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woo whom-”59.14

 

 

A PERT
MULTIP.
SCATTEI'

De:

 

A PERTURBATION METHOD FOR TRANSIENT
MULTIPATH ANALYSIS OF ELECTROMAGNETIC
SCATTERING FROM TARGETS ABOVE PERIODIC

SURFACES

By

Ahmet Kizilay

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2000

A FERIL'RE
ANALYSIS
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ABSTRACT

A PERTURBATION METHOD FOR TRANSIENT MULTIPATH
ANALYSIS OF ELECTROMAGNETIC SCATTERING FROM
TARGETS ABOVE PERIODIC SURFACES
By

Ahmet Kizilay

A cylindrical target above a simulated sea surface is illuminated by a short-pulse
electromagnetic wave to study the effect of multipath on radar measurements of
targets above the sea. A new solution technique is outlined in this thesis for both
TM and TE scattering from a cylinder above an infinite, periodic surface. A series of
time-domain experiments have been performed to verify the theoretical results.

The basis of the new solution technique is that if a target is close to the sea, the
current induced on the sea surface will be nearly identical to that induced by a plane
wave without the target present, except within a region of finite extent immediately
beneath the target. A set of coupled Electric-Field Integral Equations (EFIEs) for the
current induced on the cylinder and the perturbation current (the difference between
the current with the target present and with the target absent) on the surface has
been derived in which the perturbation current is approximated to zero outside a finite
region beneath the cylinder. The EFIEs are solved in the frequency domain using the
Method of Moments and transformed into the time domain using the inverse Fourier

transform. The time domain results are used to investigate the effects of multipath

for various sea surface profiles.

To my Mother and Father

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ACKNOWLEDGMENTS

I am very fortunate because I have had great teachers throughout my education.
I would like thank several people helping me complete this thesis. My deepest ap-
preciation is to my advisor Dr. E.J. Rothwell for his time, guidance, patience, and
being a mentor over such a long time. A special thanks must go to Dr. D.P. Nyquist
for his useful comments and suggestions which have improved the technical content
and clarity of this thesis. I would like express my sincere appreciation to Dr. K.M.
Chen and Dr. Byron Drachman for participating on my guidance committee and
their assistance.

My graduate study would never have been possible without the help and teachings
of the following peOple: Dr. Cahit Canbay who played a major role in my choosing
to study electromagnetics, Izzettin Ozgiil whose teaching made me love science and
math, Selma Arda who recognized my potential and made me also realize it.

There are several people in the electromagnetics research group at MSU have
helped me during my study here. I am grateful to Dr. A.J. Norman for his help and
advice. Also, I am greatly indebted to Chris Coleman for his help in setting up the
measurement system. I would like to thank Jungwook Suk and Garrett Stenholm for
their help and company during the measurements.

I could never give enough thanks to my father and mother who provided me
with love, support and guidance throughout my studies. I would like to thank my
brother Mehmet, my sister Hayriye for their understanding, love and support, and my

soulmate, Asiye, who supported me with her love which gave me emotional strength,

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perseverance and endurance. Without her, I could not have finished this thesis.
More thanks of a Special kind are appropriate to my friends Dr. Saban Kurucay,
Dr. Ergun Canoglu, and Eric Breiner.
I also wish to thank our graduate secretary Marilyn Shriver for keeping me up—to-

date with her instant notifications and reminders.

 

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TABLE OF CONTENTS

LIST OF FIGURES ................................ ix
LIST OF TABLES ................................. xix
CHAPTER 1
Introduction ..................................... 1
CHAPTER 2
Computation of TM Scattering from Targets above Sea Surfaces ........ 5
2.1 Introduction ................................ 5
2.2 Scattering from a Cylinder above a Finite PEC Surface ........ 5
2.2.1 Theory ............................... 6
2.2.2 Numerical Results ........................ 15
2.3 Image Technique Solution for Scatterers above an Infinite PEC Flat
Surface ................................... 29
2.3.1 Theory ............................... 29
2.3.2 Numerical Results ........................ 33
2.4 Perturbation Approach for a Cylinder above an Infinite PEC Periodic
Surface ................................... 42
2.4.1 Theory ............................... 42
2.4.2 Numerical Results ........................ 50
2.4.2.1 The Validity of the Perturbation Assumption . . . . 50
2.4.2.2 Convergence Rate Comparisons ............ 66
2.4.2.3 Multipath Observations ................ 71
2.5 Conclusions ................................ 71
CHAPTER 3
Computation of TE Scattering from Targets above Sea Surfaces ........ 74
3.1 Introduction ................................ 74
3.2 Scattering from a Cylinder above a Finite PEC Surface ........ 74
3.2.1 Electric Field Integral Equation Solution ............ 75
3.2.1.1 Obtaining Integral Equations ............. 75
3.2.1.2 Moment Method Solution ............... 81
3.2.1.3 Calculation of the Self-Terms ............. 110
3.2.2 The Scattered Electric Field in the Far Zone .......... 128
3.2.3 Numerical Results ........................ 136
3.3 Image Technique Solution for Scatterers above an Infinite PEC Flat
Surface ................................... 148
3.3.1 Theory ............................... 148

vi

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3.3.1.1 Calculation of the Induced Current and the Difference

Current ......................... 157
3.3.2 Numerical Results ........................ 159
3.4 Perturbation Approach for a Cylinder above an Infinite PEC Periodic
Surface ................................... 168
3.4.1 Theory ............................... 168
3.4.2 Numerical Results ........................ 178
3.4.2.1 The Validity of the Perturbation Assumption . . . . 178
3.4.2.2 Multipath Observations ................ 193
3.5 Conclusions ................................ 193
CHAPTER 4
Identification of the Multiple Reflections ..................... 196
4.1 Multipath Analysis for Targets above Flat Surfaces .......... 196
4.1.1 TM Polarization Case ...................... 196
4.1.2 TE Polarization Case ....................... 219
4.2 Multipath Analysis for Targets above Rough Surfaces ......... 232
4.2.1 Rough Surface Scattering in TM Polarization ......... 232
4.2.2 Rough Surface Scattering in TE Polarization .......... 234
4.3 Conclusion ................................. 246
CHAPTER 5
Experimental Measurements and Comparisons .................. 247
5.1 Introduction ................................ 247
5.2 Measurement Setup ............................ 247
5.2.1 Calibration Procedure ...................... 256
5.3 Experimental Results and Validations for TM Polarization ...... 267
5.4 Experimental Results and Validations for TE Polarization ...... 277
5.5 Conclusions ................................ 278
CHAPTER 6
Conclusions and Future Study ........................... 289
6.1 Topics for further study ......................... 290
APPENDIX A
MFIE Formulation for Two-Dimensional Perfectly Conducting Closed Surfaces
Illuminated by TEz Plane Wave .......................... 292
A1 Evaluation of Principal Value ...................... 296
A2 Moment Method Solution ........................ 299

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APPENDIX B

MFIE for Two-dimensional Perfectly Conducting Open Surfaces Illuminated by

TE, Plane Wave

BIBLIOGRAPHY

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LIST OF FIGURES

Target detection problem in a sea environment ...........

The geometry of the scattering problem for a cylinder above a finite
width sea surface ............................

Segmentation of the sea surface and the target. ..........
Evaluation of the self terms ......................

Current distribution on a finite width strip, E0 = 1.0 V/m, (15,- =
90.0°, w = 2.0 m, A = 1.0 m. ....................

Induced surface currents for finite and infinite sinusoidal surfaces
(a) amplitudes, (b) phases, kod = 2.135, d = 0.1016 m, hs/d =
0.25, d),- = 20° ..............................

Induced surface currents for finite and infinite sinusoidal surfaces
(a) amplitudes, (b) phases on one period, kod = 8.54, d = 0.1016 m,
h,/d = 0.25, <15,- = 20°. ........................

Normalized current distribution on an elliptic cylinder, ¢,- = 00°.

Parameters in the figures for (a) an elliptic cylinder (b) a cylinder.

Induced surface current (a) amplitudes (b) phases on a cylinder,
koa = 1.0675, a = 0.0127 m. .....................

Current density amplitudes for a cylinder above a finite width strip
(a) on the strip, (b) on the cylinder, koa = 1.0675, a = 0.0127 m,
mr/a = 0.0, ht/a = 1.0, w/a = 72.0, (b,- = 20°. ...........

Current density amplitudes for a cylinder above a finite sinusoidal
surface (a) on the surface, (b) on the cylinder, [cod 2 1.0675, a =
0.0127 m, arr/a = 0.0, d/a = 8.0, hs/d = 0.125, ht/d = 1.0,
w/d = 9.0, (fig = 200. .........................
Scattered electric field from a cylinder above a finite—width strip (a)

in frequency domain, (b) in time domain, a = 0.0127 m, ht/a = 8.0,
w/a = 72.0, ¢,— = 20°, ¢, = 20°. ...................

Transient scattered field for a cylinder above a finite—width rough
surface, a = 0.0127 m, d/a = 8.0, h,/d = 0.25, ht/d = 1.0, w/d =
9.0, d).- = 20°, ()3, = 20° .........................

The geometry of the scattering problem for a cylinder above an
infinite flat surface ...........................

Transient scattered E—field from a cylinder above an infinite flat
surface, a = 0.0127 m, ht/a = 8.0, ¢,- = 20°, 45, = 20° ........

ix

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Figure 2.25

Figure 2.26

Figure 2.27

Figure 2.28

Figure 2.29

Figure 2.30

The geometry of the scattering problem for a cylinder above a finite
width roughness superimposed with a ground plane .........

The geometry of the problem for a finite—width sinusoidal surface
above a ground plane. ........................

Current density amplitudes for a sinusoidal surface above an
image plane (a) on the image plane, (b) on the surface, kod = 8.54,
d = 0.1016 m, h,/d = 0.25, w/d = 1.0, ()5,- = 90° ...........

Comparison of scattered E—field calculations for a sinusoidal surface
in frequency domain, d = 0.1016 m, h,/d = 0.125, w/d = 1.0,
¢i = 200, ¢, = 200. ..........................
Scattered E—field for a target above a sinusoidal surface superim-
posed with an infinite flat surface (a) in frequency domain, (b) in
time domain, a = 0.0127 m, d/a = 8.0, MM = 1.0, h,/d = 0.25,
w/d = 9.0, 055 = 20°, 45, = 20° .....................

The geometry of the scattering problem for a cylinder above an
infinitely long periodic surface. ...................

Perturbation technique to solve the scattered E—field from a cylin-

der above an infinite two-dimensional periodically-varying surface.

Perturbation current on a flat surface for different separations be-
tween the cylinder and the surface, koa = 0.266, a = 0.0127 m,
xr/a = 0.0, ¢i = 200. .........................

Perturbation current on a flat surface for different frequencies, a =
0.0127 m, ht/a = 8.0, scr/a = 0.0, 05,- : 20° ..............

Perturbation current on a flat surface for different incidence angles,
a = 0.0127 m, ht/a = 8.0, $,/a = 0.0, law 2 2.128. ........

Normalized amplitude of scattered electric field for different trun-
cation widths, a = 0.0127 m, ht/a = 1.0, xr/a = 0.0, (/5,- = 20°,
()5, = 20°. ...............................

Normalized amplitude of scattered electric field for different trun-
cation widths, a = 0.0127 m, ht/a = 8.0, :rr/a = 0.0, <15,- = 20°,
43, = 20°. ...............................

Approximation differences for different truncation widths, a =
0.0127 m, ht/a = 1.0, xr/a = 0.0, ¢,- = 20°, (b, = 20°. .......

Approximation differences for different truncation widths, a =
0.0127 m, ht/a = 8.0, 1:,/a = 0.0, c3.- = 20°, ()3, = 20°. .......

The total approximation difference for different cylinder heights,
for Icoa ranging from 0.266 to 7.98, a = 0.0127 m, xr/a = 0.0,
¢, = 20°, 42, = 20°. ..........................

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Figure 3.2
Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Perturbation current on a rough surface for different separations
between the cylinder and the surface, koa = 0.266, a = 0.0127 m,
d/a = 8.0, h,/d = 0.125, d).- = 20°, x,/a = 0.0 ............

Perturbation current for different surface heights, koa = 0.266,
a = 0.0127 m, d/a = 8.0, 05,- = 20°, ht/a = 4.0, ctr/a = 0.0. . . . .

Perturbation current on a rough surface for different horizontal
positions of the cylinder, koa = 0.266, a = 0.0127 m, ht/a = 8.0,
d/a = 8.0, h,/d = 0.125, ¢.- = 20° ...................

Normalized amplitude of scattered electric field for different trunca-
tion widths of rough surface, a = 0.0127 m, ht/a = 1.0, 1:, /a = 0.0,
d/a = 8.0, hs/d = 0.125, air = 20°, (if, = 20°. ............

Normalized amplitude of scattered electric field for different trunca-
tion widths of rough surface, a = 0.0127 m, ht/a = 8.0, x,/a = 0.0,
d/a = 8.0, h,/d = 0.125, (15,- = 20° ...................

The total approximation difference for different solution techniques
of E—field scattered from a cylinder above a flat surface, for lead
ranging from 0.266 to 7.98, a=0.0127 m, ht/a = 8.0, x,/a = 0.0,
45.- : 20°, 45, = 20°. ..........................

The total approximation difference for different solution techniques
of E—field scattered from a cylinder above a rough surface, for lead
ranging from 0.266 to 7.98, a=0.0127 m, ht/a = 8.0, :rr/a = 0.0,
h,/d = 0.125, (15,-: 20°, d2, = 20°. ..................

The total approximation difference for different solution techniques
of E-field scattered from a cylinder above a rough surface, for [con
ranging from 0.266 to 7.98, a=0.0127 m, ht/a = 8.0, x,/a = 0.0,
h,/d = 0.250, 45,- = 20°, 43, = 20°. ..................

Transient scattered electric field from a cylinder above a flat surface
and above a sinusoidal surface, a = 0.0127 m, d/a=8.0, ht/d = 1.0,
z,/d = 0.0, w/d = 9.0, ()5.- = 20°, a5, = 20°. .............

Geometry of the problem. ......................
Segmentation of the sea surface and the target. ..........

Expansion functions for the sea surface current and the target cur-
rent. ..................................

The weighting functions (a) for the sea surface, (b) for the target
on a interior node (c) for the target on the first or the last node. .

Calculation of distance between points on the source segment and
points on the observation segment. .................

Calculation of the E—field in the far zone ...............

xi

61

62

63

64

65

68

69

70

72
76
82

84

87

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Figure 3.11
Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15
Figure 3.16

Figure 3.17
Figure 3.18
Figure 3.19
Figure 3.20
Figure 3.21

Figure 3.22

The integration contour to prove the divergence identity. ..... 132

Current distribution on a finite width strip, E0 = 1.0 V/m, <15,-

90.0°, w = 2A, /\ = 1.0 m. ...................... 139
The geometry of the problem for a cylinder above a closed surface
with a sinusoidal top .......................... 140
Induced surface currents for finite and infinite sinusoidal surfaces
(a) amplitudes, (b) phases, kod = 17.08, d = 0.1016 m, hs/d =
0.125, w/d = 5.0, t,/d = 1.0, ta/d = 0.5, 65,- = 40° .......... 141
Normalized current distribution on an elliptic cylinder, ()3.- = 00°. 142

Induced current on a cylinder by different solution techniques (a)
amplitude, (b) phase, E0 = 1.0 V/m, d),- = 180.0°, A = 1.0 m, a = 2A.143

Current density amplitudes for a cylinder above a finite width strip
(a) on the strip, (b) on the cylinder, koa = 1.0675, a = 0.0127 m,

:c,/a = 0.0, ht/a =1.0, h,/a = 0.0, w/a = 72.0, (1);: 20° ...... 144
Current density amplitudes for a cylinder above a finite sinusoidal
surface (a) on the surface, (b) on the cylinder, koa = 1.0675, a =
0.0127 In, :cr/a = 0.0, d/a = 8.0, hs/d = 0.125, ht/d = 1.0, w/d =
9.0, e.- = 20°. ............................. 145

Scattered electric field from a cylinder above a finite-width strip (a)
in frequency domain, (b) in time domain, a = 0.0127 m, x,/a = 0.0,
ht/a = 8.0, w/a = 72.0, e.- = 200.0, (1;, = 20.00, (1),, = 90.00. . . . . 146

Transient scattered field for a cylinder above a finite-width rough
surface, 45, = 200°, 43, = 200°, (15‘, = 90.0°, a = 0.0127 In, d/a =

8.0, h,/d = 0.25, arr/a = 0.0, ht/d = 1.0, w/d = 9.0 ......... 147
The solution of the scattering problem for a cylinder above an
infinite flat surface ........................... 149

Transient scattered field for a cylinder above an infinite ground
plane, (1).- = 400°, (13, = 400°, 45,, = 900°, 0 = 0.0127 m, ht/a = 8.0. 162

The geometry of the scattering problem for a cylinder above a finite

width roughness superimposed with a ground plane ......... 163
The geometry of the problem for a finite-width sinusoidal surface
above a ground plane. ........................ 164
The geometry of the problem for a semi-circle target above a
ground plane. ............................. 165
Current density amplitudes for a sinusoidal surface above an image
plane (a) on the image plane, (b) on the surface, kod = 8.54, h,/d =
0.25, w/d = 1.0, ()3,- = 90°. ...................... 166

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Figure 3.28
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Figure 3.30
Figure 3.31
Figure 3.32

Figure 3.33

Figure 3.34

Figure 3.35

Figure 3.36

Figure 3.37

Current density amplitudes for a half-circle surface above an image
plane (a) on the image plane, (b) on the surface, koa = 4.27, w / a =
2.0, d).- = 90°. .............................

The geometry of the scattering problem for a cylinder above an
infinitely long periodic surface. ...................

Perturbation technique to solve the scattered E—field from a cylin-
der above an infinite two-dimensional periodically-varying surface

Perturbation current on a flat surface for different separations be-
tween the cylinder and the surface, koa = 0.266, a = 0.0127 m,
zr/a = 0.0, (15.- = 20°. .........................

Perturbation current on a flat surface for different frequencies, a =
0.0127 m, ht/a = 8.0, arr/a = 0.0, 0’),- = 20° ..............

Perturbation current on a flat surface for different incidence angles,
a = 0.0127 In, ht/a = 8.0, xr/a = 0.0, koa = 2.128. ........

Normalized amplitude of scattered electric field for different trun-
cation widths, a = 0.0127 m, ht/a = 1.0, xr/a = 0.0, (25.- = 20°. . .

Normalized amplitude of scattered electric field for different trun-
cation widths, a = 0.0127 In, ht/a = 8.0, zr/a = 0.0, ¢.- = 20°. . .

Approximation differences for different truncation widths, a =
0.0127 m, ht/a = 1.0, :r,/a = 0.0, (f,- = 20° ..............

Approximation differences for different truncation widths, a =
0.0127 m, ht/a = 8.0, :cr/a = 0.0, (15,- = 20° ..............

The total approximation difference for different cylinder heights,
for lead ranging from 0.266 to 7.98, a = 0.0127 m, 45,- = 20°, :rr/a =
0.0 ....................................

The total approximation difference for different cylinder heights,
for [can ranging from 0.266 to 7.98, a = 0.0127 m, d),- = 40°, :rr/a =
0.0 ....................................

Perturbation current on a rough surface for different separations
between the cylinder and the surface, [can = 0.266, a = 0.0127 m,
:rr/a = 0.0, d/a = 8.0, h,/d = 0.125, 45,- = 20°. ...........

Perturbation current for different surface heights, Icoa = 0.266,
a = 0.0127 m, ht/a = 4.0, :rr/a = 0.0, d/a = 8.0, 43.- = 20°. . . . .

Perturbation current on a rough surface for different horizontal
positions of the cylinder, koa = 0.266, a = 0.0127 m, ht/a = 8.0,
d/a = 8.0, h,/d = 0.125, e. = 20° ...................

xiii

167

169

170

181

182

183

184

185

186

187

188

189

190

191

192

first 3-33 T?“
. and
8.13.

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Figure 3.38

Figure 4.1
Figure 4.2

Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17
Figure 4.18
Figure 4.19
Figure 4.20
Figure 4.21

Transient scattered electric field from a cylinder above a flat surface
and above a sinusoidal surface, a = 0.0127 m, d/a = 8.0, ht/a =
8.0, zr/d = 0.0, w/d = 13.0, 43, = 20°, 05, = 20.0°, (zip = 900°.

The incident E—Field waveform, ()5,- = 40° ...............

TM backscattered fields (a) from a cylindrical target with a =
0.0127 m (b) from a finite-flat surface of width w = 0.9144 m.
¢; = 40° .................................

The reflections from the edges of a finite flat surface. .......
Expected reflections from a cylinder above a finite flat surface. . .
Reflection directly from the cylinder. ................
Sea-Cylinder—Sea multiple reflection (Multiple C) ..........
Edge—Cylinder multiple reflection ...................
Secondary Multiple Reflections. ...................
Secondary Multiple Reflection A. ..................
Secondary Multiple Reflection B and C. ..............
Secondary Multiple Reflection D. ..................

TM backscattered field from a cylinder above a finite flat surface,
a = 0.0127 m, ht = 8a, w = 0.9144 m, 45,- = 40°. ..........

TM backscattered field from a cylinder above a finite flat surface
for time starting from multiple reflections, a = 0.0127 m, h, = 8a,
w = 0.9144 m, d),- = 40°. .......................

TM backscattered field from a cylinder above an infinite ground
plane, a = 0.0127 m, h, = 80, 43,- = 40° ................

TE backscattered fields (a) from a cylindrical target with a =
0.0127 m (b) from a finite-flat surface of width w = 0.9144 m.
¢i = 400, ¢P = 90.00. .........................

TE backscattered fields from a finite-flat surface, 10 = 0.1016 m,
(12‘, = 900° for (a) 03.- = 40° (b) 03,- = 20°. ..............

The paths for the creeping waves ...................
Creeping wave reflection A. . . g ......... ' ..........
Creeping wave reflection B. .....................

Creeping wave reflection C. .....................

TE backscattered E—field from a cylinder above a finite flat surface,
a = 0.0127 m, ht = 8a, 00 = 0.9144 m, 43,- = 40°, 45,, = 90.00.

xiv

194
205

206
207
208
209
210
211
212
213
214
215

216

217

218

223

224
225
226

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Figure 4.22

Figure 4.23

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.28

Figure 4.29

Figure 4.30

Figure 4.31

Figure 4.32

Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9

TE backscattered E—field from a cylinder above a finite flat surface,
(a) time between Cylinder and Multiple C reflections (b) time after
Multiple A and B, a = 0.0127 m, ht = 8a, 21) = 0.9144 m, 45; = 40°,

45‘, = 90.0°. .............................. 230
TE backscattered E—field from a cylinder above an infinite ground
plane, a = 0.0127 m, h; = 8a, 05,- = 40°, $9 = 90.0°. ........ 231
TM backscattered field from a finite sinusoidal surface, d =
0.1016 m, w/d = 9.0, hs/d = 0.125, 45,- = 20° ............. 237
Reflections from finitesinusoidal surface ............... 238

TM backscattered field from a cylinder above finite sinusoidal sur-
face, a = 0.0127 m, h, = 8a, d/a = 8.0, w/d = 9.0, h,/d = 0.125,
¢i = 200 ................................. 239

TM backscattered field from a cylinder above infinite sinusoidal
surface, a = 0.0127 m, h, = 8a, d/a = 8.0, h,/d = 0.125, ()5,- = 20°. 240

TM scattered field from an infinite sinusoidal surface, at =
0.1016 m, h,/d = 0.125, <15.- = 20°, x/d = 0.125, y/d = 1.25 ..... 241
TE backscattered field from a finite sinusoidal surface, 61 =
0.1016 177., w/d = 9.0, h,/d = 0.125, d),- = 20°, 90,, = 90.00 ...... 242

TE backscattered field from a cylinder above finite sinusoidal sur-
face, a = 0.0127 m, h, = 8a, d/a = 8.0, w/d = 9.0, h,/d = 0.125,
03,- = 20°, (15‘, = 90.0°. ......................... 243

TE backscattered field from a cylinder above infinite sinusoidal

surface, a = 0.0127 m, h, = 8a, d/a = 8.0, h,/d = 0.125, 45,- = 20°,

(15‘, = 90.0°. .............................. 244
TE scattered field from an infinite sinusoidal surface, at = 0.1016 m,

h,/d = 0.125, 05,- = 20°, (15‘, = 90.0°, :c/d = 0.125, y/d = 1.25. . . . 245
Geometry of the free-field arch-range scattering system at MSU. . 249
Time domain scattering measurement configuration ......... 250
Triggering step signal generated by the TDR/TDT unit. ..... 251
Step signal generated by pulse generator and pulse head. ..... 252
Ti'ansmitted pulse generated by the pulse network. ........ 253
Spectrum of the transmitted pulse. ................. 254
Location of transmitting and receiving antennas. ......... 255
Block diagram model of measurement system. ........... 258

Measured waveform from a 14 inch sphere, p = 3.5052 m, d),- =
—6.1°, ¢, = 61°. ........................... 260

XV

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Figure 5.11

Figure 5.12
Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

The measured waveform from a 14 inch sphere (a) windowed in
time domain, (b) windowed in frequency domain. .........

Theoretical scattered E-field from 14 inch sphere, [El 2 1 V/m,
p = 3.5052 m, 45.- : -6.1°, e, = 6.1°. ................

System Transfer Function .......................

Measured waveform from a 3 inch sphere (a) in time domain, (b)
in frequency domain, p = 3.5052 m, ¢,— = —6.1°, 45, = 6.1°. . . . .

Time gated measured waveform from a 3 inch sphere after the
calibration. ..............................

Comparison of theoretical and measured scattered E—field from a
3 inch sphere after completing the calibration process; (a) in fre-
quency domain, (b) in time domain, |E‘| = 1 V/m, p = 3.5052 m,
05.- : —6.1°, (p, = 6.1°. ........................

TM polarization comparison of theoretical and measured scattered
E—field from cylinder (a) in time domain and (b) in frequency do-
main, IE'I = 1 V/m, p = 3.5052 m, a = 0.0127 m, O.- = —6.1°,
3, = 6.1°. ...............................

TM polarization comparison of theoretical and measured scattered
E—field from sinusoidal surface (a) in time domain and (b) in fre-
quency domain, |E'| = 1 V/m, p = 3.5052 m, d = 0.1016 m,
w/d = 12.0, h,/d = 0.125, 05,- =17.9°, d), = 30.1°. .........

TM polarization comparison of theoretical and measured scattered
E—field from cylinder above flat surface (a) in time domain and (b)
in frequency domain, IE'I = l V/m, p = 3.5052 m, a = 0.0127 m,
ht/a = 8.0, arr/a = 0.0, w/ht = 10.5, d),- = 179°, 03, = 30.1°.. . . .

Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TM Polarization,
IE‘I = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, xr/a = 0.0,
w/h, = 10.5, 45,- = 179°, 43, = 30.1° ..................

TM polarization comparison of theoretical and measured scattered
E-field from cylinder above flat surface (a) in time domain and (b)
in frequency domain, |E'| = 1 V/m, p = 3.5052 m, a = 0.0127 m,
ht/a = 8.0, zr/a = 0.0, w/h, = 10.5, ¢,- = 31.90, (p, = 44.10.. . . .
Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TM Polarization,
|E‘| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, (cf/a = 0.0,
w/h, = 10.5, (b.- = 319°, 03, = 44.1° ..................

xvi

261

262
263

264

269

270

271

272

273

274

2 TM 1
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and

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Figure 5.22

Figure 5.23

Figure 5.24

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5.28

Figure 5.29

Figure 5.30

TM polarization comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface (a) in time domain
and (b) in frequency domain, |E‘| = 1 V/m, p = 3.5052 m, a =
0.0127 m, ht/a = 8.2, z,/a = 0.0, d/a = 8.0, 02/11 = 12.0, h,/d =
0.125, e,- = 18.9°, ¢, = 31.1°. ....................

Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above sinusoidal surface in TM Polar-
ization, IE’I = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.2,
1:,/a = 0.0, d/a = 8.0, w/d = 12.0, h,/d = 0.125, ¢,- = 189°,
0’), = 31.1°. ..............................

TE polarization comparison of theoretical and measured scattered
E—field from cylinder (a) in time domain and (b) in frequency do-
main, IE‘I = l V/m, p = 3.5052 m, a = 0.0127 m, ¢,— = —6.1°,
d), = 61°, (pp = 90.0°. ........................

TE polarization comparison of theoretical and measured scattered
E—field from sinusoidal surface (a) in time domain and (b) in fre-
quency domain, |E'| = 1 V/m, p = 3.5052 m, d = 0.1016 m,
w/d = 12.0, h,/d = 0.125, 45,- = 179°, 05, = 301°, 05,, = 90.0°. . . .

TE polarization comparison of theoretical and measured scattered
E—field from cylinder above flat surface (a) in time domain and (b)
in frequency domain, IE'I = 1 V/m, p = 3.5052 m, a = 0.0127 m,
ht/a = 8.0, arr/a = 0.0, w/h, = 10.5, 05,- = 149°, «1», = 27.1°,
¢,, = 90.00. ..............................

Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TE Polarization,
IE‘I = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, air/a = 0.0,
w/h, = 10.5, (13,-: 149°, 43, = 27.1°, ¢,, = 90.0°. ..........

TE polarization comparison of theoretical and measured scattered
E—field from cylinder above flat surface (a) in time domain and (b)
in frequency domain, IE‘I = 1 V/m, p = 3.5052 m, a = 0.0127 m,
ht/a = 8.0, zr/a = 0.0, w/h, = 10.5, ¢.- = 339°, 05, = 461°,
45‘, = 90.0°. ..............................

Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TE Polarization,
IE‘| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, arr/a = 0.0,
w/ht = 10.5, <15.- = 339°, 43, = 461°, ([2,, = 90.0°. ..........

TE polarization comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface (a) in time domain
and (b) in frequency domain, |E'| = l V/m, p = 3.5052 m, a =
0.0127 m, ht/a = 8.2, z,/a = 0.0, d/a = 8.0, w/d = 12.0, h,/d =
0.125, 45,- = 179°, 45, = 301°, 05,, = 90.0° ...............

xvii

280

281

282

283

284

285

 

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tered 1.
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Figure 5.31 Multiple reflection comparison of theoretical and measured scat—
tered E—field from cylinder above sinusoidal surface in TE Polar-
ization, |E’| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.2,
:rr/a = 0.0, d/a = 8.0, w/d = 12.0, h,/d = 0.125, (15, = 179°,
4), = 301°, 45‘, = 90.0° ......................... 286

Figure 5.32 TE polarization comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface (a) in time domain
and (b) in frequency domain, |E‘| = 1 V/m, p = 3.5052 m, a =
0.0127 m, ht/a = 8.2, zr/a = 0.0, d/a = 8.0, w/d = 12.0, h,/d =
0.125, e.- = 25.90, 3, = 38.1°, ¢,, = 90.00 ............... 287

Figure 5.33 Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above sinusoidal surface in TE Polar-
ization, IE'| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.2,
:c,/a = 0.0, d/a = 8.0, w/d :2 12.0, h,/d = 0.125, d),- = 259°,

03, = 381°, 03,, = 90.0° ......................... 288
Figure A.1 Geometry of the problem. ...................... 293
Figure A.2 Evaluation of Principal Value ..................... 298
Figure B.1 Magnetic field integral equation for open surfaces .......... 302

xviii

Table 4.1
Table 4.2

Table 1.3

Table 4.4

Table 4.5

Table 1.5

f“ T
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Table 4.10

 

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Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Table 4.7

Table 4.8

Table 4.9

Table 4.10

LIST OF TABLES

The calculated distances to be used to find Atl ........... 199

Comparison of timings in TM polarization for a cylinder above
finite flat surface, a = 0.0127 m, h, = 8a, w = 0.9144 m, (p, = 40°,
e, = 40°. ............................... 204

Comparison of timings in TM polarization for a cylinder above an
infinite ground plane, a = 0.0127 m, ft. = 80, w = 0.9144 m,

d),- = 40°, (1), = 40°. .......................... 204
Comparison of Atg in the backscattered field for different surface
widths and incidence angles ...................... 220

Comparison of timings in TE polarization for a cylinder above
finite flat surface, a = 0.0127 m, ht = 8a, w = 0.9144 m, (p, = 40°,
45, = 40°, 05,, = 90.0° .......................... 222

Comparison of timings in TE polarization for a cylinder above an
infinite ground plane, a = 0.0127 m, ht = 8a, 43,- = 40°, 65, = 40°,
dip = 90.0°. .............................. 222

Comparison of timings in TM polarization for a cylinder above
finite sinusoidal surface, a = 0.0127 m, ht = 8a, d/a = 8.0, w/d =
9.0, h,/d = 0.125, (p,- = 20°, 05, = 20° ................. 233

Comparison of timings in TM polarization for a cylinder above
an infinite sinusoidal surface, a = 0.0127 m, ht = 8a, d/a = 8.0,
w/d = 9.0, h,/d = 0.125, e,- = 20°, ¢, = 20° ............. 234

Comparison of timings in TE polarization for a cylinder above finite
sinusoidal surface, a = 0.0127 m, ft, = 8a, d/a = 8.0, w/d = 9.0,
h,/d = 0.125, (/2. = 20°, 45, = 20°, (A, = 90.0° ............. 235

Comparison of timings in TE polarization for a cylinder above an
infinite sinusoidal surface, a = 0.0127 m, ht = 8a, d/a = 8.0,
w/d = 9.0, h,/d = 0.125, 3, = 20°, 43, = 20°, (pp = 90.0°. ..... 236

xix

 

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A

CHAPTER 1

INTRODUCTION

A significant amount of research has been done to analyze scattering from periodic
structures for a variety of applications [1], [2]. In particular, periodic structures
have been used as a simple model for detection of targets near the earth [3], [4]
or the sea [5], [6]. The electromagnetic structure of the sea environment is very
complicated. Therefore, even qualitative information on the transient scattered field
from targets above the sea becomes useful. Long range detection of targets above
the sea has been an interest to researchers for several decades and the recognition of
the problem goes back to the early stage of the development of radars [7]. At low-
grazing angles energy which travels directly from the target and indirectly through
forward scattering off the sea surface may merge coherently as shown in Figure 1.1.
As a result, narrowband radar systems will be affected in either a constructive or a
destructive way and targets may fly through radar coverage undetected. The higher
range resolution of ultrawide-band (UWB) radar systems may solve this multipath
problem. This effect was investigated via an UWB radar system by Sletten et. al. [8].

The Multipath problem for targets above a rough surface has been studied pre-
viously, and the effects of surface roughness on narrow-band scattering from the sea
have been analyzed [9, 13]. Vazouras et. al. [6] computed the scattered Electric field
from a conducting cylinder above a lossy medium with a sinusoidal interface by com-
bining an integral equation approach with the extended boundary condition method.
However, the deterministic problem of transient, short-pulse scattering from an undu-
lating sea surface is not well understood. In the time domain, individual ocean swells
contribute to the scattered field which is incident on a target, making the multipath

phenomenon more complicated.

 
  
   

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In order to learn about the basic principles of transient multipath, several simple
problems can be investigated, such as a perfectly conducting (PEC) cylinder above a
finite sinusoidal surface, or above an infinite flat surface. The solution for the Electric
field from a cylinder above a finite length sinusoidal surface does not provide reliable
information about the effect of the surface structure on multiscatter, because of the
large scattered field coming directly from the surface edges and indirectly from the
cylinder via forward scatter from the surface edges. These artificial edge effects are
highly undesirable, especially at low-grazing angles. West et. a1. [10] used a hybrid
moment method/ geometrical theory of diffraction technique to avoid the reflections
from the edges of the finite sea surface. Although the electric field scattered by a PEC
cylinder above an infinite flat surface (which can be solved by the method of images)
does not have this problem, the contribution of surface structure to the scattered
electric field is lost. Consequently, to study transient multipath it is necessary to
examine the more difficult problem of a PEC cylinder above an infinite, periodic
surface.

This thesis outlines a new solution technique for both TM and TE scattering from
a cylinder above an infinite, periodic surface. In chapter 2 this solution technique will
be developed for TM polarization and numerical results will be presented. The same
analysis will be repeated for TE polarization in chapter 3. After completing the
analysis for both polarizations, the multipath phenomena will be studied in chapter
4. Finally, in chapter 5, the results from a series of time-domain experiments will be
presented to verify the theory developed.

The idea behind the new solution technique is that if a target is close to the sea, the
current induced on the sea surface will be nearly identical to that induced by a plane
wave without the target present, except within a region of finite extent immediately
beneath the target (the region of multiple interactions) where the difference between

the current with the target present and with the target absent (the “perturbation cur-

 

self] 5 large. Tb '
101 the current 11111
surface he been 11-4:

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solved by using flut

rent”) is large. Therefore, a set of coupled Electric-Field Integral Equations (EFIES)
for the current induced on a cylindrical target and the perturbation current on the
surface has been derived in which the perturbation current on the surface is approx-
imated to zero outside a finite region beneath the cylinder. The EFIEs are solved in
the frequency domain using the Method of Moments (MoM) and transformed into the
time domain using the inverse Fourier transform (IFT). This formulation requires the
knowledge of the transient scattered field produced by a short pulse interacting with
an infinite surface without the cylinder present, a problem which has been previously

solved by using Floquet analysis and the MoM [5].

Ship Radar

   
    
   

Direct Path

 

 

t\\ \\\‘ )1

\\\i\\ /' /
\\ Reflected Path/

Sea Surface

Figure 1.1. Target detection problem in a. sea environment

Coyfi’UTATlO

2.1 Introductior

Multipath prnp‘drla‘
lng radars to deter“
roughness makes tl.

Scattering prn': '.
solved for horizon: -.

Mellopoulos '4'.

 

 

 

scattering from a d;
angle frequency \‘211‘
rrltipath. this that
strates the necemr
sea. It introduces a

aperiodjc surface. c

be lied to study t'h

CHAPTER 2

CONIPUTATION OF TM SCATTERING FROM TARGETS ABOVE
SEA SURFACES

2.1 Introduction

Multipath prOpagation has been already given an important consideration when us-
ing radars to detect targets above sea. The very complex nature of the sea-surface
roughness makes this problem difficult to solve at a satisfactory level.

Scattering problem from a cylinder interacting with sinusoidal interfaces was
solved for horizontal (TM) polarization by Vazouras et. a]. [6] and by Cottis and
Kane110poulos [4]. Yet, the multipath effect is not well understood for short-pulse
scattering from a disturbed sea because the solutions have been achieved for only a
single frequency value. To help enhance the understanding of ultrawide-band (UWB)
multipath, this chapter outlines the solutions of the simple problems and demon-
strates the necessity of choosing an infinite periodically-varying surface as a model of
sea. It introduces a new solution technique for TM scattering from a cylinder above
a periodic surface, shows basic numerical results, and discusses how the results can

be used to study the multipath phenomena.

2.2 Scattering from a Cylinder above a Finite PEC Surface

Here the sea surface is represented by considering the simplest model, which is a finite-
sized, two-dimensional PEC surface. Scattering of a TM (horizontally) polarized
electric field E; as indicated in Figure 2.1 is analyzed first in the frequency domain and
then an inverse Fourier transform (IFT) is used on the spectral results. The scattered
electric field is calculated by applying boundary conditions on the sea surface and the

cylinder, and obtaining the Electric-Field Integral Equations (EFIEs) for the induced

 

current on the sea 5'11
2.2.1 Theor.V

The incident electric

horizontal
This. field will prwl'
scalar potential fur:

Where .5 l5 8 lWr'r—cl.

representing the f

since the incident

Q;
-rnCe the Scalar l

[311ng . '2 .

current on the sea surface and on the cylinder.
2.2.1 Theory

The incident electric field is assumed to be a TM plane wave with angle as, from the

horizontal

Ei(x,y) 2 E0 ejk(zcos¢;+ysin¢r)2 (2_1)

This field will produce a scattered field which can be written in terms of vector and

scalar potential functions

.0

Eco-2') = —ij(5) — we) (22)
where ,6 is a two—dimensional position vector
5 = $2“: + yr) (2-3)

representing the field points. The scattered field has only a z-directed component

since the incident field is in the z—direction. Therefore, (2.2) becomes

Em = —ij.(5) — 63f ) (2.4)

 

Since the scalar potential function <I>(fi' ) does not have 2 dependency

5M5 )

62 = o (2.5)

 

Using (2.5) , (2.4) can be rewritten as

3203' ) = -ijz(/3' ) (2-6)

PEC \,

Figure 2.1. "ft
Width Sea Swap?

PEC * ’ (p,

 

(60, Mo)

 

ed...

 

 

 

PEC

 

 

 

 

 

 

 

Figure 2.1. The geometry of the scattering problem for a cylinder above a finite
width sea surface.

where the rector pu'-

where K;i5l is the llf
given as k, : "V5
pennittitity of free ~_:

Solution of [2.7) x

and 9515') is:

he in

Here H07

I replt‘éems g}

 

 

. I .
mm 12.8,} and upon 5‘;

m be mile!) 35

Where , v
'l’lrlp,p l is a:

'Mirp‘

1'!

Her .
. e r ’
'0 Is ‘he int

ӣ2554

 

where the vector potential Az(fi' ) satisfies the two-dimensional Helmholtz equation
v2.4.0?) + leg/W) = ‘MoKzlfil (2.7)

where K,(fi’ ) is the unknown induced surface current, k0 is the free-space wave number
given as k0 = (in/co #0. Here, no is the permeability of free space, and 60 is the
permittivity of free space.

Solution of (2.7) will give the general expression for Az(fi )
A.(f2’) = #0 / rec/3' ') Go: 5') W (2.8)
r
where if ’ represents the source points

p = a: :r: + y'g} (2.9)

-o -o 1 -o -o
0w ') = —.Hé2’(ko|p — p 'l) (210)

Here H62) represents the Hankel function of the second kind. After substituting (2.10)
into (2.8) and upon substituting (2.8) into (2.6), the expression for the scattered field

can be written as

Ezra) = Marmara") (2.11)

where Mp(f)’, 5 ’) is an operator defined as
Manama ') = _’°_4_°’7"/K(p ()-" 2>(rolp— m) dl’ (2.12)

Here 770 is the intrinsic impedance of free space.

for the prelim

on the cylinder an

Where S 31,”,
Ytpresems the
curfent on the
A 56! Ol (‘01
m Condition
Wed tile

for tom] mg]

be COnsjdered'

The points 0n I

pom 0” the cr-

Subsrllulion 0

entrant on the sea

For the problem considered, (2.11) can be rewritten in terms of induced current

on the cylinder and on the surface separately. That is

E§(fi)=Ef(fi)+EI(fi)

= Mgr/259mm + Marmara) (2.13)

where S and T symbolize the surface and the cylinder respectively; accordingly K E ( fi )
represents the induced current on the surface and KIM ) represents the induced
current on the cylinder.

A set of coupled EFIEs for the induced currents can be found by using the bound-
ary condition of zero tangential total E—field over the surface and the target since the
scattered field is known in terms of the induced currents. The boundary condition

for total tangential E—field on the surface is

E3 (55) + Elms) + E2073) = 0 (2-14)

The boundary condition for total tangential electric field on the cylinder must also

be considered. It can be written as

E50?) + 133er + 19.2%) = 0 ' (2-15)

The points on the sea surface are represented by [73- Similarly, [37» represents the
points on the cylinder. Here, 53 and fir are two-dimensional position vectors to the
points on the sea surface and on the cylinder, respectively.

Substitution of (2.13) into (2.14) and (2.15) yields a pair of coupled EFIEs for the

current on the sea surface K f (55) and on the cylinder K3151)

M3055, fileflfis’) + Mrlfis: firilelfi'ri) = “Eilfisl (2-15)

M,

To solve the bill.
linear segments as

l
are then expamln.

Where Arise?“ l.
Sliflllarly‘ ll: fr

ml? and 12.1:

on lhe Cllltttler~

on the 5mm 2
lyJSlllOH (H.105
681158131] Elllniné

are written in m.

Where Am is an A

Ms(fir, 55'le(55') + NIH/3r, 5%)KI(575) = -E§(fir) (217)

To solve the EFIEs first, the surfaces 5 and T are approximated by a number of
linear segments as shown in Figure 2.2. The unknown currents K 5(55) and Kflfir)

are then expanded in a pulse basis set on these linear segments as

N5

K353) = Z arSnKSrWs) (2-18)
m=1
NT

Kfcer) = Z arm/a) (2.19)
n=l

where Kflfis) is defined as unity on segment Lm and zero on the other segments.
Similarly, Kflffi) is not zero only on segment zero Ln. After substituting these into
(2.16) and (2.17) , using point matching at N5 points on the surface and NT points

on the cylinder, the coupled EFIEs become

N5 NT
2: ariMS(fiSir fileilf’h’) + Z: aIMr(fis.-, fiIl)K1T(fi'I() = ‘EXD'SD (230)

m=l n=1

N5 NT
2 aiMscenrs'mmg) + ZaIMTcfinn—hKIcmt) = —E;(p'n) (2.21)

m=l n=1

where i = 1, 2, 3, . . . , N5 and 8 = 1,2,3, . . . , NT are the indices of the matching points
on the surface and on the target respectively, and 55,-, [in are the two-dimensional
position vectors to the matching points as shown in Figure 2.2. The method of
Gaussian elimination can be used to solve this set of linear equations; therefore, they
are written in matrix form as

'm Bi n a?" b?
A" ’ = (2.22)

Cl,m Dlm a: Her
where A5,", is an N5 x N; matrix, Bi,” is an N5 x NT matrix, Cl," is an NT x N;

10

 

 

 

 

-.."’ Segment Ln

pT Segment Lm

 

 

Figure 2.2. Segmentation of the sea surface and the target.

11

 

 

matrix. and Du

.lfrer rning the

numerical inter

The Simplest 6:

Lu: and Ln

5” lhat

matrix, and Dz,” is an NT x NT matrix. Also, bf and bf are given by

4 t
bf— E0 ejk( xrcos¢r+yasn¢r)
160770
b = E0 67 (szOSd>r+yzSIn¢.)
770

(2.23)

(2.24)

After using the definition of M, the matrix elements can be calculated by using any

numerical integration method from

 

Arm: K5.(z’,y’)H H‘Q’ckofi-- x')2+ (y—y')2)dl'

Lm

 

13,-, =/LKT('x ,'y)H(2)( kof— x')(2+ (y -y')2)dz'

 

lm,m=/I: KS( ('2: y’) (km/(13: - $02 + (31: - y’)2) dl’

 

Di," 2 / KICE’, y') Héz)(ko \/(.’E( — $02 + (y; '— y’)2) dl'
Ln

(2.25)

(2.26)

(2.27)

(2.28)

The simplest expansion is to assume the currents to be constant over surface segments

 

 

Lm and L7,
1 (2331/) 6 Ln.
K5.(z',y’) =
0 otherwise
1 (13’, y’) 6 Ln
Kflx'w') =
0 otherwise
so that
A5,". = L Hiz)(k0\/($i — 3')” + (y.- — y’)2) d"
3"" = L Hiker/(x.- — e)? + (y. — y'e) dt'

12

(2.29)

(2.30)

(2.31)

(2.32)

llhen m = t i
source points. trial
the sell-terms Slit it

armaments 15

where n, represet

1.781. Evaluation

The matching 1
distance betwee:
Therefore. the wi
method of partiti
Strips on the cyli

After the pul
matrix equation
Hammered, T

«ruminating ti.

 

04m = H§2’(ko\/(a:, — 22)? + (y, — y')2) dl’ (2.33)
Lm

De" z] 1952’th — 2:02 + (y. — we) «11' (2.34)

n

 

When m = z' or n = E (the self-terms), the integration paths go through the
source points, making the numerical integration very difficult to compute. Therefore,
the self-terms should be approximated by using the Hankel function terms for small

arguments [15]

2 ko’flh

H32’(kou,):1—j 111(7) for kouL<<1 (2.35)

ir—
where uL represents the position along segment L as shown in Figure 2.3 , and 7 =

1.781. Evaluation of the integrals containing self terms yields

A... = Asll — gunfight.) — 1)] (2.36)
D... = Arll — agenda—VAT) — 1)] (2.37)

The matching points on the sea surface are generated by making the arclength
distances between each two adjacent points on the sea surface equal to one another.
Therefore, the widths of the constant-current strips are represented by A3. The same
method of partitioning is used on the cylinder and the widths of the constant-current
strips on the cylinder are represented by AT.

After the pulse basis function amplitudes have been determined by solving the
matrix equation (2.22), the scattered E—field far away from the sea and the target
is considered. To find the E—field, a far-field approximation is used in (2.13) by

approximating the Hankel function for large argument [16]. The approximation is

13

 

 

Segt

 

 

 

 

 

  
 

 

"1}; A51;
2 0 2 u
l l r, L
7 l l
E = n
PTe fiT'
Segment Ln
_ As
2
Segment Lm x _A__5_v
._} 2
P32
UL
£13
P

 

 

Figure 2.3. Evaluation of the self terms.

14

 

 

imph‘

where 0. 5 the W

Substitullllg ‘73

which for rectanr

5:»

2.2.2 Numeri
Before generatin
formulation is d
finite flat surfacl
the result ohtair
tori = 1 m at
strip induced bi
perfect agreeme

Secondly, th
Slimmidal surfar
Partition points.
asingle period c

lith hi1: 2.13.1

simply
2j e-J'kop

yo

where d), is the scattering angle to the field point from horizontal.

Hd2)(koll7- e ’l) 2 ejk°‘$’°°‘¢’+y”‘"¢” (2.38)

Substituting (2.38) into (2.12) and using (2.29),(2.30) in (2.18),(2.19) gives

- F e.-
E§(P ) = ‘770 JB—frq‘fi

NT
+ 2 a3: [L eik°<e’°°8¢a+v’s‘"¢°) dl’] (2.39)
71:] "

 

S ‘k 'cos¢.+y’sin¢, I
m/ e10” )dl

which for rectangular rule integration becomes

 

‘9 " jok ejkP 1: s m
Ez(p)= _077 87? 7:30;? AS ejko( mco ¢s+ym8 ¢,)
+ZTaz: AT ejlco(:rn cos¢.+yn sin¢,) ] (240)
n=l

2.2.2 Numerical Results

Before generating results for a cylinder above a finite surface, some verification of the
formulation is done by considering the sea surface and the cylinder alone. First, a
finite flat surface of width or is considered, and the induced current is compared with
the result obtained by Balanis [15] for a finite strip. A two meter long strip is chosen
for A = 1 m and 250 matching points are used. The current densities on a finite
strip induced by a plane wave at normal incidence are shown in Figure 2.4. There is
perfect agreement.

Secondly, the induced current on a finite sinusoidal surface and on an infinite
sinusoidal surface is compared. The sinusoidal surfaces are approximated by using 55
partition points for each period of the surfaces. The induced current magnitudes on
a single period of the infinite sinusoidal surface and on the finite sinusoidal surface,

With kod = 2.135 are plotted in Figure 2.5a, and the current phases are compared in

15

 

    
   
   

figure tab. With it
plotted in figure ‘2 ’
away hum the ethic
lhirdly. figure .
arts of two wave let.
zero degrees from x-t
cylinder. A compart.
and odd current der:
dhtauce uorrualim‘
point on the centi-
isshowu in Pierre
ludnaseu found
to = 0.5 where t
mall? Spaced In
Finally. The
White for lion :
and F1Ellie 2.9h
mehOrlZflntal.
the induced cu
the Shit-(loving
Alter gem;
by (“EH8 at.
Width flat Str
malllltlllg p9]
Write-d [n Pi
the 5mm (

M1 Other.

Figure 2.5b. With increased frequency (lead = 8.54) the same comparison is made and
plotted in Figure 2.6a and Figure 2.6b. As expected, there is pretty good agreement
away from the edges of the finite surfaces.

Thirdly, Figure 2.7 shows the induced current on an elliptic cylinder with a major
axis of two wave lengths and an axial ratio of 4. The incidence angle is chosen to be
zero degrees from x—axis, d),- = 0.0. Point matching is used at 120 points on the elliptic
cylinder. A comparison is made with the result produced by Andreasen [17] using even
and odd current densities in an EFIE—MoM solution. In the figure, 3 is the perimeter
distance normalized to unity for the half perimeter, as measured clockwise from the
point on the center face nearest the source of the incident wave. This parameter
is shown in Figure 2.8a. There is good agreement between the results. However,
Andreasen found that additional matching points are necessary for the region close
to s = 0.5 where the shadow region starts. The scattering program in this thesis uses
equally spaced matching points, and thus the results near 3 = 0.5 are different.

Finally, The induced current on a cylinder is calculated by using 40 matching
points for lead = 1.0675. The induced current on the cylinder is plotted in Figure 2.9a
and Figure 2% for several incidence angles. In the figure 0 is the angle measured from
the horizontal. The geometry is shown in Figure 2.8b. As expected, the maximum of
the induced current occurs at the point closest to the source of the incident field and
the shadowing effect is not as significant as in the ellipse.

After gaining confidence from these verifications, the current densities generated
by targets above a sea surface can be computed. First, a cylinder above a finite
width flat strip is considered. 40 matching points are used on the cylinder and 250
matching points are used on the strip. The currents on the strip and cylinder are
plotted in Figure 2.10a and Figure 2.10b. Not surprisingly, the cylinder current and
the surface current are affected most at the points where the surfaces are closest to

each other. To check the effect of sea surface roughness on the current distributions

16

olthe cylinder and
are used on the ct
sinusoidal surface.
lllb shows the c
the induced curret
However. the indu
similarly b} the c
beneath the cyliu
For the scattt
per wavelength i
l‘li points or iii
and the larger \-
tbe current on e
computed for a
plotted in Figur.
to the thuean
Ilitre are direct
The first and s
Q'linder repec
m SlrilH‘ylinc
following the n
more Complica‘
Finally. the
Sinusoidal surfe
W GHZ resui
“(lowing the
the [FT n“. t]

of the cylinder and the sea, a sinusoidal surface is considered. 40 matching points
are used on the cylinder and 55 matching points are used for each period of the
sinusoidal surface. Figure 2.11 a shows the current density on the surface and Figure
2.11b shows the current density on the cylinder. It is seen that the amplitude of
the induced current on the cylinder is slightly different to that above a flat surface.
However, the induced currents on the sinusoidal sea and on the flat surface are affected
similarly by the cylinder. In other words, they are affected most at the points just
beneath the cylinder.

For the scattered E—field calculations, the larger value of 40 points or 10 points
per wavelength is used to represent the current on the cylinder, the larger value of
125 points or 10 points per wavelength is used to represent the current on the strip,
and the larger value of 55 points or 10 points per wavelength is used to represent
the current on each period of the sinusoidal surface. First, the scattered E—field is
computed for a cylinder above a finite width strip for several frequency points and
plotted in Figure 2.12a. Since the transient field gives more insight, the IFT is applied
to the frequency domain results and the transient E—field is shown in Figure 2.12b.
There are direct and multiple reflections coming from the strip and from the cylinder.
The first and second signals are the direct reflections from the strip and from the
cylinder respectively. The third signal comes as a superposition of cylinder-strip
and strip-cylinder multiple reflection. After that, the signal comes from the strip by
following the multipath of strip-cylinder-strip. The signals come later at time are
more complicated multiple reflections.

Finally, the transient scattered field is computed for a cylinder above a finite width
sinusoidal surface in the frequency band from 1 GHz to 30 GHz with the step size of
0.04 GHz resulting 726 data points. The time domain response is obtained by first
Windowing the frequency domain results using double cosine function then calculating

the IFT. The transient field is plotted in Figure 2.13. As seen in Figure 2.12b and in

17

  
     

Figure 2.13. multi',‘
edges of finite width
reflections. and the

Therefore. a more r.

 

Figure 2.13, multipath analysis is complicated by the presence of reflections from the
edges of finite width sea models. These reflections may be causing edge-target multiple
reflections, and these reflections are not expected in the real ocean environment.

Therefore, a more realistic sea model is necessary.

18

 

0.0

0.C

O.C

IKzI (A /m>

0.0l

0.0M

Figure 2-4- CL

 

 

 

 

 

 

 

0.0300
‘ — finite width strip
.1 ---- Solution by Balanis;
0.0225 —; .
A l
g .:
25 0.0150 if
2‘ fl é
0.0075 JL/i \J
0.0000 *'*'r""F""r"*'r
-1.00 -0.50 0.00 0.50 1.00

x-location (m)

Figure 2.4. Current distribution on a finite width strip, E0 = 1.0 V/m, 05,- : 90.00,
w=2.0m,x\=1.0m.

19

15

180

Phase (deg)

'180

o
A)

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—--—. w/d=3.0
; w/d=5.0
15 . . infinne
‘ l
O t .
E l I
"B 10 — i
S: « l
—N ‘ =
é * l
-2.50 -1.25 0.00 1.25 2.50
x-Iocation / d
(a)
I
—--—- w/d=3.0 l
q w/d=5.0 l
180 - . ~~ Infinite
8 .
o
E
m
a, .
2 0 -
o. _
180/
-2.50 -1.25 0.00 1.25 2.50
x-location/d
(b)

Figure 2.5. Induced surface currents for finite and infinite sinusoidal surfaces (a)
amflitudes, (b) phases, [cod = 2.135, d = 0.1016 m, h,/d = 0.25, 45,- = 20°.

20

E
(lel Way 0

deg)
6 (
Phas

6

1E

~1i

“glue 2 6

"l1:

a‘TQ'lltudM. {ll'

'4‘ ‘. j
wa

 

 

 

     

 

 

 

 

 

  

 

 

 

 

 

9
. —---- w/d=3.0
w/d=5.0
t ° ~ infinite
‘ !
o 6 n l
H l 2
’2 l
53' I
—~ . r
X ;
:3 3 « l
l i
o . . . , r . . - . , . . - . , . . . - ,
-2.50 -1.25 0.00 1.25 2.50
x-location / d
(a)
i —--—- w/d=3.0
4 w/d=5.0 !
j o infinite
180 _ .. 2.,
a
m
E :
a, .
w i
.2 0 ~
0. ;
480‘ ..... ?.,. ...... ..,---....-f...°...r:..
-0.50 -0.25 0.00 0.25 0.50
x-location / d
(b)

Figure 2.6. Induced surface currents for finite and infinite sinusoidal surfaces (a)
amplitudes, (b) phases on one period, kod = 8.54, d = 0.1016 m, h,/d = 0.25,
¢i = 200.

21

 

 

a 1 a . \ 4 1 . e N d a u c N . 1 N il-
0
ml. 4|. 4| 0

cm \moPLNXC

_ a 7
0. £0. 5 2
3 2 o m

mun.

 

 

 

   

 

 

 

3.0 _
Z —--—- EFlE-MoM Solution
2.5 _‘ —— Solution by Andreasen
l ay=21, ay/ax=4.0
02.0 f ------ l -
UJ .
\ < a
A 4 . y
E" 1.5 3 3J1
e ; _, a;
C 1.0 f X
0.5 -§
0.0‘....,....,....,
0.00 0.25 0.50 0.75 1.00

8

Figure 2.7. Normalized current distribution on an elliptic cylinder, 05,- = 0.00.

22

 

 

Figare 2.8. Pa

 

 

 

 

 

 

 

1.5 .. .
A y Ez
(be
2.0
1.0 0.7
S
0 5 a;
(a)
it y E’s
in.
90.0
0
180.0 0.0
270.0 (I:
(b)

Figure 2.8. Parameters in the figures for (a) an elliptic cylinder (b) a cylinder.

23

 

Figu,

£90:

)
(den
Phase

e 2.9

log“.-

[IJ.(‘

 

 

 

__ (pi = 200:

 

 

 

 

0 90 180 270 360
O-location (deg)

(a)

 

 

 

 

 

* A _ 0
360 e —" ¢i- 20
a .
£13: 180 <
a) .
(D
m .
.C .
“- 0
-130-...,fi...,-..n..m-
0 90 180 270 360
O-location (deg)
(b)

Figure 2.9. Induced surface current (a) amplitudes (b) phases on a cylinder,
koa = 1.0675, a = 0.0127 m.

24

 

 

 

 

 

 

 

 

 

 

6 m—fi
O44
lg .
’3 l
5: .
Ti
:5
:32~
f0
-0.50 -0.25 0.00 0.25 0.50
x-Iocation/w
(a)
4
3L
0 I
u\J .
"B .
{:2—
2 :
1L
oi.-..,..e.,...e,.f..
O 90 180 270 360
O-location (deg)
(b)

Figure 2.10. Current density amplitudes for a cylinder above a finite width strip (a)
on the strip, (b) on the cylinder, koa = 1.0675, 0 = 0.0127 m, 12,./a = 0.0, hg/a = 1.0,

w/a = 72.0, 4;,- = 20°.

25

 

Ill +9

 

 

 

 

 

 

0‘.o...,.s.......,...-g
-0.50 -0.25 0.00 0.25 0.50
x-location/w
(M
4
3Q

 

 

 

1r
0
0 90 180 270 360
e-location (deg)
(b)

Figure 2.11. Current density amplitudes for a cylinder above a finite sinusoidal
surface (a) on the surface, (b) on the cylinder, koa = 1.0675, a = 0.0127 m, 2:, / a = 0.0,
d/a = 8.0, h,/d = 0.125, ht/d = 1.0, w/d = 9.0, as, = 20°.

26

.Figu
m f}!

1\

-1]?

 

 

 

 

5 10 15 20 25 30
frequency (GHz)

(3)

 

 

 

O
O

 

.5
01

 

 

Scattered electric field (relative)

 

O
A
N
.. 1
.5
UI
0)

time (ns)
('0)

Figure 2.12. Scattered electric field from a cylinder above a finite-width strip (a)
in frequency domain, (b) in time domain, a = 0.0127 m, h¢/a = 8.0, w/a = 72.0,
a, = 20°, «1:. = 20°.

27

‘ 1 . 1 1 ~ g 0 I roll“;

0. 0. 0 ”0.. :
1 0 ‘_II 2 a 0 u
1 e P.“ 0.0/w
Am>.um_0...v “v.03 "Ur—«00.0 UQLQHHNOW WWW :
F . .01

 

 

.l.
O
i

 

 

 

Scattered electric field (relative)
S 2

 

 

 

‘17 T I T T Y ‘17 I 7 V V I I

4 6 8
time (ns)

C
N

Figure 2.13. Transient scattered field for a cylinder above a finite-width rough
surface, a = 0.0127 m, d/a = 8.0, h,/d = 0.25, ht/d = 1.0, w/d = 9.0, ¢,- = 20°,
¢, = 20°.

28

 

 

2.1 has”

Surface

llllhlStase. lli
indirect the re
hesie runltinat
he used to re
Cylinder rurr
23.1 The
lheineidern
lhe sum is

direction. a

on the CV1“,

Where (4 l5
Path for the

for the Cl'llrrr

2.3 Image Technique Solution for Scatterers above an Infinite PEC Flat
Surface

In this case, the sea surface is chosen to be infinitely long and flat to avoid direct and
indirect the reflections from edges of the finite-sized sea surface model. To study the
basic multipath effect, a cylinder is located above the sea. The method of images can
be used to replace the infinite flat surface with images of the incident field and the
cylinder current as it is shown in Figure 2.14.

2.3.1 Theory

The incident electric field in this case is the sum of a TM plane wave and its reflection.
The sum is a standing wave in the y-direction and a prOpagating wave in the a:-

direction, and can be expressed as
Elia, y) = 2j E0 ejk°("°s¢‘) sin (key sin 45,-) 2 (2.41)

Using (2.11) the scattered E—field can be rewritten in terms of the induced current

on the cylinder and the image current as
13:03) = MT(E5')KZ(3 ') + Mntéifi' ’)K3‘(fi ') (2-42)

where M is defined in (2.12) and 5,1)" are given in (2.3),(2.9). T,- is the integration
path for the image current; the relation between this path and the integration path

for the cylinder current is

Mr.(a:. ylx ’. y ’) = Males. ylz ’. -y ’) (2-43)

29

 

 

FlEUIe 2.14.
flat Slltfate,

A _..
y E‘

----—~----- ) cylinder 45'

Ol-

 

h
l i
r
h

T" , (152;
“rm“ Image E7:

Figure 2.14. The geometry of the scattering problem for a cylinder above an infinite
flat surface.

at.

 

30

After using relation (2.43) and
K315 )IT. = -K;’(fi Mr (244)
in (2.42), the scattered E—field becomes
Eileen!) = [MAI/6.310 '. y ') - MAID. ylx ’. -y 0] Kane '. y ' ) (2-45)

The EFIE can be obtained by applying the boundary condition on the E—field,

and is simply
[M703 ylx '. y ') - M701. ylx '. -y 0] K30: '. y ' ) = -E(% 11) (2-46)

After point matching, and using the expansion of K: in (2.19), the EFIE becomes
"T
T I I I I T I I _ i
an [MT(zla yllx 1y )— MT($la yllx I —y )] Kn (23 )y ) _ —Ez(xt1yl ) (247)
1

n:

where Z = 1,2, 3,. . . , NT are the indices of the matching points. The set of equations

in (2.47) may be written in matrix form

where D is an NT x NT matrix, aT represents the unknown the pulse basis function

amplitudes, and In can be calculated by

I)” = kf—JmE" eject“ 60W sin (koy, sin 43,-) (2.49)

For simplicity, the pulse basis function (2.30) is used; hence, the matrix elements are

31

etaluated by

The self-tern
Hankel funct
' tegration (

calculate thi:

The mat
isfound in t
and Mrfrn

fIlnctiorr in ‘

Where 5“ I ‘

[Sine 72
ll

'7 . ‘
z 111 D’Jise

evaluated by

 

D4" = (L. [Wheat/(z. — 2202 + (y. — we)

 

— H52)(k0\/($I-$')2+ (gm/let] cm (2.50)

The self-terms (K = n) should be calculated carefully because the argument of first

 

Hankel function (km/(2:; — $')2 + (y; - y’)2) in (2.50) becomes very small and the
integration of the Hankel function becomes difficult to compute numerically. To
calculate this term (2.37) should be used, and the diagonal terms in (2.50) become

.2 k
n... = 441 —J;(1n(—:1Ar)- 1)]

_ / Hiz)(kox/($n - 23')” + (yr. + 3102) 611' (2-51)

Ln

 

The matrix equation is solved for unknown currents. Then, the scattered E—field
is found in the far zone after approximating the Hankel functions in MT($, ylx’, y’ )
and MT(.’L', ylx', —y’ ) in (2.45). For large arguments the approximation of the Hankel

function in MT(x,y|:c’, —y’ ) is

~ -e 2j e-jkop ' z’coa — ’sin
H32)(kolp-p."|)= 4.7—«r» elk“ e y 4’" (2.52)

where if,- ’ = x’i: - y’g, and d), is the scattering angle to the field point from the
horizontal.
Using (2.52) in MT(:c, ylx', —y’ ), (2.38) in MT(x, y|x’,y’ ), and the expansion of

K? in pulse basis functions, the E—field may be written as

 

E‘(p‘) = ”Mi _jk0 53,: it? ejkw'mw' sin (koy'sin ¢ ) dl’ (2 53)
z 27'. fl ":1 n L" 5

32

 

A

, .
praneas rt sl

2.3-2 We

lhe infill"- W“
in will“ i
W “rel? ll":
results art’ 0‘
of 094 CH;
mdgwfil l
.tc seen f“-
refiefll‘ilnS
comes as
(the (00.19
Sim l0 f

a Signllll?
three ref
similar 1
mmmgl
ground l
to the CF
reflection:
taluahle i
roughness I
To have
technique f0

1'

as another tar
therefore. not}.

lltlllllfln.

2.3.2 Numerical Results

The image technique formulation is used and the transient scattered E—field is found
for a cylinder located above a ground plane. The larger value of 40 points or 10 points
per wavelength is used to represent the current on the cylinder. Frequency domain
results are obtained varying the frequency from 1 GHz to 30 GHz with the step size
of 0.04 GHz resulting 726 frequency data points. IFT is used on the spectral data
windowed by a double cosine function. The transient field is shown in Figure 2.15.
As seen from the transient field, the edge reflections are avoided and all three main
reflections are identified. The first reflection comes from the cylinder, the second one
comes as a superposition of cylinder-sea and sea-cylinder reflections and the third
one comes from the sea-target-sea reflection. After the main reflections, the signals
start to fade. All the reflections can be grouped as sets of three reflections. There is
a significant amplitude decrease after each set of three reflections. The second set of
three reflections are much smaller then the main three reflections. These signals are
similar to first three signals but they make an extra path to the ground plane before
coming to the observation point. The extra path starts from the cylinder goes to the
ground plane with —90° from horizontal and comes back with 90° from horizontal
to the cylinder. Similar to the second set of three reflections, the third set of three
reflections travels on this extra path twice. Although some insight is gained, very
valuable information is lost. Since the sea surface is flat, the effect of sea surface
roughness on the multipath reflections cannot be studied.

To have a roughness on the infinite flat surface and still be able use the image
technique formulation, a finite width roughness is superimposed with the ground
plane as it shown in Figure 2.16. This problem is solved by considering the roughness
as another target which has a sinusoidal shape, in addition to the cylindrical one;
therefore, nothing has be to be changed in the previous formulation to solve this

problem.

33

L /_ “—

 

The change 0
laceisohsersetll
the ground nlznn
shows the cum
harlot region)
llli? Slllllfittltlfl‘r
Sinusoidal surf,
the rough sur
flat surface l,
sinusoidal su
:lfter oi,
Sm‘tleJlt‘lal s
Wiltltm p
elmerits
Calculated
larger v3“.
llfild Slim
3 [turned
3511800.":
using the
Changes <
mghi 50."
Instead. f(
migrate H
@1999an
m9 Stat

above (be Si:

The change of the current density on the ground plane and on the sinusoidal sur-
face is observed by calculating the induced currents as the sinusoidal surface touches
the ground plane. The geometry of the problem is shown in Figure 2.17. Figure 2.18a
shows the current density on the ground plane. It is seen that the boundary of the
shadow region becomes very sharp and the induced current becomes very small below
the sinusoidal surface as it touches the ground plane. The induced current on the
sinusoidal surface changes significantly at the edges as shown in Figure 2.18b. When
the rough surface touches the flat surface, the currents on the rough surface and the
flat surface have to be continuous. Therefore, the current density at the edges of the
sinusoidal surface should reduce as the sinusoid is brought to the ground plane.

After observing the current density, the E—field scattered from one period of the
sinusoidal surface which touches the image plane calculated for different number of
partition points on the surface and using different methods to evaluate the matrix
elements (A,,,,,.) The results are plotted in Figure 2.19. First, the matrix elements are
calculated using rectangular rule and the sinusoidal surface partitioned by using the
larger value of 55 points or 10 points per wavelength at each frequency. The scattered
field shows some rapid changes after 15 GHz. To determine whether this behavior is
a numerical error or internal resonances caused by the artificial cavity between the
sinusoidal surface and the ground plane, the sinusoidal surface was partitioned by
using the larger value of 55 points or 20 points per wavelength. As a result, the rapid
changes did not disappear but the amplitudes decreased. Using more points thus
might solve this problem, but limited computer resources do not allow this approach.
Instead, for the same partitioning, the matrix elements were calculated using more
accurate Rcmberg integration routine from [18]. As a result, the rapid changes
disappeared.

The scattered E—field is plotted in Figure 2.20a for a cylindrical target located

above the sinusoidal surface which is superimposed upon an infinite flat surface.

34

 

One Of the
converge i:
frequency

cornes (lire
the sinuso:

multiple n

One of the advantages of this approach from a finite width surface is that it will
converge faster and the edge reflections will be avoided. The IFT is applied to the
frequency data and transient field is plotted in Figure 2.20b. The first four signal
comes directly from the sinusoidal surface. The other five direct reflections from
the sinusoidal surface are mixed with the direct reflection from the cylinder and the

multiple reflections between the cylinder and the surface.

35

W. «H. W ..H.
a0>2m20uv “20¢ Utuomam UmmeuuNOW

 

”Rm 2‘

5i

0: 001:).

 

.N
o
1 I

.A
O
4 l L

L

 

O
O
L r
l

 

 

 

 

-1.0 4

 

 

Scattered electric field (relative)

 

time (ns)

Figure 2.15. Transient scattered E—field from a cylinder above an infinite flat surface,
a = 0.0127 m, ht/a = 8.0, <35,- = 20°, d), = 20°.

36

Ay EV

 

 

 

 

 

 

(60) MO) *mr-J 1032'
"‘0.
PEC
PEC
m7 plane hs
0' = m r: w a m

 

Figure 2.16. The geometry of the scattering problem for a cylinder above a finite
width roughness superimposed with a ground plane.

37

 

image plane

/

0:00

 

 

 

 

 

Figure 2.17. The geometry of the problem for a finite-width sinusoidal surface above
a ground plane.

38

 

———h
————— h

/d=0.0
/d=0.25

Y

ll '\ II, \
f \ / \ y
I 'I i. \ . . 4

 

 

 

 

 

 

-5-4-3-2-1012345
x-location / w

 

 

 

 

 

(a)
* —— h /d=0.0
8 — Y
3 _____ hy/d=0.25
l r
j l l
EC 6 ‘| f
"5 l It
s: 4 t .
— 4 l l
g l i
2%
-0.50 -0.25 0.00 0.25 0.50
x-location / w
(b)

Figure 2.18. Current density amplitudes for a sinusoidal surface above an
image plane (a) on the image plane, (b) on the surface, kod = 8.54, d = 0.1016 m,
h,/d = 0.25, w/d = 1.0, 4;.- = 90°.

39

0%?qu owxrwt

0

Figure 2

in ire-quer

 

~ -------- 10 pts per wavelength, rectangular rule
0.020 — —-— 20 pts per wavelength, rectangular rule
‘ — 20 pts per wavelength, Romberg rule

 

 

 

 

 

5 10 15 20 25 30
frequency (GHz)

Figure 2.19. Comparison of scattered E—field calculations for a sinusoidal surface
in frequency domain, d = 0.1016 m, h,/d = 0.125, w/d = 1.0, 45, = 20°, d), = 20°.

40

 

 

 

 

 

 

5 10 15 20 25 30
frequency (GHz)
(80

 

.3
O

5_W._.l

 

Scattered electric field (relative)
—'~ 0
'o o

 

 

 

O
N
. 1
O)
on

time (ns)
0))

Figure 2.20. Scattered E—field for a target above a sinusoidal surface superim-
posed with an infinite flat surface (a) in frequency domain, (b) in time domain,
a = 0.0127 m, d/a = 8.0, MM = 1.0, h,/d = 0.25, w/d = 9.0, d), = 20°, ¢, = 20°.

41

2.4

.l per
ct'lind
min
the so
from a
A l
arhitrz
face as
induce
the cyl
Q'linde
Equatit
on the
imated
in the it
the mm
the km,
ifillIllte
Sillt‘ed 1}
2.4.1
lhe Sta

0055 0ft

2.4 Perturbation Approach for a Cylinder above an Infinite PEC Periodic
Surface

A perturbation approach is applied to solve for the electric field scattered from a PEC
cylinder of arbitrary cross-section above an infinite two-dimensional, periodically-
varying PEC surface. The perturbation approach simply combines the solution for
the scattered electric field from a cylinder above a finite periodic surface with that
from an infinite periodic surface without the cylinder.

A transient, short pulse TM plane wave is assumed to be incident on a cylinder of
arbitrary cross-section above a two-dimensional periodically-varying conducting sur—
face as indicated in Figure 2.21. If the cylinder is close to the surface, the current
induced on the surface will be nearly identical to that induced by a plane wave without
the cylinder present, except within a region of finite extent immediately beneath the
cylinder (the region of multiple interactions). A set of coupled Electric-Field Integral
Equations (EFIEs) for current induced on the cylinder and the perturbation current
on the surface has been derived. The perturbation current on the surface is approx—
imated to zero outside a finite region beneath the cylinder. The EFIEs are solved
in the frequency domain using the Method of Moments (MoM) and transformed into
the time domain using the inverse Fourier transform (IFT). This formulation requires
the knowledge of the scattered field produced by the plane wave interacting with an
infinite surface without the cylinder present, a problem which has been previously

solved using Floquet analysis and the MoM by Norman et. a]. [5].

2.4.1 Theory

The scattered electric field can be calculated directly by applying boundary condi-
tions on the infinite periodic surface and the cylinder, and obtaining the EFIEs for
the induced current on the infinite periodic surface and on the cylinder. However,

since the transient scattered field is desired, computations must be made for many

42

d—mr-b
—PEC 4052'

 

 

 

(€09 IU’O) “‘0'
h. «d-
PEC
<— hs ->

 

 

£13

 

Figure 2.21. The geometry of the scattering problem for a cylinder above an in-
finitely long periodic surface.

43

 

frequency t
search for

hation met
the prohiei
is to find

the pertur
to solve in
aprohlem
SOlllilUlLs g

The lllt

the horizor

The scatte
The E-fieli‘

5‘93 Sllllacr

Where 3 a,
High A’SI
z \/
l“tinted c1
It is m.

the torn n

frequency values. The amount of computational work thus becomes burdensome. A
search for a fast approximate solution for the scattered electric field led to a pertur-
bation method. This method requires less computational work, because it separates
the problem into two simpler parts, one which has already been solved. The first part
‘ is to find the scattered field produced by the induced current on the cylinder and
the perturbation current on a finite region of the periodic surface, and the second is
to solve for the scattered field from an infinite periodic surface without the cylinder,
a problem solved previously by Norman et. a1. [5]. The combination of these two
solutions gives the solution to the original problem as illustrated in Figure 2.22.
The incident electric field is assumed to be a TM plane wave with angle 45,- from
the horizontal
E‘(x, y) = Eoejkolmstfiysiwe'm (2.54)

The scattered E—field can be written as in (2.11) because the configuration and nomen-
clature is the same as the finite case, except that the sea surface is now infinitely long.
The E-field is rewritten in terms of induced current on the cylinder and on the infinite

sea surface separately. That is
Em = Ms(fi.fi’)K§(fi’) +MT(5.5')KZ‘(2") (2.55)

where S and T symbolize the periodic surface and the cylinder, respectively. Accord—
ingly K 3(5 ) represents the induced current on the surface and K503 ) represents the
induced current on the cylinder.

It is now convenient to write the electric field scattered from the surface due to

the total induced current as

E307) = E10?) + E503)

= Ms(i3. ii ')K§(fi ') + Ms(I7. 501605 ') (2-56)

44

 

rim
3}

UFJYE

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.22. Perturbation technique to solve the scattered E—field from a cylinder
above an infinite two-dimensional periodically-varying surface.

45

where E: is the perturbational field produced by the difference, or perturbation,
current K5, and E; is the field due to current K g on the surface which is impressed
by the incident field without the cylinder present. That is, K 5 = K: — K ,1 .

Ms involves a contour which has an infinite extent. Therefore, to calculate the
scattered E—field from the infinitely-long periodic surface the solution given by Nor-
man et. a1. [5] is used. That is,

E507) = Mat/7. emit/7’) (2.57)
where Mg, is a linear Operator defined as
Mg (5, 5')K’(p = —jkno fs( K’(p ')(G’ fi,fi')dl’ (2.58)

Here 8,, is the one period of the surface, and G’ (if, if ’) is the periodic Green’s function
(PGF). Integration of PGF over S, with the induced current on 8,, produces the
electric field. The PGF is given by [5]

j ":00 e-jfin (55—3,)e—an ill-y"

0! I l :__ .
($,y]$ )3!) 2d qn (2 59)

 

n=-oo

where ,6n = k0 cos 43.- + 277", qn = (Neg — 3,, and d is the period of the surface.
Using (2.55), (2.56) and (2.57) , the following equation is derived for the scattered
E—field

E‘(5)=Ms,5.( 5')K.’(5 ')+Ms(5.5')Kf(5')+Mr(5 5')KT(5 ') (2-60)

Applying the boundary condition of zero tangential total electric field to the pe-

46

 

rioc

when:

(Til?
' t

144‘er

riodic surface gives

Ez’(5s) + Ef(5s) + EH55) + 532(55) = 0 (2-51)

where Exfis) is the field on the surface due to incident plane wave, EIUXS) is the field
on the surface due to K I (5}) on the cylinder, E5075) is the field on the surface due
to the total current K g (55) on the surface which is impressed by the incident field
without the cylinder present, and Ef(fi'3) is the perturbational field on the surface
produced by the perturbation current K 5’ (5'5) = K 3(55) — K 1 (5'5).

The definition of E: (55) requires

Elvis) + E2073) = 0 (2.62)

This is the enforced boundary condition of zero tangential E—field on the periodic sea
surface for the scattering problem when the periodic sea surface is the only scatterer.
Substitution of (2.62) into (2.61) yields a relationship between the perturbational field

and field produced by the current on the cylinder given by

E5075) + E302) = 0 (2.63)

The boundary condition for total tangential electric field on the cylinder must also

be considered. It can be written as

E: (5r) + Ef(5r) = -E.' (5r) - 13; (5r) (2-64)

where Epr) is the field on the cylinder due to the total current war) on the
cylinder, E;(p”7~) is the field on the cylinder due to the incident plane wave, E; (5}) is

the scattered field on the cylinder due to K1055) on the periodic surface, and E5 (57)

47

 

II.)

(If.

th

Ha.
-1

Sign"

is the field on the cylinder produced by difference current K 5 (53) on the periodic
surface.

The boundary conditions (2.63), (2.64) yield a pair of coupled EFIEs for the
current on the cylinder Kari?) and the perturbation current K 5’ (fig) on the periodic

surface

MS(fiSafiS!)Kf(fiSI) + MT(55:5T’)KI(575) = 0 (2-65)

—M§,(i>'r. (smite?) — Bitter) (2.66)

where Kflfis’) is a known quantity as solved by Norman [19]. Therefore, there are
only two unknowns which are Kf(fis’) and “(575).

The benefit of writing the coupled EFIEs in terms of the perturbation current is
that when the cylinder is close to the surface, truncation of the perturbation current
K5055) to a finite length of the surface yields a good approximation. Using this
approximation, the system of integral equations given in (2.65), (2.66) is solved nu-
merically. For a numerical solution the truncated sea surface is approximated using
Np planar segments and the cylinder is approximated by NT planar segments. The
unknown currents K f (53) and K3951) are assumed to be constant over each segment,

and thus can be represented using piecewise constant functions as

NP

Kit/3's) = 2 am (2.67)
77121
"T

KzT(5r) = 2 am: (2.68)
11:]

Here K5, denotes the pulse expansion function which is unity over segment In on the

surface, and zero otherwise. Similarly, K3,; is the pulse function defined over segment

48

 

n on the target.
The expansion of the currents are substituted into (2.65) and (2.66) , then point

matching is used to convert the coupled EFIEs to a system of linear equations

Np NT
2 cit/vets... 55'th + Z «(EMA/75., mist = 0 (2.69)
m=l n=l

fa... MSWTtsfisflKg'l'iaTMfi/W 55m? = —M§( rim ’)Ki (p§)- Eiwn)
m=l 1121

(2.70)
where i = 1,2,3,...,Np and I = 1,2,3,...,N7~. This set of linear equations are

written in matrix form as

Ai,m Bi,n a1}; biP

= (2.71)
Cl,m Dim a:

where Aim. is an N px Np matrix, B,,,, is an Np XNT matrix, CA", is an NT xN p matrix,
and D1,, is an NT x NT matrix. Also, of, is the perturbation current amplitude for
the sea surface on segment m, a: is the current amplitude for the cylinder on segment
n, and bf and bf are given by

b? = 0 (2.72)

4
bf: k—Eo ejk°(x‘c°8¢‘+y‘s‘"¢')— —4j/ Kzl(x',y')GI(x,y]:r',y')dl' (2.73)
0770 s,

If the definition of M is used, the matrix elements can be calculated directly from
the same matrix element computation formulas (2.31)-(2.34) for the finite-size surface
problem by using a numerical integration.

The scattered E—field can be calculated after solving (2.71) for of, and a; easily
by using (2.60). f

49

2.4.2

In orde:
the trnr
ing the;
obtaine
transier.
Moll St
represer
per War
2.4.2.1
The sol
P0rtuni
EXAM pt
mined (
Li Seen
moan
the slit
along 1
With]
ln l
018 efft
Shuuld
[Bride]:
of the

..

a"L“Jllid

81,,

2.4.2 Numerical Results

In order to accurately apply the perturbation technique it is necessary to determine
the truncation width required to produce a desired accuracy. This is done by compar-
ing the perturbation and image techniques for a flat surface. A simple rule of thumb is
obtained empirically, and then applied to a surface with non-zero roughness. Finally,
transient multipath effects are identified through application of the IFT. For all the
MoM solutions, the larger value of 15 points or 10 points per wavelength is used to
represent the current on the cylinder, and the larger value of 55 points or 10 points

per wavelength is used to represent the current on each period of the surface.
2.4.2.1 The Validity of the Perturbation Assumption

The solution for a circular cylinder above an infinite flat surface gives a good op-
portunity for investigating the region of multiple interactions. Figure 2.23 shows the
exact perturbation current on the flat surface found by subtracting the currents deter-
mined using the image technique with and without the cylinder present. The current
is seen to be significant immediately beneath the cylinder, and negligible outside a
certain finite region the extent of which depends upon the cylinder’s distance from
the surface. When the cylinder height gets bigger, the perturbation current spreads
along the surface. Therefore, the truncation width must be carefully selected for the
perturbation method.

In Figure 2.24, the perturbation current is plotted for various frequencies to see
the effect of frequency on the truncation width. It is seen that the truncation width
should not be highly dependent on frequency. Similarly, Figure 2.25 shows the effect of
incidence angle on the perturbation current. As expected, the center of concentration
of the perturbational current shifts somewhat as incidence angle is changed. This
should be taken into consideration when determining the truncation width.

Since the main concern is to calculate the scattered field, it is helpful to compare

50

the ex
pertnr‘
estima
hfighh
various
defined
and in
be rho:
deterrn
the 3%?
ciderer

This is
hathIn :

error 0]

the exact scattered field found using the image technique to that found from the
perturbational method. Then, the amount of error between the two can be used to
estimate the prOper truncation width. The scattered field is found for two cylinder
heights. Figure 2.26 and Figure 2.27 plots the magnitude of the scattered field for
various truncation widths, w, as a function of frequency. The approximation difference
defined by [E’m9 — EPI for each frequency is plotted in Figure 2.28 for ht/a = 1.0
and in Figure 2.29 for h, / a = 8.0. An approximate truncation width can apparently
be chosen roughly independent of frequency to give a desired accuracy value. To
determine a rule of thumb for choosing the truncation width, it is helpful to examine
the aggregate difference between the image and perturbation techniques. The total

difference from the image technique is calculated by using

HEW-ET aw

2.74
flE’ml2 a. ( )

 

This is plotted in Figure 2.30 for different cylinder heights. As expected, the pertur-
bation solution becomes more accurate for increasing truncation width. For a relative

error of 0.02 an empirical formula for the minimum truncation width is found to be

w 20

_ = (2.75)
h‘ In (1.718 + %)

 

To see if the empirical formula holds in the presence of surface roughness, the
perturbation currents are calculated when a cylinder is placed above a finite sinusoidal
surface. The perturbation currents on the surface are shown for different cylinder
heights in Figure 2.31. Again it is concluded that the perturbation current is highly
effected by the height of the cylinder. Then, the perturbation current is plotted for

different surface roughness in Figure 2.32. It is seen that the extent of the perturbation

51

 

current
perturl
change-
figure
Chantal
The
plotted
show si
Show)

in the

current is not dramatically effected by surface roughness. It is expected that the
perturbation current will change as the location of the cylinder in the :r-direction is
changed. The perturbation current as a function of horizontal position is plotted in
Figure 2.33. It is seen that the location of the cylinder in the :v-direction does not
dramatically affect the truncation width.

The perturbational scattered field is calculated for different truncation widths and
plotted in Figure 2.34 for ht/a = 1.0 and in Figure 2.35 for ht/a = 8.0. The results
show similar convergence behavior to the cylinder above the infinite flat surface results
shown in Figure 2.26 and in Figure 2.27; as the truncation width increases, the change

in the scattered field becomes smaller.

52

All

0.
«025.0: a

20.230 COZNQLDEMwD‘

nU.

Figure

the Mr

 

 

 

 

 

1.0-
1,; : / _._.. ht/a=1.0
E. i / \ _-._. ht/a=8.0
50'8“; / ,_'l —— ht/a=32.0
Z: * / .. \ ——h/a=64.0
c * - \ t
90.6:
‘5
o .
g 0.4:
“:5
.e .
30.2-
t ..
a, .
D.
-400 -200 0 200 400

Figure 2.23. Perturbation current on a flat surface for different separations between
the cylinder and the surface, koa = 0.266, a = 0.0127 m, 1:../ a = 0.0, (b,- = 20°.

53

 

00
L3:
:00

CO.

£05.30

g

0.;

50.

A0?

 

“Eu
0 t (

 

.3
O
1 l

— ka=0.266
—— ka=2.135

.0
co
1 l 1 1 1

.O
a)
111

l
l
l
l
l

 

.0
4:.

 

l A l L l 1 l l 1

.0
N
1 l

l
l
l
l
l
l

Perturbation current (relative)

 

 

V\

\

 

Y ‘1 I r r I v I I l v '7'— I

-400 -200 0 200 400
x/a

.9
0

Figure 2.24. Perturbation current on a flat surface for different frequencies,
a = 0.0127 m, ht/a = 8.0, zr/a = 0.0, cl),- = 20°.

54

1

I

0
A®>._...m.0t a

C0LL30 COBNQthQQ

flu.

Figme .

 

o A
00 o
l 11
|

|
;.9

ll

.5

o

O

.O
on
111

 

 

Perturbation current (relative)

l
il

'1

[t

04- l
'l

lg“

 

 

 

 

 

-400 -200 0 200 400
x/a

Figure 2.25. Perturbation current on a flat surface for different incidence angles,
a = 0.0127 m, ht/a = 8.0, :rr/a = 0.0, koa = 2.128.

55

'1‘ I‘

N: «NEE ominm:

Figure

rid.

thg.

 

1 ———- image technique
4 L ----------- w/ht=4.0

—-—-- W/ht=9.0
3 _. .. _.._. w/ht=20.0
: / \., __ tht=40.0

 

 

 

Figure 2.26. Normalized amplitude of scattered electric field for different truncation
widths, a = 0.0127 m, ht/a = 1.0, zr/a = 0.0, 45,- = 20°, 45, = 20°.

56

«\RmeX owxrma:

 

 

 

 

 

 

 

 

 

   

 

- ---------- image technique
4 i —-—.- wlht=4.0
] —---— W/ht=9.0
SA 3 .‘ __ w/ht=20.0
£2 1 ,
>91 3 iii
01—. 5
g i \ \ \. '\ ,
1 1 \ l H l \
« l I \. l \. j t
0 ‘ ' .' V l W T. l
0 2 4 6 8

ka

Figure 2.27. Normalized amplitude of scattered electric field for different truncation
widths, a = 0.0127 In, h,/a = 8.0, z,/a = 0.0, 42,- = 20°, (1), = 20°.

57

 

........... WIht=4'0 —---- W/ht=20.0
_._.. w/ht=9.0 __ w/ht=40.0

 

 

 

 

Figure 2.28. Approximation differences for different truncation widths,
a = 0.0127 In, ht/a = 1.0, zr/a = 0.0, (75,- = 20°, do, = 20°.

58

m: «mxuvaominmrmfim‘t

 

m ,

 

 

 

 

 

10‘ —-—-- w/ht=4.0 __ w/ht=20.0
j —.._. w/ht=9.0
(U ‘ /‘./\'./ l/ 1‘ \ "t" \. l.
e or to it t] .. t... -t-.-~-.l,r
”a 101 — -.\t]. (t. / / \l/ \1- \l/
— h l' 1, .I
‘1 t-
”.1 102 - f
D)
g
E.
104 fl m ,
0 1 2 3 4 5 6 7 3
ka

Figure 2.29. Approximation differences for different truncation widths,
a = 0.0127 m, ht/a = 8.0, xr/a = 0.0, d),- = 20°, 43, = 20°.

59

ll

 

F1811
‘30. n

 

+ ht/a=1.0
—a— ht/a=8.0

 

 

 

 

 

 

 

 

 

 

 

01 1. .......................... \\.. ......................................................................................................................
to
0.01 ...............................................
0.001 . . . . . . . . .
0 4 81216202428323640
w/ht

Figure 2.30. The total approximation difference for different cylinder heights, for
koa ranging from 0.266 to 7.98, a = 0.0127 m, xr/a = 0.0, d),- = 20°, (0, = 20°.

60

 

 

 

 

1.0 j
a? : —-— fit/3:1 0
.g 0 8 . ,i _.._. ht/a=4 0
{a ' . ll __ ht/a=8 0
I: : j!
E) 0.6 j ] l
° * It l l= ="-
a 0'4 j l I] lll/ \
T6 ‘ = l - l
'9. . (r- l l l
g 0 2". ./ \._/j \j l/-".\-.\J'\
g: i A ./'/\ / \I \ \ \ /‘
0.0 @1591.“ J! {‘1 . ‘/i\.?'\\’/\~‘~i

Figure 2.31. Perturbation current on a rough surface for different separations
between the cylinder and the surface, koa = 0.266, a = 0.0127 m, d/a = 8.0,
h,/d = 0.125, 43,- = 20°, :cr/a = 0.0.

61

 

A

.

O
1 l
‘1.

{3] ........... hS/d=0.0
] _._.. hS/d=0.125
,il'g —.._. hS/d=0.250

.0 .o .o
«h O) 00
111 11111 111
E

Perturbation current (relative)
0
ix)

 

 

.0

o
:—
2
0
\‘2
(

‘. ./

. '\\£.

1 44'

_ Ky
7.?
\5
LL.

 

Figure 2.32. Perturbation current for different surface heights, koa = 0.266,
a = 0.0127 m, d/a = 8.0, d),- = 20°, ht/a = 4.0, $,/a = 0.0.

62

nu.

1|

00

0
202230.: a

0 0 0
C0LL30 coawmwQLDthl

AU.
0

2.:

Flglne

he

03:200

 

 

 

 

 

1.0 -
if ‘ —--—- xr /a=0.0
.3 i Xr /a=4.0
59 0.8 - .
9 1 l
S : l = : = l l
0 . ,' ll ‘/ ]
.553 ”'4? ,\ l t
.8 : , l/ \/ l
E 0 2 j \ \ s/‘-.\
0.0 TP 1 T r r

 

x/a

Figure 2.33. Perturbation current on a rough surface for different horizontal posi-
tions of the cylinder, koa = 0.266, a = 0.0127 m, hg/a = 8.0, d/a = 8.0, h,/d = 0.125,
¢; = 20°.

63

«v. a «(<va oNanC

 

 

 

 

 

[fl ........... wlht=24.0

Q 3 r I]. \ —---—-- W/ht=40.0
:1 . (=7 \ _.._. w/ht=56.0
m J/ \ 1‘
\ y \ 1'. /\
>93 2 _ f \‘5/ \\
e" : ‘3”)
tie—- .
a .
v 1 _ // /\

l / C, .

‘ / J/c-atf/

‘ / t/

l .,

0 F I r rm . 1 v I 1 . r 1 r
0 1 2 3 4 5 6 7 8

ka

Figure 2.34. Normalized amplitude of scattered electric field for different trunca-
tion widths of rough surface, a = 0.0127 m, ht/a = 1.0, zr/a = 0.0, d/a = 8.0,
h,/d = 0.125, (12,-: 20°, d), = 20°.

64

e. a $0: Wham:

 

 

 

 

 

 

 

 

Figure 2.35. Normalized amplitude of scattered electric field for different trunca-
tion widths of rough surface, a = 0.0127 m, ht/a = 8.0, 1:,/a = 0.0, d/a = 8.0,
h,/d = 0.125, 4n = 20°.

65

2.4.2.2

Several
above a
first cor
the C)’lll
.\lo.\l 211
for tacit
cylinder
images 2
found at
decrease
lfihniqu
After
a 31011503
53319 \a
nusoida;
infinite l
Yeoman]
“‘0'ng g
pennrba
(If the 011
Ma“, S
“ides, cl.

mines. c

2.4.2.2 Convergence Rate Comparisons

Several approaches have been used to solve for the scattered E—field from a cylinder
above a rough surface. Before comparing the techniques for a rough surface, the
first comparison was made for a flat surface. The length of the flat surface beneath
the cylinder is changed and the scattered field is calculated by both ordinary EFIE-
MoM and perturbation techniques. Then, the errors for both techniques are found
for various surface widths by using the exact solution of the scattered field from a
cylinder above a infinite ground plane. The exact solution is calculated by method of
images and using this solution in (2.74), the total differences for both techniques are
found and plotted in Figure 2.36. With increasing width of finite surfaces, the error
decreases for both techniques. However, the rate of convergence in the perturbation
technique is much bigger than that in the ordinary solution.

After making the comparison for a flat surface, similar comparison is made for
a sinusoidal surface. It is expected that the scattered E—field should approach the
same value with increasing surface width for the cylinder above a finite-width si-
nusoidal surface or above a finite-width sinusoidal surface superimposed upon an
infinite ground plane. Indeed, if the width goes to infinity, the solution of E—field
represents the electric field scattered from a cylinder above an infinite periodically-
varying surface in both cases. However, in order to have same accuracy as with the
perturbation technique, bigger surface widths are necessary for the ordinary solution
of the finite-width sea surface or the image technique solution of the finite-width sea
surface superimposed upon a ground plane. In other words, for the same surface
width, the perturbation technique gives the most accurate result among all the tech-
niques. Since the exact answer for a cylinder above an infinite periodically-varying
surface is unknown, the total difference for several surface widths is calculated using
the widest surface allowed by limited computer resources. All the results for various

surface widths are compared with the result from the surface 15.0 x h, meters wide.

66

techn:
f‘. .

surlac

This width is used because beyond this width the difference between all the three
techniques becomes negligible. The scattered field found using this width is called
E‘”. The total difference is calculated with a formula similar to that used for a flat

surface in (2.74)
_ f [Ew — E]2 dw

8w — f lelz dw (2.76)

 

where E is the field calculated for various surface widths. The total differences is
plotted for all techniques in Figure 2.37 and in Figure 2.38 for increasing surface
roughness. It is concluded that independent of surface roughness the perturbation

technique converges fastest.

67

 

Figure
E‘field
798' a:

 

 

—e— ordinary solution error
—£+ — Perturbation technique error

 

 

01‘ ................................................................. . ..........................................................................................................................
O . .

c
c
s
1
I
e
e
s
1
u
s
-
u
1
u
n
1
1
u
e
-
1
u
1
u
u
1
u
a
u
1
1
1
a
1
1
e
c
u
u
s
a
1
1
I
n
1
1
r
l
1
1
n
1
u
1
n
u
s
n
1
I
1
1
n
a
1
a
a
e
1
e
u
u
e
o
e
1
o
u
1
n
s
I
I
1
a
u
1
1
1
I
e
a
e
1
I
a
1
r
I
a
s
c
1

0.01

 

 

 

Figure 2.36. The total approximation difference for different solution techniques of
E-field scattered from a cylinder above a flat surface, for lean ranging from 0.266 to
7.98, a=0.0127 m, h./a = 8.0, arr/a = 0.0, 4),- = 20°, 4), = 20°.

68

 

0.1

10

 

—I-- Image technique error
+ Ordinary solution error
—v- - Perturbation technique error

 

 

 

 

 

 

 

 

 

1 n
w
01 .1 ................................................................................
0.01 , , T
w/ht

Figure 2.37. The total approximation difference for different solution techniques of
E-field scattered from a cylinder above a rough surface, for koa ranging from 0.266 to
7.98, a=0.0127 In, h¢/a = 8.0, xr/a = 0.0, h,/d = 0.125, ¢.- = 20°, ¢, = 20°.

69

ll

 

F lg
E‘fif‘lc

‘I’ nix. .
<

V.

1O

 

+ Image technique error
—+ - Ordinary solution error
—v— Perturbation technique error

 

 

 

 

 

 

 

 

to
0.01 T r I
1 3 5 7 9
w/ht

Figure 2.38. The total approximation difference for different solution techniques of
E—field scattered from a cylinder above a rough surface, for koa ranging from 0.266 to
7.98, a=0.0127 m, ht/a = 8.0, xr/a = 0.0, h,/d = 0.250, 45,- = 20°, ()5, = 20°.

70

 

2.4.2.3

Now the
scatters
turately
radius 0
resulting
ing the

IFT. Fi
the cyli
flew 1h.
59911 in.
domain
00cm.

Winn
hum Q
of these
A deta

miliipg

2.4.2.3 Multipath Observations

Now that an error bound on the scattered field has been established, the transient
scattered field for a cylinder above an infinitely-long sinusoidal surface can be ac-
curately calculated in the frequency domain. The wavenumber normalized by the
radius of the cylinder is varied from 0.266 to 7.98, with a step size of 10.64 x 10‘3,
resulting in 726 data points. The time domain response is obtained by first window-
ing the frequency domain results using a double cosine function then calculating the
IFT. Figure 2.39 shows the time domain response for the field scattered directly by
the cylinder and indirectly through cylinder/surface interactions. In order to better
view the multipath effect, the direct clutter response of the sinusoidal surface has not
been included (and must be added if the total scattered field is desired). In the time
domain response of a cylinder above an infinite flat surface, three main reflections
occur. The first signal comes directly from the cylinder, the second signal comes as
a summation of cylinder-sea and sea-cylinder reflections, and the third signal comes
from the sea-target-sea reflection. For a cylinder above a rough surface, all three
of these reflections occur along with multiple reflections from the surface roughness.
A detailed description of the relationship between surface roughness and transient

multipath will be presented in chapter 4.

2.5 Conclusions

To study the multipath problem for targets above the sea, several sea surface mod-
els have been considered. In this chapter, theoretical analysis were conducted for a
arbitrary shape target above a finite rough surface and above an infinite flat surface.
Basic numerical results are obtained for a cylindrical target above a finite flat sur-
face, a finite sinusoidal surface, an infinite flat surface, and an infinite flat surface
with finite-width roughness superimposed. It is concluded that, the finite sea models

are not suitable for multipath analysis due to reflections from the edges and interac—

71

 

.N
o

_.._. hs/d=0.125
_— hs/d=0.0

_s
C
I l

 

P
o

 

 

J.._ l

i.

Scattered electric field (relative)
\

.11
'o

 

 

 

O
_I.
N
01)

time (ns)

Figure 2.39. Transient scattered electric field from a cylinder above a flat surface
and above a sinusoidal surface, a = 0.0127 m, d/a=8.0, ht/d = 1.0, z,/d = 0.0,
w/d = 9.0, 43,- = 20°, (1), = 20°.

72

 

 

tions

rough
profit
the p:

riodic

pm 11
above

techh

tions of this reflections with the target. Although the infinite flat surface with finite
roughness superimposed did not have that unfortunate behavior, a more realistic ap-
proximation to the sea was considered to study the multipath phenomenon. Hence,
the problem of electromagnetic wave scattering from a cylinder above an infinite pe-
riodic surface excited by a TM polarized electromagnetic wave has been solved by
using a perturbation method. A rule for determining the truncation width for the
perturbation method has been developed. The scattered electric field from a cylinder
above an infinite flat surface was solved by the image technique and the perturbation
technique as a function of truncation width. It has been shown that selection of the
truncation width is highly dependent on the cylinder height from surface, but if the
pr0per width for the cylinder height is selected, the results from the perturbation
technique match well with the image technique results. The perturbation current on
a finite sinusoidal surface has been calculated and it was shown that the truncation
width is consistent with the infinite flat surface. Therefore, the truncation rule for
the infinite flat surface can be used for a sinusoidal surface as well. The perturbation
method is a useful tool for studying transient multipath for targets above a periodic
surface because of its inherent efficiency. A detailed study of multipath for infinite

periodic surfaces will be presented in chapter 4.

73

 

 

 

CO.

3.1

T0 pr.
lariza'
Shite
Yemc
above
Scam

“lib:

IQ

‘ an:
mfldi

lhda

CHAPTER 3

COIVIPUTATION OF TE SCATTERING FROM TARGETS ABOVE
SEA SURFACES

3.1 Introduction

To provide complete information about transient multipath, in this chapter TE po-
larization is considered. An experiment in an actual ocean environment was done by
Sletten [20]. It was observed that at low-grazing angles the ratio of the horizontal to
vertical signal levels can greatly exceed 0 dB. The multipath problem for a cylinder
above a sinusoidal surface has been studied previously. Cottis et. a1. [3] computed the
scattered E—field from a cylindrical scatterer buried inside a two-layer lossy ground
with sinusoidal surface for a vertically-polarized incident electric field. The problem
was solved via an integral equation approach combined with the extended boundary
conditions method. However, all analysis was done for one frequency. Therefore, the
transient multipath phenomenon was not investigated.

In order to learn about the basic principles of transient multipath, as in chapter
2, analysis has been done by considering several simple problems such as a perfectly
conducting (PEC) cylinder above a finite rough surface, above an infinite flat surface,

and above finite roughness superimposed on a flat surface.

3.2 Scattering from a Cylinder above a Finite PEC Surface

Here the sea surface is represented by considering the simplest model, which is a
finitesized, two-dimensional PEC surface. Scattering of a TE (vertically) polarized
electric field E‘(a:, y) as indicated in Figure 3.1 is analyzed first in the frequency
domain and then an inverse Fourier transform (IFT) is used on the spectral results.

The scattered electric field is calculated by applying boundary conditions on the sea

74

surla

for t]

3.2.

Th
ha

KI

surface and the cylinder, and obtaining the Electric-Field Integral Equations (EFIES)
for the induced current on the sea surface and on the cylinder.
3.2.1 Electric Field Integral Equation Solution

The Magnetic Field Integral Equation (MFIE) formulation given in Appendix A is
more suitable for TE polarization [21]. However, the MFIE can be applied to only
closed surfaces [22], and this is explained in detail in Appendix B. The problem con—

sidered here consists of both open and closed bodies; therefore, the EFIE formulation
is required.
3.2.1.1 Obtaining Integral Equations

The incident electric field is assumed to be a TB plane wave with angle ab,- from the

horizontal

E‘(:v, y) = E0 (—:i: sin d),- + g)cos ¢,-) ejk°(“°s¢"+y3i"¢‘) (3.1)

This field will produce a scattered field which can be written in terms of vector and

scalar potential functions
in?) = swim) — V<I>(fi) (3.2)
where 5 is a two-dimensional position vector
5 = 1:5: + yr] (3.3)

representing the field points. The vector potential A“? ) satisfies the two-dimensional
Helmholtz equation
V2110?) + 133(5) = molar?) (3.4)

75

PI

PEC

 

 

 

 

 

Figure 3.1. Geometry of the problem.

76

l
l
l

 

where
50?

respec

Where

andc

After
exits.

l
with»

(x;
E

where RU? ) is the unknown induced surface current. Similarly, the scalar potential

<I>(ii ) satisfies the two—dimensional Helmholtz equation.

V2<I>(fi) + k3<1>(fi) = {01(5) (3.5)

where g(ii ) is the unknown surface charge.
Solutions of (3.4) and (3.5) will give the general expressions for A(fi' ) and <I>(p" ),

respectively

”()5 =uo/K(p') CW") )dv’ (3.6)

<I>(fi)= 20 fr 101 '0) (pm W (3.7)

r7 ’ = x’i + 31’? (3-8)

and G(/')', 5 ’) is the two-dimensional Green’s Phnction given in (2.10).

The scalar potential can also be written using Lorentz’s condition

 

<I>(fi) = j v - 1(5) (3.9)

After substituting (3.6) into (3.9), upon substituting (3.9) and (3.6) into (3.2), the

expression for the scattered field can be written as

EMfi=— %[a/K(“)W (kw- pwv

+Vv - [ Km") H3 )(kolfi- MW] (3.10)
1‘

where I‘ is the perimeter of the scatterer.

Since the incident E—field has only a: and y components, the induced current on

77

Will

the

will

HE]

Pfii

the scatterer has only a: and y-directed components. Therefore, the current can be

written by using the tangential unit vector as
Km") = K03”) v (3.11)

where ’7’ is a unit vector tangent to the scatterer perimeter at the source point. After

substituting (3.11) into (3.10), the divergence term in the expression becomes
V - {WW (SW/cow— m} = Kw'w. {WW/cow— 5'0} (3.12)
where
v - {VHWOIK— 5'») = H52)(ko|fi'- fi’lW - A7 + v - VH52’(koI5— fi’l) (3.13)

Here, the unit vector ’y’ tangential to the scatterer perimeter is function of source

point if ’. Therefore,

V - ‘7’ = 0 (3.14)
Also,
ngZ’woR) .

dR R
= —ko R Hf2’(koR) (3.15)

 

VH52)(kolfi- 5'!) =
where R = |fi'— fi’l and R = R‘1(fi'— 5’). After substituting (3.14) and (3.15) into

(3.13), upon substituting (3.13) into (3.12) leads

v . {K(fi')~‘/Hé2’(kolfi- 5'0} = —ko(5/ - K)K<fi')H£”’(koR) (3.16)

78

As a result, the expression for the scattered field is

-O

53(5) = cr(5,5')K(5') (3.17)
where £p(p',p "') is an operator defined as

55(5,5')K(5') = 37—0 [4“] K (5 ') 5' H52’(kolif- 51W

—v f K '<) w K) H(2(kolfi- MW] (3.18)

For the problem considered, (3.17) can be rewritten in terms of induced current

on the cylinder and on the surface separately. That is

E’(5) = ES<5>+ET(5)

= 555(5,5.!)KS( 5') + 55(5.571)KT(5T') (3.19)

where S and T symbolize the surface and the cylinder respectively; accordingly K S ( p’ )
represents the induced current amplitude on the surface and K706 ) represents the
induced current amplitude on the cylinder. Here, the induced current amplitudes

define the induced currents as
KS(5.~'> = K5055) (3.20)

K7157!) = 5' KT(5T') (3.21)

where 12' and 13’ are two—dimensional tangent vectors to the sea surface and to the
cylinder at source points, respectively.
A set of coupled EFIEs for the induced currents can be found by using the bound-

ary condition of zero tangential total electric field over the surface and the target

79

since the scattered field is known in terms of the induced currents. The boundary

condition for total tangential E-field on the surface is
5 ~ 135(55) + a - ET(5s) + a - 13"(55) = 0 (3.22)

The boundary condition for total tangential electric field on the cylinder must also

be considered. It can be written as

a - 155(57) +0-E’T(5r) +5 - E‘"(5r) = 0 (3.23)

Here, 55 and fir are two-dimensional position vectors to the observation points on

the sea surface and on the cylinder, respectively. Also, 22 and 2“) are two-dimensional

unit vectors at observation points tangent to the sea surface and to the cylinder,
respectively.

Substitution of (3.19) into (3.22) and (3.23) yields a pair of coupled EFIEs for the

current on the sea surface K S (53) and on the cylinder KT(p'r)

a - 55(5s,5s')KS(5s') + a - 55(5s,575)KT(5T') = —a . E‘(5s) (3.24)
5 - 53(5r, 5;)KS(5S') + 5 - £1,455. 57¢)KT(57=) = —5 - E"(5r) (3.25)

After using (3.18) in (3.24) and (3.25), the EFIEs become
55(5S,5§)KS(5§) + “(53,573KTW'15) = ‘11 ' Ems) (3-26)

53(55. 5;)KS(5S') + 55(55, 57!)K"(5,=) = —5 - 13"(55) (3.27)

80

where

Here. I

Where "
”l is lh<
3.2.1.2
IO 501V
PTOXim
of Mom

where the linear operator £r(p‘, p‘ ’) is defined as

maxim/7') = £9 [5%(fi,fi’)K(/3") — %£€(fi,fi’)K(fi’)] (328)

Here, £“(p, p’) and £5(p,p ’) are
‘F(“,fi’)K(fi") = a r Km w - a) Hé2’(kow— 5'1) W (3.29)
do: “')Kw') = PKW’) (‘1’ - R) Hf2’(ko|fi- fi’l) dv’ (3.30)

where "7’ is the unit vector tangent to scatterer perimeter at source point p’ ’. Similarly,

")7 is the unit vector tangent to the perimeter at the observation point ,5.
3.2.1.2 Moment Method Solution

To solve the EFIEs for the unknown currents, the sea surface and the target are ap—
proximated by a number of linear segments as shown in Figure 3.2. Then, the Method
of Moments starts by approximating the unknown currents by a linear combination

of expansion functions as

Ns-l Ns-l

K3065) = 2 Kim)- -— m: [cix,f,(u) +d§,,y,§,(u)] (3.31)
m=1 —
NT—l NT— 1
= Z Kflv): :éc[7‘x7‘( v() + dIy,T(v)] (3.32)
n=l
where N5 is the number of points on the sea surface and m = 1, 2, 3, . . . , N3 - 1 are

the indices of the segments on the surface. Similarly, NT is the number of points on
the target and n = 1,2, 3, . . . , NT — 1 are the indices of the segments on the target.
Also, u is the arclength distance on the surface measured from U1. Similarly, v is the
arclength distance on the target measured from V1. The expansion functions Xflu)

and y,§,(u) are defined on m’th segment of the surface. Similarly, «1’: (v) and yflv)

81

 

y A s Segment Ln

 

 

 

Figure 3.2. Segmentation of the sea surface and the target.

82

 

are II

are sl

Here,

distal:

Palm

are the expansion functions on n’th segment of the target. The expansion functions

are shown in Figure 3.3, and they are given mathematically as

1
X,f,(u) = A5( +1 u) u

(3.33)
0 otherwise
1
S X—(u — Um) U E Lm
37mm) = S (3.34)
0 otherwise
—1—(Vn+1 — v) v E L,
x301) = AT (3.35)
0 otherwise
1
T ECU — V”) 'U 6 Ln
3),,(11) = T (3.36)

0 otherwise

Here, the matching points on the sea surface are generated by making the arclength
distances between each two adjacent points on the sea surface equal to one another.
Therefore, the widths of the segments are represented by AS. The same method of
partitioning is used on the cylinder and the widths of the segments on the cylinder
are represented by AT. Also, Lm is the m’th segment on the sea surface and Lu is
the n’th segment on the target.

Before doing further analysis, the boundary values of the induced currents should

be considered. The boundary values of the current on the sea surface are

K1$(U1) = 0

K§._.(U~.) = 0 (3.37)

83

 

vs /\
:VT 0’ ‘. T T I" .‘ T
n+1,’ "a yn XnI.’ ‘.. yn—l
o \. 'l \
I .\ ” ‘Q
I ‘s I‘ \‘
;-< 1/2 ~
I .t
I' ‘-
.’ ’\
I .s
Vn+1”. \. n

 

 

 

 

} ‘ l l
AT U” 0 _91
2 2
1

0‘ '\

S I. ‘\ S S ." \. S
Xm+1 '0 \‘\\J)m XTTLIO" \\ ym—l
’0' ‘\\. "" .\\

\ .' s
‘.-(\1/2 \
."' .\\
l' '\
Um+1 ." \\ Um

 

 

 

I I I
I l l

é; “mm 0 _fi £13
2 2

 

>

 

Figure 3.3. Expansion functions for the sea surface current and the target current.

84

Therefore, the expansion coefficients become

cf=0

s
st

_1 = 0 (3.38)
for the surface. Also, the induced current on the target has following relation
KlTWI) = K5,.-1(V~T) (339)

Therefore,

cg" = ((15.4 (3.40)

In other words, the induced current vanishes at the edges of the sea surface. Also,
since target is a closed surface, the last segment ends where first segment begins.
Therefore, the current at the starting point and the ending point has to be continuous

for the target. Substitution of (3.37) into (3.31), and (3.39) into (3.32) leads to

Ns—l Ns-l IVs-2
Z KiW) = Z ciXiW) + Z diyiht) (3-41)
m=1 m=2 m=1

NT—l Nf—l Nf—Z

Z K3."(v) = Z cZXflv) + Z d532,? (v) +cTy£._1(v) (3.42)
n=l 13:1 71:1

Because of the representation of the induced current on the surface, the expansion
functions X15 (u) and y,§,5_1(u) are not necessary.
The weighting functions are chosen according to Galerkin’s approach [23]. There-

fore, the weighting functions are the same as the expansion functions. The weighting

85

 

function for the sea surface is

32,501) u E Li
WiSW) = X,i,(u) u 6 L,“ (3.43)

0 otherwise

which is shown in Figure 3.4a for the sea surface. Here, 2' = 1,2, 3, . . . , N 5 - 2 are the
indices of the points on the surface. Similarly, the weighting function for the target

is

r 313(1)) 21 6 L,
1 S£< NT-l Xlrilw) 1) EL!“
0 otherwise
W,T(v) = ( (3.44)
Mfr—1(1)) v E LNT_1
£=NT—1 X1T(v) 1161/1
( 0 otherwise

 

which is shown in Figure 3.4b and Figure 3.4c for different segments of the target.
Here, Z = 1, 2, 3,. . . , NT — 1 are the indices of the points on the target.

After multiplying each side of (3.26) with Wflu) and integrating over the perime-
ter of the surface, and multiplying each side of (3.27) with Who and integrating
over the perimeter of the target, after that using the expansion of the currents, the

coupled EFIEs become a set of linear equations

[S du VViS(u) (fished) :1 K30“) + ore. 12') 1:1 K3019]

= _ / du W,S(u) a - 5‘04) (3-45)
S

86

 

 

 

 

 

 

 

 

 

L .——>
U1 UNS
A eT
l q»—
T
yNT—l
l'LNT—l
’U x - ’9
V2 Vt! VNT—l VNT

V1 Vt {4+1 l?!” {/N; V1
(1))

Figure 3.4. The weighting functions (a) for the sea surface, (b) for the target on a
interior node (c) for the target on the first or the last node.

(0)

87

wh

Adv WlT(v) [£s(v,u') Z: Kflu') +£T(v,v') Z KIM)

= —/ dv WtT(v) 13 - Ei(v) (3.46)

The integrals over the sea surface and target perimeter may be evaluated after

substituting (3.28) into (3.45) and (3.46)

4 _..
15‘; + 13% + 15‘»; + 1359, = a; [g du W,S(u) it - E'(u) (3.47)

L91~+Irflr+lr§ +I,~l9 = iv/‘dv WtT(v) fi-E‘W) (3.48)
770 7'

where the integration terms 15%, 15%, 15%, 153., I73}, 153., L139, and Ire; are

Ns—l
13‘; = [5 du W,S(u)cg(u,u') Z K5411) (3.49)

d Ns-l
Is”; = — S du MSWEEQW, u’) 2 Kim (3.50)
155} = [gdu W,S(u)[.f,’~(u,v’) Z K3;(v') (3.51)

d NT—l
15%: — / du W.S(u)g;¢(u.v'> 2: KW) (3.52)
1,9, = dev WlT(v)£%(v,v') Z K:(v') (3.53)
11% = — [T dv W?(v)d%£%(v,v’) 2 KW) (3.54)

Ns—l

‘39 = [Tdv WtT(v)£§(v,u') Z Kflu’) (3.55)

88

 

The

become

Where

   

Ns—l

17%: fdle()—L°vu)ZKS

—l

(3.56)

The terms with B superscript can be evaluated by integration by parts. Then they

become

where

Isgzlsfs'l'lsbs
Isg‘zlser +1391"
I73=Ler+1¥r

If =Irs+zxis>

Ns—l

Iss=fdu ”3‘ 35621436.!)

 

 

 

u(i+2) Ns—l u(i+2)
135 = -WS(U) [030610 2 Ki(U')])

"(0 ”=1 “(1')

NT— 1
13914 _[gdudvg 3:5") Lfiw ’U’) "E KT('U

u(i+2) NT-l 110+?)
23 = —w.S<u> (can. v) 2 14(4)]

11(3) n=1 u(i)

 

 

NT—l

Ia = fdv d—fi‘”) cam) 23 Km

v(l+2)

1,9, = _wm

 

«(+2) NT_1
[@(vw’) z K3(v’)]

v(l)

 

6(1)

Ns—l

“5:de dWi—LU)LT(U ,u')ZK,§,(u')

v(l+2) NS_1 v(t+2)
1;. = —wz'<v) [4464) 2 Km]
m=1

v(l)

 

 

v(l)

89

(3.57)
(3.58)
(3.59)

(3.60)

(3.61)

(3.62)

(3.63)

(3.64)

(3.65)

(3.66)

(3.67)

(3.68)

 

Since
with s

to I;

The

Since the weighting functions at the starting and ending points are zero, the integrals
with superscript b become zero. Hence, the integrals 753%, 132,135., and 1,1. are equal

to Is‘fg, 15$}, 1,5}, and Ir‘g, respectively. They may be written as

 

 

 

N3- 1
Isngdu dug“) u,u')ZKS(u')
Li
S( mN:- l
+ [L ldud—Wé—éim d);KS(u ’) (3.69)
4+
15 = / du 4M5“) £30. (,v')NT:1K,T(u')
ST L,- du
"N'lr— l
+ / du WSW 0,6') ZKT(’U) (3.70)
Li+l
dWT( NT- 1
1,2, = / 361M): K50)
L; d?)
dWT( "T‘ ‘
+/L dv 5+”) v,v')ZKT(v’) (3.71)
(+1
N 1
dWT( 5
IrTs— — / dv —-‘— 2220:1430!)
”‘ZCU mNs— l
+ [L (100,102 K5(u (3.72)
(+1

The derivatives of the weighting functions can be easily calculated using (3.43) and

90

 

(3.0).

Where [1

‘lfler SI]

4 “_

84ng l[

(3.44). They are

 

 

 

 

_dyflu) u E L.
du '
dW,S(u) dX-S u
_.._..du __ 2.1.5 ) u 6 [4+1 (3.73)
0 otherwise
T
( dyz (‘0) v G Ll
(17)
< —- dd” (1’)
1_€<NT 1 (cl: ”€14+1
0 otherwise
dWflv) )
= (3.74)
dv
dyT _ v
6: NT — 1 dxlT(U) 'U 6 L1
dv
k 0 otherwise
where the derivatives of expansion functions are
S -i E L
din“) = As u m (3.75)
u 0 otherwise
5 3— e L
60:7;(1‘) = A5 u m (3.76)
it 0 otherwise
T —i e L
_d": (v) _ AT 0 " (3.77)
v 0 otherwise
T 1 E L
dyn ('0) = AT 0 " (3.78)
dv 0 otherwise

After substituting the expansion of currents (3.31),(3.32) into (3.28), upon substi-
tuting into (3.69), (3.70), (3.71), and (3.72), and after that using the value of the

91

 

 

where

are

Where

tion p.

Therel

weighting function derivatives, the integrals I 5 become

 

Ns—l N3—2
15% = 2 cf. [77.15. — 7731...] + 2 df. [77.12. — 7723.....]] (3.79)
m=2 m=l
NT—l NT-l
15?, = c, [7731 71:1,,“ Z .13; [713; 773,4] (3.80)
11:1 15:1
“NT—l NT—l
7.7. = 2 c3: [77.3 - 773...] + 2: .7; [74,. - 777..,.]] (3.81)
_ nzl n=l
Ns—l N5-2
I7? = [ 2: Cf" [Him — Hithml + 2: dis); [HZm - H{+l,m]] (3'82)
m=2 m=l

where ’H’s are double integrations of the Hankel function over two segments. They

are
l

X — .—
7‘15,"; —' AS

/ xg(u')(6m . R...) H1(2)(koR(u,u')) du’du (3.83)
L,- L...

where the function R(u, 21’) measures the distance between source point and observa-

tion point. It can be written in terms of integration variables as
R(u, u’) = Iii — 11"] (3.84)

Therefore, the unit vector R“: can be written as

:1
I
:1

A

Rmu’ =

 

l (3.85)

fit
I
=17

“33" T Ail. /. yiw'xa... - 7‘7...) Hi2)(k0R(U,u’)) du’du (3.86)

743‘, = i / x3(v')(6...a,,.) Hf2’(k.R(u,v')) dv’du (3.87)
’ AS L.- L,.

"azif 3’3." (ma-1?...) H52’(koR(u.v')) dv’du (3.88)
’ A3 L.- L,.

92

775‘, = i / x3(v')(5., . it...) H§2’(k.R(v,v')) dud. (3.89)
’ AT L; I...
77;”, = -1— ] y,'{‘(u')(9,, .13..) Hf2)(koR(v, 5)) dv’dv (3.90)
’ AT Lt Ln
1

=_/ X3.(u')(7lm~172.,u7)H52)(koR(v.u')) du’dv (3.91)
Lm

HZ", = if yi(u')(71m ~ Rwy) Hf2)(koR(v,u')) du'dv (3.92)
Lm

After substituting the definition of the weighting functions (3.44) into (3.49),
(3.55) and (3.45) into (3.51), (3.53), the integrals 1373., 135)., L9,, and If; become

Ns—l

155: [L du 375(77) £§( (u ,u’) E Ks(u
m=1
N3— 1
+/ du XiiiW) £"( (u ,u') 2:1 KS(u (3.93)
Li+l
NT— 1

1.577: [duyS(U) )LTLIUU);KT(U)
NT 1

+f du X,5+1(u (')L“( (U ,v') E: KT(v (') (3.94)

”717—1
139,: L‘dvyfl cmmZKTw)

INT- 1
+ dv Xgfiv) £5}(v,v') Z K:(v') (3.95)

17:44

Ns—l

137%: Ldvyf(v)£°‘(u,u)ZKS(u')

m=1

93

 

Now I

that 5

Where

Ns—l

+ dv Xt+1(v g5,(u u') 221 Ks(u (3.96)

Lt+1

Now the expansion of the currents (3.31), (3.32) can be substituted into (3.29). After

that substitution of (3.29) into the integrations above leads to

 

 

Ns—l IVs—2
153 = k..[ z 3.32:... + 65.11,...) + :3 015.321... + e3...) (3.97)
m=2 m=l
“NT—l NT— 1 -
Isa=ko Z cZ[f23:.+ ]+ Z d: [£11. + 5:13]j (3.93)
_ n=l
TNT—1 NT- 1 1
Ir? :13 Z cZ‘[§2,’..+§ZH,].. + m [gz’,..+giiln], (3.99)
_ n=l n=1
Ns—l Ns-2 ~ ~
Ir? = ko[ 2 c3. [th + ham] + 2 d5. [th + ham] (3.100)
m=2 mzl
where
é”... = f d“ 3%) / Xflu'xa.-am)Héz’(koR(u,u’))du' (3.101)
L,- Lm

éfilm- —/ du 33%)] x,:‘( u)(u.-+1-am)Hg2>(koR(u,u'))du' (3.102)
=/ d" 3%) / yiw’xm-am)H32’(koR(u,u'))du’ (3.103)
L; L".

éfil,m=/. d“ Xiid“) L y,‘§,(u)(ui+1 um)Ho2)(koR(u, 14))(h‘ ' (3-104)

= / du MSW) / xJ(v')(a..3,)H52’(koR(u,u'))dv' (3.105)
L,- L.
31,7; = L. d“ XiidW/L- X:(v')(fi,-+1~fin)H((,2)(koR(u,v’))dv' (3-106)
1'3 = f d" 3’? (u) 3’3(v’)(fii -fin)H52’(koR(u,v’))dv’ (3-107)
L,‘ Ln

94

 

COUp

3;,"— —/L d0 x5, (u)/( y,T( )(u,+1-0,,)Hg2’(k012(u,0'))dv' (3.108)
5;”: [L d0 yflv) f1. x,’{‘(v')(0,.0,,)H32)(koR(u,v'))dv' (3.109)
t n
9:: = f dv 3,1,0) / XI(v’)('D¢+1~13n)Hé2)(k0R(v,v’))dv' (3.110)
(+1 in

92TH / dvyé’mf yflv'xm-9.)Hé”’(koR<v,v'))dv' (mm)
L; L.

9:: = / dv 291102) [n 3239391.] 339,130,130, v'))dv' (3.112)
53;, = 81d” yz"(v) 1.. xg(u')(0, . 0m)H5,2)(koR(v, u'))du’ (3.113)
555,1," = L... dv 3:1,,(0) ft. m (mt/x011, .0m)H,§2>(koR(v,u'))du' (3.114)
hem: j; dvy,(v)[anys(u')(1}t~fim)H(§2)(koR(v,u'))du' (3.115)
71211,", = L1H dv Xt+1(v)L/Lm ym( )(U(+1 um)H02 )(lcoR(v,u'))du' (3.116)

After substituting (3.97), (3.98), (3.99), (3.100), (3.79), (3.80), (3.81), (3.82), the

coupled linear equations in (3.47), (3.48) become

Ns-l

 

 

 

F -
Z cfn k0[é:},,m + gig-hm] + ”5m —Hix+l, m
m=2 ' .
+ 2: dis; [kdérm + gig-1,770] + HZm— ”:41, m
NT—l )- -
+ 2: C: k0[f?,:1+ i+1,n] + Hixm _HiX-H, n
NT-l - -
+ Edi]: kaU‘iY i,n+ i+l, n] +7137; —Hi+l,n
n=1 .1

4 S “i
=7?— du w (u )0 E (11) (3.117)
0

95

 

All:

{0 be c.

the C011

 

 

 

2: C: howl}; + gig-1,71] + Ht); _ ”ii-1,11
n=1 - J
NT—l 1
+ d: [led-6Z7: + gig-1,71] + ”Zn — 7.1;;th
n—l -
Ns-l 2'

+ 2 C51 k0[h’Zm + h£1,m]+ Him — ”£1,771
m=2 - .1
IVs—2

+ Z dfn 1“”:th + hzl,m] + HZ"; _ ”(IS-1,171
m=1 - .

_4
—770 Tdv WIT (v) v E‘(v ) (3.118)

After substituting (3.43) and (3.44) into (3.117), (3.118) and forcing the currents

to be continuous on each segment of the sea surface and the target using

S S
dm _ Cm+l
dT _ T

n _ n+1

the coupled linear equations become

Ns—l

X

77122

m

 

F
CS k0[ei, m + eix+l, m] + Hixgn

+ E2 cm-H [kdéfi m + ei+l ,m] + Him!—

:1
NT— 1
+ Z of:
71:1
NT-l
E : T
+ cn-H

=1

_fl/L .33 . E...) .. M) m

[k0[f:'},:r+ i+1, n] + 715(71—
lkotfl’,:.+- ".1 + H2.

 

96

(3.119)
_Hix+1,m

.J
HH—Lm
71591;;

J
_Hi+l, n

.1

(3.120)

 

 

After 1

NT-l .-
T VY v X X
2 Cu k0 [91,71 + 9144,11] + Htfll — HI+IJJ
n=l ‘ .
NT—l )- "
T ~Y ~ Y Y
+ Z 67““ led-qt," + gl+1,n] + ”(at - Hl+l,n
n=l - .

 

”5.1 F _,
S "Y ”X X X
m=2 - .l
—2

'.'Y ~X Y Y
+ NZ:SCm+1-k0[hl’m + hl+1,m] + ”Am — Hl+l,m

=1

 

 

_..

4 - . . '
= —[/ dv 372(1)) f1;- E‘(v) +/ dv X£1(v)vl+1~E(v)
’70 L. L... .

 

After manipulation of indexes, they become

Ns—l

Z 675;: [k0[é:m + gala-hm] + ”gm — 7.1511,?”

"1:2 ..

Ns—l .

+ Z cfn [k0[ei,m— l+éix+l,—m 1] +7{gm—l _Hi+l,m-1J
m=2

NT—l .
+ Z cZ.‘[ko[f..+ 2...] +H2,‘. -21H.,,(
n=1

NT .
X
+2671 [koLfiv 1,-n 1+ i+l,n—1]+H1n1H:+lnl
n=2

.2

 

=%[/L.d" y2(u)&.~-E"(u)+/Lmdu X.i.(u)ut+1 EM:

2 6,71: [kothm + g£+1,n] + ”in _ ”ii-1,11

.J

 

NT _
T ~Y ~ Y Y
+ 2 cu [k0[g(’n_1 + gt+1’n_l] + Hl’n_1 — ul+l,fl—1J
n=2

 

Ns-l ..

+ 2: vi [9921. + hi1.-1+ H2... — H21...

m=2 .1

Ns-l .
+ Z 6?. [ko[h}fm_, + (121..-.1+ HZ".-. — “Em-t
m=2 .

97

(3.121)

(3.122)

 

The

Aflt

equ

[ dv 372(1)) 13; ~ Ei(v) + dv Xgfiv) 131,11 - E‘(v)]
no Lt [4+1

(3123)

The unknown current amplitudes on the surface are renamed as a5, and defined as

S _ S
am_cm+l

(3.124)

After substituting (3.124) and (3.40) into (3.122) and (3.123), the coupled linear

equations become,

am [k0[ey 81' ,m+l + eiX-H ,m+l + éi,m + e1°X+1,m]
Y
+H1'X ,m+1 -H1'+l ,m-H + H1,m— Hit-1,171]
T “Y ”X “Y "X

+ C1 [kolfi,1+ fi+l,l + fi,N7~—1 + fi+1,NT—1]

+2,‘.H —H2S.. . + H. _... . —H2’.. _... .]
NT—l

T ”Y ”X ”Y ”X

+ Z Cn[k0[f1',n + f1+l,n + fi,n—1 + fi+1,n—l]
11:2

+1X,nH —H1+1,n + ”1', n— l _H1'+1,n— 1)

1

4
—[/ du y.S( (u) u,- E’( (u) +/ du Xiilm) 11,-“ E (u)
"0 L1L1+l J

 

T vY vX ~Y ~x
01 [kalgm + gum + 94027—1 + gl+l.N'r-1]

X X Y Y
+7141 — ”144,1 + ”ANT-‘1 _ Hl-HJVT—l]
NT- 1

+26 :[koqu " + gill.” + gZfl—l + gime—l]

X X Y Y
+Hl,n — ”(44,11 + Him—l — HH—lm-l]

98

(3.125)

IVs—2

+ 2:103[k0[hl,m+1 + h£+1 ,m+l + him + hl+1m1

x x Y Y
+Hl,m+1 - Hl+l,m+l + Ham “ HH—Lm

 

: %[f d” 3717(1)) ’91 ' EV”) +/ dv Xi’ldv) 17£+1 1373(1)) (3-126)
Ll 14+!

.1

Now, the coupled linear equations can be written in matrix form as

Aim: Ban 05. bis
= (3.127)
01,". D5," C}: b;

where A,” is an NS — 2 x N5 — 2 matrix, Bi," is an N3 — 2 x NT — 1 matrix, Chm
isanNT—l xNS—2matrix, and DI," isan NT—lxNT—l matrix. To make
calculations faster for the matrix elements and bf, bf, a change of variables are used

for the integrations on the target and sea surface

u=um+Um+é§i uELm

(3.128)
’U=’Un+Vn+—A—21 vELn
Then, bf and bf are
4 921
S _’ — 1
b” ’ vol/.esdu‘A—J [AS/“ml “i EW
95.
2 .0.
+/A dui+1A — :[AS/2—uH1] 11.41 °E'(u,-+1)] (3.129)
_s
4 1 _..
T _ _ __ ‘ . 3
bl — 770 /:dvt AT [AT/2+vl] ’Ug E(v()
91
2 1 . "1’
+ / dvt+1 ——[A7~/2—v,+1] vz+1°E(vz+1)] (3.130)
-921 AT

99

where the integrations can be approximated using rectangular rule. Substituting (3.1)

yields

big = 2:S[ [My 0050151) “ “2,: Sin(¢i)] ejkomcowfiyismm

+[ui+1,y cos(¢,~) — uHm sin(¢,-)] ejk°(“+‘c°s¢‘+y‘+‘3i"¢‘)] (3.131)

 

2A , . ‘
bf : ——T[ [’Ugm COS(¢,-) —v(,$ sin(¢,-)] eJko(zzcos¢2+y¢sm¢,)

770

+['U(+1,y COS(¢.~) — ”(NJ Sin(¢g)] ejk°(3‘+'°05¢i+yt+18in¢i)] (3.132)

where um and 11.3,, are the components in the a: and y-directions of the unit vector
11 tangent to the surface in segment Li, respectively. Similarly, v4; and '04,, are the
components of the unit vector 1“) tangent to the target in segment Lg. They are

calculated from

x. _ x.

ui,:z: = 1+1A 1
S

y. _ y.

U1,y = *5:

$t+1 - x;

”(as = _—
1 AT

1141/ = gig—Q‘- (3.133)

After calculating b,- and bl, the matrix elements can be written as

_ 3y ox ~Y ~X
A1,m "' k0 [ei,m+1 + ei+l,m+l + 81,171 + ei+l,m]

+H1§m+l _ Hil,m+l + ”rm — ”Bil-hm (3°134)

100

 

Her
and

brill

vY " X "Y ~X
C£,m = k0 [hl,m+l + hl+l,m+l + hl,m + hl+l,m:|

+Hfm+l _ Highm-H + ”Em — Hr+1,m (3135)

n =1 k0[f1¥1+ 1114+ iYNT—l + Liam—1]
+7155 "' ”31,14. HKNT-l - ”Lump—1
31,71 = 4 (3.136)

2 s n 3 NT —1 1422:. + "3:... + 231+ 1.1:..-)

 

Y
L +an — ”511,71 + 7{In—1 " Hi+1,n—1
( VY VX ~Y ~X
n =1 k0 g(,1+ gt+1,1+ gl’NT_1 + gl+1,NT_l

+7131 ’ ”511,1 + HINT—1 _ HELNT—l
0,," = ( (3.137)

2 S n _<_ NT —1 k0 [ézn + fig-(Hm + fizn_1+ §(X+1’n_1:l

 

X X Y Y
L +Hl’n — I’ll—+11" + thn‘1 — Hl+l,fl—1

To calculate the matrix entries of Agm, (3.33) and (3.34) are substituted into

(3101-3104), (3.83), and (3.86). Changing the variable as in (3.128) leads to

A3/2 A3/2

U 1 A A
63,2.“ = "—2011 ° um+1)/ [As/2 + “MAS/2 — um+11
AS —As/2 -As/2

H62)(koR(u.-, um+1)) dum+1 du; (3.138)

Here, the function R measures the distance between the points on the source segment
and the points on the observation segment. The geometry is shown in Figure 3.5 for

both of the segments on the sea surface. Therefore, R(u,-, um) can be written as

R(u1'1 um) = lfi— 16." = 53,1 + ”1121' — fiS,m _ "mam (3'139)

101

X 1 .. Ass/2 133/2
€1+1,':m+1 2 —(ui+l um+1)/_ [AS/2 — u,-+1][AS/2 " um+1]
As 33/2 —A5/2

H3 )(koR(u.-+1,um+1)) dum+1 an... (3.140)

~Y 1 135/2 As/2
6"", = —A—2(’&i’u m)‘/: /[ [AS/2+u,][AS/2+um]
S

A5/2 As/Z

H52>(kOR(u,-,um)) dum du. (3.141)

~x As/Z A3/2
e...,..= 32 (.1... u... )f_ /_ [As/2—ui+ll[AS/2+uml

As/2 As/2
H(2)(koR(u,-+1,um)) dum dig-+1 (3.142)

X 1 Ass/2 As/2 A ..
Hi,m+l = —/ / (um+l ° R4,m+1)[AS/2 _ um+11

A2
S -As/2 -As/2
H1(2) (koR(u.-,um+1)) dum+1 (111.- (3.143)

A5/2 A5/2 .1
Hix+1m+1=A21/_ / (am+l ' Ri+1,m+l)[AS/2 - um+l]
As/2 As/Z

H12) (k0R(ui+laum+l)) dum+1 dUi+1 (3.144)

Y 1 As/2 A3/2 .. 2
H,m_ — F f / (am-11.,m)[AS/2 + um] Hf )(kOR(u.,um)) dum du. (3.145)
3 -As/2 -As/2

Ass/2 As/2( A
H1+l,m : A21] / (um ' Rd+l,m)[AS/2 + um]

A5/2 A3/2
H§2’(koR(u.-+l,um)) dum (111,-... (3.146)

102

:31

um Um

 

)0 S,m

 

~12

53,2

b1
1

 

  

 

0

Figure 3.5. Calculation of distance between points on the source segment and points
on the observation segment.

103

Similarly, the entries of Cl,m given in (31133116) and (3.91-3.92) become

”Y 1 AT/2 AS/2
h£,m+l = —(‘91 ' 11m“) f / [A'r/2 + ”(HAS/2 - um-H]
ASAT —A-r/2 -A3/2

Hg2’(koR(v,,um11))dum., at, (3.147)

where the distance functions can be written for the other segments similar to (3.139).
When the source segment is on the surface and the observation segment is on the

target, R(v¢, um) is

(3.148)

 

13(1):, um) = ‘51:: + 11117: — 53,711 — "mam

“X 1 A A Arr/2 As/Z
ht+1,m+1 = _(WH ' um+1) / / [AT/2 " ”HIMAS/z _ um+11
ASAT —AT/2 -A3/2

H32)(koR(’U(+1,um+1)) dun,“ d’U(+1 (3.149)

"Y 1 AT/2 Ass/2
h... = —(a. - am) / f 141/2 + 2145/2 + um]
ASAT AT/2 -A5/2

H62) (IcoR(v¢, um)) dum dv; (3.150)

“X 1 A . A'r/2 ASS/2
hm... = _(.,.. mm) [ [AT/2 + 2.114312 — um]
ASAT —AT/2 —A5/2

H32)(koR(v¢+1,um)) dum dvg+1 (3.151)

 

X 1 AT/2 As/2 .. ..
Hl+l,m+l = / (um+1 ' It!-+—l,m-t-1)[AS/2 — um+l]
ASAT -AT/2 -As/2

H12)(koR(v,+1,um+1))dum+l av... (3.152)

104

A'r/2 AS/2 ..
Him-+1: ASIAT ‘/; /; (um-+1 ' R£,m+l)[AS/2 — um+l]

A'r/2 As/2
Hf )(koR(v,,um+1)) dum+1 dv, (3.153)

Y 1 AT/2 As/2(
=—_— )A 2 m
Hm ASAT/ /_ u(,,. R....)[ 5/ +u ]

-A'r/2 As/2
Hf2 )(koR(v(,um)) dum dvl (3.154)

Y 1 AT/2 As/2( A
”(+1371 = m] / (um ' R£+l.m)[AS/2 + um]

—AT/2 As/2
H12) (koR(v(+1, um» dum d’UtH (3-155)

Then, the entries Bi," given in (3.105—3.108) and (3.87-3.88) become

 

“Y 1 43/2 AT/2
2...: A A (22.2.) f / [As/2+ui11AT/2—vn]
5 T -A3/2 --A'r/2

H32) (koR(u,~, vn)) dvn du, (3.156)

When the source segment on is the target and the observation segment is on the

surface, R(u.-, u") is

R(u.-, U") = 55,; + 11221; — fir," - ’Un’Dn (3.157)

 

 

”X 1 As/2 Ar/2
1+l,n = A SAT _(u1'+1 un )/ ‘/; [AS/2 _ ui+111AT/2 - v71]

As/2 Arr/2
H62)(koR(u,-+1,vn)) dvn du,“ (3.158)
“Y 1 As/2 A'r/2
2..-. = not - a.-.) f /_ [As/2 + 4141/2 + 3,.-.)
5 T -A$/2 AT/2

105

 

M
surf

X 1 A7‘/2 A5/2
Hun.” = m] / (um-H' Rt,m+1)[AS/2 _ um+li

"AT/2 A33/2(
(2)(k0R('U(, Um+1)) dum+1 d’Ut (3.153)

 

AT/2 A5/2( A
f / (um ° Rl,m)[AS/2 + um]

“AT/2 A5/22

Y :
Him: ASAT

1(2)(koR(v(, um)) dum d'Ug (3.154)

1 AT/2 As/2( A
”Xi-hm = M] /_ (um ‘ Rl+1.m)[AS/2 + um]

—AT/2 5692/2
m(koR(’U(+1,um)) dum do,“ (3.155)

Then, the entries Bi," given in (3.105-3.108) and (3.87-3.88) become

As/2 A'r/Z
f.” = —1—(a.~o,.) / /_ [AS/2+u.][AT/2—v,]

an AsAT —As/2 —AT/2
H(2)(koR(u.-, vn)) dvn dui (3.156)

When the source segment on is the target and the observation segment is on the

surface, R(u,-, un) is

 

 

R(u.~, U”) = 55,; + an}; - 57"," - ’Un’0n (3.157)
v 1 A5/2 AT/2
34:1,” = AS —(ui+1AT 7111)] / [AS/2 _ “HIMAT/Z - vn]
A5/2 AT/2
H‘2)(koR(u,-+1,vn)) dvn du,“ (3.158)
‘Y 1 As/2 AT/2
fs,n-1 = K—A—‘(fii ' {Ln—1)] / [AS/2 + WHAT/2 + Tin-1]
S T -A.s/2 AT/2

105

 

m,

H62)(koR(u,-, vn_1)) dvn_1 (111,- (3.159)

"X 1 A3/2 A3/2
fi+l,n— l: ASAT(ui-+-l Uri—1)]: /: [AS/Q—Ui+1][AS/2+'Un_1]

A5/2 Arr/2
H32)(koR(u,-+1,vn_1)) dvn_1 du,“ (3.160)

 

Ass/2 A'r/2( (2)
Hixm = / (vn Rim. )[A'I‘/2 — 711;] H1 (150120115, 7111)) (11),; du, (3.161)
ASIA’I‘ —A5/2 Arr/2

As/2 AT/2( ..
Hix+l, n : A—SIATI. /; (v71 ' R4+1,fl)[AT/2 _ v3]

As/2 AT/2
Hf2)(koR(u.-+1, v")) (11),, dui+1 (3.162)

 

As/Z AT/2( A
[Hay—n 1 — _lAsAT/ / (vn—l ‘ Ili,n—l)[AT/2 + vn—l]

As/2 AT/2
H1(2)(koR(u.-, vn_1)) dun-) du, (3.163)

Y 1 As/2 AT/2 . .
Hum-1 = m] / (”n—1 ‘Ri+1,n—1)[AT/2 + Un—l]

—As/2 -AT/2
Hf2’(koR(u,-+1,v,,_,)) c112,.-. du.+1 (3.164)

Finally, the entries of D3,, in (3109-3112) and (3.89-3.90) become

137/2 AT/2
ggjn— _ K; (v, v.) f /_ [AT/2 + mum/2 — 3,.)

AT/2 AT/2
H32>(koR(v,, v,)) dvn do, (3165)

When both of the segments are on the target, 12(1),, v") is
R(’U(, ’0") = 57‘" + U101 — fir,” — uni)” (3.166)

106

AT/2 AT/2
gig-1,7; = A12(Ut+1 Un )1: / [AT/2 _ Ut+lllAT/2 — vnl

AT/2 AT/2)
02),(koR('U(+1 7113)) dvn dvl-H

AT/2 A'r/2
an— 1 :31; (U1 {In—1)] / [AT/2 +U£llAT/2+Un—ll
A'r/2 A'r/2

H52)(kOR(’Ub vii-1)) dun—1 d’Ug

Arr/2 Arr/2
§£1,n—1=;—%(vl+1 ”73— 01- /; [AT/2 _ vl-HMAT/2 + Un—l]
Ant/2 AT/2

H52 )(k0R(U£+1,Un—1)) dUn—l dUt+1

AT/2 AT/2
HM- -— AlT/ /( (vn Rgn)[ )[Arp/2 — vn] Hf2 )(koR(vg,vn)) d'ufl d’Ug

AT/2 AT/Z

AT/2 137/2 .
”(+1, 73 = A21]; / (vn ' Rt+1.n)lAT/2 _' vnl

AT/2 Arr/2
Hi2 )(k0R(vl+l1vn)) dun dvt+l

Y 1 Arr/2 AT/2( A
Him—l = A—Z/ /_ (Un-l 'Rt.n-1)lAT/2 + Un-ll
T

—AT/2 A'r/Z
H? )(koR(v¢,vn_1)) dvn_1 dvt

AT/2 AT/2( A
Ht+1,n- l: A212]; / (”n—l ' Rt+1,n—l)lAT/2 + vn-ll
AT/2 AA'r/Z’

Hi2 )(koR(Ut+1,Un-1)) dUn—l dUt+1

(3.167)

(3.168)

(3.169)

(3.170)

(3.171)

(3.172)

(3.173)

The integrals in Am, can be computed numerically, with great effort. Instead,

107

they can be approximated using the rectangular rule to give

 

v A2 A A -o —o
‘32ij = '—S (“i ' um“) H02)(k0lp3,i — Ps,m+1l)
v A2 A A 2 -o —o
8:5de = 75 (“i-H 'Um+1) Hci)(kolps,i+1 — Ps,m+1l)
-y A"; .. . (2) .. ..
ei,m = ’4— (Ui ' Um) H0 (kOIPS.i — PS.m|)
~x Ag . . (2) .. ..
ei+1,m = T (“HI ‘ Um) Ho (kOIPS,i+1 - PS,ml) (3-174)
x AS - . (2) .. _.
Hum“ = 7 (Um+1 °R4.m+1) H1 (kolpss - ps.m+1l)
AS A A -o -o
”3.37. = 7 (Um ~R.-,m) Himwolpss - Ps,ml)
As - - .. ..
”film“ = 7 (Um+1 'Ri+1,m+1) Hi2)(kolps,i+1 — Ps,m+1|)
As . - 2 .. ..
”gram = ’2— (um 'Ri+1,m) Hi )(kOIPS,i+1:PS,m|) (3-175)
where R”. is defined as
it,” = ‘15" " ’35” (3.176)
lpS,i "' pS,ml

Using a similar calculation, the entries of Cm, become

EZm-H
Big-1,173+]
11):,"

flig- l,m
”if... 1
HZ",

X
711+ l,m+l

 

 

 

 

AS4AT (U: . am“) H32)(ko|l3r,t "' fis,m+l l)
Asfr (am .21....1) H32’(koli>'r,z+1 — 775,6!)
MM (6, . 6m) H62’(koli>‘r,z - 55ml)
AS4AT (13”, -’&m) H32)(kolfir,z+1 — 53ml)
955 (Um+1 'Rt,m+1) Hi2)(k0lfiu - 55""“D
92—5 (6... . 1‘1...) H§”’(kolfiu - 65,...»

As

_ (flm+l ' R£+l,m+1) Hi2)(k0lfir,l+l —' fiS,m+l|)

108

Fina

A A - .. ..
HEW. = —2-S- (um 'Rt+1,m) Hl2)(kolpr,z+1- ml) (3.177)

Also, the entries of Bi," are

 

A A A A _. ..
fit/n 2 84 T (71‘3" vn) H62)(k0lp5,i — mel)

 

 

 

 

£11m = AS4AT (71:41 ‘91:) H32)(k0lfis,i+1 — final)
$4 = AS4AT (ai ' 071-1) H32)(kolfis,i — firm—1|)
align—1 = A54AT (IL-+1 ° fin—l) H32)(k0l55,i+1 — firm—1|)
H53. = 921(an-1%,")Hf"’(kolfis,i—5T,nl)
”film = £21 (22,, 'Ri+1,n) Hl2)(kolfis,i+1 — 5m )
7133.4 = 9% (fin—1 Rim—1) Hl2)(kolfis,a — mil)
Haw = 92-7"- (aw am-» Hl2’(ko|fis,i+1-fir,n—1l) (3.178)

Finally, the entries of D3," are

“Y _ A%‘ A - H(2) k -0 -o
at," — T (ve-vn) o (aim—ml)
uX _ A’?‘ A A H(2) -o -o
92“," - —4— (”2+1'vn) o (koll’r,t+1 —Pr,n|)
~ A2 A A «o 4
93:“ = 71$ (v! ' vn—l) H62)(k0lp7‘,l — me—ll)
2
aim—1 = 3;— (771+1'9n—1)Hti2)(kol5r,t+1—5T,n—1|)
' AT . A .. ..
”if" = ’2— (can) Hl2)(ko|pr.e-pr.n|)
AT A A -o -o
”til,” = T ( n ' R£+1,n) Hl2)(ko|pu+1 ‘PI‘mD
AT . ~ .. ..
“Zn-1 = —2— (”n—1 'Rl.n—1) Hl2)(ko|PrJ—Pr,n—1|)
A A " .4 —o
Him—1 = ‘21 (”n—1'R£+l,n-1)Hl2)(k0lPT.l+l—PT,n-ll) (3.179)

109

 

He

the

3.2.1.3 Calculation of the Self-Terms

When i = m, i = m — 1, or 2' = m + 1 (the self-terms of the surface) the integration
paths are the same for the source segment and the observation segment on the sea
surface. Similarly, when 5 = n, K = n — 1, or 8 = n - 2 (the self-terms of the target),
the integration paths are the same for the source segment and the observation segment
on the target. As a result of that, the self terms are difficult to compute numerically.
Therefore, the self-terms should be approximated using the Hankel function terms for

small arguments. When 2' = m, Am, becomes

_ VY vX ~Y ~X
A7757” _ k0 [Emmi-H + em+l,m+l + em,m + em+l,m:|

+Hrfi,m+l + fl;,m—’Hm+l,m+1_um+l,m (3'180)

Here, the terms can be computed from (3.174-3.l75), except for @535, +1,m +1, éfiw

Hm+l,m+1! and Him. The approximation of the Hankel function should be used for

these terms. After the approximation, they become

éri+l,m+l = 31(A3)

€37,137; = S2(AS)
”371m : S:(AS)
Hr§t+l,m+l = S§(AS) (3.181)

D/2 D/2 D D
5(1sz f (— — w) (— - w’) Hé2)(ko|w - w'l) dw'dw (3.182)
D/2 D/2 2
1 D/2 D/2(D
52(1)): —2 f [(D —--'+w) H32)(ko|w— w'l) dw’dw (3.133)
D —D/2 D/2 (2

110

X 1 D/2 D/2( A (2) I I
s, (D): “153/ — — w) (7. R.) H, (101w — w 1) dw dw (3.184)
-0/2 —(D/2
D/2 D/2 (D+ . (2)
s;’( (D): 515/ /(: 7 11,)111 (110)111—1111) dw’dw (3.185)
D/2 D/2

where R, can be written in terms of integration variables and unit vector "y tangential

to integration perimeter

 

_ I
R, -.= (w w)“ (3.186)
lw - w’l
After a change of variables (such as w = —wa and w’ = —wf,) in (3.183) , the

result of the integral becomes identical to 51(D). Therefore, only 31 (D), S? (D) and
3;, (D) have to be calculated. A change of variable

5: w’ — 11) (3.187)
is used in (3.181) to give
51(1)) = 5511:):(9 — w) 11(w) dw (3.188)
where 11(8)) is
1. (w) = [11:23? —- 5— w) H62’(kolél) d5 (3.189)

To calculate this integral, 11(w) is divided into two parts as

11(w) = If(w) + If‘(w) (3.190)
where If(w) and If‘(w) are
0
110») = [41/2— 1; — E — w) Hé2’(—ko€) d6 (3.191)

111

D/2—w
Iron) = f (3 — 5 — w) H32)(ko€) as (3.192)
0
Therefore, (3.188) becomes

_1

SI(D) 02

[33(0) + 53(0)] (3.193)

where Sf(D) and S{‘(D) are defined as

D/2

Sf(D) = ./:D/2(£2)- — w) If(w) dw (3.194)
D/2

Sf‘(D) = «/-D/2(% — w) [{‘(w) dw (3.195)

The approximation of the Hankel function in (2.35) is used for If(w). Then If(w)

becomes

11(21): [0 [(9 — w) —€] [1—1§In(—5‘9§§ ] d6 (3.196)

-D/2—w 2

After taking the integral, this becomes

mm) = (527- —w) [e—j-f;(:1n(—’°—°,1§> -£)]0

 

—D/2-w
{2 0 [.52 ( l€035 0
_ _ + J— -1+ 21n(— ) (3.197)
2 —D/2-w 27" 2 —D/2—w

 

After substituting the limits of the integration and rearranging the terms, (3.197)

becomes

D — 2w)(D + 2111)
4
+ (D +8210)2

' .2 .2 k D 2 ‘
1+J-—]—1n(07( + 10))
1 7r 1r 4 .

F 2 2
1+jl —j—ln(k07(D+ w)
_ 7r 7r 4

 

 

mm) = (

 

 

 

). (3.198)

112

After substituting (3.198) into (3.194), and using a change of variables

x .-= D + 2111 (3.199)

S? (D) can be calculated from

 

1 2” F ,2 ,2 k x'
S?(D)= —/ (213-202x 1+J;-J;1n( W )1 dx

 

 

 

 

 

 

 

16 4
+315 020(20 — x) X2 :1 + 1—11; — j%1n(kozx)? dx (3-200)
This integration yields
37(1)) = %[25j+67r —12jln(k0;D)]
+144; [133' + 67r —12jln(ko;D)]
= 112—; [73' + 27r — 4j ln(kogD)] (3.201)

To complete the calculation of Sl(D), Si‘(D) should be calculated too. Therefore,
the approximation of the Hankel function in (2.35) is used for Ir(w). After the
approximation I {‘(w) becomes

11(9)) = [OD/H [(122 — w) — s] [1— 1'—1n(’“—"—2”"E ] d: (3.202)

After taking the integral, this becomes

II‘W) = (%_w)[5- j— :D(€1 (12215 6)]?2—1»

£2

D/2—w 2 D/2-w
.E 16075
,0 "’..[.fi(-1.21(2 )l. (3.20.)

 

113

After using the limits of the integration and rearranging the terms, (3.203) becomes

 

 

 

 

 

 

(D — 2w)2 .2 .2 ko'y(D — 2111)
u z _ _ _l
11(w) 4 1+3” .771, n( 4 )
(D — 2w)2 .1 2 1907(D — 2111) J
—— — 1 - — —l
+ 8 J7r + ( 4 )
(D -— 2w)2 .3 2 ko'y(D — 2111) J
___ _ _ _ .2 4
8 1+]7r J 1( 4 ) Q 0)
After substituting (3.204) into (3.195), and using a change of variables
x = D — 2111 (3.205)
Si‘(D) becomes
1 20 3 3 2 kofyx
u = _ ._ _ ._ .2
51(0) 32/0 x [1+17r JWIM 4 )] dx (3 05)
After evaluating the integral, this becomes
1)4 ko’)’D
“ D = _ ' 2 — 4 '1 -
Si( ) ml“ 1r 1 n( 2 )] (3207)
Substituting Sf(D) and Sf(D) into (3.193) gives 51 (D)
D2
51(1)) = — [73' + 271 — 4j1n(k°70)] (3.208)
87r 2
After substituting (3.186) into (3.184), 53(D)X is calculated from
x 1 W2 x
53(1)) = 133 f I, dw (3.209)
—D/2

114

where If is defined as

D” D (111 — 111')
X _ (2)
I3 — ‘/;D/2(“2— — w')WHl (kol’w — w'|)dw' (3.210)

The second kind with first order Hankel function can be approximated for the

small arguments

kol’w — w'l 2j

 

Hf2)(k0|w — w’l) 2 for 101111 — w'l <<1 (3.211)

2 7rk0|w — w’l

Substituting (3.211) into (3.210), If becomes

k0 D/2 D 2j D/2 (2 _wl)
IX = _ _ _ I _ I I _ _2____ I . 12
3 2[0/2(2 w)(w w)dw+7rko,/_D/2(w—w’) dw (32 )

After a change variable

5' = w — 111' (3.213)

The integration becomes
w+D/2 k 6’ D 2 1 D
Ixz/ 0 [——w+']+-—J—— ——w +1] ’
3 My, [ 2 2 5 7rko 5' (2 ) d5

_ 190 I2 D 2g,
- zlé (2_w+3)]

 

w—D/2
2 . D w+D/2
+—": [(—-— — 111) In lé'l + 6'] (3-214)
7r 0 2 w—D/2

Substituting the integration limits into (3.214) givas

= 10% +11 — w) + :1 +1 91 - +1
+ g; [(1—22 - w) [111% + w) — 111—g — 111)] + D] (3.215)

115

Now, substituting (3.215) into (3.209) yields

 

 

 

s; (D) = Sg(D) + 8:;(D) (3.216)
where S:(D) and S:(D) are
D/2 D
d(D — —— 1 d
Sz(D —7rk2002 D/2 kOD—(2[ w) n(— 2 —w) w
D/2 D
_ _ 3 d 3.217
2] ”/2 D D
u —— 1 —— d
3.10) 711.122 [_0/212 w) +1, +2») w
kg ”/21 D 2 D 2 D

—— — — — — — — d 3218
+2D2fD/22(2+w) (2 w)+3(2+w) w ( )

After substituting xd = D/2 — 11) into (3.217) and xu = D/2 + 111 into (3.218), they

 

 

 

become
S“(D)= Zj [D D— In at — k” [D 3d (3219)
1! WkoD2 0 Xd Xd Xd 12D2 0 Xd Xd -
sum): [D—(D )ln d k" [Di 2(D—l )d (3220)
3 111:0? x. x. Xu+ 0—72 0 2x. 3x. x. -

Substituting (3.219) and (3.220) into (3.216) yields

 

 

D 21' ko
X = _ 2 — 3 .221
3;, (D) A [nkoDz [D + (D 2x) lnx] + 1202 [3Dx 2a: ] dx (3 )
S3" (D) can be easily calculated from

21'

S." (D) = 711.002

 

 

D
1 ’60 1134 D
[Dx( 1-1-lnx)+x(2 IDX)+DXL+12D2 [Bx 210 (3.222)

116

and after the limits of the integration, the simplified expression is

=k—0D2 + —j—- (3.223)

S}, (D) can be calculated similar way, after substituting (3.186) and (3.211) into

(3.185)
Y 1 W2 Y
—D/2
where 133’ is defined as
D/2 2- D/2 Q + w’
IY- =kO/;( D'(+w) w— w")dw +— J £2———7)—dw' (3.225)
D/2( rho —D/2 (w — w)

After a change variable given in (3.213), the integration becomes

”+0” I: 5' D 23' 1 D
x __ 0 _ _ I _ _ _ __ '
I3 —/w—D/2 [ 2 [2+w €]+7Tko[€'(2+w) 1]] d6

w+D/2
kg ,2 D 25’
= IF (SW—3)]

 

w—D/2
2 ' D MD”
+7r—IZ—0 [(—— w)ln lé I + 5'] (3.226)
w—D/2

Substituting the integration limits into (3.226) gives

13(0)=fi{;(§+w)3_1(§_w)2[(§+w)+§(2_w)]}

2 2 2
23' D D D
+ m[<3+w)[1n(—2-+w)—1n(; —w)] -0] (3227)
After substituting (3.227) into (3.224), 5;, (D) becomes
53’ (D) = 53(D) + 3;;(D) (3.228)

117

where S:(D) and S:(D) are

 

 

2j D/2 D D
S”( ) 7rkoD2 -D/2 [ + ( 2 w) n( 2 111)]
’60 fD/Z l(£_w)2 (2+w)+g(P_—w)] dw (3 229)
2D2 0/22 2 2 3 2 '
D/2( D 100 D” D 3
u( _ — d 3.230
5:10—73:12 m .2. 9m» )(_1n (_2. +w)dw+1202/1”;2 +w) w ( )

After substituting Xd = D/2 — w into (3.229) and X“ = D/2 + w into (3.230), they

 

become
S“(D)— 2]- ]D D+(D )ln d "EL/Bx: D—— d (3.231)
1" — 7rkoD2 0 Xd Xd Xd 4D2 Xd
2j D ’60 j-D 3
Sy( ) Mom/o x nx x +1202 0 x x ( )

Substituting (3.231) and (3.232) into (3.228) yields

 

 

D 23' k
Y _ _ _ 0 3 _ 2
53 (D) _ [0 [31:00? [(2X D) In x D] + 1202 [2x 30x H dx (3.233)
33’ (D) can be easily calculated from

23’

53(0) = W

 

D
1 k D
D x(1——ln x)+x2(——+ln x)—D x + 0 2 ——D)(3 (3.234)
2 0 12D 0

and after the limits of the integration, the simplified expression is

k .
sg’ (D): —§—:D— ”JED (3.235)

118

Now, Am,” may be written by substituting (3.174)-(3.175) and (3.181) into (3.180)

A2 .. ..
Am,m =kOT 2 (am um“) H62 )(k0|PS,m — PS,m+1|)

k07AS 2j koAg
+7lr[7j+21r 4jln(-——-——- 2 -——-—-—)]] 7rko 12

A - 3 3
+—5 [2... m (a... + um.1)]H§2’(ko(ps,m — p5,...“ I) (3.236)

When i = m — 1 , A,“ becomes

_ vY
Am—l,m — k0[em—1m+1+em,m+l+ém—l,m+ém,m]

+7154 ,m+1 +Hm— l,m Hm ,m+1 41%,"; (3-237)

Here, £3", and Ham have to be calculated by using the approximation of the Hankel

function for small arguments. After the approximation, they become

32:... = 5mg) (3.238)
where S}, is given in (3.235) and S4 is defined as

on D/2(D Hm
_ _/ _.. _ w) 3+ w)H 0 (k0|w — w'l) dw'dw (3.239)
—D/2 —(D/2

After using the change of variable in (3.187), (3.239) becomes

D/2
34(1)) = 1% [43/262 — w) I4(w) dw (3.240)
where I4(w) is
D/2—w
14(w) = [_W (g + c + w) H32)(kol€|) as (3.241)

119

To calculate this integral, 14(w) is divided into two parts as

I4(w) = 13(4)) + 13(4)) (3.242)

where If(w) and 151w) are

0
mm) = [_W (5— + e + w) 11523—404) 44 (3.243)
D/2—w D (2)

Law) = [0 <3 +£+w> Ho 0606) d5 (3244)

Therefore, (3.240) becomes
54(0) = 53 [53(0) + 33(0)] (3.245)

where SflD) and S};(D) are defined as

on
SflD) = «/;D/2(£2)- — w) 1,?(211) dw (3.246)
D/2
52(D) = /_ 125% — w) mm) dw (3.247)

After using the approximation of the Hankel function in (2.35), Iflw) becomes

I.:'(w)=/0 [(-123+w)+4] [1_j§1n(—5°27—5]35 (3.248)

—D/2-w

After taking the integral, it becomes

 

mm) = (g + w) [4 -j7?;(€1n(-k°27€) - ‘)lim-..
+ E; 0 + [j 52 (1— 21n(—§°27—€ no (3249)

—D/2—w 7r —D/2-w

 

120

After using the limits of the integration and rearranging the terms, (3.249) becomes

(D + 211))2

1“(w)= 87F

 

[7r + 3]“ — 2j1n(k°7(D + 2110)] (3.250)

4

After substituting (3.250) into (3.246), and using the change of variable (3.199),
SflD) can be calculated from

1
327r

 

20
S“(D)= —/ (2D —- x) x2 [7r + 33' — 231n(k°ZX)] dx (3.251)

After evaluating the integral, this becomes

 

 

SflD) = 251' + 671' — 12jln(k070)] (3.252)

144%

To calculate S4(D), 53‘ also has to be calculated. First the inner integral is written

Law) = [WM [é +40 +4] [1 —j—1n(’°"—27—é ] as (3.253)

After taking the integral, this becomes

14(10): (5+11I-)[€ j—:(€1m(k02’7€)_ END/2“”

0
£2 D/2—w £2 [€075 D/2—w
+—2—0 + [j— 2 (1—21(2 )]0 (3.254)

 

 

After substituting the limits of the integration and rearranging terms, (3.254) becomes

 

 

Ij‘(w) = (D—2w)(D+2w) [1+::— —j— 21n (km/(D- 2110)]

4 4

 

(3.255)

After substituting (3.255) into (3.247), and using the change of variables (3.205),

121

Sf,‘(D) becomes

 

 

 

 

1 20 3 . ko’YX
" = -—— - 2' d
54(0) 32” 0 X [n+1 Jln( 4 )] x
20
+——1— x2 (2D — x) 7r + 23' —— 2j1n(k°7x)] dx (3.256)
161r 0 4
After evaluating the integral, this becomes
5D4 . . ko’YD
u = _ 3.2 7
54(0) 1447r[13] +67r 12]ln( 2 )] ( 5 )

Substitution of 52(0) and SflD) into (3.245) gives S4(D), that is

2

D D
S4(D) = 8—7r [51' + 27r — 4j ln(k0; )] (3.258)

 

Now, Am_1,m may be written by substituting (3.174)-(3.175) and (3.238) into (3.237)

A2 A A .4 -o
Am—lJn = RIO—4‘5 [(um—l ‘Um+1) H62)(k0|PS,m—l _ pS.m+1l)

+ (717" ° Tim-H) Héz)(k0l53,m — fi$m+l I)

+ (1%., ~21") H32)(kolfis,m—1 - 53ml)
1
+ — [53' + 27r — 4jln(k0’;AS)]]

 

27r
As . (2) .. ..
+ '5— (Rom—l,m+l ' um+l)H1 (kOIPSJn—l — pS,m-H|)
+ (Rm-1,". 3311123353..-. — 55...!)
- _. .. ' A2
- (Rm... 'Um+1)H1(2)(k0|Ps,m — Ps,m+1|) + -’— + M (3259)
Wko 24

When i = m + 1 , A5,", becomes
_ u UX “'Y "X
Am+1,m — k0 em+l,m+l + em+2,m+l + em+1,m + em+2,m]

122

+H1§1+1 ,m-H + urn-+1, m _Hm+2 ,m+1 —Hm+2, m (3260)

Here, 33% +l,m +1 and H5, +1.m +1 have to be calculated by using the approximation of

the Hankel functions. After the approximations, they become

éri+1,m+l = 55(AS)

Hm+1,m+1 = 5:?(AS) (3-251)
where S? is given in (3.223) and S5 is defined as

1 D/2 D/2(1), (2) I I
S5(D) = 55/ [@122 — —w) H0 (kolw — w l) dw dw (3.262)

—D/2 D/2 (2

After a change of variables (such as w = —wa and w’ = —w;) in (3.262), the integral
becomes identical to 54(0). Therefore, previous results can be used to calculate

Am+1,m as

A2
Am+1,m = ko— 27r —[5j+27r-— 4jln(

 

ko’Y2As )]

+ (71mm ' 71m“) H02) (kolfis,m+2 - fis,m+1|)
+ (&m+l 'fim) H62)(k0|fi5,m+1 — 53ml)

+ (um+2° um) H52 )(kOlpS,m+2 —pS,m|)]

 

A
+ —2—5 (Rm... u...)H‘ )(kolpsm+2— p5,...»
"' (Rm+2,m+1 ' am+1)H1(2) (kOIfiS,m+2 - fiS,m+l I)
. (2) .. .. j koA§
_ (Rm+l.m ' um)H1 (kOIPSmH _ PS,m|) + FIG—0 + 24 (3263)

All the self-terms of A5,", have been considered. However, the diagonal terms on

123

the target must also be considered. When 2 = n, D1,, becomes

.4! vX ~Y ~X
D71.“ : k0 [gum + gn+l,n + gum—l + gn+l,n-l]

+7133; + ”it, n— l _Hn+1, n —Hn+1,n— l (3264)

Here, the terms can be computed from (3.179), except for 573:,” and ’Hffm. The ap-
proxim3.tion of the Hankel functions for order zero in (2.35) and for order one (3.211)

should be used for these terms. After these approximations, they become

513.1,. = 54(AT)
“in = S§(AT) (3.265)

After substituting (3.179) and (3.265) into (3.264) , Du," becomes

A2
Dmn— - 1:07 In ——[5j +21r — 4jln(

 

koAT)]

2

+ (29... ~19.) H32)(ko|fir,n+1 — m!)

+ (a. . a.-.) H62)(kolfir,n — pr..- I)

+ (177m 'fin—l) H32)(koli>'r,n+1 " finn-ln]

A
+ —T (Rn,n— 1 vfl—1)Hl(2 )(kOIPT"_fiT’"'1|)

-' (RH-lm' fin)H1(2)(k0|fiT,n+l -fiT,n|)
koA?,~

_(Rn+l,n—l Ufl-1)H1(2)(k0|pr,n+1_p7‘,—n1|)]+—+ 24 (3366)

 

When 3 = n — l, D”, becomes

Dn-l.n = k0 [fir—1,7; + grin + grY—lm—l + grin—1]

+Hn- l ,n + ”n— 1 ,ln— _Hrfrr _Hn, n— l (3'267)

124

Again, 573:”, {1,11wa Him—1: and ’Hffm have to be calculated using approximation

of the Hankel functions. They become

95:. = 51(AT)

9n—1,n—1 = 51(AT)

HZ—ln—l = {(AT)
’Hff,” — Sf (AT) (3.268)

After substituting (3.179) and (3.268) into (3.267), Dn_1,,, becomes

A2 . . _, _,
Dn—lm = kOTT 2 (”n ' vn—l) H62)(k0|p1‘,n " phi—ll)

 

 

ko’YAT)] ] _ _2_J_ _ koA'zr

1
—7' 2—4'1
+7r[‘7+7r J“ 2 wk, 12

A . - - .. ..
+ —,’"—[R,._1,n-(vn+vn_1)]H£”’(koIpr,n_1 m...» (3.269)

When 8 = n — 2, D4,, becomes

VY VX ~Y ~X
Dfl—Qfi = k0 [git—2,11 + git—1,7; + gn—2,n—1 + gn—l,n—1]

+HX '1" fly — Hf—l,” — HI—lfll-l (3.270)

n—2,n n—2,n—1

Similarly, gf_,,,,_, and Him—1 have to be calculated using the approximation of

the Hankel functions. They become

9':— l,n—l = 54 (AT)

_ sg’ (AT) (3.271)

:E
f ~<
'5'
i.

|

125

After substituting (3.179) and (3.271) into (3.270), Du-” becomes

A2 .. ..
Dir-2,71: kO— [[(vn— 2 vn)H H62)(k0|pT,n-2—p7’,nl)

+ (2H .9.) Hé”’(kolfir,n-l — final)

+(0n—2 611—1)}! [16)2(k0|fiT,n——2 finn— ll)

(W)? T ' koAi'
+ 27r —[7j+27r 4jln( —) +7rko+ 24

 

A _. ..
+ T 2[(R"_ 2n ’t)n)II(2 )(kolpl‘,n—2_p7‘,nl)

— (;—1,n'fin)Hf2)(k0lfiT,n-l _ , )

— (Rn—ma -23,,_1)H,(2)(k0|fir,,,_2 - firm—1D] (3-272)

     

Although all the diagonal terms are considered on the target, finNT and [in posi-
tion vectors go to the same point since the target is a closed surface. This condition
brings self-terms in two occasions; therefore, this should also be taken into consider-
ation.

When E = NT — 1 and n = 1, 03,, becomes

vY Vx ”Y ”X
DNT—1.l = k0 [gimp—1,1 + gNT,l + gNT—1,NT—1 + gNT,NT-l]

+71%,r _,,+’H’NT —1~T-1 H—NT, —HNTNT_, (3.273)

where

5725,) = 91,—: -- 51(AT)
fiKT—INT—l = 31(AT)
”(Sim = “(fl = S§(AT)
sg’mT) (3.274)

E
31
.L
3
.1.

II

126

After substituting (3.179) and (3.274) into (3.273), DIN/T-” becomes

A3}

”NT-1’1 = ’77 (am—1 ~01) Héz’wolmfl -f>‘r.1|)

+ (731 ‘17NT—1) H32)(k0|57,1 - FEM—1|)

 

 

 

1 _ , ko’YAT . 2j koAgw
+;[7]+27r—4]1n( 2 )] m 12

A ~ . - q .. ..
+ 7T [RM—1.1 ° (”1 + ”NT—1) H12)(ko|pr,NT-1 — Pall) (3-275)

When 2 = NT — 1 and n = 2, D5,, becomes

vY vx ~Y ~X
BAIT-1,2 = k0 [EMT—1,2 + QNT,2 + gNT—1,1 + gNTJ]

+71% —1,2 + ”KIT—1,1 - ”1’57; _ ”KIT; (3-276)

7'

where

@1631 = glxl = 54(AT)

71’an = ’HK, = S;’(AT) (3.277)
After substituting (3.179) and (3.277) into (3.276), DNT_1,2 becomes

A2 . . _, _,
012-12 = “‘41 (wt—1.222) H32)(kolpr.~T—1-Pr.2l)
+ (2 ~02) H32)(ko|iir.x — fr'ml)

+ m.-. «71) H52’(kolfir,~.,-l - m)

1 k A ' A2
+—5j+27r—4jln(07 T) -+--J—+kO T
27r who

 

 

2 24
AT . . (2) -' "
+ "‘2— (RNT-1,2 ° U2)H1 (kOIPT,NT—1 — P1321)

+ (kw. ‘01)Hl2)(kolfir.~T—1 — fimI)

127

— (R1,2'92)H1(2)(k0|5r,1— fiml) (3.278)

This term concludes all the self-terms of DAN.
3.2.2 The Scattered Electric Field in the Far Zone

To find the scattered E-field in the radiation zone, first the induced currents on the
target and on the sea are determined by solving the matrix equation (3.124). Then,
the expression for the E—field in (3.10) should be approximated for the far zone. This
is done by approximating the Hankel functions for large argument. The geometry of

the problem is shown in Figure 3.6. The expression for the scattered E—field is

135(5) )=-E[kg/K (5’)H2R)(ko) Rdl’+ +/V'K ‘ (3') W152)(koR)dl’ (3.279)
0

(2)
VHé2)(koR) = dH" (1:221?) VR

= —R ’60 Hf2)(koR)

 

l2

—/3 k0 ‘1’1(P,P')

—j R k0 vamp) (3.280)

I?

Here, the Hankel functions can be approximated for large arguments. \Iln(p, p’) is
defined as the approximation of the second kind of Hankel function with order n. It

is given as
k20‘:r e _.ikop

ejkofi'i” ’ (3.281)
f

H12)(koR)’=‘1’ mp, ’)=j"

128

 

Figure 3.6. Calculation of the E—field in the far zone.

129

vii

Therefore, the approximation of the hankel function becomes H1(2)(koR) 2 \II, (p, p’) =
j\I’o(p, p’). After substituting (3.280) into (3.279), the scattered E—field becomes

138(5) = —%./1: [k0 13(7)") ‘1’0(P,P') "j P5 ‘I’o(PaP') V, ° K453] dl' (3-282)

where
‘I'o(p,p') V' ' Kw ') = V’- [$00810 1? ()5i ')J - 305’ ') - V"1’0(p,p') (3283)
Here, V’\Ilo(p, p’) is

V"1’0(p, p’) = J'ko (3 ‘I’o(p, p') (3-284)

and

[V’ - [Wo(p, p’) 17(5)] dl’ = 0 (3.285)

Equation (3.285) can be shown starting by the definition of divergence of a two-

dimensional vector A

-o

.. _ . f0 A - ft (11
V - A _ iiToT (3.286)
This definition gives directly
(V-A'), As, = f [f-R dl (3.287)
Ci

for a very small differential surface element A3,- bounded by a contour Cj. Here,
it is the unit vector perpendicular to the integration contour. In the case of an
arbitrary surface S, the surface can be subdivided into many small differential surfaces.

Combining the contributions from all these differential surfaces for the both sides of

130

(3.287) leads to

N

2(V ' fibASj

i=1

lim
Asj -§0

(3.288)

 

 

N
=Alsijrgol:;£jA-ndl

 

The left side of (3.288) is the integration of V ~ .4 over the surface S. The line
integrals on the right side of (3.288) are summed over the surface. The contributions
from the internal contours of adjacent differential surfaces cancel each other because
the outward normals of adjacent elements are in opposite directions. Therefore, the
net contribution is the integral over the external contour bounding the surface 5. As

a result, (3.288) becomes
[(V-X) 115:}(48 dl (3.289)
S C

To prove (3.285), [f in (3.289) is replaced by \Ilo(p, p’) I?(p' ’). Then, the left side
(3.289) becomes

/ V” [WM/810') 307)] 615' = f f V’- [WM/Ad) 13072] dl'dz (3290)
s L r
Since the \Ilo and R are invariant in z-direction, combining (3.290) and (3.289) yields
I I " «I I 1 I " «I - I
[FV - [MR/2) K(p )] dl = ffiwow) K(p )-ndl (3.291)

The closed contour on the right side of (3.291) is shown in Figure 3.7.

The closed contour integration can be written more specifically as

from) 1303") '71 d1' = [1118272018578 dl'
C I‘

+ / ‘I’o(p, (0 K15”) - fin dl’
L4

131

 

 

 

 

 

 

 

2 u
D
An L y d
F
I I I .
1 z I”
i I ,m
, , f
. , e
, , .
AW] . . w
llllll FIIIIIFIIIILPIIIII
. . I
. . n.
.J. . n.
8 . . N.
Am . . u.
. . m.
_ . u.
/4 H“ _ n. .d
IIIIII IIIIVPIIIII.A
An K n. n
l. . m. v
H.
L
T ...... v ‘ ..... .II' W-
F r.
A w m.
IIIIII ” IIIILrIIIII.
. h _ w
z . . u
. . .
‘ ..................... s. ............. Q .............. s" .
a a ..
a a a
a a a
s \ x
-x -\ -\

 

 

Figure 3.7. The integration contour to prove the divergence identity.

B2

z o (3.292)

where Lu and L; are the integration paths along z-axis. Also, I‘ and [‘d are the
integration paths parallel to the :1: — 3; plane. Since I?(p‘ ’) is zero along La and Lu,
contributions from the second and last terms in (3.292) are zero. Also, I?(p‘ ’) is
perpendicular to the normals of the contour for the first term and the third term.
Therefore, thase terms also become zero. Substituting the result of the closed contour
integration in (3.292) to (3.291) proves that [1. V’ - [‘Ilo(p, p’) I?(p‘ ’)J dl’ = 0 is true.

It can be easily shown that this identity is true even when the integration path P
is a closed contour. When I‘ is a closed contour, the contour bounding the surface Cb

consist of two closed contours. Therefore (3.292) becomes

f ‘I’o(p,p')13(fi’)-fi dl’ = prm') 3079-71 011'
CI, F

+ if \I’o(p,p') 13(59'714 dl’
d

= o (3.293)

Similarly, since Kw ’) is perpendicular to the normals of the closed contours I‘ and
Pay, the contributions become zero. Therefore, (3.285) is true for closed contours.

After substituting (3.283) and (3.11) into (3.282), the E-field becomes

E35) = 4"? r [9' — 9 (9 - 7)] K91") mm) «11' (3.294)

where p is the unit vector in the direction to the observation point. That is

p = cos ¢, 5: + sin (I), 3} (3.295)

133

and 43, is the angle from the horizontal.
For the problem considered, (3.294) can be rewritten in terms of induced current

on the cylinder and on the surface separately. That is

 

1.775) = —”°4'°° [ f3 [9' — 9 (9-9’)] KS<5§> ‘I’o(P,P:9) dl'
+fT [if—mm] KTW) ‘I’O(P:P’r)dl'] (3.296)

The scattered E—field in the far zone can be calculated for a given receiving polar-

ization direction

—.
8

E297) = 9 (9*)
k.

-E
= -"° [[3 [RWY-(83$) (927)] Ks(fié)‘1'o(p,pf9)dl'

 

A

[93-17- (sé-fi) (“179] KTW) ‘I'o(p,p5r) d”) (3-297)
where 56 is unit vector in the receiving polarization direction

83 = COS(¢p + 45,) a”: + sin(¢,, + as.) g (3.298)

Here, (1),, is the angle measured counter-clockwise from scattering direction.

The induced currents have been solved previously using linear combinations of
expansion functions; therefore, the currents are known in terms of expansion func-
tion amplitudes. After the substitution of (3.31) and (3.32) into (3.297), the E—field

becomes

E29?) = Ego) + E59?) (3299)

134

where Eg(p' ) and ng ) are defined as

__ k0 Ns—l
E;(fi) = —17()‘/j——k0 8 J p fi{ 2 [COS(¢9+¢8) um,x+Sin(¢P+¢3)u m,y]

m=l

 

[As/2 . I ’ .
$.an (_A_S. _ u’) eJko($3 C08¢a+ys sm¢,) dul
--(A3/2

As/2 A . , .
+ .1 f (.2...) d.)
-A3/2

NT-l

+ n: [cos( 05‘, + 03,) 0;” + sin(q$p + (9,) firm]

A 2
_TA1[d m/( T/ (A2T _UI) ejko(zircos¢.+y’,rsin¢,) dvl

—A7~/2
AT/Z . I , .
+ dm+1 / (923"- + v') eJ'°°(7rC°S¢°+yrsm¢s) dv’ ] } (3.300)
-AT/2

 

(7.8160 e—j’m” . .
Eg(fi) = 770 _ \/— COS <15: C05(¢p + 4’s) + Sln ¢s Sln(¢P + ¢3)]

N5- 1 1
{fig—‘31[cos<;S,AS'I.I,,,,,+sinqS,umy]A
s

/‘(As/2
Cm/ _ _ u) ejk0(3’3 cos¢.+y'ssin a.) du’
—A5/2(

As/2 ' I .
+Cm+1/ 2(AS +ul) eJko(zscos¢.+y’ssm¢,) du’ ]
—A5/2

N—l
T 1

+ ":91 [00805, vin$+sin¢, 0%,];—
T

A 2
dm/‘( T/ (_A_T_ _ 1),) )ejko(:cT cos¢,+y’Tsin 4).) d'U’
—AT/2

AT/2 . I ’ .
+dm+1 / (% + v’) eJk°W°°“¢'+y'rS'"¢') dv’ ] } (3.301)
—A1~/2

135

Here, 13’s, 31%, 3:511, and yr’r are defined as

I I

I_ I I I_ I I

xS—xm+um,zu, yS—ym+um’yu
I __ I I I I _ I I I
xT-$n+vn,xv I yT-yn+vn,yv (3'302)

3.2.3 Numerical Results

In order to make some verification of the formulation, the sea surface and the cylinder
are considered alone. First, a finite flat surface of width 10 is considered, and the
induced current is compared with the rasult obtained by Balanis [15] for a finite strip.
A two meter long strip is chosen, with A = 1 m, and 250 matching points are used.
The current densitias on a finite strip induced by a plane wave at normal incidence
are shown in Figure 3.8. There is excellent agreement.

Second, the EFIE solution of the induced current on a finite sinusoidal surface
is compared with the solutions of two different cases. Using MFIE formulation, the
current on a single period of an infinite sinusoidal surface and the current on the
closed surface with a sinusoidal roughness on top is computed. The geometry of the
closed surface with a sinusoidal t0p is shown in Figure 3.9. The sinusoidal surfaces are
approximated by using 50 partition points for each period of the surfaces. Accordingly,
the closed sinusoidal surface is partitioned using 250 points on t0p and the bottom,
50 points on the sides. The induced current magnitudes are plotted in Figure 3.10a,
and the current phases are compared in Figure 3.10b. As expected, there is pretty
good agreement away from the edges of the finite surfaces.

Third, Figure 3.11 shows the induced current on an elliptic cylinder with a major
axis of two wave lengths and an axial ratio of 4. The incidence angle is chosen to
be zero degrees from the x-axis. The elliptic cylinder is approximated using 1080
points. A comparison is made with the result produced by Andreasen [17] using

even and odd current densities in an MFIE-MOM solution. In the figure, 8 is the

136

perimeter distance normalized to unity for the half perimeter, as measured clockwise
from the point on the center face nearest the source of the incident wave. There is
good agreement between the results. The difference between the results is due to
using different integral equation formulations and different methods to partition the
ellipse.

Finally, The induced current on a cylinder is calculated by using 540 matching
points for koa = 47r using both EFIE and Magnetic-Field Integral Equation (MFIE)
solutions. The results are compared with the result obtained by Balanis [15]. The
amplitude of the induced current on the cylinder is plotted in Figure 3.12a. As
expected MFIE solution is the most accurate result. The EFIE solution presented
in this chapter is different than the one given by Balanis. Therefore, the results are
slightly different between two EFIE results. However, they still agree very well with
the MFIE solution. The phase of the induced current on the cylinder is plotted in
Figure 3.12b for EFIE and MFIE solutions. There is excellent agreement between
the phases obtained by the two techniques.

After gaining confidence from these verifications, the current densities generated
by targets above a sea surface can be computed. First, a cylinder above a finite width
flat strip is considered. 100 matching points are used on the cylinder and 490 matching
points are used on the strip. The currents on the strip and cylinder are plotted in
Figure 3.133. and Figure 3.13b. Not surprisingly, the cylinder current and the surface
current are affected most at the points where the surfaces are closest to each other. To
check the effect of sea surface roughness on the current distributions of the cylinder
and the sea, a sinusoidal surface is considered. 100 matching points are used on the
cylinder and 55 matching points are used for each period of the sinusoidal surface.
Figure 3.14a shows the current density on the surface and Figure 3.14b shows the
current density on the cylinder. It is seen that the amplitude of the induced current

on the cylinder is slightly different to that above a flat surface. However, the induced

. 137

currents on the sinusoidal sea and on the flat surface are affected in a similar way by
the presence of cylinder.

For the scattered E—field calculations, the larger value of 100 points or 20 points
per wavelength is used to represent the current on the cylinder, the larger value of
125 points or 10 points per wavelength is used to represent the current on the strip,
and the larger value of 55 points or 20 points per wavelength is used to represent
the current on each period of the sinusoidal surface. First, the scattered E-field
is computed for a cylinder above a finite width strip for several frequency points
and plotted in Figure 3.15a. Since the transient field gives more insight, the IFT is
applied to the frequency domain results and the transient E—field is shown in Figure
3.15b. There are direct and multiple reflections coming from the strip and from the
cylinder. The first signal (A) comes directly from front edge of the strip, and second
signal (B) is the direct reflection from the cylinder. The third signal (C) comes as
a superposition of cylinder-strip and strip-cylinder multiple reflection. After that,
the signal (D) comes from the strip by following the multipath of strip-cylinder-strip.
The signals (E and F) coming later in time are more complicated multiple reflections.
However, the big reflection (G) coming latest in time is the reflection from the back
edge of the strip. The timing analysis will be presented in chapter 4.

Finally, the transient scattered field is computed for a cylinder above a finite width
sinusoidal surface in the frequency band from 1 GHz to 30 GHz with a step size of
0.04 GHz resulting 726 data points. The time domain response is obtained by first
windowing the frequency domain results using double cosine function then calculating
the IFT. The transient field is plotted in Figure 3.16. As seen in Figure 3.15b and in
Figure 3.16, multipath analysis is complicated by the presence of reflections from the
edges of finite width sea models. These reflections may be causing edge-target multiple
reflections, and these reflections are not expected in the real ocean environment.

Therefore, a more realistic sea model is necessary.

138

 

 

 

 

0.0300
‘ — EFIE-MoM Solution
------ Solution by Balanis
0.02259
g 1
$00150 4
Z
0.0075j
00000
-1.00 -O.50 0.00 0.50 1.00

x-location (m)

Figure 3.8. Current distribution on a finite width strip, E0 = 1.0 V/m, 45,- = 90.00,
w=2A,/\=1.0m.

139

 

 

 

 

 

 

 

 

 

 

PEC

Figure 3.9. The geometry of the problem for a cylinder above a closed surface with
a sinusoidal top.

140

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I —--—- finite-EFIE
4 j finite-MFIE
1 ° infinite-MFIE
3 1
{Jo I f).
E if ,2.
2 -< ‘ \ : f
2 "1M .’ i
V 1H": -. \ -' |
..: ' r
1!
l) ‘
0*l..,,,1-..hs,..'
-1.5 0.0 1.5
x-location/d
(8))
—~-—- finite-EFIE
finite-MFIE
‘ . infinite-MFIE
180 -
’6: .
m 9
3 l
g . _.
2 0~ /
o. 1
-180 , . 1 . , . . 1 -
-1 0 1
x-location / d
(b)

Figure 3.10. Induced surface currents for finite and infinite sinusoidal surfaces (a)
amplitudes, (b) phases, kod = 17.08, of = 0.1016 m, h,/d = 0.125, w/d = 5.0,
t./d = 1.0, ta/d = 0.5, ¢,- = 40°.

141

 

 

 

 

 

3.0 ,
j —-—-- EFIE-MOM Solution
2 5 j —-- MFIE-MOM by Andreasen
3 ay=27t, ay/ax=4.0
o 2.0 i
”J :
A 1.5 ‘
5-3 2
Y :
C 1.0 ,
0.5 i
0.0 ‘ ' ' T ' l V r v r l 1 I . r l , ,
0.00 0.25 0.50 0.75 1.00
S

Figure 3.11. Normalized current distribution on an elliptic cylinder, 45,- = 0.00.

142

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0038 T
J ------ Solution by Balanis
— EFIE-MOM Solution
—— MFIE-MOM Solution
Aooozs ,1 7"
g 1
S,
g .
0.0013 —
0.0000,,,,,,,,fi,,+,,,,,f,
0 90 180 270 360
Angle 6 (degrees) along surface
(a)
— EFIE
, ——————— MFIE
180 «
’6: ,
d)
E
m .
8 J
_c 0
o. ,
J t
-180 an
o 90 180 270 360

Angle 9 (degrees) along surface
0))

Figure 3.12. Induced current on a cylinder by different solution techniques (a)
amplitude, (b) phase, E0 = 1.0 V/m, 95, = 180.0°, A = 1.0 m, a = 2A.

143

 

—— With cylinder
—-— Without cylinder
——— Difference Current

 

 

 

 

 

 

4 I \
, I

Af/J\\'
‘I‘J\.‘\"/‘-"’" (I

I

\—
\.
\
\A/V\"\/‘J

 

 

 

 

O f Y Y Yfi f V

-0.50 -0.25 0.00 0.25 0.50
x-location /w

(a)

 

l — With strip
4 + —-- Without strip

 

 

 

 

0 90 180 270 360
9—location (deg)
(b)

Figure 3.13. Current density amplitudes for a cylinder above a finite width strip (a)
on the strip, (b) on the cylinder, Icoa = 1.0675, 0 = 0.0127 m, xr/a = 0.0, ht/a = 1.0,
h,/a = 0.0, w/a = 72.0, d}.- = 20°.

144

 

 

 

 

 

 

 

 

6 a
I ——- With cylinder
. —-— Without cylinder
5 7 ——— Difference Current
9 4 l
UJ l
A j
E a -
— f' I .
gé_ A /\ \j
2 _ "\ ./ "‘- '
1 L \/\
4 [I \\ A
0 C’f:‘?7<‘ff 5L 1, .Y ,7 ,, .1
-0.50 -0.25 0.00 0.25 0.50
x-Iocation /w
(a
, — With sinusoid
4 ~ —-— Without sinusoid

(IKI lid/Ea

 

 

 

 

o‘ffi,,,v,fi,,,e,,,,ov
0 90 180 270 360

O-location (deg)

(b)

Figure 3.14. Current density amplitudes for a cylinder above a finite sinusoidal
surface (a) on the surface, (b) on the cylinder, koa = 1.0675, (1 = 0.0127 In, 1:, / a = 0.0,
d/a = 8.0, h,/d = 0.125, ht/d = 1.0, w/d = 9.0, ¢,— = 20".

145

 

00
L l A J. A

 

 

4 A A

 

(lEsllEoXp/aWZ

 

 

 

 

 

 

 

 

 

 

1
0Lrwr,,,.,,,.YY,,,,fi,Y,fT,,,,t
5 10 15 20 25 30
frequency (GHz)
(80
“1.0,
E 4 c
E j
93 0.5 4 0
v , B
'O
E J EF 6
a: A
.g 0.0+ v
‘5
2
o
'O-O.5 ~
g :
3;: ,
«33-10 r ,,,,,,,,,,,,,, To
0 2 4 6 8 10
time (ns)
(b)

Figure 3.15. Scattered electric field from a cylinder above a finite-width strip (a)
in frequency domain, (b) in time domain, a = 0.0127 m, x,/a = 0.0, M/a = 8.0,
w/a = 72.0, 96,- = 200.0, 45, = 20.00, ()3? = 90.00.

146

 

 

 

 

 

 

 

 

A 1.0

m .

.2 *

E 2

g 0.5 —

2

a) t

‘= ‘ 1

.2 0.0 — w-vw

b t

0 t

9 A

a)

'D- . e

an 05 1

h

a) ..

3:: ,

8 .

m -1.0 q l r r r r T r r v 1 l u r v r l r r r r
0 2 4 6 8 10

time (ns)

Figure 3.16. Transient scattered field for a cylinder above a finite-width rough
surface, 4).- = 20.0”, d), = 20.00, 43‘, = 90.00, a = 0.0127 m, d/a = 8.0, h,/d = 0.25,
:cr/a = 0.0, ht/d = 1.0, w/d = 9.0.

147

3.3 Image Technique Solution for Scatterers above an Infinite PEC Flat
Surface

As in the TM polarization case, the sea surface is chosen to be infinitely long and flat
to avoid the direct and indirect reflections arising from the edges of the finite-sized sea
surface model. The method of images can be used to replace the infinite flat surface

with images of the incident field and the cylinder current as is shown in Figure 3.17.
3.3.1 Theory

The incident electric field in this case is the sum of a TE plane wave and its reflection.
The sum is a standing wave in the y-direction and a. propagating wave in the :1:-

direction, and can be expressed as

E‘(z, y) = 2E0 [ — jsin d,- sin(koy sin 49,-) :i:

+ cos ()3,- cos(koy sin 45,-) 3) ] 6’7“” mm" (3.303)

The scattered E—field can be written in terms of the induced current on the cylinder

and the image current as
5’0?) = arc-22593715) +£r.(fi,5')1?“(fi') (3.304)
where 5 and p‘ ’ are given in (3.3) and (3.8). Also, [1} (if, if ’) is an Operator defined as

£r(“,fi’)1?(b") = 2’19 [kn/1305’)H32)(kolfi-fi’l)d7/
r
—V [F Ros”) - R H52)(kolfi‘— fi’l) cw] (3.305)
In (3.304) T,- is the integration path for the image current. The magnitude of the

image current will be the same as the cylinder current. However, the image current

direction is opposite the cylinder current direction along the x-axis and in the same

148

Iy fit

T 2 i \ _..
_.._.... cylinder 9’51 E"

H-

CL‘

’

 

h
l
i

h

T" , 9%
---——~---—- lmage I?”

E2

(‘9-

 

 

Figure 3.17. The solution of the scattering problem for a cylinder above an infinite
flat surface.

149

direction along the y—axis. Therefore,

= 17! K7105.)
Ti

KT‘W)

 

 

Ti

(3.306)

 

= (‘Uzi + 0,32) KTU‘I')
7'

Now .67; (if, 5 ’ )1? T" (,5 ’) can be rewritten in terms of the cylinder current and integra-

tion path as

onw'mw') = —% [a [T K07) (-v.':r: + v.3) H32)(kolfi- fi/l) dl'

-V [T K07) [(—v.’a:~ + ugh-13’] Hf2’(kolfi— fiz’l) dl'](3.307)

where fl,’ and R’ are

 

,6‘,’ = $5: — yg (3.308)
and
~_ -or
R’ = E; 2,? (3.309)

Substituting (3.307) into (3.304), the scattered E—field becomes
5’07) = 57%;): 5')K7’(5’) (3.310)
where £9- is defined as
imfi'ww') = —%{ko [r K03") [0'H52’(koR)
+ (—v,';2—. + vy'g)H,§2’(koR’)] dl’

— v / K0?) [0' . R) H32’00R)
T

150

+ [(—v,’:i: + vy'g) - RI] H§2’(koR’)] dl’} (3.311)

The EFIE for the induced current on the cylinder can be found by using the
boundary condition of zero tangential total electric field over the cylinder. This can

be written as

a - [2'02 5')KT(5') = 2‘2 ~ Em (3.312)

After using the expansion of the cylinder current given in (3.32) and multiplying each
side of (3.312) with weighting function WlT(v) given in (3.44) and integrating over

the perimeter of the target, the EFIE becomes a set of linear equations

NT—l

fdv WtT(v) Z £75(v,v')K:(v)] = —/ dv WlT(v) i) . 312)) (3.313)
T ":1 T
where £75(v, v’) is defined as
£71(5.5')K3(v) = —% {100/ K10) [M' 33001:)
T
+ (—vx’vz + vy’vy) H32)(koR’)] dv'
0‘ a, ~ (2)
7,; [r K10) [0 10H. 0012)

+ [page + vy'g) . R’] Hf2’(koR’)] dv'} (3.314)

Substituting the definition of the weighting function in (3.74) and the expansion of

the cylinder current in (3.42) into (3.313) gives

 

 

NT—l - -
T vY vY ux ux x x x x
E : Cu [1‘70 [91,» + 0!,n + 91+”; + 01+1,n J + ”(at + Kim - Ht+l,n — (Cum
n=l J
NT—l - '-
~Y ~Y ~x ~x Y Y Y Y
+ 2 : ([11; [k0 [9cm + 0m + 91+”; + 0e+1,n + Him + Kim " Hum - Kl+l,nj
n=l .

151

= i[ dv y{(v) i); - 1337(2)) + dv Kid?!) 13t+1 °Ei(’U) (3-315)
Lt

710 Ll+1

where 02:... 0211,... 023,, and 35;,” are given in (3109—3112). The terms Hf, and

Hal," are given in (3.89-3.90). The terms 5):", 62:1,", 6):”, and 5:11;; are

3);, = L do 32%) L 333(3) (—e,,,,,. WM", my) H52l(kOR(e,e')) dv' (3.316)
t n

52:" = j]: dv 323(2)) L 31,?(11') (—v,,,,, '05,; + UM, UM) H62)(k0R(v,v')) dv’ (3.317)
t 7:

5511.7. = A dv X(:1(v)/I: XIW) ("vn,x ”(+1; + ”my ”(+134)
(+1 0

H32)(koR(e, v')) 310' (3.313)

5:113: = d” X£1(v)/ 34(7),) (“Una ”(+1; + ”my ”(+131)
L£+1 Ln

H62)(koR(v,v')) dv' (3.319)

1C3; = —/ X:(v’) [(—vn,z :i: + UN, 3)) - Rh] Hf2)(koR(v, v')) dv'dv (3.320)
L, I...

l .
1C2,” = —/ 37,?(0') l(—v,,,z :i: + ”my 3}) - Rh] H§2)(koR(v, v')) dv'dv (3.321)
3 AT Ll Ln 3

After putting (3.40) into (3.315), the set of equations may be written in matrix form
0’ c’ = b’ (3.322)

where D’ is an NT —1 x NT — 1 matrix, c’ represents the unknown expansion function

152

amplitudes, and b1 can be calculated by

 

 

A
I 4 1:
bl = 170 AT [A2611]; [AT/2 +711] vi i(vi)
£1:
2 u...
+,/£z dvl+1 [AT/2 "' ”[+1] fll+1 ' E'(:+l)] (3323)

The matrix elements are calculated as

 

The integrals i

_ vY vY vx vx ~Y
n " 1 k0 [91,1 + 03,1 + 9t+l,l + 0t+l,l + gl,N7~—1
+ 6y + + 6"
ANT-1 91X“ ,-'NT 1 (+1, NT— 1
X X X X Y
+71“ + [Cal - Ht+l,l - “(+14 + ”ANT—1

+ KENT—1 _ HL-erT-l — Egg-LNT—l
(3.324)

v VY v VX
2 S Tl S NT —1 k0 [92:11 + Ol,n + 9211;; + obi-1,11
~Y “Y "X ”X
+ Han—1 + 0!,n—1 + gl+l,n—l + 0(+1,n—1]
X X X X
+Ht,n + Kim - Hl-Hm — (bl-1,11

Y Y
+ ink—n 1+ KL—n 1 —Hl+l,n—l — Kl+l,n-—1

n Din can be approximated using the rectangular rule exactly as in

the previous section. Approximations of fig”, 57gb", g);_,, and 575,1,”4 are given in

(3.179). Similarly, 7135,, HZ,_,, 11151,,“ and 71);,“ are approximated m (3.179).

The other terms, because of the image of the cylinder, can be approximated as

2

= A4T (- ”71.x ”Ax + ”my ”(31) Ha )(kolf’nl W "D

A2
= —4—' ('— —vfl,2 vl+1’z + vn,y ”(i-1,”) H02 )(kOIfiT'J'i'l _— filinl)

A2 (2 )
= —-4— T(— ’Un_ 1,3 111,1; + 7113— 1,31 ”(’11) HO (kOIWJ_ PI,n—l|)
_ A"; (2) ~ ~
_ T (--’Un_1,z Ut+l,x + ‘Un_1’y 01.1.1”) H0 (kOIW,‘+1 — pI,fl—l|)

153

AT

K25. = 3- a.... Hf2)(ko|fir,z—fiz,nl)
[CZn—l = 'A2—T (Ian—1 Hiz)(k0l51‘,l — film—ll)
xii... = 923'— q...,.. H12’1k015r,.+1-m,.l)
Una—l = 923 qt+1,n—l Hf2’(kolifir,r+l — 61,.._r|) (3.325)

where (11,71 is defined as

th = (’vngc i + ”my I?) ' R5," (3.326)

Here, the unit vector directed from the image points to the cylinder points R5," is

 

R5,. = ’3.” _ 6”" (3.327)
’ IpTJ - pl,n

 

Similarly, bf can be approximated using the rectangular rule

2
b; = 00 [ I: — jsin 03,- 123,; sin(koyg sin 45,-)
+ cos <13,- vyy cos(koy¢¢,- sin 05.)] em“ cos ¢"
+ [ '" j sin ¢i ’Ue,t+l Sin(k0yt+l sin (15:)
+ COS 43g vy,(+1 COS(k0y(+1¢g sin 0%)] ejkoz¢+1 cos d” ] (3.328)
The diagonal terms in (3.324) should be computed more accurately. When 8 = n,

I
D”, becomes

I _ vY vX vY vY
Dru; _ Dflm + k0 [011,11 + Orr-Hm + on,n+l + 0n+l,n+l]

+IC,’,f,, + K); M, - ICX — KIM,“ (3.329)

n+1,n

where DM, is given in (3.266).

154

When I = n — 1, Din becomes

Dill—1,71:Dn—ln+ko[0n—l,fl+on,n+0fl— l,—nl+0n,n—l]

+Kn— 1,11 + (71— 1 ,n—l -K1)1(,n_K:n,n— 1 (3'330)
where Dn_1,,, is given in (3.269). When 8 = n — 2, Din becomas
Dn— 2,n=Dfl- 2,5fl+k0[n—2,n+0n—1,n+on—2,n— 1+0n— 1,n- I]
+K:n— 2,11 +Kn— 2,n— l —K:n— l,n —Kn— 1,-—n l (3'331)

where Dn_2,,, is given in (3.272).
Two more condition have to be taken into consideration because target is a close
surface as a result Em], and 5m position vectors go to the same point. When

Z = NT — 1 and n = 1, D5,, becomes

DN—T 1,1:DNT— 1.1+k0[0NT-— 1,1+0NT,1+0NT-1,—NT1+0NT,NT- I]

+KNT— 1,1 + KNT- 1 ,NT— 1 _KNT, 1 —KNT, NT— 1 (3'332)

where D~T_1,1 is given in (3.275). When 8 = NT — 1 and n = 2, D”, becomes

DNT_ 12" —DNT— 1,2+k0[0NT— 1,2+0NT,2+0NT— 1.1+0NT,1]

+ICNT—1,2+’CNT—1,1"CNT,_2 ’CNT,1 (3333)

where DNT_1,2 is given in (3.278).
The matrix equation is solved for the unknown currents. Then, the scattered E—
field is found in the far zone after approximating the Hankel functions. To find the

E—field the result in (3.294) can be used here. The scattered E—field due to the current

155

on the cylinder and the image current can be written as
58(5) = —@ko[fi [0 — Mia-17]] KW) MM» dt'
+ [T [in -i) [fi'fill] KTW') ‘I’o(p,p'z) 0W] (3-334)
where \Ilo(p,p!1~) and \Ilo(p, p’,) can be written using the definition in (3.281)
‘I'o(p, pip) = fl 5% aim/77’

2' e‘j’W’ - a“
wo(p,p',)=‘/;OJ; fl, em"! (3335)

After substituting (3.335) into (3.334) and writing the induced current using known

 

expansion function amplitudes, the scattered field in the far zone can be calculated

for a given polarization angle from

Ego?) = E30?) + 152(5) (3.336)

where E303 ) and E303 ) are defined as

 

. _. NT—l
“ 1’60 e 3"” .
E;(p) = —Z:~ 27r fl [ 2 (3,1,; [1608((15p + 453) Um,z
n=1
AT/2 A . l
/ (-2T _ v’) sin(y.'r sin ¢s) eJkozT c034,, d’U’

A'r/2 AT , ,
+sin(q§p + 45,) um,” / (— — v’) cos(y-’,~ sin (15,) CJkaT °°3¢‘ dv’]

“AT/2 2
NT-1 AT/2

+ Z cln+1[jcos(¢p + ¢,) um f (% + v’) my; sin ¢,)e""°$’7‘°3¢'dv'
n=1

—AT/2
A'r/Z

+ sin(<;3p + 43,) Um,” / (é—T -+- v') cos(yr'r sin ¢,) ej'm'T ”3"” CM] ] (3.337)

-AT/2 2

156

 

E”(fi) = — 770 jko e-jkop cosqS cos(¢ +45 ) +sin¢ sin(¢ +¢)
p AT 27f fl .9 p s s p s
”’7‘“1 AT/2 AT . i
Z Crln [j COS 45, mm] (— — v') sin(yr',- sin (15,) e’kchow' dv'
n=l ”AT/2 2
AT/z AT - I
+sin d), vmm / (— — v') cos(y.',~ sin ¢,) eJkO’T C084" dv’]
-AT/g 2

NT—l

AT/2 A . ,
+ E Gin“ jcos 45, v,” (-——T + v') sin(y.'r sin 45,) e’koxTww’ dv'
n21 —AT/2 2

AT/2

+ sin ¢, v,” / (£27; + v') cos(y3r sin 43,) ejkw'T “’8‘“ dv'] ] (3.338)

—AT/2

where ()3, is the scattering angle measured from the horizontal and (15,, is the polariza-

tion angle measured counter-clockwise from scattering direction as in Figure 3.6.
3.3.1.1 Calculation of the Induced Current and the Difference Current

The induced current and the difference current on the ground plane due to the pres-
ence of the cylinder can be calculated after calculating the induced current on the

cylinder. The vector potential can be written as
“-0 _& T I «1(2) ra IA (2) I I
A(p ) — 43' f K (v ){v H0 (koR) + (~vza: + vyy)H0 (koR )}dv (3.339)
T
The scattered magnetic field can be calculated using the vector potential as
.. .. 1 .. _.
H’(p ) = EV x A(p) (3.340)

Since the EU? ) has components only in a: and y-directions. H307 ) becomes

Ham = —1—[%A.(z,y) — 57AM]. (3.341)

157

where 5%Ay(x, y) and 5%Ax(x, y) can be written using (3.339)

 

:yAcc =%/( KT(v 35—31mm R)— Hg2’(kOR')]dv' (3.342)
3A (x y) = EB / KT(v’) v' 19— H(2)(koR) +H(2)(koR’)]dv’ (3.343)
(92: y , 4j T y 3:1: 0 0
After putting
(2) -
___dHO (“3) = _Hf2)($) (3.344)
dz
into (3.342) and (3.343), these become
0A”; (“5 —-——y—y’) =“40’7—1/KT(3(")3’”'[(R y)Hl(2)(koR)+(ygy)H1(2)(koRl)]dv' (3.345)
I
3A1,__(_ 3: y)_7~v1 (2) 1 (2) I I
62: "ijO/( K( (—a: 32') ”4E Hl (koR)+R—’Hl (koR) dv (3.346)

Substitution of (3.345) and (3.346) into (3.341) gives the expression for the scattered

magnetic field

 

 

H(fi) =)—z—/KT( v')[H(2)[ (kovR) 250—1); —————(y;iy’)]
+Hf2’(koR,) [1); 33131,) +3; (Ely/0]] dv’ (3.347)

Now the induced current on the ground plane can be calculated by using the

boundary condition of the total magnetic field

 

mm) = 3 x (37(3) +H‘(a)) (3.348)
=0
where incident field H‘U)’ ) is
_.. 2 .
H'(fi) = -—2 % cos(koy sin ¢.-) e’kflww‘ (3.349)

158

After substituting (3.349) and (3.347) into (3.348), the induced current becomes

KS(:c) = —5: { $1,115,023 H52)(koR,)i [(25 — z') 2); +y' 11;] dv'

+ E ejkoxcowe ) (3.350)

770

where R, is

 

Rx = \/(:r — x’)2 + y’ 2 (3.351)

Subtracting the current 1?? (.c) induced by the incident field without the target
present from the current induced with the target present gives the difference current

on the ground plane. The difference current can be written as

-o

KD(:1:) = KS(:z:) — K-S(1:)

 

 

 

= 11X (H‘(p')+H'(p)) —yxfi‘(5)
y=0 y=0
= 5: H’(p)
y=0
A k I 1 I I I I I
= -:E 2_; TKT( )Hf2)(k R’)E[($—$) vy-l-y v3] (1’!) (3.352)

3.3.2 Numerical Results

The transient scattered E—field is calculated for a cylinder located above a ground
plane by applying the image technique formulation. The larger value of 200 points or
20 points per wavelength is used to represent the current on the cylinder. Frequency
domain results are obtained varying the frequency from 1 GHz to 30 GHz with the
step size of 0.04 GHz resulting in 726 frequency data points. The IFT is used on the
spectral data windowed by a double cosine function. The transient field is shown in
Figure 3.18. As seen from the transient field, there are no edge reflections. All three
main reflections are identified. The first reflection (A) comes from the cylinder, the

second one (B) comes as a superposition of cylinder-sea and sea-cylinder reflections

159

and the third one (C) comes from the sea-target-sea reflection. The other reflections
need timing calculations to be identified accurately; therefore, they will be analyzed
in Chapter 4.

Although some insight is gained, the multipath information about the surface
roughness is lost. To overcome this disadvantage and be able to still use the image
technique formulation, a finite width roughness is superimposed with the ground
plane as it shown in Figure 3.19. This problem is solved by considering the roughness
as another target which has a sinusoidal shape, in addition to the cylindrical one;
therefore, nothing has be to be changed in the previous formulation to solve this
problem.

The change in the current densities on the ground plane and on the sinusoidal sur-
face is observed by calculating the induced currents as the sinusoidal surface touches
the ground plane. The geometry of the problem is shown in Figure 3.20. Figure 3.22a
shows the current density on the ground plane.

The boundary of the shadow region is recognized and the induced current beneath
the sinusoidal surface becomes smaller as the surface touches the ground plane. The
induced current on the sinusoidal surface changes significantly at the edges as shown
in Figure 3.22b. When the rough surface touches the flat surface, the currents on
the rough surface and the flat surface have to be continuous. However, the sinusoidal
surface and its image creates artificial sharp edges and this causes the induced current
to be zero at the edges. To support this reasoning, a semi circle object shown in
Figure 3.21 is lowered to the ground plane. The induced currents on the ground
plane are shown in Figure 3.23a for different target heights. The boundary of the
shadow region becomes very sharp and the induced current in this region becomes
almost zero. The induced currents on the semi-circle are shown in Figure 3.23b as
the semi-circle touches the ground plane. The current on the semi-circle is not zero

on the contact locations with the ground plane. This is because the target and its

160

image form a circle which does not have any edges. As expected, the currents on
the ground plane and semi-circle is still continuous when the semi-circle touches the

ground plane.

161

 

1.0

.O
01

l l
>
o

 

.O
o

l L 1 1
i

 

 

 

 

.C'>
U1
1

 

 

l
_L
O

l 1

 

 

Scattered electric field (relative)

 

I T T V fir I Y fiT

T 7 Y T T T T I Y 1 i I r I I

0.5 1.0 1.5 2.0 2.5 3.0
time (ns)

.0
0

Figure 3.18. Transient scattered field for a cylinder above an infinite ground plane,
<15,- = 40.00, 43, = 40.0“, (12‘, = 90.0”, a = 0.0127 m, ht/a = 8.0.

162

‘3!

I (60, #0) F332”

 

 

PEC

:‘ A 8
image plane
i

 

 

 

 

 

 

Figure 3.19. The geometry of the scattering problem for a cylinder above a finite
width roughness superimposed with a ground plane.

163

 

 

 

 

 

 

 

 

 

Figure 3.20. The geometry of the problem for a finite-width sinusoidal surface above
a ground plane.

164

 

 

 

i y X3?"
(60, #0) (pi Eli
PEC
0.
2a —/ : l
imageplane hy

,/ l

a = 00 ~ w——~ :1:

 

 

 

 

Figure 3.21. The geometry of the problem for a semi-circle target above a ground
plane.

165

 

l

 

 

 

 

 

 

 

 

 

 

 

 

 

. __hy/d=0.0
3f ........... hy/d=0.25
Lu° : . ...... ‘ l ‘
A2: '
=° 3' .
Z 4
1 -
o --~—.r=.==.-==.
-10 -8 -6 -4 -2 O 2 4 6 810
x-location/w
(a)
——hy/d=0.0
‘ ........... hy/d=0.25
4 i
o l
“\J
’3
5.-
>2
0 ‘fi= . - , . . . - T . . - . , e -
-0.50 -O.25 0.00 0.25 0.50
x-location/w
(b)

Figure 3.22. Current density amplitudes for a sinusoidal surface above an image
plane (a) on the image plane, (b) on the surface, kod = 8.54, h,/d = 0.25, w/d = 1.0,

4); = 90°.

166

 

——4L...=

lw=QO
/w =0.25

___—h
........... [1

Y
Y

 

 

 

 

 

 

o , ..==--=.....,......==-
-5-4-3-2-1012345
x-location/w

(a)

 

—— hy/w=0.0

J ........... by / w =0.25

 

 

 

 

 

 

0 I ' ff I T I ' I . T I I ' I I ' r f v I
-0.50 -o.25 0.00 0.25 0.50
x-location /w

(b)

Figure 3.23. Current density amplitudes for a half-circle surface above an image
plane (a) on the image plane, (b) on the surface, koa = 4.27, w/a = 2.0, 42,- = 90°.

167

3.4 Perturbation Approach for a Cylinder above an Infinite PEC Periodic
Surface

The perturbation technique is applied to solve for the electric field scattered
from a PEC cylinder of arbitrary cross-section above an infinite two-dimensional,
periodically-varying PEC surface. Application of this solution technique to TM po-
larization has been explained in detail in the previous chapter. In this section, the
same solution method will be used for TE polarization.

A TE plane wave is assumed to be incident on a cylinder of arbitrary cross-section
above a two-dimensional periodically-varying conducting surface as indicated in Fig-
ure 3.24. A set of coupled Electric-Field Integral Equations (EFIEs) for the current
induced on the cylinder and the perturbation current on the surface has been derived.
The EFIEs are solved in the frequency domain using MoM and transformed into the
time domain using IFT. This formulation requires the knowledge of the scattered field
produced by the plane wave interacting with an infinite surface without the cylinder
present, a problem which has been previously solved using Floquet analysis and the

MoM by Norman et. a]. [5].
3.4.1 Theory

This method separates the problem into two simple parts, one which has already been
solved. The first part is to find the scattered field produced by the induced current on
the cylinder and the perturbation current on a finite region of the periodic surface,
and the second is to solve for the scattered field from an infinite periodic surface
without the cylinder. The combination of these two solutions gives the solution to
the original problem as illustrated in Figure 3.25.

The incident electric field is assumed to be a TE plane wave with angle (1), from

168

 

 

 

all

Figure 3.24. The geometry of the scattering problem for a cylinder above an in-
finitely long periodic surface.

169

Ay

 

‘H.

O

 

AAAAAAAAK’

Wm

III
A?!

 

 

 

O

 

 

 

Figure 3.25. Perturbation technique to solve the scattered E-field from a cylinder
above an infinite two-dimensional periodically-varying surface .

170

the horizontal
Elfin, y) = E0 (—:i: sin 43.- + 3'] cos (15,-) eik°(“°9¢i+y8in ¢i) (3.353)

The scattered E—field can be written as in (3.10) because the configuration and nomen-
clature are the same. Considering that the sea surface is infinitely long, the E-field is
rewritten in terms of induced current on the cylinder and on the sea surface separately.
That is

it?) = Es(fi.fis')KS(fis’) +5r(fi.575)KT(575) (3.354)

where S and T symbolize the periodic surface and the cylinder, respectively. Accord-
ingly, K s(5's) represents the induced current amplitude on the surface and KT()5’1~)
represents the induced current amplitude on the cylinder. The induced currents can

be written as

KS( “') = 22’ KS(“’) (3.355)
WW) = 0' KTW) (3.355)

Using superposition the electric field scattered from the periodic surface can be

written as

55(5) = Em?) + E'Pw)

= ism; fis'm’ws') + 5305'. fisI)KP(5s') (3.357)

where EP is the perturbational field produced by the difference, or perturbation,
current RP, and E’ is the field due to current K" on the surface which is impressed
by the incident field without the cylinder present. Therefore, the perturbation current
can be found by RP = K's —- RI.

The Operator, 53(5, 55’) involves a contour which has an infinite extent. Therefore,

171

to calculate the scattered E—field from the infinitely-long periodic surface the solution
given by Norman et. al. [5] is used. Norman solves this problem using MOM-MFIE.
Therefore, first the scattered H—field is found by

6G1(fii 55’

Him) = K’w') ) dz' (3.358)

s, 5 an '
Here Sp is the one period of the sea surface, G’ (5', [2' ’) is the periodic Green’s function
(PGF) given in (2.59), and fig is the normal vector to the sea surface. To find
the magnetic field the derivative of the PGF with respect to the surface normal is
integrated over 8,, with the induced current. Then, the scattered E—field can be found

by using the magnetic field as

Em?) = 5307, "')K’ws') (3.359)
where Egg»; 5;) defined as
.. .., I _i I ~I 6GI(fi:/-;S,) I
£3.10 ps)K (ps’) — we ZXV: 5,.K (pr) ——,——ans d1 (3-360)

Here, V, is a two-dimensional differential operator given as

After substituting (3.357) and (3.359) into (3.354), the scattered E—field can be

rewritten as

E’(p)= £s(fi.fis')K'(ps')+£s(P P§)Kp(ps')+£rr(p'3" ')KT(5~r') (3362)

Applying the boundary condition of zero tangential total E-field to the periodic

172

surface gives
a - E”(p‘s) +21 - EP([)'§) +5 - ET(6'S) +6.E‘(5‘s) = 0 (3.363)

where E‘(,5‘s) is the field on the surface due to incident plane wave, ET (55) is the field
on the surface due to RT(,6'T) on the cylinder, Elms) is the field on the surface due
to the total current RIMS) on the surface which is impressed by the incident field
without the cylinder present, and EP (55) is the perturbational field on the surface
produced by the perturbation current RPUIS) = K S(5’s) -— K I (fig)

The definition of 1317065) requires

W

a . E’(,6‘s) + u - meg) = o (3.364)

This is the enforced boundary condition of zero tangential E—field on the periodic sea
surface for the scattering problem when the periodic sea surface is the only scatterer.
Substitution of (3.364) into (3.363) yields a relationship between the perturbational

field and field produced by the current on the cylinder given by
a - E”(6's) + a - 1.377%) = 0 (3.365)

The boundary condition for total tangential electric field on the cylinder must also

be considered. It can be written as
a - E'Pwr) + a - EWT) = _, . E’wr) — a - E"(i>'r) (3.366)

where ET(5'T) is the field on the cylinder due to the total current KT(,5'T) on the
cylinder, E‘(f>'p) is the field on the cylinder due to the incident plane wave, E’(fip) is
the scattered field on the cylinder due to R’ (5'5) on the periodic surface, and EP()5’7~)

173

is the field on the cylinder produced by difference current RPWS) on the periodic
surface.

The boundary conditions (3.365), (3.366) yield a pair of coupled EFIEs for the
current on the cylinder KT([r'r) and the perturbation current K P (53) on the periodic

surface

£s(fis. 13's )KP( ” ps') + £T(ps 5%)KT(57~)= 0 (3-367)

£s(i>‘r. fis’)K‘”(fis’) + £r(5r, WWW) --
—£§(pr ps'K) ’64 ')—v Em ) (3.368)

where Er is given in (3.28) and the linear operator cg is defined as

153(5.5.')K’(5s')= 1%.. er - v. SPK’ws') ————"G',,( ,4.) am (3369)
Here, K I (55) is a known quantity as solved by Norman [5]. Therefore, the coupled
EFIEs have only two unknowns which are K P (fig) and KT('p‘r).

The coupled EFIEs is written in terms of the perturbation current; therefore, the
perturbation approach can be applied as in TM polarization solution. Truncation of
the perturbation current K P (55) to a finite length of the surface represented with S
yields a good approximation when the cylinder is close to the surface. Simply, the
approximation is

prs) = 0 55 ¢ 5 (3.370)
KP(fiS) 33 E 3

After this approximation, the system of integral equations given in (3.367) and

(3.368) become

174

£s(5r.fis')KP(fis') + Dr(5r.fi1r')KT(/5'75) =

-£§(i>'r. fis’)K’(/'>'s') - 17 ° E‘Wr) (3-372)

After the approximation the EFIEs can be solved numerically. For a numerical
solution the truncated sea surface is approximated using NS planar segments and the
cylinder is approximated by NT planar segments. Then, the unknown currents are

approximated by a linear combination of the expansion functions

Ns—l N5— 1

prs) = 2 Kim) =2} lei. mu )+dsy.§.(u)l (3.373)
7" fir) = 2: K3,‘(c)= Z [ch,3‘(n) dTyf(u)) (3.374)

where u is the arclength distance on the surface measured from an origin U1. Similarly,
v is the arclength distance on the target measured from an origin V1. The expansion
functions X,§,(u) and 323(3) are defined on m’th segment of the surface. Similarly,
X302) and 373(7)) are the expansion functions on n’th segment of the target. The
expansion functions are shown in Figure 3.3, and they are given mathematically in
(3.33)-(3.36).

The EFIEs in (3.371)-(3.372) are almost identical to the EFIEs derived for the
finite sea surface model. Except, the forcing terms in (3.26) and (3.27) are now
modified for the truncated difference current. Hence, the solution of the previous
case can conveniently be used here. The coupled linear equations given in (3.120)
and (3.121) are modified for the forcing terms to obtain the solution of the coupled
EFIEs for the perturbation current and the target current. After making simple

modifications, they become
N5 — 1

Z of. [6323. + $131+ H. 4......)

m=2

175

IVs-2

+ Z Cfn+l [kolél’m + gig-1,771] + “Km — ”Kl-hm]

m=l
NT—l

+ 2 c3: [mm + 13...] + H3. — 113...]
11:1

NT-l

+ X 6.3.1 [MU]; + iii...) + 713;, — Him] = 0 (3.375)
n=1

NT-l
2: c: [3.33. + 32:...) + H3. - 33...]
3:.

+ Z 03.41 [koLézn + 91+1,n] + ”Zn "’ Ham]
3:.

+ Z 6.3.. [komzm + 7331”,] + 7321..., — Him]
23:3

+ Z Ci+1[k°[;lzm + hithml + ”In: — ”Xi-hm]
m=l

4 . ~e ~ ~.-
= _— {/ dv 337100) 'UI ' E301) +/ d" XIEKU) ”H1 'E (v)
770 L; Ll+l

+ [L 45 323(2)) £36.33) K’ws')

+ / dv «33.1.6) £36.33) K’s.) } (3.376)
LI+I

After renaming the unknown current amplitudes cf" on the surface as af’; in (3.127),

the coupled linear equations can be written in matrix form as

' m B; n (3,3,, (75,
A" ’ = ' (3.377)
Cl,m Dim CZ Q7;

A5,... is an N5 — 2 x N; — 2 matrix, B5,, is an N; — 2 x NT — 1 matrix, C"... is an
Nrp—lst—2matrix, andD¢,nisanNT—lxNT—1matrix.

A change of variables given in (3.128) is used to make calculations faster for the

176

S
51

matrix elements, (_2 and QC. The matrix elements Aim, BM, 04..., and D”, can
be calculated directly from the same matrix element computation formulas (3.134)-
(3.137) for the finite-size problem by a numerical integration. After the change of
variables 65, and 53 can he written as

55. = 0 (3.378)

57‘ — iUfl d0 —1—[A/2+v]f1-E"(v)
_( — 770 -921: l AT T l l l
1 . "i
+ / dUI+1 "A—[AT/2 — ”[+1] ”(+1 'E (”(+1)
_ T

1 -o I -o I
u... EMT/2 + w] £302....) M02.)

+
'|\.

+
:\
+3 '45 4: is 4.» "l?

1 .. ..
dvt+1 K;lA'I‘/2 " ”(+11 £§(vt+ltps’) KI(PS') J (3-379)

Substituting the expression for the incident electric field in (3.353) and approximating

the integrations using rectangular rule yields

53. = 0 (3.380)

2 A . .
E" = nT [ Iv... cases.) — u. sin(¢.>1 6......3........,

 

+ [UH-1,3, 003(435) _ ”(+1”: sin(¢,)] ejk0(3l+l C08¢i+yt+18in¢ii

+ 53(3rbfis') K’ws') + £s’(/3rt+1.5s') K’(b‘s')] (3.381)

where n.3,, um, Um and v)“, are the components of the unit vectors tangent to inte—

gration segments on the surface and target. These components are calculated from

(3.133).

177

The scattered E—field can be calculated after solving (3.377) for oi, and cf. To
improve the accuracy of the technique before calculating the scattered field, the per-

turbation current is weighted using a cosine function

Km.) = ”Zoos (1‘31) [mm +013. 3(a)] (3.382)

where w is the width of the surface where the perturbation current is calculated. The
weighted-perturbation current and the current on the target is used to calculate the

scattered E—field
Em?) = ism fis')K.f(fis’) + BTU»: awn/771) (3383)

3.4.2 Numerical Results

Determining the truncation width is very important to apply the perturbation tech-
nique accurately. Therefore, before applying the perturbation technique, the trunca-
tion width has to be found for a desired accuracy. To find the required truncation
width, the solutions obtained by the perturbation and image techniques for a flat
surface are compared. As a result, a simple rule of thumb is obtained empirically,
and then applied to a surface with non-zero roughness. Finally, transient multipath
effects are identified through application of the IFT. For all the MoM solutions, the
larger value of 100 points or 40 points per wavelength is used to represent the current
on the cylinder, and the larger value of 55 points or 20 points per wavelength is used

to represent the current on each period of the surface.
3.4.2.1 The Validity of the Perturbation Assumption

To investigate the region of multiple interactions, a circular cylinder is located above
an infinitely long flat surface. Then, the exact perturbation current on the flat surface

is found by subtracting the currents determined using the image technique with and

178

without the cylinder present. As is seen in Figure 3.26, the perturbation current is
significant immediately beneath the cylinder, and negligible outside a certain finite
region the extent of which depends upon the cylinder’s distance from the surface.
Similar to the TM polarization case, when the cylinder height gets bigger, the per-
turbation current Spreads along the surface. Therefore, the truncation width must be
carefully selected according to the cylinder height for the perturbation method.

The effect of frequency on the truncation width is investigated and the pertur-
bation current is plotted for various frequencies in Figure 3.27. It is seen that the
truncation width does not have to be wider for increasing frequencies. Similarly, Fig-
ure 3.28 shows the effect of incidence angle on the perturbation current. As expected,
the center of concentration of the perturbational current shifts somewhat as incidence
angle is changed. It is also observed that as incidence angle decreases, the truncation
width has to be increased to keep the accuracy at the desired level. This should be
taken into consideration when determining the truncation width.

Calculating the scattered field is the main concern; therefore, it is helpful to
compare the exact scattered field found using the image technique to that found
from the perturbational method. The amount of error between the two can be used
to estimate the pr0per truncation width. The scattered field is calculated for two
different cylinder heights. The magnitudas of the E—fields are plotted as a function
of frequency in Figure 3.29 and in Figure 3.30 for various truncation widths, w. The
approximation difference defined by |EIm9 - EP | for each frequency. This is plotted
for h,/a = 1.0 in Figure 3.31 and for ht/a = 8.0 in Figure 3.32. An approximate
truncation width can be chosen roughly independent of frequency to give a desired
accuracy value. To determine a rule of thumb for choosing the truncation width, it
is helpful to examine the aggregate difference between the image and perturbation
techniques. The total difference from the image technique is calculated the same
formula used for TM case given in (2.74). Figure 3.33 plots the total error calculated

179

for 45.- = 20° and different cylinder heights. To examine the effect of incidence angle
on truncation width, the incidence angle is increased to 400 and the total error is
plotted for different cylinder heights in Figure 3.34. As expected, the perturbation
solution becomas more accurate for increasing truncation width and incidence angle.
For a relative error of 0.033 an empirical formula for the minimum truncation width

is found to be

3 = 23 159. (3.384)

h‘ In (1.781 +21n (5)) 45"

 

where ¢o = 20° and <15,- is the incidence angle.

To see if the empirical formula holds in the presence of surface roughness, the
perturbation currents are calculated when a cylinder is placed above a finite sinusoidal
surface. The perturbation currents on the surface are shown for different cylinder
heights in Figure 3.35. Again it is concluded that the perturbation current is highly
effected by the height of the cylinder. Then, the perturbation current is plotted for
different surface roughness in Figure 3.36. It is seen that the extent of the perturbation
current is not dramatically effected by surface roughness. It is expected that the
perturbation current will change as the location of the cylinder in the x—direction is
changed. The perturbation current as a function of horizontal position is plotted in
Figure 3.37. It is seen that the location of the cylinder in the :r-direction does not

dramatically affect the truncation width.

180

 

 

 

 

AM 3 ___- ht/a=1.0
-g 2 ----------- fit/3:81)
'H 1.2 3 _
{E3 ; _._.. ht/a—32.0
:10 -‘ ,Nc. _.._. ht/a=64.0
8 3 z/ \. \..
t 0.8 5 {/2 \. \..
3 1 ; \ \__
o : l . .\ \"\..
.SCB 0'6 l l ‘\-\. \°
*" i ' \ \-\,\
g 0.4 1 [I \\
3 : ,’;<\ \\ .....................
E 0.2 ‘::________....——/ .......... ”aye \\ \\\\
o' L""";‘_':;';';;I'_'_";';....// ~~~~~

0.0 *“ . , , ,

-400 -200 O 200 400

Figure 3.26. Perturbation current on a flat surface for different separations between
the cylinder and the surface, koa = 0.266, a = 0.0127 m, 1:../a = 0.0, ¢.- = 20".

181

 

o _L
I I
on O
l L L l L L 1 .l 1
‘ - -_ a
A,

Perturbation current (relative)
.0
CD

—-—- ka=0.266
----------- ka=1.064
—-—-- ka=2.661

 

 

. l
* -'/ I! -.\\\
. . / )‘l -- \~~
0.2 ‘ ........... ,/ / .............
l ......... /,’///// l .
.;:.:_':' ~~~~ (K .\ ‘ ...........
, /
0.0 . f I I . T

—400 -200 0

 

Figure 3.27. Perturbation current on a flat surface for different frequencies,

a = 0.0127 m, ht/a = 8.0, mr/a = 0.0, d).- = 20°.

182

 

A
O
J_]_
|
I
5-
II
N
.o
O

 

 

 

 

 

 

 

'3?
.2 _
E 0.8: H) -— -— (I), = 30.0
g - H _ o
E 3 il\. ¢i 40
e 0-5 i i X \.
S : l \. \.
:04 ‘ l \ \‘
s ' 1 .‘ Ox
3 \°-\..\,
I: '\.
(D
o.

200 400

 

Figure 3.28. Perturbation current on a flat surface for different incidence angles,
a = 0.0127 m, ht/a = 8.0, zr/a = 0.0, koa = 2.128.

183

 

 

 

 

1 ——--- image technique
4i ........... w/ht=9.0
; _.._.. wlht=20.0
3A 3; _.._. w/ht=40.0
E l K
9: //"(
‘11. ’/'~/'\ W H //
1 0 ll] 3) ff" My?" -
0 1 2 3 4 5 6 7

Figure 3.29. Normalized amplitude of scattered electric field for different truncation
widths, a = 0.0127 m, ht/a = 1.0, :r,/a = 0.0, (b,- = 20°.

184

 

 

(.0
.1..

(IESI/Eo )(P/a)"2

 

———- image technique
........... W/ht=5.0

—-—-- W/ht=9.0
_.._. w/ht=17.0

 

 

Figure 3.30. Normalized amplitude of scattered electric field for different truncation
widths, a = 0.0127 m, ht/a = 8.0, z,/a = 0.0, d).- = 20°.

185

 

 

 

 

 

 

101 ,
i _._.. w/ht=9.0 ___— w/ht=40.0
~ _.._. w/ht=20.0
g 10° - /' ' 'P'/\"\ a“
a a f '- . . .«z/--~w/~~\
E. 3 V:\ 4007 " -. . \-"-. \. f
g 10-1 - i (U!
a: l
.g g *
Lu_ 10‘2 - ,
10.3 I T I I . . I ‘ ' l ' I T
0 1 2 3 4 5 6 7 3
ka

Figure 3.31. Approximation differences for different truncation widths, a = 0.0127
m, ht/a = 1.0, mr/a = 0.0, 4:; = 20°.

186

 

 

 

 

 

_._.. w/ht=50 __ w/ht=17.0
3 _.._. w/ht=9.0
g 0 _. [V‘ I" .A- \ P ’\ /"-\ ‘
75 10 (“i 1’“ //"“‘l\ I'://.""\ I” \\\//" \V "\\. l
:9; i § / \\ I: l" f, \ ll \l’/ “19] \V’
”a ‘A \ l); ’ g I, '(
EW- 3; a , , I
O. ' ,
Lu I I
ml
.§
a 10-2 ..
10.3 ff! T I 'rtr yvvurjvv..]....
o 1 2 3 4 5 6 7 8
ka

Figure 3.32. Approximation differences for different truncation widths, a = 0.0127
m, ht/a = 8.0, zr/a = 0.0, ()5, = 20°.

187

 

0.01 ~

0.1 a... ................... . ................. . ............................................................ .. ......... ....................................................................

 

+ ht/a=1.0
+ ht/a=4.0
V + ht/a=8.0

 

 

..............................................................................................................................................................................................

 

 

O 4 81216202428323640

Figure 3.33.

w/ht

The total approximMion difference for different cylinder heights, for

law ranging from 0.266 to 7.98, a = 0.0127 m, d), = 20°, 1:,/a = 0.0.

188

 

—o— ht/a=1.0
—I-— ht/a=4.0
+ ht/a=8.0
0.1 rm”... ............................... . ..................................................................................................

 

 

 

O O I _ .......................................................................................................................................................................................
I . - .

 

0 001 _ ............ . ........................................................... , ', ............ ......

 

 

 

0 4 8 12 16 20 24 28 32 36 40
MM

Figure 3.34. The total approximation difference for different cylinder heights, for
koa ranging from 0.266 to 7.98, a = 0.0127 m, o,- = 40°, zr/a = 0.0.

189

 

 

 

 

 

Figure 3.35. Perturbation current on a rough surface for different separations be-
tween the cylinder and the surface, koa = 0.266, a = 0.0127 m, x,/a = 0.0, d/a = 8.0,
h,/d = 0.125, «)3,- = 20°.

190

 

_.._.. hS/d=0.0

_.._. hS/d=0.125
__ hS/d=0.250

ent (relative)
'0 ix) '2

     

 

 

 

 

x/a

Figure 3.36. Perturbation current for different surface heights, koa = 0.266,
a = 0.0127 m, ht/a = 4.0, :cr/a = 0.0, d/a = 8,0, 45,. = 20°,

191

 

 

 

 

3:14 -} —-—-- xr /a=0.0
E 1.2 _ —~—- Xr /a=4.0
m .,
:1 0 — A .f\
at, 0 8 - 7/ \\J/ \ A
3 I i ' \I l.‘
g 0.6 i ./‘ / I .
g 0'4 T ff. '\/.--’\¥ / \"/‘\\j -’ l
:3 . -. / \/ '
E 0 2 j './-/ \" \ -'l
0. Il ‘v/

0.0 d _r l l ' T Y T l

-45 -30 -15 0 15 3O 45
x/a

Figure 3.37. Perturbation current on a rough surface for different horizontal posi-
tions of the cylinder, koa = 0.266, a = 0.0127 m, ht/a = 8.0, d/a = 8.0, h,/d = 0.125,
$5 = 200.

192

3.4.2.2 Multipath Observations

The transient scattered field for a cylinder above an infinitely-long sinusoidal surface
is calculated in the frequency domain. The wavenumber normalized by the radius of
the cylinder is varied from 0.266 to 7 .98, with a step size of 10.64 x 10‘3, resulting in
726 data points. The time domain response shown in Figure 3.38 is obtained by first
windowing the frequency domain results using a double cosine function then calcu—
lating the IFT. As in TM case solution, the direct clutter response of the sinusoidal
surface has not been included to better view the multipath effect. It must be added if
the total scattered field is desired. Three main reflections have been easily observed in
Figure 3.38 for a flat surface. The first signal (A) comes directly from the cylinder, the
second signal (B) comes as a summation of cylinder-sea and sea-cylinder reflections,
and the third signal (C) comes from the sea—target-sea reflection. These reflections
also observed for non zero roughness. However, because of the surface roughness the
amplitude of multiple reflections have been changed dramatically. A detailed analysis

is left to chapter 4 for the effect of the surface roughness on the multiscattering.

3.5 Conclusions

In this chapter, theoretical analyses were conducted for an arbitrary shape target
above several sea surface models. Numerical results are obtained for a cylindrical tar-
get above a finite flat surface, a finite sinusoidal surface and an infinite flat surface.
It is concluded that the finite sea models are not suitable for multipath analysis be-
cause of edge reflections and interactions of these reflections with the target. Hence,
a perturbation method has been used to solve the problem of electromagnetic wave
scattering from a cylinder above an infinite periodic surface excited by a TE polar-
ized electromagnetic wave. To determine the truncation width for the perturbation
method, a rule has been developed. This rule has been obtained by comparing the

scattered electric field from a cylinder above an infinite flat surface solved by the

193

 

 

 

 

 

 

 

 

 

 

A 2.0 .
a) : —--—- hs/d=0.125
> .
“7'6 1'5? B _ hs/d=0.0
E 2
b 1.0 -5 A
U :
‘5 0.5 i
‘= :
S 00 g
‘8' 2
T, -o.5 i
E -1.0 a
g i
-1.5 —
8 3
(”-2.0 - , ,,,,,,,,, , n. .
o 1 2 3

time (ns)

Figure 3.38. Transient scattered electric field from a cylinder above a flat surface
and above a sinusoidal surface, a = 0.0127 m, d/a = 8.0, ht/a = 8.0, xr/d = 0.0,
w/d = 13.0, d).- = 20°, ¢, = 20.00, (75,, = 90.00.

194

image technique and the perturbation technique as a function of truncation width.
It has been shown that selection of the truncation width is highly dependent on the
cylinder height from the surface and the incidence angle. However, if the pr0per width
is selected, the results from the perturbation technique match well with the image
technique results. The perturbation current on a finite sinusoidal surface has been
calculated and it was shown that the truncation width is consistent with the infinite
flat surface. Therefore, the truncation rule for the infinite flat surface can be used for

a sinusoidal surface as well.

195

CHAPTER 4

IDENTIFICATION OF THE MULTIPLE REFLECTIONS

Multipath propagation plays an important role when Operating radar systems at sea.
At low-grazing angles this phenomenon becomes a dominant factor for reducing the
ability to detect targets above the sea surface. There has been a great deal of research
to reduce this effect [8]. Sletten measured the ultra-wide band (UWB) response and
showed how this multiple reflections affect the detection of targets. However, in his
analysis only the most obvious multiple reflections have been considered.

In this chapter, the necessary tools developed in chapters 2 and 3 for UWB anal-
ysis are utilized to understand the multipath phenomenon. The analysis of transient
scattering for both TE and TM polarizations from cylindrical targets above flat and
sinusoidal sea surfaces are considered. The possible paths traveled by the reflected

waves are calculated and the multiple reflections are identified by using timing anal-

ysis.

4.1 Multipath Analysis for Targets above Flat Surfaces

The scattering from a cylinder above a flat surface gives the basic understanding of
the multiscattering. Therefore, the sea surface is considered for the initial analysis
to be flat because of the complexity of rough surface scattering. The analysis is made
for finite and infinitely long flat surfaces, to see how the edge scattering affects the
backscattered field.

4.1.1 TM Polarization Case

In order to differentiate the multiple reflections, first the reflections from a cylinder
and a finite flat surface are considered. The incident TM transient field is shown

in Figure 4.1. This time domain signal is constructed from a frequency Spectrum of

196

1 — 30 GHz with 726 data points. This frequency data is weighted using a double
cosine function with the values of 11 = 0.040 and 72 = 0.0625. Then, the weighted
frequency data transformed in to time domain using an inverse Fourier transform.
The backscattered field from the cylinder is shown in Figure 4.2a. In TM polar-
ization the signals having odd number reflection paths from PEC surfaces will have
negative signs. As expected the reflected field has a sign change because of a single
reflection from a PEC surface. The backscattered field from a flat surface is also
important; therefore, the response from the finite-width flat surface is calculated and
shown in Figure 4.2b. As expected, there are two reflections, one coming from the
front and one coming from the back edge of the surface. Using the geometry shown

in Figure 4.3, the time between these two reflections can be calculated using
8.3) = ___; (4.1)

where c is the speed of light. For the case considered, the incidence angle is (15,- = 400
and the width of the flat surface is w = 0.9144 m. Therefore, the time passed between
the field reflected from the front and the back is Atf,’ = 4.685 us. To compare the
time At? calculated from the path calculation with the actual time, Ato shown in
Figure 4.2b is calculated and found to be Ato = 4.682 ns, which is a good match.
After determining the direct reflections, the cylindrical target is placed above the
finite flat surface. The possible main reflection paths of the backscattered field for
this problem are shown in Figure 4.4. From Figure 4.4 the difference in the path
length of the reflection from the edge (Edge A) and the direct reflection from the
cylinder (Cylinder) is
P1 = 2 (L, + L2) (4.2)

197

The distance L1 can be calculated from

L1 = 803(3) [33’- — (z. + a)]

where the distance 2:, is calculated from

_ ht + a
3“ ‘ tan(¢,-)

 

To calculate L2, Figure 4.5 is used. From the geometry, it is found that
L2 = L3 — L9
where

L9 = a [1 - cos(¢,-) J

and L3 is found using the geometry shown in Figure 4.4

L3 = L4 cos(2¢,-)
_ (h¢+a)

_ sin(¢,) COS(2¢i)

 

Substituting (4.6) and (4.7) into (4.5) gives

— (ht + a) S ' — a _ COS '
L” ‘ sin(¢.-) 6° (W l1 W]

 

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

Using these results, the time between Cylinder reflection and Edge A reflection can

be calculated from
2 (L1 + L2)
C

At0 =

(4.9)

The values of the parameters are ()5,- = 40°, a = 0.0127m, II, = 8a and w = 0.9144 m

198

for the problem considered. Using these values in (4.3-4.8), the values of the distances
are calculated and shown in Table 4.1. Substituting the values of L1 and L2 in Table
4.1 into (4.9) gives At? = 1.766 ns. After completing all the timing calculations,

the results will be compared with that found from numerical data. The third reflec-

Table 4.1. The calculated distances to be used to find Atl.

 

 

as, = 0.136 m
L4 = 0.178 m
L9 = 2.971 10'3 m

 

 

tion comes as a summation of cylinder-sea (Multiple A) and sea-cylinder reflection
(Multiple B). Multiple A and Multiple B have a positive magnitude because they are
reflected from a PEC surface twice. This signal comes At20 seconds after the incident
field reflected directly from the cylinder. The difference time At? can be calculated

from
L9 + L4 - L2)

At€=(
C

(4.10)

 

The values of L9, L4 and L2 in Table 4.1 are substituted into (4.10) and the difference
time is found to be At? = 0.512 us.

The third multiple reflection (Multiple C) comes as a sea—cylinder—sea reflection
with a negative sign because of being reflected three times from a PEC surface. This
signals comes Atg seconds after Multiple A and B reflections. Atg can be calculated

using Figure 4.4 and Figure 4.6

2P3 — P2
C

At? =

199

 

1
= E[2(1,4 — L10 — L7 + L3 + L1) — (2L1 + L3 + 14)]
: L4—2L10—L3 (411)
C

where L7 and L3 are equal to each other and thus cancel. The distance L10 is found

from Figure 4.4
L10 = a [1 — cos(¢,-)] (4.12)

and its value is 2.971 10‘3 m for a = 0.0127 m and (13,- = 40°. The value of L10, the
values of L4 and L3 in Table 4.1 are substituted into (4.11). The difference time is
found to be At? = 0.472 ns.

It is expected that the E—field diffracted from the front edge of the flat surface will
be reflected from the cylinder. The expected path traveled by this type of reflected
signal is shown in Figure 4.7. Using this figure, the traveling time of this signal (Edge
Cylinder) can be calculated. The relative time At? to the Multiple A reflection can

be found from

1
At? = E (L, — L4 — L1) (4.13)

h

'2'

where the distance from the edge to the cylinder is L, = [(% — a)2 + (ht + a)2]
Using the values of a = 0.0127 m, ht = 8a and w = 0.9144 m, L, is found to be
0.4713 In. Le and the values of L4, L1 in Table 4.1 are substituted in (4.13) and the
relative time becomes At? = 0.150 ns.

This reflection concludes the primary multiple reflections. The expected paths
of secondary multiple reflections are shown in Figure 4.8. The first one comes as a
cylinder-sea—cylinder reflection (Secondary A). The relative time of this signal can be

calculated from Figure 4.9. At? relative to the Multiple A reflection can be found

200

using

At? =

QIH

[L3 — L, + 2 (L12 + Lll + no] (4.14)

where Lu and L12 are found using Figure 4.9

 

 

Ln = a — L13
= a [1 — sin(45 + 5:— ] (415)
L12 = L17 sin(90 — (1),)
= Fa — L15] 8111(90 — (15;)
= a, -— (L14 — [115)] 8111(90 —' ¢i)
, L .
= [a — {acos(45 — g) — tan(9013— 915:“) H Sln(90 - ¢i)
3,- sin(45 - 2,1)

 

= a [1 — cos(45 — ] sin(90 — (15,) (4.16)

3) tan(90 — ¢.—)

Lu and Lu are calculated using the parameters of the case considered in (4.15) and
(4.16). The results are Ln = 7.333 10‘3 m and L12 = 4.362 10"3 m. L3 and L4 are
already calculated in Table 4.1. Substituting the calculated values of L3,L4, Lu and
L12 into (4.14) gives At? = 0.266 us.

The next expected signal comes as a summation of cylinder-sea-cylinder-sea (Sec-
ondary B) and sea-cylinder-sea-cylinder (Secondary C) reflections. The path length
of this signal shown in Figure 4.8 is difficult to compute so will be approximated.

The approximated path is shown in Figure 4.10. The traveling time relative to the

201

Multiple A reflection can be found from

At? =

GIN

(L19 + L21 + ht — L13) (4.17)

where L13, L19 and L21 are found using Figure 4.10

L... = a [1 _ c033,] (4.18)
L19 = a— L20
= a[1—sin(¢,)] (4.19)

r

L21 -_— asin(¢,) — L22] sin(2¢,)

 

= a sin(¢i) — L18 tan(90 "' 2450] sin(2¢,-) (4-20)

The values of L18, L19 and L21 are calculated for the case considered and they are
found to be L18 = 2.9712 10‘3 m, L19 = 4.5366 10‘3 m and L21 = 7.5234 10"3 m.
Substituting these calculated values into (4.17) leads At? = 0.740 718.

The last secondary multiple reflection (Secondary D) follows the sea-cylinder-sea—
cylinder-sea path. This signal comes At? seconds after Multiple C. The difference

time can be calculated using Figure 4.11

2
At? = 2 (L23 + L24 + h.) (4.21)
where L23 and L24 are
L23 = a[1 — sin(45 + % J (4.22)

202

 

L24 = a 1— cos(45 — 521)]
= a1—sin(45+%]
= L23 (4.23)

L23 is calculated and its value is 1.1899 10"3 m. L24 is equal to L23; therefore, using
this value At? is found to be 0.695 n3.

The backscattered field calculated for this geometry using the EFIE-MOM formu-
lation developed in Chapter 2. Using the calculated times the reflections are identified
in Figure 4.12 and Figure 4.13. Then, the actual times are found from Figure 4.12
and Figure 4.13. The timings found using ray calculations and those identified from
Figure 4.12 are summarized in Table 4.2. Comparing the relative times, it is found
that the expected paths are pretty good approximations to the actual paths.

To see how the edge reflections affect the backscattered field, the flat surface is
next considered to be infinitely long, with the incidence angle and cylinder radius
kept the same. Using image theory and the EFIE-MOM formulation, the scattered
E—field is calculated and plotted in Figure 4.14. As is expected, there are no direct
edge reflections (Edge A and B) or indirect edge reflections (Edge-Cylinder). The
timings of the reflections are found from Figure 4.14 and compared in Table 4.3 with

the previously calculated timings.

203

Table 4.2. Comparison of timings in TM polarization for a cylinder above finite flat
surface, a = 0.0127 m, h, = 8a, w = 0.9144 m, 43, = 40°, (15, = 40°.

 

 

 

 

 

 

 

 

 

 

 

 

n Atn (ns) At? (ns)
0 4.682 4.685
1 1.755 1.766
2 0.513 0.512
3 0.473 0.472
4 0.266 0.269
5 0.721 0.740
6 0.696 0.695
7 0.159 0.150

 

 

Table 4.3. Comparison of timings in TM polarization for a cylinder above an infinite
ground plane, a = 0.0127 m, ht = 8a, 21) = 0.9144 m, 45,- : 40°, ¢, = 40°.

 

 

 

 

 

 

 

 

 

11 At" (ns) At? (ns)
2 0.513 0.512
3 0.470 0.472
4 0.271 0.269
5 0.720 0.740
6 0.699 0.695

 

204

 

 

 

 

0.00 ~

Incident electric field (relative)

 

 

 

 

 

-O.50 . . , , .
0 1
time (ns)

Figure 4.1. The incident E—Field waveform, 43,- = 40°.

205

 

0.50

 

 

P
o
o

-0.50 ~

 

 

 

 

Scattered electric field (relative)

 

 

 

 

 

 

-1.00*
0 1 2
time(ns)
(a)

A050
a, .
.2 ‘
«:6 1
$0.25-
2 .
m i
c J
E 000« l.
‘8 i
2 .
m
35.254
0) l E
(0.0.501l‘--f-i--fl

 

 

O 1 2 3 4 5 6
time (ns)

0))

Figure 4.2. TM backscattered fields (a) from a cylindrical target with a = 0.0127 m
(b) from a finite-flat surface of width 20 = 0.9144 m. ()5,- = 40°.

206

 

 

<——t- EdgeA
<—+ Edge B

 

 

 

 

   

‘\
‘\

\\

‘\
.\ z
\%

 

i

Figure 4.3. The reflections from the edges of a finite flat surface.

207

 

<—o— Edge A
<——--- Edge B
<—o Cylinder
<—- Multiple A

 
 
 
   

<—n- Multiple B
Multiple C

 

 

 

 

 

 

 

 

 

 

Figure 4.4. Expected reflections from a cylinder above a finite flat surface.

208

 

 

 

.--- ---------4(---

Q
P4
t.»

Figure 4.5. Reflection directly from the cylinder.

209

 

 

 

Figure 4.6. Sea-Cylinder-Sea multiple reflection (Multiple C).

210

 

 

Multiple A ,./

«——o- Edge Cylinder

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.7. Edge-Cylinder multiple reflection.

211

 

/ 4—-ce secondary A

<—- Secondary B
H Secondary C
<—-o Secondary D

/
/
I : § ' ‘g‘/ (15:;

 

 

 

 

   

 

 

 

Figure 4.8. Secondary Multiple Reflections.

212

 

 

 

 

 

 

 

 

 

Figure 4.9. Secondary Multiple Reflection A.

213

 

 

 

 

 

 

 

 

Figure 4.10. Secondary Multiple Reflection B and C.

214

 

 

a

 

 

\
\\
.. t Mm .
s 1017 \\\
s _ \
\
s .3 \\
.3 \\ \\\
¢ . 4.\
\\\ \\
ls \\\
_ \\
_ s \
\\ \\\
‘ _\\\

452'

1 L23

 

I
---

 

I
I
I
I
\
X

4%
_¢t
2

\
\
\

I

45

‘ \

¢z
45 "2

\

\
\
\

\

 

-------------1 b

Figure 4.11. Secondary Multiple Reflection D.

215

 

 

 

 

 

 

 

 

 

 

A 5 2.866 m f

g) 0.588 n: 2.368 n: 3.328 I18 5.280 n:

E 1 .00 _ Multigle A & Mugtiple B

(D 5 g

L I 3

% 0'50 .- Cylirider Mtiltiple C

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g 0.00 , if 1

o - .

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9’ -O.50 - 3 At

‘0 : F 1

93 .

a) : ' At2 ,

3: -1.00 - s

8 . I< A“ >1

m rrrrrrrrrrrrrrrrrrrrrrrrrrrrrr E rrrrrrrrrrrrrrrrrrrrrrrrrrrr
0 1 2 3 4 5 6

time (ns)

Figure 4.12. TM backscattered field from a cylinder above a finite flat surface,
a = 0.0127 m, h; = 8a, w = 0.9144 m, (l),- = 40°.

216

2.50

 

3.587 as

.3...

Secqndary B Seodndary D
:8. :

I
_3
01
O

-1.00 I

J l

l

 

 
 
 
  

  

 

 

Seooindary C

 

  
 

  

)

 

 

rd; ; 2.888 n: 3.388ns
. 53.0255 ‘ '
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« Mult" leAS 3
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e 3 I
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‘0 1
93 -0.50 '1
G)
1:
(U
U
(D

 

 

l"
01

3.0 3

5

   

 

time(ns)

Figure 4.13. TM backscattered field from a cylinder above a finite flat surface
for time starting from multiple reflections, a = 0.0127 m, ht = 8a, 111 = 0.9144 m,
d),- = 40°.

217

 

0.42:4 ns 0.937 ns 1.437 as

l

2.00 f

_3 Multiple A: . i
1.50 : Cylinder & {Multiple 2C
Multiple BE 3
; ' Secondary B
3 58:
(secondary C _
SecoindaryA ) SOOO'PW D

i

g 1.208 mi 1.657 as 2.508 as

1111

1.00
0.50
0.00

-o.5o

4.00; <—>

 
   
  

l

   

 

    

I>

5'
I>
63’

 

 

 

Scattered electric field (relative)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
time (ns)

Figure 4.14. TM backscattered field from a cylinder above an infinite ground plane,
a = 0.0127 m, h, = 8a, ¢,- = 40°.

218

4.1.2 TE Polarization Case

The same pulse used for TM polarization shown in Figure 4.1 is assumed to be an in-
cident TE pulse with angle 45,- from the horizontal. As with the previous polarization,
frequency data from 1 — 30 GHz is weighted using a double cosine function with the
values of T1 = 0.040 and 72 = 0.0625. Then, the incidence pulse is constructed from
the weighted frequency data using an inverse Fourier transform.

To analyze the cylinder response, this pulse is reflected from a cylinder. The
backscattered field from the cylinder is shown in Figure 4.15a. The first event is
similar to the TM case and this is the expected direct reflection from the cylinder.
In addition to the direct reflection, there is also a second event. This is the creeping
wave coming after travelling around the cylinder. The time relative to the direct

cylinder reflection can be calculated using the geometry shown in Figure 4.18

At? = (2 + 7r) (4.24)

Ola

Using the value of the radius of the cylinder a = 0.0127 m, the time is found to be
Atf,’ = 0.218 us. The relative time Atg found from Figure 4.15a is 0.205 ns. This is
good agreement.

The backscattered field from a flat surface is also calculated and shown in Figure
4.15b. The first reflections is coming from the front edge of the surface. Similarly, the
second one is coming from the back edge of the surface. The formula in (4.1) is used
to find the time difference (Atg’) between these two reflections and as 4.685 us. To
decide the behaviour of the third reflection the width of the finite surface w and the
incidence angle 43,- have been changed and the scattered E—fields are plotted in Figure
4.16a and Figure 4.16b. From these results, it is found that the Edge C reflection

219

comes At? seconds after Edge B reflection and Atf,’ can be calculated from

Atg’ =

ale

[1 — cos(¢,)] (4.25)

Using these inspections it is concluded that this reflection is an oscillation type of
reflection and happens when the signals travels from the front edge to the back and
radiates. Using (4.25), Atf,’ is calculated and compared with the actual values cal-
culated from the time domain results in Table 4.4 for different surface width and

incidence angles.

Table 4.4. Comparison of Atg in the backscattered field for different surface widths
and incidence angles.

 

w (m) 43,- Atg (ns) At? (17.3)
0.9144 400 0.706 0.715
0.1016 40 0.089 0.079
0.1016 200 0.034 0.021

 

 

 

 

 

 

 

 

 

Next, the cylinder is placed above a finite length flat surface. The expected
reflection paths are identical to those for the TM polarization shown in Figure 4.4,
except for the creeping wave type of reflections.

Existence of the creeping wave in the direct cylinder response leads to expectations
of this type of additional reflection when a cylinder is placed above a flat surface.
According to J ames [24] there are an infinite number of creeping rays and they reduce
to a negligible level very quickly because of exponential attenuation. Therefore, only
three creeping waves are considered and the paths are shown in Figure 4.17. The first
creeping wave (Creeping A) will come At? (given in (4.24)) after the direct cylinder

reflection. The second creeping wave (Creeping B) comes at Attoo, after the multiple

220

A and B reflections. The relative time formula is found using Figure 4.19 and given
as

2 Mt
O = — —— 9 — i
At10 c [L25+ 180( 0 (15)]

7f

180(90 — cm] (426)

2a
= —c- [cos(gbO-l-

The case considered is w = 0.9144 m, ()5,- = 40°, a = 0.0127 m and h, = 8a . The
relative time At?0 calculated for these values is found to be 0.139 n3. Similarly, the
third creeping wave (Creeping C) comes At?1 later from Multiple C reflection. The
relative time is calculated from Figure 4.20. The formula obtained from this geometry

is
Atf’, = ‘2‘ (2 + 7r) (4.27)

For the case considered, it is found to be 0.218 ns.

The backscattered E—field calculated for this geometry using EFIE-MOM formu-
lation was developed in Chapter 3. Using the calculated times, the reflections are
identified in Figure 4.21. The relative times found from path calculations and actual
values found from Figure 4.21, Figure 4.22a and Figure 4.22b are compared in Table
4.5. The scattered E—field is calculated and plotted in Figure 4.23 by using image
theory. The timings of the reflections are found from Figure 4.23 and compared in

Table 4.6 with the previously calculated timings.

221

I

Table 4.5. Comparison of timings in TE polarization for a cylinder above finite flat
surface, a = 0.0127 m, ht = 8a, 21) = 0.9144 m, d).- = 40°, (12, = 40°, (1),, = 90.00.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n Atn (ns) At? (ns)
0 4.688 4.685
1 1.758 1.766
2 0.510 0.512
3 0.473 0.472
4 0.341 0.269
5 0.747 0.740
6 0.693 0.695
7 0.152 0.150
8 0.708 0.715
9 0.204 0.218
10 0.119 0.139
11 0.238 0.218

 

 

Table 4.6. Comparison of timings in TE polarization for a cylinder above an infinite
ground plane, a = 0.0127 m, ht = 8a, ¢.- = 40°, ()5, = 40°, (1),, = 90.00.

 

 

 

 

 

 

 

 

 

 

 

 

n At" (ns) Atf,J (ns)
2 0.510 0.512
3 0.473 0.472
4 0.345 0.269
5 0.748 0.740
6 0.693 0.695
9 0.244 0.218
10 0.156 0.139
11 0.241 0.218

 

222

 

 

i Cylinder
0.50 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’5?
.2
<16
2
E 4
Q 0.00 s
c.— < 5
.9
'55 -0.50
'o 4 i
9 i
a: a
35-1 00 ~ :32;
w a, ...... 'T‘?"?-§§’r&.,.. -
0.5 1.0 1.5 2.0 2.5
time (ns)
(80
1.0 :
a 4 . 5m .
3% ‘ titans 392“"!
s * 2 i
m 4 g Edges
5 0.5 ~ 5
2 . -
“2 EdéeA Edgfec
h .L l
a: _
'U E g E
a: ' At0 I 5
h E
g -o.5 ~ '6 Mai
8 ' H ‘
‘0 - , . , , .. A
0 2 4 6 8
time (ns)
0))

Figure 4.15. TE backscattered fields (a) from a cylindrical target with a = 0.0127 m
(b) from a finite-flat surface of width w = 0.9144 m. 4),- = 40°, 43‘, = 90.00.

223

 

   

.0
01
o

 

 

 

Scattered electric field (relative)

 

 

 

 

 

 

 

 

 

 

 

 

0.0 0.5 1.0 1.5 2.0
time (ns)
(80

’5 : "uni-i
g 0.1:00ns E00000.
g 0.50 i 5ng
E ‘ L '
m s ,
‘0': E 33
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b . EpgeA
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m i i NEE
8
L ‘ : Ato :-
§ - l‘ *l.

i Me“
”3.050 :;A,."“" ”1.,

0.0 0.5 1.0 1.5 2.0

time (ns)
0))

Figure 4.16. TE backscattered fields from a finite-flat surface, w = 0.1016 m,
ab... = 90.00 for (a) 43.- = 40° (b) ¢.- = 20°.

224

 

 

a—oo Creeping A

H Creeping B

<—oo Creeping C %

 

 

 

 

 

 

$7; A 0,,

Figure 4.17. The paths for the creeping waves.

225

 

<—-o Cylinder M d”

H Creeping A

 

 

 

 

 

 

Figure 4.18. Creeping wave reflection A.

226

 

Figure 4.19. Creeping wave reflection B.

227

 

 

Figure 4.20. Creeping wave reflection C.

228

 

_x
U1
0

. 243'“;
1.923): § 2.000ns
1.00 { 5”““23'”?

1 g Multiple B§
i g Cyliniier g Mujltiplec

1 01155 11: 4.050 m 5.501 as

Edg‘e 5 Edge c

 

0.00 +1

.5
01
O

 

 

1.
o
o
111.
D
5'

 

 

 

Scattered electric field (relative)

 

I
_\
I

L 1 1

d

1

1A

.1

.

.

.

q

.1

'1

d

d

d

.1

‘

d

.4

d

-

d

d

.1

1

.

d

d

d

—

q

d

d

.

.1

q

d

d

d

d

q

d

i

q

.1

.1

d

i

.

1

q

d

.

.1

1

.1

.1

.4

time (ns)

Figure 4.21. TE backscattered E—field from a cylinder above a finite flat surface,
a = 0.0127 m, h. = 80, w = 0.9144 m, 03,- = 40°, 42;, = 90.00.

229

 

_L
O
O

O
I I
01
O
A L L A L +4 A _.L__L—L—.L _.l—_{

 
 

 

 

 

 

Scattered electric field (relative)
3
O

 

 

l A
Y— YYYYYYYY

2.0 2.5 3.0
time (ns)

(a)

 

 

 

 

 

 

 

 

 

 

 

.02.) 1.00 - 21:330- 23°?!»- gfmom 3.119...
E . .
92
2 0 50 «
m
if."
.9
1‘: 0 00 +
0
2
a)
E -0.50 -
i
a, -1.00 4
2.5 3.0 3.5 4.0
time (ns)

(b)
Figure 4.22. TE backscattered E—field from a cylinder above a finite flat surface,

(a) time between Cylinder and Multiple C reflections (b) time after Multiple A and
B, a = 0.0127 m, ht = 8a, 20 = 0.9144 m, d).- = 40°, 05‘, = 90.00.

230

A;

 

 

   
 

 

 

   
 

 

 

 

 

0.0 ' 0.5 1.0 1.5

Figure 4.23. TE backscattered E—field from a cylinder above an infinite ground

plane, a = 0.0127 m, ht = 8a, 53.- = 40°, 19‘, = 90.00.

231

’9'? 1 0.4215. 0.931§ns§ 1.275113 1.545113 2.097115
,5 2.00? 0.555115 2.105711: 40411.2 1.579119
59 1 50 : MultipleA , 5 9 5d B
2., ' J Cylinder M 8‘ ' Muléiple C1 ecoria ary
v 5 ultipleB , . .

1 00 _‘ ' ° SecondaryC
'2 ' 11Creeping A Cre5pin9181Creeping C
1.9-3 0.50 -: ' \4 i \Sewéndaw \ 367M“
0 3 § 5 5
'z 0.00 -. ; s : 11 a ;
'5' 1 1 : 1 5 1 :
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54.505 SAW #1) < g; )5
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8 -2.00 1 2 E ;< >1

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(D ‘ g a <—

 

4.2 Multipath Analysis for Targets above Rough Surfaces

After establishing the basic understanding of multipath scattering, a sea surface with
a sinusoidal shape is studied. This simple sea surface model helps to understand the
effect of surface roughness on multiple reflections. To analyze the role of roughness,
the backscattered fields from a cylinder above finite and infinite periodic surfaces are

calculated.
4.2.1 Rough Surface Scattering in TM Polarization

To see the clutter response a TM transient field is assumed to be incident on a
finite sinusoidal surface. The backscattered field is calculated and plotted in Figure
4.24. The first and last signals are the edge reflections as in the flat surface solution.
The time difference between edge reflections can be found using w = 0.9144 and
(,5,- = 20° in (4.1) and At? is found to be 5.748 ns. The time found from Figure
4.24 is Ato = 5.753. The reflections coming in between these edge reflections are the
reflections from each period (CL l-CL 9). The time between these reflections can be

found using Figure 4.25

2 d cos(qii)

At0 =
12 C

(4.28)

 

where d = 0.1016 m and 4),- = 20°. Substituting these values into (4.28) gives At?2 =
0.639 ns. The actual time is calculated from Figure 4.24 is Atlg = 0.638 ns.

After studying the clutter response a cylinder is placed above this periodic surface.
The backscattered E—fleld is shown in Figure 4.26. It is observed that although the
clutter response is too small to be seen with the figure scale, the surface roughness
affects the Multiple A, B and C reflections a great deal. From this figure only the
main reflections are identified and the relative times found from path calculations and
actual times are shown in Table 4.6

To analyze the multiple reflections in detail and avoid the edge reflections, the

232

Table 4.7. Comparison of timings in TM polarization for a cylinder above finite
sinusoidal surface, a = 0.0127 m, ht = 8a, d/a = 8.0, w/d = 9.0, h,/d = 0.125,
¢i = 200, 123, = 20°.

 

 

 

 

 

11 At" (ns) At? (ns)
0 5.753 5.748
1 2.490 2.527
2 0.272 0.267
3 0.259 0.256

 

 

 

 

 

cylinder is placed above a sea surface assumed to be infinitely-long and sinusoidally
periodic. This problem is solved using the perturbation method explained for TM
polarization in Chapter 2. The backscattered field is found after truncating the
perturbation current to 9 periods of the surface and plotted in Figure 4.27, without
including the clutter reSponse. The timings are calculated from Figure 4.27 and
compared with the ones from ray calculation in Table 4.8. The first signal is the direct
cylinder reflection. The multiple reflections AB and C are the specular reflections
observed from a flat surface. To analyze the nonspecular multiple reflections, the
field scattered from the surface is calculated at the most right edge of the cylindrical
target in the positive a: direction. The value of the field normalized by the magnitude
of the incident field is plotted in Figure 4.28. The first reflection is similar to that
expected from a flat surface. However, its magnitude is smaller than 1 because not
all the energy is reflected [19]. The second (RX) and the third (RY) reflections are
the nonspecular reflections. The time relative to the specular reflection for RX and
RY are calculated from Figure 4.28 and found to be tz = 0.058 113 and ty = 0.223 71.3,
respectively.

These timing results confirm that multiple A and B reflections in Figure 4.27 are a
result of the specular reflections following the path of target-sea and sea-target. Also,

the Multiple C reflection occurs in a similar way to that in the flat surface case as this

233

specular reflection bounces back from the target and then scatters off the surface. In
addition to these obvious multiple reflections, the sinusoidal surface brings new types
of reflections. The signals RX and RY follow the path of Multiple A and generate
RXA and RY A, respectively. RXB and RYE is similar to these reflections but instead
following the target-sea path. Also, RX and RY follow the same path as Multiple C

and resulting in RXC, RYC.

Table 4.8. Comparison of timings in TM polarization for a cylinder above an infinite
sinusoidal surface, a = 0.0127 m, h, = 8a, d/a = 8.0, w/d = 9.0, h,/d = 0.125,
433' = 200) ¢8 = 200-

 

n Atn (ns) At? (ns)
2 0.275 0.267
3 0.263 0.256

 

 

 

 

 

 

 

 

 

 

13 0.061 -
14 0.226 -
15 0.055 -
16 0.215 -

 

4.2.2 Rough Surface Scattering in TE Polarization

After analyzing the sinusoidal surface in TM polarization, the same kind of analysis is
undertaken in TE polarization. First, the clutter response is observed after calculating
the backscattered field. In Figure 4.29 the first and last signals are the edge reflections.
The time difference between edge reflections can be found using w = 0.9144 and 53,- =
200 in (4.1) and At? is 5.748 ns. The time found from Figure 4.29 is Ato = 5.741 ns.
The timing of the oscillation type of edge reflection (Edge C) is calculated from
(4.26) and found to be At? = 0.184 ns. The actual time found from Figure 4.29
is Atg = 0.229 ns. The reflections (CL l-CL 9) in between the edge reflections are

from the bumps of the surface. The time between the closest two of these reflections

234

is calculated after substituting d = 0.1016 m and ¢,- = 200 into (4.28). The relative
time is Atlo2 = 0.639 ns. The actual time is found from Figure 4.29 and to be
Atlz = 0.644 ns.

Now that the sinusoidal surface response has been examined, a cylinder is placed
above this sinusoidal surface. The backscattered E—field is calculated and shown in
Figure 4.30. It is seen that Multiple A, B and C reflections are significantly aflected
by surface roughness. These main reflections are easily identified. The timings found
from path calculations and actual times are shown in Table 4.9. There are also

additional reflections that are not observed for the flat surface.

Table 4.9. Comparison of timings in TE polarization for a cylinder above finite
sinusoidal surface, a = 0.0127 m, ht = 8a, d/a = 8.0, w/d = 9.0, hs/d = 0.125,
‘1’; = 20°, 43, = 20°, 5,9 = 90.00.

 

 

 

 

 

 

n Atn (ns) At? (ns)
0 5.744 5.748
1 2.521 2.527
2 0.275 0.267
3 0.210 0.256
8 0.229 0.184

 

 

 

 

 

In the presence of the edge reflections and interactions of these reflections with the
target, it is difficult to identify the multiple reflections, which are not obvious, due to
roughness of the surface. Therefore, the cylinder is next placed above an infinitely-
long sinusoidal sea surface. This is solved using a perturbation method explained
for TE polarization in Chapter 3. The backscattered field is found after truncating
the perturbation current to 13 periods of the surface and plotted in Figure 4.31.
The timings are calculated from Figure 4.31 and compared with the ones found from

path analysis in Table 4.10. The multiple specular reflections have similar timings

235

with those identified for TM polarization, except that their magnitude is significantly
smaller. Again, to analyze the multiple reflections that are difficult to identify, the E-
field scattered from the infinite sinusoidal surface is calculated at cylindrical target’s
right edge in the positive :5 direction. The value of the field normalized by the
magnitude of the incident field is plotted in Figure 4.28. The first reflection is similar
to that which would be expected from a flat surface. However, only half of the incident
wave is reflected. The second (RX) and the third (RY) reflections are the nonspecular
reflections. The time relative to the specular reflection for RX and RY are calculated
from Figure 4.32 and found to be t, = 0.055 ns and ty = 0.214 713, respectively.

These timing results confirm the identified signals in Figure 4.31.

Table 4.10. Comparison of timings in TE polarization for a cylinder above an infinite
sinusoidal surface, a = 0.0127 m, ht = 8a, d/a = 8.0, w/d = 9.0, h,/d = 0.125,
<2.- = 20°, ¢, = 20°, 5‘, = 90.00.

 

n Atn (ns) Atf‘J (ns)
2 0.277 0.267
3 0.321 0.256

 

 

 

 

 

 

13 0.058 -
14 0.211 -
15 0.043 -
16 0.143 -

 

 

 

 

 

236

 

 

7.800 m

 

 

Edges

E-fleld (relative)

.6 .o 9
§ § §
(')

 

 

 

 

 

 

 

Scattered electric field (relative)
2
O

 

 

 

012 3 4 5 6 7 8 9
time(ns)

Figure 4.24. TM backscattered field from a finite sinusoidal surface, d = 0.1016 m,
w/d = 9.0, h,/d = 0.125, 43,- = 20°.

237

 

 

 

 

 

 

 

 

 

 

Figure 4.25. Reflections from finite-sinusoidal surface.

238

 

l 2:.137 m clinging 5;.153 m 7.890 nip
1 -00 ‘. Multiples A; Multiple B '

0.50 j =

0.00 I)

-0.50

 

 

 

l l J 1 1

 

Scattered electric field (relative)
8
O
l
1?

At; time (ns)

 

 

 

 

time (ns)

Figure 4.26. TM backscattered field from a cylinder above finite sinusoidal surface,
a = 0.0127 m, h; = 80, d/a = 8.0, w/d = 9.0, h,/d = 0.125, d),- = 20°.

239

 

0.31:2... means; 0873mfa910m ;

 

1.50 ‘ gag-mam i0305m§
. , . 1.12503
1 00 , Cylmder a. Multiple C
' l § MultipleB ' I 1
i . {l Rl‘A RzA=
0.50 j ' 3 5A? RXC RYc

RYB:

 

 

 

0.00 j

.5
01
O

L.
'o
o

 

 

Scattered electric field (relative)

 

 

0.25 0. 50 0. 75 1 .00 1 .25 1.50
time (ns)

Figure 4.27. TM backscattered field from a cylinder above infinite sinusoidal surface,
a = 0.0127 m, ht = 8a, d/a = 8.0, h,/d = 0.125, ()5,- = 20°.

240

 

1.00j 50.304m ns.-ism

 

 

 

 

Scattered electric field (relative)
S
O

 

 

 

 

-o.5o -j V
my;
4.00 . —>; ;<—
2 i Aty s
......... <>-
0.00 0.25 0.50 0.75 1.00
time (ns)

Figure 4.28. TM scattered field from an infinite sinusoidal surface, d = 0.1016 m,
h,/d = 0.125, <15,- = 20°, x/d = 0.125, y/d = 1.25.

241

 

O
N
01
a
S
(I
a

 

0.00 *

 

.C'D
N
01

 

 

 

Scattered electric field (relative)

 

 

012 3 4 5 6 7 8 9
time(ns)

Figure 4.29. TE backscattered field from a finite sinusoidal surface, d = 0.1016 m,
w/d = 9.0, h,/d = 0.125, 43.- = 20°, (1),, = 90.00.

242

 

 

 

 

 

 

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2h: 0'50 . ;< >§ 413 35
z ' j; a“ (it?
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-100 -fifi-

 

2 3 4 5 6 7 8 9
time (ns)

Figure 4.30. TE backscattered field from a cylinder above finite sinusoidal surface,
a = 0.0127 m, ht = 8a, d/a = 8.0, w/d = 9.0, h,/d = 0.125, 45.- = 20°, 45‘, = 90.00.

243

 

} 0.373ins 0.650ins ; 0.861!“ @971”?
2 ‘ €0.708ns 5 g/1.01:‘m

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_L
O
O

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l l l l l

REYC RZXC

 

0.00

q 3 —>5 :é— : :
At14

Scattered electric field (relative)
8
O

At : 1
“2 )fi % 3.2.):

 

1 K

TTTTTTTTT I r r l' I U I I

0.25 0.50 0.75 1.00 1.25
time (ns)

 

 

 

 

Figure 4.31. TE backscattered field from a cylinder above infinite sinusoidal surface,
a = 0.0127 m, ht = 8a, d/a = 8.0, h,/d = 0.125, ¢.- = 20°, 43‘, = 90.00.

244

 

0.336 ns

1.00

.1.

0.301m 0.5%008

Specdlar

.0
01
o

 

 

 

1 L 1 l l l

 

Scattered electric field (relative)
2
O

 

 

 

-0.50
SAtxé
400 ~ —>g ;<—
5 My :
-<> ....... , .........
0.00 0.25 0.50 0.75 1.00
time (ns)

Figure 4.32. TB scattered field from an infinite sinusoidal surface, d = 0.1016 m,
h,/d = 0.125, 43, = 20°, 43,, = 90.00, z/d = 0.125, y/d =1.25.

245

4.3 Conclusion

In this chapter, the expected travelling paths of the reflected signals from a cylindrical
target above several sea surface models are calculated. Using those expected paths,
the signals in the scattered field calculations are identified. Several unobvious reflec-
tions are identified using this method. This path calculations gave physical insight

to the computed scattered field results.

246

CHAPTER 5

EXPERINIENTAL MEASUREMENTS AND COMPARISONS

5.1 Introduction

In this chapter, the technique of true time-domain measurements for scattered fields
from cylindrical targets above sea surface models will be examined. Also, the measure-
ments will be compared with numerical calculations. The time-domain measurements
were taken using MSU’s reflectivity arch range. The arch range is a 20 ft diameter
circular shaped structure allows that horn—antennas to be arbitrary located by rota-
tion about its periphery shown in Figure 5.1. Measuring the backscattered fields is
the objective; therefore, two horn antennas (each with 2-18 GHz bandwidth) are used

to simulate a mono-static radar.

5.2 Measurement Setup

Scattering targets are placed on a low density styrofoam column, which is at the
center of the arch range, at the same height as the horn antennas from the ground to
provide the most uniform incident field. The time domain response is measured by
a Hewlett Packard’s digital sampling oscilloscope (DSO) model HP54750A with the
configuration shown in Figure 5.2. This instrument has a TDR/TDT (Time-Domain-
Reflectometer / Time-Domain-Transmission) plug-in module HP54753A which pro-
vides a 20 GHz channel and an 18 GHz channel for the oscilloscope. The TDR—unit
also has an integrated step generator that sends steps from channel 3 with a rise time
of 45 ps and a magnitude of 200 mV, as shown in Figure 5.3. The step generated in
the TDR/TDT-unit triggers a Picosecond Pulse Labs (PSPL) 4015B step generator.
This instrument creates another step using a remote pulse head PSPL 4015RPH.

The step signal generated by the pulse head is shown Figure 5.4. This step has an

247

amplitude of -9 V with a fall time of 15 ps. A pulse signal is generated from sending
the step signal into a PSPL 5208-DC pulse-generating network. The generated pulse
is shown in Figure 5.5. The frequency spectrum of the pulse shown in Figure 5.6 is
obtained after using a Fourier transform.

This pulse is transmitted by an American Electronic Laboratories H-1498 horn
antenna with a bandwidth of 2 GHz to 18 GHz. The same type antenna is used for
receiving and located very close to the transmitting antenna to make the setup as
close as to mono static as possible, as shown in Figure 5.7. However, the horizontal
distance between the antennas still forms a bistatic angle of 9b = 12.2". To measure
the scattered field from a target, channel 4 of HP54753A, which samples the output
voltage of the receiving horn, is used with a typical 5—20 ns time window using 1024
sample points. The sampled signal may be averaged within the DS0 up to 4096 times
to increase the signal to noise ratio (SNR).

The measured waveform from a target is stored in one of the four memories of the
DS0. Then the background response is measured and subtracted from the waveform
stored in the memory using a math function. Using a large number of averages
is supposed to increase SNR [25]; however, when the background noise is high,
due to small time drifts in the sampling process, the subtraction of the background
measurement is worsened. Therefore, for the measurements to be shown, only 1024
averages and 1024 sample points are used.

It should be noted that the equipment should warm up for at least an hour before
measurements are taken. Otherwise the drift of the instruments, caused by heat,
disturbs the results. Also, the equipment is very vulnerable to electrostatic discharge;

therefore, one should be always be grounded when making measurements.

248

 

4 20' b

 

 

 

 

 

 

Target ‘3”...
\ / \ /
Lens “\ 1...... Lens

Receiving P Q Transmitting
Horn Horn

Figure 5.1. Geometry of the free-field arch-range scattering system at MSU.

 

249

 

Receiving Horn

 

 

 

Pulse generator

JL

CH4 CH3
T T I LJ Pulse network
Digital sampling l__ \ / Pulse head

oscilloscope with H h
TDRITDT unit '9 gm?“

 

 

 

 

 

 

 

 

 

 

 

Figure 5.2. Time domain scattering measurement configuration.

250

Voltage (V)

 

'2

 

 

 

 

d
V T T T U T I I Y 1 Y Y fir IIIIIIIIIIIIIIIIIIIIIIIIIII
0.00 l T l l l

0 20 40 60 80100120140160180200
time(ps)

Figure 5.3. Triggering step signal generated by the TDR/TDT unit.

251

Lt'al'o-L

1L11111

Voltage (V)
61

t5:

-9:

 

 

111111 L144

1 1

11.1111

 

 

 

0 20 40 60 80 100 120 140 160 180 200
time (ps)

Figure 5.4. Step signal generated by pulse generator and pulse head.

252

Voltage (V)

I
N

I
A

-3

 

 

l

1

 

 

 

 

I

0.0 0.2

Figure 5.5. Transmitted pulse generated by the pulse network.

W T r

time (ns)

253

l

0.4

 

0.6

 

P
to

1_ 1

Relative Amplitude
O
'c»

 

 

 

0.3 ' ' ' ' i 1 r . . I . . r . I . . . , I , , , T
2 4 6 8 10 1 14
frequency (GHz)

Figure 5.6. Spectrum of the transmitted pulse.

254

T T I Y I I I

T

M

I r T U U

16

18

 

29.5"

 

 

 

Figure 5.7. Location of transmitting and receiving antennas.

255

5.2.1 Calibration Procedure

To obtain the transfer function of the unknown target, it is necessary to know the
system response. This is because the scattering measurements depend on the transfer
functions of the antennas (HT( f), H R( f)), antenna coupling (HA(f)), the lenses
(HTL( f), H RL( f)), mutual interactions (Hsc(f)), and arch-range clutter (Hc(f)).
A block diagram model of the measurement system [26] is shown in Figure 5.8. The

measured waveform of an unknown target may be formulated as

em = E(f) HTU) HRU) {HA(f)
+ Hum Hell) [Han + HSCU) + Hc(f)]} + N(f) (5.1)

where H§( f ) is unknown transfer function to find and N (f) is the noise in the en-
vironment. To obtain Hf9( f) another measurement is done to find the background
response. The background measurement has to be made as soon as the target mea—
surement is taken because this measurement will be subtracted from the unknown
target response. If the time in between the measurements is long, the subtraction

process will be worsened. The measured waveform for the background is

12.0) = Em Hr(f) Hn(f) {Him + Hn(f) Hear) Hc(f)} + N(f) (5.2)

Next, the background response is subtracted from the target response. This process

gives

mm = 50) [Him + Hem]
= EU) HT(f) HR(f) HTL(f) HRLU) [H§(f) + H3000] (5-3)

256

The unknown target response H§( f) can be found if the system transfer function 8 (f)
is known. To determine S (f) a theoretically known response has to be measured. A 14
inch aluminum sphere is chosen as a known target (calibrator) because its theoretical
response can be found using the Mie series [26]. The background-subtracted sphere

measurement can be modelled as

Rc_b(f) = S(f) [HEM + Hsc(f)] (5.4)

where H§( f ) is the theoretical response of the sphere. Although the reflectivity arch
range is not a high quality chamber, absorbers have been placed where the mutual
interactions occur. Therefore, the mutual interaction term H30 (f) can be assumed
to be negligible. Any residual mutual interactions are time-windowed if they are
sufficiently separated in time from the direct target response. As a result the system

transfer function is found to be

_ Hg”) 5
S(f) — RC-b(f) (5. )

 

With the system transfer function determined, the unknown target response can be

easily found using

Him = R‘s-(”15,0 (5.6)

 

To demonstrate the calibration process, the response from a 14 inch diameter
sphere is measured and the background-subtracted time-domain response is plotted
in Figure 5.9. First, this signal is time windowed and shown Figure 5.10a. This sig-
nal is transformed into the frequency domain using the fast FT (Fourier Transform),
then windowed between 2.0 - 18 GHz. The windowed frequency spectrum is shown

in Figure 5.10b. This spectrum then is divided by the calculated theoretical sphere

257

Source: Pulse

mm

 

 

 

 

 

 

Antenna Mutual
Coupling 38“” Interactiona Clutter
Receiving
9 HRLm Antenna Lena
Receiving
Antenna
Receiver

 

+ @ Noise

Figure 5.8. Block diagram model of measurement system.

258

response shown in Figure 5.11 to obtain the system transfer function. The resulting
system transfer function is plotted in Figure 5.12. As a check on the calibration
process, a 3 inch metallic sphere was measured in the arch-range. After subtract-
ing the background the time domain response is shown Figure 5.13a. This signal is
transformed into the frequency domain using the FT and plotted in Figure 5.13b.
Then, the resulting spectrum is divided by the system transfer function. The cali-
brated signal is taken into the time domain to remove any unwanted reflections. The
windowing in the time domain is demonstrated in Figure 5.14. Then, the calibrated
and time windowed signal is transformed into the frequency domain and compared
with the theoretical response in Figure 5.15a, which has been frequency-windowed in
the same manner as the measured response. After using the inverse FT this signal is
transformed back into the time domain to compare with the theoretical time domain
response from a 3 inch diameter metallic sphere. The time domain response is shown
in Figure 5.15b. The amplitudes of the expected reflections and the timing of creep-
ing wave relative to direct sphere reflection match well. These figures show that the

measured response agrees well with the theoretical response after calibration.

259

 

.fi
1 1 1 1

 

 

 

 

 

9
E
20— W
.9
(D
8 .
54:
(D
m -1
a) e
2 .
-8-
0 2 4 6 8 10

time (ns)

Figure 5.9. Measured waveform from a 14 inch sphere, p = 3.5052 m, (13, = -6.1°,
43, = 6.1°.

260

 

 

 

.h
.-_L__1__l . .

 

Amplitude (mV)
.L c:

 

 

 

O 2 4 6 8 10

 

 

 

 

 

 

time (ns)
(a)
10-12 .
i?
l:
2.
g 1013:
g, :
a i
E .
< l
l
1044,....,H..,...W..1.,H,,,-1.q,...,.fl.l
2 4 6 8 10 12 14 16 18
frequency (GHz)
0))

Figure 5.10. The measured waveform from a 14 inch sphere (a) windowed in time
domain, (b) windowed in frequency domain.

261

 

N
O)
1 J J 1_—1 1 1 1 J
2:

 

 

 

 

 

 

Amplitude (mV/m)
N
01

N
.h

 

23,-1
2 4 6 8 10 12 14 16 18
frequency (GHz)

 

I I Y I 1' I I I Y W I I T fir V I

Figure 5.11. Theoretical scattered E—field from 14 inch sphere, IE‘I = 1 V/m,
p = 3.5052 m, e,- = —6.1°, e, = 6.1°.

262

Amplitude

 

10'1o .

10-11 -

.4
‘

 

 

 

10-12

IITTTTITfirTTjYTfTT—rTTj I Y Y

4 6 8 10 12
frequency (GHz)

Figure 5.12. System Transfer Function.

263

TITTTTTTTFYYY

14 16

18

 

 

 

 

 

 

 

 

 

 

 

2?
’>‘
E,
i
C
.9
U)
'0
2
3
m l
(0
g i
-2 s-
0 2 4 6 8 10
time(ns)
(a)
10-12
12?
£10-13
a
0)
'0
g i
3- -14.
51° 2
< .
10-15 1
2 4 6 8 10 12 14 16 18
frequency (GHz)
('0)

Figure 5.13. Measured waveform from a 3 inch sphere (a) in time domain, (b) in
frequency domain, p = 3.5052 m, d).- = —6.1°, 43, = 6.1°.

264

 

 

  

 

 

 

 

200x106 .
A 150x106 -j
E :
E 100x106
2,
a) 50x106 ‘
.0
B .
E 0.
E: :
-50x106 {
-100x106‘ .. .., ... ., ... -.. .. .1. ...
0 2 4 6 8 10
time (ns)

Figure 5.14. Time gated measured waveform from a 3 inch sphere after the cali-
bration.

265

 

 

 

 

 

 

 

 

 

 

 

) ————Measurement
A -201. 'y
E 1
3 i
m
3 1.
tn
'0
B
'5.
E
<
l
J
-100 flees
24681012141618
frequency (GHz)
(80
6 4
. —Measurement
1 —--—- Th
.. 4- l °°"
E i
> .
g 21 Creeping Wave
§ 9 \
a: 0)
"a 1
E; ;
.2i
4)--.. ......... m, .......
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
time(ns)
(b)

Figure 5.15. Comparison of theoretical and measured scattered E—field from a 3
inch sphere after completing the calibration process; (a) in frequency domain, (b) in
time domain, IE‘| = l V/m, p = 3.5052 m, ()5.- = —6.1°, d), = 6.1°.

266

5.3 Experimental Results and Validations for TM Polarization

The high confidence level due to the accuracy of the calibration applied to the response
of the 3 inch diameter sphere enabled the process of comparing the measurement
results with those established by the theory developed in Chapter 2. For this purpose,
PEC flat and sinusoidal sea surfaces are constructed by adhering aluminum foil to
flat and sinusoidally machined pieces of styrofoam. To represent the two-dimensional
cylindrical target, an aluminum cylindrical rod is used. Here, for all the experimental
results, only calibrated and time gated results will be presented.

First, the time domain response from the cylindrical target is measured and plotted
with the theoretical response in Figure 5.16a. Then, a frequency domain comparison
is made in Figure 5.16b by applying a FT to the time domain signals. It is seen that
the measurement matches well with theory in both the time and frequency domains.
Next, the response from a sinusoidal surface is measured, and compared with theory
in Figure 5.17a. The timing of every expected single event is almost identical with
that determined by theory. The first and last signals are coming from the front and
back edges, respectively. The signals in between these edge reflections are coming
from the bumps of the sinusoidal surface. It is seen that these reflections match
well with theory. Figure 5.17b compares the measured response with theory in the
frequency domain. In the frequency spectrum the first three Floquet modes conform
with the calculated response.

Since the ultimate purpose is to compare the timing of multiple reflections, the
cylindrical rod is placed above a flat surface. The measured response is displayed
in Figure 5.18a for the time domain and in Figure 5.18b for the frequency domain.
The curves in the frequency and time domains are in accord with the corresponding
calculated results. The edge reflections come as the first and last signals in the time-

domain results. To make better comparisons of the multiple reflections, the middle

267

section of Figure 5.18a was expanded and plotted in Figure 5.19. The first signal is
the direct reflection from the target. The second and third signal follows sea-target
and target-sea paths. The measurement setup has a 12.20 bistatic angle. This affects
the timing of these signals and they do not come as a summation as the way they
do in the mono static case. Similar to the mono static case the fourth signal follows
the sea-target—sea path. To see the effect of the incidence angle, the incidence angle
was increased by 140 and the measured response plotted with the theoretical response
in Figure 5.20a for the time domain and in Figure 5.20b for the frequency domain.
Using a smaller time window, the time domain results are plotted in Figure 5.21 to
make the analysis of multiple reflections easier. As expected, the time relative to
the direct cylinder response for multiple reflections is increased while target-sea and
sea-target reflections get closer to each other.

Now, after achieving a close match between theory and measurements for the
cylinder above the flat surface, the more interesting case of the cylinder above a
sinusoidal surface is considered. The measured response is shown in Figure 5.22a for
the time domain and in Figure 5.22b for the frequency domain. After focusing on the
multiple reflections, the time domain response replotted in Figure 5.23. It is noted
that the multiple reflections have lost almost half their amplitude. This is because
not all the energy being reflected from the sinusoidal surface to the target as the

specular reflection.

268

 

 

 

 

 

 

 

 

 

 

 

 

20 l
1 — Measurement
1 :- —-—-- Theory
A1015 '1
E :'
> .
E. 01
0 .
'5 I
§-10~
0- 1
g r
-2ol l
i
so?
0.0 0.5 1.0 1.5 2.0
time(ns)
(a)
o 4
‘ — Measurement
A -20 ~
é
a
3 -4
o
“O
,3 -601
'5.
E
< -8
-100 6--
2 4 6 8 10 12 14 16 18
frequency (GHz)
0))

Figure 5.16. TM polarization comparison of theoretical and measured scattered
E-field from cylinder (3) in time domain and (b) in frequency domain, IE‘I = 1 V/ m,
p = 3.5052 m, a = 0.0127 m, (15,- : -6.1°, (13, = 6.1°.

269

 

— Measurement

0')
l
!
g

OJ

 

 

1L4L41 L A A 1
~

.4.

Amplitute (mV/m)
6.:

l
(D

c'n

 

 

 

 

 

 

time (ns)
(60
O
‘ —— Measurement
A -20 «
§ ‘. . . a. I
3 -40 4 NW- qm. [Ill .J'l‘l! W .1 WWW f’
C) 3 ' ' y ' b,
.0 4 l.
,3 4504
E.
E
< 430
'100 **—'*7—**T*rw—rr—rrr—vrfi—1vvvvvavvtvvw.,...fiI
2 4 6 8 10 12 14 16 18
frequency (GHz)
(b)

Figure 5.17. TM polarization comparison of theoretical and measured scattered
E—field from sinusoidal surface (a) in time domain and (b) in frequency domain,
IE'I = 1 V/m, p = 3.5052 m, d = 0.1016 m, w/d = 12.0, h,/d = 0.125, 45.- = 17.9”,
d), = 30.1”.

270

 

.. . J...‘.

0)
O

N
O

J.

.3
O

— Measurement
ll —-~—~ Theory

 

 

 

Amplltude (mV/m)
O
5;:
l

 

 

 

 

 

 

 

 

 

 

-10 4
-20 ~ ,
I
-30 .-..- , . , -.....,..f-,.-,,
0 1 2 3 4 5 6 7 8
time (ns)
(3)
0
j — Measurement
4 —---—~ Theory
A j x .
£40 < = ”W, « "a. .,
m i" i Mi ”- “"1.
E j: ‘ --' _ 1 ' \
a: -40 -« ‘ .. a
‘8 ' l u“:
i i
E: -60 —'
1! !
-30 , .............. f +,,...
2 4 6 8 10 12 14 16 18
frequency (GHz)
0))

Figure 5.18. TM polarization comparison of theoretical and measured scattered
E-field from cylinder above flat surface (a) in time domain and (b) in frequency

domain, |E'| = 1 V/m, p = 3.5052 m, a = 0.0127 m, lu/a ==

w/h. = 10.5, 4),. = 17.90, 43, = 30.10.

8.0, :cr/a = 0.0,

271

 

30 l —- Measurement
—--—- Theory

A
O
l 1 1 1 1
t

O

1 L L _1_ l 1 1 1 1

 

 

 

 

Amplitude (mV/m)

.
A
C

All

 

 

 

 

 

.20 -j l

 

 

 

-3ol.,.-T-...
3.0 3.5 4.0 4.5 5.0 5.5
time (ns)

I I T T 7 l’ V fii T ‘r

I V

Figure 5.19. Multiple reflection comparison of theoretical and measured scat-
tered E-field from cylinder above flat surface in TM Polarization, IE‘I = 1 V/ m,
p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, xr/a = 0.0, 1.0/ht = 10.5, d).- = 17.9",
¢, = 30.10.

272

 

 

 

 

 

 

 

 

 

 

 

 

 

30 —— Measurement
j l —---—- Theory
A 20 {
E
>
E,
m
'0
a
E.
E
<
-20 j
; l
-30‘ ............................. , ---
0 2 4 6 8
time (ns)
(a)
0
‘ — Measurement
15‘
>
m
E
o
'0
3
E.
E
<

 

 

 

6 81012141618
frequency (GHz)

0))

Figure 5.20. TM polarization comparison of theoretical and measured scattered
E-field from cylinder above flat surface (a) in time domain and (b) in frequency
domain, |E‘| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, z,/a = 0.0,
w/h. = 10.5, ¢,- = 31.90, ¢, = 44.10.

273

 

 

 

 

 

 

 

 

 

 

 

 

30 , --— Measurement
; g —-°—-Theory
A20{
g 3 I
E10§ _.
v 1 ‘,
g 0: MM ..
3 ' . '4‘
z 4
E .
5-101
< I ‘ “
-20-i u
3 i

 

3.0 3.5 4.0 4.5 5.0 5.5 6.0
time (ns)

Figure 5.21. Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TM Polarization, |E'| = l V/m,
p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, :c,/a = 0.0, w/h, = 10.5, ¢,- = 31.90,
45, = 44.10.

274

 

_L
O

N
o
I #1

Amplitude (mV/m)
O

— Measurement

 

 

12345678910

 

 

 

 

0
time(ns)
(a)
0
1 —-—Measurement
4 —--—— Theory
“-204
E , l M ,.
B i I" 1'; J I o .
3 l "‘ fl.
03-40“ = “
13 . l I
e a ' i
2 1' i
<60:
3!
W.
2 4 6 8 10 12 14 16 18
frequency (GHz)
0))

Figure 5.22. TM polarization comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface (a) in time domain and (b) in frequency
domain, |E'| = 1 V/m, p = 3.5052 m, a = 0.0127 m, h,/a = 8.2, zr/a = 0.0,
d/a = 8.0, w/d = 12.0, h,/d = 0.125, 45.- = 18.9“, ¢, = 31.1°.

275

 

l A

— Measurement
—--—- Theory

-3 N
O O
14L..1l

l l l L l l

 

 

 

1 J

 

 

 

 

Amplitude (mV/m)
E5 0

1 I l

 

 

r'o
o

 

 

 

(L:
o

I I I V I I I ‘1 r fir

5 6 7
time (ns)

h

Figure 5.23. Multiple reflection comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface in TM Polarization, IE‘I = l V/ m,
p = 3.5052 m, a = 0.0127 m, ht/a = 8.2, xr/a = 0.0, d/a = 8.0, w/d = 12.0,
h,/d = 0.125, ¢,- = 18.90, ¢, = 31.1°.

276

5.4 Experimental Results and Validations for TE Polarization

To complete the analysis TE polarization is now considered. Here, the same procedure
as in TM polarization case will be followed to validate the theory obtained in Chapter
3.

The measured response from the cylindrical target is plotted with the theoretical
response in Figure 5.24a. It is seen that the first signal in Figure 5.24a is the signal
reflected from the cylinder and second smaller one is the creeping wave. Next, a
frequency domain comparison is made in Figure 5.24b by applying a FT to the time
domain signal. The measurement matches well with the theory in both the time and
frequency domains. Next, the response from the sinusoidal surface is compared with
the theory in Figure 5.25a. It is seen that the timing of the signals matches well with
the theory. In Figure 5.25b the first three Floquet modes agree with the calculated
response.

Next, the cylinder placed above the flat surface was measured, and the data is
shown along with the calculated response in Figure 5.26a. The diflerence from the
TM polarization is that now the back edge reflection is bigger than the front edge
reflection. An IFT is used on the time domain data and the frequency spectrum is
plotted in Figure 5.26b. After seeing that both the frequency and time domain data
match well with the theoretical data, the multiple reflections are plotted in a smaller
window in Figure 5.27 to focus better on those reflections. To see the incidence
angle dependence of the measured data, the incidence angle is increased by 19°. The
measured data is shown with the calculated response in Figure 5.28a for the time
domain and in Figure 5.28b for the frequency domain. The time domain results
are plotted in Figure 5.29 with in a smaller window. The timing of signals matches
very well with the theory. Finally, to observe the effect of surface roughness on the

scattered field, the cylinder was placed above the sinusoidal surface. Figure 5.30a

277

and Figure 5.30b compare the measured data with calculated response in the time
domain and in the frequency domain, respectively. The clutter reSponse is bigger
than the one for TM polarization. Therefore, the multiple reflections are mixed with
the direct clutter reflections and are examined in Figure 5.31. Clearly, the biggest
signal is the direct cylinder response. However, the other signals are not easy to
identify. To separate these signals, the incidence angle is increased and measured
data is shown with the numerical data in Figure 5.32a for the time domain and in
Figure 5.32b for the frequency domain. The multiple reflections are examined in a
smaller time window Figure 5.33. The reflection A comes directly from the cylinder.
The reflections B and C follow target-sea and sea-target paths. The last signal (D)

comes from sea-target-sea path.

5.5 Conclusions

A series of experiments was conducted to verify the theory derived throughout this
thesis. Although mono static experiments could not be conducted, comparing the
theoretical results with measured results with different incidence angles has helped

to verify the theory.

278

 

— Measurement

A N
O 0
+4 *4; L k; A

A A

 

l

 

l
-10 l \

Amplitude (mV/m)
O

 

 

 

 

 

 

 

 

 

 

 

CreepingWave
-204
. l
-3or
0.0 0.5 1.0 1.5 2.0
time(ns)
(a)
0
l —Measurement
’g-zm
S
m
E
o
u
g
'5.
E
<
-80 fi-...s.,.fi.,.fi
2 4 6 8 10 12 14 16 18
frequency(GHz)
(b)

Figure 5.24. TE polarization comparison of theoretical and measured scattered
E-field from cylinder (a) in time domain and (b) in frequency domain, |E‘| = 1 V/ m,
p = 3.5052 m, a = 0.0127 m, ¢,— = —6.1°, 43, = 6.1”, ¢,, = 90.00.

279

 

—— Measurement

 

 

 

 

15% —--—~ Theory
A l
E g l
51o: l
s . s
a: 5i "
'U
3 l ‘
‘a 0% .
E « .
< i i i i
.5j
: l
. l
-10 .............. W
0 1 2 3 4 5 6 7 8 9 10
time(ns)
(a)
0
‘ —Measurement

Amplitude (dBV/m)
L .L.
O O

61
o

 

 

 

 

 

do
0

 

f1 l r I

6 8 10 12 14 16 18
frequency (GHz)

(b)

I Y i I Y I T Y T T I

N
A .

Figure 5.25. TE polarization comparison of theoretical and measured scattered
E-field from sinusoidal surface (a) in time domain and (b) in frequency domain,
IE‘I = 1 V/m, p = 3.5052 m, d = 0.1016 m, w/d = 12.0, h,/d = 0.125, ()3,- = 17.90,
«p, = 30.10, (1),, = 90.00.

280

 

N
O

_l
O

— Measurement
—~—- Theory

 

Amplitude (mV/m)
O

 

 

 

 

 

 

 

 

-10 j
-20 §
-30‘ T YYYYYYYYY W
0 2 3 4 5 6 7 8
time (ns)
(80
0
‘ —— Measurement
E220 j 3.... s”: .
S J ‘ 1‘ ‘M, r ‘
m 4 ..Z : . . ‘ .
03-40 ~ 1 1 If
'3 ~ I I
. ~
< -60 ~ I \

 

 

 

 

 

 

 

Y Y W v I v 1 I fir I Y ‘7 ‘V

6 8 10 12 14 16 18
frequency (GHz)

(b)

V T f Y I T I 1 v I v 1 v T T v

Figure 5.26. TE polarization comparison of theoretical and measured scattered
E—field from cylinder above flat surface (a) in time domain and (b) in frequency
domain, |E'| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, mr/a = 0.0,
w/h. = 10.5, 45.- = 14.90, ¢. = 27.1°, 42‘, = 90.00.

281

 

— Measurement
—-~—- Theory

.' ‘l
1 W

N
O

_a
O

1L11411
'.

 

 

 

 

 

 

Amplitude (mV/m)
O

 

 

 

 

 

: 1
40% I”
.201 # lit
% .
430.11,,

3.0 3.5 4.0 4.5 5.0 5.5
time (ns)

Figure 5.27. Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TE Polarization, IE'I = 1 V/ m,
p = 3.5052 m, a = 0.0127 m, hg/a = 8.0, xr/a = 0.0, w/h¢ = 10.5, d),- = 14.9“,
¢, = 27.1°, ¢p = 90.00.

282

 

 

 

 

 

 

 

 

 

 

 

 

 

20 —— Measurement
% —~-—- Theory
é 1° 1 l . 4
> ‘ ' 2 l
E, 0 a : .. “'i' . ‘
° 1 ' . I
13 i " i .
3'10 i
E ,
< y w
'20 i i
i i
-30 ................... T YYYYYYYYY ,flfin T n ,
0 2 4 6 8
time (ns)
(a)
0
I —— Measurement
4 —--——- Theory
“-20 9
g r‘\ I. \ .’\ {\\ A" ‘:\ I‘ A
m . A . . . I ,1 ,‘I r- , \
1: j i , ' \
m -40 4 ' i
1: i ,
:3 ‘ I x
a j ' i
2 so .
i I
.30 .......... Trflfivflfivfiv,.fiwmaqnfi
2 4 6 8 10 12 14 16 18

frequency (GHz)
(b)

Figure 5.28. TE polarization comparison of theoretical and measured scattered
E—field from cylinder above flat surface (a) in time domain and (b) in frequency
domain, IE‘I = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, zr/a = 0.0,
w/h, = 10.5, 45,- = 33.90, ¢, = 46.10, ¢, = 90.00.

283

 

 

 

 

 

 

 

 

 

 

20 — Measurement
; —--—- Theory
A 1 i
E 10? '
S :
E .
v 0 _
m .
.0 .
g :
3'10“.
E .
< i
-20 j
-3o‘fi To
2.5 3.0 3.5 4.0 4.5 5.0
time (ns)

Figure 5.29. Multiple reflection comparison of theoretical and measured scat-
tered E—field from cylinder above flat surface in TE Polarization, IE‘I = 1 V/m,
p = 3.5052 m, a = 0.0127 m, ht/a = 8.0, $,/a = 0.0, w/ht = 10.5, 42,- = 33.90,
d2. = 46.10, 43,, = 90.00.

284

 

 

 

 

 

 

 

 

 

20 l — Measurement
1 —--—- Theory
E10: l
>
\E/ 0:
Q ‘ l r
.0 l
g :
a-10j '
E l
<
-20:
-30 ,,,,,,,,,
0 1 2 3 4 5 6 7 8 9 10
time(ns)
(a)
0
1 —Measurement
320* , r ..
\ < H. :\:' “'c. k " ‘M‘
5 i 1‘; 'H v" '1'“ V
'O i .: .
(Ia-40* ' ii .‘
'0
.2
'73 i
E450
!
-80 zfilf ,,,,,,,,, ffififi
2 4 6 8 10 12 14 16 18
frequency(GHz)
(b)

Figure 5.30. TE polarization comparison of theoretical and measured scattered
E-field from cylinder above sinusoidal surface (a) in time domain and (b) in frequency
domain, |E‘| = 1 V/m, p = 3.5052 m, a = 0.0127 m, ht/a = 8.2, zr/a = 0.0,
d/a = 8.0, w/d = 12.0, h,/d = 0.125, ¢.- = 17.90, ¢. = 30.1", ¢, = 90.00.

285

 

 

 

 

 

 

 

20 —— Measurement
* —--—- Theory
75‘
s i
g , ,
m ..
.0 .
3 .
§-1o{ '
E .
< I
-20 j _
3 !
-3o‘fi , W
3 4 5 6 7
time (ns)

Figure 5.31. Multiple reflection comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface in TE Polarization, IE‘I = 1 V/ m,
p = 3.5052 m, a = 0.0127 m, ht/a = 8.2, xr/a = 0.0, d/a = 8.0, w/d = 12.0,
h,/d = 0.125, dn = 17.90, ¢. = 30.10, «p, = 90.00.

286

 

 

 

—— Measurement

 

 

 

 

 

 

 

 

 

< —--—- Theory _
A 1 I
E « i
\ 1 .
> 0 : il
5 .. « ..-. -
m 0 I ' ‘ - ‘ - ' . ’ ;. 2‘
1: ‘ . t . ' '
a j i g l
.2: , 1
3-10 4 t l
E 1 .
<2
-20 J
-30 Y, , l, T, r 1,7, ,.,. ,j., to Y,
0 1 2 3 4 5 6 7 8
time (ns)
(a)
0
j —— Measurement
. l- A —“—' Theory
A ‘ ' .4
E -20 a 2. . 'i :5, ,
E 4 ' I ' 1:}; i - ‘ i
'o « | '
E1 Z
3 1
-30 , 7 fl vvvvvvvvvvvvvvvvvv , ,,,,,,,,,,,,,,
2 4 6 8 10 12 14 16 18
frequency (GHz)
(b)

Figure 5.32. TE polarization comparison of theoretical and measured scattered
E—fleld from cylinder above sinusoidal surface (a) in time domain and (b) in frequency
domain, IE‘I = l V/m, p = 3.5052 m, a = 0.0127 m, hg/a = 8.2, z,/a = 0.0,
d/a = 8.0, w/d = 12.0, h,/d = 0.125, d).- = 25.90, (b, = 38.10, (pp = 90.00.

287

 

—— Measurement
—---' Theory

N
O
rrLur

l l-

_s
O
1 l

O
1.1.

 

 

L l

 

l L

I
_I.
O

1 l

 

 

Amplitude (mV/m)

 

 

r'o
o
l

 

 

 

 

I
OD
O
1 L J l
.l
1
—4

l fl U f f

I Y

3.5 4.0 4.5 5.0
time (ns)

9°
C

Figure 5.33. Multiple reflection comparison of theoretical and measured scattered
E—field from cylinder above sinusoidal surface in TE Polarization, |E'| = 1 V/m,
p = 3.5052 m, a = 0.0127 m, h,/a = 8.2, xr/a = 0.0, d/a = 8.0, w/d = 12.0,
h,/d = 0.125, ¢.- = 25.90, ¢, = 38.10, 4;, = 90.00.

288

CHAPTER 6

CONCLUSIONS AND FUTURE STUDY

In this thesis, the deterministic problem of transient, short-pulse scattering from tar-
gets above the surface of sea has been studied for both TM and TE polarizations, to
investigate the feasibility of using ultrawide-band radar systems to resolve multipath
difficulties. The target is modeled using a two-dimensional cylindrical object. Sev-
eral models have been considered to represent the sea surface, including a finite flat
surface and a finite sinusoidal surface. The finite sea surface models were not good
representations of the sea because the edges produce direct and indirect reflections via
the target. Hence, to study the multipath problem, the scattered field from a cylinder
above an infinitely-long, sinusoidal surface was solved. The electric field was obtained
using the perturbation method which takes into consideration that the presence of
the cylinder changes the induced current on the sea surface only for a finite region on
the surface (region of multiple reflections).

This idea was used to solve the scattering problem of a cylinder above an infinitely-
long, sinusoidal surface for TM polarization in Chapter 2 and for TE polarization in
Chapter 3. It is found that determining the extent of the multiple regions (the trun-
cation width) is crucial for the accuracy of the technique. In order to apply the
perturbation method accurately, a rule of thumb is found for each polarization. After
the tools were deve10ped to investigate multipath problem, the events in the theoret-
ically calculated reSponses were identified by using simple method of ray calculations
in Chapter 4. It is seen that the incidence angle and the surface roughness affect
the magnitude and the timing of multiple reflections relative to the direct cylinder
reflection. At low grazing angles, the multiple reflections get close in time to the di-

rect cylinder reflection requiring a bigger frequency range to be resolved. The surface

289

roughness changes the magnitude of the specular multiple reflections.

In chapter 5, the theory was verified by the experimental results. An actual tran-
sient pulse (having a 2-18 GHz. bandwidth) was used for measurements to simulate a
realistic short pulse (SP) radar. From the results of the measurements, it is confirmed
that if a radar has sufficient time resolution, the multiple reflections can be separated

in time.

6.1 Topics for further study

Several studies can follow this research by improving the sea surface and target mod-
els. Although the sinusoidal surface approximation did provide a significant amount
of useful information, it is obvious that the waves in the sea are far from being sinu-
soidal in shape. Hence, the easiest change might be considering other periodic surface
shapes to model the sea surface. This will help to gain more information about the
effect of the surface roughness. Next, the sea surface can be removed from perfectly
conducting to being a lossy medium to study the effects of finite conductivity such as
the Brewster’s angle phenomenon. Finally, instead of using two-dimensional models,
the targets and sea can be represented in three dimensions. Solving this problem
could be very computationally expensive, but the solution could provide valuable

information for detection problems such as the scattering at oblique incidence angle.

290

APPENDICES

291

APPENDIX A

MFIE FORMULATION FOR TWO-DIMENSIONAL PERFECTLY
CONDUCTING CLOSED SURFACES ILLUMINATED BY TEz
PLANE WAVE

Consider an incident magnetic field in free space that is assumed to be a TE plane

wave with angle 45,- from the horizontal
II‘(:I:, y) = H0 ejk°(“°s¢‘+”3i"¢‘) 2 (A.1)

The geometry of the problem is shown in Figure A.1. The incident field will produce

a scattered field which can be written in terms of the vector potential function

Haw) = i v x X (A.2)
#0

where p‘ is the two-dimensional position vector
p‘ = 2:5: + yg} (A.3)

representing the field points. The vector potential A(p ) satisfies the two-dimensional

Helmholtz equation

V2507) + lei/Tm) = —uof<’(fi) (AA)

where Kw ) is the unknown induced surface current.

Solution of (AA) will give the general expression for A(p’ )

Kr) = $1 [0 RM") H52’(fi,fi’) dt' (A5)

292

 

 

 

 

 

Figure A.1. Geometry of the problem.

293

where C is the perimeter of the scatterer and p‘ ' represents the source points

“I

p = I’i + 21'!) (A-6)

After substituting (A.5) into (A.2), the expression for the scattered magnetic field

can be written as
H307): — jv x /( K (5') H(2)(ko R) dl’ (A.7)

The induced current [T(p'c) on C can be found from the tangential component of

the total magnetic field

W.) = “x W.) =fix x[lfi‘(p.)+ 13cm )]
= (n x z) H‘(p.) + ——. Pi}. [n x f v x (K(p ') H52’(k03)) dl'](A.8)

where p’ can not be set directly equal to if. for the scattered magnetic field because
the argument of Hankel function can be zero during the integration. Therefore, the
integration must be evaluated carefully.

In (A.8) the following vector identity [27] can be used
V x [K071 Hgmom] = Héi’woR) V x Rm — Rm") x VH HON/com (A9)
where the first term is equal to zero because

V x 1m; ') = o (A.10)

294

Therefore, the identity becomes

v x Ho") Hdz)(koR)] = 46(5) x VH52)(koR) (A11)
After substituting (A.11) into (A.8), the expression for the induced current becomes
Kw.) = imp.) — 21" lim [n x [C( K'(p )x VH52>(koR) dl’] (A.12)
Here, the induced current has only tangential component

K(5c) = f K(fic)

Hus") = f'Kw') (A.13)

Multiplying both sides of (A.12) with the unit vector 5 tangential to C and sub-
stituting (A.13) into (A.12) leads to

1 .. A .
K(p2)+;;.1mg / K(5')[t- (fixt'x VHd2)(koR))] HIE—Hm.) (A14)
P-‘Pc C
where

t’ x VHémUcoR) = (—fi' x 2) x VH§(koR)
-—- VH(2)(koR)X ><n('

x2)
= —z[n' VH‘2’(koR)] +fi’[2-VH52)(I¢OR)]

= —2 [71’ - VH52’(koR)] (A-15)
Here, VHé2)(koR) has only x and y components; therefore,
2 . VH32’(koR) = o (A.16)

295

After substituting (A.15) into (A.14) the term inside the brackets becomes

(5. (fix (5' x VH32)(k0R)) -_- —[f-(fix2)] [fi'-VH32’(kOR)]

= [f . i] [11' . VH32’(koR)]

 

= a'.VH,§2>(kOR) (A.17)
where
(2) . ng2)(koR)
VH0 (koR) = 1:0 R ME
= —k0 R Hf2’(koR) (A.18)
Therefore (A.17) becomes
i - (n x i' x VH32)(kOR)) = 4% cos(I/J') H§2’(koR) (A.19)
where cos(rp’) = ft’ . R
Substituting (A.19) into (A.14) yields
H(p'.) +j £9,133 f K(fi")008(¢’) H{”’(koH) dt' = Hm.) (A20)
C C

A.1 Evaluation of Principal Value

During the integration while [2‘ —* fie, H(2)(koR) will vary rapidly when p‘ ’ gets close

to 56. Therefore, integration will be divided into two parts

jfi lim K(p‘ ’)cos(1,b') Hf2)(koR)) dl’

fi‘fic C

= 1'52 lim K(p")cos(1/z') Hf2)(koR)) dl’

‘

4 fi‘k’l“)

HE K(p") cos(zpc) Hi” (how) dl’ (A.21)
4 C—AC

296

Here, the limit is easily applied to the integration over C — AC. However, the

integration over AC should be carefully evaluated since koR —-) 0. From Figure

 

 

A.2
cosh/1’): _i— (A.22)
V62 + s’2
For small arguments H 1(2)(k0R) can be approximated [17] as
2i
H”) k R = H (2) k V62 + 3'2 2 A.23
l ( 0 ) l ( 0 ) Whom ( )
Substituting (A.23) into the integral over AC gives
)5”- lim K(p')cos(1/1')H1(2)(IcoR) dl’
4 P4P: AC
~ kOK () lim /— 6 Zj ds'
_j_ 4 Km: ‘40 \/(§"’-i-s’2 7rlcox/r52+s’2
2A_____(_7_
- “"5“ H(p.)gi_13__ W ‘13
1 .. . _1 AC/2
_ —7-; K(pc) (1513,13 tan [T] (A.24)

where AC is small enough to validate the approximations of the induced current and
the Hankel function. As 6 approaches to zero, if approaches ,6 ’.

K(fic = —'I£(f£) (A25)

k01
111-Hm K(p ')cos(¢ ¢’)H1(2)(koR)dl'2-; 2

NI:

Substituting (A.25) into (A21) and upon substituting (A.21) into (A.20) gives

Hie). +j E f K07) cos(w.)H1‘2’(koH.) H = -—H"(fic) (W)
2 4 C—AC

The equation (A.26) is the Magnetic Field Integral Equation (MFIE) for the

unknown surface current induced on the closed two dimensional surface.

297

 

 

 

 

 

 

V I
l P l
I A, l
-AC/2 ACT/2
7a" 7;:

Figure A.2. Evaluation of Principal Value.

298

A.2 Moment Method Solution

To solve the MFIE, first the closed surface C is approximated by a number of linear
segments. The unknown current K (5,.) is then expanded in a pulse basis set on these

linear segments as

Kw.) = Z cum.) (A27)

where Kn(p'c) is equal to one on n’th segment and zero on the other segments.
After substituting this into (A26) and using point matching at N points on the
surface, the MFIE becomes

N pn-H .
:0" [E + (1 - amn)j 52/ COS(¢Cmn)H1 (2)(k0Rcmn) dl’] = _H'(p-;m)

n: 1 2 4 p"
(A.28)
where 6"," is the delta function defined by
l m = n
6"... = (A29)
0 m 79 n
Here, the matching points are indexed using m = 1, 2, 3,. . . , N.

The method of Gaussian elimination can be used to solve this set of linear equa-

tions; therefore, they are written in matrix form as

Amn an = bm (A.30)

where Am" is an N x N matrix given by

halt-r

Am. = (A.31)

,- E: 1"“ cos(wc...)H1‘2’(IcoR....) H m 9e n

flu

299

and bm is given by
b = —Ho 83km... cos¢.-+ym sin¢s) (A32)

300

and bm is given by
bm : —H0 ejkbm C03¢i+ym sin 4);) (A_32)

300

APPENDIX B

MFIE FOR TWO-DINIENSIONAL PERFECTLY CONDUCTING
OPEN SURFACES ILLUMINATED BY TEz PLANE WAVE

For the problem shown in Figure 8.1, the total induced current I? on a perfectly
conducting surface C will produce scattered field. Hence, the total magnetic field

consists of the incident magnetic field and the scattered magnetic field
H()?) = H()?) + H15) (8.1)
H()?) = H15) + 3,- fc K's") x VHéi’mH) H (13.2)
Approaching C in the Opposite direction of 11+ gives

Rim.) = 71+ x H(H.)

= ft" x [H‘(p'c) + lirn if lap”) x VHé2)(koR) dl’] (B.3)
PTTPC C

To find the total induced current, it is necessary to approach C in the same direction

of 11+ as well. As as a result another integral equation is obtained

if x H(H.)

= H- x [H()-2°.) + line i f H()?) x VHéi’woR) H] (BA)
P—U’c C

K (£32)

The limit over C can be written as

if KW) >< VH32)(koRc) dl’=
C

 

+11im/ 13(5') x VH32’(k.,R.) dl’ (13.5)
—AC

301

 

 

 

 

 

Figure 3.1. Magnetic field integral equation for open surfaces.

302

Total current on the surface will be equal to summation of the currents on both
sides

W.) = Him.) + H15.) (B.6)

Substituting (B.6) and (8.5) into (B.3) yields

1’96.) — K‘ (5.)) = (ii x Him) +1 (“1* x / H()-2") x vygmoa.) dl’ (B-7)
C—AC

Similarly substituting (B.6) and (BS) into (BA) yields

11‘ x / K(p')xVH32’(koR.) dl’ (B.8)
C-AC

1‘1“ = 41+ (13.9)

After substituting (B.9) into (8.8), (B.8) becomes identical to (B.7). Hence, (B.7)
is the MFIE for open surfaces. However, this equation can not be used to solve for
required total induced current I? (5,) on a perfectly conducting thin scatterer. There
are two unknown currents but only one equation. In other words, this equation is

simply an expression for the difference current in terms of total induced current I?

[28], [29].

303

BIBLIOGRAPHY

304

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

BIBLIOGRAPHY

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S. L. Chuang and J. A. Kong, “Scattering of waves from periodic surfaces,” Proc.
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P. G. Cottis, C. N. Vazouras, C. Kalamatianos, and J. D. Kane110poulos, “Scat-
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P. G. Cottis and J. D. Kane110poulos, “Scattering of electromagnetic waves from
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A. Norman, D. Nyquist, and E. Rothwell, “Transient scattering of a short pulse
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C. N. Vazouras, P. G. Cottis, and J. Kane110poulos, “Scattering from a conduct-
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M. A. Sletten, D. B. Trizna, and J. P. Hansen, “Ultrawide-band radar observa-
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