________________________________________________________________________

97

a)

10-6

10-5

10-4

10-3

10-9

10-8

10-7

10-6

10-5

10-4

10-3

dB/dt

b)

10-6

10-5

10-4

10-3

c)

10-6

10-5

10-4

10-3

-0.2

-0.1

0.0

0.1

0.2 Relative diff.

10-6

10-5

10-4

10-3

d)e)

10-6

10-5

10-4

10-3

10-9

10-8

10-7

10-6

10-5

10-4

10-3

dB/dt

f)

10-6

10-5

10-4

10-3

g)

10-6

10-5

10-4

10-3

-0.2

-0.1

0.0

0.1

0.2 Relative diff.

10-6

10-5

10-4

10-3

h)

TE

MD

DD

3D

TE

MD

DD

Half-

space

Rela

tive

Diff

ere

nce

Rx

Tx

Figu

re 9

.11:

The

EA

-fiel

d in

vers

ions

for t

he tw

o si

mpl

e re

sist

ivity

mod

els.


____________________________________________________________________________

98

differences between minus seven and plus three percent are observed. This is much

more moderate than the very large differences (>>20%) that was observed for the z-

component data in figure 9.4g. In figure 9.11d the transmitter and receiver is moved

further along the profile. The receiver is now positioned five meters from the edge of

the cube. The response does not differ significantly, though the peak of maximum

difference is shifted to slightly later times.

The plot of the EM1DINV-interpretation is seen below the true model in figure 9.11

(left side). In the 495-555m profile-interval and 0 – 25m depth-interval a area of relative

low resistivities is seen. The maximum resistivity-value in the centre of the area is

65ohmm. Below the centre of the resistive area, there is a slightly smaller area of

relatively lower resistivity-values. Resistivities in the centre of this area are around

40ohmm. Altogether the area of maximum difference seems to be concentrated around

profile-coordinate 535m, which is also shown in the difference plot. This means that the

correct plotting-point for the data is not in the middle between the transmitter and the

receiver. A more correct plotting point for this model would be at the receiver-position.

Together with the absence of negative data, this observation is a major difference from

the z-component data. Besides this, there are not a lot of differences if the EA-field

interpretation is compared to the z-field interpretation. The spatial distributions of the

resistive area in the interpretations are almost the same, though the underling

conductive area is less pronounced in the z-field interpretation.

The major differences between the z- and EA-field interpretation lies in the

residuals. The three calculated residuals are all below 0.3. For the data-residual this

indicates a good agreement between the true data and the model-data. The vertical

resistivity-residual is also below 0.3, which indicate that the vertical resistivity

constrains have not been challenged as much as in the case of the z-component

(figure 9.4, right). This is important, and one of the major reasons for using EA-field for

offset-data interpretations. It shows that it is possible for the one-dimensional model to

fit the three-dimensional data, within the specified data-errors.

In figure 9.11(right side) the result of the EA-field for the conductive cube is seen.

For the first response in plot 9.11e, a relative difference compared with the half-space

of +2 percent is observed around 3E-05 seconds. In 9.11f the positive peak has not

moved in time compared to 9.11e, but a difference of -4 percent is observed at the


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99

earliest times. In 9.11g this pattern is maintained, though the amplitude of the

differences are increased. At early times the relative difference is –8 percent, while the

positive peak is +12 percent. In the last plot the positive peak is shifted slightly in time

because of the increased distance between the conductive cube and the transmitter,

but the value of the peak maintains constant. The negative peak decreases in absolute

value to about minus two percent.

In the plot of the EM1DINV-intrepretations an area showing relatively low resistivity-

values exists in the 505 to 545 meter profile-interval and 5 to 20 meter depth-interval.

The resistivity in the centre of this area is 35ohmm. Below this an area of relative high

resistivity is present. This occupies the area in the 495 to 555 meter profile-interval,

and the 25 to 50 meter depth-interval. The resistivities in the centre of the area are

around 65 ohmm. This can also be seen from difference-plot below the EM1DINV-

intrepretation.

Like in the case of the resistive cube, the centre of highest anomaly is concentrated

around profile-coordinate 535m. This indicates that the correct plotting point is below

the receiver and not between the receiver and the transmitter, but at the receiver

position. The residuals show a pattern that is comparable to the observed for the

resistive cube. All three residual values for the 50ohmm-background model are

approximately 0.1, while the maximum values are around 0.3 at around profile

coordinate 510m. This shows that the one-dimensional EM1DINV interpretations fit the

three-dimensional EA-field data relatively well.


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100


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101

10 The influence of a shallow, random resistivity distribution

In the following the previous obtained knowledge from the small resistivity variation

is taken one step further. The single resistivity-variation is now replaced be a whole

layer of cubes with varying resistivities. This is done to try to simulate the kind of

variations that one might encounter in real-data situations. Three data sets are

presented and described. The three data sets are acquired over the same true model

using central-loop/z-component, 55m offset/z-component and 55m offset/EA-

component.

10.1 The model The model in this investigation consists of a whole layer of 20x20x20 meter cubes.

These cubes are positioned side by side in the five to twenty meter depth interval. The

background model consists of a 50ohmm half-space. To simplify the model, the cubes

are only distributed within the central area of equal nodal distribution in each TEMDDD

model-file. The central area is 160 by 160 meters, and outside this the resistivity value

is that of the half-space.

The resistivity distribution is defined by using

equation 7.4 with a mean value of 50 ohmm

and a spreading factor of 1.41, as defined in

chapter 7. A plot of the relative distribution can

be seen in figure 10.1. The value of 1.41 was

decided on because is a relatively wide

resistivity distribution that certainly would cause

some perturbation in the 50ohmm half-space.

Furthermore, the value of 1.41 did not produce

too extreme values that could cause possible

erroneous calculations in the TEMDDD

program. Such problems can arise because of

extreme resistivity contrasts close to the

receivers. The true model can be seen in the

Figure 10.1: The log-distribution ofthe resistivities calculated usingequation 7.4.


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102

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103

two upper rows of figure 10.1. The upper row shows a ‘top-view’ of the resistivity

distribution, while the second row shows the conventional ‘side-view’ at the receiver

positions. A coordinate system consisting of a letter (a-h) and a profile coordinate

define the position of the individual cubes.

10.2 Inversion-results of three different configurations Three different configurations have been used for the inversions of the data.

Central-loop and 55 meter offset, using z-component data, and 55 meter offset using

EA-field data.

10.2.1 Central loop, z-field inversion

In the left side of figure 10.1 the results from the central-loop/z-component data is

seen. From the EM1DINV interpretation it is evident that the shallow inhomogeneities

in the true model have a relatively small influence on the inverted model. Only in a

couple of areas, the presence of the shallow inhomogeneities is clearly seen. The first

position along the profile is at coordinate 370m, were inhomogeneities with relatively

low resistivities are present close to the receiver path (c-370m, c-390m, d-390m, e-

350m and e-370m). The area shows 40ohmm resistivity in the 5 to 10 meter depth-

interval. Below this is an area of larger spatial distribution that shows maximum

resistivities of 60ohmm.

At profile coordinate 580m, another area of relatively low resistivities is seen in the 0

to 10 meter depth-interval. The resistivity in this area is 40 ohmm, and below it an area

of larger spatial distribution is located in the 20 to 40 meter depth interval. The

maximum resistivity of this area is 60 ohmm. Two conductive cubes (d-570m and d-

590) positioned directly in the path of the receivers cause these two anomalies.

Besides the two described anomaly-areas, it is not possible to pinpoint other areas that

can be related to specific inhomogeneities. This does not mean that no other areas

exist. From the plot of the relative differences at the bottom of the page, areas of

relatively higher resistivities are present down to a depth of 100 meter.

The three residuals from the inversion are plotted below the EM1DINV

interpretation. The values of the total- and data-residuals are below 0.3, while the Vres-


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104

residual remains below 0.15. The general level of the residuals and absence sharp

peaks indicate a good fit between the data and the inverted model.

10.2.2 55 meter offset, z-field inversion

In the centre of figure 10.1 the results from the 55-meter offset/z-component data is

seen. From the EM1DINV interpretation it is evident that the shallow inhomogeneities

have a considerably larger influence when using the offset configuration.

The two areas that were described for the central loop configuration are also seen.

Between profile coordinate 380m and 420m resistivities which differ from the half-

space are present. In the 0 to 5-meter depth interval resistivities up to 100 ohmm are

present. A 35ohmm area follows this in the 5 to 25 meter depth interval. Below this, a

65ohmm area in the 25 to 70 meter depth interval. These anomalies are mainly caused

by the conductive inhomogeneities c-370m, c-390m, d-390m and e-390m.

The second anomaly area is in the 560m to 610m profile interval. In the upper 10

meters resistivities of up to 190 ohmm is observed. A less resistive area follows this in

the 10 to 25 meter depth interval, which show resistivity values of down to 15 ohmm.

Below this follows a 100ohmm area in the 25 to 70 meter depth interval. These

anomalies are caused by the inhomogeneities d-570 and d-590m, which contain

resistivity values of 30 and 25 ohmm, respectively.

Besides the two anomaly areas that were visible in both the central loop and offset

interpretation, other areas stand out in the offset interpretation. In the upper 5 meters in

the 650m to 690m profile interval, a 100-ohmm area is present. A 35-ohmm area

follows this in the 5 to 20 meter depth interval. Below this is a 60-ohmm area with a

larger spatial distribution exists in the 25 to 60 meter depth interval. The low-resistivity

inhomogeneities c-530m, c-550m, c-570, d-550m, d-570 and f-650m cause this

sequence of anomalies. The first five of these anomalies contain resistivities of

approximately 40 ohmm, which is a relatively low resistivity contrast to the 50 ohmm

half-space. Although the resistivity contrast is not high, their spatial distribution is

relatively large, which leads to the observed anomalies.

In the 280 to 320 meter profile interval, an anomaly not visible in the central loop

interpretation is present. In the upper 2 meters the resistivities are around 35 ohmm. A


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105

65-ohmm body follows this in the 2 to 25 meter depth interval. Below this is a smaller

area, which contain resistivities down to 40 ohmm. The anomaly in this profile interval

is most likely caused by the two relatively resistive inhomogeneities (d-290m, d-310m)

that are positioned right on the path of the receiver. The cluster of relatively conducting

inhomogeneities around c-290m, could be the cause by the relative conductive area in

the 25 to 55 meter depth interval.

The three calculated residuals clearly show the offset configurations disability to fit

the three-dimensional data with a one-dimensional model. The Vres residuals show

values ranging from 0.1 to 1.0 along the profile. The data residual range from 0.1 to 2.5

along the profile. By comparing the EM1DINV interpretation to the data residuals a

clear pattern arises. All profile intervals that show clear anomalies yield extreme

variations in the data residual. This indicates a relatively poor fit between the three-

dimensional data and the one-dimensional model whenever a relatively large resistivity

contrast between the transmitter and receiver is present. This observation is parallel to

the effects of a single inhomogeneity, described in the previous chapter.

10.2.3 55 meter offset, EA-field inversion

On the far right plots in figure 10.1 the results of the EA-field inversion is shown.

The plotting point for the data is maintained to be in the middle between the receiver

and transmitter. The EA-field inversion generally exhibits less extreme resistivity values

compared to the z-field inversion. In the 250 to 400 meter profile interval the EM1DINV

interpretation show resistivities of 60 ohmm in the upper 20 meters. This is caused by

the low-resistivity inhomogeneities at d-250m, d-290m, d-310m, d-330mm, d-350m and

d-370m.

At profile coordinate 310 conductive area is seen in the 25 to 35 meter depth

interval. This is mainly caused by the conductive inhomogeneity c-290m, but the

inhomogeneities above this one could also add to the effect. Between profile

coordinate 360m and 400m a similar conductive area is observed in the same depth

interval. This is semi-connected to a shallower conductive are in the 400 to 420 meter

profile interval. These two semi-attached areas are caused by the inhomogeneities e-

350m, e-370m, c-370m, c-390m and d-390m. The conductive below the resistive area

is also an effect of the 1D inversion. This was shown for the single inhomogeneity in


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106

figure 9.11(left side), but the complexity of the present model makes it difficult to

separate the two effects.

Between profile coordinate 470 and 500 an area of relatively high resistivities is

observed in the 5 to 30 meter depth interval. Below this is a conductive area in the 30

to 50 meter depth interval. This is caused by the presence of an inhomogeneity with a

relatively high resistivity at d-450m. The underlying conducting area is a consequence

of the 1D inversion of the 3D data. Between profile coordinate 580 and 630 an area

showing relatively large anomalies is present. In the upper 20 meters the resistivity

values are 30 ohmm. Below this, in the 25 to 65 meter depth interval, the resistivity

values are 80 ohmm. These two anomalies are caused by the two conductive

inhomogeneities at d-570m and d-590m, which are positioned right under the receiver

path.

All three residuals show values between 0.1 and 0.5. Generally the values are

below 0.3 and only in one instance they exceed this value. Like observed for the z-field

inversion, relatively high residual values coincided with relatively large variations in the

horizontal and vertical resistivity distribution. The relatively low data residuals indicate a

good agreement between the 3D data and the 1D model. Furthermore, the low residual

values for the vertical resistivity constraints indicate a relatively ‘smooth’ model.

10.2.4 Comparison of the three inversions

Compared to the central loop configuration, the anomalies from the z-component

offset interpretation show more extreme resistivity values. A strong perturbation of the

50 ohmm half-space is observed, and the z-component offset-configuration clearly

demonstrates its relatively high sensitivity to shallow resistivity variations. Compared to

the central loop, this configuration shows a stronger influence from the one-

dimensional interpretation of the three-dimensional data. The residuals are up to ten

times higher. Furthermore, the poor data-fit results in strongly resistive anomalies

above a conductive anomaly. For a resistive inhomogeneity, the poor data-fit results in

a conductive layer in the upper 1 to 8 meters.

The EA-field model shows several common features with the z-component offset

model. A conductive inhomogeneity produces a conductive anomaly followed by a


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107

resistive anomaly, and vice versa. This is not connected to the transmitter-receiver

configuration, but is a consequence of the 1D inversion of the 3D data set. Compared

to the central loop model, the EA-model shows an increased sensitivity towards

shallow resistivity variations.

Three notable differences stand out. Firstly, the thin resistive layers that appear

above a strong conductive anomaly in the z-component/offset model are gone. Parallel

to this, no thin conductive layer is seen above a strong resistive anomaly. Secondly,

the general residual level for the EA-model is lowered by a factor of ten. This

observation is closely connected to absence of the shallow artefacts. This is an

important difference, because it assures that there is a good agreement between the

model and the data. This gives the interpreter confidence in the data set, and

eliminates the need for cutting in the early time data.

Thirdly, the EA-model ‘tracks’ the resistive areas between the transmitter and

receiver better than the z-field offset model. This is especially true for resistive

inhomogeneities that create a relatively moderate resistivity contrast to the 50 ohmm

half-space. An example of this is seen in the 250 to 400 meter profile interval.


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108


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109

11 Conclusion The 3D modelling code TEMDDD has been verified for several resistivity models and

levels of complexity using 1D, 2D and 3D reference responses. The relative differences

between TEMDDD and the reference responses were described. These were found to

be approximately five percent for a homogeneous half-space and approximately seven

percent for a layered half-space. For a 3D model the relative differences were

observed to be around ten percent, though some model yielded a higher difference.

The verification of TEMDDD was followed by a introduction to the implementation of

the modelling code into a semi-graphical user interface. The projection of the

TEMDDDModelCreator model space onto the TEMDDD model space was described

and the involved approximations were derived.

This was followed by an investigation of a suite of sloping structures. 3D TEMDDD

responses were calculated and the effect of a 1D inversion of the data was analysed.

This was done for a central loop configuration, using z-component data. The question

of when a model essentially can be regarded as one-dimensional was discussed for

two specific resistivity models. For the conductive base model it was found that a slope

of 11.25 degree essentially could be regarded as one-dimensional. For the resistive

base model, the 11.25 degree slope still exhibited considerable 3D effects in the 1D

inversion of the data.

Next, the influence of a small near-surface resistivity variation on early time data

was investigated. The resolution of a small cube was described for three different

transmitter receiver configurations; central loop/z-component, 55 meter offset/z-

component and 55 meter offset/EA-component. The three data sets were analysed and

compared with special emphasis on ability of the three configurations to resolve the

small resistivity variation. The resistivity variation was found to have a minute influence

on the central loop/z-component configuration, whereas the 55 meter offset/z-

component yielded considerable influence. For the latter, the poor fit between the

inversion result and the data was discussed. Finally, the inversion of the EA-field was

described, and a comparison to the two other configurations were made. This

comparison found that a considerable improvement of the data fit was obtained using

the EA-field for 1D inversion of offset data.


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110

Finally, a complex model employing a whole layer with a random resistivity

distribution was investigated. The three different transmitter-receiver configurations

were again used to analyse the influence of the complex 3D-structure on the 1D

inversion. Considerable differences between the three inversions were observed. The

influence of the 3D-structure in the central loop/z-component inversion was relatively

small. Only areas with relatively high resistivity contrast showed signs of perturbation.

The offset/z-component inversion showed a considerable influence of the shallow

resistivity distribution and the fit between the inversion and the data was relatively poor.

The offset/EA-component inversion also showed considerably influence by the shallow

resistivity distribution. However, it showed residual values which were considerably

lower than the offset/z-component residuals.


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12 References Árnason, K., 1995: A Consistent Discretization of the Electromagnetic Field in Conducting Media and Application to the TEM Problem, Procedings of the International Symposium on Three-Dimensional Electromagnetics, Schlumberger-Doll Research, October 4-6, p.167-179. Árnason, K., 1999: A Short Manual for the Program TEMDDD, Supplied with the TEMDDD code package, 7p. Auken, E., 1995: 1-D time domain interpretations over 2-D/3-D structures, Proceedings of the Symposium on the Applications of Geophysics to Engineering and Environmental Problems (SAGEEP), Orlando 1995, p. 329-338. de Groot-Hedlin, C. and Constable, S., 1990: Occam's inversion to generate smooth two-dimensional models from magnetotelluric data, Geophysics, Soc. of Expl. Geophys., 55, 1613-1624. Effersø, F., Auken., E., Sørensen ,K.I, 1999: Inversion of band-limited TEM responses, Geophysical Prospecting, 47, p. 551-564. Farquharson, C.G., Oldenburg, D.W. and Li, Y., 1999: An approximate inversion algorithm for time-domain electromagnetic surveys, Journal of Applied Geophysics (42), p. 71-80 Flis, M. F., Newman, G. A. and Hohmann, G. W., 1989: Induced-polarization effects in time-domain electromagnetic measurements, Geophysics, Soc. of Expl. Geophys., 54, p. 514-523. Foged, Nikolaj, 2001: Inversion af lateralt sammenbundne 2-dimensionale modeller stokastiske resistivitetsfordelinger, University of Aarhus, Denmark. Gravesen, P., 1997:Three-dimensional geological modelling of the complex aquifers in an incised Quaternary valley in Jylland, Denmark, 3rd Meeting, Environmental and Engineering Geophysics, Aarhus Denmark, Environmental and Engineering Geophysical Society, European Section, Proceedings, p. 229-232. Hohmann, G.W., 1988: Numerical Modeling for Electromagnetic Methods of Geophysics, Electromagnetic Methods in Applied Geophysics, Vol. 1, Society of Exploration Geophysicists, p. 313-363. Hubing T.H., 1991: Survey of Numerical Electromagnetic Modeling Techniques, University of Missouri-Rolla, Report Number TR91-1-001.3, 20p. Kreutzmann, A., Árnason, K., 2000: 3D Modelling and Comparison of TEM and MT for Geothermal Exploration, 15th EM Induction Workshop, Brazil, August 19-26. Krivochieva, S., Chouteau, M., 2001: Improvement in 1D TDEM Interpretations by Simultaneous Inversion of Data from Two Loop Configurations, JEEG, Vol 6, Issue 1, p. 19-32.


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McNeill, J.D., 1990: Use of Electromagnetic Methods for Groundwater Studies, Geotechnical and Environmental Geophysics, Vol 1, Soc. Of Expl. Geophys, Tulsa, OK, pp 191-218. Menke, W., 1989: Geophysical Data Inverse Theory, Revised Edition, Academic Press Inc. Newman, G.A., Hohmann, G.W. and Anderson, W.L., 1986: Transient Electromagnetic Response of a Three-dimensional Body in a Layered Earth: Geophysics, 51, p. 1608-1627. Poulsen, L.H., 2000: Inversion of airborne transient electromagnetic data, Ph.D. thesis, Faculty of Science, University of Aarhus. Poulsen, L.H., Christensen, N.B, 1999:Hydrogeophysical mapping with the transient electromagnetic sounding method, European Journal of Environmental and Engineering Geophysics 3, 201-220. Raiche, A., Sugeng, F. and Xiong, Z., 1999: Accurate EM modelling for appropriate levels of geological and system complexity, 61st Mtg.: Eur. Assn. Geosci. Eng., Session:2009. Smith, R. S. and Keating, P. B., 1996: The usefulness of multicomponent, time-domain airborne electromagnetic measurements, Geophysics, Soc. of Expl. Geophys., 61, 74-81. Spies, B.R., 1974: Transient EM model studies, 1973: Bur. Minerals Res. Austra. Record 1974, 152. Spies, B.R., Frischknecht, F.C., 1991: Electromagnetic Methods in Applied Geophysics, Volumen 2, Chapter 5, p. 285-378, Soc. Of Expl. Geophys. Sørensen, K.I. & Effersø, F., 1995: Continuous transient electromagnetic sounding, 1st Meeting of the EEGS European Section, Torino, Italy, p. 472-475. Wannamaker, P. E., Hohmann, G. W. and San Filipo, W. A., 1984: Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations, Geophysics, Soc. of Expl. Geophys., 49, 60-74. Ward, S.H., Hohmann, G.W., 1988: Electromagnetic Theory for Geophysical Applications, Electromagnetic Methods in Applied Geophysics, Vol. 1, Society of Exploration Geophysicists, p. 131-250 Weedon, W. H. , Chew, W. C., Lin, J. H., Sezginer, A. and Druskin, V., 1993: A 2.5D scalar Helmholtz wave solution employing the spectral Lanczos decomposition method (SLDM), Microwave and Optical Tech. Lett., Vol. 6, pp. 587-92. Weidelt, P., 1983: The harmonic and transient electromagnetic response of a thin dipping dike, Geophysics, Soc. of Expl. Geophys., 48, 934-952.

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