ThesisFinal2

The Scattering of Waves from Randomly Rough

Surfaces

Clay Stanek

Darwin College

and

Department of Applied Mathematics and Theoretical Physics

University of Cambridge

A dissertation submitted to the University of Cambridge for the degree

of Doctor of Philosophy

17 June 2002

The Scattering of Waves from Randomly Rough Surfaces

ii

Preface

This is the presentation of my work towards a Ph.D. thesis conducted between

April 1999 and June 2002.

Chapter 5 represents the collaboration of my supervisor, Barry J. Uscinski, and

me during the period of December 1999 and December 2000. All the other

chapters in this volume contain original work carried out by myself and not done

in collaboration with another. When an already existing result is used, this is

stated clearly in the text. The graphic results in this project were produced by a

combination of the Matlab programming environment, Digital Fortran 90,

Microsoft Visual C++, and IPLab software packages.

Clay Stanek


iii

Acknowledgements

My wife Terry has been my biggest advocate and friend to me during the past

three years. This thesis belongs as much to her as to me. Barry Uscinski has

mentored me with enthusiasm and care, treating me like I was his own son.

Mark Spivak has given his time to me on many occasions and offered me

technical advice and personal friendship. This work could not have been

completed without the encouragement and expertise of Steve Bottone.

I would like to acknowledge my immediate family members and thank them for

their encouragement: Valeria, Frank, Jan, Sherry, Donna, June, Teeter, Ralph,

Janet, John P and Donna P.

Finally, gratitude is extended to the members of ANZUS. When professional

colleagues are such a large part of one’s life, they become extended family.

Within this family, let me mention specifically Philip Y and Janet B, Steve D,

Robert T, and Patrick M. Thanks for your patience with me during the last

several years.


iv

Abstract

This thesis investigates acoustic wave scattering from rough surfaces using the

paraxial wave equation in both differential and integral forms. The principal

physical mechanism is forward scatter. Applications include the propagation of

sound over rough terrain subject to varying weather conditions including

temperature, humidity, and wind. This is of interest to those in city planning,

airport planning, meteorology, and munitions testing to name a few. Still other

applications include the propagation of sound in the ocean in deep and shallow

water. In all these cases, the slope of the surface is assumed to be gentle but the

depth of the surface relative to the wavelength of the radiation can be large.

In the first third of the work, we develop numerical solutions to acoustic fields in

both Cartesian and cylindrical coordinate systems using the differential form of

the paraxial wave equation. In the Cartesian case, we extend the numerical

method to handle a varying refractive index profile over the surface and a

varying, complex reflection coefficient. In the cylindrical formulation, we

propagate a true point source and compare this to the Gaussian source of the

Cartesian case.

The bulk of the thesis uses the integral equation method to find an analytic

solution for the mean acoustic field for randomly rough surfaces with Gaussian

statistics. The method of Laplace transforms plays a pivotal role. A solution to

the one-surface problem in an isovelocity medium is derived and shown to agree

with accepted solutions in the limiting case of a flat surface. The effect of surface

roughness on the field is also characterized. Finally, the technique is generalized

to address the mean field for the rough waveguide, or two-surface problem.


Symbol List

v

The symbols defined here are the ones used throughout the thesis and have the

meaning described here unless stated otherwise. The order of presentation is

alphabetical with Roman before Greek, lower case before upper case, vectors

before scalars, normal before calligraphic, symbols without subscripts before

symbols with subscripts.

f< > ensemble average of f

β (1) Fourier shift operator (Chapter 2)

β (2) linear medium parameter (Chapter 9)

c speed of sound

0c reference speed of sound; usually 350 meters per second

C covariance estimate

δ Dirac delta function

D differential operator

E paraxial acoustic field

0E initial acoustic field (Chapter 2)

, imIE E image acoustic source (marching technique, Chapter 2)

, reRE E real acoustic source (marching technique, Chapter 2)

totE< > ensemble average of total field in scaled coordinates

ε wind speed slope parameter in [sec-1]

ε refractive index slope parameter in [m-1]

( )0, ,inc Zλ γε Laplace transform of incident acoustic field

( ), ,s Zλ γε Laplace transform of acoustic field

( )ˆ ,F λ γ Laplace transform of ( )( ),E x S xz

∂ ′ ′< >′∂

Φ power spectrum function

G Green function for parabolic wave equation

G Green function for full wave equation

( )ˆ ,G λ γ Laplace transform of surface parabolic Green function

( )ˆ , ,G Zλ γ Laplace transform of parabolic Green function of medium ˆ

jkG Laplace transform of two-surface parabolic Green function

γ surface roughness parameter


Symbol List (Con’t)

vi

Γ Gamma function

H Hankel function

H Laplace transform of derivative of parabolic Green function of

medium ˆ

jkH Laplace transform of derivative of two-surface parabolic

Green function

η coordinate in direction normal to rough surface at each point

i 1− J surface current

0k primary wavenumber [m-1]

L horizontal scaling length [m]

λ Laplace transform variable

0λ primary wavelength [m]

n total refractive index, including constant and varying

components

n ′ varying component of refractive index

0n constant component of refractive index

N (1) number of mesh points in z direction

( ),N x z (2) refractive index definition with real and image media

ν frequency [s-1]

( ),p x z acoustic pressure field, includes envelope and rapidly varying

component

P covariance matrix

Q cumulative distribution function

r radial curvilinear coordinate in the direction of propagation [m]

R reflection coefficient

ρ correlation coefficient ( )S x rough surface [m] 2σ dispersion of rough surface [m2]

T transmission coefficient

( ),u r θ full acoustic field in curvilinear coordinates (Chapter 3)


Symbol List (Con’t)

vii

θ polar coordinate [rad], [deg] ( )v z wind speed profile [m-sec-1]

V sample variance

x direction of propagation [m]

X scaled direction of propagation [m-m-1]

ξ Fourier transform of acoustic field

w source width [m]

ω angular frequency [rad-sec-1]

z vertical direction [m]

0z source position above or below surface [m]

Z scaled vertical direction [m-m-1]


Table of Contents Page

viii

1 INTRODUCTION _________________________________________________ 1-1

2 CARTESIAN WAVE PROPAGATION OVER ROUGH SURFACES WITH VARYING REFRACTIVE INDEX PROFILES __________________ 2-1 2.1 THE MARCHING TECHNIQUE............................................................................ 2-2 2.2 2-D PROPAGATION IN CARTESIAN COORDINATES ............................................. 2-4

2.2.1 Finite Difference Method for Propagation ............................................ 2-5 2.2.2 Fourier Method for Propagation............................................................ 2-8

2.3 2-D PHASE MODULATION IN CARTESIAN COORDINATES ................................ 2-12 2.4 EXAMPLES OF CARTESIAN PROPAGATION USING THE

MARCHING TECHNIQUE ................................................................................. 2-13 2.4.1 Free Space Propagation with Flat, Absorbing Surface........................ 2-13 2.4.2 Source at 20 m with Reflecting and Absorbing Surface ..................... 2-16

2.5 CASES INVOLVING ARBITRARY PROPAGATION MEDIA AND ROUGH SURFACES.......................................................................................... 2-19 2.5.1 Comparing Numerical Results in Varying Medium to Theory ........... 2-20

2.6 VARYING REFLECTION COEFFICIENT OVER THE TERRAIN............................... 2-22 2.7 EXTENSION OF MARCHING TECHNIQUE TO 3-D.............................................. 2-24

3 CURVILINEAR WAVE PROPAGATION OVER ROUGH SURFACES WITH VARYING REFRACTIVE INDEX PROFILES __________________ 3-1 3.1 2-D PROPAGATION IN CURVILINEAR COORDINATES ......................................... 3-2 3.2 2-D MODULATION IN CURVILINEAR COORDINATES.......................................... 3-4

3.2.1 Mapping a Wind Velocity Profile in Cartesian Coordinates to Curvilinear Coordinates......................................................................... 3-5

3.3 EXAMPLES OF CURVILINEAR PROPAGATION USING THE MARCHING TECHNIQUE ................................................................................... 3-8 3.3.1 Curvilinear Solutions in Partial Form.................................................... 3-9 3.3.2 Curvilinear Solutions in Full Form...................................................... 3-13

4 COMPARISON OF GAUSSIAN SOURCE TO POINT SOURCE _________ 4-1

4.1 COMPARISON FOR ε = .00134 ........................................................................... 4-5 4.2 COMPARISON FOR ε = 2.58E-3.......................................................................... 4-7 4.3 COMPARISON FOR ε = 5.795E-3........................................................................ 4-9 4.4 COMPARISON FOR ε = 9.86E-3 ....................................................................... 4-12

5 SOLUTIONS TO THE SCATTERED FIELD USING THE INTEGRAL EQUATION METHOD _________________________________ 5-1 5.1 INTRODUCTION TO THE INTEGRAL EQUATION................................................... 5-2 5.2 THE PARAXIAL POINT SOURCE AND GREEN FUNCTION .................................... 5-4 5.3 VOLTERRA EQUATIONS OF THE FIRST AND SECOND KIND ................................ 5-6

5.3.1 Pressure-Release Surface....................................................................... 5-6 5.3.2 Reflecting (Hard) Surface...................................................................... 5-7

5.4 AN INTEGRAL EQUATION INVOLVING THE FIRST MOMENT OR MEAN FIELD .... 5-7 5.4.1 Gaussian Random Process..................................................................... 5-9


Table of Contents (Con’t.) Page

ix

5.4.2 The Ensemble Averages ...................................................................... 5-10 5.4.3 Scaling the Ensemble-Averaged Equations......................................... 5-11

5.5 THE LAPLACE TRANSFORM TECHNIQUE......................................................... 5-12 5.5.1 Definition of the Laplace Transform................................................... 5-12 5.5.2 Relation of the Laplace Transform to the Fourier Transform ............. 5-13 5.5.3 Laplace Transform Solution for Mean Field with Volterra Equation

of the First Kind, Isovelocity Profile, and Rough Surface .................. 5-13

6 ELEMENTS OF THE SOLUTIONS TO ENSEMBLE-AVERAGED INTEGRAL EQUATIONS USING THE LAPLACE TRANSFORM_______ 6-1 6.1 THE POINT SOURCE REPRESENTATION AND ITS LAPLACE TRANSFORM ............ 6-1

6.1.1 Visualizing the Point Source and Its Laplace Transform...................... 6-3 6.1.2 Asymptotic Results................................................................................ 6-5

6.2 LAPLACE TRANSFORM OF ENSEMBLE-AVERAGED GREEN FUNCTION FOR THE SURFACE ............................................................................................ 6-8

6.3 THE LAPLACE TRANSFORM FOR THE GREEN FUNCTION OF THE MEDIUM ...... 6-11 6.4 SOLUTION TO THE MEAN FIELD PROBLEM VIA LAPLACE TRANSFORMS ......... 6-12

6.4.1 The Solution in the Laplace Domain................................................... 6-12 6.5 THE SOLUTION IN THE SPATIAL DOMAIN ........................................................ 6-13

6.5.1 Example with Simple Poles................................................................. 6-15 6.5.2 Example with Branch Cuts.................................................................. 6-16 6.5.3 The Scattered Field Inverse Transform ............................................... 6-17

7 ACOUSTIC SCATTERING FROM A ROUGH SEA AND BOTTOM SURFACE. THE MEAN FIELD BY THE INTEGRAL EQUATION METHOD FOR SHALLOW WATER ________________________________ 7-1 7.1 WAVE PROPAGATION IN SHALLOW WATER: THE MEAN FIELD BETWEEN

TWO SURFACES................................................................................................ 7-1 7.2 BOTH SURFACES ARE PRESSURE-RELEASE SURFACES...................................... 7-3 7.3 ONE PRESSURE-RELEASE SURFACE AND ONE HARD SURFACE......................... 7-4 7.4 ENSEMBLE AVERAGING OF INTEGRAL EQUATIONS ........................................... 7-5

7.4.1 Previous Ensemble Averages Performed .............................................. 7-6 7.4.2 New Ensemble Averages to Perform .................................................... 7-8

7.4.3 Ensemble Averaging ( )( );G x x z S xz

∂ ′ ′< − − >′∂

............................... 7-8

7.4.4 Ensemble Averaging ( ) ( )( ); jk

G x x S x S xz

∂ ′ ′< − − >′∂

when k j= . 7-9

7.4.5 Ensemble Averaging ( ) ( )( ); jkG x x S x S x′ ′< − − > and

( ) ( )( ); jk

G x x S x S xz

∂ ′ ′< − − >′∂

when k j≠ .................................. 7-10

7.5 NOMENCLATURE ............................................................................................ 7-12 7.6 THE SCALED INTEGRAL EQUATIONS FOR THE MEAN FIELD............................ 7-13



x

7.7 THE SOLUTION TO THE TWO-SURFACE MEAN FIELD IN THE LAPLACE DOMAIN ......................................................................................... 7-16 7.7.1 Laplace Representation for One Hard and One Pressure-Release

Surface................................................................................................. 7-17 7.7.2 Additional Laplace Transforms to Perform......................................... 7-19

7.8 LOOKING AHEAD ........................................................................................... 7-21

8 INDEPENDENCE OF THE FIELD DERIVATIVE AT THE SURFACE FROM THE GREEN FUNCTION AFTER SEVERAL CORRELATION LENGTHS _______________________________________________________ 8-1 8.1 ENSEMBLE AVERAGING THE FIELD DERIVATIVE AND THE GREEN FUNCTION ... 8-2

8.1.1 Independence......................................................................................... 8-3 8.1.2 Expectation and Averaging ................................................................... 8-3 8.1.3 Independence and Correlation of Normal Variables ............................. 8-5

8.2 DEDUCING THE CORRELATION COEFFICIENT BETWEEN RANDOM VARIABLES . 8-7 8.2.1 Transforming Correlated Random Variables to Uncorrelated Ones ..... 8-9 8.2.2 Estimating the Correlation Coefficient Between the Surface and

the Field Derivative ............................................................................. 8-11 8.2.3 Estimating the Correlation Coefficient Between the Green Function

and the Field Derivative at the Surface ............................................... 8-15 8.2.4 Estimating the Correlation Coefficient Between the Surface and

the Field at One Correlation Length Above the Surface ..................... 8-16 8.2.5 Distribution of the Sample Correlation Coefficient ............................ 8-17

8.3 CONCLUSIONS REGARDING THE ENSEMBLE AVERAGING OF GREEN FUNCTION AND FIELD DERIVATIVE ................................................................ 8-18

8.4 GENERATION OF ROUGH SURFACES WITH KNOWN STATISTICS....................... 8-19

9 FINDING ANALYTIC SOLUTIONS TO ACOUSTIC SCATTERING FROM A ROUGH SEA AND BOTTOM SURFACE AND OTHER FOLLOW-ON WORK _____________________________________________ 9-1 9.1 FINDING ANALYTIC SOLUTIONS TO THE TWO-SURFACE PROBLEM: TWO

PRESSURE-RELEASE SURFACES WITH 0γ = ................................................... 9-1 9.1.1 A Flat-Surface Solution via Eigenfunction Expansion ......................... 9-2 9.1.2 The Laplace Transform Approach to Two Pressure-Release

Surfaces with 0γ = ............................................................................. 9-3 9.2 LAPLACE TRANSFORM APPROACH FOR NON-ISOVELOCITY MEDIUMS ............. 9-7 9.3 NUMERICAL COMPUTATION OF INTEGRAL EQUATION SOLUTIONS.................. 9-11

9.3.1 Solutions Via Wavelet Transforms ..................................................... 9-11

10 CONCLUSIONS _________________________________________________ 10-1 10.1 RECOMMENDATIONS FOR FOLLOW-ON WORK RELATED TO CHAPTERS 2-4 .... 10-2 10.2 RECOMMENDATIONS FOR FOLLOW-ON WORK RELATED TO CHAPTERS 5-9 .... 10-4



xi

11 APPENDICES ___________________________________________________ 11-1 11.1 APPENDIX A THE PARABOLIC WAVE EQUATION IN CARTESIAN

COORDINATES ................................................................................................ 11-2 11.2 APPENDIX B THE IMAGE METHOD FOR PARAXIAL WAVE PROPAGATION

WITH 1R = ................................................................................................... 11-4 11.3 APPENDIX C THE PARABOLIC WAVE EQUATION IN CYLINDRICAL

COORDINATES ................................................................................................ 11-7 11.4 APPENDIX D THE INTEGRAL EQUATION ......................................................... 11-9 11.5 APPENDIX E RELATION OF THE LAPLACE TRANSFORM TO THE FOURIER

TRANSFORM................................................................................................. 11-11 11.6 APPENDIX F ................................................................................................. 11-13 11.7 APPENDIX G................................................................................................. 11-15 11.8 APPENDIX H................................................................................................. 11-17 11.9 APPENDIX I .................................................................................................. 11-20

12 BIBLIOGRAPHY ________________________________________________ 12-1

ENDNOTES _____________________________________________________ 12-6


Table of Figures Page

xii

Figure 1 The image method requires the sources to be symmetric about the surface..................................................................................................... 2-4

Figure 2 Results from absorbing, flat surface with constant refractive index and source at 0 10z = ................................................................................. 2-14

Figure 3 Numerical to analytical solution comparison for the source height at 10 m and absorbing surface. ................................................................... 2-16

Figure 4 Reflecting and absorbing surfaces with source at 20 m. ............................ 2-17 Figure 5 Multiple slit interference demonstrated by moving source to 200 m

of the ground............................................................................................... 2-17 Figure 6 Plot of intensity with distance from source for 0z = meters.................... 2-18 Figure 7 Acoustic intensity for flat, hard surface with wind speed profile of

( ) , .0105v z zε ε= = and wind direction of 90°................................................ 2-20 Figure 8 Acoustic intensity shown for isovelocity medium and rough surface. ....... 2-21 Figure 9 Flat, hard surface with constant reflection coefficient throughout

propagation. ................................................................................................ 2-23 Figure 10 Flat, mostly hard surface with reflection coefficient change 78%

through propagation.................................................................................... 2-23 Figure 11 Flat, mostly hard surface with change to absorbing surface 78%

through propagation.................................................................................... 2-23 Figure 12 Mild wind profile and resultant refractive index, .00258ε = ..................... 3-6 Figure 13 Strong wind profile and resultant refractive index with .00989ε = ........... 3-6 Figure 14 Linear wind profile results in θ asymmetry................................................. 3-8 Figure 15 Field intensity with progressively stronger refractive index profiles. ........ 3-10 Figure 16 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 1. ............. 3-11 Figure 17 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 0. ............ 3-11 Figure 18 Full curvilinear solutions with ε increasing from left to right,

top to bottom............................................................................................... 3-14 Figure 19 The field from a point source (left) and from a Gaussian source (right)

are shown in the double-half plane. The surface is flat and reflecting and the source is positioned on the surface. ................................................. 4-1

Figure 20 Numerical results of curvilinear and Cartesian field intensities for case ε = 1.34e-3 ............................................................................................ 4-5

Figure 21 Comparison of field intensities along ground for case ε = 1.34e-3. ............. 4-6 Figure 22 Comparison of field intensities at 500 m for case ε = 1.34e-3. .................... 4-6 Figure 23 Numerical results of curvilinear and Cartesian field intensities for

case ε =2.584e-3. .......................................................................................... 4-8 Figure 24 Comparison of field intensities along ground for case ε = 2.58e-3. ............. 4-8 Figure 25 Comparison of field intensities at 500 m for case ε = 2.58e-3. .................... 4-9 Figure 26 Numerical results of curvilinear and Cartesian field intensities for

case ε = 5.795-3. ......................................................................................... 4-10 Figure 27 Comparison of field intensities along ground for case ε = 5.795e-3. ......... 4-11 Figure 28 Comparison of field intensities at 500 m for case ε = 5.79e-3. .................. 4-11 Figure 29 Numerical results of curvilinear and Cartesian field intensities for

case ε = 9.86e-3. ......................................................................................... 4-12


Table of Figures (Con’t.) Page

xiii

Figure 30 Comparison of field intensities at 500 m for case ε = 9.86e-3. .................. 4-13 Figure 31 Comparison of field intensities at 500 m for case ε = 9.86e-3. .................. 4-13 Figure 32 The acoustic intensity along the ground in the curvilinear case is

compensated by w

πθ

.................................................................................... 4-14

Figure 33 The closed surface, s , includes the real surface, ( )S x , but excludes the source at 0z ............................................................................................. 5-3

Figure 34 Point source field, ( )0, ,incE X Z γ , for several source positions................... 6-3

Figure 35 Point source field, ( )0, ,incE X Z γ , for several source positions over a smaller range 0Z . ....................................................................................... 6-3

Figure 36 Laplace transform integrand for point source at several source positions with 1λ = and 2 1γ = ................................................................. 6-4

Figure 37 A plot of point source Laplace transform for several 0Z and fixed 2γ . ...... 6-5 Figure 38 The Laplace transform of the point source agrees well with theory

for small and large λ . At the crossover 1λ ∼ , the real part of the integral (Laplace transform) becomes negative, but close to zero. .............. 6-5

Figure 39 The complex error function exhibits Stokes phenomenon, which implies different asymptotic representations depending on location within the complex plane.............................................................................. 6-7

Figure 40 Exact expression and approximation as function of X . ............................ 6-10 Figure 41 Green function with exponential autocorrelation function. ........................ 6-10 Figure 42 Numeric versus analytic Laplace transform of surface Green function. .... 6-11 Figure 43 Several level curves for mean scattered field in Laplace domain

with fixed source depth, surface roughness, and desired depth.................. 6-13 Figure 44 A line integral in the complex plane with c right of all singularities. ....... 6-14 Figure 45 Bromwich contours closed to left and right in complex plane. .................. 6-14 Figure 46 Integration contour with branch points....................................................... 6-16 Figure 47 Searching for zeros of ( )ˆ ,G λ γ ................................................................... 6-18 Figure 48 Interpretation of the scattered field from a flat surface. ............................. 6-24 Figure 49 Total field as scattered off a flat surface for a source at 0 20Z = (left)

and 0 2Z = (right)...................................................................................... 6-25 Figure 50 Laplace inversion integrand real and imaginary parts above and

below the branch cut (corresponding to (6.68)).......................................... 6-26 Figure 51 Curves of ( ),sE X Z< > , the mean scattered field, as a function of

scaled range X at different scaled distances Z from a rough surface with ACF (6.27) . A point source is situated at 0 2Z = . The effect of increasing surface roughness is evident from the following different values of surface roughness, 2γ : 2 0.0,γ = [ ];

[ ]2 1.0,γ = −−−− ; [ ]2 5.0,γ = −⋅−⋅−⋅−⋅ . ................................. 6-27



xiv

Figure 52 Effect of surface roughness on the fringe pattern of the mean total field shown as ( ) ( )( )log , ,sincE X Z E X Z< > + < > . In all cases, the source is at 0 2Z = . Red corresponds to higher field strengths. ......... 6-28

Figure 53 The sea surface and bottom are represented along with an acoustic source to the left............................................................................................ 7-2

Figure 54 Venn diagram explains possible combinations............................................. 8-6 Figure 55 Left: One realization of rough surface with 40 sample points marked

in red. First sample point value, 1Y , is then subtracted from the remaining 39 points to generate 2Y at different separations. Right: Correlation coefficient for 1Y , 2Y as a function of these separations. ........... 8-8

Figure 56 Gaussian probability density function in two variables. Uncorrelated on left, correlated on right to 1

2ρ = − with 2 1

2Y Yσ σ= in both cases. ... 8-9

Figure 57 Field beneath reflecting rough surface for point source at –75 m shown as Sound Pressure Level (related to decibels). The depth is shown as negative here due to a plotting package technicality. We still maintain a positive-down sense to our coordinate system................... 8-13

Figure 58 Instantiation of rough surface and corresponding field just above and below the surface along with interpolated derivative at the surface........... 8-13

Figure 59 Family of 400 rough surfaces ( )S x with 0 5000, 100x dx= = ,

and 300nx = samples. If sample is in red, then the correlation coefficient being nonzero is significant to 2.5%. Blue points on the correlation coefficient plots imply we accept the hypothesis

that 0ρ = with 97.5% confidence. Here, ( )( ),E x S x

z∂

′∂ is

correlated with ( )S x as a function of x . ................................................... 8-15 Figure 60 Sample correlation coefficient between ( ) ( )( );G x x S x S x′ ′− −

and ( )( ),E x S x

z

′ ′∂′∂

as a function of x x ′− . Blue points on the

correlation coefficient plots imply we accept the hypothesis that 0ρ = with 97.5% confidence.................................................................... 8-16

Figure 61 Sample correlation coefficient between ( )( )0, /E x S x L k+ and ( )S x

as a function of x . Blue points on the correlation coefficient plots imply we accept the hypothesis that 0ρ = with 97.5% confidence. ........ 8-17

Figure 62 A uniform distribution mapped to a Rayleigh distribution using the inverse cumulative distribution function. ................................................... 8-21

Figure 63 Real and imaginary random Fourier components and corresponding surface......................................................................................................... 8-21

Figure 64 Unfiltered surface and three subsequent filtered surfaces with different Gaussian correlation lengths. ....................................................... 8-24



xv

Figure 65 Level curves of real part of Airy function exponent (left) and exponent of equation (9.27) (right)............................................................... 9-9

Figure 66 Real and imaginary parts for Airy-like function. Different values of λ correspond to different colored curves. β3 = 4.17e-8 (left) β3 = 4.17e-6 (right)........................................................................................ 9-9

Figure 67 Example of wavelet transform matrix for DB2 (left), scaling and wavelet functions for DB4 (right)............................................................... 9-13

The Scattering of Waves From Randomly Rough Surfaces

1-1

C h a p t e r 1

1 Introduction

From a propagation point of view, each encounter of light or sound with matter

can be viewed as an event where a wave interacts with an array of atoms. The

fact that both transverse and longitudinal fields share much in common in their

mathematical description is a tribute to the power of mathematical abstraction in

describing complex phenomena. For the electromagnetic wave, the journey of the

field through the matter determines the appearance of objects, the color of the

sky, the translucency of glass, and the reason snow is white and water clear. For

the acoustic wave, the sound generated from freeway and airport traffic and

propagated to a nearby village or the ability of whales to communicate over long

distances are described as well by this interaction with matter. The propagation

and scattering processes are fundamental.

Lord Rayleigh (1871) analyzed scattering in terms of molecular oscillators and

correctly concluded that the intensity of scattered light was proportional to the

fourth power of the wavelength of the light ( 4λ−∼ ) in the upper atmosphere.

The red end of the spectrum is mostly undeviated whereas the blue, high-

frequency scattered light reaches the observer from many directions. Before this

work, it was widely believed that the sky was blue because of scattering from

dust particles. Today, Rayleigh’s treatment of the dipole is still a powerful tool

in understanding many aspects of scattering.

One rule of thumb in scattering states that the denser a substance through which

a field advances, the less the lateral scattering, and that applies to

electromagnetic propagation through much of the lower atmosphere.

M. Smoluchowski (1908) and A. Einstein (1910) independently provided the basic

ideas for the theory of this type of scattering as a result of the density

fluctuations on local scales. Their results are similar to those of Rayleigh [1].

In the Rayleigh theory, each molecule is independent and randomly arrayed in

space so that the phases of the secondary wavelets scattered off to the side have


1-2

no particular relationship to one another and no sustained pattern of interference.

This situation (such as a rarified gas like the upper atmosphere) occurs when the

spacing of the scatterers is roughly a wavelength or more. In the forward

direction, the scattered wavelets add constructively with each other.

In a denser medium, the scattered wavelets cannot be assumed to arrive at a

point P with random phases and interference will be important. Again,

scattered wavelets will interfere constructively in the forward direction, but

destructive interference dominates in all other directions and little or none of the

field ends up scattered laterally or backwards in a dense homogeneous medium.

Thus, in many acoustic and electromagnetic applications, the forward direction is

of prominent importance. Keeping this in mind, we will find great utility in

using the parabolic form of the wave equation. For volume propagation, this

form of the wave equation is natural for forward scattering. For interactions with

rough surfaces, the parabolic form of the wave equation limits us to the case of

forward scattering, when backscattering may occur in some situations.

Other effects of scattering include those that involve the irregularities of the

medium through which the waves propagate. The effect, termed scintillation,

explains (to name a few) the twinkling of stars, certain aspects of sonar operation

in the ocean, and the effect of turbulence on sensors including radar and

microwave devices. Scattering by random media is often undesirable, but can be

used to deduce properties about the medium itself. In this way, the intensity

patterns observed through an acoustic or electromagnetic measurement can be a

means of remote sensing. Deducing properties of the medium through indirect

measurement has been accomplished by Jakeman (1978) and Uscinski on several

occasions. Mostly recently, Uscinski used this technique to study vertical water

motion in the Greenland Sea [2].

Acoustic applications form the main area of study for this thesis, but many of the

results will prove useful in electromagnetic applications as well. Our study of the

acoustic applications fall into two main categories: (1) the propagation of sound

in the ocean, or underwater acoustics, and (2) the propagation of sound in the

atmosphere.


1-3

In the next three chapters, we examine the propagation of sound through varying

media and over rough terrain. We develop a numerical solution to the parabolic

wave equation and demonstrate how this technique can be used to solve for the

acoustic field over randomly rough terrain with arbitrary wind profiles over the

terrain. Here, the application is in the prediction of sound intensities along the

ground produced by quasi-point source explosions. One parameter of interest was

the wind profile when the explosion occurred. Everyday experience tells us that

when we are downwind of sound, we tend to notice more of the acoustic

disturbance. If testing munitions within several kilometers of habitat, a windy

day can mean the difference between harmless, acoustic background noise and the

destruction of property such as glass windows.

Chapter 2 begins with a discussion of the parabolic wave equation (PWE) [3] and

its numerical solution via the marching technique [4]. We solve the PWE in

Cartesian coordinates given an initial field condition of that due to a Gaussian

source. We discuss several numerical issues and formulate the solution using

spectral methods. After a fairly thorough discussion of the Fourier aspects of the

numerical solution, we provide examples of PWE solutions for different surface

roughness and wind profiles. The marching technique is extended to surfaces of

varying reflection coefficient and variable wind profiles along the surface. Finally,

we explain some of the difficulties in extending the marching technique to three

spatial dimensions.

Chapter 3 is parallel to Chapter 2 in many ways, except we formulate the

marching technique in curvilinear coordinates and we use a true point source in

the numerical solution for the acoustic field. We are unable to deal with

anything but flat surfaces, but are able to accommodate various refractive index

profiles.

Finally in Chapter 4, we are able to compare the numerical solutions using a

Gaussian source propagated in Cartesian coordinates and those using a true point

source propagated in curvilinear coordinates. In doing so, we are able to discuss

differences and explore what information we may lose when using a Gaussian


1-4

source. The Gaussian source only approximates that of the point source. Even

where it does approximate it reasonably well, it is only over a relatively small

field of regard. Of particular interest is the effect of these errors on acoustic

propagation in the atmosphere when there is some form of ducting caused by the

sound speed profile. The profile will be a function of wind, temperature and

humidity. All of the work in the first three chapters relies on the differential

form of the PWE.

In Chapter 5, we begin our study of scattering from randomly rough surfaces

using the integral form of the PWE. When the full wave equation is used, the

integral formulation begins as an exact solution to the surface scatter problem.

In the parabolic form, we will be well equipped to handle scattering at low

grazing angles ( )20< from a deeply modulated, rough surface. We approach the

problem in terms of an acoustic source with an incident field interacting with the

underside of the sea surface. However, the approach is applicable to other classes

of acoustic problems and electromagnetic problems too. The goal of this work is

to derive an expression for the mean acoustic field when the incident field is that

due to a point source and the surface(s) have certain statistical properties.

The problem of rough surface scattering has led to a variety of approximate

techniques including: Kirchhoff and perturbation theories, operator expansion

technique, ray theory, perturbation theory extensions such as the smoothing

method and phase perturbation [5], and composite methods that combine others

for various regions. All of these techniques have been examined with respect to

the exact theory. In general, approximations can be divided into two categories:

small surface slopes and small surface heights. The small surface height

approximation contains the variations of perturbation theory. If the surface

height variation is characterized by σ and the wavelength of the radiation is given

by 0λ , then the theory is usually valid when 0

1σλ . We should note that this is

a guideline as the actual regime of validity also depends on angle of incidence and

surface slope to some degree [6]. The theory is accurate for backscatter, but can

only handle small surface modulations.


1-5

The Kirchhoff theory can be viewed as considering the surface to be locally flat at

each point and neglecting multiple scatter [7]. The field derivative at the surface

is found by replacing the surface with its tangent plane at each point. It is also

seen from the viewpoint of uniform illumination of the rough surface. This

method will break down for small angle incidence, or quasi-grazing incidence as

multiscatter effects become more important. Certainly, shadow regions do not

meet the uniform illumination condition [8]. The Kirchhoff theory is accurate

when applied to the condition 0

1Lλ ≥ where L is the correlation length scale on

the surface. It can also be applied in the case of deeply modulated surfaces so

long as the surface slope remains small. One can only have deeply modulated

surfaces with 0

1Lλ when the propagation is taken over many L . The

approach in this thesis handles deeply modulated rough surfaces and multiple

scattering, but is restricted to the case of forward scattering, which is implicit in

the PWE.

Chapters 5 through 9 contribute to the body of knowledge in scattering from

rough surfaces by offering a solution to the mean acoustic field given a surface of

known statistics. To date, no one has provided a useful, analytical expression for

the mean field. Current attempts at determining this quantity rely on performing

numerous simulations that are averaged to approximate the true ensemble

average. Numerically, we can only estimate the true population mean from the

sample mean; the variance on the estimate is a function of the number of

samples. Thus, to get an accurate estimate of the population mean, a large

number of samples are needed. Furthermore, new insight is often found in

analytic solutions that is missed from the pure simulation point-of-view.

Chapter 5 describes the specific problem of the scattering of an acoustic signal

incident from below at low angles on a rough sea surface. The problem is treated

by the integral equation method in the parabolic approximation. First, we obtain

equations allowing the mean scattered field to be calculated. We show how the

equations can be scaled into a more simplified form and then offer a general

solution via Laplace transforms.


1-6

In Chapter 6, we use the general solution of Chapter 5 to demonstrate the power

of the technique. Here, we develop expressions for the Laplace transform of a

point source and Green functions necessary to solve the set of integral equations.

After developing the transform expressions, we use them to express the full

solution to the mean field in the Laplace domain for a surface with Gaussian

statistics and exponential-like autocorrelation function (ACF). At this point, we

examine the inversion of this expression and discuss some of the difficulties.

Knowledge of branch cuts is necessary to develop a more simplified expression for

the mean field. We are able to show how this solution approaches the accepted

solution in the limiting case of a flat surface. Then we provide solutions in the

case of increasing surface roughness. The effect of surface roughness on the field

is clearly visible.

Chapter 7 further extends the analytic treatment for the mean acoustic field by

examining the two-surface problem. We examine the case where both surfaces

are pressure-release surfaces and the case where one surface is hard and the other

is pressure-release. The new Laplace transform expressions needed in the two-

surface problem are derived. Finally, the general solution to the two-surface

problem using the method of Laplace transforms is provided.

Chapter 8 is devoted to the study of statistical independence and correlation.

Central to the derivation of the analytic expression for the mean field is

performing ensemble averaging of various expressions. Chapter 8 examines, and

answers, how justified we are in our treatment of the ensemble averaging of the

Green functions and the derivative of the acoustic field at the surface.

Finally, Chapter 9 discusses the way ahead. We examine the full solution to the

two-surface problem, discuss using this technique in other than isovelocity media,

and then completely change topics to the wavelet solutions of integral equations.

The final topic of wavelets brings much promise to the fast solution of integral

equations and is an important topic of current research in many disciplines.


2-1

C h a p t e r 2

2 Cartesian Wave Propagation Over Rough Surfaces

with Varying Refractive Index Profiles

There has been great interest in the numerical solution to the propagation of

acoustic and electromagnetic sources over rough terrain in various media.

Terrain can include areas that are not earth covered, such as lakes and rivers.

Furthermore, terrain may contain different types of vegetation, including

coniferous and deciduous trees, cultivated land, natural grassland, and others.

The different types of surfaces lead to different reflection and transmission

coefficients for an acoustic or electromagnetic wave incident on the surface. In

the electromagnetic case, these coefficients, and R T , are determined by the

angle of incidence for the plane wave component at wavenumber 0k , the two

refractive indices of the media at the interface, and the state of polarization of the

incoming plane wave.

In the acoustic case, the field is not transverse, but longitudinal. Ultimately, in

either case, the coefficients must be derived from the parameters above and the

conservation of energy at the interface. For an acoustic field, the reflection

coefficients for many types of surfaces are known empirically and in some cases

closely approximate certain familiar boundary conditions. For example, the

reflection coefficient for water is often taken as 1R = , which is complete

reflection. A simple model for other types of non-water surfaces might use a

completely absorbing surface with 0R = . Here, the implication of the term

“absorbing” is the interface acts as if it were an open window— the sound goes

straight through. At the other extreme, a pressure-release surface with 1R = −

implies the field goes to zero on the surface.

Other types of surfaces and their interface to the adjacent media [i] might be

represented with a complex reflection coefficient R iR e φ= . In this form, we can

[i] Usually air to some other medium, water to some other medium, or water to air in the domain of study

here.


2-2

take into account amplitude as well as phase changes between the incident and

reflected fields. Complex reflection coefficients can lead to evanescent waves [9].

We will be examining propagation cases when the paraxial form of the wave

equation applies:

( )2

202

0

, 12 2

ikE i E n x z Ex k z

∂ ∂ = + − ∂ ∂ (2.1)

where ( ),E x z is the acoustic field, 0k is the primary wavenumber, and ( ),n x z is

the refractive index of the medium. We provide a derivation of equation (2.1) in

Appendix A.

2.1 The Marching Technique

The technique has been described in detail in the work of Sheard [10] and

Hatziioannou [11]. A brief summary of the method is provided here. There are

three important elements to the marching technique:

1. The surface scatter, or boundary, problem can be mapped to an extended

medium and solved as a volume problem with real and image media.

2. The field can be solved numerically by considering the propagating part of the

equation to be independent of any scattering during a step, while the phase

modulation can be considered to occur in a single ‘screen’ before the next

propagation. This is known as the split-step method.

3. The initial condition, field ( )0 00,E x z= with source at ( )00,z , requires that

an auxiliary source exist in the image medium at ( )0 0 00,2 ( )E x S x z= − . The

field is propagated in both media, with the portion of the field in the real

medium representing the desired part. The location of the image source with

respect to the real source is symmetric about the surface 0( )S x .

Thus, the marching technique is an adequate name for it captures how the effect

of distance and modulation are essentially decoupled and enacted separately as

the numerical scheme steps forward in the direction of propagation.


2-3

Consider a rough surface ( )S x . It has been shown that the rough surface can be

replaced by a medium in the space ( )z S x< with an artificial refractive index.

The refractive index is defined as

( ) ( ) ( )

( ) ( )( ) ( ) ( )[ ] ( )

1

1

, , ,

, ,2 2 ,

N x z n x z z S x

N x z n x S x z S x z S x z S x

= >

′′= − − − < (2.2)

where

( ) ( )0 01, , , 1n x z n n x z n= + = (2.3)

For the initial condition, the real source and an image source must be defined as

to satisfy

( ) ( )( )0, 0,2 0I R

E z RE S z= − (2.4)

RE is the real source,

IE is the image source, and R is the reflection coefficient

at the surface. For the case of a Gaussian source, the initial condition becomes

( ) ( ) ( )

( ) ( )( ) ( ) ( )[ ] ( )

2 20

2 20 0

/212

2 0 /2 2 0 012

0, , 0

0, e , 0

z z ww

S z z w ik S z Sw

E z e z S

E z R e z S

π

π

− −

′− − + −

= >

= < (2.5)

where w is the source width. We give more detail in Appendix B.

Finally, this is subject to the parabolic wave equation (PWE):

( )2

2 00

,2

E i E ik N x z Ex k z

∂ ∂= +∂ ∂

(2.6)

with the initial field condition, ( )0 ,E x z . There is no fundamental difference

between (2.6) and (2.1). A plausible rough surface and location of the real and

image sources are shown in Figure 1.


2-4

Figure 1 The image method requires the sources to be symmetric about the surface.

Once the initial field is configured, the marching technique begins and the field is

propagated/modulated in a series of steps. The solution obtained in the real

medium is used and that of the image medium discarded.

2.2 2-D Propagation in Cartesian Coordinates

In the next few sections, we will examine some of the issues involved in

numerically solving the PWE in Cartesian coordinates. We examine some

standard techniques that might be used when solving a parabolic partial

differential equation and then describe the spectral method for propagation. The

latter technique offers several advantages and was used in the work presented

here.

The general expression for paraxial, rectilinear wave propagation in two

dimensions is given by

2

202

E i Ex k z

∂ − ∂=∂ ∂

(2.7)

There are two general procedures applicable to the numerical simulation of wave

propagation: one is to work in the spatial domain and use finite difference

approximations to the derivatives to propagate the field in discrete steps, the

other is to transform the equation into spatial frequency space via the Fourier

transform and perform the propagation in the Fourier domain.


2-5

2.2.1 Finite Difference Method for Propagation

The 2-D form of the PWE is identical in structure to the Schrödinger equation:

( )( )2

2i iV xt xψ ψ ψ∂ ∂= + −∂ ∂

(2.8)

Both are parabolic partial differential equations. The difference between them is

the parabolic form of the Helmholtz equation in (2.6) relates two spatial

derivatives to each other (the propagation), which is balanced by scintillation of

the field (2nd term, right-hand side), while (2.8) looks like a diffusion equation and

accompanying source term with time being the ‘propagated’.variable

An obvious numerical scheme might try to approximate the derivatives in x and

z by

( )

( )

1

21 1 2

2 2

0

0

2

k kj j

k k kj j j

k

j

E EE O xx x

E E EE O zz z

x x k x

z z j z

+

+ −

−∂ = + ∆∂ ∆

− +∂ = + ∆∂ ∆

= + ∆

= + ∆

(2.9)

such that the full, differenced equation looks like

( )1

21 1 02

0

2, 1

2 2

k k k k kj j j j j k

k j j

E E E E Ei ik n x z Ex k z

++ −− − + = + − ∆ ∆

(2.10)

Using a von Neumann stability analysis, we can test for unstable eigenmodes of

this difference scheme by allowing k k imj zjE eζ ∆= . We find that

( )( ) ( )2 20

20

21 sin 12 2

ik xi x m zm nk z

ζ∆ ∆ ∆ = − + − ∆

(2.11)

Let’s assume that the refractive index of medium is unity: ( ), 1n x z ≡ . Then, the

stability criterion requires that all the modes meet ( ) 1 for allm mζ ≤ such that


2-6

( )

[ ]2220

21 1 sin 1m zi xk z

∆∆− ≤ − ≤∆

(2.12)

Taking the nontrivial case, we must have ( )20

12

xk z

∆ ≤∆

for a numerically stable

scheme, as 2sin 1m z∆ ≤ [12].

This shows some of the issues when a numerical scheme is conceived on a spatial

grid. The stability requirements are strenuous on the acceptable mesh fidelity as

we more closely approximate a true point source. For instance, when a Gaussian

source is used, the grid size must be fine to pick up the features on a narrow

source. This is often the case as the narrower the source, the more closely it

approximates a point source and the greater the angle of regard that contains

useful information. But the stability requirements insist that z∆ at least be

equal to 0

2 xk

∆ . Arguably the most widely used differencing scheme is the

famed Crank-Nicholson method [13], but the variety and extent of research and

application into different differencing methods is enormous.

In the next section, we formulate derivative operators for numerical analysis. Of

key importance to the stability and truncation error of the method is the

approximation of the Laplacian term in (2.6).

2.2.1.1 Approximating the Laplacian Operator

Our first step will be to analyze the accuracy and fidelity of the spatial second

derivative. Define the following finite difference operators [14]

The shift operator, ( ) 1k kz zβ += (2.13)

The forward difference operator, ( ) 1k kkz z z+ +∆ = − (2.14)

The backward difference operator, ( ) 1k kkz z z− −∆ = − (2.15)

The central difference operator, ( ) 1 12 2

0 k kkz z z

+ −∆ = − (2.16)

The differential operator ( ) ( )kDz z kh′= (2.17)


2-7

In the above definitions, z is a real or complex sequence indexed by all the

integers, k . In the last definition, h refers to the sampling on equally spaced

points so that ( ); 0kz z kh h= > .

Iserles shows that the differential operator D can be expressed as

( ) ( ) ( ) ( )1 22 2 2 2 61 112 452 0 0 0

1 , 0s s ss

sD h hh

+ + = ∆ − ∆ + ∆ + Ο → (2.18)

The definition above applies for even derivatives (e.g. second, fourth, etc.). Some

basic manipulation shows that 1/2 1/20 β β−∆ = − and therefore 2 1

0 2β β−∆ = − + .

Applying 2D to z and retaining the first two terms on the right hand side

( ) ( )2 22 41 1

01 1 122k k k kh h k

D z z z z z+ − = − + − ∆ (2.19)

A more heuristic derivation involves the use of Taylor’s Theorem [15]. Suppose

that the function, ( )z k , has at least m continuous derivatives. Then by Taylor’s

Theorem

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 3 4 5

2 3 4 5

2! 3! 4 ! 5!

2! 3! 4 ! 5!

iv vh h h h

iv vh h h h

z k h z k h z k z k z k z k z

z k h z k h z k z k z k z k z

ζ

ζ

+

−

′ ′′ ′′′+ = + + + + +

′ ′′ ′′′− = − + − + − (2.20)

The existence of ζ+ and ζ- is an extension of the mean value theorem; there must

exist a number on the interval in question that makes the expansion an equality.

By adding the two relations, we obtain an approximation for the second

derivative of ( )z k .

( )( ) ( ) ( )

( )2 6122

2,ivh

z k h z k z k hz k z k h k

hζ ζ−+

+ − + −′′ = − + Ο = = (2.21)

We now write a form of the parabolic propagation in z and x coordinates with a

discrete right-hand side:


2-8

( )

( ) ( )

20

20

1

20

2

2

2

EE ix k z

EE ix k z

β β−

∆∂ =∂ ∆

− +∂ =∂ ∆

(2.22)

2.2.2 Fourier Method for Propagation

The second option is commonly used because it makes use of the discrete Fourier

Transform and Fast Fourier Transform (FFT) algorithm. This allows the field to

be propagated in a smaller number of computing cycles; results are almost

immediately available. It has the additional benefit of reducing the number of

previous iterations that must be accessible for propagation at the current

location. This brings large computational savings. The idea using the continuous

Fourier transform will be easily demonstrated.

One defines the continuous Fourier transform pair as

( ) ( )

( ) ( )

12

i x

i x

F f x e dx

f x F e d

ω

ω

ωπ

ω ω

∞

−∞

∞−

−∞

=

=

∫

∫ (2.23)

and can apply this transform to the two-dimensional, rectilinear or curvilinear

propagation equation. In this definition, the variable ω is assumed to be angular

frequency with dimensions of radians per second.

A second definition, which can be more intuitive when working in a system where

time or space is the independent variable, is to define the continuous Fourier

transform pair as

( ) ( )

( ) ( )

2

2

i x

i x

F f x e dx

f x F e d

π ν

π ν

ν

ν ν

∞

−∞

∞−

−∞

=

=

∫

∫


2-9

where ν has dimensions of cycles per meter or inverse wavelength. ν and ω are

related by

2ω πν= (2.24)

When implementing this scheme numerically, care must be given in applying this

technique. The typical approach is to discretize the problem onto a grid that

contains 2N points and employ the Fast Fourier Transform (FFT) algorithm.

While variants of the Cooley-Tukey algorithm can accommodate non-powers-of-2

grid sizes, their performance requires more than the 2logN N steps the

power-of-2 algorithm uses [16].

A few subtleties that arise in applying this technique are:

• Mapping the θ (polar) coordinate space to angular frequency

space, ω , or the z coordinate to wavenumber space, ν .

• Understanding the errors involved in the discrete Fourier

transform of the Laplacian operator.

2.2.2.1 Discrete Fourier Transform of the Laplacian Operator

To take the Fourier transform of this equation, we need to recognize a

fundamental property of the shift operator, β . That is ( )[ ]( ) ( )i hf x e Fωβ ω=F

when F denotes the Fourier transform and h the step size in x . We will now

define the Fourier transform of our basic operators:

The shift operator ( )( )[ ] ( )i hf x e Fωβ ω=F (2.25)

The forward difference operator ( ) ( ) ( )1i hf x e Fω ω+ ∆ = − F (2.26)

The backward difference operator ( )[ ] ( ) ( )1 i hf x e Fω ω−−∆ = −F (2.27)

The central difference operator

( ) ( ) ( ) ( ) ( )2 220 2 sin

i h i h hf x e e F i Fω ω ωω ω− ∆ = − = F (2.28)

Applying the discrete Fourier transform to equation (2.22) and explicitly noting

the components with subscript n :


2-10

( ) ( )

( ) ( )( )( )

20

2

2

20

, 2 ,2

2 sin,,

2

n n

n

i z i zn

n

z

nn

x i e e xx k z

ix i xx k z

ω ω

ω

ξ ωξ ω

ξ ωξ ω

∆ − ∆

∆

∂ − +=∂ ∆

∂=

∂ ∆

(2.29)

The term, ( )( )22

20

2 sin

2

n zi

k z

ω ∆−

∆, is part of a quantity known as the kernel for this

equation. ( ),xξ ω is the Fourier transform of the field, ( ),E x z .

A little more insight can be gathered by expanding each expression in a series:

( ) ( ) ( )

( ) ( ) ( ) ( )( )( ) ( )

( ) ( )

2 4 2

2

2 2

122! 4 ! 2 !

2 4 2 6 41 112 360

2 cos 12 , ,

1 1 ,

... ,

n n

mmn n n

i z i zn

n n

z z znmz

n n n n

ze e x xz z

x

z z x

ω ω

ω ω ω

ωξ ω ξ ω

ξ ω

ω ω ω ξ ω

∆ − ∆

∆ ∆ − ∆∆

∆ −− + = ∆ ∆

= − + − + −

= − + ∆ − ∆ +

… (2.30)

or with the second expression

( )( )( )

( ) ( ) ( )( ) ( )( ) ( )

( ) ( )

2 1

2

2 4 4 6

2

2

2

23 5 2 114 1 12 2 2 23! 5! 2 1 !

212 360

2 sin,

... ,

... ,

n

mn n n n

n n

z

n

mz z z znmz

z zn n

ix

z

x

x

ω

ω ω ω ω

ω ω

ξ ω

ξ ω

ω ξ ω

−

∆

−−∆ ∆ ∆ ∆−−∆

∆ ∆

∆

= − + −

= − + − +

(2.31)

In each case, we see that this is the Fourier transform with respect to z of

( ) ( ) ( ) ( ) ( )2 42 4 6

2 4 6

, , ,...

12 360z zE x z E x z E x z

z z z∆ ∆∂ ∂ ∂− + −

∂ ∂ ∂ (2.32)

with an error of ( )6zΟ ∆ .


2-11

An excellent, discrete approximation to the Laplacian operator can be made by

retaining the fourth order term:

( ) ( )( ) ( )( )

( )[ ]2 4

22 2

2 2 2

2 sin 2 sin,,

12

n nz zi iE x zE x z

z z z

ω ω∆ ∆ ∂ = + ∂ ∆ ∆ F F (2.33)

And this is the expression implemented in the simulations with an error of

( )4zΟ ∆ .

2.2.2.2 From the Spatial to the Frequency Domain in Discrete Systems

Consider a mesh in z upon which there are N samples of spacing z∆ . Then the

spatial frequency components will be defined as

nn

N zν =

∆ (2.34)

with n varying between 0 and 1N − in the discrete Fourier transform. Strictly

speaking, the important point is that the samples in Fourier space will be periodic

in n with period N . The second definition ν of the Fourier transform with the

2 iπ factor in the exponential is explicitly saying that the principal range is 0 to

2π . However, there is no reason why one can’t consider the period [ ],π π− with

n ranging on [ ]/2, /2N N− with the Nyquist frequency occurring for 0n = .

How is ω Defined?

As an example, consider an N point mesh in the z direction with a grid spacing

of z∆ . Then the nω ’s will be

22 , 0,... 1nn n N z n Nπω πν ∆≡ ≡ = − (2.35)

Going back to the kernel, we can now see what the actual values to be evaluated

will be

( )( ) ( )( ) ( )

22 2 222

2 2 2

2 sin2 sin 2 sin, 0, 1,... 1

2 2

nn z nz

N z Nii

n Nz z z

π πω ∆∆∆ −

≡ = = −∆ ∆ ∆

(2.36)


2-12

Here is the elegance and efficiency of the method. The Fourier transform has

essentially decoupled the original spatial relation into its constituent modes, each

orthogonal to the other. The propagation operation in the spatial domain now

becomes multiplication by the kernel for each constituent plane wave component

in the field. It should be noted that these pseudo-spectral methods work well

when the field does not have discontinuities. In the presence of discontinuities

the results from the method may not be as expected. However, the behavior of

the Gaussian source is well defined and has excellent convergence properties

towards infinity. This makes the method an excellent choice in propagating our

field numerically.

Now the propagation is a simple algorithm. To advance the field from one

position in nx to 1nx + , simply transform the field at nx with respect to z . For

each grid point in z , evaluate the kernel for that mode and multiply it by the

value of the transformed field for that n . Take the inverse Fourier transform of

this product, and the result is the new field at the next position, 1nx + for all z .

2.3 2-D Phase Modulation in Cartesian Coordinates

The general expression for paraxial, rectilinear phase modulation in two


( )0 ,E ik N x z Ex

∂ =∂

(2.37)

This is also known as the scattering component of the equation. The general

form of this equation is E AEx

∂ =∂

, which can be rewritten

as ( ) ( )( , )

( , )

lnE x x z x x

E x z x

d E d Ax+∆ +∆

=∫ ∫ . The general solution is

( ) ( )0, ,ik N xE x x z e E x z∆+∆ = (2.38)

Spivak and Uscinski have shown that for constant step size, x∆ , that small

scattering per step with this independent phase screen model is valid when the

scale size in the direction of forward propagation is less than x∆ [17].


2-13

2.4 Examples of Cartesian Propagation Using the

Marching Technique

In this section, we examine the use of the marching technique for numerically

solving various initial field conditions, rough surface types, and propagation

media. The purpose is to demonstrate that the numerical solutions appeal to

physical intuition. Also, in the cases where analytic solutions exist, we can

directly compare theory to numerical work to serve as a useful benchmark to the

technique’s accuracy. For instance, a flat surface with a constant refractive index

medium or a linear refractive index medium can be solved exactly and with the

marching technique as well.

2.4.1 Free Space Propagation with Flat, Absorbing Surface

In this particular example, the ground is taken to be flat, with no wind profile,

and the surface to be absorbing ( )0R = . The analytic solution for the field

intensity is

( ) ( ) ( ) ( )( )220

22 2 404 2 2

0

1, , exp /2 /

xkI x z E x z w z z w

w x kπ= = − − +

+ (2.39)

when the initial field is defined as

( ) ( )( )2 202

10, exp /22

E z z z wwπ

= − − (2.40)

The initial field condition is known as a Gaussian source. Notice that w is the

source width parameter and not ω . The result in (2.39) is derived by

considering the solution to be a summation of point sources along an aperture. In

this case, the aperture can be taken as along the 0x = axis. All points in the

aperture plane may be thought of as secondary point sources by the Huygens-

Fresnel principle. The aperture plane is divided into elementary segments, dz , so

small that each infinitesimal segment can be thought of as a secondary source.

Thus, we may write the contributing element to the acoustic field, dE , at P as


2-14

( ) ( )( )2

0

1Cnst 0, exp /2dE E z dz ik z z xx

′ ′ ′= ⋅ ⋅ − (2.41)

where the observation point P is at ( , )x z .

We may integrate over the entire aperture plane as shown in equation (2.42) to

produce the result of (2.39)

( ) ( ) ( )( )2

00

1 1, 0, exp /22 2

iE x z E z ik z z x dzk xπ

∞

−∞

′ ′ ′= ⋅ −∫ (2.42)

Carrying out the integration proves that (2.42) is tantamount to (2.39).

The numerical solution was computed and is shown in Figure 2 with the key

parameters in Table 1. A color map was chosen so that areas of highest intensity

appear in red, followed then by orange, yellow, green, blue, and black.

Figure 2 Results from absorbing, flat surface with constant refractive index and source at

0 10z = .


2-15

Table 1 Absorbing surface input parameters for marching technique solution.

Parameter Result Parameter Result

dx 100 m Terrain flat

dz (initial) 0.1 m Refractive index n constant

nz 2048 Reflection coefficient R 0

Source Height 0z 10 m Primary wavenumber 0k 1 m-1

Source width w 10 m

Table 1 lists several of the important inputs into the numerical model. We have

not yet stated the equation that relates the refractive index of the medium to the

speed of sound profile through it, but it follows from the definition of refractive

index in terms of the ratio of the reference to local speed of sound. A constant

refractive index profile implies a constant wind or no wind conditions for all

heights. From standard diffraction theory, one can envision the acoustic source

impinging a slit and having a characteristic spreading angle. This angle is

proportional to 0D

λ where D is usually the diameter of the aperture and 0λ the

wavelength. Here 0λ is 01 k and D is ( )O w . Thus, one would expect θ to be

proportional to 01 k w . In fact, at any point x , the width of the field in z is 122

2 40

412w

w xzk w

= + . For x large, this becomes

0

2w

xzk w

= (i.e.

0

2,wz xk w

θ θ= = ).

Figure 3 provides the accuracy of the results compared to the analytic solution.

For fixed step size, accuracy decreases locally as the wave propagates. Here the

numerical solution is well within 1% for the first 100 steps and continues to stay

within 4% over the full 200 steps of the simulation. Much of this can be

attributed to accumulation of round-off error in comparison to small field

intensities as we move farther and farther from the source. The round-off error

becomes, as a percentage, a larger part of the field intensity far from the source.


2-16

Figure 3 Numerical to analytical solution comparison for the source height at 10 m and

absorbing surface.

2.4.2 Source at 20 m with Reflecting and Absorbing Surface

The solution for the reflecting surface case in a constant index of refraction

medium is found by the superposition principle: the field for a point source at 0z

is added to the solution to the field for a point source at 0z− .

If the initial field is defined as ( )( )2 202

1 exp /22

z z wwπ

− − , then the solution will

be

( ) ( )( ) ( )( )

( )( ) ( )( )

0

0

2 2 202

0 0

2 2 202

0 0

1, exp /2 2 /

1 exp /2 2 /

ixk

ixk

E x z z z wik w ix k

z z wik w ix k

= − − + ++

− + ++

(2.43)

and the field intensity will be

( ) ( )2 2024 24

2 200

22

22 042 4 2 2 4

0 0

22, exp cos4 / x

k

z z

w xk

zz wE x z

wk w x k+

+

= − + + (2.44)


2-17

Figure 4 Reflecting and absorbing surfaces with source at 20 m.

This result is understood in the context of Young’s fringes, or interference from

multiple slits. Diffraction theory tells us that the far-field pattern from two slits

will be the Fraunhofer diffraction pattern from one of the slits modulated by a

cosine term whose frequency depends on the separation of the two slits. We

demonstrate this in Figure 5. The plot on the right side shows the interference

term generated by showing the fringe intensity.

Figure 5 Multiple slit interference demonstrated by moving source to 200 m of the ground.

2.4.2.1 Forward Maximum on Ground

With the reflecting case (left side of Figure 4) and the source noticeably off the

ground, theory predicts the occurrence of a local maximum away from the line


2-18

0x = for 0z = . This local maximum will occur when 202

21

zw> where w is the

source width and 0z is the positional height of the source. This will occur on the

ground at

2

2 020

21

Max

zx k w

w= − (2.45)

For this example, 0 1k = , the source height is 0 20z = meters, and the source

width 10w = meters. Evaluating (2.45) shows that Max

x should occur at

265 meters. In Figure 6, the plot shows a maximum occurring at step 26. With

10 meters between steps, the numerical solution agrees very well with the

analytical result.

Figure 6 Plot of intensity with distance from source for 0z = meters.

For the absorbing surface on the right side of Figure 4, we are no longer in the

multiple-slit situation and the fringe should disappear. The figure confirms this.


2-19

2.5 Cases Involving Arbitrary Propagation Media and

Rough Surfaces

The software to perform the marching technique was written for predicting the

acoustic fields due to a point source over rough terrain in the presence of various

meteorological conditions. One of the chief parameters that affect the sound

patterns away from the source is the wind profile. Common experience tells us

that we better hear things when we are so-called ‘downwind’ of them.

Consider an acoustic source at a height 0z off the ground and let it be a Gaussian

source such as that in (2.40). We might imagine it to be an explosion of some

kind. Based upon the wind patterns around the source, can we predict what the

acoustic intensity will be along the ground away from the source? This same

type of approach can be used in urban planning when designing airports near

cities or habitats, motorway and road planning through villages, and others.

The marching technique offers a computationally efficient means to answer real-

world problems in a timely fashion. The software written here was used with real

terrain data and meteorology data to predict the acoustic intensities over a range

of 30 km from the source. In the next two figures, we provide examples of the

marching technique to handle a linear wind profile blowing in the x+ direction

with a flat surface and the same wind profile blowing over a rough surface.

In Figure 7, we demonstrate the effect of the medium on the acoustic field with a

flat surface. Just like a lens serves to refract an electromagnetic wave in a

manner governed by Snell’s law, the atmosphere can function similarly to a lens.

Here we see the effect of a linear wind profile. The field tends to bend in the

direction of higher index of refraction. In Chapter 3, we will explore the

generation of caustics caused by the bending of the sound field toward the ground

and the trapping of it there. This figure helps explain why we hear things so

much better ‘downwind’ — there is more of the sound energy along the ground.


2-20

Figure 7 Acoustic intensity for flat, hard surface with wind speed profile of ( ) , .0105v z zε ε= =

and wind direction of 90°.

2.5.1 Comparing Numerical Results in Varying Medium to Theory

For a Gaussian source propagating through a medium with refractive index

( )

0

1 ,n z zcεε ε= − = , the Green function has been shown to be [18]:

( )

( )( ) ( )( ) ( )

2 230

0

, ; ,

1 exp2 2 2 12

G x x z z

z ziki z z x x x xk x x x x

εεπ

′ ′ =

′− ′ ′ ′− + − − − ′ ′ − −

(2.46)

The subtlety between and ε ε is discussed in Chapter 4. By using the Green

function and integrating over all source points along the 0x = aperture, the

acoustic field can be derived analytically. The intensity of the acoustic field is

found to be

( )( )2

222200

2 220

44

1, exp2

x

xxkk

z z wI x z

ww

ε

π

− + = − ++ (2.47)


2-21

when ( ) ( )( )2 202

10, exp /22

E z z z wwπ

= − − and the surface is absorbing, e.g.

0R = . The theory was compared to the numerical results and showed that the

even after 200 steps, the error in intensity is less than 1.7% when following the

locus 2

2xz ε= − and 0 0z = . After 50 steps, the error in intensity is less than

1/1000th of a percent [19].

In Figure 8, we give the example of a rough surface with constant reflection

coefficient of unity. In equation (2.2), notice how the curvature of the surface

affects the refractive index. With a surface of negative curvature, the refractive

index term for the image medium will tend to increase. At points of inflection of

the surface, the refractive index term in the image medium is symmetric with the

refractive index in the real medium.

Figure 8 Acoustic intensity shown for isovelocity medium and rough surface.


2-22

2.6 Varying Reflection Coefficient over the Terrain

The marching technique can be extended to handle variable terrain in a

straightforward manner. In fact, Hatziioannou did related work in dealing with

electromagnetic radiation incident on a rough surface [20]. Given the transverse

nature of the field, one must consider the horizontal and vertical polarisation

cases distinctly. He showed that for a horizontally polarised electromagnetic

plane wave incident on a rough surface at angle iθ going from a non-conducting

to a conducting medium has a reflection coefficient of

( )( )

2 2

2 2

cos 1 sin

cos 1 sin

nni i

nni i

n nR

n n

θ θ

θ θ

′

′

′− −=

′+ − (2.48)

where ,n n ′ represent the refractive indices of the two media. He went on to

show how a variable reflection coefficient could then be accommodated in the

marching technique for electromagnetic radiation. (2.48) is known as the Fresnel

equation for the interface of two media.

In this work, the marching technique was expanded to handle the same type of

situations for acoustic radiation. The propagation part of the marching technique

can be thought of as generating a new source at each step, which is then

propagated to the next step. When the reflection coefficient does not change, we

can retain the image field from step to step and just propagate it along with the

real field above the surface. However, when the reflection coefficient changes,

then we have to always satisfy (2.4). Thus, when the reflection coefficient is

variable, at each new step we discard the image field and reapply (2.4) with the

new value. This technique is demonstrated in Figure 9 through Figure 11 on the

next page.

In all three figures, we start with 1R = . In Figure 10 and Figure 11, we change

the reflection coefficient after about 230 steps. The effect on the field is clear

when compared to the perfectly reflecting case for all 300 steps. The right half of

each figure provides a plot of the complex reflection coefficient at each point

along the surface. The location of the change in reflection coefficient and change

in the field are in agreement.


2-23

Figure 9 Flat, hard surface with constant reflection coefficient throughout propagation.

Figure 10 Flat, mostly hard surface with reflection coefficient change 78% through propagation.

Figure 11 Flat, mostly hard surface with change to absorbing surface 78% through propagation.


2-24

Also, the technique was expanded to handle a three-dimensional refractive index

profile.

2.7 Extension of Marching Technique to 3-D

One potential prize in this work is the extension of the image method and

marching technique to three dimensions. The analogue seems to peer from the

page, yet the derivation is much more difficult than one would expect, and to

date has not been discovered.

The governing parabolic equation in three dimensions is

( )2 0

0

2 22

2 2

, ,2 2T

T

ikE i E N x y z Ex k

y z

∂ = ∇ +∂

∂ ∂∇ = +∂ ∂

(2.49)

One might expect the image method to cleanly map to the three-dimensional case

with the initial field defined as

( ) ( )

( )( ) ( )( ) ( )[ ] ( ){ }0 0 0

0 0 0 0

0, 0, 0, 0,

ˆ0,0,2 0,0 0,2 0,0 exp 2 0,0 0,0

R

R

E z E z

E S z RE S z ik z S S

=

− = − − ∇ ⋅ u (2.50)

with ( ), ˆdS x y

Sds

= ∇ ⋅ u being the derivative in the direction of the wavefront at

( ),x y with the refractive index defined as

( ) ( ) ( )

( ) ( )( ) ( ) ( )[ ] ( )

1

2

21

, , , , ; ,

, , , ,2 , 2 , , ; ,

N x y z n x y z z S x y

d SN x y z n x y S x y z x y z S x y z S x yds

= >

= − − − > (2.51)

and

2

2

2 22 2 2

2 2 2 2xx x yy y xy

d S d S x S yds ds x s y s

d S x x y y y xS S S S Sds s s s s s s

∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ = + + + + ∂ ∂ ∂ ∂ ∂ ∂

(2.52)


2-25

However, the derivation quickly leads to a conundrum: the technique used in two

dimensions to replace REx

∂∂

with the right hand side of the 2-D paraxial equation,

in this case can not be done for both the x and y components of the field. It is

almost as though the physics of the 3-D paraxial equation is resisting this

multidirectional approach.

Recall the defining equation on the image medium: ( ) ( ), , , ,reimE X Y Z RE x y z= ,

with ( ); ; 2 ,X x Y y Z S x y z= = = − the coordinates defined for the image

medium. We have let im IE E≡ and re R

E E≡ for clarity in notation. The image

field differential can be written two ways:

ˆ ˆ ˆ ˆ ˆ ˆ;

r reim

r

dE R E d

d dx dy dzx y z

= ∇ ⋅

∂ ∂ ∂= + + ∇ = + +∂ ∂ ∂

r

r i j k i j k (2.53)

or

ˆ ˆ ˆ ˆ ˆ ˆ;

im imR

R

dE E d

d dX dY dZX Y Z

= ∇ ⋅

∂ ∂ ∂= + + ∇ = + +∂ ∂ ∂

R

R i j k i j k (2.54)

Relating dR to dr via the chain rule: ˆ ˆ ˆ2 1 2 1S Sd dX Y

∂ ∂ = + + + − ⋅ ∂ ∂r i j k R ,

one can show that in the first case the equation becomes

ˆ ˆ ˆ2 2re re re re reim

E S E E S E EdE R dx X z y Y z z

∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + − ⋅ ∂ ∂ ∂ ∂ ∂ ∂ ∂ i j k R (2.55)

Using the fact that reimE ERZ z

∂ ∂= −∂ ∂

and equating the differential components

from the two relations one can get an expression for imE directly:

2

2

reim im

reim im

E ES ERX X Z x

E ES ERY Y Z y

∂ ∂∂ ∂= − +∂ ∂ ∂ ∂

∂ ∂∂ ∂= − +∂ ∂ ∂ ∂

(2.56)


2-26

At this point in the 2-D derivation, one would only have the x component

equation. One could then substitute the original paraxial equation for the real

source in place of reEx

∂∂

and the derivation is nearly completed except for an

integrating factor.

But attempting the same approach in three dimensions leaves two uncoupled

equations with no means to directly substitute for reEy

∂∂

and there is no obvious

way to close the equations.

In the next chapter, we formulate the marching technique in cylindrical

coordinates and propagate a point source. We will be unable to handle rough

terrain, but can accommodate arbitrary refractive index profiles.


3-1

C h a p t e r 3

3 Curvilinear Wave Propagation Over Rough Surfaces

with Varying Refractive Index Profiles

In this section, we extend the marching technique to curvilinear coordinates and

in doing so perform some simulations of wave propagation with a varying

refractive index.

The numerical solution to the PWE in curvilinear coordinates has several useful

applications. For example, the study of the moments of the field (such as the 4th

moment) for a medium containing weak variations in refractive index and the

study of ocean sound ribbons have both been carried out in curvilinear

coordinates. Its utility comes from the ability to more adequately describe a true

point source in the numerical solution. As we stated in the last section, the use

of a Gaussian source requires a very narrow source if it is to approach a point

source over an appreciable range of angles. This is the method proposed by

Tappert [21].

In this work, the curvilinear marching technique has not been able to handle

varying terrain, but flat terrain with varying refractive index profiles can be

accommodated. Thus, in the Cartesian cases when the surface slopes of curvature

were important, all these terms will be zero for the cylindrical cases presented

here. The conditions on the refractive index will reduce to

( ) ( )

( ) ( )1

1

, 1 , ; 0

, 1 , ; 2

N r n r

N r n r

θ θ θ π

θ θ π θ π

= + < <

= + − < < (3.1)

And the initial field will be taken as a delta function in the real medium,

0 θ π< < , with the usual definition in the image medium. Any reflection

coefficient is still valid, but the examples will be restricted to reflecting and

absorbing surfaces.


3-2

( ) ( ) ( )

( ) ( )

0

0

0, 0, ; 0

0, 0, ; 2

R

R

E E r

E RE

θ θ δ θ π

θ θ π θ π

= = < <

= − < < (3.2)

The first section provides a brief outline of the derivation of the paraxial wave

equation in circular coordinates.

3.1 2-D Propagation in Curvilinear Coordinates

Uscinski [22] has shown that the Helmholtz form of the 2-D wave equation in

curvilinear coordinates

( )( )

( ) ( ) ( )

22

2 2 0

1 1 ,

, , exp

u ur k n r ur r r r

U r u r i t

∂ ∂ ∂ θ∂ ∂ ∂θ

θ θ ω

+ = −

= (3.3)

can be reduced to an approximate, paraxial form. The approach is similar to

that for the rectilinear case as the solution is assumed to be separable into the

product of a field term and a function, ( ),E r θ . Said another way, the rapidly

varying phase term and geometrical, amplitude fall-off necessary for energy

conservation are implicit in the solution to the field. Therefore

( ) ( )( )0exp

, ,ik r

u r E rr

θ θ= (3.4)

The paraxial approximation implies that ( ),E r θ is a slowly varying function that

can only change over the distance on order of the scale size of the medium.

Namely 2

20

E Ekr r

∂ ∂>>∂ ∂

, which implies the second order term can be neglected.

The final approximate equation for E(r, θ) is

( )2

2 2 0 10

,2

E i E ik n r Er k r

θθ

∂ ∂= +∂ ∂

(3.5)

with ( ) ( )1, 1 ,n r n rθ θ= + . We derive (3.5) in Appendix C.


3-3

Focusing on the propagation term in the equation above, we will examine

methods for the numerical simulation of this part of the equation. We start with

2

2 202

E i Er k r θ

∂ ∂=∂ ∂

(3.6)

and begin by examining the Fourier technique applied to equation (3.6).

One can apply the Fourier transform to the two-dimensional, curvilinear

propagation equation above.

2

2 20

12 2

i E i Ee dr k r

ωθ θπ θ

∞

−∞

∂ ∂ = ∂ ∂ ∫ (3.7)

As examined in Chapter 2, this technique can be used to avoid the direct

computation of the second derivative (or Laplacian operator in multiple

dimensions) on the right side of the equation. Instead, the rule for Fourier

transforms of derivatives is used. Namely,

( ) ( )2

22

f i Fx

∂ ω ω∂= (3.8)

Denoting the Fourier Transform of ( ),E r θ with respect to θ as

1( , ) ( , )2

ir e E r dωθξ ω θ θπ

∞

−∞

= ∫ (3.9)

Then in Fourier space the propagation equation is

( ) ( ) ( )2

20

,,

2r ii rr k r

∂ξ ω ω ξ ω∂

= (3.10)

Separating terms and rearranging

( )( )2

0

1ln2id dk rωξ = (3.11)


3-4

Finally, evaluating the integrands yield

( )( )( )

1 1

2

0 1

2

0

1 12

1

1ln ,2

( , ) ( , )

n n

n n

nn

r r

r r

ik r r

nn

id r dk r

r e rω

ωξ ω

ξ ω ξ ω

+ +

+

− +

=

=

∫ ∫ (3.12)

In (3.12), we see how the propagation between any two concentric rings of radius

1nr + and nr can be done in the Fourier domain. The last step is to take the

inverse Fourier transform of the equation so that the result is in ( ),r θ

coordinates.

2

10 1

1 1( , ) exp ( , )2

inn

nn

iE r e r dk r r

ωθ ωθ ξ ω ω∞

−+

+−∞

= − ∫ (3.13)

Therefore, the technique used for the Cartesian approach has a close analogue in

curvilinear coordinates. The major difference is in the kernel term, 2

0 1

1 1exp2 nn

ik r rω

+

− .

3.2 2-D Modulation in Curvilinear Coordinates

The general expression for paraxial, curvilinear wave modulation in two


( )0 1 ,E ik n r Er

θ∂ =∂

(3.14)

This is analogous to the Cartesian case. Furthermore, the general form of this

equation, E AEr

∂ =∂

, has the solution:

( ) { } ( )0 1, exp ,E r r ik n r E rθ θ+∆ = ∆ (3.15)


3-5

3.2.1 Mapping a Wind Velocity Profile in Cartesian Coordinates to

Curvilinear Coordinates

In section 2.4.1, we stated that the acoustic refractive index is always calculated

with respect to a reference speed of sound and in section 2.5 we showed an

example of a Cartesian propagation with a linear wind profile. In this section, we

complete the definition of the refractive index for acoustic applications.

The refractive index with regard to sound is usually calculated with respect to a

stationary frame:

( )( )0

0 sinc

n zc v z φ

=+

(3.16)

where, in general, 00 z zc c == is the ambient speed of sound (which can be a

function of temperature or humidity), 0z taken as representing sea level and

standard temperature and pressure conditions, v is the wind speed, and φ is the

direction of the wind. The convention on wind direction is:

Wind blowing from the north in a southerly direction is 0 degrees. Wind blowing from the west in an easterly direction is 90 degrees.

In Figure 12 and Figure 13, a linear wind speed profile was imposed and

zε=v (3.17)

where the wind is expressed in vector notation in (3.17). In the top half, the

resultant refractive index profile was computed with a wind direction of –90°

degrees. In the bottom half, the same wind speed profile was used, but blowing

in the opposite direction. The same calculations were performed for Figure 13, a

stronger profile. Even though the wind speed profile is linear, only when ε is

very small is the resultant refractive index profile also very linear.


3-6

Figure 12 Mild wind profile and resultant refractive index, .00258ε = .

Figure 13 Strong wind profile and resultant refractive index with .00989ε = .

This general result is true, independently of the coordinate system of choice.

However, the situation is further complicated in polar coordinates.


3-7

First imagine a ray coming from the source with angle θ . After traveling a

distance r , the height above the ground will be sinz r θ= . Based upon a linear

profile of wind speed, the ray will experience a wind speed

( )sinz rε ε θ= (3.18)

Now, the ray will only experience the portion of the wind speed at that height

projected into the direction of propagation, r . So the effective refractive index

the ray will experience at any distance r from the origin at any angle θ is

( ) ( ) ( )sin cosrv z rε θ θ= (3.19)

when a linear wind profile is used. A plot of (3.19) can be seen in Figure 14.

The radial axis is labeled in meters, but the curves in blue are scaled to show

what refractive index the field would experience at that radial for all θ . For

example, take the curve at 10 km. If the refractive index were unity at all

positions, the curve would coincide with the 10000r = contour. But instead, we

see that the refractive index is unity at precisely 4 positions: 30, , ,2 2π ππ . In the

first and fourth quadrants, the refractive index is greater than unity; for the

second and third quadrants, it is less than unity. The general shape of the

refractive index curve at any particular radial distance is that of a cardoid.

Recognize that the cardoid in Figure 14 arises from imposing (3.1) in conjunction

with (3.19) and (3.16). We show the real and image media in Figure 14 through

Figure 18, the image medium on the bottom.


3-8

Figure 14 Linear wind profile results in θ asymmetry.

3.3 Examples of Curvilinear Propagation Using the

Marching Technique

In this section we present solutions of the marching technique in polar

coordinates with flat terrain and varying strengths of linear wind speed profiles.

For solutions demonstrated in partial form, that is, without the geometrical radial

fall-off factor necessary for energy conservation, we will be displaying images

where the mean paraxial field is 1.0. This is to be expected as the δ function

representing the point source on the surface is initialized to one there. As this

field propagates, the energy that is bent away from one region must be accounted

for in another region. Bear in mind that the colormap in the partial solution

figures is highly contrasted to provide richer detail and the field variations for

most of the image may only be a few percent, except for close to the ground.

In the full solution cases, the radial decay dominates the detail in many instances.

But this has the advantage of displaying only those features where a substantial

change in field intensity is occurring.


3-9

3.3.1 Curvilinear Solutions in Partial Form

By partial, we mean the exclusion of the field fall-off by r so that we may gain

insight that may otherwise be dominated by geometrical factors.

In the following figures, the scale size is a radius of 10000 m. Physical data for a

reflecting flat surface lay in the 0 < θ < π region. The numerical simulation has

been performed with 2048 samples in θ and radial step size of r∆ = 100 m.

The primary wavenumber is taken as unity for convenience.

The initial value of the point source was taken as 1 with a radius of .01 m to

avoid the singularity at the origin. Finally, a reflection coefficient of 1R = was

used.

The major feature of the following figures is the tendency of the field to bend

‘toward the normal’ when traversing from a region of lower to higher index of

refraction. In Figure 15, the wind is blowing right to left in the images, or

90ϕ = − . Therefore, the refractive index is increasing from the surface up, for

θ between 0 and 45 in the ( )0x > region. Similarly, the refractive index is

decreasing from the surface up for theta between 135 and 180 degrees ( )0x < .

As the refractive index profile in z becomes stronger, the breaks toward the

normal are more severe. One feature unobserved before is this tendency for

ducting to occur along the 4πθ = radial. The ducting becomes more and more

focused with a stronger refractive index profile, but does not change radials as

1r >> .


3-10

Figure 15 Field intensity with progressively stronger refractive index profiles.

To get a better look at potential caustics, the curvilinear data was spline fit to a

Cartesian grid, and the scale in z magnified by a factor of 5. The images in

Figure 16 and Figure 17 correspond to 20 km across and 4 km in the vertical, or

2000 m above the ground.

In Figure 16 and Figure 17 (left-hand images) the refractive index profile is

strong enough to duct the field along the ground, but not strong enough to

introduce many reflections off the surface ( 0x < ). However, the right-hand side

images show multiple bounces for 0x < . There appear to be asymptotes in the

field along 3,4 4π πθ θ= = . The explanation is as follows: imagine the rays

emanating near the 3,4 4π πθ θ= = radials. Examining Figure 16, one sees that a


3-11

local maximum or minimum in refractive index occur along these radials. Rays

will tend to bend toward this radial for the set of rays 4πθ ≈ . Once they

attempt to bend past this radial, they encounter a smaller index of refraction at

the next step and therefore tend to converge along this radial. The rays may

very well oscillate about the 4πθ ≈ but cannot diverge from it.

Figure 16 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 1.

Figure 17 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 0.


3-12

For the 34πθ = radial, a local minimum in refractive index is encountered for

those rays near it and will always tend to break from it. But notice from Figure

14 that the refractive index profile is also locally very flat along 34πθ = . Those

rays will not see a local change in refractive index from one step to the next.

3.3.1.1 The Presence of Caustics

The presence of caustics when the field is bending towards the ground can be

derived from three main principles:

1. When a ray strikes the ground at position 1x , its next contact with the

ground occurs at 13x , etc.

2. A ray’s trajectory is unique and depends only on the launch angle from the

source.

3. As the angle of launch is increased, each subsequent ray will intersect the

preceding ray on its second arc.

Consider a beam centered on the source position 2

0 2xz z ε= − . The ray

intersects the ground where 0z = or at 01

2zx

ε =

and subsequent intersections

with the surface occur at ( ) 12 1n x+ where n is a whole number.

In terms of launch angle, 0 10x

z xx

∂ θ ε∂ =

= = such that the first maximum in peak

height after a reflection occurs when 20

0 2z z

θε

= = . One can build a series of these

trajectories and show that for small θ , the caustic has the form 2

18cxz ε= . This

compares to the full solution of ( )

21tan

32

c

c

x

z

ε

ε

− = and 3 tancx θ

ε= [23].


3-13

3.3.2 Curvilinear Solutions in Full Form

Here the full form of the solution is used by returning to the equation:

( ) ( )( )0exp

, ,ik r

u r E rr

θ θ= (3.20)

That is, we now include the geometric radial term in the denominator and the

rapidly varying phase term. In the following figures we plot

( )( ) ( )*

2 , ,,

E r E ru r

r

θ θθ = (3.21)

This is the intensity of the acoustic field in a vertical cross section of r and θ .

In the four subplots of Figure 18, the scale size is a radius of 10000 m. Physical

data for a reflecting flat surface obviously lay in the 0 θ π< < region. The

numerical simulation has been performed with 2048 samples in theta and radial

step size of 100m. The primary wavenumber is taken as unity for convenience.

As explained in section 3.2.1, the wind speed is stipulated by the equation (3.17)

and then the refractive index is calculated after a number of geometric factors are

accounted for.

Radial attenuation tends to dominate much of the structure displayed in earlier

pictures. But the most noticeable characteristics are very much present: bending

toward or away from the ground and regions of focusing.

In the next section, we will compare some of the curvilinear results to the results

achieved with the Cartesian approach.


3-14

Figure 18 Full curvilinear solutions with ε increasing from left to right, top to bottom.


4-1

C h a p t e r 4

4 Comparison of Gaussian Source to Point Source

In this chapter, we compare the methods of using a Gaussian source and using a

point source in numerical simulations. If accurate simulations are required over a

wide range of angles then the Gaussian source must have a very small width.

Even with relevant parameters identical, such as wavenumber, step size,

refractive index profile, reflection coefficient, etc., there will be differences. Many

of these differences can be traced to this statement: a Gaussian source, no matter

how narrow, is not a true point source.

In Figure 19, we show a Gaussian source and point source for the same quadrant.

The color scaling is over different intensity ranges from the left plot to the right

plot as it was chosen to convey the difference in angular information that will be

available from the two different sources.

Figure 19 The field from a point source (left) and from a Gaussian source (right) are shown in the

double-half plane. The surface is flat and reflecting and the source is positioned on the

surface.


4-2

Of particular interest is the effect of model differences on the acoustic

propagation in the atmosphere when some form of ducting is present. This type

of profile can be caused by wind shear and/or temperature and humidity

gradients. As mentioned in the previous chapter, we will be able to compare

various refractive index profiles but cannot accommodate rough terrain.

With regards to the point source, the primary definition of the Dirac delta

function is

( ) ( ) ( )0f dA fδ∞ ∞

−∞ −∞

=∫ ∫ 0r r - r r (4.1)

where dA is a differential element of area, ( )δ 0r - r is the Dirac delta function,

and ( )f r is a function. Because of its singular nature, we have some issues to

consider when implementing a delta function numerically. Therefore, in the

curvilinear solutions we assign it finite radial width and non-infinite amplitude

over this region. Finally, we must match the energy contained in the Gaussian

source of the Cartesian case to the point source of the curvilinear case. This

allows for direct intensity comparisons as well as relative ones for the two sources.

We should briefly examine the differences in the two forms of the PWE. In the

curvilinear case, we note that equation C.8 in Appendix C states 2

20

E Ekr r

∂ ∂>>∂ ∂

. In the Cartesian case, we are subject to the same

approximation 2

20

E Ekx x

∂ ∂>>∂ ∂

. This is the primary assumption in both

parabolic derivations and is interpreted as: the variation in the envelope of the

field is small in the direction of propagation, while in the transverse direction can

vary rapidly. Said another way, the envelope of the field is smooth. We are

subject to these restrictions in both cases. Furthermore, in the curvilinear form,

the derivation imposes the condition 2 20

1 04k r

≈ . This means we must be at least

several wavelengths from the source as it is only beyond this region that the

asymptotic form of the Hankel function is valid. Finally, the conditions on the

refractive index profile are identical: we ignore the ( )( )2,n x z′ term.


4-3

One effect still not described, which is different between the two cases, is what

the forward scatter approximation imposes on the field. In the Cartesian case,

this is intuitive enough as there is one natural direction to the propagation. We

can interpret this spatially as a ‘left-to-right’ sense, an ‘east-to-west’ sense, or a

‘top-to-bottom’ sense, etc. With the curvilinear case, the forward direction really

varies depending on the region as defined by and r θ . We can imagine this as

several pie-shaped sub-regions in which the curvature condition is imposed.

Therefore, even though we have information over a greater angular region, in

each sub-region we are restricted by the paraxial approximation. For example,

even in a high amount of shear wind, we cannot expect rays beginning along the

/2π radial to be able to quickly break toward the ground as this would require

us to precipitously cross pie-shaped regions that subtend roughly 20 degrees each.

The polar plots require only one run for each plot as an entire π radians of

information is obtained. However, to capture the similar type information in the

Cartesian case, two separate runs were performed and the results joined together

to create the output figure. That is, one run took 90ϕ = − and the second run

took 90ϕ = . We used the definition of section 3.2.1 for φ . Even with the

two runs joined, the PWE coupled with the Gaussian source restrict our

information to approximately the 0 /8θ π≤ ≤ and 7 /8π θ π≥ ≥ sectors.

There are four series of results, each corresponding to a different ε when the

wind speed profile is defined as

( )z zε=v (4.2)

When the refractive index is given as equation (3.16): ( )( )0

0 sinc

n zc v z φ

=+

, then

if 90φ = ± , we have

( )

2

0 0

1 z zn zc cε ε = + + +

(4.3)


4-4

when 90φ = − . If 0

1zcε , a linear wind speed profile results in a linear

refractive index profile and equation (4.3) becomes

( )

0

1 zn zcε+ (4.4)

This is the origin of the definition for 0/cε ε= in Chapter 2. Also, when

90φ = + ,

( )

2

0 0

1 z zn zc cε ε = − + −

(4.5)

Note that 0

1zcε < for the expansion to converge. The different ε (in units of

[s-1]) used in the sections below correspond to maximum refractive index values

(at 10000 meters) of MAX

n = 1.04, 1.08, 1.21, and 1.4 respectively when the

reference speed of sound is 0 350c = meters per second. This corresponds to the

positive x region of the plots in Figure 20, Figure 23, Figure 26, and Figure 29.

And similarly, the negative x region of the plots in these figures represent the

opposite effect as the wind is now blowing in the direction of propagation. Thus,

the refractive index is decreasing with altitude from a unity value on the ground

in this direction. The images represent acoustic intensity where the colour map

has been scaled with field intensity. Black is the lowest intensity with red being

the highest.

The graphical plots in Figure 21, Figure 22, Figure 24, Figure 25, Figure 27,

Figure 28, Figure 30, and Figure 31 provide the intensity profiles computed on

the ground and at 500 meters for both the polar and Cartesian cases for the

various refractive index profiles.

We have now outlined some of the points of consideration for the comparison and

we must bear these in mind when examining the results of the next few sections.

The figures in sections 4.1 through 4.4 are intended to provide some qualitative

and quantitative comparisons between the two approaches.


4-5

4.1 Comparison for ε = .00134

From Figure 20, we can see that the wind is blowing right-to-left. The refractive

index profile is not particularly strong as the rays are gently bent either toward

or away from the ground depending on whether they are upwind or downwind.

The source width in the Gaussian case is 10 m and the wavelength is about 6 m.

One of the main points to recognize is not only the difference in field of regard as

mentioned repeatedly, but also the distribution of energy. Notice that the energy

from the source spreads over a smaller angular range in the Gaussian case.

Thus, if the sources start with the same amount of energy, we must have a

greater amount of energy in the pie-shaped sectors of the Gaussian source than in

the point source.

Figure 20 Numerical results of curvilinear and Cartesian field intensities for case ε = 1.34e-3


4-6

Figure 21 and Figure 22 provide intensity profiles along the ground and at a

height of 500 m for both the point and Gaussian source cases. On the right side

of Figure 21, we see the power-law fall-off is the same for both the Gaussian and

point source cases on the upwind side with a one decade attenuation in 7000 m.

Figure 21 Comparison of field intensities along ground for case ε = 1.34e-3.

Figure 22 Comparison of field intensities at 500 m for case ε = 1.34e-3.


4-7

The downwind side shows similar shape for the intensity profile along the ground,

however the intensity levels are higher in the Gaussian case. Again, we must

consider this in terms of the propagation of energy away from the source and its

concentration in the Gaussian case. In Figure 22, we see the intensity profile at a

height of 500 m from the surface. This provides clear evidence of the slit-like

behaviour of the Gaussian source as we can see the field drop precipitously out of

the primary direction. On the bottom plot of Figure 22, it occurs between

-2000 m and 2000 m in x .

4.2 Comparison for ε = 2.58e-3

In Figure 23, we nearly double the parameter ε and see that the field bends more

noticeably on the upwind side. On the downwind side, it appears that caustics

are beginning to develop close to the ground. Looking at Figure 24, the acoustic

intensity along the ground for the two sources is still similar both upwind and

downwind. As in Figure 21, we notice a higher field intensity close to the source

( )0x = for the point source case. However, the Gaussian source intensity

downwind along the ground is higher as previously. The power law fall-off

upwind is consistent between the different cases with a one decade attenuation in

4500 m.

One of the interesting features of Figure 24 is the local minimum on the

downwind side. In the Gaussian source case, it occurs near –8000 m. While in

the point source case, it occurs around –7000 m. Furthermore, the Gaussian case

shows a larger change in intensity, 2I E∆ = ∆ , near the –8000 m mark than in

the point source case at its –7000 m mark. One explanation for the change in

intensity near with local minimum is offered in terms of the energy perspective.

Because the energy is initially confined to this 20≈ wedge in the downwind

direction, more rays will tend to reflect from the ground in the same area. While

in the point source case, it is more spread out in θ . The change in position of

this local downwind minimum between the different cases is surprising. Figure 25

shows that we still have the absence of field information in the Gaussian case at

angles greater than the slit angle. The field intensities in both plots of Figure 25

are asymmetrical and this is to be expected.


4-8

Figure 23 Numerical results of curvilinear and Cartesian field intensities for case ε =2.584e-3.



4-9


4.3 Comparison for ε = 5.795e-3

Now in Figure 26, the wind speed profile has increased considerably again and we

see strong upward ducting for the field propagating into the wind. The

development of caustics is clear and we see noticeable refracting of the fields in

both cases. One numerical issue arises in the Gaussian case that can be seen in

the bottom image of Figure 26. As described in Chapter 2, the propagation part

of the PWE is done in the Fourier domain and we use the FFT algorithm to

transform in the z direction. We use a finite number of samples and choose a

range of z to sample over. The problem arises as the field continues to

propagate and spread. Because the process is periodic on a lattice, we must be

careful to avoid wrap-around by the image field. Therefore, we always need to

discretize a region in z that extends beyond the current edge of the field. Thus,

we must increase z∆ as a function of x if the number of samples is to stay

constant. In practice, the sample number remains fixed as a programming

convenience. z∆ can easily be adjusted based on the slit analogy of Chapter 2.

However, if the field is being bent as well by the medium, it becomes much more

difficult to anticipate what an acceptable z∆ must be to encompass the field. As


4-10

can be seen in Figure 26, the upward refraction of the field is strong enough to

encounter this wrap-around problem at about 8000 mx = . We can control this

problem by making z∆ larger to begin with, but this does negatively affect the

resolution in z for a fixed sample size.

Figure 26 Numerical results of curvilinear and Cartesian field intensities for case ε = 5.795-3.

In Figure 27, the intensity patterns along the ground become more irregular as

the rays bend more violently toward the ground downwind and experience

multiple bounces. As in previous plots, we note the shift in local minimum with

the point source field leading the Gaussian field minimum by roughly 500 m

where 4000mx = − . Both fields attenuate about one decade over about 2500 m

in the positive x direction. Notice that the fall-off has increased as the upward-

bending of the field is more severe.


4-11



In Figure 28, the asymmetry is becoming quite pronounced in the Gaussian case.

Much of this is due to the strong upward refraction of the field. It is fairly easy

to see the yellow lobe on the bottom of Figure 26 and correlate that with the

intensity profile for 2000 m 4000 mx< < in the Gaussian case. In the


4-12

curvilinear case in Figure 28, we do see how the field near 0x = is washing out

along the 500 mz = line as compared to the ground profile in Figure 27.

4.4 Comparison For ε = 9.86e-3

The last set of figures represent the results of a physically unrealistic refractive

index profile. All of the issues discussed in the previous results are grossly

magnified here. In Figure 30, the fall-off of the acoustic intensity in the upwind

direction is precipitous and there is now evidence that the point source and

Gaussian source fields fall off at slightly different rates for this profile.

Downwind, the fields of Figure 30 share many similar features out to

4000 mx = − , but then the local intensity peaks and valleys do not correlate as

well. We see yet again that the Gaussian source case leads to higher field

intensities along the ground. Figure 28 and Figure 31 are clearly similar.

However, the Cartesian case on the bottom plot of Figure 31 shows a large gain

in acoustic intensity along the 500 mz = when 9000 mx = − . This is due to the

line of constant z crossing the caustic on display in Figure 29.

Figure 29 Numerical results of curvilinear and Cartesian field intensities for case ε = 9.86e-3.


4-13




4-14

In summary, we have noted both differences and similarities in the results

between the two sources. The upwind behaviour in both cases is very consistent

within a multiplicative factor. The downwind behaviour shares many similar

features in terms of the overall structure of the local intensity peaks and valleys

along the ground, but the locations of these local minima/maxima do not line up

perfectly. The most prominent differences are clearly the lack of any field

information in a large wedge region in the Gaussian case and the overall higher

acoustic intensities along the ground in the Gaussian case. Because this

difference is reasonably well corrected by a constant multiplicative factor, one

explanation is simply to examine the angular distribution of power going away

from the source.

In Figure 32, we compensate our original Figure 21 by a multiplicative factor of

w

πθ

where wθ is the characteristic angle for a Gaussian source of wavelength 0λ

and source width w . This factor is only applied to the curvilinear case. For the

parameters used in our problem, this factor has a value of about 8 since

/8wθ π≈ . From visual inspection, the correction brings the acoustic intensities

along the ground to nearly identical values over a large range of x .

Figure 32 The acoustic intensity along the ground in the curvilinear case is compensated by w

πθ


5-1

C h a p t e r 5

5 Solutions to the Scattered Field Using the

Integral Equation Method

An acoustic wave incident on the underside of the sea surface is both reflected

and, at the same time, scattered by the irregular surface formed by waves and

ripples. For many sonar frequencies, the acoustic wavelength is less than the

height of the water waves and so the surface roughness can lead to very large

phase modulations in the reflected wave. For example, almost all sea surfaces

range between several millimeters and several meters in variation. For an

11 KHz signal, this range of sea heights would be on the order of, to ~1000 times

greater than, the wavelength. Also, in many practical situations, the acoustic

energy is incident at very low angles on the under surface of the sea. In short,

the problem that arises is one of scattering at low grazing angles from a deeply

modulated pressure-release surface [24].

There are several accepted approximations to the integral equation itself that

have various regimes where they are successful. However, the two main

approximations relate to that of small surface slopes and small surface heights.

The first leads to the Kirchhoff approximation and the latter to perturbation

theory [25]. Even these methods break down for small angles of incidence due to

the inability to handle shadowing and multi-scatter effects. Another

approximation used in much computer rendering today is ray theory, which is the

zero-wavelength limit. Since diffraction effects are potentially significant with

acoustic wavelengths and typical surfaces, the ray theory is usually inappropriate

here.

In this chapter and beyond, we will use the parabolic form of the Integral

Equation approach to derive expressions for the mean scattered acoustic field for

the situation above. The expressions will be valid for grazing incidence and

multiple scattering, but are restricted to the case of forward scattering. Finally,

we will restrict the discussion to one-dimensional surfaces.


5-2

5.1 Introduction to the Integral Equation

Let ( )p r be the acoustic pressure at the general point ( ),x z=r . The integral

equation approach to finding the field off a rough surface relies on this

fundamental relation which is the Helmholtz form of the wave equation:

( ) ( ) ( ) ( ) ( ), ,pp r p dsν ν

∂ ∂ ′ ′ ′ ′= − ∂ ∂∫r r r r r rGG (5.1)

where ds is an element of the surface defined by an outward facing normal at

that place and ( )r r ′−G is the Green function for the Helmholtz form of the

wave equation [26]. A brief derivation of this equation is provided in

Appendix D.

With gentle surface slopes, the normal to the surface, η , can be approximated

with j and derivatives with respect to the surface normal can be substituted with

z∂∂ . That is, ( ) ˆ ˆˆ ˆf f ff

x z η ∂ ∂ ∂∇ ⋅ = + ⋅ = ∂ ∂ ∂

i jη η where fη

∂∂

is the directional

derivative of the scalar function ( ),f x z in the outward normal direction to the

surface and ˆ ˆ≈j η . The Helmholtz integral equation then becomes

( ) ( ) ( ) ( ), , ,E GE x z G E dsz z

∂ ∂′ ′ ′= − ′ ′ ∂ ∂ ∫ r r r r r (5.2)

where ( ) ( ), and , ; ,E x z G x z x z′ ′ are discussed below.

From Figure 33,

( )1 4 2 3

, [ ] [ ] [ ] [ ]S S S S

E x z ds ds ds ds= + + +∫ ∫ ∫ ∫ (5.3)


5-3

x x0

z0

xz

S2

S1

S4S3

ε

Figure 33 The closed surface, s , includes the real surface, ( )S x , but excludes the source at 0z .

First, consider 4S over 0x x ε= + . The contribution from this surface is zero

since there is no back-propagation from 0x to x , i.e. the backward Green

function vanishes. Next, let both surfaces 1 2,S S move to ±∞ respectively. In

this case, the contributions from 1 2,S S vanish leaving

( ) [ ] ( )3

, ,incS

E x z ds E x z= =∫ (5.4)

while the contribution from 4S is zero as there is no backscatter. We can now

write (5.2), on restoring 2S to its original position, as

( ) ( )

( )( ) ( )( ) ( )( ) ( )( )0

, ,

; , ; ,

inc

x

E x z E x z

E GG x x z S x x S x x x z S x E x S x dxz z

− =

∂ ∂′ ′ ′ ′ ′ ′ ′ ′ ′− − − − − ′ ′ ∂ ∂ ∫ (5.5)

where the integration is over 2S , the rough sea surface ( )S x .


5-4

5.2 The Paraxial Point Source and Green Function

In practice, the present method here is applicable for all values of surface wave

height and for those acoustic wavelengths that are shorter than the correlation

lengths of important surface features. This restriction ensures that surface

scattering will be predominantly in the forward direction. In this case, the

forward scatter, or parabolic approximation can be employed. Under the

parabolic approximation, the acoustic pressure, p , is defined as

( ) ( ), , ikxp x z E x z e= (5.6)

and the parabolic form of the Green function is defined as

( , ; , ) ( , ; , ) ikxx x z z G x x z z e′ ′ ′ ′=G (5.7)

In both of these definitions, ( ) ( ), and , ; ,E x z G x z x z′ ′ , the rapidly varying phase

term of ikxe is absorbed from the full expressions. For the pressure, this leaves

the slowly varying envelope, ( ),E x z . As mentioned in the opening paragraphs,

implicit in this approximation is that the angle of incidence between the incident

field and the surface is small (say 15 20− ).

As is well known from theory, the Green function in (5.2) represents the solution

to the parabolic wave equation when there is a delta function source at ( ),x z′ ′ .

That is, the Green function tells us how the field is influenced at ( ),x z from a

point source at ( ),x z′ ′ or due to Maxwell’s reciprocity, vice versa.

Mathematically we express the equation describing the Green function as

( ) ( ) ( )2

2 00

, , , , ;2i ik n x z G x x z z x x z z

x k zδ

∂ ∂ ′ ′ ′ ′ ′− − = − − ∂ ∂ (5.8)

where ( );x x z zδ ′ ′− − is a point source, 0k is the wavenumber, and ( ),n x z′ is

the variation in refractive index. Note that had we not made the paraxial

approximation, equation (5.8) would represent the Green function for the full,

time-independent wave (or Helmholtz) equation:


5-5

( ) ( ) ( )2 2

2 22 2 0 , , , , ;k n x z G x x z z x x z z

x zδ

∂ ∂ ′ ′ ′ ′ + + = − − ∂ ∂ (5.9)

where ( ) ( ) ( )0, , ,n x z n x z n x z′= + and 0 1n ≡ typically.

As mentioned in Chapter 4, the solution to equation (5.8) for the Green function

is

( )

( )( )( ) ( )( ) ( )2 32

0

0

;

1 exp2 2 2 12

G x x z z

z z x xiki z z x xk x x x x

εε

π

′ ′− − =

′ ′− − ′ ′− + − − ′ ′ − −

(5.10)

when the medium has a linear profile ( 0ε ≠ ) and

( ) ( )( )( )

2

0

0

1; exp2 2 2

z zikiG x x z zk x x x xπ

′− ′ ′− − = ′ ′ − − (5.11)

for an isovelocity medium ( )0ε = [27]. The latter is the case we will mostly

consider in this work.

We are using the paraxial form of the wave equation and therefore must use the

paraxial approximation to the field due to the source

( )( )( )

20

0 020 0

0

, e eik z z

ik x xinc

E EE

x x

− −− − −= ≈

−−r r

0

0

r rr r

(5.12)

Typically, the coordinate system is chosen with 0 0x = . As we discuss the

analytic solution to integral equations in the next few chapters, bear in mind that

they have been evaluated analytically in certain conditions before and are the

subject of broad research. The primary condition is that the Green function is

known and the boundary conditions can be handled. An instance where these

types of problems have been solved analytically include regions containing the

upper half-plane (flat surface) where the refractive index is varying linearly or is


5-6

constant. When the refractive index of the medium no longer has a simple

relation or the surface is no longer flat, analytical evaluation of the integral

becomes very difficult. Some attempts have been made to transform the problem

into a modified coordinate system to handle surfaces that aren’t flat (e.g.

Gaussian hills) but the method is of limited utility [28].

5.3 Volterra Equations of the First and Second Kind

The integral equation technique here is a specific example of a special set of

integral equations known as Volterra equations. In two specific cases of interest,

either the derivative of the field vanishes at the surface (Neumann boundary

condition) or the field itself vanishes at the surface (Dirichlet boundary

condition).

5.3.1 Pressure-Release Surface

In electromagnetics, this case is known as the perfectly conducting surface. It

might be encountered for an acoustic source underwater when the sound wave

reaches the underside of the sea surface.

In this case, the field ( )( , )

0x S x

E = on the surface and the reflection coefficient,

1R = − . From (5.5),

( ) ( ) ( )( ) ( )( )0

, , , ,x

tot inc

EE x z E x z x S x G x x z S x dxz

∂ ′ ′ ′ ′ ′= + − −′∂∫ (5.13)

where this can be thought of as an equation relating the total field to the sum of

the incident and scattered fields.

On the surface, ( ),E x z vanishes and we have an integral equation where

( )( ),E x S xz

∂ ′ ′′∂

is unknown.

( )( ) ( )( ) ( ) ( )( )0

, , ; 0inc

x EE x S x x S x G x x S x S x dxz

∂ ′ ′ ′ ′ ′ ′+ − − =′∂∫ (5.14)


5-7

(5.13) and (5.14) form a set of equations. Since ( )( ),inc

E x S x and

( ) ( )( );G x x S x S x′ ′ ′− − are known, one can solve for ( )( ),E x S xz

∂ ′ ′′∂

. This

quantity is needed for the more straightforward computation of ( ),totE x z . The

solution for the unknown in (5.14), when set into (5.13), gives an integral

expression for the total field in the space above the surface. It is a Volterra

Integral Equation of the first kind with respect to ( )( ),E x S xz

∂ ′ ′′∂

.

5.3.2 Reflecting (Hard) Surface

In this case the field condition is ( )( ), 0E x S xz

∂∂

′ ′ =′

, the reflection coefficient

1cR = + , [29] and the equation becomes

( ) ( ) ( )( ) ( )( )0

, , , ,x

tot inc

GE x z E x z E x S x x x z S x dxz

∂∂

′ ′ ′ ′ ′= − − − ′ ∫ (5.15)

An analogous method to that in section 5.3.1 can be used to express the field

above the surface to that on the surface with

( )( ) ( )( ) ( )( ) ( ) ( )( )0

, , , ,x

tot inc

GE x S x E x S x E x S x x x S x S x dxz

∂∂

′ ′ ′ ′ ′= − − − ′ ∫ (5.16)

valid on the surface.

5.4 An Integral Equation Involving the First Moment or Mean

Field

Of interest is the ensemble average of the total field, also known as the mean

field. With scattering off a rough surface (which has some probability

distribution), one can numerically solve a Volterra equation to produce a solution

to the scattered field for a given set of boundary conditions. However, this only

provides one instantiation of the field. To gather meaningful statistics on

properties of the field, many such simulations must be run (with the surface


5-8

generated from the same probability distribution each time) to try and come up

with the quantity known as the ensemble average.

If we were to take the ensemble average of the Volterra equation (say for a

perfectly conducting or pressure-release surface), in its most general form it would

look like

( ) ( ) ( )( ) ( )( )0

, , , ,x

tot inc


∂ ′ ′ ′ ′ ′< > = < > + < − − >′∂∫ (5.17)

anywhere in the field and

( )( ) ( )( ) ( ) ( )( )0

, , ; 0inc

x EE x S x x S x G x x S x S x dxz

∂ ′ ′ ′ ′ ′< > + < − − > =′∂∫ (5.18)

on the surface, where <> denotes the ensemble average.

Suppose that the terms under the integral were uncorrelated, then we could

express these relations as

( ) ( ) ( )( ) ( )( )

( )( ) ( )( ) ( ) ( )( )0

0

, , , ,

, , ; 0inc

x

tot inc

x


EE x S x x S x G x x S x S x dxz

∂ ′ ′ ′ ′ ′< >=< > + < >< − − >′∂

∂ ′ ′ ′ ′ ′< > + < >< − − > =′∂

∫

∫ (5.19)

The proposition of equality in equation (5.19) related to the ensemble averaging

is explored in depth in Chapter 8 and Uscinski has argued in favor of (5.19) based

on very physical arguments about the process itself. He has stated:

“The structure in ( ),E x z′ has been caused by the accumulated

interactions with the surface up to that range. Now suppose that the

surface contour ( )S x is altered only in the vicinity of x ′ over a distance

of about one correlation length. The change to ( ),E x z′ will be small

because its structure has been determined by all the preceding

irregularities, so that the effect of the change in ( )S x in the final step will

be negligible by comparison.” [30]


5-9

We then use these statements to draw the conclusion that in this type of low

angle propagation the complex acoustic amplitude ( ),E x z′ is statistically

independent of the local form of the surface ( )S x at any range x ′ , if several

correlation lengths have been traversed. Specifically, both ( ),E x z′ and ( ),E x zz

∂ ′∂

are statistically independent of ( )S x ′ locally, and are therefore statistically

independent of ( )( ), , ,G x z x S x′ ′ since it is a function of ( )S x ′ .

5.4.1 Gaussian Random Process

Let S be a random variable with zero mean and variance 2 2S σ< > = obeying

the normalized Gaussian probability distribution [31]

( ) { }2 21

1 exp /22

P S S σπσ

= − (5.20)

Then the ensemble average of a function ( )Y S is defined as

( )[ ] ( ) ( ) ( )E Y S Y S Y S P S dS∞

−∞

= < > = ∫ (5.21)

It is assumed that the surface height is a function, ( )S S x= , and is a stationary

random process with Gaussian statistics, normalized spatial autocorrelation

function of

( )( ) ( )

2

S x S xx xρ

σ′< >

′− = (5.22)

and a one-point distribution of that shown in (5.21). The two-point distribution

is given by

( ) ( ) ( ){ }( ) ( ) ( )

2 2

2 2 2 212 1 2 1 1 2 22 1

1 2

, exp 2 /2 1

, ,

P S S S S S S

S S x S S x x xπσ ρ

ρ σ ρ

ρ ρ−

= − − + −

′ ′= = = − (5.23)


5-10

when the two random variables have the same variance. The two-point

distribution has application for this problem. For instance, when we ensemble

average the surface Green function in equation (5.19)b, we must use the two-

point distribution. This is because the quantity ( ) ( )S x S x ′− is correlated over a

distance L , where L is the typical correlation length in the direction of

propagation. Therefore, we need to know both the joint probability density

function on the surface (specified as Gaussian in (5.23)) and the correlation

function for the surface (model is arbitrary).

5.4.2 The Ensemble Averages

The ensemble-averaged Green functions and incident field are computed using

(5.21) with respect to the appropriate probability distribution [32]:

( )( ) ( )( ) ( )1, ,inc incE x S x E x S x P S dS∞

−∞

< > = ∫ (5.24)

( )( ) ( )( ) ( )1; ;G x x z S x G x x z S x P S dS∞

−∞

′ ′ ′ ′< − − > = − −∫ (5.25)

( ) ( )( ) ( ) ( )( ) ( )2 1 2 1 2; ; ,G x x S x S x G x x S x S x P S S dS dS∞

−∞

′ ′ ′ ′< − − > = − −∫ (5.26)

When evaluating these integrals and using the expressions in (5.11) (with 0ε =

for an isovelocity medium), (5.20), and (5.23) we get

( )( ) ( )( )

( )2

0 02

0 02

0 0

2,ik z

E x ikinc inc k x ik

E x S x E x e σ

σ

−

−< > = < > = (5.27)

( )( ) ( ) ( )( )( )

20

20

20 0

21 12 2; ;

ik z

x x ikik x x ik

G x x z S x G x x z e σπ σ

′− −′− −

′ ′ ′< − − > = < − > = (5.28)

( ) ( )( ) ( ) ( ) ( )( )2

0 0

1 12 2 2 1

; ik x x ik x x

G x x S x S x G x x π σ ρ′ ′− − − −′ ′ ′< − − > = < − > = (5.29)


5-11

Note that the statistical averaging is done as an integral over the random

variable and its probability distribution so that the final result is no longer a

function of the Gaussian random variable. The details of these integrals are not

provided, but similar ones can be found in Appendix F through Appendix I in

relation to the two-surface problem formulated in Chapter 7.

5.4.3 Scaling the Ensemble-Averaged Equations

In the vertical, distance is scaled like 0/f L k= , the Fresnel radius for an

observer at a range L , where L is the correlation distance of the rough surface

[33]. In the horizontal, the correlation distance is its own scaling factor. This

scaling follows from the basic paraxial equation: 20

1 1x k z∆ ∆∼ . So the scaled

spatial dimensions are

/

/

X x L

Z z f

=

= (5.30)

When we apply these scalings to (5.14) and (5.13) we get

( ) ( ) ( )

( ) ( ) ( ) ( )

00

00

, 0,

, , ,

x

inc

x

tot inc

E X Z F X G X X dX

E X Z E X Z F X G X X Z dX

′ ′ ′< > + < >< − > =

′ ′ ′< > = < > + < >< − >

∫

∫(5.31)

which are both independent of the rough surface ( )S X . The rescaled versions of

(5.27), (5.28), and (5.29) become

( ) ( ) ( )( )

20 0

20

2

20 0, , ,

ik ZE X i

inc inc X iE X Z E X Z e γ

γγ −

−< > = = (5.32)

( ) ( )( )( )

2

2

20

21 12 2;

iZX X ii

k L X X iG X X Z e γ

π γ′− −

′− −′< − > = (5.33)

( ) ( ) ( )( )20

1 12 2 2 1

ik L X X i X X

G X X π γ ρ′ ′− − − −′< − > = (5.34)


5-12

with

2

2 kLσγ = (5.35)

00

EEkL

= (5.36)

( ) ( )E XF X

Z

′∂′< > = < >

′∂ (5.37)

5.5 The Laplace Transform Technique

One technique used to express solutions to integral equations involves the Laplace

transform. This transform is a natural fit due to the bounds of integration for

this problem (see equation (5.31) for example). Convolution in the Laplace

domain is well-suited to the bounds of integration of the integral equations. The

same is not true in the more general Fourier domain.

5.5.1 Definition of the Laplace Transform

Let’s take a moment to state the definition of the Laplace transform. Many of

the problems typically solved by this technique are systems where the time

domain is important: electronic circuits, feedback and control systems are two

good examples. In this work, the independent variable is not associated with

time or a function of it, such as ( )f t . Since we are dealing with time-

independent propagation problems, we will make the independent variable

familiar and consistent with our spatial propagation problems by denoting the

Laplace transform as

( )[ ] ( ) ( )

0

ˆ xf x F f x e dxλλ∞

−= = ∫L (5.38)

For the Laplace transform to exist, the integral in (5.38) must converge.

Therefore, for many functions λ has a restricted domain. Note that if λ is

complex, then [ ] [ ]( ) [ ]( )( )Re cos Im sin Imx xe e x xλ λ λ λ− −= − and [ ]Re 0λ > for the


5-13

integral to converge. From the values of the integrands, the function, ( )f x , is

only needed for 0x > and ( )f x is usually defined as zero for ( ) 0f x < .

5.5.1.1 The Convolution Theorem

One relation of particular significance in this work is the convolution theorem,

which states that

( ) ( ) ( ) ( )0

ˆ ˆx

y x h x x dx Y Hλ λ ′ ′ ′− = ∫L (5.39)

5.5.2 Relation of the Laplace Transform to the Fourier Transform

The Laplace transform is just a special case of the Fourier transform. In

Appendix E, it is shown that if we let ( ),i d idλ γ ω λ ω= − = − , then ( )F λ is the

Fourier transform of

( )

( ) ( )

0 0

2 0x

g x x

g x f x e xγπ −

= <

= > (5.40)

5.5.3 Laplace Transform Solution for Mean Field with Volterra Equation

of the First Kind, Isovelocity Profile, and Rough Surface

We will transform the Volterra equation and yield a simple algebraic solution.

Using (5.39) (the convolution theorem) and applying the Laplace transform

defined in (5.38) to (5.31), we get

( ) ( ) ( )

( ) ( ) ( ) ( )0

0

ˆ ˆ, , , ,

ˆ ˆ,, , , , , ,

ˆˆ ˆ

inc

tot inc

Z F G

Z F G Z

λ γ λ γ λ γ

λ γ λ γ λ γ λ γ

εε ε

= −

= + (5.41)

The beauty of this technique is the ability to recover the derivative of the mean

field at the surface, ( ) ( )E XF X

Z

′∂′< > = < >

′∂, in terms of two known quantities

all in the Laplace domain. Attempting to do this in the spatial domain is much


5-14

more difficult. We can eliminate ( )ˆ ,F λ γ from (5.41)b and take the inverse

transform to express the solution to the mean field problem as

( ) ( ) ( )( )

10

ˆ , ,ˆ ,, , , 1ˆtot inc

G ZGE X Z Z λ γ

λ γλ γε− < > = − L (5.42)

In the next chapter, we explore analytical and numerical solutions to (5.42). This

is dependent upon finding analytic expressions for the Laplace transform terms

and then an inverse as shown in (5.42). As we will demonstrate, there are

analytic expressions for ( )0, ,inc Zλ γε , ( )ˆ , ,G Zλ γ , and we have found an

approximate analytic form for ( )ˆ ,G λ γ when the autocorrelation function is

( ) X XX X eρ ′− −′− = . We have also been able to express the final solution in

integral form, which then has to be integrated numerically for every point in the

field where the value of ( ),totE X Z< > is desired.


6-1

C h a p t e r 6

6 Elements of the Solutions to Ensemble-Averaged Integral

Equations Using the Laplace Transform

In this chapter, we examine the analytic elements involved in using the Laplace

transform to solve integral equations. Because of the way a Laplace transform is

defined, when we perform an inverse transform we are really doing a contour

integration in the complex plane. Thus, much of Laplace transform inversion

theory is centered on Cauchy’s theorem and residue theory. After the elements of

the Laplace technique are described mathematically, we will present some results

for the mean acoustic field given a rough surface with Gaussian statistics and

roughness parameter γ .

6.1 The Point Source Representation and Its Laplace Transform

To solve the rough surface problem with Laplace transforms, we will first need to

define nomenclature. The mean incident field will be defined by ( )0, ,incE X Z γ in

the spatial domain. If the surface roughness parameter is present as a function

argument, then one can infer that the quantity in question has been statistically

averaged and is the computed expectation of that quantity. Thus,

( ) ( )0 0, , ,inc incE X Z E X Z γ< > ≡ (6.1)

are the same quantity. 0andX Z are the scaled, spatial variables that account

for the source position and γ is the scaled surface roughness parameter. The

coordinate system is in keeping with the definition in Chapter 5, Figure 33. In

the Laplace domain, we will use the variable λ as our independent variable and

express the Laplace transform of the mean incident field as ( )0, ,ˆ Zλ γε . Here we

have dropped the “inc” subscript for compactness, but may occasionally refer to

it as ( )0, ,inc Zλ γε if the meaning is not clear.


6-2

In the spatial domain, we expressed the mean incident field in scaled variables

(see equation 5.32) as

( ) ( ) ( )( )

20

20

2

20 0, , ,

ikZE X i

inc inc X iE X Z E X Z e γ

γγ −

−=< > = (6.2)

Using equation 5.38, the Laplace transform of the paraxial point source can be

expressed in integral form as

( ) ( )20

22 20 2

0

1, ,îZ

XX iZ e dX

X i

λγλ γ

γε

∞ −−=

−∫ (6.3)

To attempt an analytic evaluation of this integral, an appropriate transformation

is 2

22 , dXy X i dy

X iγ

γ= − =

− that converts the integral to a more workable

form:

( )2 2 2

02 2

2

2 22 4

0

2

, ,î iZ y

i y

i

Z e e dyγ λ

λγ

γ

λ γε∞−

−−

−

= ∫ (6.4)

A further variable change of 2

Y yλ= gives

( )22

2 22

2

20

2, ,ˆ bY

iiY

i

eZ e dYλ γλγ

λγ

λ γλ

ε∞−−

−

−

= ∫ (6.5)

The upper integration limit can be taken as ∞ since the square root branch points

effectively move to the real axis as the real part of the radical approaches infinity.

This leaves

( )2 2

2 22

2

22 2 0

0

2, , ; 1,2

î ba Y

Y

i

i ZeZ e dY a bλγ

λγ

λλ γ

λε

∞−− +

−

= = = ∫ (6.6)


6-3

However, there is an indefinite integral solution to the problem [34]

( ) ( ) ( )

( ) ( ) ( )( )

22 22 2 2 2

2

2 2

/

4

erfc 1 erf

b a xa x b x b bx x x

z z

e dx e w iax w iax Ca

w z e iz e iz

π −− +

− −

= − + + − +

= − = +

∫ (6.7)

and we must make use of the complex error function.

Evaluating this for ,a b and the bounds of integration 2 ,iλγ − ∞ yields

( ) ( ) ( )

( ) ( )

02

02

20

0

10

1

12

12

1, , 1 erf2

1 erf

ˆ i i Z

i Z

i

i

ZZ e e

Ze

λγ

λγ

λγ λ

λ

γ

γ

πλ γλ

ε − ± −

−

± ±

±

= − + −

∓

∓

∓ ∓

(6.8)

as the Laplace transform of the point source.

6.1.1 Visualizing the Point Source and Its Laplace Transform

Figure 34 and Figure 35 provide a series of curves in 0Z for the real and

imaginary components of the incident mean field.

Figure 34 (left) Point source field, ( )0, ,incE X Z γ , for several source positions.

Figure 35 (right) Point source field, ( )0, ,incE X Z γ , for several source positions over a smaller

range 0Z .


6-4

Notice that when 1X ∼ , the mean field begins to oscillate quite rapidly as a

function of X . This is noticeable in part because of the semi-logarithmic plot

style; but eventually, the 1/ X behaviour kills the integrand for large X .

In Figure 36, for a choice of 1λ ∼ we can see how the Laplace transform for

certain λ will kill the integrand while the field is in the negative portion of its

oscillation. These two factors can lead to the Laplace transform for certain λ

and 0Z giving a negative value.

Figure 36 Laplace transform integrand for point source at several source positions with 1λ =

and 2 1γ = .

The negative real part cannot be shown on the logarithmic plot of Figure 37 and

the data point is excluded from the plot, but it can be seen in the semi-log plot of

Figure 38. While the real part of the transform is not large in magnitude for

1λ ∼ , it is negative in this range as the circles do dip below the zero marker in a

few instances. Notice in Figure 38 that the λ axis is only two and a half decades

wide.


6-5

Figure 37 (left) A plot of point source Laplace transform for several 0Z and fixed 2γ .

Figure 38 (right) The Laplace transform of the point source agrees well with theory for small and

large λ . At the crossover 1λ ∼ , the real part of the integral (Laplace

transform) becomes negative, but close to zero.

The consistency of the numerical results to the theoretical results is generally

quite high. For instance, when 610 1λ− < < the difference between the

theoretical and numerical results is no greater than .1% and when 6100 10λ< <

the difference is no more than .8%. In the transition region where 1 100λ< < ,

the differences are as much as 6%. The general remarks about the Laplace

transform for the point source are as follows:

• Both the real and imaginary parts of the transform behave like 1λ

for large λ

• The real part behaves like 1

λ for small λ

• The imaginary part behaves like a constant for small λ

The asymptotic results in the next section further quantify these remarks.

6.1.2 Asymptotic Results

6.1.2.1 Case for Small λ

The Laplace transform can be represented in just a few terms for 0λ → . This

parameter expression is contingent upon Taylor series expansions for the

exponential function and the error function.


6-6

( ) ( ) ( ) ( )2 2 22 13erf erf 1 2 1zz z z z e z z z zπ ε− +∆ = +∆ − ∆ + ∆ − + (6.9)

Specifically, one needs to expand for small λ :

( ) ( ) ( )

( ) ( ) ( )( )( ) ( ) ( )

00

2

20

20 0

112!0

2 2 2

1 1 2

2 2 2

1 1

cos sin 1

erf erf , 0Z

i Zi Z

i

i iZ Z

e i Z

e i i i i

e γ

λλ

λγ

λ γ λ γπγ γ

λ

λγ λγ λγ

λ

−−

−

− − −

+ − + +

= − −

+ + →

…

(6.10)

and use these results in the full solution to the Laplace transform of the point

source.

Substituting these into the full expression before any reduction of terms gives

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( )

2020 2

2020 2

220 0 22

2

0 2

0, , 1 1 1 1 erf 1

1 1 1 erf 1

ˆ Z

Z

Z

Z

Z i i Z i e

i Z i e

γ

γ

λ γππγλ

λ γπγ

λ γ λγ λ

λ

ε −

−−

→ − + − − − − + − − − − −

(6.11)

After days of arduous, laborious, onerous algebra the final result can be obtained.

( ) ( ) ( )2 20 02 20 02 2

0 0 02 20, , erf 2 erf 2ˆ Z ZZ ZZ Z e i Z eγ γπγ γλλ γ π γ π γε − −

→ − − + + (6.12)

We see that ( )0, ,ˆ Zλ γε 0

1 iCλ

+ ∼ as 0λ → where 0C is a constant.

6.1.2.2 Case for Large λ

Now consider λ as λ → ∞ . To do so, we will use the asymptotic form of the

error function for large z :

( ) ( ) ( )

( ) ( )2

2

1 3 2 142

1

1erf 1 1 , , argz

mm

m

zm

e

zz z z π

π

− ∞⋅ −

=

− − − → ∞ < ∑ …∼ (6.13)


6-7

While this representation diverges, the first few terms are useful and are an

accurate representation for large z [35]. In the integral solution, there are two

error functions that must be addressed for λ → ∞ .

Because the error function possesses different asymptotic series in different sectors

of the complex plane, it exhibits what is called Stokes phenomenon due to a non-

analytic essential singularity at infinity [36]. (See Figure 39.) In fact,

( ) ( ) ( )

( )

2

2

1 3 2 1 3 54 42

1

1erf 1 1 as with argz

mm

m

zm

e

zz z zπ π

π

− ∞⋅ −

=

− − − − → ∞ < < ∑ …∼ (6.14)

should be of no surprise as for real ( ) ( ), erf erfz z z− = − . But finding that the

error function grows without bound in the other two quarters of the complex

plane might be a surprise. Evaluating the error function in the top and lower

quadrants shows that

( )

2

34 4

754 4

erf as with arg

arg

ze

zz z z

z

π π

πππ

−

− → ∞ < <

< <

∼ (6.15)

but the expression in (6.13) will work equally well for large z .

Figure 39 The complex error function exhibits Stokes phenomenon, which implies different

asymptotic representations depending on location within the complex plane.


6-8

We will approximate the error functions for large λ as

( )( ) ( )( )

( )( )( )( ){ }( )( )

0

0

0

0

2 2 2

2

2 2

2 2

2 2

erf 1 erf 1 1

exp 1erf 1 1

1

Z

Z

Z

Z

i i

ii

i

λγ λγγ

λγγ

λγγ λγ

γ

δ

π

± ± ± +

− ±± −

±

∓ ∓

∓ ∓∓ ∓

∓ ∓

(6.16)

In the first asymptotic approximation, we neglect the higher-order term as the

argument is 4π< . δ represents the portion related to 0Z that keeps the

argument 4π< . While in the second case, we must keep it as the argument is

4π> . Substituting these asymptotic forms into the integral solution and reducing

( ) ( ) ( )( )

( ) ( )( )

( )( )

2 2 20 0

220

2 020 0

0

1 10

11 1

2 2

1, ,2

11

ˆZ

i i Z i Z

i i Zi Z i Z

Z

Z e e e

ee ei

γ

λγ λ λ

λγ λλ λ

λγγ

πλ γλ

π

ε − ± + +

− + ± −± + +

+ − + − ±

∓

∓

∼

∓ ∓

(6.17)

All the terms cancel except for the last one, which leaves

( )( )( )

( ) ( )

202

02

202

20

20

1 1, , ,2 1

1, , 1 as 2

ˆ

ˆ

Z

Z

Z

Z ei

Z e i

γ

λ γ

γ

λ γλγ

λ γ λλγ

ε

ε

−

−

±

→ + → ∞

∼∓ ∓

(6.18)

The results for both small and large λ are consistent with Figure 37.

6.2 Laplace Transform of Ensemble-Averaged Green Function

for the Surface

In the last chapter we showed that the solution to the rough surface problem

requires the solution to two integral equations. In the second of these, we restrict

one coordinate to always be on the surface and when the ensemble average of the

so-called “surface” Green function is taken (as described by equation 5.33) we got


6-9

( )( ) ( )( )2

0

1 12 2 2 1

iG X Xk L X X i X Xπ γ ρ

′< − >=′ ′− − − −

(6.19)

To further our analytic cause, let us start with an exponential autocorrelation

function of

( ) X XX X eρ ′− −′− = (6.20)

for the rough surface and consider the Laplace transform of (6.19). We condense

notation by redefining X X ′− as X ; it does not change our result.

( )( )2

0 0

1ˆ ,2 2 2 1

X

X

i eG dXk L X i e

λ

λ γπ γ

∞ −

−=

− −∫ (6.21)

Using an exponential autocorrelation function with scaled variable X implies the

correlation length is ( )1O X = . To facilitate the transformation in equation

(6.21), a further simplification can be made by approximating the exponential as

[ ]

[ ]

1 , 0,1

0, 1,

X

X

e X X

e X

−

−

= − ∈

= ∈ ∞ (6.22)

The use of (6.22) in defining the autocorrelation function (ACF) allows for us to

have an exponential-like ACF that we can analytically transform. It maintains

similar properties of the spectrum for an exponential ACF.

( )0

112 2 2 2

0 1

1ˆ ,1 2 2

X Xik L

e eG dX dXXi X i

λ λ

πλ γγ γ

∞− − ≈ + − −

∫ ∫ (6.23)

Figure 40 shows how the exact integrand of (6.21) and its composite approximate

compare. The different curves correspond to different values of λ. Note the

independent variable is X , not λ. The exact expression and its approximation

closely agree. Again, we are not computing the transform here; we are merely

examining what the integrand looks like in each case to establish how much of a

difference there ends up between an exponential correlation function and its

approximate.


6-10

Figure 40 (left) Exact expression and approximation as function of X .

Figure 41 (right) Green function with exponential autocorrelation function.

Figure 41 plots the exact Green function of (6.19) in a linear way for several

values of the scaled surface roughness parameter, γ .

The first integral of (6.23) is easily evaluated through the substitution

y Xλ= and yields

( ) ( )1

1 2 20

1 1ˆ , erf1 2 1 2

XeG dXXi i

λ πλ γ λλγ γ

−

= =− −∫ (6.24)

The second integral is transformed in a similar manner with ( )22y X iλ γ= − as

this converts the integral into error function form,

( )2 2

2

22 2

1 2

2ˆ ,2

Xi y

i

eG dX e e dyX i

λλγ

λ λγ

λ γλγ

∞ ∞−− −

−

= =−∫ ∫ (6.25)

Using the complex error function, we can express the solution as

( ) ( )( )22 22 , 1 erf 1 2iG e iλγπλ γ λ γ

λ−= − − (6.26)


6-11

When we combine the two parts of the integral and insert the constant, we have

( ) ( ) ( )

( ) ( )( )2

1 2

2 2

20

12

ˆ ˆ ˆ, , ,

1 erf 1 erf 1 22 1 2

i

G G G

i e ik L i

λγ

λ γ λ γ λ γ

π λ λ γπ λ γ

−

≈ +

= + − − −

(6.27)

For comparison, Figure 42 shows the results when the approximate solution is

plotted together with the numerical integration of equation (6.24).

Figure 42 Numeric versus analytic Laplace transform of surface Green function.

6.3 The Laplace Transform for the Green Function of the Medium

Having found the Laplace transform for the field due to a point source, we can

directly use those results for expressing the transform of the Green function.

From equations 5.32 and 5.33, we notice the only difference between the point

source field and the Green function is a constant term and variable re-definitions

of X to X X ′− and Z to 0Z . By inspection:


6-12

( ) ( ) ( )

( ) ( )

2

2

2 1

0

1

12 1

2

12

ˆ , , 1 erf2

1 erf

i Zi

i Z

i

i

i ZG Z e ek L

Ze

λγ

λγ

λλγ

λ

γ

γ

πλ γπ λ

± −−

−

± ±

±

= − + −

∓

∓

∓ ∓

(6.28)

6.4 Solution to the Mean Field Problem via Laplace Transforms

6.4.1 The Solution in the Laplace Domain

In the last chapter (equation 5.41), we showed that through the rescaling,

ensemble averaging, and Laplace transforms that the set of the integral equations

for the mean field on the surface and the field in the medium, expressed in the

Laplace domain, are

( ) ( ) ( )0ˆ ˆ, , , ,inc Z F Gλ γ λ γ λ γε = − (6.29)

( ) ( ) ( )ˆ ˆ, , , , ,s

Z F G Zλ γ λ γ λ γε = (6.30)

These can be algebraically combined and solved for the scattered field in the

Laplace domain as

( ) ( ) ( ) ( )0ˆ ˆ, , , , , , / ,ˆ

incs Z Z G Z Gλ γ λ γ λ γ λ γε ε= − ⋅ (6.31)

We have included the subscript “inc” for clarity. This technique eliminates the

need to compute ( )ˆ ,F λ γ , or in the spatial domain, the derivative of the mean

field on the surface. Equation (6.31) differs from equation 5.42 only in that we

have not added the mean incident field.

Figure 43 shows what the scattered field equation (6.31) in the Laplace domain

looks like. First, the number of independent variables requires some explanation.

To express the scattered field, we need to specify a source position 0Z , a surface

roughness γ , and a depth Z where we wish to evaluate the scattered field. The

most natural way to express these is to make the surface roughness and source


6-13

position constant in each figure and plot a set of curves corresponding to a set of

depths. In the Laplace domain, λ is an independent variable. Upon inversion of

the transform, X will be the independent variable. Also, since the Laplace

transform of the scattered field is a complex expression it is necessary to visualize

the real and imaginary parts separately. As the plot is done on a logarithmic

scale, when scatε takes on a negative value it does not appear on the plot. We

have chosen a depth of 0 1Z = for the source.

Figure 43 Several level curves for mean scattered field in Laplace domain with fixed source

depth, surface roughness, and desired depth.

6.5 The Solution in the Spatial Domain

The solution will require the evaluation of the inverse Laplace transform, which is

expressed as

( ) ( )12

ˆ , 0c i

xi

c i

f x F e d xλπ λ λ+ ∞

− ∞

= >∫ (6.32)

where c is a real constant chosen greater than the real part of the rightmost

singularity of ( )F λ as pictured in Figure 44. This is a line integral in the

complex λ -plane. As long as c is right of all the singularities of ( )F λ , then the

line integral is independent of c .


6-14

c

x

x

c + iR

c - iR

c + iR

c - iR

Figure 44 (left) A line integral in the complex plane with c right of all singularities.

Figure 45 (right) Bromwich contours closed to left and right in complex plane.

The fundamental tool of inverse Laplace transforms is Cauchy’s theorem, which

states that if a function ( )g λ is analytic in the complex plane and specifically,

within a closed contour C (Figure 45), then the closed line integral along C is

zero.

( ) 0C

g dλ λ =∫ (6.33)

The integral is only nonzero due to singularities of ( )g λ . The residue theorem

states that in the presence of singularities in the absence of branch cuts (such as

square roots or logarithmic functions) the closed line integral can be evaluated as

( ) ( )2 res nnC

g d i sλ λ π= ∑∫ (6.34)

where the residue is the coefficient of the 1λ term in the Laurent series expansion

for ( )g λ evaluated at the singularities, ns .

In many situations, it is easy to evaluate the residues. In fact, if ( )g λ can be

written as a quotient of two functions, ( ) ( )/R Qλ λ , and ( )g λ has simple poles at

the simple zeros of ( )Q λ , then the residues at ns are

( ) ( ) ( )res /n n ns R s Q s′= (6.35)


6-15

The inversion integral is really along a straight line from [ ],c i c i− ∞ + ∞ and we

utilize a closed contour subject to the following interpretation. We will close this

contour by considering two large semicircles and let the radius of these contours

approach ∞ . The entire closed contour is called a Bromwich contour. For

convergence, we require the line integral along the circular arc to vanish for both

cases. Closing the contour to the right implies that 0λ > and for xeλ to decay

(see (6.32) ) we must have 0x < . Since there are no singularities to the right of

c , the line integral along the right-closing Bromwich contour is 0. And by

construction, Laplace transforms are defined as 0 for 0x < .

For the left-closing contour we have 0λ < and for the xeλ to decay we must

have 0x > . The contribution to the integral will come from any singularities

( ) ( ) ( ) ( )1 1ˆ ˆ res2 2

c ix x

nnc i C

f x F e d F e d si i

λ λλ λ λ λπ π

+ ∞

− ∞

= = =∑∫ ∫ (6.36)

If the Laplace transform can be written as a quotient in (6.35) with simple poles

at the simple zeros of ( )Q λ then

( )( )( )

nn s x

n n

R sf x e

Q s=

′∑ (6.37)

6.5.1 Example with Simple Poles

Consider the Laplace transform of ( )cos xω and its inverse Laplace transform.

The Laplace transform is well known and ( )( )2 2cos x λω

λ ω=

+L . Defining the

numerator as R and the denominator as Q in (6.37) we can show that

( ) ( )

( ) ( ) ( )

2

21

12

, 1

cos

n ns xn

n

i x i x

f x e s i

f x e e x

λλ

ω ω

ω

ω

=

−

= = −

= + =

∑ (6.38)


6-16

6.5.2 Example with Branch Cuts

When we must make a slit in the complex plane to define a domain of

analyticity, this is called a branch cut. Unfortunately, the evaluations of inverse

Laplace transforms with branch cuts are significantly more complicated. The

essential step in the method is the contributing parts of the contour integration

along the branch cut do not cancel. This is demonstrated in finding the inverse

Laplace transform when

( )ˆ , 0 1aF aλ λ−= < < (6.39)

We will take our branch cut along the negative real axis and the integration

contour will look as depicted in Figure 46.

c + iR

c - iR

λ = r exp{iπ}

λ = r exp{-iπ}

RaC

RbC

Cε

Figure 46 Integration contour with branch points.

By Cauchy’s theorem (6.33) we have

( )( )ˆ

02i i

Ra Rb

x

C C Cre re

Ff x e d

iπ πε

λ

λ λ

λ λπ−= =

+ + + + + = ∫ ∫ ∫ ∫ ∫ (6.40)

By definition, we recover ( )f x as the portion of the contour integral from

,c iR c iR − + as R → ∞ and must deal with the remainder of the terms. The

integral along the contour C ε (taken around the branch point) tends to zero given

( ) ( )ˆ 0 as 0o Fλ λ λ ε− → → . This proof is shown in many complex variable

texts, e.g. [37].


6-17

The line integral along Ra

C is parameterized by 12 1e ; , iR Rφ πλ φ π= < < → ∞

and the line integral along Rb

C is parameterized by 2

21e ; - , iR Rφ πλ π φ= < <− → ∞ . Both of the integrals along these contours go

to zero. This is intuitive given the exponential term, xeλ , will tend to zero as the

( )Re 0λ < along these quarter-circle arcs. Proofs are offered in [24].

Equation (6.40) then reduces to

( )

( ) ( )

01 1

2 2

0

12

0

0

or

a ia rx a ia rxi i

r r

ia ia a rxi

r

f x r e e dr r e e dr

f x e e r e dr

π ππ π

π ππ

∞− − − − −

=∞ =

∞− − −

=

− − =

= −

∫ ∫

∫

(6.41)

The general solution to this integral requires the gamma function, ( )xΓ , and it

can be shown that

( ) ( )1sin 1aaf x x aππ

− = Γ − (6.42)

The case of relevance here is when 12a = . Then the integral in (6.41) is much

easier to evaluate and can be done using the variable substitution ,r xr′ =

4xrdr dr′ = followed by the double integral trick of evaluating

/22

0 0

rxer

r

drIπ

θ

−∞

= =

= ∫ ∫

to give

( ) ( )12

1ia iaif x e e

xIπ π

π π− −= − = (6.43)

6.5.3 The Scattered Field Inverse Transform

Having shown the solution to the scattered field problem to be (6.31) and

discussed some of the basics of performing inverse Laplace transforms we can now

look at this case more explicitly. In the Laplace transform solution to the

scattered field problem define


6-18

( ) ( ) ( ) ( )

( ) ( ) ( )

21 2

223 4

2

2

1 , 1 ,

1 , 1 2

Z

Za

a Z i Z a Z i

Z i a i

λγ

λγ

γ

γ

λ

γ γ

= − = − +

= − − = − (6.44)

then

( ) ( ) ( )( )( ) ( ) ( )( )( )20 0

0 01 1

0 2 31

2, , 1 erf 1 erfˆ i Z Za aa Z a ZZ e e einc

λγπ

λλ γε − − = − + − (6.45)

( ) ( ) ( )( )( ) ( ) ( )( )( )0

21 1

2 31

2 2, , 1 erf 1 erfˆ Z Zi a ai

ka Z a ZZ e e eG λγπ

π λλ γ −− = − + − (6.46)

and

( )( ) ( ) ( )( )( )22

44

1, erf 1 erfˆ ie aa

G λγπλ γ λ λ γλ γ

− = + −

(6.47)

The inverse Laplace transform will depend upon any singularities of (6.31) and

therefore we must look for zeros of ( )G λ . Figure 47 shows that in the complex

plane of unit radius, there are no places where both the real and imaginary parts

are zero and therefore no zeros. Some simple analysis can show that as λ grows

where ( ) ( )3 54 4 4arg or argπ π πλ λ< < < , ( )G λ tends to a constant and where

( )3 54 4argπ πλ< < , ( )G λ oscillates. In the rest of the complex plane, ( )G λ grows

without bound. Therefore, there are no zeros.

Figure 47 Searching for zeros of ( )ˆ ,G λ γ .


6-19

When we take the equations in (6.45), substitute them into (6.31), and apply

(6.32), we get

( )

( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )( )

( ) ( ) ( )( )( )

0 01 1 1 12 0 02 3 2 3

224

4

1 erf 1 erf 1 erf 1 erf24 1 erf 1 erf

,

1 a Z a Z a Z a Z

i

scat

c iRe a Z e a Z e a Z e a Zi X

e ac iR a

E X Z

e e dλγ

λγ λπ

λ λ γγ

λλ

− −

−

+ − + − − + − − + − −

=

∫ (6.48)

Certainly, this is a complicated expression and finding an analytic inversion

requires a temerity, moxie, and assiduousness seldom found in the natural world.

We will, nevertheless, attempt it. There are several initial observations to make

regarding this inversion. They are:

• Around a suitable closed contour we can apply Cauchy’s theorem.

• The contributions to this integral will come from the branch point

considerations.

• There appear to be no other singularities in the expression.

To perform this line integral we will follow the method of section 6.5.2 almost

exactly and use Figure 46 as our guide. In analogy to (6.40) we have

( ) ( )1, 02

î i

Ra Rb

Xscat

C C Cre re

E X Z e di π π

ς

λ

λ λ

λ λπ

ε−= =

< > + + + + + = ∫ ∫ ∫ ∫ ∫ (6.49)

Along C ς we parameterize and i ie d i e dθ θλ ς λ ς θ= = . The contour integral is

bounded by

( ) ( )1 12 2 1 0 as 0ˆ iX e X

i iscatC

e d i e dθ

ε

πλ ς

π ππ

λ λ ς θ ςε−

≤ → →∫ ∫ …∼ (6.50)

The ellipsis in parentheses represents the other terms, which are all exponentially

small or constants in the numerator and the denominator, ( )G λ (without πλ

which has already been accounted for in ς term), which goes to 1 as 0ς → .


6-20

The integrals along and Ra Rb

C C should also tend to zero as we know the

mathematics represent a physical process and its solution must exist. However,

the proof of this is more difficult. Along Ra

C we parameterize iRe θλ = and

id iRe dθλ θ= with 2π θ π< < ; along

RbC we parameterize iRe θλ = and

id iRe dθλ θ= with 2ππ θ− < <− . One heuristic way to demonstrate they tend

to zero is to consider the asymptotic forms for large λ as represented in (6.18).

We know that for large λ

( ) ( )( )2 20

22211 1

2 2 2/2

0 as RˆZ Z i

i

Rb

iX Re X ii iscat Re

C

e d e e iRe dθ

γθ

πλ θ

π π γπ

λ λ θε− +

+≤ → → ∞∫ ∫ (6.51)

Notice that in (6.51) we have included the asymptotic forms for ( )0, ,ˆ Zinc λ γε

( )ând , ,G Zλ γ in our integral estimate but have set ( )ˆ , 1G λ γ ≡ . This was done

without loss of generality as it represents a worst case as far as convergence is

concerned and the integral does tend to zero even without a more rigorous

estimate for ( )ˆ , 1G λ γ ≡ , which would include different representations for

( )ˆ , 1G λ γ ≡ depending on the location of λ in the complex plane due to Stokes

phenomenon.

For the branch cuts we will parameterize λ . Along the top of the branch (+) we

let [ ]e , , 0iR Rπλ = ∈ ∞ and must append our terms as follows:

( ) ( ) ( ) ( )

( ) ( ) ( )

21 2

223 4

2

2

1 , 1 ,

1 , 1 2

R

R

Z

Za

a Z i RZ a Z i

Z i a i

γ

γ

γ

γγ γ

+ +

+ +

= + = + +

= + − = − (6.52)

Along the bottom of the branch (-) [ ]e , 0,iR Rπλ −= ∈ ∞ and

( ) ( ) ( ) ( )

( ) ( ) ( )

21 2

223 4

2

2

1 , 1 ,

1 , 1 2

R

R

Z

Za

a Z i RZ a Z i

Z i a i

γ

γ

γ

γγ γ

− −

− −

= − + = − + +

= − + − = − (6.53)


6-21

Comparing these to their original definitions we see that terms originally

involving λ now show a 2π± rotation depending on which side of the branch cut

we are taking into consideration. Note that is real and non-negativeR . Lastly,

let us express the proper form for ( )ˆ ,G λ γ on each side of the branch cut.

( )( ) ( ) ( )( )( )

( )( ) ( ) ( )( )( )

2

2

24

4

24

4

1, erf 1 erf

1, erf 1 erf

ˆ

ˆ

iR

iR

i i R e i R aR a

i i R e i R aR a

G

G

γ

γ

πλ γ γγ

πλ γ γγ

+

−

= − + −

= − + − −

(6.54)

We have now considerably simplified the inversion integral that is now expressed

in (6.55)

( )( )

, 02

î i

Xscatscat

Re Re

E X Z e diπ π

λ

λ λ

λλ

πε

−= =

< > + + = ∫ ∫ (6.55)

Using (6.52), (6.53), (6.54), and (6.55) we get:

( ) ( ) ( )( )( ) ( ) ( )( )( )1 12 31 1a aM e erf a e erf aϑ ϑϑ ϑ ϑ± ±−

± ± ±= − + − (6.56)

and

( )( ) ( ) ( )( )( )22

44

1 erf 1 erfiN i e i aa

ϑγϑ ϑ ϑ γγ± = ± + − ± (6.57)

then

( ) ( ) ( ) ( )

( )( )( )

( ) ( ) ( )

( )( )( )

20

20

02 2

2

2 22

0

1,2

1 02

M Z M ZiR RX iiRs i N R

RR

M Z M ZiR RX iiR i N R

RR

E X Z e e e dRi

e e e dRi

γ πππ

γ πππ

π

π

+ +

+

− −

−

−

−=∞

∞− −

=

< > + −

+ + =

∫

∫ (6.58)

Simplifying as much as possible gives:


6-22

( ) ( ) ( ) ( )

( )( )

( ) ( ) ( )( )( )

20

20

2

0

2

0

1,8

18

M Z M ZiR RXRs N R

R

M Z M ZiR RXR N R

R

E X Z e e dR

e e dR

γπ

γπ

π

π

+ +

+

− −

−

∞−

=

∞−

=

< > =

+

∫

∫ (6.59)

It is the expression in (6.59) that we will use when performing our numerical

quadrature for the calculation of the mean scattered field. This is further

discussed, with results, in 6.5.3.2

6.5.3.1 Inversion for 0γ =

We will try an analytic inversion for 0γ = as a case study of the method and

also as a check on the results until this point. There will be considerable

simplification of the expression in (6.59) and we should be able to perform the

whole inversion analytically.

First, we will directly simplify the expressions for ( ) ( ), , , , ,ˆ ˆZ ZG Gλ γ λ γ+ − ,

( ) ( ), , ,ˆ ˆG Gλ γ λ γ+ − , ( )0, ,inc Zλ γε+ and ( )0, ,inc Zλ γε− . The ± denotes which side

of the branch these functions are referenced. As 0γ → , the error function’s

behavior is governed by ( ) ( ) ( )2 2 3, , a Z a Z a Z+ − +→ ∞ → ∞ → −∞ , and

( )3a Z− → −∞ . In these cases ( )( )2erf 1, 0 a Z γ± → → and

( )( )3erf 1, 0a Z γ± → − → . When we substitute these simplifications using (6.52),

(6.53), (6.54), we have

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

0

0

01

1

10

1

, ,

, , and ,

, ,

, , and ,

ˆˆ

ˆˆ

ˆ

ˆ

i RZ

i RZ

i RZ

i RZ

inc

inc

inc

inc

R

R

R

R

Z

Z

Z

i e

i e iR

Z i e

i e iR

G G

G G

π

π

π

π

λ γ

πλ γ λ γ

λ γ

πλ γ λ γ

ε

ε

− +

− +

+

+

+

+

−

−

+

−

= −

= − = −

=

= =

,

,

(6.60)

as 0γ → .


6-23

When substituting these into (6.55), we get

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

0

0

01

1

0

1,2

1 02

i R Z Z RX iRs

R

i R Z Z RX iR

R

E X Z i e e dRi

i e e dRi

ππ

ππ

π

π

− + + +

=∞

∞+ + − −

=

< > + −

+ =

∫

∫ (6.61)

Completing these integrals analytically will require completing the square in the

exponent and a few variable transformations. First we rewrite the exponents as

( )( )( )

( )( ) ( )( )

( )( )( )( )( ) ( )( )

0 0

0

0 0

0

2 21 1

2 2

2 21 1

2 2

1

1

i Z Z i Z ZX X

i Z Z i Z ZX X

RXi Z Z R RX

RXi Z Z R RX

e e

e e

+ + + +

+ + + +

− + −− + + +

− − −−− + + +

=

=

(6.62)

and rewrite (6.61) as

( ) ( )( )( )( )

( )( )( )( )

0

0

0

0

21

2

21

2

021

2

21

20

1, exp2

1 exp 02

i Z ZX

i Z ZX

i Z ZRs X

R

i Z ZRX

R

RX

RX

E X Z i dRi

i dRi

e

e

π

π

π

π

+ +

+ +

+ +

=∞

∞ + +

=

− +

− −

< > +

+ − =

∫

∫

(6.63)

Next we use the substitution of 12 and Rp R dp dR= = so that

( ) ( )( )( )( )

( )( )( )( )

0

0

0

0

21

2

21

2

021 2

2

21 2

20

1, exp2

1 exp 02

i Z ZX

i Z ZX

p Xi Z Z

s Xp

p Xi Z Z

Xp

E X Z i dpi

i dpi

e

e

π

π

+ +

+ +

+ +

=∞

∞ + +

=

− +

− −

< > +

+ − =

∫

∫

(6.64)

With one more substitution, let ( )( )012 22 , i Z Zp X X

Xs ds dp+ += ± = depending on

which side of the branch cut we are on ( )± and (6.64) becomes


6-24

( ) ( )( ) ( )

( )( )

( )( ) ( )

( )( )

0

1 02

0

1 02

2

2

21

2

21

2

1, exp2

1 exp2

i Z Z

X

i Z Z

X

i Z Zs X

s

i Z Z

X

s

s

s

E X Z dsX

dsX

e

e

π

π

+ +

+ +

∞+ +

=

∞+ +

=−

−

−

< > =

+

∫

∫

(6.65)

Finally, we see that the integrals left are in the form of complementary error

functions, which cancel as shown here:

( ) ( )( ) ( )( )( ) ( )( )( )

( ) ( )( ) ( )( )( ) ( )( )( )

0 0 0

0 0 0

21 1 1

2 2 2

21 1 1

2 2 2

1, exp erfc erfc2 2

or

1, exp 1 erf 1 erf2

i Z Z i Z Z i Z Zs X X X

i Z Z i Z Z i Z Zs X X X

E X ZX

E X ZX

ππ

+ + + + + +

+ + + + + +

< > = + −

< > = − + +

(6.66)

And the final solution for the scattered field in the flat surface case is

( )( )201< , exp , 0

2s

i Z ZE X Z

X Xγ

+ > = = (6.67)

This is the field that would be produced by an image source at 0Z Z= − [38] and

demonstrated in Figure 48 and Figure 49.

x

z

0z z+real source

image source

0z

0z−

Figure 48 Interpretation of the scattered field from a flat surface.


6-25

Figure 49 Total field as scattered off a flat surface for a source at 0 20Z = (left) and 0 2Z =

(right).

6.5.3.2 Inversion for 0γ ≠

To perform the inversion of most interest to us, we will have to perform

numerical quadrature of (6.59). First we will use the same type of variable

transformation as that of (6.64); namely 12and Rp R dp dR= = . This will

remove the singularity at 0 for our quadrature scheme. Applying this

transformation to (6.59) gives

( ) ( ) ( ) ( )

( )( ) ( )

( )2 2

0 02 2

2

0

,1

4M Z M Z M Z M Zi p p X

N ip N ipp

s X Z e e dpE γ

π− −+ +

−+

∞−

=

> = + < ∫ (6.68)

Note that equation (6.44) becomes

( ) ( )

( ) ( )

( ) ( )

1

2

3

2

2

1 ,

1 , 2

12

Z

Za

a Z i pZ

pa Z i

pZ i

γ

γ

γ

γ

±

±

±

= ± +

= ± + +

= ± + −

(6.69)

under the transformation.


6-26

In Figure 50 we see the real and imaginary parts for the integrand of the first

term on the right-hand side of (6.68) as well as the real and imaginary parts for

the integrand on the second term on the right-hand side. These correspond to

above and below the branch cut, respectively.

Figure 50 Laplace inversion integrand real and imaginary parts above and below the branch cut

(corresponding to (6.68)).

Finally, some results of evaluating (6.68) for the mean scattered field are shown

in Figure 51 for a rough surface with the exponential-like autocorrelation function

in (6.27). In order to illustrate the main effects of the surface roughness we show

( ),sE X Z< > as a function of range X at different distances from the surface for

several values of γ . The region excluded from the curves near 0X =

corresponds to the field propagating at angles greater than /4π relative to the

X direction, where the parabolic approximation becomes very inaccurate.


6-27

5 10 15 20 25 30

100z

= 2

0

5 10 15 20 25 30

100

z =

12

5 10 15 20 25 30

100

z =

8

x

Figure 51 Curves of ( ),sE X Z< > , the mean scattered field, as a function of scaled range X

at different scaled distances Z from a rough surface with ACF (6.27) . A point

source is situated at 0 2Z = . The effect of increasing surface roughness is evident

from the following different values of surface roughness, 2γ : 2 0.0,γ = [ ];

[ ]2 1.0,γ = −−−− ; [ ]2 5.0,γ = −⋅−⋅−⋅−⋅ .

The effect of surface roughness is clearly visible in the total field, i.e., the sum of

scattered and unscattered fields, as in Figure 52.


6-28

Figure 52 Effect of surface roughness on the fringe pattern of the mean total field shown as

( ) ( )( )log , ,sincE X Z E X Z< > + < > . In all cases, the source is at 0 2Z = .

Red corresponds to higher field strengths.

Increasing surface roughness leads to the disappearance of the interference fringes

characteristic of reflections from a flat surface. In Figure 52, the top left-hand

plot is the same as the analytical solution for a flat surface. The remaining plots

present our integral solution evaluated numerically. The black region corresponds

to angles greater than /4π relative to the forward X direction where the

method becomes inaccurate. Finally, since the source was close to the surface

( )0 2Z = , there is only one fringe; the minimum of which lies close to the /4π

limit. We see that this minimum is progressively eroded as the surface becomes

rougher.


7-1

C h a p t e r 7

7 Acoustic Scattering from a Rough Sea and Bottom

Surface. The Mean Field by the Integral Equation

Method for Shallow Water

This problem has been approached with acoustic applications for a number of

years. From the results of Chapter 6 in dealing with the one-surface problem and

extending the equations from Chapter 5, we can formulate the two-surface

problem.

Once again, the beauty in this approach is its ability to analyze the mean field

directly. While we speak of this problem with regards to shallow water, in

general we are discussing a rough waveguide that has applications well beyond

acoustics.

In this chapter, we use familiar notation when possible and hope to extend

notation in a natural way.

7.1 Wave Propagation in Shallow Water: the Mean Field Between

Two Surfaces

The essence of the problem is shown in Figure 53. We will again use the integral

equation approach with the field at any point given as (7.1).

( ) ( ) ( ) ( ), , ,E GE x z G E dsz z

∂ ∂′ ′ ′= − ′ ′ ∂ ∂ ∫ r r r r r (7.1)

( ) ( ), and , ; ,E x z G x z x z′ ′ are still the parabolic variables; the rapidly varying

phase term of ikxe is absorbed from the full expressions. For the pressure, this

leaves the slowly varying envelope, ( ),E x z , and the Green function, ( ), ; ,G x z x z′ ′ ,

left in its parabolic form (see Chapters 5 & 6). Still implicit in this

approximation is that the angle of incidence between the incident field and the


7-2

surfaces is small (say 15 20− ) and the scattering is in the forward direction.

With respect to Figure 53, the Helmholtz integral equation is still

( ) [ ] [ ] [ ] [ ]

1 4 2 3

,E x z ds ds ds dsΓ Γ Γ Γ

= + + +∫ ∫ ∫ ∫ (7.2)

where ds is an element along the surface.

(0,z0) -source position

Γ3

Γ1 S1(x)

Γ2

S2(x)

Γ4

ηj

z

x

Figure 53 The sea surface and bottom are represented along with an acoustic source to the left.

It has been previously shown that the integral along 4Γ is zero as there is no

backscatter [39]. Also, by moving 1 2and Γ Γ to ±∞ , the integral along 3Γ is

equal to the incident field ( ),incE x z . Thus we can express (7.2) as

( ) ( )

( )( ) ( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( )

2 2 2 2

1 1 1 1

0

0

; , ; ,

; , ; ,

, ,inc

x

x

E GG x x z S x x S x x x z S x E x S x

z z

E GG x x z S x x S x x x z S x E x S x

z z

E x z E x z

dx

dx

∂ ∂′ ′ ′ ′ ′ ′ ′ ′− − − − −′ ′∂ ∂

∂ ∂′ ′ ′ ′ ′ ′ ′ ′− − − − −′ ′∂ ∂

− =

′

′+

∫

∫

(7.3)


7-3

where no boundary conditions have been yet imposed. This is the general

integral equation with the paraxial approximation implicit in the notation. For a

two-surface problem, we should be clear on the unknown terms of equation (7.3)

and list them in Table 2 with brief descriptions.

Table 2 Unknown quantities in the full integral equation.

Unknown Term(s) Explanation

( )( )1;G x x z S x′ ′− − ,

( )( )2;G x x z S x′ ′− −

The Green function, which expresses the influence of a

point source on the surface, ( )( )1,x S x′ ′ or ( )( )2,x S x′ ′ , to

the field at ( ),x z

( )( )1;G x x z S xz

∂ ′ ′− −′∂

,

( )( )2;G x x z S xz

∂ ′ ′− −′∂

The derivative of the Green function with respect to z ′ evaluated at 1 2and S S

( )( )1,E x S x′ ′ ,

( )( )2,E x S x′ ′

The value of the field on 1 2and S S

( )( )1,E x S xz

∂ ′ ′′∂

,

( )( )2,E x S xz

∂ ′ ′′∂

The derivative of the field with respect to the surface

normal evaluated on 1 2and S S

Essentially everything is unknown at this point. However, we will evaluate the

field explicitly on the surface and impose some boundary conditions that will

provide two additional equations and also allow us to eliminate some of these

variables.

7.2 Both Surfaces are Pressure-Release Surfaces

We take the pressure-release definition of Chapter 5 and impose it on both

surfaces. In this case, equation (7.3) becomes

( ) ( ) ( )( ) ( )( )

( )( ) ( )( )

2 20

1 10

, , ; ,

; ,

x

inc

x

EE x z E x z G x x z S x x S x dxz

EG x x z S x x S x dxz

∂′ ′ ′ ′ ′− = − − ′ ∂ ∂′ ′ ′ ′ ′− − − ′ ∂

∫

∫ (7.4)


7-4

and we are left with two equations for the surfaces. Since the total field is zero at

both surfaces under these conditions, we have on 1S :

( )( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

1 1 2 20

1 1 10

, ; ,

; ,

x

inc

x

EE x S x G x x S x S x x S x dxz

EG x x S x S x x S x dxz

∂′ ′ ′ ′ ′= − − − ′ ∂ ∂′ ′ ′ ′ ′+ − − ′ ∂

∫

∫ (7.5)

And on 2S :

( )( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

2 2 2 20

2 1 10

, ; ,

; ,

x

inc

x



∂′ ′ ′ ′ ′= − − − ′ ∂ ∂′ ′ ′ ′ ′+ − − ′ ∂

∫

∫ (7.6)

The unknowns are ( )( )1,E x S xz

∂ ′ ′′∂

and ( )( )2,E x S xz

∂ ′ ′′∂

. Once the surface

equations are solved, these terms can be substituted into (7.4) to find ( ),E x z .

7.3 One Pressure-Release Surface and One Hard Surface

The obvious choice is to keep 1S the pressure-release surface and make 2S the

hard (non-conducting) surface. The main equation (as compared to (7.4)) will

change as a result to

( ) ( ) ( )( ) ( )( )

( )( ) ( )( )

2 20

1 10

, , ; ,

; ,

x

inc

x

GE x z E x z x x z S x E x S x dxz


∂ ′ ′ ′ ′ ′− = − − ′ ∂ ∂′ ′ ′ ′ ′− − − ′ ∂

∫

∫ (7.7)

The equations on the two surfaces become, on 1S

( )( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

1 1 2 20

1 1 10

, ; ,

; ,

x

inc

x

GE x S x x x S x S x E x S x dxz


∂ ′ ′ ′ ′ ′= − − − ′ ∂ ∂′ ′ ′ ′ ′+ − − ′ ∂

∫

∫ (7.8)


7-5

and on 2S :

( )( ) ( )( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

2 2 2 2 20

2 1 10

, , ; ,

; ,

x

inc

x

GE x S x E x S x x x S x S x E x S x dxz


∂ ′ ′ ′ ′ ′− = − − ′ ∂ ∂′ ′ ′ ′ ′− − − ′ ∂

∫

∫ (7.9)

The unknowns are ( )( )2,E x S x and ( )( )1,E x S xz

∂′∂

. With two equations and two

unknowns in (7.8) and (7.9), these coupled equations can be solved and then used

to find ( ),E x z in equation (7.7).

7.4 Ensemble Averaging of Integral Equations

Because the situation of a pressure-release surface and hard surface on the sea

bottom more closely resembles the actual condition in the sea, we will primarily

deal with the case of section 0. The claims concerning the evolution of the mean

field and its statistical independence from the surface were presented in the one-

surface case of Chapter 5 and are rigorously discussed in Chapter 8. For now, we

naturally extend the one-surface case to include conditions on both surfaces with

the following statements:

• ( ),E x z , ( )( )1,E x S xz

∂′∂

, and ( )( )2,E x S x are statistically independent of 1S

and 2S locally and are therefore statistically independent of

( )( )2;G x x z S xz

∂ ′ ′− −′∂

, ( ) ( )( )2 2;G x x S x S xz

∂ ′ ′− −′∂

,

( ) ( )( )1 2;G x x S x S xz

∂ ′ ′− −′∂

, ( )( )1;G x x z S x′ ′− − , ( ) ( )( )1 1;G x x S x S x′ ′− − ,

and ( ) ( )( )2 1;G x x S x S x′ ′− − .

• ( ) ( )( )2 2;G x x S x S xz

∂ ′ ′− −′∂

and ( ) ( )( )1 1;G x x S x S x′ ′− − are functions of S

and S ′ , two quantities that are correlated. One case we have dealt with

explicitly; the other (involving the derivative) is a slight twist.

• ( ) ( )( )2 1;G x x S x S xz

∂ ′ ′− −′∂

and ( ) ( )( )2 1;G x x S x S x′ ′− − are the functions

of two Gaussian random variables that are statistically independent. We

require them to have zero mean with possibly different variances. These

two terms completely characterize the probability distribution.


7-6

7.4.1 Previous Ensemble Averages Performed

In order to ensemble average three of the relevant terms in (7.7), (7.8), and (7.9),

we will use the results of Chapter 5. First, recall the statistical averaging of a

function ( )( ),f x S x , where ( )S x is a random variable, is:

( )( ) ( )( ) ( )1, ,f x S x f x S x P S dS∞

−∞

< > = ∫ (7.10)

We will need expressions for ( )( );G x x z S x′ ′< − − > and ( )( );incE x S x< > at

both surfaces. Previous results will be readily available to us for those cases.

Furthermore, for the surface Green function ( ) ( )( );G x x S x S x′ ′< − − > , which is

a function of multiple, possibly correlated random variables, we have

( ) ( )( ) ( ) ( )( ) ( )2; ; ,G x x S x S x G x x S x S x P S S dSdS∞ ∞

−∞ −∞

′ ′ ′ ′ ′ ′< − − > = − −∫ ∫ (7.11)

As before, S is a random variable with variance 2 2 2S S σ< >−< > = and mean

µ obeying the normalized Gaussian probability distribution

( )( )2

221 2

12

SP S e

µσ

πσ

− −= (7.12)

and the two-point Gaussian distribution (bivariate) of

( ) ( )( ) ( )( ) ( )

( ) ( ) ( )

22

2 22

212 1

2 2

1,2 1

, ,

S S SS

P S S e

S S x S S x x x

ρ µ µ µµσσσ σρ

πσσ ρ

ρ ρ

′ ′− − − ′ ′−− − + ′ ′ − ′ =′ −

′ ′ ′= = = −

(7.13)

with a normalized spatial autocorrelation function of

( )( ) ( )

2

S x S xx xρ

σ′< >

′− = (7.14)

and mean and µ µ′ and variance 2 2and σ σ ′ .


7-7

For a surface at depth H we have an equivalent relation to (7.12):

( )( )2

221 2

12

S HP S e σ

πσ

− −= (7.15)

where H is the mean. For two-point Gaussian distribution when and S S ′ have

zero mean and the same variance, we have

( ) ( )

( ) ( ) ( )

2 2

2 2

2 2

2

2 112 2 1

,

, ,

S SS S

P S S e

S S x S S x x x

ρ

σ ρπσ ρ

ρ ρ

′ ′′− +−

−−

′ =

′ ′ ′= = = −

(7.16)

We will use the extended form in (7.13) when we consider the ensemble average

of the Green function between two surfaces with possibly different variances, one

with mean of zero and one at a depth, H , and (7.16) for the ensemble average of

the Green function for a single surface.

Our previous work with these quantities has shown that for either surface, ( )jS x ,

when these integrals are carried out we get

( )( ) ( ) ( )( )

20 0

20 0

20 0

20 0, , , , j

j

ik z

E x ikinc j inc j k x ik

E x z S x E x z e σ

σγ −

−< > = = (7.17)

for the ensemble average of the incident field,

( )( ) ( )

( )( )( )

20

20

20 0

21 12 2

; ; ,

j

j

j j

ik z

x x ikik x x ik

G x x z S x G x x z

e σπ σ

γ

′− −′− −

′ ′ ′< − − > = < − >

=

(7.18)

for the ensemble average of the Green function for the medium, and

( ) ( )( ) ( )

( ) ( )( )20 0

,

1 12 2 2 1

; ;

j j

jk j k

ik x x ik x x

G x x S x S x G x x

π σ ρ

γ

′ ′− − − −

′ ′ ′< − − > = < − >

= (7.19)


7-8

when k j= for the ensemble average of the Green function for the surfaces.

They are merely restated here for convenience. k j= is along the same surface,

while k j≠ is between the two surfaces. Although we have not scaled these

relations yet, keep in mind that ( )γ γ σ= .

7.4.2 New Ensemble Averages to Perform

From our three governing integral equations and ensemble averages presented

until this point, we see there are still some quantities that have not been

statistically averaged; namely ( )( );G x x z S xz

∂ ′ ′< − − >′∂

,

( ) ( )( ); jkG x x S x S x′ ′< − − > , when k j≠ , and ( ) ( )( ); jk

G x x S x S xz

∂ ′ ′< − − >′∂

for any and k j . We will now discuss how the ensemble average of these new

quantities is performed.

7.4.3 Ensemble Averaging ( )( );G x x z S xz

∂ ′ ′< − − >′∂

We use the primary definition for an ensemble average

( )( ) ( )( ) ( )1; ;G Gx x z S x x x z S x P S dSz z

∞

−∞

∂ ∂′ ′ ′ ′< − − > = − −′ ′∂ ∂∫ (7.20)

and from the actual Green function, we can evaluate the partial derivative as

( ) ( )( )

( )( )

2

0 03; exp

28

z zik ikG x x z z i z zz x xx xπ

′− ∂ ′ ′ ′− − = − − ′ ′ ∂ −′ − (7.21)

by simple differentiation. From the definition of the ensemble average, we can

express it as an improper integral of the product of (7.21) and the one-point

probability distribution of the surface.

( )( ) ( ) ( )

( )( ) ( )( )( )

( )2 2

0

2

02

0

2 2

1 1 1;2 2 2

exp ik z S x S x

x x

ikG ix x z S xz k x x x x

z S x dSσ

π πσ

∞

−∞

′ ′−−

′−

∂ ′ ′< − − > = − ×′ ′ ′∂ − −

′− ∫

(7.22)


7-9

This integral is carried out in Appendix F and has the value of

( )

( ) ( )( ) ( )( )2

0 02 2 2

0 0 0 0

; ;

1 1 exp2 2 2

G x x zz

ik z ik zik x x ik x x ik x x ik

γ

π σ σ σ

′∂ −=

′∂ − ′ ′ ′− − − − − −

(7.23)

Note that the averaging makes the ensemble average a function of ( ); ,x x z γ′−

only. Also, it is shown that ( ) ( )( ); , ; ,G x x z G x x zz z

γ γ∂ ∂′ ′< − > = − < − >′∂ ∂

.

7.4.4 Ensemble Averaging ( ) ( )( ); jk

G x x S x S xz

∂ ′ ′< − − >′∂

when k j=

Once again, we refer to the definition of the ensemble average and note that we

are dealing with a function that depends on correlated terms. We must explicitly

use the two-point distribution to do the averaging because when k j= we are

talking about the same random process -- the same surface, which has a given

correlation length. That is, we are either interested in the two-point distribution

on 1 2 or Γ Γ .

( ) ( )( )

( ) ( )( ) ( )2

;

; ,

G x x S x S xz

G x x S x S x P S S dS dSz

∞ ∞

−∞ −∞

∂ ′ ′< − − > =′∂

∂ ′ ′ ′ ′− −′∂∫ ∫

(7.24)

Substituting into (7.24) with (7.21), and (7.13) we get

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )( )( ) ( )

2 22

0

2 2

02 2

0

2

2 2 1

1 1 1;2 2 2 1

expS SS Sik S x S x

x x

ikG ix x S x S xz k x x x x

S x S x dSdSρ

σ ρ

π πσ ρ

∞ ∞

−∞ −∞

′ ′′− +′−−

′− −

∂ ′ ′< − − > = − ×′ ′ ′∂ − − −

′ ′− ∫ ∫ (7.25)


7-10

Note that the averaging makes the ensemble average a function of ( );x x γ′−

only. However, this integral is performed in Appendix G and shown to be zero.

The mathematical argument is as follows: the exponent under the double integral

is completely symmetric with respect to and S S ′ . The term out front gives us

the difference of two integrals that must have the same value and therefore the

answer is zero.

7.4.5 Ensemble Averaging ( ) ( )( ); jkG x x S x S x′ ′< − − > and

( ) ( )( ); jk

G x x S x S xz

∂ ′ ′< − − >′∂

when k j≠

In Chapter 5, it was discussed how ( ) ( )( ); jkG x x S x S x′ ′< − − > is calculated

when and j kS S are the same surface, i.e. when k j= . This led to needing the

two-point probability distribution when computing the ensemble average. Here

and j kS S are rough surfaces that are independent, identically distributed

Gaussian processes, which we will assume to be uncorrelated. So the

computation of the ensemble average will involve a double integral, but the joint

density function can be expressed as the product of one-point density functions.

We have for both ensemble average quantities:

( ) ( )( )

( ) ( )( ) ( ) ( )

( ) ( )( )

( ) ( )( ) ( ) ( )

1 2

1 2 1 1 1 2 1 2

1 2

1 2 1 1 1 2 1 2

;

;

;

;

G x x S x S x

G x x S x S x P S P S dS dS

G x x S x S xz

G x x S x S x P S P S dS dSz

∞ ∞

−∞ −∞

∞ ∞

−∞ −∞

′ ′< − − > =

′ ′− −

∂ ′ ′< − − > =′∂

∂ ′ ′− −′∂

∫ ∫

∫ ∫

(7.26)

where the one-point probability distributions are used. Each surface 1 2 and S S is

a stationary Gaussian random process that can have different variance 2 2 and j kσ σ , but is assumed to be uncorrelated from the other. The assumption of

different variances is necessary considering one surface is a sea bottom and the

other is the sea surface.


7-11

Using the defining relation in (7.26)a and collecting terms, we have an explicit

integral relationship to evaluate:

( ) ( )( ) ( ) ( )( )( )

( )2 22 2 2

0

0

2 22 2

1 1;2 2 2

exp k j k j j k

j kj k

j kj k

ik S S S S H

x x

iG x x S x S xk x x

dS dSσ σ

σ σ

π π σ σ

∞ ∞

−∞ −∞

− + −−

′−

′ ′< − − > = ×′−

∫ ∫

(7.27)

This quantity represents the mean field on the surface k

S due to a point source

at x ′ on the surface jS . The general notation of and k j has been used to

represent surfaces 1 2 and Γ Γ in Figure 53. The evaluation of this integral is

discussed in Appendix H:

( )

( ) ( )( ) ( ) ( )2

02 22 2

0 00

;

1 exp2 2

j k

j kj k

G x x S S

iki Hk x x ikx x ikπ σ σσ σ

′ ′< − − > =

′′ − − +− − +

(7.28)

The result of this integral is a function only of ( ); ,j kx x γ γ′− .

For the condition that arises on the hard (non-conducting) surface, we have a

similar type of integral to perform as that in (7.27). The only difference will be

an additional term due to differentiating the exponent in the Green function.

( ) ( )( ) ( ) ( )

( ) ( )( ) ( )( )

( )2 22 2 2

0

2 2

0

0

2 2

1 1 1;2 2 2

' exp k j k j j k

j k

j kj k

j jk k

ik S S S S H

x x


S x S x dS dSσ σ

σ σ

π πσ σ

∞ ∞

−∞ −∞

− + −−

′−

∂ ′ ′< − − > = − ×′ ′ ′∂ − −

− ∫ ∫

(7.29)

Note that ( ) ( )( ); j k

G x x S x S xz

∂ ′ ′< − − >′∂

now becomes a function of

( ); ,j kx x γ γ′− only. The evaluation of this integral is discussed in Appendix I

and shown to be


7-12

( )

( ) ( )( ) ( ) ( )( )

( ) ( )

0

2 2 2 20 0 0

20

2 20

;

12

exp2

j k

j jk k

j k

G x x S Sz

ik Hik x x ik x x ik

ikHx x ik

π σ σ σ σ

σ σ

∂ ′< − − > =′′∂

′ ′− − + − − +

′− − +

(7.30)

7.5 Nomenclature

For notational convenience in the following work, we will define

( ) ( )( ) ( ); ; ,j jk jk k jk

G x x S x S x x xz

γ γ∂ ′ ′ ′< − − > = − ≡′∂

H H (7.31)

for terms involving the various ensemble averages of the Green function

derivatives. When the ensemble average of the Green function is only a function

of one surface, the following notation will be used:

( )( ) ( ); ; ,j j j j

G x x z S x x x zz

γ∂ ′ ′ ′< − − > = − ≡′∂ H H (7.32)

Similarly, ( ) ( )( ); jkG x x S x S x′ ′< − − > is defined as

( ) ( )( ) ( ); ; ,j jk jk k jkG x x S x S x x x γ γ′ ′ ′< − − > = − ≡G G (7.33)

and

( )( ) ( ); ; ,j j j jG x x z S x x x z γ′ ′ ′< − − > = − ≡G G (7.34)

We will use a slightly different notation when dealing with the fields

( )( ) ( ), ,j jE x S x E x γ< > = (7.35)

so we can account for which surface the averaging was done with respect to. This

affects our evaluation of terms like equation (7.17) where the variances on each


7-13

surface are allowed to be different. If the variances were the same, there would

be no need for the additional notation in equation (7.35). Notation parallel to

that of (7.32) would be adequate. However, this helps to differentiate them as

unknowns in the problem.

7.6 The Scaled Integral Equations for the Mean Field

In our previous results of Chapter 5, we used a scaling based on the correlation

length, L , in the horizontal direction and the Fresnel radius in the vertical, L

kf = . The scaled variables became

/

/

X x L

Z z f

=

= (7.36)

To help express the equations in a more compact form, we will introduce the

scaling as done previously.

We have shown in the Chapter 5 that the average for the point source and Green

function in scaled form are

( ) ( ) ( )( )

20

20

2 0

2 10 0 0, , , ,

iZE X i

k Linc j X iE X Z E X Z e Eγ

γγ −

−< > = = = (7.37)

and

( )( )( )

2

2

20

21 1 12 2

j

j

iZ

X X iik Lj X X i

e γπ γ

′− −′− −

=G (7.38)

where

2

2 0kL

σγ = (7.39)

These are the scaled forms of equations (7.17) and (7.18). Now we must scale

some additional relations for the two-surface problem with hard and non-hard

boundary conditions.


7-14

Applying the above scaling to (7.28), the two-surface Green function, gives

( ) ( )( )

( ) ( )( )

2 20

2

2 2

1 1 12 2

exp2

jk

j k

j k

ik L X X i

H i

X X i

π γ γ

γ γ

=′− − +

′− − +

G

(7.40)

where

2

02 , 1,2jj

kj

L

σγ = ∈ (7.41)

2

2 0 , 1,2kk

kk

Lσ

γ = ∈ (7.42)

and

/H H f= (7.43)

for the ensemble average of the Green function between two rough surfaces

separated by a depth H , with Gaussian statistics and arbitrary surface roughness

for each surface. In the case that the surfaces approach flatness, so , 0j kγ γ → ,

the interpretation of the Green function reduces to that of the field at location

( ),X H′ induced by a paraxial point source at ( ), 0X . As this Green function

relates the field mutually induced on each surface and this separation is fixed for

all X when the surfaces are flat, (7.40) does not depend on Z . Also, it is a

symmetric function with respect to surface labeling so it matters not which

surfaces we call 1 and 2.

In a similar fashion, we can scale (7.23) and get

( )( ) ( )( ) ( )( )H

2

2 2 20

1 1 1 1 exp2 2 2j

j j j

i iZ iZk L f X X i X X i X X iπ γ γ γ

= ′ ′ ′− − − − − − (7.44)


7-15

which represents the derivative of the Green function with respect to the scaled

z ′ coordinate, Z ′ , for the influence of a point source at ( )( ),X S X′ ′ on anywhere

in the medium, ( ),X Z . It is worth mentioning that ( )Z S X′ ′≡ in this case.

Also, taking a derivative with respect to the scaled vertical coordinate Z ′ will

introduce a factor of /f L k= into the scaling of the equation.

Finally, we must scale the derivative of the Green function between two

independent, identically distributed surfaces with Gaussian statistics. This was

the derivation of Appendix I and given in (7.30). It becomes

( ) ( )( ) ( ) ( )

( ) ( )( )

H2 22 2

0

2

2 2

1 1 1 12 2

exp2

jkj kj k

j k

i iHk L f X X iX X i

iH

X X i

π γ γγ γ

γ γ

=

′ ′ − − +− − + × ′− − +

(7.45)

Note again how differentiation of the Green function leads to an additional

scaling of f through this parameterization, as expected from (7.36).

Also given the conclusions of Chapter 8 regarding statistical independence of the

field locally from the surface, the added statements of section 5.4, and equations

(7.17) through (7.29), we can write the mean integral equations for the two-

surface problem as

( ) ( ) ( ) ( )0 2 2 1 10 0

, , , ,X X

inc

EE X Z E X Z E X dX X dXZ

γ γ∂′ ′ ′ ′< >− = −′∂∫ ∫H G (7.46)

( ) ( ) ( )0 1 12 2 11 10 0

, , , ,X X

inc

EE X Z E X dX X dXZ

γ γ γ∂′ ′ ′ ′= − + <′∂∫ ∫H G (7.47)

( ) ( ) ( ) ( )02 2 22 2 21 10 0

, , , , ,X X

inc

EE X E X Z E X dX X dXZ

γ γ γ γ∂′ ′ ′ ′− = −′∂∫ ∫H G (7.48)


7-16

In this scaled, mean form, the unknown quantities are ( )2,E X γ′ , ( )1,E XZ

γ∂ ′′∂

,

and ( ),E X Z< > . Once again the typical approach would be to solve the two

coupled equations, (7.47)and (7.48), for ( )2,E X γ′ and ( )1,E XZ

γ∂ ′′∂

, then

substitute these into (7.46) to find the full mean field. In the next section, we

outline how this solution can be done using Laplace transforms.

7.7 The Solution to the Two-Surface Mean Field in the

Laplace Domain

We have previously defined the Laplace transform as

( )[ ] ( ) ( )

0

ˆ xf x F f x e dxλλ∞

−= = ∫L (7.49)

and the inverse Laplace transform as

( ) ( ) ( )ˆ ˆc i

x

c i

F f x F e dλλ λ λ+ ∞

− ∞

= = ∫-1L (7.50)

with c a constant that is to the right of the real part of the right-most

singularity of ( )F λ . As long as this condition is met the value of c is

unconstrained. We will also make use of the convolution theorem, which states

( ) ( ) ( ) ( )0

ˆ ˆx

f x g x x dx F Gλ λ ′ ′ ′− = ∫L (7.51)

With these transforms in hand, let us make the following definitions:

( ) ( )

( ) ( ) ( )

H H

G G

22 22 12 120 0

21 21 11 110 0

ˆ ˆ,

ˆ ˆ,

X X

X X

H e dX H e dX

G X e dX G e dX

λ λ

λ λ

λ λ

λ λ

∞ ∞− −

∞ ∞− −

= =

= =

∫ ∫

∫ ∫ (7.52)

for the various Green functions involving the two surfaces,


7-17

( ) ( )G H2 2 1 10 0

ˆ ˆ,X XG e dX H e dXλ λλ λ∞ ∞

− −= =∫ ∫ (7.53)

for the two Green functions describing how the field is affected by a point source

on the surfaces, and

( ) ( )

( ) ( )

1 10

2 20

ˆ , ,

ˆ , ,

X

X

EF X e dXZ

F E X e dX

λ

λ

λ γ γ

λ γ γ

∞−

∞−

∂ ′=′∂

′=

∫

∫ (7.54)

to describe the terms from the two surface equations that are coupled. We hope

to solve for, and eliminate, without directly computing them. Finally, for the

transform of the incident field and the total field define

( ) ( )

( ) ( )

( ) ( )

( ) ( )

0

1 10

2 20

0

, ,

, ,

, ,

, ,

ˆ

ˆ

ˆ

ˆ

Xinc inc

Xinc inc

Xinc inc

X

Z E X Z e dX

E X e dX

E X e dX

Z E X Z e dX

λ

λ

λ

λ

λ

λ γ γ

λ γ γ

λ

ε

ε

ε

ε

∞−

∞−

∞−

∞−

=

=

=

= < >

∫

∫

∫

∫

(7.55)

7.7.1 Laplace Representation for One Hard and One Pressure-Release

Surface

Applying the Laplace transform notation above and using the convolution

theorem we can rewrite equations (7.46), (7.47), and (7.48) in the λ domain as

( ) ( ) ( ) ( ) ( ) ( )0 2 2 1 1ˆ ˆ ˆ ˆ, , , , ,ˆ incZ Z Z H F G Fλ λ λ λ γ λ λ γε ε− = − (7.56)

( ) ( ) ( ) ( ) ( )0 1 11 1 12 2ˆ ˆ ˆ ˆ, , , ,inc Z G F H Fλ γ λ λ γ λ λ γε = − (7.57)


7-18

and

( ) ( ) ( ) ( ) ( ) ( )2 2 22 2 21 1ˆ ˆ ˆ ˆ ˆ, , , ,incF H F G Fλ γ λ γ λ λ γ λ λ γε− = − (7.58)

In the Laplace domain it is not too difficult to get expressions for

( ) ( )1 2ˆ ˆ, and ,F Fλ γ λ γ from the coupled equations, (7.57) and (7.58). The

solution to this set of coupled equations is

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )

( ) ( )( )

( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )

( )( )

11 2 21 12

11 22 12 21

11 2 21 1 1121

11 1111 22 12 21

ˆ ˆ, ,ˆ ,ˆ ˆ ˆ ˆ1 ,

ˆ ˆˆ , , ,ˆ , ˆ ˆˆ ˆ ˆ ˆ1

ˆ ˆ

ˆ ˆ ˆ

inc inc

inc inc inc

G GF

G H H G Z

G GHF

G GG H H G

λ λ γ λ λ γλ γ

λ λ λ λ

λ λ γ λ λ γ λ γλλ γ

λ λλ λ λ λ

ε ε

ε ε ε

−=

− +

− = + − +

(7.59)

To simplify notation, noting that ( )22ˆ 0H λ = , let us define

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

11 2 21 11 2

11 12 21

ˆ ˆ, ,ˆ , , ˆ ˆ ˆˆ înc incG G

KG H G

λ λ γ λ λ γλ γ γ

λ λ λε ε−

=+

(7.60)

Taking ( ) ( ) ( )0, , , ,ˆ ˆ ˆs incZ Z Z Zλ λ λε ε ε= − we can then write the solution to the

mean scattered field in the Laplace domain as

( ) ( ) ( ) ( )( )

( ) ( ) ( )102 1 2 12 1 2 1

11

ˆ ,ˆ ˆ ˆ ˆ, , , , , , , ,ˆˆ ˆs inc

G ZZ H Z K H K Z

Gλ

λ λ λ γ γ λ λ γ γ λ γλ

ε ε = − + (7.61)

using equation (7.56).

The final step remains taking the inverse Laplace transform so that the mean

scattered field can be recovered, as well as the total field:

( ) ( ) ( )0, , , ,ˆ ˆs incE X Z Z Z Zλ λε ε < > = + -1L (7.62)

An analytic solution is unlikely to exist for either case of boundary conditions,

but it can likely be simplified using some complex integration analysis around a


7-19

suitable inversion contour and turning the full solution into an improper integral

along the real axis.

7.7.2 Additional Laplace Transforms to Perform

To formulate the problem in the Laplace domain, we will need the Laplace

transforms of certain quantities discussed above. Luckily, most of the work has

been accomplished in solving the one-surface problem and we will need to only

slightly modify the results for a two-surface application.

7.7.2.1 Two-Surface Green Function, ,jk j k≠G

Laplace Transform for ( ) ( )G0

ˆ Xjk jk

G X e dXλλ∞

−= ∫ has been computed previously

as it is of the same form for the Laplace transform of a paraxial point source and

Green function. Thus, by inspection we have

( ) ( ) ( ) ( )( )

( )

( ) ( )( )

( )

2 2

2 2

2 22 2

2 2

10 2

12

12

12

1ˆ 1 erf2

1 erf

j k

j k

j kj k

j k

i i Hjk

i H

Hi

Hi

G G e e

e

λ γ γλ γ γ λ

λ γ γλ

γ γ

γ γ

πλλ

+− + ± −

+−

± ±+

±+

= − + −

∓

∓

∓ ∓

(7.63)

when the two surfaces are separated by a mean depth, /H H f= and 0G is just a

constant with 0

12 20

ik LG π= .

7.7.2.2 Single-Surface Green Function with Arbitrary Autocorrelation Function

Laplace transform for ( ) G11 110

ˆ XG e dXλλ∞

−= ∫ has been computed for a simple case

of surface autocorrelation function approximating that of exponential in

Chapter 6. For more complicated expressions, the analytic result for the Laplace

transform is difficult to find and is usually approximated by polynomial functions.


7-20

7.7.2.3 Single-Surface Green Function Derivative,Hjj , with Arbitrary Correlation

Function

As shown in Appendix G, the ensemble average of this quantity, Hjj , is zero so

we must have

( ) H0

ˆ 0Xjj jjH e dXλλ

∞−= =∫ (7.64)

in this model using Gaussian probability density functions on the surfaces.

7.7.2.4 Green Function Derivative for the Medium, Hj

The form is very similar to that discussed in the above section 0, but has an

additional term that complicates the transform. The general form of the

transform is

( ) ( ) ( )

3/22

2 20

1ˆ , exp2j

iZH Z X dXX i X i

λ λγ γ

∞ = − − − ∫ (7.65)

and the details are similar to the integral of Chapter 6:

( ) ( ) ( )( )( )

( ) ( )( )( )

2 1 12

1 12

4ˆ , 1 erf 1

1 erf 1

i i Z Zyj

i Z Zy

i

ZLH Z e e i

e i

λγ λ

λ

λ λγ

λγ

− ±

− ±

= − − + ± + − − ±

∓

∓

∓

∓ (7.66)

The functional form is very close to that for the Laplace transform of the Green

function for the medium, which is almost identical to the Green function for the

paraxial point source. The important functional difference is the lack of a leading

order term out front that is proportional to 1λ and we should expect the

asymptotic behavior to go ( )1 1 ,i λλγ

+ → ∞∼ .


7-21

7.7.2.5 Two-Surface Green Function Derivative, Hjk

This one also looks possible to complete analytically, and in fact is analogous in

form to (7.65). Thus, if we find ( )ˆ ,jH Zλ , then we have ( )ˆjkH λ in hand. By

taking equation (7.66) and letting ( )2 2j kγ γ γ γ= + ≡ , Z H= , we can find the

Laplace transform for Hjk immediately:

( ) ( ) ( )( )( )

( ) ( )( )( )

2 1 14 2

1 12

ˆ 1 erf 1

1 erf 1

i i Hi HZLjk

i H H

H e e i

e i

λγ λγ

λγ

λ λγ

λγ

− ±−

− ±

= − + ± +

− − ±

∓

∓

∓

∓ (7.67)

7.8 Looking Ahead

We have now outlined the two-surface problem and explained all the terms,

including their Laplace transforms and developed the solution in the Laplace

domain. In the next chapter, we will look closely at the ensemble averaging of

the integral equations. We present several important ideas from probability

theory and statistics that are necessary to the analysis. We will attempt to

provide some guidance as to whether this independent averaging is

mathematically and/or physically justifiable.

In the final chapter, we discuss using the results of this chapter to find an

analytic solution to the two-surface problem in the spatial domain. In doing so,

we outline follow-on work to that presented in this thesis.


8-1

C h a p t e r 8

8 Independence of the Field Derivative at the Surface from

the Green Function after Several Correlation Lengths

In this chapter we examine some of the statistical aspects of this work in greater

detail. As mentioned previously, this work is novel in the ability to deal not only

with integral equations in an analytical manner, but also to deal with integral

equations that are functions of random variables in an analytical fashion.

As in all statistical approaches, the probability distributions really dominate the

problem. For instance, in statistical pattern recognition applications, where we

are trying to classify observations, those distributions are usually difficult to find.

Without the distributions, then the problems become ones in classifying based on

other types of decision rules (rather than hypothesis testing with a Bayes

classifier) and using parametric and nonparametric techniques to do clustering,

feature extraction, and the like [40].

Sometimes, even when we know the distributions, we aren’t even in as powerful a

position as one might originally believe. With scattering off of rough surfaces

with known statistics, we know that the Helmholtz equation can be used to

completely describe the field within a closed region, whether we use the paraxial

form for the propagation or not. Then, with rough surfaces generated by

prescribed statistics and a source at a given location, this would seem to be all

the information needed given we have the Green function. Indeed it is enough for

any one realization. However, to say anything about the average field requires

averaging quantities with respect to the probability density functions of random

variable(s). And while the probability density function (pdf) for a single random

variable might well be prescribed as Gaussian, this does not necessarily tell us the

form of the joint probability density function. Furthermore, we must be careful

to call attention to what quantities are stochastic.


8-2

Thus, this section is devoted to the discussion of the ensemble averaging of the

integral equation for paraxial propagation and the potential approximations we

make in doing so.

8.1 Ensemble Averaging the Field Derivative and the Green

Function

We have shown the solution to the mean field for the integral equation

(one-surface case) is analytically tractable when we can express the ensemble

average of the integral equation

( ) ( ) ( )( ) ( )( )0

, , ; ,x

inc

EE x z E x z G x x z S x x S x dxz

∂′ ′ ′ ′ ′< − >=< − − > ′ ∂ ∫ (8.1)

with a right-hand side of

( )( ) ( )( )

( )( ) ( )( )

?

0

0

; ,

; ,

x

x



∂′ ′ ′ ′ ′< − − > =′∂

∂′ ′ ′ ′ ′< − − >< > ′ ∂

∫

∫ (8.2)

Similarly, by moving to the surface, (8.1) becomes

( )( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )0

?

0

, ; ,

; ,

x

inc

x



∂′ ′ ′ ′ ′< >= −< − − > ′ ∂ ∂′ ′ ′ ′ ′=− < − − >< > ′ ∂

∫

∫(8.3)

The use of the ?= is meant to call attention the ensemble averaging, for it is not

obvious that one is justified in separately averaging the two quantities under the

integral sign. We now will discuss the subtleties of the equality in equations (8.2)

and (8.3). This will require us to make careful use of the definitions of

independence and correlation of random variables.


8-3

8.1.1 Independence

Statement 1

Two random variables, 1Y and 2Y are independent when their joint

probability density function is equal to the product of their one-

point density functions.

( ) ( ) ( )1 2 1 21 2 1 2,

,Y Y Y Y

p y y p y p y= (8.4)

In terms of the cumulative distribution functions (the probability

that the random variable takes on a value less or equal to some

number), we have

( )

( ) ( )1 2

1 2

21 2,

1 21 2

,Y Y

Y Y

F y yp y p y

y y

∂=

∂ ∂ (8.5)

when the random variables 1Y and 2Y are independent.

Simply, the value of one random variable does not depend on the value of the

other random variable. This is the essence of independence.

Be aware in this notation ( )1 1Y

p y denotes the probability that the random

variable 1Y takes on the value 1y . We could have also said that

independence requires the joint probability density function of the random

variables to be equal to the product of the marginal probability density

functions. In N variables, this statement becomes

( ) ( ) ( ) ( ) ( )1 2 3 1 2 31 2 3 1 2 3, , ,

, , ,N NY Y Y Y N Y Y Y Y N

p y y y y p y p y p y p y=… (8.6)

8.1.2 Expectation and Averaging

Every expectation is defined as a weighted integral over the probability

distribution of the random variable. The weighting tells us what type of

expectation we are calculating. When we deal with the expectation of powers of

random variables, we are referring to moments.


8-4

For example,

[ ] ( )YY E Y yp y dy y

∞

−∞

< > = = =∫ (8.7)

This is the average value taken on by the random variable Y . Expectation of

products of random variables is a measure of the cross-correlation between the

random variables. For instance,

( )1 21 2 1 2 1 2 1 2,

,Y Y

E YY YY y y p y y dxdy∞ ∞

−∞ −∞

= < > = ∫ ∫ (8.8)

is the cross-correlation while

( ) ( )( )

( )( ) ( )1 2

1 2 1 1 2 2

1 1 2 2 1 2 1 2,

cov

,Y Y

YY Y Y Y Y

y y y y p y y dy dy∞ ∞

−∞ −∞

= < −< > −< > >

= − −∫ ∫ (8.9)

is known as the covariance between 1Y and 2Y . The covariance can also be

expressed as

( )( )1 1 2 2 1 2 1 2Y Y Y Y YY Y Y< −< > −< > > =< >−< >< > (8.10)

Notice that the cross-correlation and covariance are closely related with one being

a mean-removed computation.

As stated in section 8.1.1, when 1Y and 2Y are independent, then the joint

probability density function can be expressed as the product of the marginal

distributions. Using this notion of independence and averaging the product of 1Y

and 2Y gives

( ) ( )1 21 2 1 2 1 1 1 2 2 2 1 2Y Y

E YY YY y p y dy y p y dy E Y E Y∞ ∞

−∞ −∞

= < > = = ∫ ∫ (8.11)


8-5

Statement 2

Thus if two random variables are independent, they are

uncorrelated, as the definition of uncorrelated is

1 2 1 2E YY E Y E Y = (8.12)

Clearly, independence implies uncorrelation, but the converse is not true.

Independence is a much stronger requirement.

One claim regarding independence that is somewhat remarkable is the

requirement that any measurable function of an independent random variable

must be uncorrelated from any other measurable function of another independent

random variable. At first glance, it would seem that there must exist some

function for which we break the correlation requirement, but it is not so.

Statement 3

Statistical independence of the random variables 1 2, ,N

Y Y Y…

requires

( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 1 1 2 2N N N Ng Y g Y g Y g Y g Y g Y< > = < >< > < > (8.13)

for all functions 1 2, ,N

g g g… .

8.1.3 Independence and Correlation of Normal Variables

Statement 4

However, uncorrelation and independence are equivalent if we have

normal distributions. More precisely, if two variables are jointly

normally distributed (meaning that any linear combination of the

two variables is normally distributed), then these two variables are

independent if and only if they are uncorrelated. This is an

exceptional property of the normal distribution that cannot be

extended to other distributions.


8-6

Figure 54 summarizes the possible combinations regarding correlation and

independence when the random variable is normal and also when it is not. It is

not possible for two random variables to be jointly normal, uncorrelated, but not

independent. Thus, class 7 is empty. It is not possible for two random variables

to be independent but correlated. Thus classes 3 and 5 are empty. All the other

classes are possible [41].

Figure 54 Venn diagram explains possible combinations.

Let us examine (8.3) in more detail. We have an expression in (8.3),

( )( ) ( ) ( )( )0

, ;x E x S x G x x S x S x dx

z∂ ′ ′ ′ ′ ′< − − >

′∂∫ , that is comparable to an

ensemble average of the form ( ) ( )1 1 2 2g Y g Y< > where 2g G= is the surface

Green function and 1

Egz

∂=′∂ is the derivative of the field at the surface. But

notice that while ( )1Y S x ′= is certainly a Gaussian random variable of zero

mean and variance 2σ (by our definition of the random surface statistics), it is

not clear what 2Y should be! After all, we can only test for correlation and

independence between two random variables; to test against one random variable

is meaningless.

For the Green function, the term ( ) ( )2Y S x S x ′= − does serve as a second

random variable. When L

x x ξ′− > , by the definition of autocorrelation and

correlation length scale, we know that two points on the surface are uncorrelated

when they are more than a distance Lξ , the scale length, apart. When

Lx x ξ′− > , ( ) ( )21 0,Y S x σ′= = , ( ) ( ) ( )22 0,2Y S x S x σ′= − = and

( ) ( ) 0S x S x ′< > = .


8-7

That 2Y is Normal but twice the variance follows at once from

( ) ( ) ( ) ( ) ( )2 1 1 1 1Y Y Y Y Y

p z p y p z y dy p z y p y dy∞ ∞

−∞ −∞

= − = −∫ ∫ (8.14)

which is the convolution theorem for probability density functions. In this case,

we are dealing with Gaussian probability density functions, so under the

convolution operation, the variances add.

8.2 Deducing the Correlation Coefficient Between

Random Variables

Now that we have established what the random variables are in (8.3), the next

question to ask is, “Are 1Y and 2Y uncorrelated random variables?” If they are

uncorrelated, then by the fact that they are also Gaussian, they must be

independent by statement 4. We could heuristically argue along the following

lines. Both 1Y and 2Y are Gaussian random variables of zero mean. So, they are

both equally likely to be positive or negative. Case in point, if 1Y is positive for a

given sample, even though 2Y is a linear function of 1Y , it is equally likely to still

be positive or negative. Thus, 1Y and 2Y will be uncorrelated.

And we would be heuristically wrong. It is easy to simulate these matters and

deduce a correlation coefficient numerically or perform the calculation outright.

The techniques described later in this section were used to generate a rough

surface with Gaussian statistics, a prescribed correlation length, and of a given

number of samples.

For each realization of a rough surface, a particular point on the rough surface

was taken as a reference. Then 39 other points were sampled equidistantly from

the original reference point. Thought of in a matrix, each row represented an

observation or realization, and each column one of the 40 points on the rough

surface. A large number of realizations were generated, say 500, resulting in a

500x40 matrix. To simulate our random variables 1Y and 2Y , the samples of 1Y

were taken as the first column of our matrix. Then to construct 2Y , we merely

subtracted column 1 ( 1Y ) from each of the remaining 39 columns in the matrix.


8-8

This gives us 500 samples for 2Y at 39 different spacings that cover lengths less

than and greater than the implicit correlation length scale. Next, the covariance

matrix for this 500x40 matrix was generated. The correlation coefficient is just

( )( )

1 2

1 2

1 1 2 2YY

Y Y

Y Y Y Yρ

σ σ< −< > −< > >

= (8.15)

We know that 2 1

2Y Yσ σ= by equation (8.14). Furthermore, the numerator of

(8.15) becomes 1

2Yσ− and

1 2

12YY

ρ = − .

The correlation coefficient ρ should tend to zero if 1Y and 2Y are independent.

However, Figure 55 tells us that there is correlation in the variables even as

( ) Lx x ξ′− >> as the plot covers up to ( ) 8

Lx x ξ′− ∼ . Note that asymptotically

1 2

12YY

ρ → − in Figure 55.

100 200 300 400 500 600 700 800 900 1000-5

-4

-3

-2

-1

0

1

2

3

4

5

Typical Rough Surface and 40 Sampled Points for one Realization

Y1

0 5 10 15 20 25 30 35 40-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

Coe

ffici

ent,

<Y1Y

2>/ σ

Y 1σY 2

Correlation Coefficient versus Separation Distance for Y1 and Y2

The correlation length scale ρ = 5

Separation Figure 55 Left: One realization of rough surface with 40 sample points marked in red. First

sample point value, 1Y , is then subtracted from the remaining 39 points to generate

2Y at different separations. Right: Correlation coefficient for 1Y , 2Y as a function of

these separations.

Ideally, we would expect the joint probability density function for 1Y and 2Y

independent to be of the form

( ) ( ) ( )112

1/2

12

T Pp ePπ

−−= y-y y-yY y (8.16)


8-9

with 1

2

2

2

0

0Y

Y

Pσ

σ

=

the covariance matrix. However, we have found that P has

off-diagonal entries in our results, 1 1 2

1 2 2

2

2

Y Y Y

Y Y Y

Pσ ρσ σ

ρσ σ σ

=

. A color plot of the

difference between the pdf’s for the correlated and uncorrelated case is below in

Figure 56.

Figure 56 Gaussian probability density function in two variables. Uncorrelated on left,

correlated on right to 12ρ = − with

2 12Y Yσ σ= in both cases.

8.2.1 Transforming Correlated Random Variables to Uncorrelated Ones

However, we can always find a transformation into a set of uncorrelated random

variables. Letting 1 2U AY BY= + and 1 2V CY DY= + , the covariance of U and

V will be zero when

( )( ) 0uvC U u V v= − − = (8.17)

This condition is satisfied when

( )1 2 1 2

2 2 0Y Y YYAC BD AD BC Cσ σ+ + + = (8.18)


8-10

Further letting cosA D θ= = and sinB C θ= − = , we can introduce

uncorrelated random variables U and V when

1 2 1 2

1 2

112 2 2

2tan YY Y Y

Y Y

ρ σ σθ

σ σ−=

− (8.19)

recognizing that 1 2 1 2 1 2YY YY Y Y

C ρ σ σ= . Thus, 1Y and 2Y become

1

2

cos sin

sin cos

Y U V

Y U V

θ θ

θ θ

= −

= + (8.20)

Certainly this can also be seen as an orthonormal transformation between

coordinate systems by rotation of angle θ . The rotation matrix just being

( )cos sin

sin cosT

θ θθ

θ θ

− =

(8.21)

from [ ],U V to 1 2,Y Y .

Thus, using equation (8.19) coupled with the variable transformation introduced

by (8.20) will give us uncorrelated random variables. Furthermore, we have

stated that when dealing with Gaussian random variables, if two random

variables are uncorrelated, they are independent by statement 4. So we can

always choose a coordinate reference where 1 2,Y Y are independent random

variables when they are Gaussian. Furthermore, measurable functions of the

random variables are also independent by statement 2.

Bear in mind it is possible to have the covariance be zero and still be dealing with

two random variables that are not independent with other, non-Gaussian random

variables as shown in the Venn diagram. There are several examples where two

variables are very much dependent, yet their product expectation is still zero.

This relates more to orthogonality over the averaging than it does independence.

We can use these facts to further numerically estimate the implications of the


8-11

averaging shown in (8.3) and how justified we are in doing so after we use the

decorrelation transformation.

To this end, we use the definition of covariance in (8.9) and correlation coefficient

(8.15) to test the degree of correlation between the rough surface and the

derivative of the field at the surface. We also estimate the degree of correlation

between the Green function and derivative of the field at the surface. This will

be a tangible measure of the success of the decorrelation transformation.

8.2.2 Estimating the Correlation Coefficient Between the Surface and

the Field Derivative

We compute the covariance of the rough surface and the derivative of the field on

the surface using the formula

( ) ( ) ( )( )( ) ( )

,Cov

E x S x E ES x S x S x S xz z z

′ ′∂ ∂ ∂ ′ ′− = < − − > ′ ′ ′∂ ∂ ∂ (8.22)

Notice that ( ) ( ) 0S S x S x ′= < − > = and is not included on the right side of

(8.22).

This expression is for the cross-covariance of two quantities evaluated at lag 0.

Note that the general form here is of ( )( )2 1 1Cov ,Y g Y where ( ) ( )1

Egz

∂⋅ ≡ ⋅′∂

and

should tend to zero by statements 3 and 4 for independent 1Y and 2Y .

In practice, we must estimate the covariance from a finite sample. For this

purpose, we define the sample mean

1

1ˆN

ii

x xN =

= ∑ (8.23)

and define one estimator (maximum likelihood estimator) of the cross-covariance

as


8-12

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )( )1

,ˆ ,

ˆ, ,1 Ni i i i

i ii

E x S xC S x S x

z

E x S x E x S xS x S x

N z z=

′ ′∂ ′− ′ ∂

′ ′ ′ ′∂ ∂ ′− − ′ ′∂ ∂ ∑

(8.24)

Here again, ˆ 0S = . For the cross-covariance at zero lag, (8.24) is unbiased.

However, over all other lags it is a biased estimator although asymptotically

unbiased because it is consistent with the actual cross-covariance sequence as the

number of samples approaches infinity.

If equation (8.3) is to be believed, then we should find

( )( ) ( ) ( ) ( )( )2 1

,ˆˆ , , 0E x S x

Y g Y C S x S xz

ρ ′ ′∂ ′∝ − → ′ ∂

(8.25)

where C is the covariance estimate.

The process for doing this has a few steps. For a family of N realizations of a

rough surface of given correlation length and roughness, we numerically calculate

the field everywhere, estimate the derivative of the field on the surface, then

evaluate (8.25) for our family of surfaces together with the derivative at the

surface.

Figure 57 is an example of the first part: calculate the field everywhere using our

propagation model described in Chapter 2. In this case, the ratio of the

correlation length to the step size along the surface is 50. By the arguments in

Chapter 5, we should expect the correlation between the field derivative and the

surface to go to zero after traversing several correlation lengths.

Figure 58 gives an example of estimating the derivative at the surface. In the

center plot, we have the magnitude of the field at the closest samples above and

below the surface plotted together. In the bottom of the figure, we form the

difference of these field values divided by the step size in the vertical direction.


8-13

0 0.5 1 1.5 2 2.5

x 104

-1800

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

Dept

h [m

]

distance from source [m]

SPL, cst RI, Rough Surface of Gaussian Spectrum, X0 = 5000 ∆X = 100

Figure 57 Field beneath reflecting rough surface for point source at –75 m shown as Sound

Pressure Level (related to decibels). The depth is shown as negative here due to a

plotting package technicality. We still maintain a positive-down sense to our

coordinate system.

0 0.5 1 1.5 2 2.5 3

x 104

-100

0

100Rough Surface With Gaussian Spectrum

S(x)

0 0.5 1 1.5 2 2.5 3

x 104

-10

-5

0

5

|E(x

,z)|

0 0.5 1 1.5 2 2.5 3

x 104

0

0.05

0.1

|dE(

x,z)

/dz|

x

Magnitude of Field At Closest Sample Above and Below Surface

Estimate of Derivative of Field on the Surface via Interpolation

Figure 58 Instantiation of rough surface and corresponding field just above and below the

surface along with interpolated derivative at the surface.


8-14

To make us less sensitive to the potential floating-point errors in differencing

small quantities, we use a surface with larger variance. Here, 0 1000φ = is the

rms surface variation for the unfiltered (white) surface. To scale the covariance

appropriately with the variances of the individual quantities, we compute the

correlation coefficient as described by equation (8.15) with sample values

replacing the population values

( )( )( ) ( )( )( )( )

( ) 21 1

1 1 1 1 2 2

1 1 2

ˆ ˆˆˆ ,

ˆYg Y

C g Y g Y Y Yg Y Yρ

σ σ

− −= (8.26)

We will be testing the hypothesis, 0H , that ( )( )1 1 1ˆ , 0g Y Yρ = . We use ( )

1Y S x=

and ( ) ( )( )1 1 ,EY x S xz

g ∂≡′∂

so that the reader sees the relation to the earlier

sections of this chapter. When we can accept 0H with 97.5% confidence, this

point is marked in blue on the plots. When we must reject 0H with 2.5%

significance, we mark that point as red on the plots. It is important to realize

0H is really a re-statement of one of the conditions in Chapter 5, section 4, which

we are testing here. We will discuss the nature of the statistical hypothesis test

in 8.2.5.

In Figure 59, we have six correlation lengths’ worth of sample correlation

coefficient, ( )( )1 1 2ˆ ,g Y Yρ , plotted as a function of x . We demonstrate the

correlation coefficients and their statistical significance in the second and third

plots of Figure 59. The field is complex on the surface; therefore, we have two

components of the field to plot. Here, the results provide strong evidence that

even ( )[ ]( )( ),ˆ , 0

E x S xS x

zρ ∂ → ′ ∂

after 3 or 4 correlation lengths. One way to

think about this test is to imagine we have tested ( ) ( )( ) ( )( ),,

E x S xS x S x

zρ ∂ ′− ′ ∂

as ( ) 0S x′ → . So we are in the regime where ( ) ( )( )S x S x L′− < , but still

uncorrelated after several correlation lengths are traversed


8-15

Figure 59 Family of 400 rough surfaces ( )S x with 0 5000, 100x dx= = , and 300nx =

samples. If sample is in red, then the correlation coefficient being nonzero is

significant to 2.5%. Blue points on the correlation coefficient plots imply we accept

the hypothesis that 0ρ = with 97.5% confidence. Here, ( )( ),E x S x

z∂

′∂ is correlated

with ( )S x as a function of x .

8.2.3 Estimating the Correlation Coefficient Between the Green Function

and the Field Derivative at the Surface

Here we compute the correlation coefficient of the Green function and the

derivative of the field on the based on the formula

( ) ( )( ) ( )( )[ ]2 2

,;

ˆ ˆ

E EG GE x S x z zG x x S x S x

z EGz

ρσ σ

∂ ∂ < − − > ′ ′∂ ′ ′ ∂ ∂ ′ ′− − = ′ ∂ ∂ ′ ∂

(8.27)

employing an estimator, ρ , based on (8.22) and the sample variances. In this

instance, the general form is of ( ) ( )( )1 1 2 2ˆ ,g Y g Yρ where ( ) ( )

1

Egz

∂⋅ ≡ ⋅′∂

,

( ) ( )2g G⋅ ≡ ⋅ . This quantity should tend to zero by statements 3 and 4 for

independent 1Y and 2Y . We should be able to achieve this for all separations

greater than a correlation length.


8-16

Again in Figure 60, we estimate the correlation coefficient (real and imaginary

parts) as described by (8.27) instead of the covariance so that the variance of the

two quantities is normalized in the result. Here, there is no strong evidence of

correlation when the Green function and field derivative are referenced one

correlation length apart. This is true across the vast majority of the surface,

except the first few positions. This has been the case for all range of lags

examined greater than one correlation length.

Figure 60 Sample correlation coefficient between ( ) ( )( );G x x S x S x′ ′− − and

( )( ),E x S x

z

′ ′∂′∂

as a function of x x ′− . Blue points on the correlation coefficient

plots imply we accept the hypothesis that 0ρ = with 97.5% confidence.

8.2.4 Estimating the Correlation Coefficient Between the Surface and

the Field at One Correlation Length Above the Surface

For the results in Figure 61, we moved above the surface one Fresnel radius in

the vertical and looked at the correlation coefficient estimate for the field at the

point at the surface below it. This was also one of the statements of Chapter 5

that should be true after a few correlation lengths have been traversed, based on

the physical arguments. We cannot unanimously accept the hypothesis for all x


8-17

as evidenced by the red markers in the plots, but have evidence for 0H after a

few correlation lengths.

Figure 61 Sample correlation coefficient between ( )( )0, /E x S x L k+ and ( )S x as a

function of x . Blue points on the correlation coefficient plots imply we accept the

hypothesis that 0ρ = with 97.5% confidence.

8.2.5 Distribution of the Sample Correlation Coefficient

It is well known that the variable 1 1ˆ ˆ

ˆ/

Y E YT

V n

− = has a t distribution, where 1Y

is the sample mean and V the sample variance. The t distribution has the

following functional form

( )( )[ ][ ]

( )1 /221 /21

/2

n

n

n tf tn n nπ

− + Γ + = + Γ (8.28)

It approaches a Gaussian as n → ∞ . What is less well known is that

( )22

ˆ2

ˆ1 nn f tρρ −− =

− (8.29)


8-18

is also t-distributed with 2n − degrees of freedom. Thus, we can do a hypothesis

test on the statement 0ρ = using the t statistic with a given confidence. Note

that the significance of the test depends on the sample size. This is precisely the

statistic we used to establish which points were red and which were blue in

Figure 59 through Figure 61.

8.3 Conclusions Regarding the Ensemble Averaging of Green

Function and Field Derivative

The results lead us to conclude that (8.3) deserves the question mark above the

equal sign because 1Y and 2Y have a correlation coefficient of 1 2

12YYρ → − and

are not independent. However, with a linear transformation of 1Y and 2Y , we can

decorrelate the Gaussian variables and hence the averaging of the quantities in

(8.3) is legitimate. The results in Figure 59 and especially Figure 60 support this

conclusion.

Regarding equation (8.2), clearly ( )( );G x x z S x′ ′− − and ( )( ),E x S xz

∂ ′ ′′∂

cannot

be independent as they are functions of the same, and only, random variable.

There can be no test for independence because it fails even the most basic criteria

for performing the test: being a function of different random variables. The

argument, ‘Well isn’t a function of a random variable another random variable?

So then there are two random variables!’ misses the point as well. Taking that

path as amusement, let ( )( )1 ;Y G x x z S x′ ′= − − and ( )( )2 ,EY x S xz

∂ ′ ′=′∂

be two

random variables. We could then calculate their covariance ( 1Y and 2Y )

according to (8.9) if we knew their joint probability density, which we don’t.

Even calculating the marginal distribution for 1Y involves expressing the inverse

Green function for ( )S x and calculating the Jacobian, ( )1Y

S x∂∂ , to complete the

transformation. We don’t analytically know the probability density function for

the derivative of the field at the surface. So the only recourse to us is to

numerically calculate values for 1Y , 2Y and estimate the covariance. Let us

assume that this covariance estimate always returns a number close to zero. We

might have evidence then, that these two quantities were uncorrelated. But this

is not enough to make them independent. They are surely not linearly related


8-19

(uncorrelated), but this does not make them independent. Perhaps they are very

much related, but in some nonlinear way. And then, one would be forced to

remember that even to generate numerical values for 1Y and 2Y , we needed a

value for ( )S x , and that this value was required for both 1Y and 2Y . This effort

would only serve to postpone the inescapable: 1Y and 2Y must be dependent.

Therefore, the separate averaging of the quantities for any ( )z S x≠ must always

be an approximation because we can never perfectly decorrelate the quantities

through any linear transformation or otherwise. Thus, we must always bear in

mind that

( )( ) ( )( )

( )( ) ( )( )0

0

; ,

; ,

x

x



∂′ ′ ′ ′ ′< − − > ≈′∂

∂′ ′ ′ ′ ′< − − >< > ′ ∂

∫

∫ (8.30)

based on the arguments of Chapter 5 and presented in [42].

8.4 Generation of Rough Surfaces with Known Statistics

Here, we discuss the generation of rough surfaces with known statistics. There

are several methods available and we outline one of the methods [43].

The three important steps are listed below [44]:

1. The Fourier components of a realization of a (band-limited) discrete

Gaussian white noise process are generated.

2. The output of 1. is filtered to produce the Fourier components of a

realization of the desired process.

3. The output of 2. is inverse Fourier-transformed using the FFT to

generate the desired process.

We will choose parameters such that the properties of our stochastic process will

be a good approximation to an actual realization of a rough surface with the

desired statistics. Let ( ) ( ) ( ){ }, : , 1 , 1S x x l x l n n n∈ ∆ = − − − −… be a discrete

Normal, white noise process, with zero mean and variance 2σ . By Normal, we


8-20

mean that the distribution of the surface heights is Gaussian and by white noise,

we mean that the surface is uncorrelated. Then the complex Fourier components

of S are

( ) ( ){ }1

2 /1 , , 1 , 1n

ilj NNj l

l n

c S e j n n nπ−

−

=−

= ∈ − − − −∑ … (8.31)

If the real and imaginary parts of jc are and j jA B , then

( ) ( )1 1

2 21 1cos , sinn n

lj ljN N N Nj jl l

l n l n

A S B Sπ π− −

=− =−

= = −∑ ∑ (8.32)

Furthermore, it is easy to show that the components equidistant from the

Nyquist frequency are related by

*j jc c− = (8.33)

Each l

S has the ( )20,σ distribution and the autocovariance sequence is a delta

function. This is practically what is meant by the definition of white noise! 2

ml lmS S δ σ< > = , where lmδ is the Kronecker delta.

It is not difficult to show that ( )arg jc and jc are independent for all j with jc

having a Rayleigh distribution, a chi-squared statistic. By using the inverse

cumulative function, we have a way of mapping a uniformly generated random

variable to any type we desire. Since the Rayleigh cumulative distribution has

the form of

( ) 2 20 /2

0 1 xQ x e σ−= − (8.34)

with ( ) [ ]0 0,1Q x ∈ , we can generate the appropriate distribution from

( )22

11log

jyjcσ

− =

(8.35)


8-21

Figure 62 is an example with sequence of 122 points having a uniform

distribution. Here is the distribution of that sequence when mapped from the

equation (8.35) above.

Figure 62 A uniform distribution mapped to a Rayleigh distribution using the inverse

cumulative distribution function.

Fourier components jA and jB should have Normal distributions, which we show

in Figure 63, where ( )( )exp argj j j jA B c i c+ = − .

Figure 63 Real and imaginary random Fourier components and corresponding surface.


8-22

In stage 2), we filter the frequency components with a white spectrum generated

in stage 1) into a new spectrum of interest. We will call the spectrum we desire

( )ˆjνΦ and find a transfer function jH such that

( ) ( ) ( )2ˆwj j jHν ν νΦ = Φ (8.36)

and

( ) ( ){ }ˆ , , 1 , 1j j jc H c j n n n= ∈ − − − −… (8.37)

where ( )j jH H ν= determines the filter. Simply, the filter is determined by the

ratio of the desired to actual spectrums:

( ) ( )2 2ˆ /j jH xν ν σ= Φ ∆ (8.38)

There are many ways to understand the origin of (8.36) through (8.38). Perhaps

the easiest way is through the analogous steps of Weiner filtering

( ) ( ) ( ) ( )ˆ ,b

a

y t y t h t x dξ ξ ξ≈ = ∫ (8.39)

In filtering language, this tells us that y can be expressed as a linear combination

of the inputs x , or, we can filter x is such a way as to produce a good estimate

for y . This equation can be ensemble averaged to give

( ) ( ) ( )b

yx xxa

t h t dρ ξ ρ ξ ξ ξ ξ′ ′ ′− = − −∫ (8.40)

where

( ) ( ) ( )

( ), Cross-correlation of ,

For stationary processes: ,yx

yx

y t x t y x

t y x

ξ ρ ξ

ρ ξ

=

= − (8.41)


8-23

Here, we have used the Wiener-Khinchin Theorem for random functions. Stated

in words:

For a stationary, random process, the autocorrelation function is related to the power

spectrum of the process. That is, they are Fourier transform pairs.

Mathematically, we have

( ) ( )

( ) ( )12

i

i

e d

e d

ωτ

ωτπ

ρ τ ω ω

ω ρ τ τ

∞

−∞∞

−

−∞

= Φ

Φ =

∫

∫ (8.42)

We can take the Fourier transform of (8.40) to get

( ) ( ) ( )yx xxHω ω ωΦ = Φ , (8.43)

which relates the spectrums of the two quantities x and y .

Note that (8.43) relates one power spectrum in one variable to a cross-power

spectrum in two variables. But directly transforming (8.39) gives us ( ) ( ) ( )Y H Xω ω ω= so

( ) ( ) ( )2yy xxHω ω ωΦ = Φ (8.44)

In the final stage 3), we need to just take the inverse Fourier transform of our

filtered components shown in (8.37) so that ( )ˆ ˆl lS S x= has the desired spectrum,

( )ˆjνΦ , and corresponding autocorrelation function of interest.

In Figure 64, the generation of 3 rough surfaces with Gaussian statistics and

autocorrelation function are shown. They are generated from a surface with

Gaussian white noise (shown in upper left plot). The three filtered surfaces all

have different correlation lengths, which results in a different bandpass of the

frequency spectrum. The longer the correlation length scale, the more the

spectrum is filtered (passing lower frequencies only).


8-24

Figure 64 Unfiltered surface and three subsequent filtered surfaces with different Gaussian

correlation lengths.


9-1

C h a p t e r 9

9 Finding Analytic Solutions to Acoustic Scattering

from a Rough Sea and Bottom Surface and Other

Follow-on Work

In this chapter we outline future work related to the two-surface problem and

make some general observations about the form of the solution. We also mention

other areas for further research. These include combining effects of the medium

to go beyond the isovelocity medium studied in this work. Also, we point to

promising numerical work regarding the use of wavelets that also seek to solve

the integral equation through transformation into a new basis. This approach

shares some key concepts from analysis that are used with the Laplace transform

approach.

9.1 Finding Analytic Solutions to the Two-Surface Problem: Two

Pressure-Release Surfaces with 0γ =

A useful point of analytic exploration is to seek a solution to equation (9.1) when

both surfaces are flat ( 0γ = ). Notice that (9.1) is similar in structure to

equation 6.62:

( )

( ) ( ){ } ( )( )

( ) ( ){ } ( )( )102 1 2 12 1 2 1

11

,

ˆ ,ˆ ˆ ˆ ˆ, , , , , , ,ˆ

ˆ

ˆ

s

inc

Z

G ZG Z K G K Z

G

λ

λλ λ γ γ λ λ γ γ λ γ

λ

ε

ε

=

− + (9.1)

with

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

11 2 21 11 2 2

11 22 21

ˆ ˆ, ,ˆ , , ˆ ˆ ˆˆ înc incG G

KG G G


λ λ λε ε−

=− +

(9.2)

for the two-surface, pressure-release, boundary conditions. It describes the

scattered field in the Laplace domain. In equation (9.1), we bracket the solution

to call out explicitly the solution for the scattered field from the upper surface

with an infinitely deep bottom (the one-surface solution in the Laplace domain).


9-2

The one-surface solution is in the square brackets on the right of (9.1), the

additional terms due to adding the second surface, with given boundary

conditions, are shown in curved brackets.

9.1.1 A Flat-Surface Solution via Eigenfunction Expansion

This point of analytic inchoation is useful because there are standard techniques

available for the flat-surface problem, including eigenfunction expansion and

method of images. As mentioned very early in the thesis, the paraxial form of

the wave equation is a SchrÖdinger equation. In (9.3), we display it with a source

term ( )0 ,Q x z .

( )2

2 00

,2

E i E Q x zx k z

∂ ∂+ =∂ ∂

(9.3)

Recall that the Green function satisfies

( )2

20

,2

G i G x x z zx k z

δ∂ ∂ ′ ′+ = − −∂ ∂

(9.4)

when ( ),n x z′ = 0 for the paraxial wave equation and ( ) ( )0 , ,Q x z x x z zδ ′ ′= − − .

In an infinite domain, we know that the Green function for this equation is

closely related to the diffusion equation Green function and has a solution

( ) ( )( )( )

2

01; exp2 2 2

z zikiG x x z zk x x x xπ

′− ′ ′− − = ′ ′ − − (9.5)

On an infinite strip, such as that defined by the rough waveguide problem in the

limit of no surface roughness, we have a domain given by [ ] [ ]0, , ,z H x∈ ∈ −∞ ∞ .

We can solve for the field given any initial condition and boundary conditions of

( ) ( ), 0 0, , 0E x E x H= = by using the method of eigenfunction expansion or the

method of images. Using the method of eigenfunction expansion implies finding a

solution of (9.3) subject to our given boundary conditions and an incident field,

or nonhomogeneous source term of


9-3

( ) ( )( )200

0 00

, , exp2inc

z zEQ x z E x z ik

k x x

− = = (9.6)

The Green function for this situation can be found via a method of separation of

variables applied to (9.3) subject to strip boundary conditions and it results in

( ) ( )2

01

2; sin sin exp 2n

n z n z nG x x z z i k x xH H H H

π π π∞

=

′ ′ ′ ′− − = − ∑ (9.7)

as the Green function via eigenfunction expansion [45]. And we can immediately

write down the solution for the field as

( ) ( ) ( )00 0

, , ;H x

E x z Q x z G x x z z dx dz′ ′ ′ ′ ′ ′= − −∫ ∫ (9.8)

Similarly, by the method of images, we can satisfy the boundary conditions on

the strip through an infinite series summation:

( )

( )( )( )

( )( )

2 2

0 0

;

2 21 exp exp2 2 2 2n

G x x z z

z z Hn z z Hnik ikik x x x x x xπ

∞

=−∞

′ ′− − =

′ ′− − + − − ′ ′ ′ − − − ∑

(9.9)

with the general solution still given by (9.8). This alternative representation

arises by symmetry arguments. We can satisfy the boundary conditions along

0z = and z H= by placing positive sources at 0 2z z Ln= + and negative

sources at 0 2z z Ln= − + . If ( )20k H x x ′>> − , the expansion in (9.9) converges

very quickly, needing only a few terms [46].

9.1.2 The Laplace Transform Approach to Two Pressure-Release Surfaces

with 0γ =

The Laplace transform of the point source ensemble averaged over the flat surface

is ( ) ( ) 010 01, , 0ˆ i Z

inc Z E e λπλ γλ

ε −= = ∓ on the upper and


9-4

( ) ( ) ( )010 02, , 0ˆ i Z H

inc Z E e λπλ γλ

ε − −= = ∓ on the other. Similarly, the Green function

for the medium becomes ( ) ( )101 1

ˆ , , 0 i ZG Z G e λπλ γλ

−= = ∓ on the upper and

( ) ( ) ( )102 2

ˆ , , 0 i Z HG Z G e λπλ γλ

− += = ∓ on the lower, and for the surface

( ) 0ˆ , 0jjG G πλ γ

λ= = . Finally, ( ) ( )1

0,ˆ , 0 i H

jk j kG G e λπλ γ

λ−= = ∓ represents the

Green function between the two surfaces.

We now will outline the steps for the full solution in the limiting case of two flat

pressure-release surfaces. Once this is satisfied, the forms of the Laplace

transforms for nonzero roughness can be inserted in the scattered field equation

and inverted numerically or approximated analytically.

First, the numerator and denominator for ( )1 2ˆ , ,K λ γ γ are given by:

( ) ( ) ( ) ( ) ( ) ( ) ( )( )01 1 10 011 2 21 1

ˆ ˆ, ,ˆ ˆ i Z i H i Hinc incG G G E e e eλ λ λπλ λ γ λ λ γ

λε ε − ± − −− = −∓ ∓

( ) ( ) ( ) ( )( )2 2 2 1011 22 21

ˆ ˆ ˆ 1 i HG G G G e λπλ λ λλ

−− + = − ∓ ,

which we combine into

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( )( )( )

( ) ( )

0

0

11 2 21 11 2 2

11 22 21

1 1 10 0

2 2 10

10

0

ˆ ˆ, ,ˆ , , ˆ ˆ ˆ

1

ˆ înc inc

i Z i H i H

i H

i Z H

G GK

G G G

G E e e e

G e

E eG

λ λ λ

λ

λ


λ λ λ

πλ

πλ

ε ε

− ± − −

−

− −

−=

− +

−=

−

=

∓ ∓

∓

∓

(9.10)

We look at the first term on the right-hand side of (9.1) and denote ( )1 2ˆ , ,K λ γ γ

by ( )K λ for brevity.


9-5

( )( ) ( ) ( ) ( ) ( )( )( )

( )( )( ) ( )

( )( ) ( )

0

00

1 1 1 10

2 2 1

11

0 01

ˆ ˆ1

i Z H i Z i H i H

i H

i Z Z Hi Z Z

i H

e E e e eG K

e

eE E ee

λ λ λ λπλ

λ

λλπ π

λ λλ

λ− + − ± − −

−

− + +− +

−

−=

−

= =

∓ ∓ ∓

∓

∓∓

∓

(9.11)

The second term in (9.1) is

( )( )

( ) ( ) ( ) ( ) ( ) ( )01 1 11012

11

ˆˆ ˆ

î Z i H i Z HG

G K E e e eG

λ λ λπλ

λλ λ

λ− − − − =

∓ ∓ ∓ (9.12)

and the third term

( )( )

( ) ( ) ( )01101

11

ˆ, 0ˆ ˆ i Z Z

inc

GE e

Gλπ

λλ

λ γλε − += = ∓ (9.13)

Letting ( )1A i H= ± − , we should expect to collect terms and express the

solution in the general form of

( ) ( ) ( )( )

210

sinh,

sinh

c i Ai B X

sc i

E X Z C e dA

λ λπλ

λλ

λ

+ ∞− +

− ∞

< >=

∫ ∓ (9.14)

with ( )0, ,B B Z Z H= and 0C just a complex constant. To find the inverse

Laplace transform, we can use the convolution theorem directly.

9.1.2.1 Applications of the Convolution Theorem in the Inverse Transformation

Given the form of (9.14), let us symbolically express the scattered field in the

Laplace domain as

( ) ( ) ( )0ˆ ˆ, , ,s Z C S Z T Zλ λ λε = (9.15)

We will drop the Z in our notation as it is not a transform variable, but shall be

implicit. Then if we can find the individual inverse Laplace transforms of ( )S λ


9-6

and ( )T λ , we can write the general solution in the spatial domain via

convolution.

We know that the inverse Laplace transform of ( )( )( )

2sinhˆsinh

A

SA

λλ

λ= can be

expressed in terms of the modified θ functions [47]:

( )( )( )

2124

sinh 1ˆ |2sinh

A xSA A AA

λ ϑλ θϑλ

− ∂ = = ∂

L (9.16)

where 2Aϑ = and

( ) ( ){ }

( ) ( ){ } ( )

2124

2

0

1| exp /

1 exp cos 2

n

nn

n

x n xx

n x n

θ ϑ ϑπ

ε π π ϑ

∞

=−∞

∞

=

= − + +

= − −

∑

∑

(9.17)

with nε being the Neumann number. We differentiate (9.17) with respect to ϑ

and scale (9.17) by the arguments in (9.16)

( ) ( ) ( ) ( ){ } ( )( )

( ) ( ) ( ) ( ){ } ( )

222

2 2

24

0,

22 24

0

| 1 2 exp sin 2

| 1 2 exp sin 2

A XA A

nn

nx

nX XnA AA A

n

x n n x n

n n n

ϑϑϑ

ϑ

ϑ ϑ

θ ϑ ε π π π ϑϑ

θ ε π π πϑ

∞

= == =

∞

=

∂ = − − − ∂

∂ = − − −∂

∑

∑ (9.18)

remembering that 2Aϑ =

( ) ( ) ( ) ( ){ } ( )2 21 24 24

0

| 1 2 exp sinnX X nnA A

n

n n πθ ε π πϑ

∞

=

∂ = − − −∂ ∑ (9.19)

We have already computed the integral for ( ) ( )1ˆ i BT e λπλλ −= ∓ in the one-surface

problem ( 0H = ) and found that it becomes

( ) { }212

1ˆ exp iBXT

Xλ− − = L (9.20)


9-7

Therefore, the final solution can be expressed as

( ) ( ) ( )0, , ,sE X Z C S X Z T X Z= ⋅ ∗ (9.21)

We substitute in our terms at this point and the solution begins to take shape,

{ } ( ) ( ) ( ){ } ( )2

22

22000

1 exp 1 2 exp sinX

niB X ns nX A

n

E C n n dXX

πε π π∞

−′

=

′= − − − ′

∑∫ (9.22)

to that given by equations (9.6) through (9.8). However, it should be noted that

the precise equivalence of (9.22) and (9.8) has not been proven yet.

Having (9.22) to guide us will be a tremendous benefit in further analytical and

numerical work on the two-surface problem with this method.

9.2 Laplace Transform Approach for Non-isovelocity Mediums

In principle there is no reason why the method can’t be extended to more

generalized mediums. For example, the Green function given in Chapter 4 has

some extra terms in the exponent to deal with a linear refractive index profile,

governed by the parameter ε

( )

( )( )( ) ( )( ) ( )2 32

0

;

1 exp2 2 2 12

G x x z z

z z x xiki z z x xk x x x x

εε

π

′ ′− − =

′ ′− − ′ ′− + − − ′ ′ − −

(9.23)

Take the integral equation for the one-surface, pressure-release problem when z

moves to the surface

( )( ) ( ) ( )( ) ( )( )0

, ; ,x

inc


∂′ ′ ′ ′ ′= − − ′ ∂ ∫ (9.24)

Then the Green function in a linear medium will simplify from (9.23) into

( ) ( )( )32

0

0

1; exp2 2 2 12

x xikiG x x z zk x x

επ

′− ′ ′− − = ′ − (9.25)


9-8

when the surface is flat, yielding ( )( ) ( );G x x z S x G x x′ ′ ′< − − > = − . If we wish

to work in the Laplace domain, even for a flat surface, then we are faced with an

integral of the form

( ) ( ) ( )( )33

02

3 0

0

1ˆ exp

,24 2 2

G C i x x x x dxx x

k iCk

λ β λ

εβ

π

∞

′ ′= − − − −′−

= =

∫ (9.26)

where ( ) ( )G G x xλ ′= − L .

We remove the singularity with the variable substitution ( )z x x ′= − .

( ) ( )3 6 2

0

ˆ 2 expG C i z z dzλ β λ∞

= − −∫ (9.27)

(9.27) resembles the Airy function, which is defined as

( ) ( )31 12 3

ˆ expi

C

Ai z z dzπλ λ= − −∫ (9.28)

with the contour C following 2arg3π= − to 2arg

3π= along the path to and

from infinity.

These are integrals of the general class

( ) ( ) , is real and positivef z

C

e g z dzλ λ∫ (9.29)

and λ may have asymptotic values such as 0, or λ λ→ → ∞ . Take the case

when λ is real, positive, and very large. The integrand is largest where the real

part of ( )f z is largest along the contour.

The method of steepest descents exploits the path up and away from the saddle

point (Figure 65) so that the imaginary part of ( )f z is constant along it. This

will make the estimate for the integral in (9.29)


9-9

( ) ( )( )sf zs sO e g z zλ ∆ (9.30)

where sz is the saddle point, a good one with

( ) 1/2

s sz f zλ − ′′∆ ≈ − (9.31)

In fact, Hinch [48] shows the integral estimate to be

( ) ( ) ( ) ( )( ) ( )( )

1/2

1/22 1sf z f zs

sC

e g z dz e g z Of z

λ λ π λλ

− = + ′′ −

∫ (9.32)

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500Level Curves For Real Part of Airy Function Exponent

x x

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20Level Curves for the Real Part of the Quasi-Airy Function Exponent

Figure 65 Level curves of real part of Airy function exponent (left) and exponent of equation

(9.27) (right).

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

X

Re(

Y)

Real and Imag Parts of Airy-like Function. Lambda: 0, .001, .1, 1, 10, 100; eps = .001

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

X

Img(

Y)

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

X

Re(

Y)

Real and Imag Parts of Airy-like Function. Lambda: 0, .001, .1, 1, 10, 100; eps = .01

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

X

Img(

Y)

Figure 66 Real and imaginary parts for Airy-like function. Different values of λ correspond to

different colored curves. β3 = 4.17e-8 (left) β3 = 4.17e-6 (right).


9-10

At the saddle, ( ) ( )3 6 2 0f z i z zβ λ ′′ = − − = and 3

1/48 2

30, ; 0,1,2,3

kisz e k

ππλβ

+ = =

.

The width of the peak at the saddle scales like (9.31) and here

( ) ( ) ( )1/2 1/2 1/23 40 2 30sf z i zλ β λ− − − ′′− = = + ≈ (9.33)

for the saddle at zero. Typically, the variable is rescaled with respect to the width

of the peak at the saddle or

1sz z ξ

λ= + (9.34)

and (9.27) rescales as shown in Figure 66 to

( ) ( )( )3 6 22

0

ˆ expG i dβλλλ ξ ξ ξ

∞

= − −∫ (9.35)

The contribution to the integral occurs over a region of width 1λ and we can

expand the one term in a Taylor series such as

( ) ( )2 222

0

ˆ 1G e dξ ςλλ ς ξ

∞−= + + +∫ (9.36)

with ( )3 6i βλς ξ= − .

Using the identity ( )2 21

0

1 3 5 2 12

nn

ne dξ ξ ξ π∞

−+

⋅ ⋅ ⋅ −=∫ , we can express ( )G λ as a

series, which looks very much like πλ

when keeping only first term as λ → ∞ .

This brief outline of a steepest descent approach for a linear medium and flat

surface introduces some of the mathematical aspects to be addressed in follow-on

work and highlights the elements in any approach of this type.


9-11

9.3 Numerical Computation of Integral Equation Solutions

The integral equation technique is a specific example of a special set of integral

equations known as Volterra equations. For example, we mentioned equation

6.22 is a Volterra Integral Equation of the first kind with respect to

( )( ),E x S x

z

′ ′∂′∂

. Volterra equations are really special cases of the more general

Fredholm equations in the sense that the upper limit of integration is the

independent variable x and has a matrix formulation of

⋅G f = e (9.37)

when ( ) ( ) ( )x

a

E x G x x f x dx′ ′ ′= −∫ [49]. G is often referred to as the kernel.

Volterra equations are generally less complex than the Fredholm equations

because the kernel, G , is lower triangular and there are many strategies for their

numerical solution. Typical methods for solving Volterra equations numerically

rely on quadrature rules. One simple rule that is used in practice is the

trapezoidal rule.

9.3.1 Solutions Via Wavelet Transforms

While these equations in general have been numerically solvable and tractable for

a long time, there has been renewed interest in them with the advent of wavelet

applications. Wavelet transforms are not only suited to applications involving

data compression and signal processing, but can also be used to transform some

classes of integral equations into sparse linear problems.

The main idea is to take ⋅ =G f e into an analogous wavelet basis such that ˆ,≡ ⋅ ⋅ ≡ ⋅TG W G W f W f , and ≡ ⋅e W e . Then solve ⋅ =G f e and finally

transform to the answer by the inverse wavelet transform = ⋅Tf W f .

Use of wavelets as a solution basis produces a sparse interaction matrix which

may be solved rapidly as a wavelet solution can be obtained in ( )logO N N

operations, where N is the number of unknowns in the discretized integral


9-12

equation. This is in contrast to ( )3O N through an inversion method or ( )2O N

for a conjugate-gradient technique. For example, the numerical integrations are

performed by dividing the spatial extent of interest into bins, leaving several sub-

integrals to perform. In each of these bins, the unknown part of the integrand is

assumed to be constant. The result of this discretization is a matrix equation

with the matrix containing integrals of the Green function. The inversion

procedure for the full matrix is ( )3O N .

Consider the problem of computing the scattering of a TM ( )zE polarized

electromagnetic wave from a two-dimensional conducting cylinder with boundary

contour C . Through the Fresnel equations at the surface, we have

( ) ( ) ( ) ( )0z z z

scatincC

E E i G J dlωµ ′ ′ ′= − = − −∫r r r r r (9.38)

where ( )G ′−r r is the Green function and has the form of a Hankel function:

( ) ( ) ( )14 0 0iG H k′ ′− = −r r r r (9.39)

with wavenumber 0k and surface current J [50].

We can easily express (9.38) as

z zinc = ⋅e G j (9.40)

and through the use of the wavelet transform matrix, we transform (9.40) into

z zinc = ⋅TWe WGW Wj (9.41)

For clarity, recall that the field and surface current in this case are discretized

into vectors of length N , so we can view G as a tensor relating two N -length

vectors and is thus an N N× matrix. To transform with respect to a wavelet

basis, W must also be an N N× matrix. In the case of the surface current

distribution, G can be thought of as the wavelet transform of the impedance

matrix. ijG represents the field radiated by a unit amplitude current pulse j ,


9-13

sampled at the midpoint of the receiving pulse i . The sampled field is a linear

combination of these field components.

In Figure 67, we give a concrete example of a wavelet transform matrix, the Haar

matrix. In common with all wavelet transforms is a multi-resolution

representation derived from a mother wavelet. Dilations and translations of this

mother wavelet are used to construct an orthonormal basis. Translations of the

finest resolution wavelet comprise 1/2 of the basis set, translations of the next

finest resolution wavelet make up 1/4 of the set, and so on down the hierarchy.

The finest resolution basis vectors (rows in Figure 67 (left), top-half) have the

most compact support (implying extent) while those rows towards the bottom of

the figure are basis vectors with the broadest support. Figure 67 (right) provides

an example of the Daubechies 4 wavelet and scaling function.

Daubechies 2 Wavelet Transform Matrix, 2562

50 100 150 200 250

50

100

150

200

2500 5 10

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Wavelet Function db4

0 5 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Scaling Function db4

Figure 67 Example of wavelet transform matrix for DB2 (left), scaling and wavelet functions for

DB4 (right).

We can construct a wavelet transform matrix like that on the left of the figure for

any Daubechies p wavelet, where p is the number of vanishing moments. This

refers to the degree of polynomial that is suppressed under the wavelet transform.

Mathematically, vanishing moments implies that a wavelet basis vector of N

elements with p vanishing moments must satisfy the relation:

1

0

0, 0,1,2, 1N

ji

i

w i j p−

=

= = −∑ … (9.42)


9-14

Also, because the wavelet transform is an orthonormal transform, 1T −=W W .

After transformation to a wavelet basis, the new impedance matrix G is found to

be approximately sparse. That is, many of its matrix elements are negligibly

small. To take advantage of this, elements of G which are smaller than an

element threshold t are discarded, and the remaining elements of G are stored in

sparse matrix form. This allows matrix−vector multiplications to be carried out

in a time proportional to the number of non−zero elements in the sparse matrix,

speeding the iterative solution of the transformed matrix equation. One threshold

that can be used to control the sparsity of G is

tNκ

∞= G (9.43)

where κ is a constant and the infinity norm is defined as

1

maxN

i ijj

G∞=

= ∑G (9.44)

In the wavelet basis (such as DB2, DB4, or DB12 where DB is short for

Daubechies), the number of non-negligible elements in a typical transformed

representation scale like N , the number of rows or columns in G . And a

number of sparse linear systems solvers exist, especially for the hierarchically

band diagonal form typical of G .


10-1

C h a p t e r 1 0

10 Conclusions

In the first half of the thesis, we discussed the theory of wave propagation over

slowly varying terrain and refractive media under the paraxial, or forward scatter,

approximation. With this technique, we do not specifically find ourselves in small

surface slopes or small surface height regimes as the paraxial approximation

accommodates multi-scatter, deeply modulated surfaces provided the correlation

length along the surface is large compared to the wavelength, but is restricted to

forward scatter. The utility of the marching technique in handling this problem

is unquestionable; its main shortcoming being the difficulties of implementing full

three-dimensional propagation over rough terrain. By implementing the paraxial

wave equation in both Cartesian and curvilinear forms, we were able to directly

compare differences in Gaussian source to point source models. In the curvilinear

case, the parabolic equation appears almost identical to the Cartesian case except

for the necessary radial scaling terms from geometrical considerations. However,

there is one fundamental fact to keep in mind: while both PWE’s approximate a

curvature term as zero, which removes the field’s ability to react precipitously to

likewise precipitous changes in media or surfaces, there is a difference in the

implication of the approximation. For the curvilinear case, the radial

components’ restriction comes in the field’s inability to quickly bend along a

radial in the direction of propagation. It is not a function of the angle the ray

makes with respect to the surface. In the Cartesian case, there is the same

restriction on the envelope of the field in the direction of propagation. Also,

plane wave components satisfying 1tan 20y

x

kk

− ≥

are not consistent with the

forward scatter approximation of the model and results in this region must be

viewed with cynicism due to the phase errors incurred.

Results demonstrate that propagation in mildly refracting media were consistent

between the Gaussian Cartesian case and point source curvilinear case within a

scaling factor to accommodate the different energies in the sources. There were

intensity variation differences along the ground that grew more pronounced as

the media refracted more heavily.


10-2

10.1 Recommendations for Follow-on Work Related to Chapters 2-4

Specific recommendations for follow-on work to this particular effort should be

• Plane wave analysis to strengthen the insight from simulation

As is discussed by Goodman, the field distribution across any plane of

propagation can be envisioned as a superposition of plane waves traveling in

different directions away from that plane. The field amplitude at any other point

along any other plane can be calculated by the superposition principle. The plane

wave contributions each undergo a different phase shift depending on the angular

frequency of the wave; how we calculate the phase shifts of each contributing

plane wave will determine the accuracy of the result.

If the 2-D angular spectrum of the field distribution along the plane 0z = is

given as the Fourier transform of the field distribution:

( ) ( ) { }, 0 , 0 exp 2x xA E x z i dxν πν∞

−∞

= = −∫ (10.1)

then we could consider the propagation process by propagating the angular

spectrum. However we need a relation between the angular spectrum across a

reference plane and the plane of interest. Using the basic Helmholtz equation, we

find that the angular spectrum relation needed is

( ) ( ) { }2, , 0 exp 1x xA z A ik zν ν α= − (10.2)

where the wave vector is ( )2 ˆ ˆk x zπλ α γ= + and 21γ α= − is the relation for the

direction cosines. Notice that xαλ ν= . Combining (10.1) and (10.2), we can

express the field distribution along any plane z as

( ) ( ) ( ) ( )22, , 0 exp 1 exp 2x x xE x z A i z i x dπλν α πν ν

∞

−∞

= −∫ (10.3)

which defines the propagation of the angular spectrum to determine the field at

z .


10-3

Since ( )221 1 xα λν− = − , then we can view the parabolic approximation as

the Fresnel approximation where ( ) ( )2221 1 x

xλνλν− ≈ − + . This of course is

highly accurate when ( )2 1xλν , which occurs when the plane waves are

propagating at small angles with respect to the forward direction. Thus, small

diffraction angles are going to retain the proper phase relationships under the

approximations while large diffraction angles will not.

We can use the same approach to understanding the propagation of the angular

spectrum in curvilinear coordinates. One can compare the results of the

simulation of the fields by looking at the implications for the phase errors in the

parabolic equations. By equation (10.3), these effects are clearly separated from

the initial field distribution and thus the phase errors can be studied from a

transfer function point of view.

• Extending the curvilinear simulation to encompass rough terrain as well as

varying media

As we saw in the case of the Cartesian marching technique, the image source is

placed in a reflective position about the rough surface. Furthermore, the location

of the source is not central to the origin of the coordinate system. We were able

to place the initial source at any position relative to the rough surface. In the

curvilinear case, there appears to be no easy way to satisfy the boundary

condition on the rough surface unless the surface is flat. However, there may be

other ways of using a bilinear transformation to handle the boundary condition

on the rough surface. For example, we know that in complex coordinates, the

bilinear transformation maps a circle to a line. Thus, we could solve the problem

for the field distribution on a corrugated circle around the source. The

corrugated circle would represent the randomly rough surface. Then under a

bilinear transformation, we can map the solution to a rough surface along a line.

• Extending marching technique for fully three-dimensional propagation

In Chapter 2, we discussed the intriguing possibility of a fully three-dimensional

marching technique. This would be a powerful method that could handle rough


10-4

terrain and arbitrary refractive index profiles. As an added benefit, the

propagation algorithm is incredibly fast in this form and can be used to simulate

many situations nearly instantly. The critical point in the derivation of the

marching technique relies on the chain rule and substituting one side of the

parabolic equation for the other. However, in three dimensions these terms are

coupled and this substitution appears to not be possible.

10.2 Recommendations for Follow-on Work Related to Chapters 5-9

In the second half of the thesis, we developed a solution for the mean field

scattering from a randomly rough surface in simplified, analytical form using the

Laplace transform method. This result is useful for calculation of higher

moments. For instance, the mean field is openly used in the second moment,

where one needs it not only for the cross-moment between fields at two different

places, but also between fields at different frequencies. Also, the mean field itself

is often needed when calculating how much non-fluctuating signal remains after

varying ranges of propagation. In general, the mean field is needed whenever any

higher central moment is of interest; therefore, the mean field would not be

necessary for computing a quantity such as the scintillation index.

In this thesis, the analytic solution was compared to the known result in the

limiting case of a flat surface and shown to correspond identically. Furthermore,

the analytic approach was analyzed using a numerical simulation for computing

the scattered field. The comparison between results from the Laplace transform

solution and the numerical solution was necessary to justify the primary

assumption of the analytic approach: the derivative of the field at the surface

and the Green function evaluated at the surface can be ensemble averaged

independently.

The Laplace transform method is quite general for the solution of integral

equations and was used here to formulate solutions to both one- and two-surface

problems under the paraxial wave approximation. That is, the integral equation

method still applies to conditions of forward scatter with gently varying envelopes

in the direction of propagation. The statistics of the rough surfaces were taken to

be Gaussian, but the method could be extended to accommodate other statistics


10-5

and surface correlation functions. Because the method is so general in

application, there are several suggested areas for future work:

• Solution to the two-surface problem using a more realistic boundary

condition on the bottom surface

Chapter 7 describes the two-surface solution using Laplace transforms for mixed

boundary conditions in the sense that one surface was Neumann and the other

Dirichlet. However, to most realistically represent a surface such as the ocean

floor, it would be necessary to use something other than these two boundary

conditions. Certainly this is possible using the Laplace technique, but it

considerably complicates the algebraic solution in the Laplace domain, which

must then be inverted to attain the solution. However, the theory of Chapter 7

does provide a starting point.

• Analytic solutions with Gaussian statistics and polynomial approximations

to other autocorrelation functions

In Chapter 6, we provided an approximate expression for an exponential

autocorrelation function. The goal was to find a Laplace transform

representation for the surface Green function, which depended on the

autocorrelation function of the surface. For the method to be used for surfaces

with other autocorrelation functions, a simplified representation of that function

will be needed. This is because the autocorrelation function works itself into the

square root of a denominator, and complicated functions can not easily be

Laplace transformed analytically. However, with a polynomial approximation to

the autocorrelation function, an analytic Laplace transform becomes more

feasible.

• Application of the method to solving integral equations that are not being

averaged

The use of Laplace transforms for solving integral equations is a well known

technique. Their main value in wave propagation stems from the bounds of


10-6

integration on the convolution integral; the bounds are not suited to standard

Fourier transform and convolution theorem application. However, in the Laplace

domain, the convolution theorem is directly applicable. Therefore, whether we

seek a solution to a mean field or the actual field, we can still express the solution

algebraically in the Laplace domain. The Laplace transform for the Green

function and derivative of the Green function are relatively straightforward in the

non-averaged case. Furthermore, the Laplace transform of the Green function for

the non-isovelocity medium is easier to perform analytically.

• Investigation of the method to the solution of three-dimensional integral

equations

Many of the interesting aerospace problems today require solving the three-

dimensional wave equation. For example, calculating the radar cross-section of a

body from various angles requires a solution to the three-dimensional wave

equation. Also, many surface scattering phenomena concern field interactions

with three-dimensional surfaces such as the ground or the sea floor. Thus, there

is no shortage of applications for a fully three-dimensional representation. Since

most modern applications are now solved numerically, there is benefit in seeking

representations of the equations that can be solved more efficiently in terms of

processing resources. Using the Laplace technique on one or more of the

dimensions may have advantages in processing time.

• Investigation of the method for two surfaces with a varying refractive

index profile as discussed in section 9.2

This would be the most complicated 2-D scenario we could hope to model.

Coupling the results of Chapter 7 describing the equations for the rough

waveguide with the basic theory outlined in section 9.2, we have the ability to

handle the rough waveguide propagation problem for a non-isovelocity medium.

This would give us a powerful approach to realistically modeling situations such

as the ocean with salinity and temperature variations that lead to more complex

speed of sound fluctuations as a function of depth.


11-1

11 Appendices


11-2

11.1 Appendix A

The Parabolic Wave Equation in Cartesian Coordinates

Let ( ),p x z be the complex acoustic pressure field, containing both amplitude and

phase. The Helmholtz equation for the monochromatic field in the space above

the surface ( )z S x> is

( )2 2

2 22 2 0 0p p k n p

x z∂ ∂+ + =∂ ∂

(A.1)

Here, 2 2

2 2 00

0

ck n

c cω =

with 0k the wavenumber, n the refractive index, and ω

the angular rate of change of the time-dependent portion of the acoustic field.

This time-dependent portion has been separated out from the full wave equation

to yield the Helmholtz equation.

The basic assumption in the paraxial formulation is that the propagation occurs

principally in the forward direction so that

( ) ( ) 0, , ik xp x z E x z e= (A.2)

( ),E x z is referred to as the slowly varying envelope and 0ik xe is the rapidly-

varying spatial oscillation due to forward propagation. In the parabolic

approximation,

2

2 0Ex

∂ ≈∂

(A.3)

When we substitute (A.2) into (A.1) and use the approximation of (A.3), we get

( )2

202

0

12 2

ikE i E nx k z

∂ ∂= + −∂ ∂

(A.4)


11-3

When the refractive index is defined as

( ) ( )0 01, , , 1n x z n n x z n= + = (A.5)

then the paraxial equation of (A.4) is often used in this form:

( ) ( ) ( ) ( )

2

2 0 10

, ,, ,

2E x z E x zi ik n x z E x z

x k z∂ ∂= +

∂ ∂ (A.6)

when the ( )21nΟ term is neglected.


11-4

11.2 Appendix B

The Image Method for Paraxial Wave Propagation with 1R =

The method consists of replacing the space below the surface ( )z S x< with an

auxiliary medium in which the field also propagates. The fields above and below

the surface must satisfy

( ) ( ), ,R I

RE x z E x Z= (B.1)

where R

E is the field in the real medium, I

E is the field in the image medium, R

is the reflection coefficient, and ( )2Z S x z= − . From (B.1), we see there are two

ways to state the differential of the field in the image medium: ( ),I

dE x Z or

( ),R

RdE x z .

Expanding out ( ),I

dE x Z in terms of ( ),R

E x z :

( )( )

( ),

, 2R R RI

E x z E EdE x Z R dx R S x dx R dZ

x x z∂ ∂ ∂′= + −

∂ ∂ ∂ (B.2)

or in terms of ( ),I

E x Z only:

( )( ),

, I II

E x z EdE x Z dx dZ

x Z∂ ∂

= +∂ ∂

(B.3)

Comparing (B.2) and (B.3) shows

RIEE

RZ z

∂∂= −

∂ ∂ (B.4)

and therefore

22

2 2RI

EER

Z z∂∂

=∂ ∂

(B.5)


11-5

Looking at the other terms, (B.2) and (B.3) imply

( )

( )( ) ( ), ,,

2 R RIE x z E x zE x z

RS x Rx z x

∂ ∂∂ ′= +∂ ∂ ∂

(B.6)

The first term on the right-hand side of (B.6) can be replaced by (B.4) and the

second term can be replaced by the parabolic wave equation. At this point we

use (B.1) and (B.5) to get a relation in the image medium:

( )

( )( ) ( )

( ) ( )2

2 0 10

, , ,2 , ,

2I I I

I

E x Z E x Z E x ZiS x ik n x Z E x Zx Z k Z

∂ ∂ ∂′= − + +∂ ∂ ∂

(B.7)

Equations (A.6) and (B.7) now provide the basic equations that described

propagation in both the real and image media.

By using one equation with slightly different definitions of terms depending on

the media, we can handle the rough surface problem in one equation. Let

( ) ( ) ( ) ( )

2

2 00

, ,, ,

2H x z H x zi ik N x z H x z

x k z∂ ∂= +

∂ ∂ (B.8)

with

( ) ( ) ( )

( ) ( ) ( ) ( )[ ]{ } ( )0

, , ,

, , exp 2 ,

H x z E x z z S x

H x z E x z ik S x z S x z S x

= >

′= − < (B.9)

and

( ) ( ) ( )

( ) ( )( ) ( ) ( )[ ] ( )

1

1

, , ,

, ,2 2 ,

N x z n x z z S x

N x z n x S x z S x z S x z S x

= >

′′= − − − < (B.10)

The initial condition (typically Gaussian, quasi-point source) must conform to

(B.9). That is,

( ) ( ) ( )

( ) ( ) ( ) ( ){ } ( )

0 0 0

0 0 0 0 0

0, 0, , 0

0, 0, exp 2 0 0 , 0

H z E z z S

H z E z ik S z S z S

= >

′= − < (B.11)


11-6

Thus, we use the paraxial equation in both media with different definitions for

refractive index ( ),N x z depending on the medium. The field is propagated

according to (B.8). The field above the surface is retained as the solution. This

derivation is more fully discussed in [51].


11-7

11.3 Appendix C

The Parabolic Wave Equation in Cylindrical Coordinates

Here we derive the parabolic wave equation in cylindrical coordinates. We start

with the Helmholtz form of the wave equation in cylindrical coordinates. In (C.1)

we make the time-separable assumption for the field and work with the spatial

components alone.

( )( )

( ) ( ) ( )

22

2 2 0

1 1 ,

, , exp

u ur k n r ur r r r

U r u r i t

∂ ∂ ∂ θ∂ ∂ ∂θ

θ θ ω

+ = −

= (C.1)

The general form of the solution is well known and expressed through standard

separation of variable techniques as in (C.2).

( ) ( ) ( )0, expp pp

u r a H k nr ipθ θ= ±∑ (C.2)

where ( )0pH k nr is a Hankel function, p is an integer eigenvalue. The solution

will be periodic in θ with 2π radians. The constant terms, pa , are determined

from the boundary conditions.

( ) ( ) ( )( ) ( ) ( )

10 0 0

20 0 0

p p p

p p p

H nk r J nk r iN nk r

H nk r J nk r iN nk r

= +

= − (C.3)

These combinations of Bessel functions represent outgoing and incoming wave

components for circular geometry and the incoming component is neglected in the

assumption of forward scatter. These functions have a singularity at the origin.

At reasonable distances from the source, the Hankel function has an asymptotic

form

( ) ( ){ }1/2

10 0

0

2 exp 2 14pH nk r i nk r p

nk rπ

π

≈ − + (C.4)


11-8

The conditions for (C.4) to hold are

0 01,nk r nk r p (C.5)

Since one of these conditions essentially reduces to 0

1rλ , we interpret this as

a requirement to be many wavelengths from the source.

Combining (C.2) and (C.4), we can write

( ) ( )( )0exp

, ,ik r

u r E rr

θ θ= (C.6)

where ( ),E r θ contains the summation with respect to p and is valid for all θ .

With (C.6) used in (C.1), we have to look carefully at two of the terms. First, we

see that

( )2

22 2 20 0 0

0

1 1 12 1 exp4

u E Er ik k E ik rr r r r r k r r

∂ ∂∂ ∂

∂ ∂ = + − − ∂ ∂ (C.7)

Just as with the Cartesian case, the paraxial approximation implies that the

curvature of the envelope in the forward direction is small and can be neglected:

2

20

E Ekr r

∂ ∂>>∂ ∂

(C.8)

Furthermore, we are limited to the case of several wavelengths from the source at

a minimum, and 2 20

1 04k r

≈ . With ( ) ( )0 1, ,n r n n rθ θ= + and neglecting ( )21nΟ

terms:

( )2

2 2 0 10

,2

E i E ik n r Er k r

θθ

∂ ∂= +∂ ∂

(C.9)


11-9

11.4 Appendix D

The Integral Equation

The classic Sturm-Liouville eigenvalue problem is the one-dimensional equivalent

of the topic here. It defines a class of solutions for self-adjoint operators, L ,

through use of Lagrange’s identity.

Recall the vector identities below:

( )

( )

2u u

a a a

∇ = ∇⋅ ∇

∇⋅ = ∇⋅ + ∇ ⋅B B B (D.1)

where u and a are scalars and B is a vector.

Then it follows

( )

( )

2

2

u v u v u v

v u v u v u

∇⋅ ∇ = ∇ + ∇ ⋅∇

∇ ⋅ ∇ = ∇ + ∇ ⋅∇ (D.2)

By subtracting these and integrating over the entire two-dimensional region R ,

( ) ( )2 2

R R

u v v u dxdy u v v u dxdy∇ − ∇ = ∇⋅ ∇ − ∇∫∫ ∫∫ (D.3)

and then applying the divergence theorem to the right hand side:

( ) ( )2 2

R

R

dxdy ds

u v v u dxdy u v v u ds

∇⋅ = ⋅

∇ − ∇ = ∇ − ∇ ⋅

∫∫ ∫

∫∫ ∫

A A n

n (D.4)


11-10

Consider a Helmholtz equation of the form below with L the appropriate linear

operator.

( )

[ ]

2 2

2 2 0

k

E E k E

= ∇ +

= ∇ + =

L

L (D.5)

If we take u as E and v as the Green function, G , then Green’s formula implies

that

[ ] [ ]

( )E G G E dxdy

E G G E ds

− = ∇ − ∇ ⋅

∫∫∫ n

L L (D.6)

with G satisfying

[ ] ( )0 0,G x x y yδ= − −L (D.7)

The second term on the left-hand side is zero by the defining equation and

therefore Green’s formula becomes

( ) ( )0 0 0 0( , ) ,E x y E x x y y dx dy E G G E dsδ= − − = ∇ − ∇ ⋅∫∫ ∫ n (D.8)

and some additional simplifications make this expression identical to equation 5.1.


11-11

11.5 Appendix E

Relation of the Laplace Transform to the Fourier Transform

It can be shown that the Laplace transform is just a special case of the Fourier

Transform. And with careful choice of parameters, we can show how this will

relate to the ensemble average of the full field.

Recall the definition of the Fourier transform pair

( ) ( )

( ) ( )

1ˆ2

ˆ

i x

i x

F e f x dx

f x e F d

ω

ω

ωπ

ω ω

∞

−∞∞

−

−∞

=

=

∫

∫ (E.1)

Suppose we are looking to transform a function such that

( )

( ) ( )

0 0

2 0x

g x x

g x f x e xγπ −

= <

= > (E.2)

The exponential is chosen so that ( )g x automatically decays rapidly enough to

meet the convergence requirements as x → ∞ . For this function, the Fourier

transform pair are

( ) ( ) ( )

( ) ( )

0

ˆ

ˆ2

i x

x i x

F f x e dx

f x e F e d

ω γ

γ ω

ω

π ω ω

∞− − +

∞− −

−∞

=

=

∫

∫ (E.3)

And we let

( ),i d idλ γ ω λ ω= − = − (E.4)


11-12

and ( ) ( )ˆ ˆ ˆF G Gi

λ γλ ω − ≡ = − we can construct the Laplace transform relation

( ) ( )

( ) ( )

0

ˆ

1 ˆ2

x

ix

i

F f x e dx

f x F e di

λ

γλ

γ

λ

λ λπ

∞−

+ ∞

− ∞

=

=

∫

∫ (E.5)

So we have shown that ( )F λ is the Fourier transform of

( )

( ) ( )

0 0

2 0x

g x x

g x f x e xγπ −

= <

= > (E.6)


11-13

11.6 Appendix F

Here we ensemble average the derivative of the Green function that relates the

field at any point to a point source on one of the surfaces. This was discussed in

section 7.4.3 and stated as equation 7.20:

( )( ) ( ) ( )

( )( ) ( )( )( )

( )2 2

0

2

02

0

2 2

1 1 1;2 2 2

exp ik z S x S x

x x

ikG ix x z S xz k x x x x

z S x dSσ

π πσ

∞

−∞

′ ′−−

′−

∂ ′ ′< − − >= − ×′ ′ ′∂ − −

′− ∫

(F.1)

To simplify the notation define the following constants:

( ) ( )0

20

1 1 12 2 2

ikiAk x x x xπ πσ

= −′ ′− −

, ( )

0

2

ik

x xB

′−

= , and 2

1

2C

σ= so that

( ) ( )( ){ }22expG A z S CS B z S dSz

∞

−∞

∂ ′ ′ ′ ′< >= − − − −′∂ ∫ (F.2)

Completing the square gives

( ) ( ) ( ) ( ){ }2 21 expB

C BBz BZC BC B S

G Ae z S dSz

−

∞+

−−∞

′− +∂ ′ ′< >= − − ′∂ ∫ (F.3)

Let ( )

( )2

022

x x ikD C B

x xσ

σ′− −

= − =′−

and BzDX DS ′= + so that 2

X BzD D

S ′ = − .

Substituting into (F.3) gives

( ) ( ) ( )( )2 22

2

11

BD

Bz XA B XD DD

G e z e dXz

∞+ −

−∞

+∂< >= −

′∂ ∫ (F.4)


11-14

We can think of this as two separate integrals. The integral over the second term

with 2XX e

D− will be zero because it is an odd function. So we are left to evaluate

( ) ( )2 22

2

11

BD

Bz XA BD D

G e z e dXz

∞+ −

−∞

+∂< >=

′∂ ∫ (F.5)

We can evaluate ( ) ( )( )20

20

12

BD

ikB

x x ik σ+ =

′− −, ( ) ( )

( )( )2 20

1 BD

x x

x x ik σ

′−+ =

′− −, and

( ) ( )( )

( )2

022

0

21 1 12 2 2

x xikA iD k x x x x x x ik

σπ σπσ

′− = − ′ ′ ′ − − − − .

This integral contributes π so that the answer upon plugging back in our terms

and combining where possible gives

( )

( ) ( )( ) ( )( )2

0 02 2 2

0 0 0 0

;

1 1 exp2 2 2

G x x zz

ik z ik zik x x ik x x ik x x ikπ σ σ σ

′∂ −< > =

′∂ − ′ ′ ′− − − − − −

(F.6)

This answer should be confirmed by differentiating ( );G x x z′< − > directly as

shown in (F.7)a

( ) ( )

( )( )

( )( ) ( )( )( ) ( )( )

20

20

20 0

20

20 0

2 20 0 0

21 12 2

21 12 2

;

;

ik z

x x ikik x x ik

ik zik z x x iki

k x x ik x x ik

G x x z e

G x x ze

z

µπ σ

σπ σ σ

′− −′− −

′− −′ ′− − − −

′< − >=

′∂ < − >=

∂

(F.7)

They differ by a factor of minus one because of the difference of differentiating

with respect to and z z ′ in (F.7) and (F.6).


11-15

11.7 Appendix G

Here we will perform the integral for the ensemble average of the derivative of the

Green function between two points on a single surface with Gaussian statistics.

From the definition in equation 7.24 we have:

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )( )( ) ( )

2 22

0

2 2

02 2

0

2

2 2 1

1 1 1;2 2 2 1

expS SS Sik S x S x

x x


S x S x dSdSρ

σ ρ

π πσ ρ

∞ ∞

−∞ −∞

′ ′′− +′−−

′− −

∂ ′ ′< − − > = − ×′ ′ ′∂ − − −

′ ′− ∫ ∫

(G.1)

where and S S ′ are two heights on the rough surface at the points and x x ′ ,

which are correlated over a correlation length L with autocorrelation function is

( )( )x xLx x eρ

− − ′′− = .

To simplify the notation define the following constants:

( ) ( )0

2 20

1 1 12 2 2 1

ikiAk x x x xπ πσ ρ

= −′ ′− − −

, ( )

0

2

ik

x xB

′−

= , ( )2 2

1

2 1C

σ ρ−= ,

and ( )2 22 1

D ρ

σ ρ−= so that

( ) ( ) ( ) ( ){ }2 2 2expG A S S B S S C S S D S S dSdSz

∞ ∞

−∞ −∞

−∂ ′ ′ ′ ′ ′< >= − − + − ⋅

′∂ ∫ ∫ (G.2)

Notice that a shorthand notation is used to express ( );G Gx x S Sz z

∂ ∂′ ′< − − > ≡ < >′ ′∂ ∂

.

By completing the squares in the exponent, equation (G.2) can be rewritten as:

( ) ( )( ) ( )( )( )2 2

2 22 2 2 2exp 2 D S S D S SG A S S B C C dSdSz

∞ ∞′ ′

−∞ −∞

∂ ′ ′< > = − − + + − + − ′ ∂ ∫ ∫ (G.3)


11-16

With a variable transformation into sum and difference coordinates:

( )( )

12

12

S S

S S

ξ

η

′= −

′= + (G.4)

and S S ′ are expressed from (G.4) as

2

2

S

S

ξ η

ξ η

+=

− +′ = (G.5)

In changing to the sum and difference coordinates, we need to calculate the

differential element of area change via the Jacobian:

1 12 2

1 12 2

1

S S

S SJ

ξ η

ξ η

∂ ∂

∂ ∂′ ′∂ ∂

∂ ∂

= = = −

(G.6)

Then (G.3) can be rewritten as

( )( ) ( )( )( )0

2 22 2

0

2 exp 2 D DG A B C C d dz

ξ η ξ ξ η∞

−∞

∂ < > = − + − + ′∂ ∫ ∫ (G.7)

We use one final rescaling to get the integral into appropriate form:

( ) ( )2 2

02 22 1

20

exp exp 0D DA

C C B

G d dz

ξ ξ ξ η η∞

+ − −−∞

∂< > = − − =′∂ ∫ ∫ (G.8)

The bounds of integration get compressed to the same point and the answer is 0.

One can argue this solely from symmetry of the integral as well.


11-17

11.8 Appendix H

Here we will perform the integral for the ensemble average of the Green function

between two surfaces with Gaussian statistics. These surfaces are allowed to

have possibly different variances. Furthermore, it is assumed that the mean of

the top surface is zero while the mean of the bottom surface is the average depth,

H . Let the lower surface then be ( ) ( )j jS x H S x′= + and the upper surface

( )kS x ′ . From the definition in equation 7.26a we have:

( ) ( )( ) ( )

( )( )

( )2 22 2 2

0

2 2

0

2 2

1 1 1;2 2 2

exp j j jk k k

j k

j kj k

jk

ik S S S S H

x x

iG x x S x S xk x x

dS dSσ

σ

π πσ σ

σσ

∞ ∞

−∞ −∞

− + −−

′−

′ ′< − − >= ×′−

∫ ∫

(H.1)

Note the two surfaces are assumed to be uncorrelated.

Define ( )0

1 1 12 2 2 j k

iAk x xπ πσ σ

=′−

, ( )

0

2

ik

x xB

′−= , 2

12 j

Cσ

= , 2

12 k

Dσ

= , and

2j

HEσ

= then

( ) ( )( )

( ) ( ){ }2

22 2 2 2

;

expj

j k

H

j k j k j k jS C B S D B BS S ES

G x x S x S x

Ae dS dSσ∞ ∞−

−∞ −∞

− − −

′ ′< − − > =

− − +∫ ∫ (H.2)

Let ( ) jX C B S= − with ( )

( )

20

22j

j

x x ikC B

x xσ

σ′− −

− =′−

and ( ) kY D B S= − , with

( )( )

20

22k

k

x x ikD B

x xσ

σ′− −

− =′−

, ( )( )( )0 2

022 j j

x xE HXx x ikC B σ σ

′−= =

′− −− and


11-18

( ) ( )

0

2 20 0

22 1k j

k j

ikBFC B D B x x ik x x ik

σ σ

σ σ

= = − − ′ ′− − − −

then

( ) ( ){ }202

22 2 2

0exp; j

H XA

C B D BG x x X Y e X X Y FXY dYdXσ

∞ ∞− +

− −−∞ −∞

−′< − − > = − − −∫ ∫ (H.3)

We continue with the algebra expressly written:

( ) ( ) ( ) ( )( ){ }202

22 222

0 2 2exp; j

H XA FX FX

C B D BG x x X Y e X X Y dYdXσ

∞ ∞− +

− −−∞ −∞

′< − − > = − − + − +∫ ∫

( ) ( ) ( ) ( )( )202

22 222

0 2 2exp; expj

H XA FX FX

C B D BG x x X Y e X X dX Y dYσ

∞ ∞− +

− −−∞ −∞

′< − − > = − − + − + ∫ ∫

( ) ( ) ( )( )202

22 22

0 2; expj

H XA FX

C B D BG x x X Y e X X dXσ π

∞− +

− −−∞

′< − − > = − − − ∫

( ) ( )( )( )202

22 2 2 2

0 02; exp 1 2j

H XA F

C B D BG x x X Y e X XX X dXσπ

∞− +

− −−∞

′< − − > = − − − + ∫

( ) ( )( ) ( )

202

2222

22

22 21 1

0 02 11; exp 1 1j

FF

H XA F

C B D BG x x X Y e X X X dXσπ

∞− +

− − −−−∞

′< − − > = − − − + − ∫

( ) ( )( )

( )( )

2 2022

2222

22

22 21

0 02 11; exp 1

Fj

FF

H XA F

C B D BG x x X Y e X X X dXσπ

∞− +

− − −−−∞

′< − − > = − − − − ∫

( ) ( ) ( )( )

202 2

22

2

22

1 221 1

02 1; exp 1

Fj

F

H XA F

C B D BG x x X Y e X X dX

σπ

− + ∞−

− − −−∞

′< − − > = − − −

∫

( )( )

( ) ( )202 2

22

2

22

1211

1; exp

Fj

F

H XA

C B D BG x x X Y e Z dZ

σπ

− + ∞−

− −−−∞

′< − − > = − ∫

( )( )

( )202 2

22

2

22

1

11

1;

Fj

F

H XA

C B D BG x x X Y e

σπ

− +−

− −−′< − − > =


11-19

We evaluate the non-exponential term below

( ) ( )( )2 20 0

2 1 12 22 2

j k

A ikF F C B D B x x ikπ π σ σ

− = − − + − − ′ − − +

(H.4)

and the exponential term

( )( ) ( )

( ) ( )( )( )

2 20 0

22 200

2 22

22 202

11

2 21j k

jj kF

j j

ik x x ik x x x x

ik x xx x ik x x

H HXσ σ

σσ σσ σ

− − − −′ ′ − ′= − −

− − ′− + − −′ ′− −

−

( )( )

( )2 2200

2 2 20 0

2 2

2 211 12 2

j kk

j jk kj j

ik x xik x x

ik x x ik

H H σ σσ

σ σ σ σσ σ

+ − − ′− − ′= − − = −

+ − − +′ ( )( )( )

( ) ( )

20

2 2 20

220

2 2 20

2

k

j k

j

j j k

ik x x

x x ik x x

ikHx x ik

σ

σ σ

σσ σ σ

− − ′−

− − + − −′ ′

= −

′− − + ( ) ( )

20

2 20

2j k

ikHx x ik σ σ

= −

′− − +

(H.5)

The final answer is

( )

( ) ( )( ) ( ) ( )2

02 22 2

0 00

;

1 exp2 2

j k

j kj k

G x x S S

iki Hk x x ikx x ikπ σ σσ σ

′ ′< − − > =

′− − +′− − +

(H.6)


11-20

11.9 Appendix I

Here we ensemble average the derivative of the Green function that relates the

field on one surface due to a point source on the other surface. This was

discussed in section 7.4.5 and stated by definition as equation 7.26b:

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( )( )

( )2 22 2 2

0

2 2

0

0

2 2

1 1 1;2 2 2

' exp j k j k k j

j k

j kj k

j k j kik S S S S H

x x


S x S x dS dSσ σ

σ σ

π π σ σ

∞ ∞

−∞ −∞

− + −−

′−

∂ ′ ′< − − > = − ×′ ′ ′∂ − −

−

∫ ∫

(I.1)

Define ( )10

1 1 12 2 2 j k

iAk x xπ πσ σ

=′−

, ( )0

2

ikA

x x= −

′−,

( )0

2

ik

x xB

′−= , 2

12 j

Cσ

= ,

2

12 k

Dσ

= , and 2j

HEσ

= then

( )

( ) ( ) ( ){ }2

2 2 221 2 2expj

H

j j j j jk k k kS B C S B D BS S ES

G x xz

AAe S S dS dSσ∞ ∞−

−∞ −∞

− − −

∂ ′< − >=′∂

− + +∫ ∫

Let ( ) jX C B S= − with ( )( )

2022

j

j

x x ikC B

x x

σσ

′− −− =

′− and ( ) kY D B S= − ,

with ( )( )

2022

k

k

x x ikD B

x xσ

σ′− −

− =′−

, ( )

( )( )0 20

2 2j j

x xE HXC B x x ikσ σ

′−= =

− ′− − and

( ) ( )0

2 20 0

22 1jk

jk

ikBFC B D B x x ik x x ik

σ σ

σ σ

= = − − ′ ′− − − −

then


11-21

( )

( ){ }2

2 02

2 20

2 expjX

AC B D B

H

G x xz

X Ye X X Y FXY dYdXC B D B

σ∞ ∞

− +

− −−∞ −∞

−

∂ ′< − > =′∂

− − − − − −∫ ∫ (I.2)

This can be factored into

( )

( ) ( ) ( )( )2

2 01 2

22 22

2 202 expj

XA A FX FXC B D B

H X Y

B C B D

G x xz

e X X Y dXdYσ∞ ∞

− +

− −−∞ −∞

−− −

∂ ′< − > =′∂

− − + − + ∫ ∫ (I.3) or in two integrals explicitly, (I.3) becomes

( )

( ) ( ) ( )( )

( ) ( ) ( )( )

22 01 2

22 01 2

2

2

2 222 20

2 222 20

2

2

exp

exp

j

j

XA A FX FXC B D B

XA A FX FXC B D B

H

H

X

C B

Y

D B

G x xz

e X X Y dXdY

e X X Y dXdY

σ

σ

∞ ∞− +

− −−∞ −∞

∞ ∞− +

− −−∞ −∞

−

−

∂ ′< − > =′∂

− − + − +

− − − + − +

∫ ∫

∫ ∫

(I.4)

Taking the second integral in (I.4) first

( ) ( ) ( )( )

( ) ( )[ ] ( )( ) ( )( ) ( ) ( )( )[ ]

( ) ( )

2 2 2 21 1

2 2 2 2 20

22 01 2

22 01 2

22 01 2

2

2

2

2 222 20

2 220

2

2

2

exp exp exp

exp

exp

FX FX FX FX FX

D B D B

j

j

j

XA A FX FXC B D B

XA A

C B D B

XA A FXC B D B

H

H

H

X X dX Y Y Y dY

Y

B De X X Y dXdY

e

e X X

σ

σ

σ

∞ ∞

− −

−∞ −∞

∞ ∞− +

− −−∞ −∞

− +

− −

− +

− −

− − + − + − + + − +

−−

− − + − + =

− − +

∫ ∫

∫ ∫

( ) ( )( )212 2expFX FX

D BdX Y dY∞ ∞

−−∞ −∞

− + ∫ ∫

Performing the integral with respect to y and gives π , and then completing the

square with respect to x gives

( ) ( ) ( )( )

22 2 0

1 2 22

2

2 1 221 1 1

2 2 01

2 exp 1Fj

F

XA A FX F

C B D B D B

H

e X X dXσ π∞− +

−− − − −

−∞

− − − ∫ (I.5)


11-22

To finish this calculation, we must make a variable substitution that will make

the symmetry clearer.

Let ( )( )

( )( ) ( )22 2

22 2

2 21 1 12 20 011 1

1 , 1 ,FF F

F FZ X X dZ dX X Z X−− −

= − − = − = + then

(I.5) becomes

( )( ) ( )

( ) ( )

22 2 0

1 2 222 2

22 2

2 1211 1 1

02 11 1

2 expFj

FF F

XA A F

C B D B D B

H

e Z X Z dZσπ∞− +

−− − − −− −

−∞

− + − ∫ , which

evaluates to

( )

( ) ( )( )

[ ]2

2 2 01 2 2

2222

2 1

11 102 11

2 Fj

FF

XA A F

C B D B D B

H

e Xσπ π− +

−− − − −−

− (I.6)

Now for the first integral of the two in (I.3)

( ) ( ) ( )( )2

2 01 2

22 22

2 202 expexpj

XA A X FX FXC B D B C B

H

e X X dX Y dYσ∞ ∞

− +

− − −−∞ −∞

− − + − + ∫ ∫

We can do the y integral immediately

( ) ( ) [ ]2

2 01 2

22 2

202 expj

XA A X FXC B D B C B

H

e X X dXσ π∞

− +

− − −−∞

− − + ∫

( ) ( )( )

22 2 0

1 2 22

2

2 1 221 1

2 01

2 exp 1Fj

F

XA A X F

C B D B C B

H

e X X dXσπ∞− +

−− − − −

−∞

− − − ∫

Let ( )( )

( )( ) ( )22 2

22 2

2 21 1 12 20 011 1

1 , 1 ,FF F

F FZ X X dZ dX X Z X−− −

= − − = − = +

( )( )

( ) ( )

22 2 0

1 2 222 2

22 2

2 1211 1 1 1

011 1

2 expFj

FF F

XA A

C B D B C B

H

e Z X Z dXσπ∞− +

−− − − −− −

−∞

+ − ∫

The first term will integrate to zero and the second term is just a constant.

( )( )

( )[ ]

22 2 0

1 2 222

22

2 1

11 1 1011

2 Fj

FF

XA A

C B D B C B

H

e Xσπ π− +

−− − − −−


11-23

Combining the two results gives

( )

( )( )

22 2 0

1 2 0222

22

2 1

11 1211

2 Fj

FF

XA A X F

C B D B C B D B

H

e σπ− +

−− − − −−−

− (I.7)

We evaluate the expressions which appear new from the derivation in

Appendix E:

( )

( )( )( )

( )( )( ) ( )( )

02

2

2 20 0

1 2 20

2F

X j k

j j k

ik x x ik x xHx x ik x x

σ σσ σ σ−

′ ′− − − − = ′− ′+ − −

( )( )( )( )

( )( )( )

02

2

12 221

00

2F

X jFC B D B

kj

x xx x

ik x xik x x

σσσ− − −

′′ − −− − = = ′′ − −− −

and come up with the final expression

( )( ) ( )( )

( ) ( ) ( ) ( )

2 20 0

20 0

2 2 2 20 0

1;2

exp2

j k

j k

j jk k

G ix x S Sz k x x ik

ik ikHHx x ik x x ik

π σ σ

σ σ σ σ

∂ ′< − − >= ×′′∂ ′− − +

′ ′− − + − − +

(I.8)


12-1

12 Bibliography

M.J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and

Applications, Cambridge University Press (1997)

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover

Publishing (1964)

B. Alpert, G. Beylkin, R. Coifman and V. Rokhlin, Wavelet-like bases for the fast

solution of second-kind integral equations, SIAM J. Sci. Comput. 14, pp 159-

184 (1993)

B. Alpert, A class of bases in L^2 for the sparse representation of integral

operators, SIAM J. Math. Anal. 24, pp 246-262 (1993)

K. Attenborough, Acoustical impedance models for outdoor ground surfaces,

J. Sound and Vibration 99, pp 521-544 (1985)

C.T. Baker, The Numerical Treatment of Integral Equations, Claredon Press:

Oxford (1977)

F. Bass and I. Fuks, Wave Scattering from Statistically Rough Surfaces,

Pergamon Press: Oxford (1979)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from

Rough Surfaces, Pergamon Press: New York (1963)

G. Beylkin, R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical

algorithms I, Comm. Pure and Appl. Math. 44, pp 141-183 (1991)

G. Beylkin, On the representation of operators in bases on compactly supported

wavelets, SIAM J. Numer. Anal. 29, pp 1716-1740 (1992)

L. Brekhovskikh and Y. Lysanov, Fundamentals of Ocean Acoustics,

Springer-Verlag: New York (1982)

R. Burden and J. Faires, Numerical Analysis, 5th Edition, PWS-Kent Publishing:

Boston (1993)

Z. Chen, C.A. Micchelli and Y. Xu, The Petrov-Galerkin methods for second kind

integral equations II: multiwavelet scheme, Adv. Comput. Math. 7,

pp 199-233 (1997)

M. Collins, A self-starter for the parabolic-equation method, J. Acoust. Soc.

Am. 92, Issue 4, pp. 2069-2074 (1992)


12-2

W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for

pseudodifferential equations II: Matrix compression and fast solution,

Adv. Comput. Math. 1, pp 259-335 (1993)

W. Dahmen, Wavelet and multiscale methods for operator equations, Acta

Numerica, pp 55-228 (1997)

J. DeSanto, Green’s function for electromagnetic scattering from a random rough

surface, J. Math. Phys. 15, No. 3, pp 283-288 (1974)

D.F. Elliot and K.R. Rao, Fast Transforms: Algorithms, Analyses, Applications,

Academic Press: New York (1982)

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd Ed, Academic

Press: New York (1990)

R. Haberman, Elementary Applied Partial Differential Equations, Prentice-Hall

(1987)

R.D. Harding, Fourier Series and Transforms, Adam Hilger Ltd (1985)

Y. Hatziioannou, The Scattering of Electromagnetic Waves by Rough Surfaces,

Ph.D. Thesis, University of Cambridge (1996)

E. Hecht, Optics, 3rd Ed, Addison-Wesley (1994)

E.J. Hinch, Perturbation Methods, Cambridge University Press, p 20 (1991)

A. Iserles, A First Course in the Numerical Analysis of Differential Equations,

Cambridge University Press (1996)

A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic

Press: New York (1978)

A. Ishimaru, J.S. Chen, P. Phu and K. Yoshitomi, Numerical, analytical, and

experimental studies of scattering from very rough surfaces and backscattering

enhancement, Waves in Random Media 1, S91-S107 (1991)

I.A. Joia, Electromagnetic Propagation Through Controlled Turbulence, Ph.D.

Thesis, University of Cambridge (1993)

I.A. Joia, B.J. Uscinski, R.J. Perkins, et al., Intensity fluctuations in a laser beam

due to propagation through a plane turbulent jet, Waves in Random Media 7,

pp 169-181 (1997)

D. Jones, Methods in Electromagnetic Wave Propagation, Vols. I and II,

Clarendon Press: Oxford (1987)


12-3

C.C. Macaskill and B.J. Uscinski, Propagating in waveguides containing random

irregularities: the second moment equation, Proc. R. Soc. Lond. A377,

pp 73-98 (1981)

W.C. Meecham, On the use of the Kirchhoff approximation for the solution of

reflection problems, J. Rat. Mech. Anal. 5, pp 323-334 (1956)

C.A. Micchelli, Y. Xu and Y. Zhao, Wavelet Galerkin methods for systems of

multivariate integral equations, J. Comput. Appl. Math. 86, pp 251-270

(1997)

D.M. Milder, An improved formalism for wave scattering from rough surfaces,

J. Acoust. Soc. Am. 89, pp 529-541 (1991)

F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer-Verlag

(1973)

J.A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Adam

Hilger (1991)

A. Papoulis, Probability, Random Variables, and Stochastic Processes,

McGraw-Hill (1965)

T. von Petersdorff and C. Schwab, Wavelet approximations for first kind

boundary integral equations on polygons, Numer. Math. 74, pp 479-516 (1996)

T. von Petersdorff, C. Schwab and R. Schneider, Multiwavelets for second-kind

integral equations, SIAM J. Numer. Anal. 34, pp 2212-2227 (1997)

W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes In C;

2nd Edition reprinted, Cambridge University Press (1999)

A. Rathsfeld, A wavelet algorithm for the solution of the double layer potential

equation over polygonal boundaries, J. Int. Equations Appl. 7, pp 47-98

(1994)

A. Rathsfeld, A wavelet algorithm for the boundary element solution of a geodetic

boundary value problem, Comput. Methods Appl. Mech. Eng. 157, pp 267-287

(1998)

D. Roberts, Statistical Properties of Waves Diffracted by a Random Phase

Screen, Cambridge University Press (1984)

J.D. Sheard, Acoustic Propagation in Ice Covered Oceans, Ph.D. Thesis,

Cambridge University (1993)

M. Spivak and B.J. Uscinski, The split-step solution in random wave propagation,

J. Comput. and Appl. Math. 27, pp 349-361 (1989)


12-4

M. Spivak, A numerical approach to rough-surface scattering by the parabolic

equation method, J. Acoust. Soc. Am. 87, pp 1999-2004 (1990)

M. Spivak and B.J. Uscinski, Numerical solution of scattering from a hard surface

in a medium with a linear profile, J. Acoust. Soc. Am. 93, pp 249-254 (1993)

F.D. Tappert, The parabolic equation approximation method, Wave Propagation

and Underwater Acoustics, Lecture Notes in Physics, Vol. 70, eds. Keller and

Papdakis, Springer-Verlag: Heidelberg (1977)

E.I. Thorsos, The validity of the Kirchhoff approximation for rough surface

scattering using a Gaussian roughness spectrum, J. Acoust. Soc. Am. 83,

pp 78-92 (1988)

E. Thorsos and D.R. Jackson, The validity of the perturbation approximation for

rough surface scattering using a Gaussian roughness spectrum, J. Acoust.

Soc. Am. 86, pp 261-277 (1989)

B.J. Uscinski, The Elements of Wave Propagation in Random Media,

McGraw-Hill (1977)

B.J. Uscinski, The multiple scattering of waves in irregular media, Phil. Trans.

Roy. Soc. Lon. A 262, No. 1133, pp 609-643 (1968)

B.J. Uscinski, Numerical simulations and moments of the field from a point

source in a random medium, J. Mod. Optics 36, No. 12, pp 1631-1643 (1989)

B.J. Uscinski, Sound propagation with a linear sound-speed profile over a rough

surface, J. Acoust. Soc. Am. 94, pp 491-498 (1993)

B.J. Uscinski, Waves in Random Media, Lecture notes given at University of

Cambridge, DAMTP Part III (1994)

B.J. Uscinski, I.A. Joia, R.J. Perkins, et al., Optical properties of a planar

turbulent jet, Appl. Optics 34, No. 30, pp 7039-7053 (1995)

B.J. Uscinski and C.J. Stanek, Acoustic scattering from a rough sea surface: the

mean field by the integral equation method, Waves in Random Media 12

pp 247- 263 (2002)

B. Uscinski, A. Kaletzsky, C. Stanek and D. Rouseff, An Acoustic Shadowgraph

Trial to Detection Convection in the Arctic, Waves in Random Media (2002)

A.G. Voronovich, Wave Scattering from Rough Surfaces, Springer-Verlag (1994)

R. Wagner, Shadowing of randomly rough surfaces, J. Acoust. Soc. Am. 41,

pp 138-147 (1966)


12-5

R. Wagner and W. Chew, A study of wavelet for the solution of electromagnetic

integral equations, IEEE Trans. Ant. Propag. 43, No. 8 (1995)

D.P. Winebrenner and A. Ishimaru, Application of the phase-perturbation

technique to randomly rough surfaces, J. Opt. Soc. Am. A2, Issue 12

pp 2285-2294 (1985)


12-6

Endnotes

1 E. Hecht, Optics, 4th Ed, Addison-Wesley, p 87 (2002)

2 B. Uscinski, A. Kaletzsky, C. Stanek and D. Rouseff, An Acoustic Shadowgraph Trial to Detection

Convection in the Arctic, awaiting publication (2002)

3 F. Tappert, The parabolic equation approximation method, Wave Propagation and Underwater

Acoustics, Lecture Notes in Physics, Vol. 70, Springer-Verlag: Heidelberg (1977)

4 M. Spivak and B. Uscinski, Numerical solution of scattering from a hard surface in a medium with a

linear profile, J. Acoust. Soc. Am. 93, pp 249-254 (1993)

5 D.P. Winebrenner and A. Ishimaru, Application of the phase-perturbation technique to randomly rough

surfaces, J. Opt. Soc. Am. A2, Issue 12, pp 2285-2294 (1985)

6 E.I. Thorsos and D.R. Jackson, The validity of the perturbation approximation for rough surface

scattering using a Gaussian roughness spectrum, J. Acoust. Soc. Am. 86, pp 261-277 (1989)

7 E.I. Thorsos, The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian

roughness spectrum, J. Acoust. Soc. Am. 83, pp 78-92 (1988)

8 R. Wagner, Shadowing of randomly rough surfaces, J. Acoust. Soc. Am. 41, pp 138-147 (1966)

9 [1], pp 124-126 (2002)

10 D. Sheard, Acoustic Propagation in Ice Covered Oceans; Ph.D. Thesis, Cambridge University (1993)

11 Y. Hatziioannou, The Scattering of Electromagnetic Waves By Rough Surfaces; Ph.D. Thesis,

Cambridge University, pp 11-16 (1996)

12 W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes In C, 2nd Ed reprinted,

Cambridge University Press, p 836 (1999)

13 R. Burden and J. Faires, Numerical Analysis, 5th Ed, PWS-Kent Publishing: Boston, p 300 (1993)

14 A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University

Press, p 106 (1996)

15 [13], pp 625-647 (1993)

16 D.F. Elliot and K.R. Rao, Fast Transforms: Algorithms, Analyses, Applications, Academic Press:

New York (1982)

17 M. Spivak and B. Uscinski, The split-step solution in random wave propagation, J. Comput. Appl.

Math. 27, pp 349-361 (1989)

18 B. Uscinski, Sound propagation with a linear sound-speed profile over a rough surface, J. Acous. Soc.

Am. 94, p 493 (1993)

19 R. Horgan, The Tests of the 1+1 Propagation Model, DAMTP Cambridge, May (1999)

20 [11], pp 22-33 (1996)

21 F.D. Tappert, Wave Propagation and Underwater Acoustics, Vol. 70, Springer-Verlang: Heidelberg,

pp 224-284 (1977)

22 B. Uscinski, Numerical simulations and moments of the field from a point source in a random medium,

J. Mod. Optics 36, No. 12, pp 1632-33 (1989)


12-7

23 L. Brekhovskikh and Y. Lysanov, Fundamentals of Ocean Acoustics, Springer-Verlag: New York,

p 111 (1982)

24 B.J. Uscinski and C.J. Stanek, Acoustic scattering from a rough sea surface: the mean field by the

integral equation method, Waves in Random Media 12, pp 247-263 (2002)

25 [11], p 4 (1996)

26 [18], p 497 (1993)

27 [18], p 497 (1993)

28 B.J. Uscinski, private communications, Feb (2001)

29 [11], p 30 (1996)

30 [24], p 252 (2002)

31 A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, p 100 (1965)

32 [24], p 253 (2002)

33 B.J. Uscinski, private communications, March (2001)

34 M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover Publishing, pp 297-305

(1964)

35 E.J. Hinch, Perturbation Methods, Cambridge University Press, p 20 (1991)

36 [35], p 32 (1991)

37 M.J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge

University Press, p 239 (1997)

38 B.J. Uscinski, private communications, December (2000)

39 B.J. Uscinski and M. Spivak, private communications, March (2001)

40 K. Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press: New York (1990)

41 J. Peixoto, Recitation Notes, September (1999)

42 B.J. Uscinski and C.J. Stanek, The solution by the integral equation method, Waves in Random Media,

p 7 (2002)

43 D. Roberts, Statistical Properties of Waves Diffracted by a Random Phase Screen, Cambridge

University Press (1984)

44 [43], (1984)

45 R. Haberman, Elementary Applied Partial Differential Equations, Prentice-Hall, pp 320-325 (1987)

46 [45], pp 413-414 (1987)

47 F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer-Verlag, pp 245-267 (1973)

48 [35], p 32 (1991)

49 W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in C, 2nd Ed, Cambridge

University Press, p 790 (1992)

50 R. Wagner and W. Chew, A study of wavelet for the solution of electromagnetic integral equations,

IEEE Trans. Ant. Propag. 43, No. 8, pp 802-810 (1995)

51 [11], pp 17-19 (1996)

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