close

Вход

Забыли?

вход по аккаунту

?

A laboratory investigation of microwave backscattering from non-tenuous dense media with and without rough surface boundaries

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may
be from any type of computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete
manuscript and there are missing pages, these will be noted. Also, if
unauthorized copyright material had to be removed, a note will indicate
the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in
reduced form at the back of the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6" x 9" black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact UMI directly
to order.
U niversity Microfilms In tern atio n al
A Bell & Howell Inform ation C o m p a n y
3 0 0 N orth Z e e b R oad. A nn A rbor. Ml 4 8 1 0 6 -1 3 4 6 USA
3 1 3 /7 6 1 -4 7 0 0 8 0 0 /5 2 1 -0 6 0 0
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O rd er N u m ber S5097S6
A laboratory investigation of microwave backscattering from
non-tenuous dense media with and without rough surface
boundaries
Porco, Ronald L., Ph.D.
The University of Texas at Arlington, 1994
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A LABORATORY INVESTIGATION OF MICROWAVE BACKSCATTERING
FROM NON-TENUOUS DENSE MEDIA WITH AND
WITHOUT ROUGH SURFACE BOUNDARIES
The members of the Committee approve the doctoral
dissertation o f Ronald L. Porco
Jonathan W. Bredow
Supervising Professor
o
' •
/?
0
Adrian K. Fung
Brian L. Huff
John II. McEIroy
Theresa A. Maldonado
Dean o f the Graduate School
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Copyright® by Ronald L. Porco 1994
All Rights Reserved
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A LABORATORY INVESTIGATION OF MICROWAVE BACKSCATTERING
FROM NON-TENUOUS DENSE MEDIA WITH AND
W ITHOUT ROUGH SURFACE BOUNDARIES
by
RONALD L. PORCO
Presented to the Faculty o f the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT ARLINGTON
August 1994
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
ACKNOWLEDGMENTS
I am eternally indebted to my advisor, Dr. Jonathan W. Bredow, for his guidance,
technical advice, and friendship during the research compiled in this dissertation. The
work presented here is as much a result o f his hard work as it is mine.
I would also like to thank the other members o f m y graduate committee, Dr. Adrian
Fung, Dr. Brian Huff, Dr. John McElroy, and Dr. Theresa Maldonado for taking the time
to review this dissertation. It is an honor to have you on my committee.
In addition, I would like to give special appreciation to the staff o f the Automation
and Robotics Research Institute for allowing me to use their facilities. Most notably, Brian
Huff, Dave Vanacek, and Scott Livingston were of invaluable assistance during my robotic
training. I also wish to express my appreciation to the staff of the Byrd Polar Research
Center and the ElectroScience Laboratory at The Ohio State University for allowing us to
use their antenna during our saline-ice experiment. Elias Nasser has been particularly
helpful in answering my questions about the antenna and was very cooperative in sharing
the antenna pattern data with me.
This w ork is supported by the O ffice o f Naval Research under ONR Grant
N00014-90-J-1329. The numerical evaluations were supported by the Center for High
Performance Computing o f The University of Texas system.
August 4, 1994
iv
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
ABSTRA CT
A LABORATORY INVESTIGATION OF MICROWAVE BACKSCATTERING
FROM NON-TENUOUS DENSE MEDIA WITH AND
WITHOUT ROUGH SURFACE BOUNDARIES
Publication No._________
Ronald L. Porco, Ph.D.
The University of Texas at Arlington, 1994
Supervising Professor: Jonathan Bredow
There have been numerous studies in the past o f dense medium scattering effects.
These media are o f interest since many natural media such as sea-ice, snow, and soils can
be classified as dense, having volume fractions exceeding a few percent. Several models,
such as the dense medium transfer integral equation method (DMT-IEM) developed at The
University o f Texas at Arlington and the dense medium radiative transfer theory (DMRT)
developed at The University o f W ashington, have been used to compare theoretical and
experimental scattering results from naturally occurring dense media targets. However,
since the statistical param eters o f these targets can not be assured, there is still much
uncertainty as to the im portant dense medium scattering mechanisms. In order to study
these mechanisms, innovative methods for fabricating dense medium targets with known
distributions and scatterer locations have been developed. The research is unique in this
respect. Other studies to date have made use o f fabrication techniques such as kneading
spheres into a clay-like substance and impacting spheres into holes formed by manual
v
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
force. N either o f these techniques lend to precise control over target parameters. The
volume fabrication method presented here involves a Monte Carlo simulation for scatterer
location and the use o f a robot to accurately position scatterers into suitable materials (in
this case foam which is transparent at microwave frequencies). This approach to dense
medium target fabrication fully controls the electrical and geometrical parameters needed for
model inputs, and provides the ability for repeatable measurements in a controlled indoor
chamber environment. These synthetic targets have planar surfaces.
Since measurements from real world targets such as sea-ice and snow involve both
surface and volume scattering, a technique is presented here for the construction o f an
actual saline-ice target with a known surface roughness. In the past, researchers have
attempted to fit sea-ice measurements with various m odels by varying input parameters
defining the physical structure o f the ice. In the saline-ice fabrication technique presented
here, all input param eters, with the exception o f albedo, are known before the RCS
measurements are even made. Comparisons are made between smooth and known rough
ice for constant albedo.
M onostatic radar cross section measurements from 6 to 15 GHz are performed on
the synthetic volum e-controlled dense medium targets to test the importance o f volume
fraction, particle correlation, and distribution type on the scattering coefficient.
M easurem ents o f both like- and cross-polarizations are com pared with theoretical data
obtained using a single scattering approximation as well as the DMT-IEM. Results indicate
very similar trends as the single scattering approximation and scattering levels comparable
to those predicted by the DMT-IEM . It is also noticed that for ka values close to one,
scattering coefficient decreases with increasing volume fraction after 14%.
B ackscatter HH polarized RCS measurements were perform ed on the saline-ice
target for frequencies from 7 to 17 GHz and 10 to 50° incidence. Results indicate very
strong agreem ent between the m easurem ents and theory based on the IEM surface
vi
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
scattering model and a first-order volume scattering m odel based on the radiative transfer
formulation.
vii
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
TABLE OF CONTENTS
ACKNOWLEDGMENTS................................................................................................................ iv
A BSTRA CT........................................................................................................................................v
LIST O F ILLUSTRATIONS...........................................................................................................xi
LIST OF TABLES............................................................................................................................xv
CHAPTER ONE
INTRODUCTION...................................................................................1
1.1 Background...................................................................................... 1
1.2 Past Work: Theory.........................................................................3
1.3 Radiative Transfer Models............................................................ 4
1.4 Past Work: Measurements............................................................6
1.5 Objectives o f Research...................................................................9
1.6 Summaiy of W ork........................................................................ 10
CHAPTER TW O
THEORETICAL BACKGROUND................................................... 12
2.1 Rayleigh and Mie.......................................................................... 12
2.2 Scattering from a Single Sphere................................................. 12
2.3 Independent Scattering.................................................................14
2.4 Radiative Transfer.........................................................................16
2.4.1 First-order Solution........................................................ 17
2.5 Phase Matrices...............................................................................30
2.5.1 Mie Phase Matrix............................................................ 31
2.5.2 Rayleigh Phase Matrix................................................... 34
2.6 Surface Backscattering.................................................................36
2.7 Bottom Surface Contribution......................................................39
CHAPTER THREE
TARGET FABRICATION................................................................. 41
3.1
Statistically Defined Volume Targets....................................... 41
viii
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
3.1.1 Choice of Materials........................................................ 41
3.1.2 Generation of Data for Target Fabrication..................42
3.1.3 Uniformity T esting........................................................47
3.1.4 Fabrication Facilities...................................................... 51
3.1.5 Sample Target................................................................. 55
3.1.6 Fabricated Target Parameters........................................58
3.2 Statistically Defined Surface Target...........................................58
CHAPTER FOUR
MEASUREMENT SYSTEMS AND CALIBRATION.................. 62
4.1 Measurement Systems.................................................................62
4.1.1 Background.....................................................................62
4.1.2 Anechoic Chamber......................................................... 63
4.1.3 Network Analyzer System............................................ 65
4.1.4 Chamber Modifications................................................. 71
4.1.5 CRREL Measurements..................................................76
4.2 Calibration..................................................................................... 77
4.2.1 One-port Error Model.................................................... 77
4.2.2 Single Reference, Three Target Calibration............... 79
4.2.3 Calibration Verification..................................................90
4.2.4 Calibration of Saline-ice Target Data...........................93
CHAPTER FIVE
BACKSCATTER RESULTS..............................................................98
5.1
Backscattering from Volume Targets...................................... 98
5.1.1 Initial Volume Target, Setup, and Results................. 98
5.1.2 Backscatter Results...................................................... 100
5.1.3 Coherent Interference...................................................108
5.1.4 DMT-IEM with Coherent Interaction........................ 110
5.1.5 Measurements vs. DMT-IEM with
Coherent Interaction..................................................... 113
ix
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
5.2 Bistatic Measurements (Preliminary Results)........................ 115
5.2.1 Fabrication and Target Parameters.............................115
5.2.2 Measurement Results................................................... 116
5.3 Saline-ice Measurements........................................................... 119
CHAPTER SIX
CONCLUSIONS AND RECOMMENDATIONS..........................126
6.1 Conclusions................................................................................. 126
6.2 Recommendations for Future Work........................................127
6.2.1 Saline-ice T argets..........................................................127
6.2.2 Volume Targets.............................................................128
6.2.3 Synthetic Sea-ice Targets............................................ 128
APPENDIX A
ISEED SELECTION PROGRAM.................................................... 130
APPENDIX B
TARGET DATA GENERATION PROGRAM.............................. 137
APPENDIX C
CHI-SQUARE VALUES....................................................................151
APPENDIX D
ROBOTIC DRILLING PROGRAM.................................................153
APPENDIX E
CALIBRATION PROGRAM............................................................ 157
APPENDIX F
TIME GATING AND DATA SMOOTHING PROGRAM
175
APPENDIX G
SURFACE AND VOLUME SCATTERING PROGRAM
192
APPENDIX H
ANTENNA PATTERNS.................................................................... 198
REFERENCES
216
x
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
LIST OF ILLUSTRATIONS
2 .1 .
Intensity components.........................................................................................................19
2.2(a). First-order intensity components (first term)................................................................28
2.2(b). First-order intensity components (second term)...........................................................28
2.2(c). First-order intensity components (third term)...............................................................29
2.2(d). First-order intensity components (fourth term)............................................................ 29
2 .3 .
Zero-order for plane boundary........................................................................................ 30
2 .4 .
Scattering geometry o f a single sphere...........................................................................32
2 .5 .
Scattering geometry demonstrating conversion o f unprimed
coordinate system to primed coordinate system..........................................................33
3 .1 .
Coordinate system used in data generation....................................................................44
3 .2 .
Overlapping holes..............................................................................................................46
3 .3 .
Z-cuts for chi-square goodness-of-fit test..................................................................... 48
3 .4.
W edge-cuts for chi-square goodness-of-fit test........................................................... 48
3 .5.
Ring-cuts for chi-square goodness-of-fit test............................................................... 49
3 .6.
Target fabrication setup.....................................................................................................53
3 .7.
Robotic drilling procedure................................................................................................54
3 .8.
Z, wedge, and ring histogram......................................................................................... 56
3.9.
Pair distribution function.................................................................................................. 57
3.10.
X-Y view o f 10% target for z=6.35 cm to 8.0 cm.......................................................57
3.11 (a) Saline-ice target with smooth surface............................................................................. 60
3.11 (b) Saline-ice target with rough surface added.................................................................... 61
4 .1 .
Transmitter configuration................................................................................................. 64
4 .2 .
Bistatic receive antenna configuration............................................................................ 64
4 .3 .
HP8510 Network Analyzer System............................................................................... 68
4 .4 .
H P 8510 IF detector........................................................................................................... 69
xi
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
4 .5 .
Synchronous (IQ) detector.............................................................................................69
4 .6 .
Configuration o f measurement system.........................................................................72
4 .7.
Antenna pattern o f dual-polarization 2-18 GHz conical hom
antennas used in experiment (4 GHz).......................................................................... 73
4 .8.
Antenna pattern o f dual-polarization 2-18 GHz conical hom
antennas used in experiment (18 GHz)........................................................................ 73
4 .9 .
One-port error model....................................................................................................... 77
4.10.
M easured response o f a 2" sphere.................................................................................91
4.11.
Like-pol magnitude error................................................................................................ 92
4.12.
Cross-pol magnitude error..............................................................................................92
4.13.
Time domain response of 8" sphere and absorber
at twenty degree incidence (70 to 80 ns)......................................................................94
4.14.
Time domain response of 8" sphere and absorber
at twenty degree incidence (72 to 74 ns)......................................................................94
4.15.
Time domain response o f saline-ice target and absorber
at twenty degree incidence..............................................................................................95
4.16.
Comparison of antenna response o f probe compared to the
measured target response................................................................................................97
5.1.
Original measurement setup........................................................................................... 98
5.2.
Measurement vs. theory as a function o f frequency...................................................99
5 .3.
Measurement results: 5% volume fraction................................................................ 101
5.4.
Measurement results: 8% volume fraction................................................................ 101
5.5.
Measurement results:
11 % volume fraction............................................................ 102
5.6.
Measurement results:
14% volume fraction............................................................ 102
5.7.
Measurement results: 20% volume fraction............................................................. 103
5.8.
Like-polarized comparison (ka = 0.92 to 1.38)........................................................ 104
5.9.
Cross-polarized comparison {ka = 0.92 to 1.38)......................................................104
5.10.
Backscattering coefficient vs. volume fraction (6 GHz)..........................................106
5.11.
Backscattering coefficient vs. volume fraction (10 GHz)........................................106
xii
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
5.12.
Backscattering coefficient vs. volume fraction (12 GHz)........................................ 107
5.13.
Destructive interference................................................................................................. 109
5.14.
Constructive interference...............................................................................................110
5.15.
Measurements vs. DMT-IEM (Cl) (6G H z).............................................................. 113
5.16.
Measurements vs. DMT-IEM (Cl) (10 GHz)............................................................114
5.17.
Measurements vs. DMT-IEM (Cl) (12 GHz)............................................................ 114
5.18.
VV polarized measurements vs. single scattering theory
(6, = 45°, 6r = 55°, A0 = 70°)....................................................................................117
5.19.
HV polarized measurements vs. single scattering theory
(6, = 4 5 °,0 r = 55°, A$ = 70°)....................................................................................117
5.20.
W polarized measurements vs. single scattering theory
(6, = 4 5°,0r = 7O°,A0 = 3O°)....................................................................................118
5.21.
HV polarized measurements vs. single scattering theory
(0, = 4 5 °,0 r =7O °,A 0 = 3O°)....................................................................................118
5.22.
M easurements vs. theory at 7.5 GHz..........................................................................122
5.23.
M easurements vs. theory at 10.7 5 GHz..................................................................... 122
5.24.
M easurements vs. theory at 12.5 GHz....................................................................... 123
5.25.
Measurements vs. theory at 15.25 GHz..................................................................... 123
5.26.
Measurements vs. theory at 16.5 GHz....................................................................... 124
5.27.
Percent contribution due to surface scattering...........................................................124
5.28.
Percent contribution due to volume scattering.......................................................... 125
B . 1.
2 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 199
B .2.
3 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna..............................................................................200
B . 3.
4 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna..............................................................................201
B .4.
5 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna..............................................................................202
xiii
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
B .5.
6 GHz Horizontal polarization, near-field horizontal scan
o f lm offset-fed reflector antenna,............................................................................. 203
B .6.
7 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 204
B .7.
8 GHz Horizontal polarization, near-field horizontal scan
o f lm offset-fed reflector antenna...............................................................................205
B . 8.
9 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 206
B .9.
10 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 207
B . 10.
11 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 208
B. 11.
12 GHz Horizontal polarization, near-field horizontal scan
o f lm offset-fed reflector antenna............................................................................. 209
B . 12.
13 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 210
B . 13.
14 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna...............................................................................211
B. 14.
15 GHz Horizontal polarization, near-field horizontal scan
of lm offset-fed reflector antenna.............................................................................. 212
B. 15.
16 GHz Horizontal polarization, near-field horizontal scan
o f lm offset-fed reflector antenna.............................................................................. 213
B. 16.
17 GHz Horizontal polarization, near-field horizontal scan
o f lm offset-fed reflector antenna.............................................................................. 214
B . 17. 18 GHz Horizontal polarization, near-field horizontal scan
o f 1m offset-fed reflector antenna............................................................................. 215
xiv
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
LIST OF TABLES
3.1.
Chi-square values..............................................................................................................50
3.2.
Parameters used in target fabrication............................................................................. 58
4.1.
Test equipment frequency range.................................................................................... 70
5.1.
Parameters for spherical targets.................................................................................... 116
C.
1.
Critical values o f the chi-square distribution.............................................. 152
xv
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
CHAPTER ONE
INTRODUCTION
1.1
Background
Scattering o f waves from media composed o f densely populated scatterers is an area
o f increasing study since many geological materials can be classified as dense m edia, i.e.
the volum e fraction o f scatterers is greater than approximately 0.1% [1]. For example, dry
snow is a mixture o f ice particles and air, rocks are a combination o f rock grains and gas or
fluid filled pores, and sea-ice consists of ice plus air and brine pockets as inclusions [2],
By studying the scattering behavior of such m aterials, theoretical models to predict
scattering as a function o f parameters such as volume fraction of scatterers, permittivity of
the background medium as well as the scatterers themselves, and thickness o f the medium
can be validated and improved. Knowledge o f these scattering behaviors may allow us to
retrieve the parameters from remote sensing data and track naturally occurring phenomena
such as global warming as well as improve our knowledge of the environment.
Perhaps the most commonly studied dense media in remote sensing are sea-ice and
snow. M icrowave remote sensing of polar regions has been an area o f intensive study for
over thirty years due to the role o f sea-ice in global energy balance and to its navigational
importance in the polar regions [3]. The polar climate is greatly influenced by radiative heat
loss and oceanic and atmospheric heat transport. Since snow covers and ice layers alter
this energy exchange between the ocean and the atm osphere, the snow and sea-ice
thickness and extent are considered as important indicators o f the greenhouse-gas-induced
1
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
2
global climate change [3-6]. Navigation on and below the ocean surface is concerned with
forecasting ice conditions and identifying hazards such as very thick ice [3].
In the study of wave scattering from random media, numerous models to predict
both surface and volume scattering have been presented, but uncertainty remains as to
which provide the m ost accurate predictions over a variety o f surface, volume, and
m easurem ent system parameters. Unlike sparse media, where existing Rayleigh and Mie
theories have accurately predicted scattering behavior, questions still remain as to the
im portant scattering mechanisms involved in dense media [7]: Does surface scattering or
volume scattering dominate? Is positional correlation between scatterers important? What
is the effect o f close particle spacing? Are there other mechanisms o f primary importance?
In order to answ er these questions and test existing theories, it is helpful to build and
perform m easurem ents on targets with statistically known surface roughness, scatterer
locations, and distribution types.
Past efforts to construct volume scattering targets include manually inserting
scatterers into a low dielectric medium or spraying polyurethane foam over a collection of
scatterers, with little control over the precision o f scatterer positions [8,9],
Other
techniques used include molding spheres in clay, im mersing scatterers in liquid, and
im bedding spheres into sand [10-13]. These techniques do not offer much control over
scatterer location in the x, y, and z directions. A new technique proposed in Section 3.1
will discuss a feasible method o f fabricating dense medium targets while maintaining these
vital parameters.
W hile construction o f statistically known volume targets is being presented here for
the first time, a thorough study on perfectly conducting, pre-defined surface roughness
targets has already been performed [14], A sim ilar technique presented in Section 3.2
demonstrates a method for constructing saline-ice targets with known surface roughness.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
3
1.2 Past Work: Theory
The first known studies of the radiation field in a light scattering atmosphere were
performed by Lord Rayleigh in 1871 on the sunlit sky. However, it wasn't until 1949 that
the fundamental equations governing absorption and scattering from a radiating medium
were completely examined [15]. Since Radiative Transfer is a mathematically complex
problem and com puters were not available at the time, it becam e standard practice to
approxim ate the equation o f transfer with the zeroth and first-order solutions. W ith
improved computational facilities becoming available in the late 60's and 70's, it became
possible to num erically solve the equation o f transfer for orders higher than first. Two
methods, the eigen-analysis technique and matrix doubling were applied [2,16]. In 1981,
the matrix doubling m ethod was expanded by Leader [17] to include irregular boundary
effects as well as dense medium effects from small scatterers. In 1992, a matrix doubling
method for multilayered inhomogeneous media with irregular boundaries was presented by
Tjuatja [4], This form ulation has shown excellent agreement with the US Army Cold
Regions Research and Engineering Laboratory (CRREL) and Coordinated Eastern Arctic
Experiment (CEAREX) data, but has not yet been verified with measurements from targets
with known parameters.
In the 1960's measurements from land and sea surfaces were modeled using one of
two formulations: the Kirchhoff model and the small perturbation model [18-22], While
the Kirchhoff m ethod was shown to produce reliable values for surfaces with large scale
roughness, the small perturbation model gives good results for surfaces which are slightly
rough [23-25]. By the end of the 60's, two-scale models were developed in an attempt to
obtain better agreem ent with m easurem ents [26-28]. These m odels consider surface
statistics as the sum o f two independent random processes. Although these models did
result in better agreement with measurements, they are flawed in the sense that real random
surfaces are actually composed o f a continuous roughness spectrum [14]. Hence, dividing
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
4
the surface roughness into a large and sm all com ponent m ay not be an accurate
representation o f the actual surface roughness. A method to predict scattering behavior for
surfaces whose roughness lay between the large and small scale was needed. In the 70's
and 80's, many attempted to formulate models with a wider range of validity than the twoscale m odel. In 1986, Fung and Pan [29] presented a scattering model for perfectly
conducting surfaces.
This m ethod was expanded by Fung and Li [30] in 1989 to
encompass finitely conducting surfaces o f any roughness and came to be known as the
Integral Equation M odel (IEM). This model was shown to accurately model surfaces o f
varying scales o f roughness by Chen in 1990 [31,32]. However, in the com parison of
theory and real-world experimental data, the statistical parameters o f the target can not be
assured. In order to prove the validity o f the IEM, rough surface targets with known
statistical param eters were generated and coated with a conductive paint to assure only
surface scattering would be present. This experim ent was com pleted by Nance in 1992
[14,32],
1.3 Radiative Transfer Models
The radiative transfer theory governs the propagation o f energy through a scattering
m edium [15]. If the medium is em bedded with discrete scatterers, then the scattering
intensity is obtained by linear summation o f intensity reflected by each o f the scatterers
since the radiative transfer theory assumes that the scattered fields from each particle are
uncorrelated [4], The theory therefore accounts for all incoherent effects but ignores
possible coherent effects. This method does not have a closed form solution, i.e. no
analytic solution, but zeroth and first-order analytic approxim ations can be obtained via
iteration o f the radiative transfer equations. However, if the scattering loss within the
medium is large, higher order than first m ust be considered. Solving for this problem is
done using numerical techniques such as the eigenvalue-eigenfunction technique and the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
5
m atrix doubling m ethod. Since the eigenvalue-eigenfuncion technique can not ensure
stability in matrix inversion for large optical thicknesses, the matrix doubling formulation is
the method o f choice in the model developed at The University of Texas at Arlington. This
model also incorporates the integral equation m ethod to account for direct scattering from
the surfaces as well as all surface/volume interactions. This model has been labeled the
dense m edium transfer integral equation m ethod (DM T-IEM ) and has been shown to
provide excellent agreement with multifrequency, multipolarization, multiangle, active and
passive measurem ents [4,32].
In classical radiative transfer, the phase function is evaluated in the far-field.
However, this assumption may be inaccurate for dense media. Since the spacing between
particles in a dense medium is small, the particles are not necessarily in the far-field o f each
other. The dense m edium effect for a random distribution o f spheres is incorporated by
redefining the phase matrix to account for close spacing effects. The elements o f the phase
matrix are functionally dependent upon scatterer size, dielectric properties of the scatterers
and background m edium , frequency, polarization states, and volum e fraction o f the
scatterers [4], In the DMT-IEM the effective permittivity o f the layer is calculated using
em pirical relations determined from measurements [33], and the sphere size is an effective
value. M ultiple scattering and interaction between the volume and surfaces are accounted
for using the matrix doubling method.
The radiative transfer formulation does not account for phase interference effects
involved in multiple scattering. One model which partially accounts for the coherent effects
is the dense m edium radiative transfer theory (DMRT) developed at The University of
W ashington [34], As the name implies, this method is also based on the radiative transfer
theory and accounts for scatterer position correlation. Unlike the DMT-IEM, however, this
method does not account for full surface/volume interaction. Until recently, the DM RT has
been unable to accurately model naturally dense m edia.
Previous com parisons of
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
6
theoretical values with measurements from naturally occurring dense media showed that the
theory severely underpredicts the scattering [35]. In an attempt to increase the theoretical
predictions to the level o f the measurements, m odifications were made to the theory to
allow for a distribution o f scatterer sizes instead o f an effective size [36], Although this
technique does increase the scattering level close to the level o f the measurements, the size
param eters input to the model are unrealistic. The newer model uses a distribution of
scatterers with a maximum size much larger than would be encountered in a real situation.
Although the number o f such occurrences is small, scattering can be dependent upon the
radius o f the sphere to the sixth power. Hence, these very large values can dominate the
scattering and force the overall level upward.
1.4 Past Work: Measurements
There have been numerous works in the past on the dense medium problem [1,1013,37,38]. One o f the more intensive studies was done by Kuga in 1983 at The University
o f W ashington [1], Kuga studied both the backscattering and attenuation of various sized
latex spheres im m ersed in water. His experim ent involved the use o f a HeNe laser
(A = 0 .6 3 2 8 /im ) to illuminate a sample cell composed o f water and latex spheres. The
polarized intensity was received in the forward direction using a photo diode and the
backscatter response was collected from a detector. The particle diameters studied were
0.091, 0.109, 0.481, 1.101, 2.02, 5.7, and 11.9/m i, implying ka values in water o f 0.60,
0.72, 3.17, 6.67, 13.33, 37.64, and 78.58 respectively.
Num erous volum e fractions
ranging from 0.001% to 40% were examined. The lower volume fractions were obtained
by adding deionized water to concentrations supplied by Dow Chemical, and the higher
volume fractions were obtained by letting the particles settle and decanting some o f the
excess water. Note that if the particle sizes were too small, they did not settle thus making
it impossible to obtain the higher volume fractions desired.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
7
The results o f Kuga's attenuation experiments show that the normalized attenuation
constant for the 0.091 iim particles increases up to 10% volume fraction and then decreases
for higher concentrations. This agrees with Twersky's formulation in his study o f paircorrelated scatterers [39,40]. For particles larger than 0.481 /t m , however, the attenuation
increases throughout the full range of concentrations.
In his backscattering experiment, Kuga observed that for all particle sizes the
backscattering coefficient increases rapidly up to approximately 5% volume fraction. For
ka = 0.529 the backscattered intensity shows a decrease between 10 and 20% volum e
fractions whereas for a ka = 36.238, the backscattered intensity increases very slightly over
this region.
Kuga’s results were verified in a sim ilar experiment conducted by Gibbs at The
U niversity o f Texas at A rlington [11].
G ibbs perform ed experim ents on Dow
manufactured Teflon particles o f average diameter 0.15^m (ka = 0.99) as well as on
polyvinyl acetate particles (Vinac881) with an average diam eter o f 0.18 fxm (ka = 1.19).
Gibbs' experim ents indicate the same trend as did Kuga's, i.e. for ka <1 (Teflon), both
attenuation and backscattering decrease for volum e fractions greater than 10%, and for
particles with k a > \ (Vinac881), backscattering saturates at 10%. Information on the
attenuation from the Vinac881 particles is not available for volum e fractions greater than
10%.
Kuga's experiments showed very interesting trends which merit consideration for
other investigations. In order to build on Kuga’s study, ways to improve the experiment
need to be noted. First, the particle sizes provided by Dow represent the mean diam eter
within that sample rather than a single size for all particles. For example, for the particles
listed as 5.7 jim , the mixture is actually made up o f particles with diameters ranging from
2.056 to 14.01 fj.m. Secondly, it appears that the main area of interest is between 10 and
20% volum e fractions for particles with ka values from about 0.5 to 1.5 since the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
experim ents show that som ewhere within this region there is a transition in both the
backscattering and attenuation behavior. The two particle sizes closest to this region,
however, could not be obtained in volume fractions higher than 10% since these particles
did not settle and therefore excess water could not be removed. Another problem was that
coagulation tended to occur, creating doublets, triplets, etc. [41,42]. It also would have
been beneficial to perform cross-polarized backscatter m easurem ents as well as co ­
polarized m easurements. Cross-pol can be a valuable tool since single scattering from a
sphere results in zero cross-pol in the backscatter direction. Thus, any return from a crosspol backscatter m easurem ent is due strictly to m ultiple scattering.
A final area for
im provem ent in K uga's experim ents w ould be studying non-tenuous m edia.
M ost
naturally occurring dense media are non-tenuous, i.e. the permittivity o f the scatterer is
much different from that o f the background medium. The latex spheres used in the
experiment, however, have a permittivity fairly close to that of the water (2.50 compared to
1.77 at the observed frequency), thus creating a tenuous medium.
Two of the more recent controlled dense media experiments were published by Koh
[13] and Mandt [38] in 1992. In Koh's experiment, a mixture of spherical glass scatterers
and a background m aterial com posed o f linseed oil-based clay were kneaded until the
spheres were distributed thoroughly throughout the clay as determ ined visually. The
mixture was allowed to harden and low-millimeter wave measurements were performed.
Koh used scatterer sizes o f 0.4 mm, 1.2 mm, 3.0 mm, and 4 mm and volume fractions o f
5, 10, 20, 30, and 40% over a frequency range o f 26.5 to 40 GHz. A t 26.5 GHz, the
corresponding ka values within the medium are 0.14, 0.43, 1.06, and 1.42 and at 40 GHz,
these values become 0.21, 0.65, 1.6, and 2.14. Trends similar to those obtained by Kuga
resulted, but many o f the same measurement uncertainties remain. Once again the scatterer
distribution is unknown, and the areas o f interest, i.e. volume fractions between 10 and
20% and ka's very close to one, are not examined.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
9
The experiment most closely resembling the one examined in this work was the one
perform ed by M andt [38].
em bedded in Styrofoam .
M andt built targets com posed o f 5.73 mm glass beads
The locations o f these beads were selected by creating a
rectangular grid on thin sheets o f Styrofoam and using a M onte Carlo simulation to
determine at which grid points the scatterers would lie. Mandt built volume fraction targets
o f 0.55, 1.18, 5.6, and 10.9% and performed attenuation m easurem ents on these targets
from 18 to 20.5 GHz, im plying a ka range o f 1.08 to 1.23. Again the region from 10 to
20% for ka 's around one was unexamined. Also, although the distribution o f the scatterers
was m ore controlled than the previously described experiments, improvements can still be
made by allowing scatterers to lie anywhere within the volum e o f the target rather than
being limited by grid locations and the thickness o f the Styrofoam sheet.
1.5 Objectives of Research
The main objective o f this work is to experimentally determine answers to the dense
medium problem . This will be accom plished in two phases, i.e., by studying volume
scattering with little boundary contribution and examining com bined surface and volume
scattering for targets with know n surface roughness.
In the volum e portion o f the
experiment, m icrowave backscatter measurements are performed on precisely fabricated
targets over a wide range o f volum e fractions, frequencies, and polarizations. For these
targets, all physical param eters are controlled and electrical properties are measured. The
scatterers are o f uniform size, thereby making it unnecessary to determine an effective size
when modeling. The background medium is chosen such that it is essentially transparent at
microwave frequencies, and scatterers with dielectric constants much different than the
background are chosen. Therefore, the overall backscattering return will be dominated by
scattering from the spheres within this non-tenuous medium. The scatterer locations are
defined according to a known distribution type thus removing the possibility o f scatterer
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
10
clustering. The physical scatterer locations are created by a robotic arm, thereby limiting
the possibility o f human error. Maintaining strict control over these parameters allows for
an accurate comparison o f measurements with results from theory. This will also enable us
to locate any possible shortcomings o f the theories studied.
The com bined surface and volume scattering phase o f the experim ent involves
perform ing RCS backscatter measurements for various incidence angles on a saline-ice
target with a known surface roughness. This surface is created by freezing water onto a
m old shaped by a com puter controlled m illing m achine.
M easurem ents w ere also
performed on a saline-ice target whose surface is relatively flat. Thus, measurements from
saline-ice with and w ithout surface roughness can be com pared. This enables us to
determ ine at what angles and frequencies surface scattering is dom inating the return, and
when volume scattering is the important parameter.
1.6 Summary of Work
Chapter Two gives a brief overview of the theoretical approaches used, including
scattering from a sphere, the integral equation model, and radiative transfer. A combined
surface-and-volum e scattering model is presented based on the IEM surface scattering
model and a first-order volume scattering model based on the radiative transfer formulation.
A M athem atica program em ploying this model is given in Appendix G. The complex
solutions for the IEM and DMT-IEM can be found in [32].
C hapter Three describes the unique target fabrication processes developed and the
facilities used to construct these targets. A new method for fabricating volumetric targets
with known statistical parameters is described in depth, as are the methods for testing the
distribution type. Values corresponding to the chi-square goodness-of-fit tests performed
in this chapter are given in Appendix C. A technique for creating saline-ice targets with
known surface roughness is also presented in this chapter. The two FORTRAN programs
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
11
used in the choice o f scatterer locations for the volumetric targets are given in Appendices A
and B. The VAL2 program used to perform the actual drilling of the holes for these targets
is given in Appendix D.
C hapter Four concerns itself with the m easurem ent system s em ployed and
describes the calibration techniques used. This is followed by an analysis as to the
accuracy o f the calibration processes used. The data processing algorithms used are also
described in this chapter. Appendices E and F contain the FORTRAN code for some of
these algorithms. Appendix H contains the antenna patterns for the antenna used during the
measurements o f the saline-ice targets.
Chapter Five presents the results o f the experiments, as well as comparisons with
model predictions. Improvements to the DMT-IEM are also presented in this chapter. In
Section 5.2, a technique for building spherical shaped targets is discussed, and preliminary
bistatic measurements on two such targets are presented.
Conclusions and suggestions for future work are given in Chapter Six. One of
these suggestions involves performing bistatic measurements on spherical-shaped dense
medium targets, as was briefly discussed in Chapter Five.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
CHAPTER TWO
THEORETICAL BACKGROUND
2.1 Rayleigh and Mie
There have been a number o f studies performed on scattering from two-dimensional
objects. However, there are very few cases where an exact or classical solution can be
obtained for scattering from finite three-dimensional bodies [43]. Due to the geometrical
symmetry o f a sphere, it was one o f the first to receive successful theoretical treatment for
the case o f scattering in a hom ogeneous m edium , such as free space.
Studies o f
backscatter for non-spherical particles show that the reflectivity varies as the shape of the
particles become more distorted from that o f a sphere, as the refractive index changes, and
as the particle positioning becomes less random [44,45].
W hen scattering occurs from a particle that is much smaller than the wavelength, it
is called Rayleigh scattering. However, if the diam eter is greater than about one-tenth of
the wavelength, the Rayleigh theory proves to be inadequate, and it is necessary to use the
more complex M ie theory [46]. Although this theory is exactly applicable only for true
spheres, a wide variety o f particles can be approxim ated by spheres when particle
orientation is sufficiently random; hence the Mie theory can be used even though the
particles are irregular in shape.
2.2 Scattering from a Single Sphere
In order to understand how m ultiple scattering occurs, it is first necessary to
understand the scattering from a single sphere. These results are presented in a number of
12
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
13
texts [46-49], so they will be covered here only briefly. The exact solutions derived by
Mie for the scattering and extinction efficiencies are
& «.*) = - * - X
( 2 / + l ) ( | a / p + | 6 / p) = - %
* 2 /=i
^ 2
U n ,X ) = ^ i ( 2 l + l ) R e { a l + b,} =
.
71T*
X
i= i
^
(2.1)
(2.2)
where X ~ K r = ~ r ~ ^ r b > w i£h £ri) being the real part o f the relative dielectric constant
A0
o f the background medium, and a, and bt are known as the Mie coefficients [46]. In
literature x is more commonly referred to as the "ka value" where k is the wavenumber in
the background medium and the variable a represents the radius o f the scatterer. The value
n is the index o f refraction o f the spherical particle with reference to the background
medium, i.e. n = ^ £ p/ £ b . Qs and Qt are the scattering and extinction cross sections.
Sim plifying M ie’s equation o f scattering efficiency for the special case of
backscattering yields the norm alized radar backscattering cross section (backscattering
efficiency)
X ( - l p ( 2 / + l )(a i -b ,)
^ = Jnr2
T =x
J2
?
i= l
(2.3)
If the particle size is much smaller than the wavelength o f the incident wave such that
\nx\ < 0.5 , this expression can be reduced by the Rayleigh approximations [46], namely
Zb = 4 x 4 \K\2
(2.4)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
14
where K =
n2 - \
n2+2
. This is called R ayleigh’s backscattering law.
If the Rayleigh
conditions are met, then backscatter cross section o f individual spherical particles can be
calculated according to
c% = a r 2 & = M ^ L r 6|K|2
Ao
(2.5)
The precision o f this approximation was tested on water particles and found to be accurate
within 1 ± 0.3 times the Mie value for water particles at all wavelengths in the microwave
region if \nw%\ < 2 [46].
2.3
Independent Scattering
Most practical cases concern total scattering from all particles contained in a certain
volume o f space [48]. If the average separation o f particles of a random distribution is
several times greater than the radius of the particles, independent scattering occurs. This
means that each particle is considered to scatter independently o f all others, and each
scattering pattern is unaffected by the other scatterers. It m ust be noted that in the case
where there are m any particles, especially ones which are very close together, single
scattering very rarely occurs. That is, each particle is not exclusively exposed to energy of
a direct incident beam. Each particle also reflects a small amount o f energy already
scattered by one or m ore o f the other particles. This is called secondary o r multiple
scattering [48].
Although researchers often make the assumption that multiple scattering does not
occur in their experiments with random distributions o f spheroids, this is never actually the
case. The best that can be said for a given distribution is that m ultiple scattering is
negligible [50]. This assumption is a good one as long as the particle cloud containing
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
15
free-floating spheres in air transmits at least ninety percent o f the incident radiation.
Otherwise, it is possible that some waves are being scattered several times before they leave
the cloud, implying that multiple scattering can not be neglected.
A nother im portant consideration in scattering by a collection of targets is the
randomness o f their relative positions. If the particles are located at predictable points, the
phases o f the scattered waves are well defined and coherent scattering dominates [50]. In
order to find the intensity, the wave amplitudes must be added and squared. If the particles
are random ly positioned the phases are also random, implying that the scattering is
incoherent and that intensities can be added directly [50].
In the case o f incoherent scattering, the results for a single particle may be extended
to encompass a cloud made up of similar particles. For example, if there are N identical
particles per unit volume of space scattering independently, the power scattered by this unit
volume is
( 2 .6 )
Dividing by the incident power density, S; , gives the scattering cross section
Q s«= ^L= W «
(2.7)
Sim ilarly, the absorption cross section is
(2 . 8)
and the extinction cross section is defined as [46,50]
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
16
(2.9)
Q' = Q. + Qs
Hence, if there are N identical particles which undergo independent scattering, then
the total backscatter cross section is given by the sum o f their individual backscatter cross
sections, that is [46]
c bN = N o b = N 64 S - [ /f p r 6
Ao
(2.10)
2.4 Radiative Transfer
Although the Mie and Rayleigh theories are accurate for measurements o f freefloating particles in air, a more accurate model can be used to predict the response o f
layered dense medium targets, such as a snow layer. Normally, when taking scattering
measurem ents from a layer, it is necessary to consider the backscattering contributions
from the top surface, the bottom surface, and the volum e o f the layer [16].
The
contribution from the top surface is a direct contribution from the surface.
The
contributions from the bottom surface and volum e, however, need to be m odified to
account for transm issivity, reflectivity, and attenuation (loss) from the layer and upper
surface. The radiative transfer solution has the ability to account for this interaction.
For rough boundaries, it is possible to expand the surface phase function in a
Fourier series in <p, i.e. the azimuth angle. Then, using quadrature, the 6 integral is
converted to matrix representation for each Fourier component. As a result, the boundary
conditions are in algebraic form similar to the case o f a plane layer. The same solution
technique is applicable to treat each Fourier com ponent of the rough layer problem. The
next section will present the first-order solution for the radiative transfer equation. Higher
order solutions can be obtained by numerically solving the radiative transfer equation using
techniques such as the matrix doubling method [4,16].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
17
2.4.1 First-order Solution
R adiative transfer deals with the level o f intensity and not with the com plex field
quantities which are related by Maxwell's Equations. The equation of transfer describes
the variation o f intensities within a medium that absorbs, em its, and scatters radiation.
Consider intensity propagating through a cylinder o f unit cross section and length dl. As it
passes through the cylinder, the intensity will experience some loss due to absorption by
the cylinder and scattering not in the direction o f propagation. Therefore, this change in
intensity, dl, can be expressed by the loss due to absorption and scattering plus the increase
in intensity due to thermal emission by the cylinder and scattering into the direction o f
propagation from surrounding sources. The mathem atical representation o f the transfer
equation is given by [32]
d l = -(*•„ +
where
Ka
and
Ks
k s)I
dl + {KaJa + KsJ s)dl
(2.11)
are the volume absorption and scattering coefficients, and J a and J s are
the absorption (or em ission) and scattering source functions.
The scattering source
function is defined as
1 2icic
j j p { e s,<i>/,d,<p)i(d,(t>)sm6d9d<j)
(2 . 12)
00
where P (6s,<ps;d,<f>) is the phase function accounting for scattering within the medium.
The first term on the right hand side of (2.11) represents absorption loss and loss due to
scattering away from the direction of propagation. The second term represents the intensity
emitted and scattered into the direction o f propagation. Note that although the term
KaJ a
represents an emission source, the subscript a is used since an object's ability to absorb is
the same as its ability to emit. This term is neglected in an active problem since the source,
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
18
i.e. a radar transm itter, comes from outside the medium instead o f from the medium itself.
Although emission is still present, it is much weaker than the scattered signal. The term
kJ s
com es from the energy scattered by another scatterer where this energy is now
incident on the observed scatterer. Simply put, it is the result o f m ultiple scattering.
Therefore, if multiple scattering does not occur, this term drops from the equation.
T he extinction coefficient, Ke, is the sum o f the absorption and scattering
coefficients. Therefore, combining (2.11) and (2.12):
jj
2icx
— = - K j + - z - j f p ( e s, <t>s-,e, 0 ) /( 0 , <p)sineded<t>
dl
4^ QQ
( 2 . 13)
or in vector notation
^
^
dl
= - K j { e t , ^ ) + ^ T \ p , { e t ,< i > M ) i ( e ^ ) a n e d 0 d 4 >
4 n J0 J0
q .u
)
By splitting the intensity into upward and downward components as shown in Figure 2.1,
the problem can be formulated along with the boundary conditions.
Letting
(1 = COS0
dfi = - s i n 6 d 6
/ns = cosd3
dz = dl cos 6
(2.15)
(2.14) can be separated into two equations involving the upward intensity, I*, and the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Figure 2.1: Intensity components.
downward intensity, / ':
|
2 *1
+7- J
71 0 0
I 2ff1
J
/ +(z. ^. 0) dll dtp
(2.16)
+ t z J J KfA v ^ ,< P s - < P ) r ( z ,ii,tp ) dii dtp
n 00
- 7“ Jj
KsPs{ -^ ^ < t> s ~ <t>)
d ll dtp
(2.17)
n 00
j
2*1
- 7 - J J ksF s(-H s -H,<I>s -<P) I'{z,n ,tp ) dll dtp
A ft
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
20
Note that in the original equation of transfer, 6 goes from 0 to n and <p goes from
0 to 2 n . However, dividing the intensity into upward and downward portions results in
6 from 0 to ^
f°r 1* and ^
to n for / ' while <j> still covers from 0 to 2 n .
when 6 = 0,jj. = 1 and when 9
Also,
,H = 0 implying that the initial limits o f integration
o
were J , but since djx = - s i n 9 d 9 , the negative sign allows us to invert the limits o f
i
i
integration to j as shown in (2.16) and (2.17).
o
Letting
k« = k ./c o s 6S
Ka = K j c o s d ,
I +(z) = l(z,n„<t>3)
r ( z ) = / ( z , - / i J,0 J)
(2.18)
00
(2.16) and (2.17) can now be written
(2.19)
( 2 . 20 )
(2.19)
and (2.20) are o f the first-order and have standard solutions of
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
21
r (2) = J * (-d ) e- K“(‘*d) + J F*{z')e-K“u' l)dz'
( 2 . 21 )
v
/-(z ) = 1 ( 0 ) e K“l + ]F -(z')e * “{z- ' )dz'
( 2 . 22 )
In order to put these equations into a form suitable for an iterative solution, the boundary
conditions need to be incorporated. The boundary conditions at the upper boundary, i.e.
z=0 , are
.
2*1
r ( 0 ./* ..* ,) = 7
- 0 ) / +(O ,/i,0) dll dtp
00
2 2*1
+ 7 - J S s A - V . - M s - <p)r(0,li,<p) d n dtp
" 00
w here S R( - l l s,n,tps - tp )
and
(2.23)
S T(~ n t ,-fj.,tps - tp) are the surface scattering and
transmission phase matrices and / ', the intensity o f the incident plane wave, is defined as
V = I° 8 ( n -lii)8 {tp - h )
(2.24)
where S( ) is die Dirac delta function and (0, ,0, ) denotes the direction o f propagation of
the incident wave. The first term on the right-hand side of (2.23) is due to reflection o f the
upward intensity and the second term is due to transmission of incident intensity. If the top
surface is a plane boundary, the boundary condition at the upper medium becomes
I-(0 ) = T J ‘ + R J +
(2.25)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
22
where Tu is the pow er transmission coefficient (transmissivity) through the medium, and
is the pow er reflection coefficient (reflectivity) at the upper boundary. In the case o f
perpendicular (horizontal) polarization,
* * ~ |rou| —
T], cos 60 - r)0 cos 9,
tj1cos 60 + r/0 cos
(2.26)
and for parallel (vertical) polarization
7), COS6X- Tj0cos 60
= T 01, =
where
i]l cos 8X+ t]qcos d0
(2.27)
is the intrinsic im pedance and r0l is the field reflection coefficient o f a
wave in medium 0 incident on medium 1. Note that r01 = - r 10 and thus Ru = |r0I|2 = |r10|2.
This relationship is also true at the lower boundary. The transmissivity is easily found as
T u = l - R u.
At the low er boundary, i.e. z=-d, the upward intensity is due to intensity coming
down through the medium and getting scattered upward:
1 2x\
I \ - d , n s, ^ s) = — J J G ( /i„ - /z , 0 , - <t>)r(-d,n,<t>) dii dip
(2.28)
where G is the scattering phase m atrix for the low er surface boundary. If the lower
boundary is also planar, the boundary condition becomes
n -d )= R j-
(2.29)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
23
where Rd is the reflectivity at the lower boundary and is given by
r]2 c o s
fl, -
T?, COS 6 ;
~ ri2A — t]2 cos 61 + 77, cos d2
(2.30)
for horizontal polarization and
=
r|2cos6) - r i , cos82
r\2 cosG, + rj, cos9j
(2.31)
for vertical polarization.
Com bining (2.21) and (2.22) with (2.25) and (2.29), we obtain
7+(z) =
e~K“lt*d)Rir{-d) +} F+(z')e-K^ - !)dz'
(2.32)
and
v
/ - ( z) = e '-'fr/* ' + fl„/+(0)] + j F -{z')e-K^ - z]dz'
(2.33)
For the zeroth-order solution, there is no scattering, only reflection. Therefore, we
can ignore the contribution o f F and the zeroth-order equations can be written as
r0 { z ) = e - K" (l+i)Rdr0{ -d )
(2.34)
r0(z) = e ^ [ T ur + R ui ; ( o)]
(2.35)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
24
Evaluating (2.34) at z=0 and (2.35) at z= -d, we can solve for the intensities at the
boundaries, i.e.
ro (0) = e -K“dRdro( - d )
(2.36)
r0{-d)=e-K-d[Tur+Rj;{ o)]
(2.37)
and solving algebraically yields
p~
/n (0) = —
0
1-
R T
e - 2K“dRuRd
(2.38)
r
e~K“dT
rg( - d ) = — - ^ r r -— / '
1 - e - 2K“dRuRd
(2.39)
where e~K“d represents the loss through the layer, and the denom inator accounts for
multiple scattering between the layers. From equations (2.24), (2.34), and (2.39),
r 0 ( z ) = e - K"{!'+d]Ra
T jt
-2 k
l-e
■K„d
(2.40)
K Ri
Letting
T ue K“d
C, =e~K M R,
1
- e 2K“dRuRd
(2.41)
the zeroth-order intensity is defined as
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
25
Ig (z' ) = 4 n e~ K“(i)C ,5(/i -/r,)<S(0 - 0 ,)/°
(2.42)
Similarly,
■K„d
r 0(z') = e
1 - e~2K',dRuRd J
4 # 5(/i
- 0,.)/°
(2.43)
which can be written as
/o(z’) = 4 ^ e ~ ’r“ (l)C25(/z - / r ,) 5 ( 0 - & )/°
(2.44)
Rewriting (2.32) and (2.33) in terms o f first-order yields
i ; ( z ) = e - '" {z+d)R j ; ( - d ) + s ; ( z )
(2.45)
I](z) = e K“!R j ; ( 0 ) + S -(z )
(2.46)
where
(2.47)
Note that the 7T is dropped from (2.46) since it only goes with zero order. Evaluating
(2.45) at z=0 and (2.46) at z=-d, we can solve for the intensities at the boundaries, i.e.
I l ( 0 ) = e~K"dRdi ; ( - d ) + SnO)
(2.48)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
26
I](-d) = e - ^ R j : ( 0 ) + S -(-d )
(2.49)
Solving (2.48) into (2.49) and vice-versa yields
7
/ , (
(2.50)
\ - e - 2K“dR„R,
(2.51)
1 e~2K“dR„R.
d)
-
where
u
S l(0 ) = j e -K“z'F ;(z’)dz'
(2.52)
and
2*1
F + o { z ' ]
U
= 74 ft?
/fc o o
-* )W
) dM d$
(2.53)
2 ffl
47T 0 0
If we let
c, = 1 + _ w
'
1 e~2K“d
—
K.
K U ) = - f [ P s( l l ^ < P s - < P ) C ^ + Pt { n , - M , - <P)C2e ^ ]
f^S
Sf ( 0 ) =
—
Ms
l-e ~ 2
C \ P , { p , M , - <t>)d + C 2P 3(M s-M ,< t> s - <t>)
2 k .,
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
27
M.L
'
:
(2.54)
After substituting SJ (0) and Sj (—d) into / / (0) we obtain the first-order intensity inside
the layer, namely
+ Rdde-2K“dPs{ns,LiA-<l>)
(2.55)
+ Rdde-2K“dPs(-lis-H > <Ps ~ <t>)
All four term s o f (2.55) are shown graphically in Figure 2.2. Two common substitutions
in (2.55) are that the albedo is the ratio o f the scattering coefficient and the extinction
coefficient ( K s / K t ), and the optical depth is defined as the product o f the extinction
coefficient and the depth o f the target
Ked .
The first-order intensity outside the layer is T UI *(0) and the total solution up to
first-order is
/+ (0) = Tu( i ; ( 0 ) + I,+(0))
(2.56)
and the scattering coefficient from the volume can be found from
4 g /;( 0 ) c o s f l,0
(2.57)
K
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
28
Layer 0
(air)
z=0
Layer 1
z=-d
Layer 2
(a) First Term
Layer 0
(air)
Layer 1
Layer 2
(b) Second Term
Figure 2.2: First-order intensity components.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
29
Layer 0
(air)
z=0
Layer 1
z=-d
Layer 2
(c) Third Term
Layer 0
(air)
Layer 1
Layer 2
(d) Fourth Term
Figure 2.2: First-order intensity components.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
30
where 9s0 is the scattering direction in the upper medium, i.e. the direction o f the receiver,
p is the receive polarization, and q is the transm it polarization. Note that if the plane
boundary assumption is used, zeroth-order can be ignored in non-specular directions since
Snell's Law must be obeyed and reflection will only occur in the specular direction, as
shown in Figure 2.3.
Layer 0
(air)
Layer 1
Layer 2
Figure 2.3: Zero-order for plane boundary.
2.5 Phase Matrices
In determ ining the phase functions, it is first noted that those terms o f the Stoke's
matrix involving only phase properties can be ignored since they are generally unimportant
in intensity calculations. Therefore, only Mn ,A/12,M 21, and M2 from the Stoke's matrix
provided by Ishimaru [49] and shown in (2.58) are considered.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
31
M =
|5vJ2
tSv/J2
I^ A v f2
\S*f
2 R e ( 5 ,w 5 * v )
2 Im(SwShv)
-Im(SvX h)
-lm(SkXkk)
Re (S.X *)
Re (ShXhh)
R e ( 5 va 5 a a ) / ? e ( S w» S * a + 5 v* 5 * v) - It n( SvvShh- SVhShV)
2Im(SvhShh) Im(SvvShh + SvhSh*) R e (S wS hh - 5 » / , S a » )
2
(2.58)
2 5 .1 M ie Phase Matrix
C onsider a time harmonic plane wave incident on a sphere along the z-axis, as
depicted in Figure 2.4. The sphere has a relative perm ittivity o f e r = e' +je " and a
permeability equal to that o f the background medium. The phasor representations o f the
incident electric and magnetic field components are [47]
E ‘ = a xE0eib
(2.59a)
H ' = a — e ik*
y V
(2.59b)
where k is the wavenumber and T) represents the intrinsic impedance. The scattered fields
due to the sphere are [47]
E s = E sg + E ;
(2.60a)
H ‘ = H ‘e + H ;
(2.60b)
where
6
Eoco s0 y
kr
, (2n + l)J ^ . a)
d pl
n(n + 1)1 n n K ’ d 6 nK
'
jb nH " \ k r ) P >
J " n v ' " sin 0 j
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
32
=
f o S
i n ^
£
kr
j / ' _ £ 0 sin <p y
■nkr ~
,
( 2 «
+
l l j
a f fm y a r ) j L
/>>
+
dO " sin 0
n(n + l ) l
+1 (2w + l)
n(« + l)
jb HH ^ ( k r ) — P\(ca&&)
" " v J dO "v
'
(cos 0) + ja nH ™ {kr)-^-P\
dd
sin 6 J
rf p l (cos 0)
d 0 “ sin 0
jL .p i /
'r f0
(2.61)
P \(co s0) represents the Associated Legendre polynomials, H ^ \k r ) and H } '\k r ) are the
Ricatti-Hankel function of the first kind and its derivative, and an and bn are the Mie
coefficients given by [46]
a. =
(2.62)
X
Figure 2.4: Scattering geometry of a single sphere [32].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
33
where J„(u) and J H{u) are the Ricatti-Bessel function and its derivative, u = ka , and
v = k a ^ e l , with a being the radius o f the sphere.
The results described above are valid when the incident direction is along the z-axis
as shown in Figure 2.4. However, in order to construct a phase matrix, a coordinate
transformation is required to convert (2.61) to permit arbitrary incident angles for both
vertical and horizontal polarizations [32], This transformation converts the scattered field
components in xyz-system described above into a primed system for arbitrary incidence as
shown in Figure 2.5 and explained in detail in [51].
Z
E‘
Figure 2.5: Scattering geometry demonstrating conversion of unprimed
coordinate system to primed coordinate system [32].
The phase matrix associated with the first two Stokes parameters is configured from
the scattered fields as
(2.63)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
34
where «o is the scatterer num ber density,
ks
is the effective volume scattering coefficient,
, ,
Ivolume o f one sphere J / l 710? ri-~n
and the range r = 3 ----------- --------= 3-4^----- [52].
\
volume fraction
y vf
(2.63)
is a phase m atrix for single-scattering calculations and can be used to
approxim ate the m ultiple scattering phase matrix for an infinitesimally thin layer in the
matrix-doubling method [4].
2 5 .2 Rayleigh Phase Matrix
If the size o f the scattering particles is small compared to wavelength, the Rayleigh
phase m atrix m ay be used instead o f the more complex Mie phase matrix. Ignoring the
third and fourth Stoke's parameters, the Rayleigh phase matrix [15] is listed below in terms
o f its Fourier components (which terminate at two).
p o(
^
w
2 sin2 8S sin2 0 + cos2 6S cos2 0
cos2 6S
cos2 0
1
0s - 0 ) = -
3 sin 0 r sin 0 4cos 6S cos 6cos(<f>s -<j>)
p \{ ^ ^ 0 s ~ 0 ) =
0
cos2 8Scos2 6cos2(<f)J - <j>)
- c o s 2 9cos2(<ps - <p)
0
0
- c o s 2 6S co s2 (0 t - <j>)
cos
(2.64)
(2.65)
( 2 . 66 )
The polarization com ponents can be found by sum m ing up the corresponding
Fourier components as shown below:
- 0) =
- 0)
= 0.75[2sin2 6Ssin2 8 + cos2 8Scos2 0]
+ 0.75[cos2 6Scos2 0 co s2 (^ - 0)]
(2.67)
+ 3[sin 6Ssin 8 cos 8Scos8cos(0s - 0)]
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
35
-< !> ) =
-
<t>)
= 0.75[2sinz 9Ssin2 8 + cos2 9S cos2 0]
( 2 . 68 )
+ 0.75[cos2 9Scos2 9cos2(<ps - 0)]
- 3[sin&3sin 0 c o s 9s cos 0cos(0, - 0)]
Pu =
-<P)
= P J t H .’M
m-
= ~ 0)
<t>) = P j j l , - l l , 4 , - <p)
(2.69)
= 0.75[l + c o s 2 (& -</>)]
Ph. =
-<!>) = P k . { - V . - M , - <t>)
=
-<P) = PhM s ~H,<Ps ~ <t>)
(2.70)
= 0.75cos2 0[l - cos2(<ps - 0)]
P * = P j j l , * M , ~<t>) = P d - V s - H ^ s ~ <t>)
= p J t ^ M , -<P) = p J p . - M , ~ <t>)
(2.71)
= 0.75cos2 0,[l - cos 2(0, - $)]
For the special case o f backscattering ( $ , - $ = n, 6S = 9) these equations reduce to
P vvl = 1.5(l - 4 sin 2 0 co s2 0)
/*w2= 1.5
^ m =1-5
P /IV
=
P*h =
0
(2.72)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
36
2.6
Surface Backscattering
In order to accurately model real-world dense m edia, the surface scattering
contributions m ust be considered. Therefore, the basic IEM equations governing likepolarized backscattering from randomly rough surfaces are presented.
W hen comparing field measurements with a surface scattering model, the correct
surface param eters (correlation length and rm s height) and correlation function are
necessary. The calculation o f the backscattering coefficient is accomplished according to
the following equation [16,32]:
^
= Y e x p ( - 2 ^ 2q 2) £ o 21 / ; f U/< -(~ f x’0)
Z
A= l
(2.73)
nl
where kt = k cos 6, kx = k sin 8 , ^ " ^ ( - 2 ^ ,0 ) is the Fourier transform o f the «th power of
the surface correlation coefficient, p (£ ,C ), and
c
= ( 2 o x e x p K V ) ^ - )' [F - K
;0 ) t F ” ( t - 0)]
2 r,,
f » = — UZ
c o sy
(2.74)
(2-75)
(2.76)
cos 9
2
F „ ( - k x, 0) + Fvv{kx, 0) = 2Sin-- ( 17 ° 1//)
1-1
COS&
(2.77)
j i r£r - sin2 6 - e r cos2 8
£? cos2 8
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
37
1 -COS0
Mr
(2.78)
fj.rer - sin2 6 - n r cos2 6
+
f i 2r cos2 6
Note that the local angle o f incidence is approximated with the angle o f incidence. Also,
the K irchhoff coefficients, f n and /**, should be evaluated at norm al incidence for
surfaces satisfying the tangent plane approximation where kL is large. Although the
transition point from the local angle o f incidence to normal incidence has not been defined
to date, it is expected that, for a Gaussian correlated surface, the Kirchhoff terms should be
evaluated at normal incidence when kL is greater than four [16,32].
(2.77)
and (2.78) can be simplified by letting fj.r = 1 yielding
.
.
.
.
2sin2 0(l + r,,y
^ K . 0 ) + F „ (kx, 0 ) = -------- i - J d
cos u
2
i - l
(2.79)
£ , - s i n 2 0 - e ,.c o s 2 fl
£2 cos2 6
F .K .
The general expression for
COS
<2 -8°)
&
is
W ^ ( U ,V ) = j - ] e - m + vi)p ’'^ ,C ) d ^ d C
(n = 1,2,...)
(2.81)
where U = ka - k x = jfc(sin0,cos(j>s - s i n 0) and V = k ^ - k y = £ ( s in 0 js in ^ J . (2.81) can
be written as the Fourier-Bessel transform
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
38
W w (K ) = \ v p ,'{ v ) J Q{ v K )d v
0
where K = sJU2 + V 2 and v =
(n = 1,2,...)
(2.82)
+ £2 .
Two o f the more commonly encountered correlation coefficients are the Gaussian
and exponential correlations, namely p G( £ ,Q = e~(v/^
and P £ (£>C) = e~(v/L). Hence, for
the Gaussian case,
Ww ( / 0 = Jv < f',(v/i)J J 0{vK )d v
0
{n = 1,2,...)
(2.83)
which, using a table o f integrals [53], can be written
W [n\ K ) = — eHLKfl*n
2n
(n = 1,2,...)
(2.84)
Note that in the backscatter case, U = - 2 k x = - 2 k sin 6 and V = 0 resulting in the
backscatter solution of
W {n)(- 2 k I ,0) = ^ e - {-2iLsine)tl*''
2n
(n = 1,2,...)
(2.85)
If the correlation coefficient is exponential, then
W ^ { K ) = J ve-K[vlL) J 0(v K )d v
o
(n = 1,2,...)
(2.86)
which is the same as
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
39
Un
W {K)( K ) =
(2.87)
[n2 + ( K L ) 2]
implying a backscatter solution of
iyM (—2 ^ , 0 ) = —
[n + (-2 k L sin 0 ) J
(« = 1,2,...)
(2.88)
The scattering coefficient from the top surface can be calculated directly from (2.73),
i.e. <Jjop = <ysurface
, however the contribution from the bottom surface needs to be
( Upper Boundary)
modified as described in the next section.
2.7
Bottom Surface Contribution
The backscattering from the bottom surface follows the same format as that o f the
top surface, with a few modifications. The wavenumber incident on the lower boundary is
now the w avenum ber o f the inhomogeneous medium, and the angle o f incidence on the
bottom layer needs to be calculated using the angle of transmission and Snell's Law. Since
the wave had to travel from medium 0 through medium 1, we must multiply the backscatter
by the transmissivity o f the upper medium and the attenuation through the layer, e~K“d, and
divide by the m ultiple scattering effect between the boundaries, 1 - RuRde~2K“d. Since the
measurements will come in the upper medium, the backscattering must again be multiplied
by the loss and transmissivity at the upper medium, resulting in a final expression for the
bottom surface contribution (attenuated by the layer) of [32]
T 10ee~2K- d{T°
1 01J
u Surface
(LowtrBoundary)
T
„0
^
_
l - R uRde '2' “d
00\
•
(~ 89)
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
40
Finally, the total backscattering is found by summing the contributions of the top surface,
volume, and bottom surface, namely
+< i +< ,
(2.90)
which can be expressed in decibel form by taking ten times the logarithm of this value.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
CHAPTER THREE
TARGET FABRICATION
3.1
Statistically Defined Volume Targets
3.1.1 Choice o f Materials
In order to study the effects of scattering amongst a large num ber of scatterers, the
background m edium would ideally be air. Since it is not possible to suspend a large
num ber o f scatterers in air at defined locations, it is necessary to choose a background
m edium whose dielectric constant is low enough such that it is essentially transparent at
m icrowave frequencies yet dense enough to suspend the scatterers. In the past, similar
experim ents were performed using one-pound-per-cubic-foot polystyrene [9,37,38,54],
Using the H P 85070 dielectric probe, the dielectric constant o f this material was found to
be very close to that o f air, or more precisely e = 1.05 with a negligible loss tangent. The
second criteria for the background medium is that it must be rigid enough to withstand the
drilling process described at the end of this chapter. The one-pound density polystyrene
failed to meet this test, so it was necessary to increase the density of the polystyrene to two
pounds per-cubic-foot. Although the dielectric constant is slightly larger, e = 1.08 as
measured with the dielectric probe, it is still essentially transparent. Drilling into the twopound density polystyrene produced a well-defined hole, so this material was chosen as the
background medium.
The size o f the target is determined by the antenna parameters o f the measurement
system. At 18 GHz, the maximum frequency o f the available system, the beamwidth of the
antennas is roughly ten degrees, and at 2 GHz, the minimum frequency o f the system, the
41
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
42
beamwidth is sixty degrees. In order to avoid edge effects, it is desired that the target
should fully encompass the beam. However, this implies that the target would have to be
at least 4.4 m in order to satisfy the beam constraints at 2 GHz. This size is far too large to
fabricate using the robots available, as well as being too heavy to manipulate in the WSRC
anechoic cham ber. Since the background medium is o f such low dielectric value, it is
assumed that the edge effects will be minimal as long as the target faces lie at nearly normal
incidence to the beam. Hence, the targets were built such that the target would lie entirely
within the beam over the available frequency range. At 18 GHz, the beamwidth implies a
maximum target radius of 34 cm. The targets were therefore chosen to be 11 inches, or
27.94 cm, in radius.
The validity o f some previous works has been questioned due to the fact that the
scatterers used were not spheres although they were modeled as spheres [9], In order to
avoid this problem from the onset and to study deviations in behavior from existing Mie
theories, the first specification is that the scatterers be smooth spheres. Secondly, the
refractive index needs to be much different from the background medium in order to create
a non-tenuous m edium . Third, the size o f the spheres should be such that both the
Rayleigh and M ie regions will be covered for the available frequency band. Finally, the
scatterers need to be in ample supply. Clear, 9/16" diam eter lead-glass marbles, m ade by
Marble King o f Paden City, West Virginia, were chosen since they satisfy both the shape
and size constraints (ka = 0.29 to 2.65 for the available m easurem ent system). The
dielectric constant o f the m arbles was determ ined to be £ = 6 . 9 - yO.10 by using the
dielectric probe on a flat piece of glass made from the same material as the marbles.
3.1.2 Generation o f Data fo r Target Fabrication
The first step in generating data is to choose the physical parameters o f the target.
These include the diam eter o f the scatterer, diam eter o f the target, volume fraction, and
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
43
height o f the target. The diam eter of the selected scatterer has already been stated to be
9/16", or m ore exact, 1.4072 cm diameter as measured with a caliper. The overall target
diameter, as previously specified, is 55.88 cm. The height o f the target is chosen such that
even for sm all volum e fractions the signal hitting the back o f the target will be weak. A
thickness o f 25.4 cm was found to be acceptable. The volume fractions selected were 5, 8,
11, 14, and 20%.
In order to obtain a uniform distribution o f scatterers throughout a cylindrical
volum e o f 27.94 cm radius, a M onte Carlo simulation was performed to create a uniform
distribution o f scatterers in the x, y, and z directions throughout a box o f dimensions 55.88
cm by 55.88 cm by 25.4 cm. The desired cylindrical volume was then cut from the center
o f this box to obtain the original data for the dense medium simulation. Several trials are
run where thousands o f numbers are input as the starting point (iseed value) for the random
num ber generator, and those values which yield a chi-square goodness-of-fit in the zdirection greater than 99% are output to a file using the program c h isq ra .f given in
Appendix A. The iseed value which yields the best result from the goodness-of-fit test is
then input to a second program, where the x-y locations are changed and various other
goodness-of-fit tests as well as a pair distribution check are performed. If any one o f the
tests fails, another iseed value is chosen from the c h isq ra .f output file, until a value
passing all tests is found. These tests will be discussed in further detail in Section 3.1.3.
A fter choosing a starting point for the random number generator, this value is input
into the main data generation program, targfab.f, given in Appendix B. The first step of
the program is to generate a uniform volume o f scatterers throughout a box o f specified
dim ensions and cut a cylinder from the center of this box. Figure 3.1 shows how this is
done and demonstrates the coordinate system used.
It is apparent that generation o f this original data is very easy.
However,
fabrication o f the physical target is not so simple [55,56]. To this point, the x,y, and z
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
44
Z
► X
Y
Figure 3.1: Coordinate system used in data generation.
coordinates have been generated for a given number o f points. These coordinates represent
the centers o f the scatterers. Since the scatterers to be used are solid, there can not be any
two center point locations which lie within a scatterer diameter of each other.
A second problem can occur as a result of the z-directed holes drilled by the robot.
If two particles have the same or nearly the same x-y coordinates but different z locations,
no matter what the z-values o f the two scatterers are, they will be placed one on top o f the
other since they are inserted into the same hole. This effect is reduced by building the
target in layers. The thinner the layers, the less frequently this problem will occur.
However, building the target in layers, where only one layer is drilled at a time, results in a
reduction o f control in the z-axis positioning since there will be gaps where two layers meet
and no scatterers are centered within ± the radius o f a scatterer from this location. This
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
45
problem can be overcome by drilling through one layer into the next for particles that lie in
the intermediate region.
Since the target is built in z-directed layers, the data is sorted from lowest to highest
z. The data will then be divided into equally thick layers, where some data will overlap
between two adjacent layers. As stated previously, it is desired to make these layers as thin
as possible and still be able to hold a scatterer within its boundaries. Hence, the layers
should be slightly m ore than a scatterer diam eter thick. Since the scatterers chosen are
9/16" in diameter the layers are chosen to be 10/16" thick.
Starting with the bottom layer, each layer is now checked for center locations which
are not more than one diameter from the next closest center point. Actually, it was decided
to make the separation distance between all particles slightly more than the diameter o f the
scatterer, as well as making it a function o f the average distance between particles. This
elim inates the situation where particles cluster together and form one large effective
scatterer. The minimum spacing is defined as
min. spacing = dKal + ^{a v g . d is t.-d scat)
(3.1)
where dscal is the diameter of a scatterer and the average distance between particles is given
by V v o r u m e ™ U o fIh eT ig et [52]' A11 scatterer coordinates within a Siven layer
compared with the other data points within that layer as well as those points which lie
within a scatterer radius of the top and bottom layer boundaries. If the distance between the
center points in the x -y plane is less than stated by (3.1), new x-y coordinates are chosen
and again compared with the rest o f the data until all scatterer positioning satisfies (3.1).
Figure 3.2 demonstrates the need for the overlapping algorithm. Consider a 10%
volume fraction target fabricated as described above with two particles centered at exactly
the same x -y coordinates but at z-values separated by about 2.5 cm.
Each layer in
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
46
Figure 3.2 represents 1.6 cm. For these two scatterers the minimum spacing o f 1.66 cm is
clearly satisfied. However, the fabrication process requires drilling through layer 2 into
layer 1 to position scatterer A, and then drilling through layer 3 into layer 2 to position
scatterer B. Hence, the two scatterers are placed one on top of the other since there is no
foam between the two scatterers to prevent the scatterer B from being pushed onto scatterer
A. Therefore, an overlapping algorithm is required to prevent this from occurring. Note
that the dark regions of Figure 3.2 represent the volume that has been removed by the drill.
3
■
1A 1
1
Figure 3.2: Overlapping holes [7].
O ne problem that occurs when the overlapping algorithm searches for valid
locations for a scatterer is that scatterers on the outer ring o f the target are not surrounded
by other scatterers and hence there is a tendency o f the algorithm to place an excessive
number o f scatterers in this region. It is therefore necessary to define a maximum number
o f scatterers allowed to lie within this outer ring. If more than this number occur, the
program randomly selects scatterers within this outer ring and sends those x-y coordinates
into a second overlap routine. This routine will find new x-y coordinates which do not lie
in the outer ring and don't overlap any other scatterers. In the end, the number of scatterers
in the outer ring will be less than or equal to the number specified at the beginning o f the
program.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
47
3.1.3 Uniformity Testing
The scatterer locations are now all defined. All that rem ains is to determ ine the
statistical uniform ity of the locations. This is checked by perform ing three chi-square
goodness-of-fit tests as well as a two-particle correlation check. The chi-square tests are
performed by dividing the target into K equal-volume intervals and counting the number of
scatterers falling within each o f the ith class intervals. The observed number o f scatterers is
denoted by f„ where i is the ith interval, and the number o f scatterers expected to fall within
the ith interval is called the expected frequency and is denoted by Fi. In order to assess the
discrepancy for all class intervals, the squares o f the discrepancies in each interval, ft - F„
are norm alized by the associated expected frequencies and summed [57], The resulting
sample statistic is thus given by
*=i
(3.2)
For example, the target was divided into 16 equally long sections in the z-direction,
and it was noted how many scatterers, fi, were within each section. Since the desired
distribution is uniform, there should be the same number of scatterers within each section.
Hence, F\ = F 2 = ... = Fm = l-otaf ^ sf scatterers
16
number Qf degrees o f freedom , n,
is given by K - 1. H ence, by consulting a chart (such as the one given in Appendix C)
2
containing the critical values o f X and using 15 degrees o f freedom and the value obtained
from (3.2), the percentage o f acceptance can be determined. A view o f the sixteen layers
involved in a z-cut test is shown in Figure 3.3.
In addition to the z-directed test, two other chi-square goodness-of-fit tests were
performed for each target. In one case, sixteen equally spaced wedges were cut throughout
the volume. A three-dimensional view of the wedge cut method is shown in Figure 3.4.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
Figure 3.3: Z-cuts for chi-square goodness-of-fit test.
Figure 3.4: Wedge-cuts for chi-square goodness-of-fit test.
The final chi-square test involves dividing the target into sixteen equal-volume
cylindrical rings. This is done by first determining the necessary radius of the innermost
ring in order to achieve one-sixteenth the volume of the entire target, or
V t o ta l
=
K r fiA g
1 h ta r g e t
(3 3)
which can be rewritten as
r
.
n"g
/ V to ta l
V
16
7T
h t a r g 't
(3
4 )
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
49
Since cylinderl with a radius as defined above composes a volume one-sixteenth the entire
target, it is now necessary to determine what size radius cylinder/ should be so that the
volume o f cylinder/ minus the volume o f cylinder//-1) would be one-sixteenth the size of
the target. So the equation to determine the necessary radii o f the rings is
_
/V
r~ivt
total
""•‘"V
16 n ,ari‘t
(3.5)
Once the sizes o f the rings are determined, the number o f scatterers lying within ring 1 are
counted, as are those which lie in subsequent rings. Figure 3.5 shows a two-dimensional
view of three ring cuts. Notice that although the rings get thinner the further away from the
center they are, the volume contained by each ring is the same.
Figure 3.5: Ring-cuts for chi-square goodness-of-fit test.
Note that three different chi-square tests are performed because it is desired that the
target be uniform in every view possible. For example, it is possible for the scatterer
locations to pass the uniformity tests for both the z- and wedge-cuts but appear Gaussian
according to the ring-cuts. This would occur if cylindrical coordinates were used to
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
50
generate a uniform distribution. A view o f this target would show a large concentration of
scatterers at the center.
Table 3.1 gives a listing of the chi-square values for each target. Comparing these
values with those listed in Appendix C, it can be noticed that even the worst case fit, i.e.
the ring cut for the 5% volume fraction target, a 99.2% fit to a uniform distribution has
been achieved.
Table 3.1: Chi-square values.
Volume Fraction
Z Cuts
Wedge Cuts
Ring Cuts
5%
3.422
4.118
4.978
8%
3.892
2.743
3.743
11 %
3.607
4.017
3.273
14%
3.524
1.616
2.642
20%
2.717
2.515
4.281
A final uniformity test is performed by checking the pair distribution function. This
is done by random ly selecting numerous reference scatterer locations and counting the
num ber o f occurrences o f scatterers different distances from the reference points. That is,
once a scatterer location is chosen as the reference point, the num ber of times other
scatterers occur within a shell radius of two diameters, between two and three diameters,
between three and four diameters, etc., is counted. Note that since a finite size target is
used, the boundaries must also be considered. Hence, the distance from the reference point
to the closest edge is the limit o f testing for that particular reference location. Then all the
shells are tested and the num ber o f occurrences in each shell is noted. This process is
perform ed using various scatterers as the reference point. In the end, the num ber o f
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
51
scatterers found in the z'th shell is divided by the num ber o f times a sphere large enough to
contain this shell size was used. This gives the average num ber o f scatterers in the z'th
shell, which is then divided by the volume of the ith shell to obtain the num ber density
within that shell. This value is then divided by the num ber density o f the target to obtain
the pair distribution function. Note that as the shell size gets larger, the accuracy will
decrease due to the boundary conditions. This is due to the fact that the number o f times
these large shells can be realized is small.
3.1.4 Fabrication Facilities
C lear lead-glass marbles purchased from Marble King and Berry Pink Industries of
Paden City, W est Virginia, w ere used as scatterers. The marbles are listed as 9/16",
however using a caliper to measure the diam eter shows a m ore exact value o f 1.4072 cm.
The polystyrene sheets, which were obtained locally, are slightly thicker than the diam eter
of a scatterer. Since the foam company's equipm ent is set in terms o f inches, each sheet
was specified to be 5/8" thick with the exception o f the top and bottom layers which were
1" thick.
Drilling holes into the polystyrene sheets was performed at The University of Texas
at Arlington's Autom ation and Robotics Research Institute with the aid o f an Adept One
M anipulator. This robot has the capability to cycle one inch up, twelve inches over, one
inch down, and back to the starting point in under one second with a one-pound load [58],
It has a m obility in the x -y plane o f 57 cm and a maximum vertical stroke o f 30 cm. The
input pressure to the Rodac M odel 132K pneum atic drill is 90 psi, im plying a rotation
speed o f 22,000 RPM. In order to ensure that the marbles will remain in their respective
holes, the holes are drilled slightly smaller than the marbles, so the marbles must be forced
into their proper positions. The drill therefore uses a 1/2" drill bit.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
52
There are three aspects to consider in the robot's m ovem ent and drilling: (1) the
speed to move in the air above the target from one x-y location to the next; (2) the speed the
drill enters the foam; and (3) the speed the drill withdrawals from the foam. In order to
limit the amount o f vibrations caused by the robot moving from one point to the next and
coming to a complete stop as well as limiting the wear on the robot, the robot will operate at
65% of its maximum capability. Note that since the target is built from lowest to largest yvalue and the target diam eter is only 55.88 cm, operating at this speed outside the foam
rather than at 100% causes very little change in the overall fabrication time of the target.
However, the same can not be said for the actual drilling portion o f the robot's movement.
W hen approaching a point to be drilled, it is important to approach at a slow enough speed
so the drill bit will actually drill into the foam, making a smooth hole and not destroying the
holes closest to it. If the location is approached too quickly, the hole will not actually be
drilled but pushed in. For the material used and the operating speed o f the drill, it was
necessary to operate at 3% of the maximum speed capability of the robot. This is by far the
m ost time consum ing portion o f the drilling process. In order to reduce this time, the
VAL2 program for controlling the robot's actions is written such that the drill will move in
the x-y plane just barely above the target, thereby limiting the amount o f distance the drill
m ust travel in the z-direction. The speed o f departure from the hole can once again be
65%. Considering these criteria, the robot drills holes at a rate o f about 22 holes/minute
with a precision of ±0.13 mm.
The table used to support the foam sheets during the drilling process is shown in
Figure 3.6. Brackets one and two are fixed to ensure that all sheets are aligned to the same
points. The third bracket is attached to the table with wing nuts so once the sheets are in
place, the bracket can be tightened to restrict m ovement in the x-y plane. Two braces lie
over the edge o f the target to prevent the sheets from lifting during the drill's withdrawal.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission .
53
These braces are also attached with wing nuts so the pressure on the foam sheets can be
increased or decreased as needed.
Due to the constraints o f the robot in the ^-direction, it is not possible to drill the
entire layer in one sweep. For this reason, it is necessary to drill each layer in two halves.
The first step is to stack the foam sheets one on top o f the other and draw a thin vertical line
from the top sheet to the bottom sheet. The table itself has a line drawn on it through which
the center point of the cylinders will lie. The line drawn on the sheet is aligned with the line
on the table, the first half of the layer is drilled, the sheet is rotated 180 degrees by again
aligning the two lines, and the second half is drilled. Note that by drilling this way, brace
two can be and was placed closer to the center of the target than shown in Figure 3.6.
ROBOT
Brace 1
polystyrene sheet
Bracket 1
Bracket 3
drilling area
Bracket 2
Brace 2
Figure 3.6: Target fabrication setup.
Before beginning the drilling process, it is necessary to calibrate the robot to the
coordinate system defined in the data generation program. Once this is done and the foam
sheets are marked as described in the previous paragraph, the actual drilling process can be
started. Layer 1 is aligned with the markings on the table, and any hole location which will
contain a scatterer that completely lies in this layer is drilled. Once the first half of this layer
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
54
is drilled, the target is rotated 180 degrees and the second half is drilled. The first layer is
again rotated 180 degrees so it will be back at its original position, and the second layer is
placed on top by aligning the marks on the two layers. Then holes are drilled for all
scatterers which lie simultaneously in layers one and two and for those which lie entirely
within the second layer. Once this is accomplished, the first layer is removed, the second
layer is placed on the bottom, and the third layer is placed on top. This procedure is
continually repeated until all layers have been drilled. The drilling process is demonstrated
by Figure 3.7.
Figure 3.7: Robotic drilling procedure.
Once all layers have been built, the marbles are inserted by hand by pushing the
marbles into the holes until the bottom o f the hole is reached. Note that this method of
manual insertion can be replaced by program ming the robot to perform the scatterer
insertion. Once all the holes for a given layer are drilled, the drill end-effector can be
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
55
replaced with a second end-effector containing a strong tube which is slightly thinner than
the diam eter o f a sphere. The robotic arm can then move to a receptacle containing a
collection o f scatterers and, using suction, pick up a scatterer. Then, using the same
coordinates used during the drilling process, the arm goes to the location o f a previously
drilled hole and forces the scatterer to the bottom of the hole. Once the scatterer reaches the
bottom o f the hole, the robot will shut the air off, thus leaving the scatterer in its pre­
defined location. Next, the robot withdraws, goes back to the receptacle, picks up another
scatterer and repeats the process until all holes are filled. Further simplifications can be
made by using a hopper to feed the scatterers directly to the suction device and to use a ball
plunger to place the scatterer in its proper location.
5.7.5
Sample Target
T o further understand the fabrication technique, a sample run for a 10% volume
fraction target is shown in depth. The first step is to determ ine the number to be input to
the random num ber generator. The FORTRAN program c h isq ra f is run to give the values
which produce good uniformity in the z-direction. The results of this program indicate that
an iseed value o f 456 produces an excellent chi-square value for the z-direction. This will
be the first number tested in the main program.
It is noticed that if an iseed value o f 456 is used, the program will generate 4252
scatterer locations throughout the defined cylinder. As explained in Section 3.1.2, it is
necessary to set a limit on the number o f scatterers which may lie in the outer ring. Since
sixteen rings are used, there is an average o f 4252/16 = 265.75 scatterers per ring. Hence,
266 will be input as the maxim um num ber o f scatterers in the outerm ost ring. So the
values input to targfab.f are:
volume fraction:
10
iseed:
456
maximum # of
scatterers in outer ring
266
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
56
If any one o f the uniformity tests is failed, a new num ber is tried for the iseed value until
one passing all tests is found.
The results o f chi-square tests are as follows:
z-cuts:
2.762
wedge cuts:
1.600
ring cuts:
4.068
Since each o f these values is lower than the 4.6 necessary to obtain a 99.5% agreement, the
target can be said to be over 99.5% uniform for each test region. Figure 3.8 shows a
histogram o f the num ber o f scatterers found within each cut.
399
2 6 6 - 56565659565656565659625656
e3
133 -
z
1
2
3
4
5
6
7
8
9 10111213141516
Cut N um ber
Figure 3.8: Z, wedge, and ring histogram.
The results o f the pair distribution function also show excellent agreement, with
Figure 3.9 dem onstrating the results. This plot strongly resem bles the Percus-Yevick
approxim ation used by numerous authors [2,10,34,59]. Note that the x-axis o f the plot
represents the average o f the distance range. For example, a value o f 2.5 im plies those
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
57
scatterers whose distance from the reference scatterer is anywhere between two and three
scatterer diameters.
3.0
2.5
2.0
u
01
0. 5
0.0
1.5
2.0
2.5
3.0
3.5
4.0 4.5
d istance/sphere
5.0
5.5
6.0
6.5
d iam eter
Figure 3.9: Pair distribution function [7].
In order to give the reader a visual image o f what a section of the fabricated target
looks like, Figure 3.10 shows a two-dimensional view of the x-y locations of scatterers
located in layers 4 and 5, that is from z=6.35 cm to 8.0 cm.
oo o
Oo'
o % o ° ou
D
oo
Oo
o°o«
Oo® ooo
o
oo
° —OO 0 °O
oo°0 oooo
OO
,oo
Figure 3.10: X -Y view of 10% target for z = 6.35 cm to 8.0 cm.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
58
3.1.6 Fabricated Target Parameters
The iseed values input into targfab.f for each o f the fabricated targets are shown in
Table 3.2 along with the number o f scatterers used for each target, the average spacing
between particles, and the m inim um spacing between particles. The machine used to
generate the random numbers was a Convex 220.
Table 3.2: Parameters used in target fabrication.
Volume
Fraction
Iseed
Value
# of
Scatterers
Average Distance
Between Particles
Minimum Spacing
Between Particles
5%
1680
2168
3.079 cm
1.825 cm
8%
67
3425
2.632 cm
1.713 cm
11 %
774
4691
2.367 cm
1.647 cm
14%
326
5961
2.184 cm
1.601 cm
20%
3341
8551
1.939 cm
1.540 cm
Note that these iseed values actually represent the number of times three coordinates
were chosen for each point in the rectangular box. Since it takes approximately 5436
scatterers to obtain a 10% volume fraction within the defined box, an iseed value of 456
actually im plies that there are approxim ately 3 x 5 4 3 6 x 4 5 6 = 7.5 m illion num bers
generated before the first number involving the iseed of 456 is generated.
3.2
Statistically Defined Surface Target
In the past, statistically known highly conductive rough surface targets have been
fabricated for the purpose o f comparing surface scattering with predictions made by the
IEM surface scattering model [14], Results from these studies indicate excellent agreement
at various transmit/receive angle and polarization combinations. Using the same technique
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
59
o f target fabrication, a statistically known rough surface target can be created for use as a
mold to generate a layer o f ice with the same surface roughness characteristics. This target
can then be used to study the effects of both surface and volum e scattering since the top
interface is not highly conductive.
The construction o f the mold involved the application o f a digital filter to a set o f
independent random variables for the purpose o f producing the desired statistical surface
roughness [60]. This data was then transformed into instructions for a computer controlled
milling machine which milled the desired surface contours onto a large block o f 16 poundsper-cubic-foot density polyurethane foam purchased from Kayco o f Grand Prairie, Texas.
The milling was performed at Special Products Manufacturing in Fort Worth, Texas.
The mold was selected to have a single scale Gaussian surface with a standard
height deviation o f 0.25 cm and a correlation length o f 2 cm. The target diam eter was
chosen to be six feet to ensure that the beam of the antenna used lies fully within the target
at all incidence angles. Once the foam was milled to the desired surface roughness, it was
painted with several coats o f white latex prim er to fill the small air holes present in the
foam. Next, the mold was painted with silver particle paint containing 56.3% silver with a
conductive polym er base, thus producing a highly conductive surface. This ensures that
future surface scattering studies using the mold as the target can easily be perform ed
w ithout m aking any additional changes. These m easurem ents can be helpful during
calibration in future saline-ice studies by including the effects of variation in amplitude over
antenna aperture.
The desired target is a thick layer o f saline-ice with the above described surface
geometry. This was obtained by growing an ice sheet on top o f the foam mold. In order to
avoid damaging the mold, it was covered with an epoxy paint and a thin layer o f petroleum
jelly. The m old was then placed in a circular six foot tank with water poured on top and
allowed to freeze for approximately one week. Meanwhile, a deeper tank was filled with
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
60
saline water and allowed to freeze. These ice layers were prepared at the US Army Cold
Regions R esearch and Engineering Laboratory (CRREL). After one week, the tank
containing the mold was moved to a 60°F room and allowed to thaw for approximately six
hours. This allowed the ice layer to be removed without damaging its newly formed rough
surface. This ice sheet was then placed outdoors and allowed to completely freeze over
once again. The overall thickness o f this sheet was approximately three to four inches.
W hen the measurements were performed, this rough ice sheet was placed on top of the flat
surface o f the saline-ice which was forming at the same time in the deeper tank. Although a
slight discontinuity occurred at the boundary between the saline-ice layer and the rough ice
sheet, the effects on the measurements were negligible. Measurements were performed on
the flat ice target and then on the combined target in order to determine the effect of volume
scattering. Figure 3.11 shows the target before and after the rough surface boundary was
added.
Figure 3 .1 1(a): Saline-ice target with smooth surface.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
61
Figure 3.11(b): Saline-ice target with rough surface added.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
CHAPTER FOUR
MEASUREMENT SYSTEMS
AND CALIBRATION
4.1
Measurement Systems
4.1.1 Background
W hen making radar cross section (RCS) measurements, the goal is to characterize
the scattering from a particular target. In order to achieve accurate results, it is necessary to
rem ove, as much as possible, the energy scattered by the targ e t’s surrounding
environment. By using an anechoic chamber, it is possible to greatly reduce the amount of
unwanted reflections by absorbing energy incident on the walls, floor, and ceiling o f the
chamber. However, no anechoic chamber is truly “anechoic,” so there will always be
some unwanted energy reflected. These residual reflections in the chamber, along with
leakage between transmit and receive antennas, are referred to as isolation errors [61]. It is
also likely that there w ill be some error in tracking between the test and reference
measurement channels o f the network analyzer. This error, referred to as response error,
results in errors in the m agnitude and phase o f the measured signals. However, both
isolation and response errors are systematic in nature, i.e., they occur in the test set-up and
measurement equipment during calibration and target measurements, and can therefore be
measured and their effects largely removed using calibration techniques which will be
discussed later. Random errors, on the other hand, are variations that occur in the test
setup between calibration and the actual m easurem ent o f the target. They can not be
62
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
63
accurately m easured and therefore can not be removed. This section briefly describes the
measurement equipment and the calibration procedures used.
4.1.2 Anechoic Chamber
The W ave Scattering Research Center's anechoic chamber has floor dimensions of
42 feet by 27 feet [62]. The bistatic receive antennas are mounted on a 26 foot diameter,
one-quarter section o f a geodesic dome. A one-quarter dom e section was used so as to
position all o f the bistatic receive antennas at a uniform distance, 13 feet in this case, from
the target being measured. The chamber houses 27 receive antennas and three transmit
antennas, which translates into 3 x 27 = 81 possible transmit/receive bistatic antenna pairs.
Each pair is capable o f acquiring measurements o f the four different linear polarization
states: VV, VH, HV, and HH, where the letter V represents vertical polarization, and an H
stands for horizontal polarization.
Note that the first letter represents the received
polarization, and the second represents the transmitted polarization.
The chamber has three monostatic transmit/receive antenna combinations, as shown
in Figure 4.1.
W ith the target support pedestal in the horizontal position shown,
Transmitter 1 yields an incidence angle of 90°, Transm itter 2 is at 45°, and Transm itter 3
is at 0° incidence angle with the target. Note that any elevation angle from 0° to 90° can
be attained by tilting the pedestal. However, because tilting the target pedestal too much
may cause difficulties in keeping the target in place, the pedestal is usually only tilted a
maximum o f about 22°. For instance, Transmitter 3 is used to cover incidence angles from
0° to 22°, Transm itter 2 from 23° to 67°, and Transm itter 1 from 68° to 90°. These
changes in the target's elevation, as well as its azimuth position, are controlled through the
use o f the Scientific Atlanta Positioner Control Unit.
Figure 4.2 shows the bistatic receive antenna configuration. The longitudinal (0)
and latitudinal (0 ) lines are measured with reference to the normal o f the target mount.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
64
Transm itter 3
T ransm itter 2
Bistatic
Receivers
T ransm itter 1
Absorber
Figure 4.1: Transm itter configuration.
0°
9 = 25'
0 = 40'
0 = 70'
0 = 85
<t>= 150'
<)>= 110'
<t>= 70'
<t>= 30'
<J>= 10'
<i) = o '
Figure 4.2: Bistatic receive antenna confguration [62],
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
65
Bistatic receive antenna #1 is directly above the pedestal. It should also be noticed that
antennas are more highly concentrated in the 0 = 0° to 30° longitudinal region, and there is
an additional latitudinal line at 6 = 85°. The antennas in this region are used for forward
scattering measurements.
The antennas used in the chamber are quadridge dual-polarization conical horn
antennas, m odel num ber A6100, made by the Dalm o V ector Division o f the Singer
Corporation. The horns have an aperture of 5.5 inches and the rated frequency response of
the antennas is 2 to 18 GHz. They are guaranteed to have a minimum o f 30 dB o f crosspolarization isolation across the entire rated frequency band. Since the cross-polarization
response o f a target is usually much weaker than the like-polarization response, this
isolation is very important when cross-polarization measurements are taken. That is, if the
antennas did not have good cross-polarization isolation, the dominant like-polarization
response would leak through and corrupt any cross-polarization measurements.
4.1.3 N etw ork A nalyzer System
An H P8510 netw ork analyzer system is at the heart o f RCS m easurem ent
acquisitions. O f particular interest for RCS measurements is the mathematical analysis o f
frequency swept transmission and reflection data [62], The HP8510 has the capability to
com pute the inverse Fourier transform o f the m easured data in near real time, so as to
display the time domain response o f the target, i.e., to indicate the position and magnitude
o f the reflections.
The specific components o f the HP8510 are as follows [61]:
1. HP8341A swept signal source: This RF source is controlled by the network analyzer
via the H P8510 system bus.
The use o f the system bus gives the control o f other
necessary source functions. It can operate in the Ramp Sweep Mode (the source sweeps in
a continuous linear ram p over the selected frequency range) using the “Lock and Roll”
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
66
tuning technique. In this technique, the first frequency o f the sweep is set with synthesizer
accuracy, and a linear analog sweep proceeds to the stop frequency. It can also operate in
the Step Sweep mode. In this mode, synthesizer-class frequency accuracy and repeatability
are obtained by phase-locking the source at each o f the up to 401 frequency steps over the
selected frequency range. This mode provides the highest accuracy and can lower the noise
floor by as much as 30 dB, but operates at a much slower speed.
2. HP 8511A frequency converter: O f the five available test sets in the HP851_ product
family, this is the best for a dedicated RCS system because it is the most flexible and
provides the greatest dynamic range. It is a general purpose, phase-locked receiver that
operates from 45 M Hz to 26.5 GHz and compares the transmitted and reflected signals in
amplitude and phase.
3. HP8510A network analyzer: This, the hub of the system, is composed of two parts:
the IF detector and the display/processor. Operation o f the entire system can be controlled
from the front panel o f the 8510A via the system bus.
The best configuration o f the HP8510 for RCS m easurem ents is as a general
purpose, four-channel, phase locked receiver displaying both the frequency and time
domain responses o f a test target [63], The HP8341A provides the source o f the radar
signal. Its output is amplified by an HP8349B amplifier, which has a small signal gain of
15 dB. This amplification is necessary to increase the transmitted signal such that the
signal-to-noise ratio o f the received signal is high enough for accurate detection and
processing [14]. The am plified signal is now split by a HP11692D directional coupler
between the transm it antenna and, via an 10 dB attenuator, the reference port a i of the
HP8511A. The attenuator is necessary to lower the signal power to a level below the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
67
maximum pow er restriction o f the reference port. The actual received signal coming off of
the receive antenna is fed back to the b i port of the HP8511A where it acts as the test
signal. The HP8510A is used to display Su , the ratio of the received and reference signal
(6 ,/a ,). This synthesized frequency modulated continuous wave (FM-CW) configuration
is shown in Figure 4.3.
Figure 4.4 demonstrates the path of the test and reference signals once they are
input to the HP 8511A test set. The test set down converts the high frequency signals to an
initial IF of 20 MHz. The signals are then sent to the HP 8510A where an IF detector (HP
85102) down converts the signals to an IF of 100 kHz and separates both signals into an
in-phase signal (I) and a 90° phase shifted signal (Q). This is done through the use o f two
synchronous detectors, one for the reference signal and one for the test signal. Figure 4.5
displays the operation o f the synchronous detectors. The detector output can be written as
[64]
o{t) = I + j Q
(4.1)
The I,Q pairs are sequentially converted to digital values and read by the Central Processing
Unit [61]. Next, digital techniques are used to correct for the pre-amplification o f the down
converted signals and other drift errors which may have occurred [14]. Finally, the test
and reference pairs are ratioed, averaged, and stored in the raw data array. This process is
repeated for each step in the discrete frequency sweep (51,101,201, or 401 points).
The frequency range of all the components is shown in Table 4.1. Notice that even
though the HP8510A is capable o f performing RCS measurements from 45 MHz to 26.5
GHz, some o f the other components, particularly the antennas, decrease this range to 2 to
18 GHz. In this range, the amplifier is rated for a minimum o f 15 dB gain.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
68
HP 8349B
Microwave Amplifier
HP 11692D
Directional Coupler
To Transmit
Six-Pole Switch
Receive Six-Pole
Switch
From the
Receivers
HP 8495B
Attenuator
E
>
HP 8511A
Frequency Converter
Return Path
Selector
mm
HP8510A
Display/Processor
O •s s si1
IffW in • • •••B
SB
HP 8510A
Network Analyzer
HP 9000
Series 300
Computer
HP 7475A
Plotter
Switch
Controller
Box
HP 8341A
Synthesized Sweeper
Pedestal
Controller
DC Power
Supply
Figure 4.3: H P 8510 Network A nalyzer System [9].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
69
Down
a ,0 -
Converter
Keterence
Synchronous
Detector
Ratio
and
Averaging
HP 8511A
Diown
t>! O—
Correction
Converter
Raw Data
Arrays
Test
Synchronous
Detector
Correction
Figure 4.4: HP 8510 IF detector.
IF Input
M ixer
Low-pass
Filter
► In-phase
Output
Reference
Oscillator
Mixer
Low-pass
Filter
Quadraphase
Output
Figure 4.5: Synchronous (IQ) detector [64].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
70
Another major component of the system is the HP9133 com puter which contains
forty megabytes of disk space and runs HP Technical Basic. This computer is connected to
the HP8510A via another HP-IB bus, more commonly known as an EEEE 488.1 interface
bus. Since the computer is capable of controlling the HP8510A, which in turn can control
the HP8511A and HP8341A, programs written on the com puter can control the entire
system. It can be seen from Figure 4.3 that the HP-IB bus connecting the computer to the
HP8510A operates independently o f the HP-IB bus connecting the HP8510A to the
HP8511A and HP8340B. The test set IF interconnect is another interface bus, but is not a
standard HP-IB bus.
Table 4.1: Test equipment frequency range.
Part Number
Name
Frequency Range
HP 8510A
Network Analyzer
0.045 - 26.5 GHz
HP 8341A
Synthesized Sweeper
0.010 - 20.0 GHz
HP 8511A
Frequency Converter
0.045 - 26.5 GHz
HP 8349B
Microwave Amplifier
2.000 - 20.0 GHz
HP 8494B
Attenuator (10 dB)
HP 11692D
Dual Directional Coupler
DC -1 8 .0 GHz
2 .0 0 0 - 18.0 GHz
Although not shown in Figure 4.3, there is a complex system o f switches linking
all transm it and receive antennas. A switch control box is responsible for connecting a
given transmit antenna to the HP11692D coupler and a given receive antenna to the b i port
of the HP8511A. This switch box also selects the antenna polarization. Since the switch
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
71
box also has an HP-IB interface, it can be controlled by the HP9133 com puter and the
same software used to control the HP8510.
Once the data is collected, the hard drive from the HP9133 is physically transported
to the com puter room o f the engineering department. Once here, it is mounted into the file
system o f an H P Apollo 425t workstation. The workstation contains 220 megabytes of
disk space and 16M RAM. The collected data is then downloaded onto the workstation and
the necessary calibration and data reduction are performed.
4.1.4 Chamber Modifications
The anechoic cham ber set-up described in this chapter has been shown to give
excellent results for measurements performed on large, perfectly conducting surface targets
and num erous other targets [14,65], However, experim ents perform ed on the dense
medium targets were im proved by using a stand with a very low dielectric constant and
which allowed the targets to be mounted on edge, thus reducing coupling with the target
support. The m ain disadvantage is that the new stand has no automatic positioning
capability.
The new stand was built from Styrofoam, whose dielectric constant was measured
to be e = 1.06. In order to make the target stable for measurements, the stand was made
with an arc at the top to match the curvature o f the target sides. The stand is portable and
transm it and receive antennas were place on a tripod in a backscatter m easurem ent
configuration. The configuration for the measurement set-up is shown in Figure 4.6.
The experim ents presented in this paper were perform ed at 10° incidence with
respect to the target faces. In order for both the transm it and receive antennas to fully
encompass the target at all frequencies, the distance between the antennas and the target
must be at least
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
72
d.a n t to
tar
(4.2)
tan(B W /2)
where d M is the separation distance between the transm it and receive antennas (9.25
sep
inches for this case), BW is the beamwidth of the antenna, and rlar is the radius o f the
target. U sing Figures 4.7 and 4.8, it is noticed that the antenna beamwidth at 4 GHz is
38° while at 18 GHz it is approximately 10°. If the beam w idth at 18 GHz is used in
equation 4.2, then the distance calculated will be acceptable at all frequencies. In this case,
the distance from the target to the antennas must be at least
Transmitter & Receiver
Target
Figure 4.6: Configuration of measurement system.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
73
CQ
T3
U,
<u
i£
•»•••• 0 . 8 m
o — 1.0 m
A — 1.2 m
■s— 1.4 m
-15
-45
-22.5
0
Angle (degrees)
22.5
45
Figure 4.7: Antenna pattern of dual-polarization 2-18 GHz conical
horn antennas used in experiment (4 GHz) [9].
-10
S§
-20
-30
•»•••• 0.8
-o— 1.0
A - 1.2
s — 1.4
-40
m
m
m
m
-50
-45
-22
0
Angle (degrees)
22
45
Figure 4.8: Antenna pattern of dual-polarization 2-18 GHz conical
horn antennas used in experiment (18 GHz) [9].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
74
Recall that the size o f the targets was determined such that the antenna beam would
fully encompass the target at all frequencies. Although edge effects were determined to be
small, one problem that does occur in this scenario is that the target is perceived by the
system to be one scatterer rather than a collection o f scatterers. From past experiments
described in Section 5.1.1, it appears that measurements performed on these targets for
frequencies above 6 GHz appear to act as a collection of scatterers, as desired. In order to
allow 1 GHz for windowing, the measurements will therefore begin at 5 GHz. The upper
end o f the frequency span will be 16 GHz to ensure that the entire target lies within the
beam even when it is rotated to an incidence angle of 10°. Therefore, the actual frequency
range which will be examined is 6 to 15 GHz although measurements are performed from 5
to 16 GHz.
The second criteria for the location o f the target with respect to the antennas is that
the maximum unambiguous range requirements should be met. The target range, R, can be
determined in a CW radar by measuring the relative phase difference, A<p, between the
transmitted and received waves since the phase o f an electromagnetic wave is a function of
the distance d traveled by the wave [64]. The relative phase difference is given by
A<j>= 2l!d = 4nR
X
X
(4.4)
which implies a target range of
_ AA0
4K
(4.5)
The maximum unambiguous range occurs when A0 = 2 n , implying
Rmai = X /2
(4.6)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
75
or 9.375 mm at 16 GHz. Obviously, this value is much too small for practical use. This
value can be increased, however, by using multiple CW signals with different frequencies.
It can be shown [64] that the maximum unambiguous range using varying frequencies can
be calculated from
Rmax = ^
2A f
(4.7)
where Af is the unaliased frequency, i.e. the interval between each o f the frequencies, and
can be found from
ar
fslop ~f start
# o f p o in ts - !
(4 g)
If the frequency range used is 5 to 16 GHz for 401 points, the unaliased frequency is 27.5
MHz, and the maxim um unambiguous range is about 17.9 feet. Hence, for a 10" thick
target with a 22" diameter, the front face o f the target must be between 15 and 17 feet from
the antennas to satisfy both criteria stated above. Range ambiguity, as stated, is not for our
case as important as it seems. M any of the potential interference effects will cancel out
during the calibration even if the targets are located greater than 17 feet away. It can
become crucial, however, if the back of the chamber lies at one range ambiguity from the
target, causing aliasing o f shadow effects.
The minimum observation range from the antennas to the target is determined
according to the far-field requirement, i.e., [66]
2D 2
r2 p -r-
(4.9)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
76
where p = 2 for the standard far-field requirement, i.e. a phase variation o f 22.5°, and D is
the m aximum dimension o f the target or the antenna aperture, whichever is larger. Note
that for a determ inistic target, D refers to the total size o f the target. However, for a
random target such as the one dealt with here, this dimension refers to the correlation length
between scatterers, which should be no m ore than the diam eter o f one of the scatterers,
namely 1.41 cm, implying r > 2.11 c m (0.83in). Since this value is so small, the far field
due to antenna size is the determining factor in the far field calculations. Therefore, since
the wavelength at the upper frequency limit o f this experiment (15 GHz) is 2 cm, and the
antenna aperture is 14 cm, the minimum observation range is r > 1.96m (6.43 f t ) .
4 .1 5 CRREL Measurements
M easurements in a controlled environment, such as an anechoic chamber, are ideal
since the environment remains stable thus permitting coherent subtraction for calibration.
However, when talcing measurements in outdoor surroundings this is not usually an option
since the environment is constantly changing. M easurements performed on the saline-ice
targets fall into this category.
Radar backscattering measurements were performed on the saline-ice targets using a
network analyzer system very similar to that described in Section 4.1.3, with the exception
that the system used in the saline-ice measurements can perform 1601 frequency steps over
the selected frequency range rather than the 401 steps used in the measurem ents o f the
volum etric targets. There are, however, two very im portant differences between the
measurement setup used on the volumetric targets and that used on the saline-ice targets.
The first is that the volume targets are measured in an anechoic chamber whereas the salineice targets are measured in an outdoor laboratory where the environment is uncontrollable.
The second m ajor difference is the antenna used. As described in Section 4.1.2, the
anechoic chamber uses quadridge conical horn antennas emitting a spherical wave which is
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
77
approximated as a plane wave in the far-field. The antenna used to measure the saline-ice
targets, however, is a 42" offset-fed reflector operating in the near field. This antenna was
designed by the ElectroScience Laboratory at The Ohio State University according to the
com pact range antenna concept. According to this technique, an offset-fed parabolic
reflector propagates a uniformly phased plane wave over an area equivalent to the reflector
size for ranges up to the ratio of the reflector diam eter squared and two wavelengths, i.e.
£>2/2X. [67]. The feed for the reflector is a 2 to 18 GHz AEL horn located at the focal
point o f the antenna so that the reflected rays are both parallel and in phase across the
reflector's aperture [67].
4.2
Calibration
4 2 .1 One-port Error Model
A s stated previously, in order to elim inate systematic errors it is necessary to
perform a calibration.
T he HP8510 Network A nalyzer contains internal calibration
capabilities which can be used to produce error correction for RCS measurements [61].
The H P8510 system offers three types o f error m odels:
one-port, tw o-port, and
frequency-response-only. Combining the use o f programs run on the external computer
and the internal calibration registers of the HP8510A, it is possible to remove effects o f
most o f the unwanted reflections.
Sum
11 A
Er
Figure 4.9: One - port calibration model [61].
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
78
Figure 4.9 shows the Su
S um
measured response
uncalibrated (raw) data of the target
S\IA
actual response
calibrated data of the target
Ed
directivity error
empty chamber measurement
Es
source match error
zero for RCS measurements
Er
reflection tracking error
chamber with sphere - empty chamber
theoretical sphere calculation
For RCS measurements, a metal sphere is used as the reference target since it is one
o f the few shapes for which the theoretical RCS is known, and its complete symmetry
makes it insensitive to positioning and alignment. Note that ED, the isolation error, is used
to model the empty anechoic chamber, and ER, which is a measurement o f the normalized
response error (tracking error), is used to model the correction necessary as determined
from the reference sphere.
From the flow chart it can be seen that
~ Ed +
S iu to )
1 —EsSUA
(4.10)
which can also be written
S ua —
*^11M~ &D
E r + E s (Sum Eq )
(4.11)
which becomes
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
79
(4.12)
since Es = 0 for RCS measurements. Rewriting this equation from its HP-type format into
a more conventional format shows
r u r
pp
41PP Spp
M ppp
(4.13)
where S p" is the calibrated target RCS data, Mppr is the m easured target data with
background subtraction already perform ed, M ppp is the m easured sphere data with
background subtraction already performed, and SPP
P is the theoretical data of the sphere.
Note that this calibration can be performed on external computers if the raw data of all
measurements is saved to the disk. This allows the user to reprocess the data with different
calibration techniques and time gates.
4.2.2 Single Reference, Three Target Calibration
A lthough there are num erous techniques for calibration o f fully polarim etric
backscatter data, perhaps the best available is that developed by W iesbeck and Riegger
[68]. This technique presents a model for the systematic errors in polarimetric free space
m easurem ents, including errors of frequency response, channel imbalance, coupling
between transmit channels, coupling between receive channels, coupling from transmit to
receive, and residual reflections o f the environment [68]. The measured scattering matrix
o f the target is subject to twelve error terms, with the errors contained in three 2 x 2
matrices; the isolation matrix [I], the transmit matrix [T], and the receive matrix [R]. The
measured scattering matrix is therefore given by
[ S ‘"] = [/] + [ * ] [5] [f]
(4.14)
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
80
or
Cm1
5 vm
v tJvh
c**
Sm
hv
Ohki
Rvh
Rkh.
Svv
lSav
Svh
Shh.
Tvv 7*1
LTav T j
(4.15)
[S] represents the corrected scattering matrix and through inversion can be written as
[S] = [R]-1{ [ S l - [ / ] } [ 7 ’r
(4.16)
The isolation matrix is composed of background (isolation) measurements, so the measured
scattering matrix with background subtraction is given by
[M ]
= [5 m] - [/]
(4.17)
Also, all m easurements result in integrated effects o f both the transm itter and receiver
R 'P 'T ,p, so it is helpful to create a new m atrix consisting o f all possible RqpTqp
com binations and order the matrix in vector form [68,69]. This new matrix is known as
the distortion or error coefficient matrix [C] and is written as
[C]
rc n
C21
= C31
-C41
C \2
C 22
C 32
C42
C13
C23
C33
C43
C14
C24
C 34
C44-
f?vvTvv
R vvT ,k
R h vT vv
-R hvT vA
RvvThv
RvvThh
RhvThv
R h vT AA
RvhTvv
RvhTvh
RhhTvv
RhhTvh
RvhThv
RvhThh
RhhThv
RhhThh-
(4.18)
Thus, the scattering matrix o f a target is given as
[S] = [C]-'[M ]
(4.19)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
81
Note that only eight o f the sixteen elements o f [C] are independent, implying that
the sixteen term s can be written in terms o f 8 terms. In order to limit the amount o f
possible round off error during the solution, it is suggested to choose the eight terms with
the largest m agnitude [68]. The four largest terms are those containing like-polarized
responses for both the transmission and reflection terms, i.e., cu ,c22,c33, and cM, hence
these terms will be determined directly. These elements represent the frequency response
error o f the system and can be m easured with small error. The four smallest terms are
those containing cross-polarized responses for both transm ission and reflection terms.
These terms lie along the opposite diagonal as the like-polarized values, i.e., c4l,c32,c23,
and c14, and may be more than 50 dB below the cu terms. They result from the cross­
coupling o f the two orthogonally polarized channels and will be calculated from eight other
elements o f the matrix. Next, four elements must be chosen to represent the independent
elem ents and the remaining four will represent the dependent elements. All eight of the
remaining elements contain both a like- and cross-polarized element and can be measured
with reasonable accuracy. They are usually no more than 30 dB below the cu terms. The
four independent samples used here will be c21,c3pC^, and cM, although the other four
elem ents could just as easily have been chosen. The eight dependent elem ents can be
calculated by:
C 12 —RyvThv
C \2
“
(/?,,T,v) (RhhThv) _ cu C34
(RvvThh) (RhhThv) _ C22 C34
—RyvThv —
(4.20)
(RyyTvv) (RvhThh) _ Cll C24
c 13 —R VhTVv —
C13 = RyhTyy =
(RkhTyy) (RyhThh) _ Cjj C2A
RhhThh
Caa
(4.21)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
82
(/? v/,T„„) _ C34 C24
RhkTkh
C44
cm = RvhThv =
C 23
= R,kTvh = {R
(4.22)
'" T '" ') ( R '»'T >'>') _ C2LC24
RwThh
c22
(4.23)
C32 = A avT*,, = ( £ * Z ” )_(/ ? **7 *1’) _ C3L.C34
c 33
RhhTn
C41 = * * * 7 * = (R ^ ) ( R ^
)
= C3LC2L
Z?„,rVv
C42
= RtoTu, = {Rl"'T^
C42
= *A,7** =
C43 = ZJmTv* =
R m T vv
iRrT^(R^ )
C43 = RkkTvH =
Cu
R - T^ =
R vvTw
/?v»Taa
(4.24)
(4.25)
CJLC22
^11
=C44.C3L
c 33
(4.26)
= £^C 33
Cu
_ C2LC44
C22
(4.27)
Note that the terms involving a like-polarized
c
possible solutions. This is due to the fact that
C22
term with a cross-polarized
c
— = — . Therefore, either
term have two
equation listed
C44
can be used.
In order to solve for the independent components of the [C] matrix, three calibration
targets, one with known theoretical values, are used. In order to simplify calculations, the
three reference targets are chosen such that Svh = Shv. Some o f the more commonly used
targets are listed below:
conducting sphere
[S p] = ^ p [ q
(4.28)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
83
[ s rf0°] = S d0° [q
vertically oriented dihedral
com er reflector
Te ^5"! _
J
dihedral rotated 45° from
vertical
[ c d22.50~[
dihedral rotated 22.5° from
vertical
>-
q<ms°
0]
(4.29)
TO 1]
Ll 0J
r.J22.5° F 1
1 15
(4.30)
-ll
L-l -1J
(4.31)
Using (4.19) with any o f the above listed reference targets produces
C11 Cl2
[M refr\ = C 21 c 22
C31 C32
CAl CA2
c 13 C14
c 23 CtA
C33 C34
C43
[s re!!\
C44-
(4.32)
Since S $ ‘ = SK f\ the above equation can be written as
M&r
M lf
M lf
cu
C21
(C 12+ C 13)
Cl4
(C22 + C 23)
C24
C31
(C 32 + C 33 )
C34
Cai
(C42 +C 43 )
C44_
where r e f i represents the reference target being used. Substituting r e f i with notations for
the sphere (sp), vertical dihedral (<^0°), and rotated dihedral (^0°) and grouping the same
polarized measured responses gives
'M i r
Mi°° =[V ]
.M tr.
'M
m
i r
$ °
.M tr .
Cu
C 12+ C 13
Cl4
(4.34)
C21
= [V ]
C 22+C 23
C 24
(4.35)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
84
r mss i
cu
M g r = [ v ] C32 + C 33
M gr.
.
C 34
.
(4 .3 6 )
C41
C42+C43
C44
(4.37)
where
(4.38)
The elements of [V] represent the theoretical responses o f the three reference targets. Note
that the like-polarized backscattering coefficients from a sphere are equal {SZ5 = SfH, = Ssp)
and the cross-polarized terms are zero (Stf = S tf = 0). Therefore, solving (4.33) for the
like-polarized measured values o f the sphere and using the theoretical response as the
scattering matrix yields
MJf = S ip (cu + ch) = CnSsp
(4.39)
M & = S*p (C41 +
(4.40)
C44 ) =
CuSsp
Hence,
(4.41)
and
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
85
S lp
(4.42)
Therefore, solving (4.34) for the theoretical response for the vertically oriented dihedral
yields
M n — CwSvv +(cn+Cu)Sfh + ChSa*
(4.43)
Since there is no cross-pol for a vertical dihedral and the like pols differ by a sign, this can
be rewritten as
Mn
—
Sv° ( c u
- C u) =
Ci\Sv°
^4 4 4
)
or
S do° _ Mt°° _ Mt°° S X
Cu
MS
(4.45)
Note that this equation is in the same format as (4.13). Similarly, the horizontal likepolarized theoretical response for the vertically oriented dihedral can be calculated as
odo°
l'h
_Mir _Mir sx
c22
MW.
(4.46)
Now th at the theoretical response from the vertically oriented dihedral com er
reflector is known, these values are used to determine the theoretical response for a dihedral
rotated 6°. M easuring 6° clockwise as seen from the antennas, the following equation is
used:
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
86
ad0°
*3w
c dv
OAv
nd$°'
»Jvh
c d0°
Ohh
J
cos 0
-sin 0
sin 9
cos 0.
Svf°
o
0 '
s r .
cos 0
sin 0
-sin 0
cos 0.
(4.47)
which, using the trigonometric identities cos2 0 - sin2 0 = cos 20 and 2cos 0 sin 0 = sin 20
and the fact that Sf°° = -S i* \ reduces to
S i r = SvTcos2 0 + S r s i n 2 0 = S ^ t c o s 2 0 - sin2 ©) = S M° cos 20
(4.48)
sir
(4.49)
=
sar =
( s r -
sir) c o s
0 s m 0 = - s -0 * s i n 2 0
S i r = s i r cos2 0 + Svf'sin2 0 = S ^ t s in 2 0 - cos2 0) = - S " cos 20
(4.50)
Using the above equations, it can be shown that
SO 5’ = S if° = 0
(4.51)
c* d4S°
r*<M5°
OyA — *>Av
r>dO°
o
(4.52)
Svf2'5° = S*° COS 45°
(4.53)
SiZZ5° = S S 2-5’ = SH2y = -S *"cos 45°
(4.54)
Inverting (4.34) through (4.37) yields
Ci 1
C 12+ C 13
Cl4 .
'M U '
= [ v y Mi°°
M ir .
(4.55)
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
87
'Mis'
C 21
C22+C23
C 24
■■[ VV
.
C 31
C 32+ C 33
C34
< v y
M%°
LMvT-
(4.56)
Mi*
M ir
l M tr.
(4.57)
MU'
M tr
C41
C 42 + C 4 3
C44
iMir.
(4.58)
The above equations can be solved for cu ,c2i,c2A,c34, and cM. They can also be solved for
the summations b\ =
C 22+ C 23
and b i =
C 32+C 33.
The next step is to solve for the terms
involved in the summations. From (4.18), it can be noticed that
r
p
t
_ ( R v J T v h ) ( R v h T hh) _ C21 C24
C22_/<vv/---------- ^
Replacing c 23 by b\
- C22
- " ^ r
(4.59)
yields
ch. -b\C22 + C21 C24 = 0
(4.60)
which can be solved for C22 by
bi ± I b t + 4c2i c24
C22 = ■
(4.61)
where the sign in front o f the quadratic is determined from the requirement that IC22) > |c2sj
with
C23 —
C33
b\ -
C 22.
is solved in a similar manner, with
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
88
_
n
T
_
(R kvT *v}{R kkT h v)
C 3 3 — A A A i v v ----------------r
=
_
R avT *.
C31 C34
“
C 32
(4.62)
Replacing c 32 by 62 - c 33 yields
c ?3 -bzC33 + C31 C34 = 0
(4.63)
which can be solved for C33 by
6 2 ± Y 6 2 + 4 C 31 C 34
C 33 = -------------------- r ---------------------
,,
,
where the sign in front o f the quadratic is determined from the requirement that |c3a| > IC32I
with
C 32
=
b z - C33.
Finally, C12, C13, cm, c*1, ce.2, and C43 are solved using (4.20) through (4.27). Now
that the distortion matrix is complete, the measurem ent o f the unknown target can be
performed. The procedure used is as follows:
1) Calculate the theoretical scattering matrix of the sphere:
‘asp
[ 5 S»] = tJVV
SX
5,;jei
Si,',
=\ssp
U)
ssp.
(4.65)
2) Measure the scattering matrix o f the sphere with background subtraction. This value is
determined by subtracting the values obtained in the measurem ent of the sphere with the
background used during the sphere measurement. This includes the empty chamber and
whatever support is used for the sphere:
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
89
r y i f =
L
J
3)
\M*
^ j f i _ r c m V l UsP
-]
lm * ? M issJ l
j ' l
\s,vp s?hspi r m / a n
.s;r/p s r H ' wp nsi
(4 .6 6 )
M easure the scattering m atrix o f a vertically oriented dihedral with background
subtraction.
This value is determ ined by subtracting the values obtained in the
m easurem ent o f the vertical dihedral with the background used during the dihedral
m easurem ent. This includes the em pty cham ber and whatever support is used for the
dihedral.
M ir
T = [S 'nd0° ] - [ / ‘f] =
S mkvd0°
M ir j
m
mdO'
5 vh
SmdD
iik
fd
Ihh
kh
(4.67)
4) M easure the scattering matrix o f a dihedral rotated 6° with background subtraction.
This value is determ ined by subtracting the values obtained in the m easurement o f the
rotated dihedral with the background used during the dihedral measurement. This includes
the empty chamber and whatever support is used for the dihedral:
[M de°] = M ir
M ir
M t r = [S mde°] - [ / “*] =
md(P
5 kv
M tr
i de°‘
fd
Jkv
fd '
ivk
fd
/WiJ
(4.68)
5) Measure the scattering matrix o f the unknown target with background subtraction. This
value is determined by subtracting the values obtained in the measurement of the target with
the background used during the target measurement. This includes the empty chamber and
whatever support is used for the target:
[M ,ar] = M i
= [ s mur] - [ / “”] =
Ovk
cm tar
/ tar
w
i tar
Jkv
Ilf
liar
IkkJ
(4.69)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
90
The calculated scattering matrix of the target,
tar
v
[S 'flr] = 5Cvtar
OAv
O vh
Cta r
Ohh.
(4.70)
can be calculated from:
o ta r
*3yv
C ta r
JvA
C ta r
•JAv
C ta r
Loaa J
'M l f
M ir
= [ c r M ir
IMIV.
(4.71)
where [C] is calculated according to (4.20) through (4.27) and (4.55) through (4.64).
4 2 .3 Calibration Verification
In order to test the validity o f the calibration technique, a second sphere o f different
size than the calibration sphere is considered as the target o f interest. The reason for using
a different size sphere can be easily inferred from (4.13). Note that if the measured target
and the m easured sphere are one in the same, the calibrated target is set equal to the
theoretical sphere data. Hence, you would be forcing the target to look like whatever data
is input for the theoretical sphere.
In this test, an eight-inch sphere is used as the reference sphere, and a two-inch
sphere is used as the target. Applying a 1.4 ns gate centered at 0 ns yields the results
shown in Figure 4.10. Note that the like-pol plots follow the expected pattern o f a twoinch sphere and the plots for the cross-pols are down around the noise floor since there is
no cross-pol backscatter for a sphere.
In order to better see the error from the measurements, the measured values of the
like-pol data are compared with the theoretical values for a two-inch sphere and plotted in
Figure 4.11. It is noticed that the trends of both the VV and HH plots very closely
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
91
-20
-30
_
-40
I
3
-50
t/i
U
-60
o s
2"
2"
2"
2"
VV
HH
HV
VH
-70
-80
-90
6
7
8
9
10
11
12
13
14
15
F re q u e n c y (G H z)
Figure 4.10: Measured response of a 2" sphere.
resemble the oscillatory behavior of the theoretical sphere. However, the theoretical plot
shows slightly larger peaks. This is most likely due to the fact that the 2" sphere is not
quite perfectly conducting. In fact the 2" sphere is actually a ball-bearing and not machined
to such accuracy as the 8" sphere used for calibration. With this is mind, the co-polarized
calibration demonstrates excellent behavior.
Since a sphere has no cross-polarized backscatter return, it can not be used to
determine the accuracy of the cross-pol calibration. However, it is known that a dihedral
rotated forty-five degrees from vertical should have identical cross-pol values. Hence, in
order to determine the accuracy o f the cross-pol measurements, the measured responses
from the two cross-pols are subtracted to somewhat dem onstrate the accuracy o f the
calibration. The results are shown in Figure 4.12 and demonstrate a maximum error o f
0.20 dB.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
92
-23
-24
RCS (dBsm)
-25
-26
-27
-28
-29
-30
-31
6
7
8
9
10
11
12
13
14
15
F re q u e n c y (G H z)
Figure 4.11: Like-pol measurements vs. theory.
1
Magnitude (dB)
0.8
0.6
0.4
0.2
0-
0.2
-0.4
-
0.6
-
0.8
-1
6
7
8
9
10
11
12
13
14
15
F re q u e n c y (G H z)
Figure 4.12: Cross-pol magnitude error.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
93
4 2 .4 Calibration o f Saline-ice Target Data
W hen perform ing m easurem ents in an outdoor environm ent, the coherent
subtraction described in the preceding sections can not be em ployed due to the constant
changing o f the surrounding environment. Hence, a different calibration technique such as
the one described in this section must be used. For the m easurem ents reported here, a
sphere w as placed on m icrow ave absorbing m aterial laying on the ice surface.
M easurem ents were taken with the sphere present and without the sphere so that the
location o f the sphere could be identified in the time domain response. The next step is to
locate the calibration sphere amongst the background. Figure 4.13 shows the time domain
response o f an eight-inch sphere as compared to the response of a thin piece o f absorbing
m aterial.
Upon close inspection, a double peak is noticed on the sphere plot at
approximately 73 ns which does not appear on the plot o f the absorber. M agnifying this
region in Figure 4.14 shows a time dom ain response very sim ilar to what w ould be
expected from a sphere, i.e. is a large peak resulting from the front surface o f the sphere
followed by a smaller peak due to the creeping wave reflecting from the back o f the sphere.
Therefore, tim e gating the sphere data from approximately 72.3 ns to 73.4 ns produces a
return very strongly dom inated by the 8" sphere. The gating locations for the rough
surface target and the smooth ice returns are found in the same manner, as shown in Figure
4.15, with the gates for these targets being from approximately 73 ns to 78 ns.
Since the backscatter return from a sphere is independent o f incident angle, only
one sphere m easurem ent needs to be performed to calibrate any angle. In order to avoid
possible interference from the specular ice return at low incidence angles as well as a
possible significant contribution from the sidelobes interacting with the ice at large angles,
the sphere return at 20 degrees is chosen as the calibration for all measurements. Once the
sphere data is time gated, the normalized sphere response is calculated by dividing by the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
94
-60
-70
§
N
—
✓
t>
1
I
I
-80
-90
-100
-110
8" Sphere
Absorber
-120
70
72
74
76
Time (ns)
78
80
Figure 4.13 Time domain response of 8" sphere and absorber
at twenty degree incidence (70 to 80 ns).
-60
-70
«
-80
3
-90
%
I -100
-110
8" Sphere
Absorber
-120
72
72.4
73.2
72.8
Time (ns)
73.6
74
Figure 4.14: Time domain response of 8" sphere and absorber at
twenty degree incidence (72 to 74 ns).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
95
-70
-80
/-“N
CO
T3
«
•a
-90
2
I)
-100
03
s
-110
Absorber
Target 20
-120
70
72
74
76
Time (ns)
78
80
Figure 4.15: Time domain response o f saline-ice target and
absorber at twenty degree incidence.
theoretical response o f the sphere. Finally, the calibrated target data is found by dividing
the gated ice response by this value.
In order to facilitate the measurements, a parabolic reflector with an offset feed was
borrowed from the ElectroScience Laboratory of The Ohio State University. At the time of
the measurem ents, it was believed that the near-field antenna pattern had a flat behavior
over the aperture. However, this proved to be an invalid assumption as can be seen from
the patterns shown in Appendix H. It can be noticed from these plots that at some
frequencies the magnitude peaks at the center of the aperture while at other frequencies a
m inim um is present. Since a point target located at the center of the beam is used for
calibration, the response at the center is very important. If an area extensive target is
measured using this antenna configuration and the average magnitude across the aperture is
not the same as that found at the center, a calibration error will result. Note that in the
saline-ice data analysis frequencies lower than 7 GHz are not examined due to an inherent
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
96
property o f the offset-fed reflector which apparently creates a large null due to diffraction
problems from the edge o f the reflector [70]. These frequencies are not critical in our
analysis, however, since it is assumed that scattering from these frequencies will be almost
entirely dominated by surface scattering at all incidence angles observed.
Figure 4.16 shows a frequency plot of the m agnitude o f a horizontally polarized,
horizontal scan from 2 to 44" across the antenna aperture normalized to the response at 23",
the beam center. Also on the same graph is a plot o f the backscattering coefficient as a
function of frequency (with 1 GHz increments) for the rough boundary saline-ice target as
measured at 50° incidence. The left vertical axis applies to the normalized scan response
while the right vertical axis corresponds with m easurem ents m ade from the saline-ice
target. W hen comparing the two curves, it is noticed that the trends are almost exactly the
same, indicating that the point target did indeed cause problems in the calibration of the area
extensive targets. This problem can be eliminated in future experiments by using an area
extensive target with a known scattering behavior in conjunction with a sphere to perform
the calibration. For example, a perfectly conducting Kirchhoff-scale roughness target o f
the same dimensions as the target to be measured could be used for calibration.
In examining the angular behavior of the smooth saline-ice and saline-ice with a
rough layer, we must choose which frequencies to study. Examining the magnitude plot in
Figure 4.16, we wee that the frequencies which cross the 0 dB line are approximately 7.5,
8.5, 11, 13.5, 15.5, and 16.25 GHz.
Also, an expanded frequency plot o f the
backscattering coefficient shows another peak in the data at 12.25 GHz, indicating that
between 12 and 13 GHz two more frequencies whose magnitudes are 0 dB are present.
Therefore, choosing frequencies within ±0.25 GHz o f 11, 12.5, 13.5, 15.5, and 16.25
GHz should provide accurate results. 7.5 GHz and 8.5 GHz should provide reasonable
results as well.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
97
4 — A vg (2 to 44")/23"
— €►— 50 Target
/3
-10 C
O
s
QQ
■O
(D
-15
T3
3
a.
3
ere
-20
-25
-30
2
.
c5‘
3
a)
-35
7
8
9
10
11 12 13 14
F requency (G H z)
15
16
17
Figure 4.16: Comparison of antenna response of probe compared
to the measured target response.
Note that since the scattering coefficient is defined as the radar cross section divided
by the effective area, the area illuminated by the antenna needs to be calculated for final data
reduction. As stated in Section 4.1.5, the antenna illuminates an area approxim ately the
same size as the reflector at nadir. Since measurements are performed at angles off-nadir,
the effective area can be calculated according to
cos 9
(4.72)
where, in this case, r = 21"= 0.5334 m and 6 ranges from 10 to 50°.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
CHAPTER FIVE
EXPERIMENTAL RESULTS
5.1.
Backscattering from Volume Targets
5.1.1 Initial Volume Target, Setup, and Results
T he sample target described in Section 3.1.5 was built in February 1993. At this
time, the target stand consisted o f two large wooden supports with a 4' by 4 ’ by 6"
rectangular piece o f dense polyurethane between them. A 3' diameter hole was cut from
the center o f the polyurethane, and the target to measured was to be placed here. Conical
absorber was then glued to the front face o f the polyurethane block in an attempt to remove
possible edge effects from the target as well as the polyurethane support.
Polyurethane
Target
Absorber
Figure 5.1: Original measurement setup.
U sing this m easurem ent setup, thirty equally spaced azim uthal sweeps were
performed on the 10% volume fraction target, covering a frequency range of 5 to 17 GHz
98
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
99
for VV polarization at five degrees incidence. Figure 5.2 compares these results with
theoretical values obtained using the DMT-IEM and the single scattering approximation.
Comparison o f the shape o f the measured data with the single scattering prediction indicates
that the trend o f the curve as a function of frequency follows the Mie behavior, although it
is not as dynamic. In comparing with the DM T-IEM , it is noticed that this model still
follows the trends o f the single-scattering theory, but to much lesser extent than the
measured data. It is believed that the DMT-IEM prediction shown in Figure 5.2 deviates
from the trend o f the measured data at higher frequencies because the number o f Fourier
components used when running this program was too low for the higher frequencies.
c
>ou^4
10
-
8
u
00
C
-1 0 -
10% Measured
10% DMT-IEM
Indep Scat Ther
1
1
cd
CQ
-2 0 -3 0 5
6
7
8
9 10 11 12 13 14 15 16 17
Frequency (GHz)
Figure 5.2: Measurement vs. theory as a function o f frequency.
Although the data from this experiment will not again be used directly in the balance
o f this paper, the results provided the basis for the rem ainder o f the experiments. It is
apparent from Figure 5.2 that the measured frequency behavior, i.e. the peaks and dips, is
sim ilar to the single scattering trend, although it is damped due to the effects o f multiple
scattering. Below 6 GHz, the shape o f the m easured data appears to deviate from that
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
100
predicted by the Mie scattering behavior. A possible reason for this is that the beamwidth
is wide enough such that edge effects can not be ignored. Hence, future measurements will
start at 6 GHz. In order to reduce coherent effects caused by the specular return from the
effective boundary, the incidence angle for the rest o f the data to be shown in this section is
increased to ten degrees.
Also, to reduce edge effects from the target support, the
measurement setup described in Section 4.1.4 is employed. Finally, to obtain better data
averaging, 90 sweeps are performed instead o f 30.
5.1.2 Backscatter Results
M easurem ents on the volum e targets described in the preceding chapters were
perform ed at ten degrees incidence from 6 to 15 GHz for all four linear polarizations.
Figures 5.3 through 5.7 show the backscattering coefficients obtained for like and cross
polarizations as compared with the single scattering approximation. The like-polarized
response is an average o f VV and HH, and the cross-polarized response is the average o f
VH and HV. The results are presented in this m anner since results from both likepolarizations were nearly identical, as were those obtained for the two cross-polarizations.
It is noticed in Figures 5.3 through 5.7 that the measured backscattering coefficients
dem onstrate the same peaks and dips as the single scattering approxim ation, but the
deviation between the two generally increases as volume fraction increases. Since there is
no cross-pol backscattering from a single sphere, the cross-polarized plots are the result of
m ultiple scattering. These plots can therefore provide some insight as to the scattering
behavior o f the co-polarized measurements by indicating the strength o f multiple scattering
at various frequencies and volum e fractions. A nother trend which is apparent from
viewing Figures 5.3 through 5.7 is that although the single scattering theory continues to
increase with increasing volume fraction, the measurements show that the co-polarized data
saturates from the 11% volume fraction on up for frequencies greater than 7 GHz.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
101
w>
-10
5 like meas
5 cross meas
5 single scat ther
-15
-20
6
7
8
9
10
11
12
Frequency (GHz)
13
14
15
Figure 5.3: Measurement results: 5% volume fraction.
CQ
<D
^
I - io
J*
o
c
« -15
8 like meas
8 cross meas
8 single scat ther
CQ
-20
6
7
8
9
10
11
12
Frequency (GHz)
13
14
15
Figure 5.4: Measurement results: 8% volume fraction.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
102
Backscattering Coefficient (dB)
20
15
10
5
0
-5
-10
11 like meas
11 cross meas
11 single scat ther
-15
-20
6
7
8
9
10
11
12
Frequency (GHz)
13
14
15
Figure 5.5: Measurement results: 11% volume fraction.
Backscattering Coefficient (dB)
20
15
10
5
0
-5
-10
14 like meas
14 cross meas
14 single scat ther
-15
-20
6
7
8
9
10
11
12
Frequency (GHz)
13
14
15
Figure 5.6: Measurement results: 14% volume fraction.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
103
•u
•
20 like meas
20 cross meas
20 single scat ther
-15
-20
6
7
8
9
10
11
12
Frequency (GHz)
13
14
15
Figure 5.7: Measurement results: 20% volume fraction.
As stated in Chapter One, past experiments performed by other authors indicate a
decrease in scattering coefficient for volume fractions greater than 10% when the ka value
is about one.
In order to exam ine this region m ore closely, Figures 5.8 and 5.9
demonstrate the co-polarized and cross-polarized returns for all five volume fraction targets
from 6 to 9 GHz, i.e. ka = 0.92 to 1.38. It is apparent from Figure 5.8 that for frequencies
less than 7 GHz (ka = 1.07) the co-polarized plots show an increase in the backscattering
coefficient as a function o f volume fraction from the 5% target to the 14% target, and a
decrease in the backscattering coefficient from 14% to 20%. For frequencies above 7
GHz, however, the backscattering coefficients for the 11, 14, and 20% targets converge
and m aintain approximately the same level as one another for the rest o f the frequency
range used.
In the case o f the cross-polarized data, a sim ilar trend occurs with the
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
104
exception that this apparent saturation o f the data does not occur until about 9 GHz (ka =
Backscattering Coefficient (dB)
1.38).
5 like meas
8 like meas
11 like meas
14 like meas
20 like meas
-10
6
7
8
Frequency (GHz)
9
Backscattering Coefficient (dB)
Figure 5.8: Like-polarized comparison (ka = 0.92 to 1.38)
5 cross meas
8 cross meas
11 cross meas
14 cross meas
20 cross meas
-12
-16
-20
6
7
8
Frequency (GHz)
9
Figure 5.9: Cross-polarized comparison (ka = 0.92 to 1.38)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
105
Figures 5.10 through 5.12 better demonstrate the trends mentioned in the preceding
paragraph.
N otice that at 6 G H z (ka = 0.92) both the like- and cross-polarized
measurements show rises in backscattering coefficient with increase in volume fraction for
volume fractions up to 14%. However, both the like-polarized and the cross-polarized
returns demonstrate a decrease in backscattering coefficient between 14 and 20% volume
fractions. The plots of the theoretical predictions made by the DMT-IEM, however, do not
show this drop in scattering coefficient. Also, although the theoretical and measured likepolarized responses are relatively close, the predicted cross-polarized values are much
greater than the measured values.
The drop in backscattering coefficient at higher volum e fractions and low er
frequencies can be explained by destructive coherent interference, as will be discussed in
the following section. In other words, as the volume fraction is increased, the spacing
between particles decreases and coherent effects come into play, thus causing scattering to
no longer be proportional to the number of scatterers. This would also explain the large
disparity between the measured and predicted cross-polarized data, whereas the likepolarized measurem ents and predictions are very close in level. That is, since the cross­
polarized return is solely due to multiple scattering, it shows more dependence upon the
phase coherency than the co-polarized data.
In addition to the drop in backscattering coefficient at 20% volume fraction shown
in Figure 5.10, it is noticed that the scattering level rises quickly from 5 to 14% volume
fractions. This is probably due to the fact that at lower volume fractions the spheres are in
the far-field o f each other and if the frequency is low enough such that the penetration depth
is large, backscattering is proportional to the number of scatterers.
Figure 5.11 shows that at 10 GHz {ka = 1.53) the like-polarized data seems to
saturate at about 3 dB for volume fractions greater than 11% while the cross-polarized data
saturates at approximately -2.5 dB for these volume fractions. This saturation of the data is
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
106
10
S3
3
5
C
.22
o
{G
t+-<
0
j
-5
oo
c
•c
S -10
£
oCO -15
o
▲
S3
6
6
6
6
like meas
cross meas
like DMT-IEM
cross DMT-IEM
-20
5
8
11
14
Volume Fraction (%)
17
20
1
1
o
10
O
Figure 5.10: Backscattering coefficient vs. volume fraction (6 GHz).
1
i
4
1
S3
w
5
C
.22
’5
£
0
a
-5
oo
c
■c
£CO -10
8
-15
£
S3
-20
A
4 -
3
1
“
o
10 like meas
A
10 cross meas
----------10 like DMT-IEM
--------- 10 cross DMT-IEM
------ 1----------- 1
' ”1------------- 1------------11
14
17
20
Volume Fraction (%)
Figure 5.11: Backscattering coefficient vs. volume fraction (10 GHz).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
107
10
CQ
X>
5
cIO
0
8
u
-5
oo
c
■g -10
g
-15
CQ
o
A
12 like meas
12 cross meas
12 like DMT-IEM
12 cross DMT-IEM
-20
11
14
Volume Fraction (%)
17
20
Figure 5.12: Backscattering coefficient vs. volume fraction (12 GHz).
also dem onstrated in Figure 5.12 for 12 GHz (ka = 1.84). At these frequencies, the likepolarized predictions m ade by the DM T-IEM com pare reasonably well with the
m easurement values obtained while the predicted cross-polarized returns are once again
greater than the actual values obtained through measurements.
The DM T-IEM agrees much better with the 10 and 12 GHz cases than for the 6
GHz data due to the fact that coherent effects increase with increasing wavelength
(decreasing frequency). The amount o f coherent interference is also dependent on the
volum e fractions studied. Hence, coherent effects are a function of kd, where k is the
wavenumber and d is the average distance between scatterers. Therefore, as the number of
scatterers per wavelength increases, the coherent effects also increase.
Figures 5.11 and 5.12 show a saturation of data because at these frequencies and
the volume fractions studied, the scatterers are approaching the far-field o f each other. The
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
108
plots don't continue to rise with increasing volume fraction, however, since as the volume
fraction is increased the albedo also increases while the penetration depth decreases.
Therefore, the decrease in the depth of penetration is offset by the corresponding increase
in albedo and an apparent saturation point is reached. However, eventually a volume
fraction will be reached where the coherent effects are strong and the curve as a function of
volume fraction will begin to drop.
5.1.3
Coherent Interference
Consider two waves with the same frequency and speed given by
£,
=
E0] sinfour - k (x + Ax)]
(5.1)
and
E 2 = £ 02 sin(ct)f - kx)
(5.2)
and overlapping in space. The resultant wave is therefore the linear superposition o f these
two, namely
£ = £ ,+ £ ,
If
we
le t
£ 01 = £ 02 and
(5.3)
use
the
trig o n o m e tric
id e n tity
sin ( a - { 3 ) =
sin a c o s /3 -s in /J cos a , (5.3) can be written
E = £ 01{sin ct>r[cos(&(x + Ax)) + cosfct] - cos cot[sin(ife(x + Ax)) + sin fcx]}
(5.4)
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
109
Then, using the identities c o s a + cos/1 = 2 c o s ^ -(a + / J ) c o s ^ ( a - / J ) and s i n a + sin/J =
2 s in ^ -(a + /J )c o s ;j(a ! -/J ), (5.4) can now be written
' = £01jsinct)r^2cos^-(2x4- A x)cos^^- - cos cot 2sin^-(2x + A x)cos-^y^
(5.5)
= 2£01cos^^ jsin<yrcos^x + ^j-cos<ursin& ^x + - ^ j j
which is simplified using the previously mentioned trigonometric identities as [72]
Ax
„
k Ax .
E = 2E n, cos
sm cot - k \ x +
(5.6)
Hence, if the path-length difference Ax is much smaller than the wavelength, i.e. Ax «
A
constructive interference results and the resultant amplitude is approximately 2 £ 01, whereas
if Ax = A/2 destructive interference occurs and the resultant wave has zero magnitude [72].
Visual representations o f constructive and destructive interference are shown in Figures
5.13 and 5.14.
Figure 5.13: Destructive interference.
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
110
Ax
E2\
Figure 5.14: Constructive interference.
5.1.4 DM T-IEM with Coherent Interaction
The coherent interactions presented in the preceding section can be accounted for as
follows.
C onsider a unit volum e divided into M ,N , and P cells along the x , y , and z
directions respectively. Within each cell lies a scatterer whose location is within a deviation
S r ^ about the center, where
= mdx + ndy + p d z . If the average linear distance
between cell centers is d, then the total scattered field due to this collection o f scatterers is
[73]
E sulal= A F -E ;ingU
(5.7)
where E*ingU is the scattered field due to a single scatterer and the antenna factor A F is
defined as
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
I ll
AF = X X X exP[X*, - k ^ m d ] ■exp[y(fc, - k ) ynd\
m=0 n = 0 p=0
•exp[X *J - * 1),P d ] ' 5 r *v’
(5-8)
M - l tf- 1 /> -l
=
A:, and
X X X exp [ ^ wrf] • e
x p [ y * ,m
/ ]
• exp[y*z/jrf] • 5
r m v
ki are the propagation vectors in the scattered and incident directions, respectively,
and are given by
ks = &0(sin 6Scos <psx + sin 6S sin <psy + cos 9 J )
kt -
£0(sin 0, c o s<ptx + sin 6t sin <pty + c o s0:z)
(5.9)
Multiplying A F by its complex conjugate yields
M - 1 M - 1 N - 1 W -l P - 1 P- 1
\A F f = X X X X X X
m=0 m'=0«=0n’=0 p=0
ex p [A (™ “ m ’)rf]exp[y'X(« “ n' )<*]
^ 10)
p =0
exp[y*,(P “ P')d ] ^ p [ j ks,{s ^
~
)]
with the primed coordinates indicating values obtained from AF* and where
= ( * , - * i)||5 r J
C0S?)
(p is the angle between the vectors
(ks - k t) and
(5.11)
Sr
.
8 m»p *s a random variable assumed to have a Gaussian probability density function
with variance a 2 . The joint probability density function o f the two Gaussian variables a 2
and 8m.n.p. is a bivariate Gaussian density function. The joint probability density function
o f (‘L V - ‘W
) is a Gaussian distribution given by [73]
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
112
p ( A * ~ 5mV,') =
eXp[ (^wy ~ < W ) 7 2<^ ]
(5‘12)
where the variance o £ = 2 < r2( l - p ) with p representing the correlation coefficient
betw een the two random variables.
Hence, by using the averaging of the random
variables, (5.10) can be written as
(lA/72|) = X X X X X X
J exp [ A (m - n i)d ] e x p [ jk y{ n - r i ) d \
m=0 m'=0 n=0 n’=0 p=0 p'=0
e x p ^ lp - p O ^ e x p f A t^ -„.„•)]
P { 5 mrp ~
A „>' )
(5.13)
)
Using the identity [53]
‘j e x p ( - p 2x 2 ± q x )d x = — QXTp(q2/ 4 p 2},
P
[p>0]
(5.14)
(5.13) can be re-written as
A f-l M - l N - 1 tf-1 />-! P - l
(lA F1) = X X X X X X
m=0 m'=0 «=0n'=0 p=0 p'=0
exp [ y ^ ( i» - m ,)rf]e x p [y ^ (« -« ,)d]
(5.15)
exp[ A (p - P' ) d \ t \ v [ - k sia 2Duwl2 \
where
~ 2(T2(l
p uyw)
« = |m —m'|
v = |n-n'|
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
113
w = \ p - p '\
(5.16)
In the past, the DM T-IEM calculated the phase m atrix for n scatterers by
multiplying the single-scattering phase matrix Ps , as defined in (2.63), by the number of
scatterers. In order to include the effect o f phase correction for a collection o f scatterers,
the model can be easily modified by multiplying the single-scattering phase matrix by the
phase correction factor, i.e. [73]
d 3(M - 1)(7V —1)(P - 1 )
'
5.7.5 M easurements vs. DMT-IEM with Coherent Interaction
M odifying the dense medium transfer integral equation m ethod to account for
coherent interaction as described in the preceding section produces the results shown in
Figures 5.15 through 5.17.
10
I
5
c
o
*5
c:
<*4
0
3
00
c
•e
<D
-5
S3
-10
u
-15
tS
O
CO
03
6 like meas
6 cross meas
6 like DMT-IEM (Cl)
6 cross DMT-IEM (Cl)
-20
11
14
Volume Fraction (%)
17
20
Figure 5.15: Measurements vs. DMT-IEM (Cl) (6 GHz).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
114
Backscattering Coefficient (dB)
10 -------------!-------------- \-------------!-------------!----------5 ................. 1..................1................. !.................!...............
>f
<>------------ ;------------ <
A
-g-----------:>
<
!
o i - ........T
............,................. I................x................i..............
___—
--------1
----"Tk
— I—
~4
I
^i
................. }................. {................. | ...............
-5 it................f
k................4 ................
_10 ................. 1
o
10like
likemeas
meas
10
10
10cross
crossmeas
meas
---------- 10
—
10like
likeDMT-IEM
DMT-IEM(Cl)
(Cl)
10 cross DMT-IEM (Cl)
A
15
i
-2 0
------------------------j--------------------------i------------------------j------------------------i---------------------
5
8
11
14
Volume Fraction (%)
17
20
Figure 5.16: M easurements vs. DM T-IEM (Cl) (10 GHz).
Backscattering Coefficient (dB)
10
1---------------- 1--------------- 1--------------- 1-------------
5>
0
L
-5
12like
likemeas
meas
12
12 cross meas
,------------------------- ----------12
12like
likeDMT-IEM
DMT-IEM(Cl)
(Cl)
................. I
--------- 12
12cross
crossDMT-IEM
DMT-IEM(Cl)
(Cl) "
_J0 ..................1
o
A
-2 0
r
------------------------j----------------------- j------------------------j------------------------j-----------------------
5
8
11
14
Volume Fraction (%)
17
20
Figure 5.17: M easurements vs. DMT-IEM (Cl) (12 GHz).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
115
All three Figures indicate a much better comparison between the model and the
measured data than the DM T-IEM without the coherent interaction shown in Figures 5.10
through 5.12. Although all three data fits are very good, the m easurem ents at 6 GHz
compare extremely favorably with the model prediction. One possible explanation for this
is that the permittivities o f the background medium and the scatterers used in the model
were those values measured at 6 GHz.
5.2
Bistatic Measurements (Preliminary Results)
In order to learn m ore about the scattering behavior o f dense m edia, more
measurements need to be performed on targets sim ilar to the ones manufactured for this
experim ent.
M ore transm it and receive angles should be used to obtain additional
information about the scattering behavior o f the target. The bistatic configuration described
in Section 4.1.2 is ideal for these measurements, but methods o f reducing the positionertarget interaction effects should be improved. This section briefly describes a fabrication
technique for constructing targets which look the same at any bistatic angle. In addition,
several preliminaty measurement results are presented on two such targets.
5.2.1 Fabrication and Target Parameters
In order to be certain that measurements m ade from the various transmit/receive
com binations could be accurately compared with one another, the desired target needs to
look identical at all angles. Hence, the target should be in the shape o f a sphere. Note that
if the background material used is transparent at microwave frequencies, the physical target
does not need to be spherical, but the area where the scatterers lie must be spherical.
The FORTRAN program targfab.f in Appendix B can easily be modified to account
for placem ent o f scatterers within a spherical region rather than a cylindrical one.
Uniformity can be checked by dividing the sphere into equally cut wedges and equal-
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
116
volume spherical shells, similar to the wedge and ring cut verifications described in Section
3.1.3. In order to demonstrate the construction method, two spherical shaped targets were
built. Both targets have a diameter of 20 inches and all uniformity tests had fits greater than
99.5%. The iseed values used and some of the corresponding parameters obtained are
shown in Table 5.1.
Table 5.1: Parameters for spherical targets.
Volume
Fraction
Iseed
Value
# of
Scatterers
Average Distance
Between Particles
Minimum Spacing
Between Particles
15%
16
6886
2.135 cm
1.589 cm
25%
1
11,427
1.800 cm
1.506 cm
5 .2 2 M easurement Results
Thirty azimuthal sweeps were performed on the above described targets for VV and
HV polarizations at two bistatic angles. The transmitter used was Transmitter 2 (6, = 45°)
show n in F igure 4.1 and the bistatic receivers chosen w ere R eceivers
(8r = 55°,0 r = 110°) and 20 (dr = 7 0 ° ,0r = 150°).
15
Figures 5.18 through 5.21 show
scattering coefficient o f the two targets as compared with the single scattering theory for the
25% volum e fraction target. Note that a plot o f the single-scattering theory for the 15%
volume fraction target is the same as that for the 25% target but 2.2 dB lower. For all
transm it-receive-polarization combinations examined, the trends of both targets closely
resemble those exhibited by the single scattering theory. It is also noticed that in the case of
bistatic receiver 15, the 15% target exhibits a stronger scattering level than the 25% target
for both polarizations and all frequencies up to 15 GHz, indicating strong coherent
interaction. In the case o f receiver 20, however, the 15% target has a larger backscattering
coefficient only up to about 7 GHz for both polarizations. After this frequency, it appears
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
117
i
20
CO
15
3
4—
»
a
10
o
U
<4Q
-H
5
3
00
c
-5
i
i
i
i
• j
i
i
..... y
\
i
-
*
\
i
i
i
........h ....!
: V :
. . . i. .
:
#•••>....... >.......<•........................ J........
....
i
i
§
§
V
0
•c -10
Si
eS
u
co
-15
---------- 15% VV Meas.
■ -—»»» z d 7 0 v v oing. ocat. iner.
-20
-25
5
1
1
i
6
7
8
1
1
i " i
i
i
i
i
9 10 11 12 13 14 15 16 17
Frequency (GHz)
Figure 5.18: VV polarized measurements vs. single scattering theory
(0, = 45°, 0r = 55°, A0 = 70°).
CQ
3
w
c
o
<4-1
3
00
c
■c
C3
u
CO
■25% HV Meas.
15% HV Meas.
25% HV Sing. Scat. Ther.
9 10 11 12 13
Frequency (GHz)
Figure 5.19: HV polarized measurements vs. single scattering theory
(0, = 4 5 ° ,0 r = 55o,A0 = 7O°).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
5
6
7
8
9 10 11 12 13 14 15 16 17
Frequency (GHz)
Figure 5.20: VV polarized measurements vs. single scattering theory
(6, = 45°, = 70°, A0 = 30°).
CO
3
c
u
U
c
*C
0o3
C/3
25% HV Meas.
15% HV Meas.
25% HV Sing. Scat. Ther.
i ----- 1— r
9 10 11 12 13 14 15 16 17
Frequency (GHz)
Figure 5.21: HV polarized measurements vs. single scattering theory
(9, = 45°, 9r = 70°, A0 = 30°).
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
119
that the scattering coefficient for both the 15 and 25% volume fraction targets is equivalent,
i.e. a saturation point has been reached. Note that slight deviations in the level may be due
to the fact that only 30 sweeps were performed or to target-pedestal interaction.
5.3
Saline-ice Measurements
B ackscattering m easurem ents perform ed on sea-ice have been m odeled using
various approaches. However, since the statistical parameters o f the sea-ice targets are not
known but rather estimated, there is disagreement as to which models correctly predict the
scattering; in particular, it remains unclear as to when the dominant scattering contribution
is due to surface scattering and when it is due to volume scattering. Since the parameters
are unknown, at the extremes some researchers have been able to fit a given data set using a
surface scattering m odel while other researchers fit the same data set using a volume
scattering model.
In order to better understand surface and volume scattering contributions, a thin ice
layer with a statistically known top surface roughness and flat bottom boundary was
constructed according to the method discussed in Section 3.2. Also, a thick saline-ice
medium with a relatively smooth top boundary was grown at the US Army Cold Regions
Research and Engineering Laboratory. This "smooth-ice" target was m easured from 7 to
17 G H z for backscatter angles o f 10, 20, 30, 40, and 50 degrees. Since it has only a
slightly rough surface, backscattering m easurem ents at the larger incidence angles and
higher frequencies are expected to be dominated by the volume scattering. Therefore, the
sm ooth-ice m easurem ents at 50 degrees are used to determ ine the albedo at various
frequencies.
Once measurements were completed on the smooth-ice target and the backscattering
behavior was determ ined, the thin rough surface layer was placed on top o f the thicker
saline-ice sheet. This effectively generated a saline-ice sheet with statistically known
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
120
random roughness. Although a slight discontinuity existed at the boundary o f the two
targets, its effect was found to be negligible. For this target, measurements were repeated
over the same incident angles and frequencies.
Although the smooth-ice target was modeled using estimated surface roughness and
volum e parameters, matching the data from the rough surface target does not allow this
freedom since the roughness parameters are pre-defined. The only adjustable parameters in
the case o f the rough surface target, therefore, are the optical depth and albedo. Since the
overall target is very thick, the optical depth will be large enough such that interaction with
the bottom boundary will be negligible, and the albedo has already been estimated from the
m easurem ents perform ed on the sm ooth-ice targets.
H ence, all model input
param eters f o r the rough surface target are defined in advance.
G ood
agreement between measured and predicted data will therefore verify the applicability of the
model as well as indicate relative contributions from surface and volume scattering.
Figures 5.22 through 5.26 show com parisons between the combined scattering
model described in Sections 2.4 through 2.7 and backscattering measurements from the
saline-ice medium with and without the added surface roughness. Note that the measured
data also includes error bars for a 95% confidence interval. This is the interval over which
the average scattering coefficient could occur given the sample size and standard deviation
o f the observations.
P ast C RR EL m easurem ents indicate that a sm ooth saline-ice surface is
exponentially correlated and has an rms height less than 0.05 cm and a correlation length
between 1 and 2.5 cm [71]. The parameters determ ined for this experiment are an
exponential correlation with an rms height o f 0.0434 cm and a correlation length of 1.8 cm.
The added rough surface is Gaussian correlated with an rms height of 0.25 cm and a
correlation length o f 2.0 cm. A ll surface param eters m ust remain the same for all
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
121
frequencies.
The albedo o f the saline-ice is determ ined through com parisons with
measurements before adding the rough ice top boundary.
Figure 5.22 shows very good agreement between the measurements and predictions
at 7.5 G H z (C band) with the exception of the 10 degree m easurem ent o f the smooth
saline-ice. This is most likely due to a specular return present at low incidence angles since
there m ay be a slight variation in the actual incidence angle. Even though this return is
expected to be weaker at 10 degrees, the theoretical return is only -20 dB at this frequency
and incident angle; a specular return (due to a low incidence angle) o f -20 dB would
increase the total backscattering coefficient to the measured -17 dB. Notice that the surface
scattering alm ost com pletely dom inates the return at all incidence angles except at 50
degrees, i.e. measurements from the rough surface target and the smooth-ice target do not
converge until 50 degrees.
Figures 5.23 shows excellent agreement between measurements and theory for all
incidence angles at 10.75 GHz (X band).
It is also noticed that there is increasing
im portance due to the volum e scattering, where now the return at 40 degrees is being
strongly influenced by volume scattering.
Figures 5.24 through 5.26 all show excellent agreement for Ku band data, namely
12.5, 15.25, and 15.5 GHz. It is seen that the volume scattering becomes increasingly
important: as frequency increases, the scattering behavior versus incidence angle becomes
flatter and the level increases. To show this concisely, Figures 5.27 and 5.28 show the
contributions to the overall scattering coefficient due to scattering from the top surface and
due to volume scattering. Viewing these graphs at 50% show that at 7.5 GHz the surface
and volum e scattering contribute equally at approxim ately 47.5° incidence.
As the
frequency increases, the incidence angle where surface and volum e scattering contribute
equally decreases. At 16.5 GHz, this point occurs around 27.5°. Hence, the decrease in
penetration depth at higher frequencies is compensated for by the increase in albedo.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
122
Backscattering Coefficient (dB)
X
1—■■“! I"I I • I I I |
...|.... 7,5 GHi
I
dlbedo = 0J0025
o
x
7.5
7.5
7.5
7.5
7.5
7.5
7.5
7.5
Tot Ther
Smooth Ther
Tot Meas
Smooth Meas
Tot Meas Low
Tot Meas High
Smooth Meas Low
Smooth Meas High
i
r
25
30
35
Theta (Degrees)
Backscattering Coefficient (dB)
Figure 5.22: M easurements vs. theory at 7.5 GHz.
1 10.75 GHz...
plbedO = 0.018
10.75 Tot Ther
10.75 Smooth Ther
10.75 Tot Meas
10.75 Smooth Meas
10.75 T ot Meas Low
10.75 Tot Meas High
10.75 Smooth Meas Low
10.75 Smooth Meas High
-30 .......... | ....... | ..............j .......... } ............. | ...... j ............ j. . . . . . . .
I
1 1 1ij I 1 11 »j-{ 1 1 1j 1 I 11 1| | ! ' 1| 1 ' i ij ' 1 1 j |11 1 1 I i» | '
10
15
20
25
30
35
Theta (Degrees)
40
45
50
Figure 5.23: Measurements vs. theory at 10.75 GHz.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
123
Backscattering Coefficient (dB)
T
I' I I ! I I I I j I I I I j 1—I - T T 'I
..12 .5 .(xHz.......
albedo = 0.021
■ ■ ■ 111
10
15
- 12.5 Tot Ther
■ 12.5 Smooth Ther
o
12.5 Tot Meas
X
12.5 Smooth Meas
—
12.5 Tot Meas Low
—
12.5 Tot Meas High
—
12.5 Smooth Meas Low
—
P 5 Smooth Meas High
■ i ■ i i■ ■ i i ■ i ■ 1 1 ■ ■ 1 1 1 1
20 25
30
35
Theta (Degrees)
■ 111
>i ■ i i i ■ 11
40 45
50
Figure 5.24: Measurements vs. theory at 12.5 GHz.
Backscattering Coefficient (dB)
! 111
i ! 11
j
i ! 1 1 1 1 ! i
.] 15.25G.Hz.....
albedo = 0.035
10
15
20
25
30
35
Theta (Degrees)
15.25
15.25
15.25
15.25
15.25
15.25
15.25
15.25
40
T ot Ther
Smooth Ther
Tot Meas
Smooth Meas
T ot Meas Low
Tot Meas High
Smooth Meas Low
Smooth Meas High
45
50
Figure 5.25: Measurements vs. theory at 15.25 GHz.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Backscattering Coefficient (dB)
124
- 16.5 Tot Ther
■ 16.5 Smooth Ther
o
16.5 Tot Meas
X
16.5 Smooth Meas
—
16.5 Tot Meas Low
—
16.5 Tot Meas High
—
16.5 Smooth Meas Low
—
16.5 Smooth Meas High
i 16.5 .(jiJ«f1*.......
albedo = 0.055
Q >.
■ i
10
■ ■ I
i
15
■ '
i
I
20
■ i
i
■ I
i
■ i
i
I
■ i
i
■ I
i
25
30
35
Theta (Degrees)
i
i
'
j
40
i
i
i
i
I
i
i
45
i
i
50
Figure 5.26: Measurements vs. theory at 16.5 GHz.
7.5 Surf%
10.75 Surf%
12.5 Surf%
15.25 Surf%
16.5 Surf%
10
15
20
25
30
35
Theta (Degrees)
40
45
50
Figure 5.27: Percent contribution due to surface scattering.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
% Contribution
125
7.5 Vol%
10.75 Vol%
12.5 Vol%
15.25 Vol%
16.5 Vol%
Theta (Degrees)
Figure 5.28: Percent contribution due to volume scattering.
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
CHAPTER SIX
CONCLUSIONS AND RECOMMENDATIONS
6.1
Conclusions
Construction techniques for two types of targets, volum e scattering targets and
surface/volum e interaction targets, were presented and m icrow ave m easurem ents
performed on these targets. In the case o f the volume scattering targets, all physical and
statistical parameters are measured before the target is built, so model input parameters are
defined in advance. From these experiments the following were determined:
•accurate scatterer positioning into their pre-defined locations can be effectively
accomplished through the use of a robot. Using this technique, volume fractions o f
at least 25% can be attained.
•the effects o f multiple scattering cause the scattering level to deviate considerably
from the single scattering prediction for volume fractions at least as low as 5%.
•volume scattering from targets consisting o f spheres exhibit very similar trends to
the scattering behavior of a single sphere for volume fractions up to at least 25%.
•coherent interaction effects can be neglected only if kd is much greater than a value
o f approxim ately two, where d is the average center-to-center spacing between
particles.
•DMT-IEM with coherent interaction can accurately m odel backscattering from
volume scattering targets consisting o f spheres.
In order to better understand the effects o f both surface and volume scattering from
sea-ice, a technique for fabricating saline-ice targets with known surface roughness was
presented. Backscatter measurements from smooth ice allows for the determination o f the
126
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
127
albedo. Hence, surface and volum e parameters are defined. Results from this portion o f
the experiment indicate:
•a saline-ice target with known surface roughness can be accurately fabricated
through the use o f a mold generated by a computer-controlled milling machine.
•com bining surface and volum e scattering m odels to predict total scattering is
correct to the first-order for saline-ice.
•the contribution due to volume scattering gradually increases with frequency and
incident angle, indicating that the lower penetration depth is compensated for by the
higher albedo.
6.2 Recommendations for Future Work
6 2 .1 Saline-ice Targets
T he fabrication technique employed in the use o f the saline-ice target produced a
physically realizable target with a known surface roughness. Construction of future targets
with different surface roughness will serve as further proof to the accuracy of models used.
The way in which the m easurem ents are performed could be also improved. Although
reliable data was obtained at certain frequencies, it is desired that the entire frequency span
produce accurate results. Hence, an improved calibration technique is required to include
the non-uniform illumination effects o f the antenna used. Since the target to be measured is
an area extensive target, it is suggested that an area extensive target with known scattering
behavior be used in conjunction with a point target for calibration. For example, a perfectly
conducting Kirchhoff-scale roughness target could be used as a calibration target, as could
the m old used to fabricate the rough ice boundary, if the mold was coated with highly
conductive paint as described in Section 3.2. This method will double the amount o f time
required to perform measurements but will greatly improve the results.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
128
6 .2 2 Volume Targets
The prelim inary results presented in Section 5.2 indicate that coherent effects are
dependent not only upon kd, but also on the transm it/receive angular combination. By
building more targets o f varying volume fractions and performing measurements on these
targets at various transmit/receive angular combinations, coherent interaction can be studied
m ore closely and its effects can be better defined as a function o f frequency, volume
fraction, scatterer size, polarization, and transm it and receive angle. In particular, it is
suggested that the W SRC build 5 ,1 0 , and 20% volume fraction targets in the same manner
as the 15 and 25% volume fraction targets and measure all five targets at various bistatic
combinations.
6 2 .3 Synthetic Sea-ice Targets
Since much o f the interest in dense media scattering is related to scattering from seaice, it would useful to apply the fabrication technique presented here for materials which
more closely resemble the electrical properties exhibited by sea-ice. Hence, a material with
a real dielectric constant around 2.5 to 4 and a small loss tangent should be used as a
background m aterial; foam spheres can simulate air pockets and brine pockets can be
approximated by metal coated spheres. Some o f the materials which may potentially serve
as a suitable background include epoxies and room temperature vulcanizations. It may be
possible to mix these materials with powders such as iron or carbon to obtain the desired
real and imaginary dielectric values.
Construction of the target would be very similar to the method described in Chapter
3. For example, an epoxy material would be poured into a thin container and allowed to
harden. H oles would be drilled and the scatterers put into place. Epoxy will then be
poured on top o f this layer, filling the holes and creating a second layer o f thickness.
Holes are then drilled in this layer, and the process is repeated until the entire target is built.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
129
A second possibility would involve creating a mold consisting o f cylinders o f various
height. The epoxy would be poured onto the mold and allowed to harden. The m old is
then removed and spheres would be placed in the areas vacated by the cylinders. The holes
are then filled with epoxy, the mold is rotated, and a second and subsequent layers would
be built. An approximate amount of rotation will be determined so as to maintain a uniform
distribution and independent scatterer positions.
W hen performing measurements from these targets, diffraction from the edges of
the target play an important role in the overall scattering. In order to eliminate this problem,
either a very large target is needed or a narrow beam antenna needs to be acquired. Since a
large target would be very costly as well as extrem ely heavy, a narrow beam antenna
should be obtained to perform the measurements.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
APPENDIX A
ISEED SELECTION PROGRAM
130
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
131
PROGRAM CHISQRA
Q* • k 'k 'k 'k 'k 'k 'k 'k 'k 'k * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
C
C
C
C
0*
c
WRITTEN BY: RON PORCO
DATE:
DECEMBER 15, 1992
THIS PROGRAM CREATES A UNIFORM DISTRIBUTION OF SCATTERERS
THROUGHOUT A CYLINDER. THE PROGRAM WILL TEST VARIOUS
ISEED VALUES IN SEARCH OF THE ONES THAT PRODUCE THE BEST
CHI-SQUARE TESTED Z LOCATIONS.
*
*
*
*
*
*
*
*
*
*
*
parameter par=300000
dimension x (30000),y (30000),z (30000), dif(100)
dimension count(30000),zdiv(30000),wksp(30000),iwksp(30000)
real*8 volfrac,rscat, rtarg,h,volscat,voltargl,volscats
real*8 r,voltarg,hlayr,zlayr
real*8 minspac, distavg, dscat,avglayr, chisqr
integer numscatsl,numscats, countera, i,numpts, count2,numlayr
integer numcyls,count3,lrgnum,counti,countil,counti3
c
pi=3.14159265358978
volfrac=10.0
rscat=0.7036
rtarg=27.94
h=25.4
numlayr=16
Cdo 11 lrgnum=l,171
C WRITE(*,*) 'INPUT ISEED FOR RANDOM NUMBER GENERATOR1
c READ(*,*)iseed
iseed = 9999
C
do 11 lrgnum=l, iseed
hlayr=h/numlayr
volscat=(4*pi*rscat**3)/3
voltargl=h*2*rtarg*2*rtarg
volscatsl=volfrac*voltargl*0.01
numscatsl=int(volscatsl/volscat)
voltarg=pi*h*rtarg**2
volscats=volfrac*voltarg*0.01
numscats=int(volscats/volscat)
dscat=2.*rscat
distavg=(100*volscat/volfrac)**(0.333333333333)
minspac=dscat+(distavg-dscat)/4.
C
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
CREATE THE RANDOM NUMBERS IN X-Y-Z COORDINATES AND
C
CUT OUT A CYLINDER FROM THE SQUARE SPECIFIED
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
^
countera=0
do 20 jj=l,numscatsl
z (jj)=ran0(lrgnum)*h
x (jj)=ran0(lrgnum)*2*rtarg
y(jj)=ran0(lrgnum)*2*rtarg
r=sqrt((x(jj)-rtarg)**2+(y(jj)-rtarg)**2)
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
132
20
c
if(r.gt.rtarg) then
z (jj)=99999.0
x (jj)=99999.0
countera=countera+1
endif
continue
numcyls=numscatsl-countera
cCwrite(*,*)'NUMBER OF POINTS USED IN CYLINDER = ',numcyls
C
C
C
SORT THE ARRAY FROM LOWEST TO HIGHEST Z AND WRITE TO
TARG.OUT.
0************************************************************
c
c
cl6
c
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
open (8, file=1chi.dat1,status='unknown1,form='formatted1)
call sort3(numscatsl,z,x, y, wksp,iwksp)
do 16 jj=l,numcyls
write(8, *)x(jj),y(jj),z(jj)
continue
close(8)
WE WISH TO DIVIDE THE CYLINDER INTO SECTIONS
OF EQUAL THICKNESS. FIRST WE MUST LOCATE WHERE THE
DIVISIONS IN THE TARGET TAKE PLACE, THAT IS, HOW MANY
SCATTERERS LIE IN EACH SEPARATE SECTION.WE ALSO NEED TO
FIND WHICH POINTS LIE IN TWO LAYERS AND DETERMINE IF AN
OVERLAP OCCURS IN ONE OR BOTH LAYERS.
EXAMPLE:
5 LAYERS, 4CM THICK, SCATTERER RADIUS=1CM
OVERLAPPING POSSIBILITIES OCCUR BETWEEN:
Z=0 TO 5CM
Z=4-1=3CM TO 8+l=9CM
Z=8-1=7CM TO 12+1=13CM
Z=12-1=11CM TO 16+1=17CM
Z=16-1=15CM TO 20CM
SO WE MIGHT HAVE TWO DIFFERENT OVERLAPS FOR, SAY Z=3.2CM
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
do 18 ii=l,numlayr-l
zlayr=hlayr*ii
zdiv(ii)=zlayr
Q ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
C
ZDIVO GIVES THEZ VALUE LIMITS WHERE THEOVERLAPS
MAY OCCUR
COUNT() GIVES THE RUNNING COUNT OFTHE LOCATION
OF THE LAST
POINT BEFORE EACH ZDIV().
18
continue
zdiv(numlayr)=h
count (numlayr) =numcyls
k=l
do 19 j=l,numcyls
if(z(j).gt.zdiv(k))then
count(k)=j-1
0*
**** *
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
133
k=k+l
endif
19
continue
avglayr=numcyls/numlayr
dif (1) =count (1)
chisqr=((dif(1)-avglayr)**2)/avglayr
do 22 j=2,numlayr
dif(j)=count(j)-count(j-1)
chisqr=chisqr+((dif(j) -avglayr)**2)/avglayr
22
continue
if(chisqr.le.4.0)then
write(*,*)lrgnum,numcyls, chisqr
ctype*,lrgnum,numcyls, chisqr
endif
11
continue
close (8)
c
step
end
C
C
C
Q*
11
12
SUBROUTINES
★ Vr * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
function ranO(idum)
dimension v(97)
data iff /0/
if(idum.lt.0.or.iff.eq.O)then
iff=l
iseed=abs(idum)
idum=l
do 11 j=l,9
dum=ran(iseed)
continue
do 12 j=l,97
v (j) =ran (iseed)
continue
y=ran(iseed)
endif
j=l+int (97.*y)
if(j.gt.97.or.j.It.1)pause
y=v(j)
ran0=y
v(j)=ran(iseed)
return
end
c
subroutine overlap(rscat, kstart,kstop,x,y,z,par
*, rtarg, nlim,space)
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
dimension x(par),y(par),z(par)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
134
26
27
17
integer kstop,kstart,nlim
real*8 rtarg,rscat,space, delx, dely, delz, dscat, distxy,distxyz
dscat=2.*rscat
do 17 k=kstart+l,kstop+l
do 27 n=0,k-nlim-l
delx=abs(x(k+1)-x(k-n))
dely=abs (y (k+1) -y (k-n))
distxy=sqrt(delx**2+dely**2)
delz=abs(z(k+1)-z(k-n))
distxyz=sqrt(delx**2+dely**2+delz**2)
if(distxy.le.dscat.or.distxyz.It.space)then
call numgen(k,par, x,y,rtarg)
goto 26
endif
continue
continue
return
end
c
c
subroutine numgen(k,par, x, y, rtarg)
£***************************************************************
dimension x(par),y(par)
integer k
real*8 rtarg
355
x(k+1)=ranO(lrgnum)*2*rtarg
y(k+1)=ranO(lrgnum)*2*rtarg
r=sqrt((x(k+1)-rtarg)**2+(y(k+1)-rtarg)**2)
if(r.gt.rtarg) then
goto 355
else
endif
return
end
c
subroutine sort3(N,ra,rb,rc, wksp,iwksp)
c
c
c
c
sorts an array ra of length n into ascending numerical order
using Heapsort algorithm, while making the corresponding
rearrangement of the arrays rb and rc.
An index table is
constructed via the routine INDEXX.
c
11
12
dimension ra(N),rb(N),rc(N),wksp(N),iwksp(N)
call indexx(n,ra,iwksp)
do 11 j=l,N
wksp(j)=ra(j)
continue
do 12 j=l,N
ra(j)=wksp(iwksp(j))
continue
do 13 j=l,N
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
135
13
14
15
16
wksp(j)=rb(j)
continue
do 14 j=l,N
rb(j)=wksp(iwksp(j))
continue
do 15 j=l,N
wksp(j)=rc(j)
continue
do 16 j=l,N
r c (j)=wksp(iwksp(j))
continue
return
end
c
subroutine indexx(N,arrin,indx)
£ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
c
c
indexes an array ARRIN of length n, i.e., outputs the array
indx such that arrin(indx(j)) is in ascending order for
j=l,2,..n.The input quantities n and arrin are not changed.
Q ***** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
11
10
dimension arrin (N),indx (N)
do 11 j=l,N
indx(j)=j
continue
if(l.eq.N)return
l=n/2+l
ir=n
continue
if (l.gt.l)then
1= 1-1
indxt=indx(1)
q=arrin(indxt)
else
indxt=indx(ir)
q=arrin(indxt)
indx(ir)=indx(1)
ir=ir-l
if(ir.eq.l)then
indx(1)=indxt
return
endif
20
endif
i=l
j=l+l
i f (j.le.ir)then
if(j.lt.ir)then
if(arrin(indx(j)).It.arrin(indx(j+1)))j=j+l
endif
if (q. It. arrin (indx (j)))then
indx (i) =indx (j)
i=j
j=j+j
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
136
else
j=ir+l
endif
go to 20
endif
indx(i)=indxt
go to 10
end
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
APPENDIX B
TARGET DATA GENERATION PROGRAM
137
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
138
PROGRAM TARG
0*★**★**★*★★★★************************************************
C
C
C
C
C
C
C
WRITTEN BY: RON PORCO
DATE:
MARCH 31, 1993
LAST UPDATE:SEPTEMBER 7, 1993
THIS PROGRAM CREATES A UNIFORM DISTRIBUTION OF SCATTERERS
THROUGHOUT A CYLINDER. THE PROGRAM WILL CUT THE CYLINDER
INTO EQUAL SECTIONS WITH THE CORRESPONDING SCATTERERS IN
THEIR PROPER SECTIONS.
*********************************************
integer max
parameter (max=30000)
double precision x (max),y (max),z (max),xx (max),yy (max)
double precision zz(max),phi(max)
double precision rcir (max),zdiv (max),wksp (max),iwksp (max)
double precision volfrac,rscat,rtarg,h,volscat,voltargl,volscats
double precision r,voltarg,hlayr,zlayr,vollcut,pi,f2
double precision minspac,distavg, dscat, wedgemax, wedgemin, rfoam
double precision avglayr, chisqrw,chisqrz,chisqrr, zmax, zmin, rmax
double precision hlayr2,drl2tbl, edge, offset, reduc, rminl6,maxcir
double precision ypick, zpick, xpick, censep, rmin, numdens, redge
double precision r r (20),shlvol(20), shlsct(20)
double precision gr (20),loc2,loc3,volsp, dist(max),sphvol(max)
integer zcut(20),rcut(20),wedge(20)
integer numscatsl,numscats,countera, i,numpts,count2,numlayr
integer numcyls,count3,lrgnum,counti, countil,counti3,evnodd
integer numcuts, j, k, count (2000), num, countr, teirp (1000)
integer ntot,loc,lamda,kk,numedge,numsl(20) ,n(20),begin
character*10 filenamea,filenameb
c
open(7, file=lrobot.out1,status='unknown1)
open(4,file=1chisqr.out',status='unknown1)
c
c
list constants and some parameters here!!!!
pi=3.14159265358978
C*****THE FOLLOWING PARAMETERS (IN CM) ARE: RADIUS OF THE SCATTERER
C
RADIUS OF THE TARGET AREA, HEIGHT OF THE TARGET, RADIUS OF THE
C
FOAM CYLINDERS, NUMBER OF CUTS USED IN THE CHI-SQUARE TESTS,
C
DISTANCE BETWEEN THE BOTTOM OF THE DRILL BIT
C
AND THE TOP OF THE TABLE, THE EXCESS EDGE OF FOAM AROUND THE
C
CIRCUMFERENCE OF THE TARGET, THE EXTRA THICKNESS OF THE TOP
C
AND BOTTOM FOAM SHEETS, THE NUMBER OF LAYERS, AND THE DISTANCE
C
BETWEEN SCATTERER CENTER POINTS WHICH THE OVERLAP ALGORITHM
C
WILL CHECK (USUALLY THE DIAMETER OF THE SCATTERER FOR SMALL
C
VOLUME FRACTIONS AND THE RADIUS OF A SCATTERER FOR LARGE
C
VOLUME FRACTIONS) .
rscat = 0.7036
rtarg = 27.94
h = 25.4
rfoam = 31.75
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
139
numcuts = 1 6
drl2tbl =9.53
edge =3.81
xtra = 0.9525
numlayr = 1 6
censep = rscat
c
WRITE(*, *) 'INPUT VOLUME FRACTION PERCENTAGE1
READ(*,*)volfrac
WRITE(*,*) 'INPUT ISEED FOR RANDOM NUMBER GENERATOR1
READ(*,*)iseed
WRITE(*,*) 'THE MAXIMUM NUMBER OF PTS IN CIRCLE 16 IS:'
READ(*,*)maxcir
C
do 20 lrgnum=l, iseed
volscat=(4*pi*rscat**3)/3
voltargl=h*2*rtarg*2*rtarg
volscatsl=volfrac*voltargl*0.01
numscatsl=int(volscatsl/volscat)
C
C
C
C
10
20
c
c
GO TO DESIRED ISEED VALUE AND
CREATE THE RANDOM NUMBERS IN X-Y-Z COORDINATES AND
CUT OUT A CYLINDER FROM THE SQUARE SPECIFIED
***************
countera=0
do 10 jj=l,numscatsl
z (jj) =ran0 (lrgnum) *h
x (jj)=ran0(lrgnum)*2*rtarg
y (jj)=ran0(lrgnum)*2*rtarg
r=sqrt((x(jj)-rtarg)**2+(y(jj)-rtarg)**2)
if(r.gt.rtarg) then
z(jj)=99999.0
x(jj)=99999.0
countera=countera+l
endif
continue
continue
hlayr=h/numlayr
voltarg=pi *h*rtarg* *2
volscats=volfrac*voltarg*0.01
numscats=int(volscats/volscat)
write(*,*)'NUMSCATS FOR RECTANGLE = \numscatsl
write(*,*)'NUMSCATS FOR CYLINDER = ',numscats
dscat=2.*rscat
distavg=(100*volscat/volfrac)**(0.333333333333)
minspac=dscat+(distavg-dscat)/4.
write(*,*)'AVG. DIST BETWEEN PARTICLES = ',distavg
write (*,*)'MIN. SPACING BETWEEN PARTICLES = ',minspac
c
numcyls=numscatsl-countera
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
140
write(*,*)'NUMBER OF POINTS USED IN CYLINDER =
numcyls
C
0
***********************************************************
C
C
SORT THE ARRAY FROM LOWEST TO HIGHEST Z AND WRITE TO
TARG.OUT.
c
c
c
cl6
c
C
open(8,file='targ.out',status='unknown1)
call sort3(numscatsl, z, x, y, wksp, iwksp)
do 16 jj=l,numcyls
write(8, *) x (jj) ,y (jj),z (jj)
continue
close(8)
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
C
C
C
C
C
C
C
C
C
C
C
C
0*
30
WE WISH TO DIVIDE THE CYLINDER INTO SECTIONS
OF EQUAL THICKNESS. FIRST WE MUST LOCATE WHERE THE
DIVISIONS IN THE TARGET TAKE PLACE, THAT IS, HOW MANY
SCATTERERS LIE IN EACH SEPARATE SECTION.WE ALSO NEED TO
FIND WHICH POINTS LIE IN TWO LAYERS AND DETERMINE IF AN
OVERLAP OCCURS IN ONE OR BOTH LAYERS.
EXAMPLE:
5 LAYERS, 4CM THICK, SCATTERER RADIUS=1CM
OVERLAPPING POSSIBILITIES OCCUR BETWEEN:
Z=0 TO 5CM
Z=4-1=3CM TO 8+l=9CM
Z=8-1=7CM TO 12+1=13CM
Z=12-1=11CM TO 16+1=17CM
Z=16-1=15CM TO 20CM
SO WE MIGHT HAVE TOO DIFFERENT OVERLAPS FOR, SAY Z=3.2CM
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
do 30 i=l,2*(numlayr-1) ,2
zlayr=hlayr*(i+1)/2
zdiv(i)=zlayr-rscat
zdiv(i+1)=zlayr+rscat
continue
0*********************************************************************
C
ZDIV () GIVES THE Z VALUE LIMITS WHERE THE OVERLAPS MAY OCCUR
C
COUNT() GIVES THE RUNNING COUNT OF THE LOCATION OF THE LAST
C
POINT BEFORE EACH ZDIV().
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
k=l
do 40 j=l,numcyls
if(z(j).gt.zdiv(k))then
count (k) = j-1
c
type*,k, zdiv (k),count (k)
k=k+l
endif
40
continue
C
C
C
C
FIND THE OVERLAPPING POINTS IN THE X-Y PLANE
AND REMOVE.KEEP THE SAME Z VALUES AND GENERAATE
A NEW X AND Y VALUE UNTIL THE OVERLAP CEASES TO
OCCUR
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
141
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
par=30000
c
c
c
do 50 i=-l,2*(numlayr-1),2
if(i.eq.-l)then
counti=0
countil=0
counti3=count(2)
else if(i+2.gt.2 * (numlayr-1))then
counti=count(i)
countil=count(i+1)
counti3=numcyls
else
counti=count(i)
countil=count(i+1)
counti3=count(i+3)
endif
numpts=count3-count (i)
*
call overlap(censep,countil,counti3,x,y,z,par
,rtarg, counti, minspac)
c
50
continue
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
SEARCH FOR EXTRA POINTS ALONG THE OUTERMOST RING OF THE
C
TARGET AND PUT THEM SOMEWHERE ELSE WITHIN THE TARGET
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
countr=0
i=l
RMIN16=SQRT ((numlayr-1) *V0LTARG/ (NUMCUTS*PI*H))
do 60 j=l,numcyls
r=sqrt ((x (j) -rtarg) **2+ (y (j) -rtarg) **2)
if(r.ge.rminl6)then
teirp (i) =j
xx(j)=2000
i=i+l
countr=countr+l
endif
60
continue
c
if(countr.gt.maxcir)then
do 80 j=l,countr-maxcir
61
pick=int (ranO (numcyls+j) * (countr))+1
if(xx(temp(pick)).It.1000)then
goto 61
endif
xpick=0.0
ypick=0.0
zpick=z(temp(pick))
do 70 k=l,2*(numlayr-1)
evnodd=k-int(k/2)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
142
70
80
c
if (zpick.le.zdiv (1)) then
counti=0
counti3=count (2)
else if(zpick.gt.zdiv(l).and.zpick.le.zdiv(2))then
counti=0
counti3=count(4)
else if(zpick.gt.zdiv(2).and.zpick.le.zdiv(3))then
counti=count (1)
counti3=count(4)
else if(zpick.gt.zdiv(2*numlayr-3))then
counti=count(2*numlayr-5)
counti3=numcyls
else if(zpick.gt.zdiv(k).and.zpick.le.zdiv(k+1).and.
*
evnodd.ne.int(k/2))then
counti=count(k-2)
counti3=count(k+3)
else if(zpick.gt.zdiv(k).and.zpick.le.zdiv(k+1).and.
*
evnodd.eq.int(k/2))then
counti=count(k-1)
counti3=count(k+2)
endif
continue
call overlap2(censep,counti,counti3,x,y,z,xpick,ypick,zpick,
*
par, rminl6, minspac, rtarg)
x (temp (pick))=xpick
xx (temp (pick))=xpick
y (temp (pick))=ypick
continue
endif
c
90
do 90 j=l,numcyls
write (7, *) x (j),y (j),z (j)
continue
close(7)
c
0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
C
C
C
C
THIS PORTION COMPUTES THE PDF OF THE TARGET BY RANDOMLY
CHOOSING 400 SCATTERER LOCATIONS AND COUNTING HOW FREQUENTLY
SCATTERERS LIE WITHIN ONE TO TWO SCATTER DIAMETERS, TWO TO
THREE SCATTERER DIAMETERS, ETC. OF THE REFERENCE SCATTERER'S
CENTER POINT. THIS NUMBER IS NORMALIZED TO THE EXPECTED NUMBER
LYING IN THOSE REGIONS.
Q ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
numdens = numcyls/voltarg
rr(l) = minspac
numsl(1)=0
ntot = 0
do 91 lamda = 2,15
rr(lamda) = dscat*(lamda*0.2+1)
rr (lamda) = dscat*larrda
numsl(lamda)=0
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
143
91
92
93
94
95
96
97
98
c
99
1000
shlvol (lamda) =1.33333333*pi* (((rr (lamda))**3) - ((rr (lamda-1))**3))
shlsct(lamda)=0.01*volfrac*shlvol(lamda)/volscat
n(lamda) = 0
gr(lamda) = 0
continue
begin = 999991
do 98 j = 1,400
loc = int (numcyls*ranO (begin))
redge = min(x(loc),y(loc),z(loc))
if(redge.le.dscat)goto 92
do 94 lamda=2,15
loc2 = redge-rr(lamda)
if(loc2.It.0.0)then
loc3 = rr(lamda-1)
volsp = 1 .3333*pi*(loc3)**3
do 93 kk=lamda-l,l,-l
numsl(kk)=numsl(kk)+1
continue
goto 95
endif
continue
numedge = numcyls
do 97 i = 1,numcyls
xx (i) =
x(i)-x(loc)
yy(i) =
y(i)-y(loc)
zz (i) =
z (i)-z (loc)
dist(i) = sqrt ((xx(i))**2 + (yy(i))**2 + (zz(i))**2)
if (dist(i).le.redge)then
do 96 lamda=2,15
if(dist(i).gt.rr(lamda-1).and.dist(i).le.rr(lamda))then
n(lamda) = n(lamda)+1
ntot = ntot+1
endif
continue
else
numedge = numedge-1
endif
continue
sphvol(j) = 3*numedge/(4*pi*(redge**3))
ntot = 0
continue
write(4,*)' dist/diam
g(r) '
write ( 4 , *) 1------------------- 1
do 99 lamda=2,8
gr(lamda) = n(lamda)/ (numsl(lamda)*shlsct(lamda))
vldnsl = numsl(lamda)/ (n(lamda)*shlvol(lamda))
type*, shlsct (lamda) ,n (lamda),numsl (lamda),shlvol (lamda)
write ( 4 , 1000) lamda, gr (lamda)
continue
format(5x, i2,7x, f5.3)
£ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
CALCULATE THE NUMBER OF SCATTERERS IN 16 CUTS IN THE
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
144
C
C
Z DIRECTION, WEDGE CUTS, AND EQUAL VOLUME CIRCLES. THEN
CALCULATE THE CHI-SQUARE GOODNESS OF FIT FOR EACH CASE
c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
100
C
110
120
130
V0L1CUT=SQRT (VOLTARG/ (NUMCUTS*PI*H))
CHISQRW=0.0
CHISQRZ=0.0
CHISQRR=0.0
DO 100 1=1,NUMCUTS
WEDGE(I)=0.0
ZCUT(I)=0.0
RCUT(I)=0.0
CONTINUE
DO 120 J=l,NUMCYLS
XX(J)=X(J)-RTARG
YY(J)=Y(J)-RTARG
RCIR(J)=SQRT((XX(J))**2 + (YY(J))**2)
if(xx(j).gt.O.O.and.yy(j).gt.0.0)then
P H I (J)=ATAN(YY(J)/XX(J))*180.0/pi
else if(xx(j).gt.O.O.and.yy(j).It.0.0)then
PHI(J)=ATAN(YY(J)/XX(J))*180.0/pi+360.0
else
PHI(J) =ATAN(YY(J)/XX(J))*180.0/pi+180.0
endif
DO 110 1=1, NUMCUTS
WEDGEMAX=(360.0/NUMCUTS)*1
WEDGEMIN=(360.0/NUMCUTS)*(1-1)
ZMAX=HLAYR*I
ZMIN=HLAYR*(1-1)
f2=I
RMAX=VOLlCUT *SQRT(f2)
RMIN=VOLlCUT*SQRT(f2-l)
IF (PHI (J) .GE.WEDGEMIN.AND.PHI (J) .L T .WEDGEMAX) THEN
WEDGE(I)=WEDGE(I)+1
ENDIF
IF (Z (J) .GE.ZMIN.AND.Z (J) ,LT.ZMAX)THEN
ZCUT (I) =ZCUT (I) +1
ENDIF
IF (RCIR (J) .GE.RMIN. AND. RCIR (J) .LT.RMAX) THEN
RCUT (I) =RCUT (I) +1
ENDIF
CONTINUE
CONTINUE
WRITE(4,*)1
WEDGE
Z
CIRCLE'
WRITE (4, *) '
CUTS
CUTS
CUTS'
WRITE (4, *) '------------------------------------- '
DO 130 1=1,NUMCUTS
WRITE(4,2000)I,WEDGE(I),ZCUT(I),RCUT(I)
CONTINUE
avglayr=numcyls/numcuts
do 140 j=l,NUMCUTS
chisqrw=chisqrw+((wedge(j)-avglayr)**2) /avglayr
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
145
140
C
2000
chisqrz=chisqrz+((zcut(j)-avglayr)**2)/avglayr
chisqrr=chisqrr+((rcut (j)-avglayr)**2)/avglayr
continue
WRITE(4,*)'CHI-SQUARE OF WEDGE CUTS = ',CHISQRW
WRITE(4,*)'CHI-SQUARE OF Z CUTS = ',CHISQRZ
WRITE(4,*)'CHI-SQUARE OF CIRCLE CUTS = ',CHISQRR
FORMAT(3X,13,3x,i4,5X,i4,5X,i4)
close(7)
close(4)
c*
150
do 150 i=l, numlayr-1
count (i)=count(2*(i—1)+1)
continue
count(0) = 0
count (16) = numcyls
Q ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C*
C*
C*
C*
C*
C*
C*
C*
C*
PUT THE COORDINATES INTO 32 DIFFERENT FILES
IN THE ROBOT'S COORDINATES. THERE IS ONE FILE
FOR EACH HALF OF EACH LAYER. ALLOW FOR AN EDGE AROUND
THE TARGET. IF A SCATTER LIES BETWEEN TOO LAYERS, IT
WILL DRILL THE FIRST LAYER AND INTO THE LOWER LAYER.
NOTE THAT THE FIRST AND LAST LAYERS MUST BE THICKER THAN
THE REST. THE EXTRA THICKNESS OF THE FIRST LAYER MUST
BE ACCOUNTED FOR WHEN CONVERTING THE Z-COMPONENTS OF THE
FIRST TWO LAYERS.
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
hlayr2=2 *hlayr
offset=drl2tbl-hlayr2
reduc=hlayr-xtra
c*
do 180 j=l,16
filenames(1:3) = 'rob'
filenameb(l:3) = 'rob'
if (j.le.9) then
filenames(4:4) = char(j+48)
filenameb(4:4) = char(j+48)
filenames(5:5) = 'a'
filenameb(5:5) = 'b'
filenames(6:9) = '.out'
filenameb(6:9) = '.out'
else
filenames(4:4) = '1'
filenameb(4:4) = '1'
filenames(5:5) = char(j—10+48)
filenameb(5:5) = char(j—10+48)
filenames(6:6) = 'a'
filenameb(6:6) = 'b'
filenames(7:10) = '.out'
filenameb(7:10) = '.out'
endif
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
146
open(unit=j,file=filenamea,status=1unknown1)
open(unit=j+numlayr,file=filenameb,status='unknown1)
c*
num=count (j)-count{j—1)
do 160 i=l,num
if(j.eq.l)then
z(i) = (j*hlayr-z(count(j—1)+i))+offset+rscat+reduc
else i f (j.eq.2)then
z(i) = (j*hlayr-z(count(j-l)+i))+offset+rscat-xtra
else
z(i) = (j*hlayr-z(count(j-l)+i))+offset+rscat
endif
c*
160
170
180
if(y(count(j-1)+i).gt.rfoam-edge)then
x(i) = 2*rfoam-x(count(j-1)+i)-edge+99.0
y (i) = 2 *rfoam-y(count(j-1)+i)-edge
else
x(i) = x(count(j-1)+i)+edge
y(i) = y(count(j-l)+i)+edge
endif
continue
call sort3(num,y,x,z, wksp, iwksp)
do 170 i=l,num
if(x(i).ge.99.0)then
x(i)=x(i)-99.0
write (j+numlayr, *)x(i) ,y(i) ,z(i)
else
write(j,*)x(i),y(i) ,z(i)
endif
continue
close(j)
close(j+numlayr)
continue
step
end
Q* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
-k
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
C
C
SUBROUTINES
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
11
* * * * * * * * * * * * *
function ranO(idum)
double precision v(97)
data iff /0/
if(idum.It.0.or.iff.eq.0)then
iff=l
iseed=abs(idum)
idum=l
do 11 j=l, 9
dum=ran(iseed)
continue
do 12 j=l,97
v(j)=ran(iseed)
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
147
12
continue
y=ran(iseed)
endif
j=l+int(97.*y)
if(j.gt.97.or.j.It.1)pause
y = v ( j)
ranO=y
v(j)=ran(iseed)
return
end
c
subroutine overlap(censep,kstart, kstop, x, y,z,par
*,rtarg, nlim, space)
******************************************************
c
double precision x(par),y(par),z(par)
integer kstop, kstart, nlim
double precision rtarg,censep, space, delx, dely,delz
double precision distxy, distxyz
do 17 k=kstart+l,kstop+l
do 27 n=0,k-nlim-l
delx=abs(x(k+1)-x(k-n))
dely=abs(y(k+1)-y(k-n))
distxy=sqrt(delx**2+dely**2)
delz=abs (z (k+1) -z (k-n))
distxyz=sqrt(delx* *2+dely**2+delz **2)
if(distxy.le.censep.or.distxyz.It.space)then
call numgen(k,par, x,y, rtarg)
goto 26
endif
continue
continue
return
26
27
17
end
c
subroutine numgen (k,par, x, y, rtarg)
q*
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
355
double precision x(par),y(par)
integer k
double precision rtarg
x(k+1)=ranO(lrgnum)*2*rtarg
y(k+1)=ranO(lrgnum)*2*rtarg
r=sqrt((x(k+1)-rtarg)**2+(y(k+1)-rtarg)**2)
if(r.gt.rtarg) then
goto 355
endif
return
end
c
*
subroutine overlap2 (censep, kstart, kstop, x, y, z, xpick, ypick,
zpick, par, rminl6, space, rtarg)
0****************************************************************
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
148
c
26
17
double precision x (par),y (par),z (par)
inteyer kstop,kstart
double precision rtarg,censep,space,delx,dely,delz
double precision distxy, distxyz
double precision xpick,ypick,zpick
double precision m i n i 6
call numgen2 (xpick, ypick, m i n i 6, rtarg)
do 17 k=kstart+l,kstop
delx=abs(xpick-x(k))
dely=abs(ypick-y(k))
delz=abs(zpick-z(k))
distxy=sqrt(delx**2+dely**2)
distxyz=sqrt(delx**2+dely**2+delz**2)
if(distxy.le.censep.or.distxyz.It.space)then
call numgen2(xpick,ypick,rminl6, rtarg)
goto 26
endif
continue
return
end
c
c
subroutine numgen2 (xpick, ypick, rminl6, rtarg)
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
355
double precision xpick,ypick
double precision minl6, rtarg
xpick=ranO(lrgnum)*2*rtarg
ypick=ranO(lrgnum)*2*rtarg
r=sqrt((xpick-rtarg)**2+(ypick-rtarg)**2)
if(r.gt.minl6) then
goto 355
endif
return
end
c
subroutine sort3(N,ra,rb,rc, wksp,iwksp)
Q *******************************************
c
c
c
c
sorts an array ra of length n into ascending numerical order
using Heapsort algorithm, while making the corresponding
rearrangement of the arrays rb and rc. An index table is
constructed via the routine INDEXX.
£ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
11
12
double precision ra(N),rb(N),rc(N),wksp(N),iwksp(N)
call indexx(n,ra, iwksp)
do 11 j=l,N
wksp(j)=ra(j)
continue
do 12 j=l,N
ra(j)=wksp(iwksp(j))
continue
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
149
13
14
15
16
do 13 j=l,N
wksp(j)=rb( j)
continue
do 14 j=l,N
rb(j)=wksp(iwksp(j))
continue
do 15 j=l,N
wksp( j)=rc( j)
continue
do 16 j=l,N
rc(j)=wksp(iwksp(j))
continue
return
end
c
subroutine indexx (N,arrin, indx)
£***********************************************
c
indexes an array ARRIN of length n, i.e., outputs the array
c
indx such that arrin (indx (j)) is in ascending order for
c
j=l/2, ..n.The input quantities n and arrin are not changed.
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
11
10
double precision arrin (N),indx (N)
do 11 j=l,N
indx (j) =j
continue
if(1.eq.N)return
l=n/2+l
ir=n
continue
if (l.gt.l) then
1= 1-1
20
indxt=indx(1)
q=arrin(indxt)
else
indxt=indx(ir)
q=arrin(indxt)
indx(ir)=indx(1)
ir=ir-l
if(ir.eq.l)then
indx(1)=indxt
return
endif
endif
i=l
j=l+l
i f (j.le.ir)then
if(j.lt.ir)then
if(arrin(indx(j)).It.arrin(indx(j+1)))j=j+l
endif
if (q. It. arrin (indx (j)))then
indx(i)=indx(j)
i=j
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
150
j=j+j
else
j=ir+l
endif
go to 20
endif
indx(i)=indxt
go to 10
end
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
A P P E N D IX C
C H I-S Q U A R E V A L U E S
151
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Table C. 1: Critical Values o f the
Chi-square Distribution [74],
a
V
.995
.99
.98
.975
I
2
3
4
5
.0*393
.0100
.0717
.207
.412
.0J157
.0201
.115
.297
.554
.0*628
.0404
.185
.429
.752
.0*982
.0506
116
.484
.831
.95
.90
.80
.75
.70
.50
.00393
.103
.352
.711
1.145
.0158
111
.584
1.064
1.610
.0642
.446
1.005
1.649
2143
.102
175
1113
1123
1675
.148
.713
1.424
1195
3.000
.455
1186
2166
3157
4151
6
7
S
9
10
.676
.989
1.344
1.735
2.156
.872
1139
1.646
1088
1558
1.134
1.564
1032
1532
3.059
1137
1.690
1180
1700
3147
1.635
1167
1733
3.325
3.940
1204
1833
3.490'
4.168
4.865
3.070
3.822
4194
5180
6.179
3.455
4155
5.071
5.899
6.737
3.828
4.671
5127
6193
7167
5148
6146
7144
8.343
9.342
11
12
13
14
15
2.603
3.074
3.565
4.075
4.601
3.053
3.571
4.107
4.660
5129
3.609
4.178
4.765
5168
5.985
3.816
4.404
5.009
5.629
6162
4.575
5126
5.892
6.571
7161
5178
6104
7.042
7.790
8147
6.989
7.807
8.634
9.467
10107
7.584
8.438
9199
10.165
11.036
8.148
9.034
9.926
10.821
11.721
10.341
11140
12140
13139
14.339
16
17
18
19
20
5.142
5.697
6.265
6.844
7.434
5.812
6.408
7.015
7.633
8160
6.614
7155
7.906
8.567
9137
6.908
7.564
8131
8.907
9.591
7.962
8.672
9190
10.117
10.851
9112
10.085
10.865
11.651
11443
11.152
11002
11857
13.716
14.578
11.912
11792
13.675
14.562
15.452
11624
13131
14.440
15152
16166
15138
16138
17.338
18138
19137
21
22
23
24
25
8.034
8.643
9.260
9.886
10.520
8.897
9.542
10.196
10.856
11.524
9.915
10.600
11193
11.992
11697
10.283
10.982
11.688
11401
13.120
11.591
12138
13.091
13.848
14.611
13140
14.041
14.848
15.659
16.473
15.445
16.314
17.187
18.062
18.940
16144
17140
18.137
19.037
19.939
17.182
18.101
19.021
19.943
20.867
20137
21.337
22137
23.337
24.337
26
27
28
29
30
11.160
11.808
11461
13.121
13.787
11198
11879
13.565
14.256
14.953
13.409
14.125
14.847
15.574
16.306
13.844
14.573
15.308
16.047
16.791
15.379
16.151
16.928
17.708
18.493
17192
18.114
18.939
19.768
20199
19.820
20.703
21188
21475
23.364
20.843
21.749
21657
23.567
24.478
21.792
21719
23.647
24.577
25.508
25.336
26.336
27.336
28.336
29.336
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
A P P E N D IX D
R O B O T IC D R IL L IN G P R O G R A M
153
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
154
.PROGRAM drill()
;AUTHOR:
Ron Porco
;LAST REVISION: June 23,1993
LOCAL $filename
LOCAL $list[]
LOCAL card
$filename = "a:robl6b.out"
CALL filelist($filename, $list[], rows)
i= 1
WHILE i < rows DO
CALL stringdec($list[i], trf[], 3)
x = 10*trf[l]
y = 10*trf[2]
z = 10*trf[3]
SET gotoplace = wsrcb.frame;TRANS(x,y,z,0,0,57.691)
offset = z-38
APPRO gotoplace, offset
SPEED 2
MOVE gotoplace
BREAK
SPEED 50
DEPART offset
i = i+1
END
.END
PRO G RA M filelist($file, $list[], row s); IMPORT DATA FILE
;AUTHOR: TIMOTHY PHILIPP
;REVISION: 06-14-1992
;ASSUMPTIONS: The necessary disk is mounted to offer the
;
specified file.
;USAGE CALL FELfilelist($file,$list[],rows)
•PROGRAM STARTS HERE:
LOCAL i
rows = 1
DETACH (8)
ATTACH (8)
FOPENR (8) $file
READ (8) $list[rows]
WHILE rows < 400 DO
rows = rows+1
READ (8) $list[rows]
TYPE $list[rows]
END
FCLOSE (8)
.END
.PROGRAM stringdec($string, numm[], ca rd ); DECODE $STRING INTO ARRAYf]
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
155
;AUTHOR: TIMOTHY PHILIPP
;REVISION 06-14-1992
;ASSUMPTIONS: The string should be a list o f numbers seperated by spaces.
;USAGE: CALLstringdec($stringofnumbers,arrayofnumbers[],cardinalityofarray)
;RESULTS:
;
;INPUT:
;
;OUTPUT:
;
Replaces the first given elements of array [] with the
the first given numbers found in and removed from Sstri*
$string o f numbers separated by spaces
cardinality o f the array or how many numbers specified to*
array [] o f numbers that came from the Sstring
Sstring is reduced by the numbers removed
•PROGRAM STARTS HERE:
LOCAL i, Stemp
i=0
DO
Stemp = $DECODE($string," ,",0)
numm[i] = VAL(Stemp)
Stemp = $DECODE($string," ,",1)
i = i+1
UNTIL i == card+1
TYPE num m [l], numm[2], numm[3]
RETURN
.END
.LOCATIONS
fix.ffame
-0.98815488 -0.15346005 0 -0.15346005 0.98815488 0
0 0 -1 729.16381 -123.57437 857.46856
gotoplace
0.84517794 0.5344851 0 0.5344851 -0.84517794 0
0 0 -1 367.51501 -334.19799 876.78973
joint4
0.79438984 0.60740828 0 0.60740828 -0.79438984 0
0 0 -1 577.89385 -191.26196 876.79968
junk
0.79797864 0.6026858 0 0.6026858 -0.79797864 0
0 0 -1 518.16656 -81.860839 876.77319
junk2
0.88605833 0.46357387 0 0.46357387 -0.88605833 0
0 0 -1 503.55755 -177.93557 876.69714
junk3
0.88912338 0.45766761 0 0.45766761 -0.88912338 0
0 0 -1 445.23422 -88.919311 876.72692
junk4
0.79438984 0.60740828 0 0.60740828 -0.79438984 0
0 0 -1 577.89385 -191.26196 876.79968
junk5
0.67196303 0.74058473 0 0.74058473 -0.67196303 0
0 0 -1 609.07135 -95.30603 876.71038
junk6
0.6408323 0.76768094 0 0.76768094 -0.6408323 0
0 0 -1 608.56744 -63.496002 876.73681
wsrca.ffame
0 1 0 10 0
0 0 -1 370.73498 -337.617 876.78997
wsrcb.frame
0 1010 0
0 0 - 1 367.51501 -334.19799 876.78973
.END
.REALS
1
386
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
156
offset
rows
trffO]
trfll]
trf[2]
trf[3]
x
y
-38
400
0
0
0
0
0
0
z
0
.END
.STRINGS
$list[l]
"1.070759
"
$list[2]
"0.9465142
"
$list[3]
"1.559276
"
$list[4]
"0.2227426
"
$list[5]
"2.397186
"
$list[6]
"7.5838774E-02"
$list[7]
"1.070479
"
$list[8]
"1.040921
"
$list[9]
"0.9927556
"
$list[10]
" 1.448488
31.06258
.END
1.751018
"
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
A P P E N D IX E
C A L IB R A T IO N P R O G R A M
157
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
158
The FILENAME for the TARGET:
V0L5B
The number of FREQUENCY POINTS used:
401
The number of AZIMUTHAL SWEEPS performed:
180
The STARTING FREQUENCY:
4
The STOPPING FREQUENCY:
16
The angle of rotation of the dihedral:
22.5
The FILENAME for the output CALIBRATED DATA:
vol5b.cal
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
159
program sr3tc
q*
C
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SINGLE REFERENCE, 3 TARGET CALIBRATION PROGRAM
Q ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
The following program performs a single reference, three
target calibration and error correction for monostatic
RCS measurements. It will read binary files produced by
HP BASIC and perform a fully polarimetrica calibration
(W,VH,HV, and HH) on a target using measured data from a
conducting sphere, a vertical dihedral, a rotated dihedral,
and theoretical values of the sphere. The calibrated data
will be output in ASCII in real and imaginary form. This
program is based on the paper by Wiesbeck, et. al., Proc
Proc of the IEEE vol. 79 #10 Oct 1991. The program also offers
the option to perform a 1-port calibration on like-polarized
data. The program will also perform a time domain analysis
on all four polarizations using the techniques described by
Brian Jersak in his thesis "Time Domain Analysis of Measured
Frequency Domain Radar Cross Section Data." The input file
for this program is xcal.parm.
Written by:
Last Modified:
Ron Porco
November 1,1993
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
parameter (maxpts=4096)
character*1 recal
character*8 ii, ij
character*10 infile,filename(70), outfile
real*8 rreal, imag
character*20 tarfile
character*20 calfile
character*80 charexplain
integer freqpts,anglpts,ang, freq, start, stop, caltype, incmnt
integer j,ll,kk,mm,np, filenum
integer i,n,indx(3),indx2(4)
complex*8 M0w,Md0w,Mdthw,tar(4,73000) ,M0hh,Md0hh,Mdthhh
complex* 8 MOvh, MdOvh, Mdthvh, MOhv, MdOhv, Mdthhv
complex* 8 SdOw, Sdthw, SdOvh, Sdthvh
complex*8 SdOhv, Sdthhv, SdOhh, Sdthhh, SO
complex*8 V(3,3) ,M(3) ,C(4,4),S (4),file(24,401)
complex*8 ell (401),cl2(401),cl3(401), cl4(401),c21(401), c22 (401)
complex*8 c23(401),c24 (401),c31(401), c32(401),c33(401),c34(401)
complex*8 c41(401),c42(401),c43(401),c44(401),cc41(401),ccl4(401)
corrplex*8 bistcal,denom(401)
double precision ts_r,ts_i,bistcal_r,bistcal_i,wdatr (maxpts)
double precision wdati(maxpts),vhdatr(maxpts),vhdati(maxpts)
double precision hvdatr(maxpts) ,hvdati(maxpts),hhdatr(maxpts)
double precision datar(maxpts),datai(maxpts),theta
double precision hhdati(maxpts)
complex*8 bl,b2,b3,b4,c22a,c22b,c23a,c23b,c33a,c33b,c32a,c32b
double precision strtfr,stopfr
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
160
double precision pi,tpi
pi = 3.141592653589793238462643D0
tpi = 6.283185307179586476925286D0
open(8,file='xcal.parm1,status=1old')
read(8,1000)charexplain
read(8, *)tarfile
read(8,1000)charexplain
read (8, *) fneqpts
read(8,1000)charexplain
read(8,*)anglpts
read(8,1000)charexplain
read(8,*)strtfr
read(8,1000)charexplain
read(8, *)stopfr
read(8,1000)charexplain
read(8,*)theta
read(8,1000)charexplain
read(8, *)calfile
close(8)
theta=theta*pi/180.0
filename (1) = 'EMPCHW0'
filename (2) = 'EMPCHVH0'
filename (3) = 'EMPCHHV0'
filename (4) = 'EMPCHHH0'
filenane (5) = 1E M P EMW01
filename (6)='EMPEMVH0'
filenane (7) = 1E M P EMM)'
filenane (8) = 1EMPFMHH01
filenane (9)=' SPHR8W01
filenane (10) = 'SPHR8VH0'
filenane (11) = 'SPHR8HV01
filenane (12) = 1SPHR8HH0'
filenane (13) = 'DIHD0W01
filename (14) = 'DIHD0VH0'
filename(15)='DIHD0HV0'
filename(16)=1DIHD0HH01
filename (17) = 'DIHDTW0'
filenane (18) = 'DIHDTVH01
filename (19) = 'DIHDTHV0'
filename (20) ='DIHDTHH0'
filenane (21) =' D I H DSW0'
filename(22)='DIHDSVH01
filename(23)='DIHDSHV0'
filename(24)=1DIHDSHH0'
filename(55)=1thersph1
*
write(*,*)'Do you wish to perform a single target
(1) calibration'
write(*,*)'or a three target (3) calibration?'
read(*,*)caltype
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
161
if (caltype.eq.l)then
write(*,*)'What is the polarization:'
write (*,*)
' W = > 1'
write(*,*)
'HH =>4'
read(*,*)start
incmnt=4
stop=12
else
start=l
incmnt=l
stop=24
endif
do 3 n=start,step,incmnt
open(65,file=filename(n),access='direct',
*
recl=l 6, fontt='unformatted',readonly)
c
HEADER***************************
do 5 i=l,48
read(65,rec=i)ii,ij
continue
5
13
3
c
c99
do 13 j=l,freqpts
read(65,rec=j+48)rreal, imag
file(n,j)=cmplx(rreal, imag)
continue
close(65)
continue
do 17, n=start+50,54,incmnt
do 17, n=start,4,incmnt
c******'pARGET*************************************
Q*
c*
c*
c*
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
There are angltps number ofbinary targetfiles. Each
of these are read and converted to asciiand then stacked
one on top of the other.
£*****************************************************************
infile(1:5)=tarfile
if(n.eq.l)then
infile(6:7)='W'
else if(n.eq.2)then
infile(6:7)='VH'
else if(n.eq.3)then
infile(6:7)='HV'
else
infile(6:7)='HH'
endif
do 16 j=0,anglpts-l
if(j.le.9)then
infile(8:8)=char(48+j)
infile(9:9)=char(0)
infile(10:10)=char(0)
else if(j.le.99)then
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
162
c
*
ll=int(j/10)
kk=j-ll*10
infile(8:8)=char(48+11)
infile(9:9)=char(48+kk)
infile(10:10)=char(0)
else
mm=int(j/100)
ll=int((j-mm*100)/10)
kk=j-mm*100-ll*10
infile (8:8) =char (48+rrm)
infile(9:9)=char(48+11)
infile(10:10)=char(48+kk)
endif
write(*,*)infile, j
open(65,file=infile,access='direct',
recl=16,form='unformatted',readonly)
c * * * * * * * * * * * * * * * * * * * * p g j^ g y E
h rrd rr* ************************ ******
do 7 i=l,48
read(65,rec=i)ii,ij
continue
7
£ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
15
16
17
do 15 i=l,freqpts
read(65,rec=i+48)rreal, imag
tar(n, (freqpts*j)+i) = crrplx (rreal, imag)
continue
close(65)
continue
outfile(1:5)=calfile
if(n.eq.l)then
outfile (6:10) ='w.ca'
filename (56)=outfile
else if(n.eq.2)then
outfile(6:10)='vh.ca'
filename(57)=outfile
else if(n.eq.3)then
outfile(6:10)='hv.ca'
filename(58)=outfile
else
outfile(6:10)='hh.ca'
filename (59)=outfile
endif
continue
£***** ******************** *************** *******************************
C
Q*
END OF BINARY-TO-ASCII CONVERSION
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
open(55,file=filename (55),status=lunknown1,form=!formatted')
if(caltype.eq.l)then
open (start+55, file=filename (start+55), status=Iunknown',
*
form=' formatted')
do 18, freq=l,freqpts
read(55,*)ts r,ts i
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
163
18
SO = cmplx (ts_r, ts_i)
denom(freq) = (file(start+8,freq) - file(start+4,freq))/SO
continue
close(55)
c
19
20
do 20 ang=0,anglpts-1
do 19 freq=l,freqpts
bistcal = (tar(start,(freqpts*ang)+freq) *
file(start, freq))/denom(freq)
bistcal_r = real(bistcal)
bistcal_i = aimag (bistcal)
write (start+55,900) bistcal__r,bistcal_i
if(ang.eq.O)then
datar(freq) = bistcal_r
datai(freq) = bistcal_i
endif
continue
continue
close(start+55)
filenum=start+24
call time(filenum,freqpts,strtfr, stopfr, datar, datai)
else
do 35, freq=l,freqpts
MOvv = file(9,freq) - file(5, freq)
MOvh = file(10, freq) - file(6, freq)
MOhv = file(11,freq) - file(7, freq)
MOhh = file (12, freq) - file (8, freq)
M d O w = file (13, freq) - file (21, freq)
MdOvh = file(14,freq) - file(22,freq)
MdOhv = file(15,freq) - file(23,freq)
MdOhh = file(16,freq) - file(24, freq)
M d t h w = file (17, freq) - file (21, freq)
Mdthvh = file(18,freq) - file(22, freq)
Mdthhv = file(19,freq) - file(23,freq)
Mdthhh = file(20, freq) - file(24,freq)
read (55, *) ts_r,ts_i
SO = cmplx (ts_r, ts_i)
c
SdOw
SdOhh
SdOvh
SdOhv
=
=
=
=
SO*MdOw/MOw
S0*Md0hh/M0hh
amplx(0,0)
cmplx (0,0)
c
Sdthw
*
Sdthhh
*
Sdthvh
Sdthhv
= ((cos (theta))**2) *SdOw +
((sin(theta))**2)*SdOhh
= ((cos(theta))**2)*Sd0hh +
((sin (theta))**2) *Sd0w
= cos (theta) *sin (theta) * (SdOhh - SdOw)
= Sdthvh
c
n=3
rp=3
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
164
V(l,l)
V (2,1)
V(3,l)
V (1,2)
V (2,2)
V(3,2)
V(l,3)
V(2,3)
V (3,3)
=
=
=
=
=
=
=
=
=
so
SdOw
Sdthw
aiplx (0,0)
SdOvh
Sdthvh
SO
SdOhh
Sdthhh
M(l) = MOvh
M(2) = MdOvh
M(3) = Mdthvh
call ludarp(V,n,np, indx,D)
call lubksb(V,n,np, indx,M)
c21(freq) = M(l)
bl = M(2)
c24 (freq) = M (3)
M(l) = MOhv
M(2) = MdOhv
M(3) = Mdthhv
call lubksb(V,n,np,indx,M)
c31(freq) = M(l)
b2 = M(2)
c34(freq) = M(3)
M(l) = MOhh
M(2) = MdOhh
M(3) = Mdthhh
call lubksb(V,n,np, indx,M)
c41(freq) = M(l)
b3 = M(2)
c44 (freq) = M (3)
M(l) = M O w
M(2) - M d O w
M(3) = M d t h w
call lubksb(V,n,np,indx,M)
cll(freq) = M(l)
b4 = M(2)
cl4 (freq) = M(3)
c22a = (bl + sqrt(bl**2 - 4*c21(freq)*c24(freq) ) ) 1 2
c22b = (bl - sqrt(bl**2 - 4*c21(freq)*c24(freq)))/2
c23a = c21(freq)*c24(freq)/c22a
c23b = c21(freq)*c24(freq)/c22b
if(cabs(c22a).gt .cabs(c23a))then
c22(freq) = c22a
c23(freq) = c23a
else
c22(freq) = c22b
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
165
c23(freq) = c23b
endif
c
c33a = (b2 + sqrt (b2**2 - 4*c31(freq)*c34(freq)))/2
c33b = (b2 - sqrt (b2**2 - 4*c31(freq)*c34(freq)))/2
c32a = c31(freq)*c34(freq)/c33a
c32b = c31(freq)*c34(freq)/c33b
if(cabs(c33a).gt.cabs(c32a))then
c33(freq) = c33a
c32(freq) = c32a
else
c33(freq) = c33b
c32(freq) = c32b
endif
c
cl2 (freq) = c22 (freq) *c34 (freq) /c44 (freq)
ctype*,cl2(freq)
cl2(freq) = ell(freq)*c34(freq)/c33(freq)
ctype*, cl2 (freq)
cl3(freq) = ell(freq)*c24(freq)/c22(freq)
c42(freq) = c22(freq)*c31(freq)/ell(freq)
c43(freq) = c21(freq)*c44(freq) /c22(freq)
ctype*,c43(freq)
c43(freq) = c21(freq)*c33(freq)/ell(freq)
ctype*, c43(freq)
c41(freq) = c21(freq)*c31(freq)/ell(freq)
cl4(freq) = c24(freq)*c34(freq)/c44(freq)
c
35
continue
close(45)
c
open (56, file=filename (56), status='unknown1,form=1formatted1)
open (57, file=filename (57), status=1unknown1,form=1formatted1)
open (58, file=filename (58), status= 'unknown', form=1formatted')
open (59, file=filename (59), status='unknown', form='formatted')
c
n=4
np=4
do 25, ancpO,anglpts-1
do 36, freq=l,freqpts
c
C(l,l) = ell(freq)
C (2,1) = c21(freq)
C(3,1) = c31(freq)
C (4,1) = c41 (freq)
C (1,2) = cl2(freq)
C (2,2)
c22(freq)
C (3,2)
c32(freq)
C (4,2) - c42 (freq)
C (1,3) = cl3(freq)
C (2,3) = c23(freq)
C(3,3) = c33(freq)
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
166
C(4,3)
C (1,4)
C (2,4)
C (3,4)
C (4,4)
=
=
=
=
=
c43(freq)
cl4 (freq)
c24 (freq)
c34 (freq)
c44(freq)
c
S (1) =
tar(l, (freqpts*ang)+freq)- filed,freq)
S(2) =tar(2,(freqpts*ang)+freq) - file(2, freq)
S (3) =tar(3,(freqpts*ang)+freq) - file(3, freq)
S (4) =tar(4,(freqpts*ang)+freq) - file(4, freq)
call ludcnp (C,n,np, indx2,D)
call lubksb(C,n,np,indx2,S)
write (56,900) S(l)
write (57,900) S(2)
write(58,900) S (3)
write(59,900) S(4)
if (ang.eq.O) then
w d a t r (freq) = real (S (1))
w d a t i (freq) = aimag (S (1))
vhdatr(freq) = real(S(2))
vhdati(freq) = aimag(S(2))
hvdatr(freq) = real(S(3))
hvdati(freq) = aimag(S(3))
hhdatr(freq) = real(S(4))
hhdati(freq) = aimag(S(4))
endif
36
continue
25
continue
call time (25, freqpts, strtfr, stopfr, wdatr, wdati)
call time(26,freqpts,strtfr,stopfr,vhdatr, vhdati)
call time(27, freqpts,strtfr,stopfr,hvdatr, hvdati)
call time(28, freqpts,strtfr,stopfr,hhdatr, hhdati)
close(56)
close(57)
close(58)
close(59)
endif
900
format(f14.8,3x,f14.8)
1000
format(80a)
cwrite(*,*)'Do you wish to calibrate another target?1
cread(*,*)recal
cif(recal.eq.'Y'.or.recal.eq.'y')then
c write(*,*)'NOTE: If single ref used, must use same polarization'
c write(*,*)'What is the basename(5 char) of the targ input file(s)?'
c read(*,*)tarfile
c write(*,*)'What is the basename(5 char) of the calibrated output?'
c read(*,*)calfile
c goto 99
cendif
write(*,*)'Program Completed'
c
step
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
167
end
c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
SUBROUTINES
c
SUBROUTINE LUDcnp (A,nsize, nuse, Indx, d)
C
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * £
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Purpose:
Matrix LU Decomposition
Parameters:
A : nuse*nuse matrix with physical
dimension nsize. Replaced with
the LU decomposed on return
nsize : physical matrix size
nuse : matrix size used
Indx : row permutation record vector
d : number of row interchanges, even
orodd (+/- 1.0)
Reference:
Numerical Recepies
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c
C—
IMPLICIT CHARACTER (A-Z)
INTEGER nsize, nuse, NSZ, indx (nsize),d
PARAMETER ( NSZ=32)
real*8 aamax,dum,W(NSZ)
complex*8 A (nsize,nsize),tiny
local variables
complex*8 sum,cdum
INTEGER i,j,k,imax
tiny=cmplx(1.0e-35,0.0)
C
11
12
C—
d=l
DO 12 i=l,nuse
AAmax=0.0
DO 11 j=l,nuse
IF (cabs (A(i, j)) .GT.AAmax) AAmax=cABS(A(i,j))
CONTINUE
IF (AAmax.EQ.0.0) THEN
PRINT*,1 SINGULAR MATRIX !!!!! 1
RETURN
ENDIF
W(I)=1.0/AAmax
CONTINUE
loop over columns of Crouts method
DO 19 j=l,nuse
DO 14 i=l,j-1
sunf=A(i, j)
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
168
13
14
C—
15
16
17
18
19
DO 13 k=l,i-1
sum=sum-A(i,k) *A(k, j)
CONTINUE
A(i, j)=sum
CONTINUE
search for the largest pivot element
AAmax=0.0
DO 16 i=j,nuse
sum=A(i,j)
DO 15 k=l,j-1
sum=sum-A(i,k) *A(k, j)
CONTINUE
A(i, j)=sum
dum=W(i) *cABS (sum)
IF (dum.GE.AAmax) THEN
imax=i
AAmax=dum
ENDIF
CONTINUE
IF (j.NE.imax) THEN
DO 17 k=l,nuse
cdum=A(imax,k)
A(imax,k)=A( j,k)
A(j,k)=cdum
CONTINUE
d=-d
W(imax)=W(j)
ENDIF
Indx(j)=imax
IF(cabs(A(j,j)).EQ.O.0) THEN
PRINT*,1 SINGULARITY RESULTED, FIXED WITH TINY !!'
A (j,j)=TINY
EM) IF
IF (j.NE.nuse) THEN
cdum=1.0/A(j, j)
DO 18 i=j+l,nuse
A(i, j)=A(i, j) *cdum
CONTINUE
ENDIF
CONTINUE
C
non
RETURN
END
SUBROUTINE LUBksb (A,nsize, nuse, Indx, B)
C
0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * £
C
C
C
Purpose:
Solve a set of 'nuse' linear equations
A*X=B.Where A is in LU form.
C
C
C
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
169
c
C
C
C
C
C
C
C
C
C
C
C
c
Parameters:
A : LU decomposed matrix
nsize : physical matrix size
nuse : matrix size used
Indx : permutation vector return by
subroutine LUDcmp
B : right hand side vector
Reference:
Numerical Recipies
C
C
C
C
C
C
C
C
C
C
C
Q *** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * £
c
C—
IMPLICIT CHARACTER (A-Z)
INTEGER nsize, nuse
complex*8 A (nsize,nsize),B(nsize),sum
integer Indx(nsize)
local variables
INTEGER i,j,ii,ll
C
11
12
ii=0
DO 12 i=l,nuse
ll=Indx(i)
sum=B(ll)
B(ll)=B(i)
IF (ii.NE.O) THEN
DO 11 j=ii,i-l
sum=sum-A(i, j) *B(j)
CONTINUE
ELSE
IF (cabs(sum),NE.0.0) ii=i
ENDIF
B(i)=sum
CONTINUE
C
13
14
DO 14 i=nuse,1,-1
sum=B (i)
IF (i.LT.nuse) THEN
DO 13 j=i+l,nuse
sum=sum-A(i, j)*B(j)
CONTINUE
ENDIF
B(i)=sum/A(i,i)
CONTINUE
C
RETURN
END
C
SUBROUTINE TIME(filenum,nfqpts,strtfr, stopfr,datar,datai)
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C*****PARAMETERS FOR THE TIME DOMAIN DATA GENERATION
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
170
0*
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
parameter (maxpts=4096)
character*10 filename(40)
integer nfqpts,ntipts,wtype
integer fftsze,nzeros,cpoint,n,ptwo, filenum
double precision datar(maxpts),datai(maxpts)
double precision strtfr,stopfr,frqinc, frqsft
double precision strttm, stoptm,timinc,maxtm
double precision tempr, tempi, temp, corfct
double precision pi, tpi,arg, wvalue, datn2d
pi = 3.141592653589793238462643D0
tpi = 6.283185307179586476925286D0
fftsze=2048
wtype=3
strttm=-10.0
stoptm=10.0
ptwo=ll
c***note:2**11=2048*******
THREE TYPES OF WINDOWS AVAILABLE ARE:
c
2=> Hanning window (minimum)
c
3=> 3-sairple Blackman-Harris (normal)
c
4=> 4-sample Blackman-Harris (maximum)
0*
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
filename (25) = 'timew'
filename (26) = 'timevh1
filename(27) = 'timehv'
filename(28) = 'timehh1
frqinc=(stopfr-strtfr)/dble(nfqpts-1)
frqsft=(strtfr+stopfr)/2.0D0
maxtm=l.ODO/frqinc
timinc=maxtm/dble(fftsze)
ntipts=int(((stoptm-strttm)/timinc) +1. 0D-5)+1
cpoint=(nfqpts+1)/2
if(ntipts.gt.maxpts)then
write(*,*)'TOO MANY TIME POINTS!!!1
write(*,*)'Program will continue with 4096 points'
ntipts=maxpts
endif
c******PERFORM WINDOWING ON THE FREQUENCY DATA**********
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
*
50
if (wtype .eq. 2) then
corfct = 0.548501251d0 - 0.00795454170d0 *
dlog(dble(nfqpts))
do 50 n = 1, (cpoint - 1)
wvalue = 0.5d0 * (l.OdO - dcos(tpi * n / dble(nfqpts)))
datar(n) = datar(n) * wvalue / corfct
datai(n) = datai(n) * wvalue / corfct
continue
datar(cpoint) = datar(cpoint) / corfct
datai(cpoint) = datai(cpoint) / corfct
do 60 n = (cpoint + 1), nfqpts
wvalue = 0.5d0 * (l.OdO - dcos(tpi * (nfqpts + 1 - n)
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
171
*
60
*
*
*
70
*
*
*
*
80
/ dble(nfqpts)))
datar(n) = datar(n) * wvalue / corfct
datai(n) = datai(n) * wvalue / corfct
continue
else if (wtype .eq. 3) then
corfct = 0.428399637d0 + 0.00347545132d0 *
dlog(dble(nfqpts))
do 70 n = 1, nfqpts
wvalue = 0.44959d0 - 0.49364d0 * dcos(tpi * (n - 1) /
dble(nfqpts - 1)) + 0.05677d0 * dcos(2.0d0 *
tpi * (n - 1) / dble(nfqpts - 1))
datar(n) = datar(n) * wvalue / corfct
datai(n) = datai(n) * wvalue / corfct
continue
else if (wtype .eq. 4) then
corfct = 0.341355266d0 + 0.00285276577d0 *
dlog(dble(nfqpts))
do 80 n = 1, nfqpts
wvalue = 0.35875d0 - 0.48829d0 * dcos (tpi * (n - 1) /
dble(nfqpts - 1)) + 0.14128d0 * dcos(2.0d0 *
tpi * (n - 1) / dble(nfqpts - 1)) - 0.01168d0 *
dcos(3.0d0 * tpi * (n - 1) / dble(nfqpts - 1))
datar(n) = datar(n) * wvalue / corfct
datai(n) = datai(n) * wvalue / corfct
continue
end if
c
c******pj2PPOPM TIME-SHIFT PRE-MULTIPLICATION************
90
if (dabs (strttm) .gt. 1.0d-10) then
temp = tpi * strttm
do 90 n = 1, nfqpts
arg = temp * (strtfr + (n - 1) * frqinc)
tempr = datar(n) * dcos(arg) - datai(n) * dsin(arg)
datai(n) = datai(n) * dcos(arg) + datar(n) * dsin(arg)
datar(n) = tenpr
continue
end if
c
C******ZERO PAD THE WINDOWED FREQUENCY DATA***********
Q ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
100
nzeros = fftsze - nfqpts
do 100 n = 1, (cpoint - 1)
temp = datar(n)
datar(n) = datar(n + cpoint - 1)
datar(n + cpoint - 1) = temp
tenp = datai(n)
datai(n) = datai(n + cpoint - 1)
datai(n + cpoint - 1) = temp
continue
tenpr -- datar(nfqpts)
tempi = datai(nfqpts)
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
172
do 110 n = fftsze, (fftsze - cpoint + 2), -1
datar(n) = datar(n - nzeros - 1)
datai(n) = datai(n - nzeros - 1)
continue
datar(cpoint) = tenpr
datai(cpoint) = tempi
do 120 n = (cpoint + 1), (cpoint + nzeros)
datar(n) = O.OdO
datai(n) = 0.OdO
continue
110
120
c
C*****TAKE THE INVERSE FAST FOURIER TRANSFORM***********
do 130 n = 1, fftsze
datai(n) = -datai(n)
continue
call fft(datar, datai, fftsze, ptwo)
do 140 n = 1, fftsze
datai(n) = -datai(n)
continue
130
140
c
C**PERFORM FREQUENCY-SHIFT POST-MULITPLICATION**********
if (dabs(frqsft) .gt. 1.0d-10) then
temp = tpi * frqsft * timinc
do 150 n = 1, ntipts
arg = dble(n - 1) * temp
tenpr = datar(n) * dcos(arg) - datai(n) * dsin(arg)
datai(n) = datai(n) * dcos(arg) + datar(n) * dsin(arg)
datar(n) = tenpr
continue
end if
150
c
C**WRITE OUTPUT DATA TO DISK****************
q*
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
open(filenum, file=filenane (filenum),status='unknown'
,form=1formatted1)
write(filenum,*)1
Time(ns)
Magnitude(db) Phase(deg)1
write (filenum, *) 1--------------------------------------------1
do 160 n = 1, ntipts,4
terrp = strttm + (n - 1) * timinc
tenpr = 20.OdO * dloglO(dsqrt(datar(n) * datar(n) +
*
datai(n) * datai(n)) / dble(nfqpts))
tempi = datn2d(datai(n), datar(n), pi)
write (filenum,2000) temp, tenpr,tempi
160
continue
temp = stoptm
tenpr = 20. OdO * dloglO (dsqrt (datar (ntipts) * datar (ntipts) +
*
datai(ntipts) * datai(ntipts)) / dble(nfqpts))
tempi = datn2d(datai(ntipts), datar(ntipts), pi)
write (filenum,2000) temp, tenpr,tempi
2000
format (Ix,f9.4,3x,f9.4,3x, f9.4)
*
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
173
close(filenum)
return
end
c
double precision function datn2d(y, x, pi)
c
c
c
c
c
c
c
this function returns the arctangent of the two arguments
x represents the real value and y represents the imaginary value
the resulting angle will be in degrees and range from -180 to +180
implicit none
double precision x, y, pi
c
c
c
datn2d = O.OdO
if ((x .eq. O.OdO) .and. (y .eq. O.OdO)) datn2d = O.OdO
if ((y .eq. O.OdO) .and. (x .gt. O.OdO)) datnd2 = O.OdO
if ((x .gt. O.OdO) .and. (y .ne. O.OdO)) then
datn2d = datan(y / x) * 180.OdO / pi
else if ((x .It. O.OdO) .and. (y .gt. O.OdO)) then
datn2d = datan(y / x) * 180.OdO / pi + 180.OdO
else if ((x .It. O.OdO) .and. (y .It. O.OdO)) then
datn2d = datan(y / x) * 180.OdO / p i - 180.OdO
else if ((x .eq. O.OdO) .and. (y .gt. O.OdO)) then
datn2d = 90.OdO
else if ((x .eq. O.OdO) .and. (y .It. O.OdO)) then
datn2d = -90.OdO
else if ((y .eq. O.OdO) .and. (x .It. O.OdO)) then
datn2d = 180.OdO
end if
return
end
c
c ------------------------------------------c
subroutine fft(x, y, n, m)
n n
c
c
c
c
c
c
c
this is a standard fast fourier transform routine
the real input array is in the x array and the imaginary
input array is in the y array
the real and imaginary output data is placed in these same
two arrays
implicit none
double precision x(*), y(*), e, c, cl, s, si, xt, yt, t
c
integer i, j, k, 1, m, n, nl, n2
c
n2 = n
do 10 k = 1, m
nl = n2
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
174
n2 = n2 / 2
e = 6.283185307179586d0 / dble(nl)
c = l.OdO
s = O.OdO
cl = dcos(e)
si = dsin(e)
do 20 j = 1, n2
do 30 i = j, n, nl
1 = i + n2
xt
= x(i) - x(l)
x(i)
= x (i) + x(l)
yt
= y(i) - y (1)
y (i)
= y(i) + yd)
x(l) = c * x t + s * y t
yd) = c * y t - s * x t
30
continue
t = c
c= c*c l - s * s l
s = t*sl + s*cl
20
continue
10 continue
j = 1
nl = n - 1
do 40 i = 1, nl
if (i .It. j) then
xt
= x(j)
x (j) = x (i)
x(i) = xt
yt
= y(j)
y (j) = yd)
yd) = yt
end if
k= n/ 2
60
if (k.ge. j) goto 50
j =j - k
k =k / 2
goto 60
50
j = j+ k
40 continue
return
end
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX F
TIME GATING AND DATA SMOOTHING PROGRAM
175
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
176
The # of frequency points per sweep =>
401
The Start Frequency =>
4.0
Stop Frequency =>
16.0
The Number of Sweeps or Angular Samples =>
180
The Time Center for Gating (ns):
- 0 .2
The Time Span for Gating (ns):
2.8
The Smoothnum Value =>
6
The Index for the File Sampling (l=all files, 2=every other file, etc.)
1
Outflag Value => (0=every 50th point, l=25th point, 2=4th point, 3=all)
2
Confidence Interval Chosen (0 = 90%, 1 = 95%, 2 = 99%)
1
The FILENAME for the Vector Corrected INPUT DATA =>
vol5bw.ca
The FILENAME for the GATED OUTPUT DATA:
vol5bw.ga
The FILENAME for the SMOOTHED OUTPUT DATA =>
vol5bw.rs
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
177
program gatsmth
c
TIME GATING AND DATA SMOOTHING PROGRAM
c
c
c
c
c
c
c
c
c
c
The following program will perform time gating on frequency
domain data provided in real and imaginary form and then
smooth the data. This program is combination of past programs
written by Dr. Brian Jersak in his Master's Thesis, "Time
Domain Analysis of Measured Frequency Domain Radar Cross Section
Data," and Dr. Eric Nance in his PhD Dissertation, "Scattering
and Image Analysis of Conducting Rough Surfaces."
Last Modified:
Modifications made by:
February 1, 1994
Ron Porco
Q ************************************************
c
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
freqr(401), freqi(401), smoothr(401), smoothi(401)
datar(4096), datai(4096), gater(4096), gatei(4096)
LPcoefr(75), LPcoefi(75), futurer(150), futurei(150)
startfreq, stopfreq, centertime, timespan
tenpr, tempi, temp, wvalue, cutofffreq
pi, tpi, maxtimerange
fdatar(401), fdatai(401), magsqsweep(401)
avgdata(401), savgdata(401)
magsqsmooth(401), smagsqavgdata(401)
powersq(401), avgpowersq(401), stdev(401)
savgpowersq(401), smstdev(401)
lower(401),upper(401), freqincr, stacknum
rselect,flag,bandsmooth,freq,confvalue
c
integer
integer
integer
integer
integer
integer
integer
numfreqpoints, windowtype, gatetype
fftsize, powertwo, numcoef, m
centerpoint, numzeros, i, j, n, numsweeps
numpoles, numfut, default, totnunpoints
smoothnum
scounter, fileincr, iselect
outflag,numprint,outstep, conflag
c
character*1 ht,skip
character*80 infile,gatefile, smoothfile, charexplain
c
complex f (401),data(101)
integer kk
c
c
initialize variables and set defaults
ht = char(9)
fftsize = 2048
powertwo = 11
! 2048 = 2^11
pi = 3.141592653589793238462643
tpi = 6.283185307179586476925286
gatetype = 2
windowtype = 2
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
178
nurnpoles = 30
numfut = 50
numcoef = 1000
m = 7
default = 1
c
c***********pEM) i n p u t d a t a *************
open (1, file='gatsmth.parm', status='old', readonly)
read(l,1000) charexplain
read (1,*) numfreqpts
read (1,1000) charexplain
read(1,*) startfreq
read(1,1000) charexplain
read(1,*) stopfreq
read (1,1000) charexplain
read(1,*) numsweeps
a d GATING INFORMATION**********
read(l,1000) charexplain
read(l,*) centertime
readd, 1000) charexplain
read(l,*) timespan
SMOOTHING INFORMATION**********
read(l,1000) charexplain
read(1,*) smoothnum
read(1,1000) charexplain
read(1,*) fileincr
readd, 1000) charexplain
read(l,*) outflag
readd, 1000) charexplain
read(1,*) conflag
c*************pg^D f t t.
f m &m r r **********
read(1,1000) charexplain
read(1,1000) infile
read(1,1000) charexplain
read(1,1000) gatefile
read(1,1000) charexplain
readd, 1000) smoothfile
c
close (1)
write(*,*) 'Do you wish to skip the gating portion?1
read(*,79) skip
if (skip.eq.'y'.or.skip.eq.'Y') then
write(*,*)'What is the input file to be smoothed?'
read(*,1000)gatefile
goto 77
endif
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C*********CALCULATE VARIABLES
totnumpoints = numfreqpts + 2 * numfut
centerpoint = (totnumpoints + 1) / 2
c
C
GENERATE GATE COEFFICIENTS
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
179
o o
TOTAL OF (2 * NUMOOEF + 1) GATE VALUES
maxtimerange = (numfreqpts - 1) / (stopfreq - startfreq)
cutofffreq = pi * timespan / maxtimerange
gater(l) = timespan / maxtimerange
do 10 n = 2, (numcoef + 1)
if (gatetype .eq. 2) then
wvalue = 0.5 * (1.0 + dcos(tpi * (n - 1) / (2 * numcoef)))
else if (gatetype .eq. 3) then
wvalue = 0.54 + 0.46 * dcos (tpi * (n - 1) / (2 * numcoef))
else if (gatetype .eq. 4) then
wvalue = 0.42 + 0.5 * dcos(tpi * (n - 1) / (2 * numcoef))
*
+ 0.08 * dcos(2.0 * tpi * (n - 1) / (2 * numcoef))
else
wvalue = 1 . 0
end if
gater(n) = wvalue * dsin(cutofffreq * (n - 1)) / (pi * (n - 1))
gater(fftsize - n + 2) = gater(n)
10 continue
do 20 n = (numcoef + 2), (fftsize - numcoef)
gater(n) = 0 . 0
20 continue
do 30 n = 1, fftsize
gatei(n) = 0 . 0
30 continue
call fft(gater, gatei, fftsize, powertwo)
99
c
c
c
70
60
90
open(2,file=infile,status='old1,readonly)
do 40 i = 1, numsweeps
do 99,n=l,numfreqpts
read(2,*)freqr(n),freqi(n)
continue
smooth data for use in the maximum entropy method
(2m+l points)
do 60 n = 1, m
tempr = 0 . 0
tempi = 0 . 0
do 70 j = 1, (n + m)
tempr = tenpr + freqr(j)
tempi = tempi + freqi(j)
continue
smoothr(n) = tempr / dble(n + m)
smoothi(n) = tempi / dble(n + m)
continue
do 80 n = (m + 1), (numfreqpts - m)
tempr = 0.0
tenpi = 0 . 0
do 90 j = (n - m), (n + m)
tempr = tempr + freqr (j)
tempi = tempi + freqi (j)
continue
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
180
80
110
o o
100
smoothr(n) = tempr / dble(2 * m + 1)
smoothi(n) = tempi / dble(2 * m + 1)
continue
do 100 n = (numfreqpts - m + 1), numfreqpts
tempr = 0 . 0
tempi = 0 . 0
do 110 j = (n - m), numfreqpts
tempr = tempr + freqr(j)
tenpi = tempi + freqi(j)
continue
smoothr(n) = tempr / dble(numfreqpts - n + m + 1)
smoothi(n) = tenpi / dble(numfreqpts - n + m + 1)
continue
extend data
call memcoef(smoothr, numfreqpts, nunpoles, LPcoefr)
call memcoef(smoothi, numfreqpts, nunpoles, LPcoefi)
call predictor(smoothr, numfreqpts, LPcoefr, nunpoles,
*
futurer, numfut)
call predictor(smoothi, numfreqpts, LPcoefi, nunpoles,
*
futurei, numfut)
do 120 n = (numfut + 1), (numfut + numfreqpts)
datar(n) = freqr(n - numfut)
datai(n) = freqi(n - numfut)
120
continue
do 130 n = (numfut + numfreqpts + 1), totnumpoints
datar(n) = futurer(n - numfut - numfreqpts)
datai(n) = futurei(n - numfut - numfreqpts)
130
continue
do 140 n = 1, ((numfreqpts - 1) / 2)
temp = smoothr(n)
smoothr(n) = smoothr(numfreqpts - n + 1)
smoothr (numfreqpts - n + 1) = t o p
tenp = smoothi (n)
smoothi(n) = smoothi(numfreqpts - n + 1)
smoothi(numfreqpts - n + 1) = tenp
140
continue
call memcoef(smoothr, numfreqpts, nunpoles, LPcoefr)
call memcoef(smoothi, numfreqpts, nunpoles, LPcoefi)
call predictor(smoothr, numfreqpts, LPcoefr, nunpoles,
*
futurer, numfut)
call predictor(smoothi, numfreqpts, LPcoefi, nunpoles,
*
futurei, numfut)
do 150 n = 1, numfut
datar(n) = futurer(numfut + 1 - n)
datai (n) = futurei (numfut + 1 - n)
150
continue
c
c
perform windowing on the frequency data
if (windowtype .eq. 2) then
do 160 n = 1, (centerpoint - 1)
wvalue = 0.5 * (1.0 - dcos(tpi * n / totnumpoints))
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
181
datar(n)
datai(n)
continue
do 170 n =
wvalue =
160
*
170
*
*
180
*
*
*
190
*
200
c
c
= datar(n) * wvalue
= datai(n) * wvalue
(centerpoint + 1), totnumpoints
0.5 * (1.0 - dcos(tpi * (totnunpoints + 1 - n)
/ totnunpoints))
datar(n) = datar(n) * wvalue
datai(n) = datai(n) * wvalue
continue
else if (windowtype .eq. 3) then
do 180 n = 1, totnunpoints
wvalue = 0.44959 - 0.49364 * dcos(tpi * (n - 1) /
(totnunpoints - 1)) + 0.05677 * dcos(2.0 * tpi *
(n - 1) / (totnunpoints - 1))
datar(n) - datar(n) * wvalue
datai(n) = datai(n) * wvalue
continue
else if (windowtype .eq. 4) then
do 190 n = 1, totnunpoints
wvalue = 0.35875 - 0.48829 * dcos(tpi * (n - 1) /
(totnunpoints - 1)) + 0.14128 * dcos(2.0 * tpi *
(n - 1) / (totnunpoints - 1)) - 0.01168 *
dcos(3.0 * tpi * (n - 1) / (totnunpoints - 1))
datar(n) = datar(n) * wvalue
datai(n) = datai(n) * wvalue
continue
else if (windowtype .eq. 5) then
do 200 n = 1, totnunpoints
wvalue = dexp(-0.5d0 * (3.0 * (n - centerpoint) /
centerpoint)**2)
datar(n) = datar(n) * wvalue
datai(n) = datai(n) * wvalue
continue
end if
zero pad the windowed frequency data
numzeros = fftsize - totnunpoints
do 210 n = 1, (centerpoint - 1)
tenp = datar(n)
datar(n) = datar(n + centerpoint - 1)
datar(n + centerpoint - 1) = tenp
tenp = datai(n)
datai(n) = datai(n + centerpoint - 1)
datai(n + centerpoint - 1) = tenp
210
continue
tenpr = datar(totnunpoints)
tenpi = datai(totnunpoints)
do 220 n = fftsize, (fftsize - centerpoint + 2), -1
datar(n) = datar(n - numzeros - 1)
datai(n) = datai(n - numzeros - 1)
220
continue
datar(centerpoint) = tenpr
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
182
datai(centerpoint) = tenpi
do 230 n = (centerpoint + 1), (centerpoint + numzeros)
datar(n) = 0 . 0
datai(n) = 0 . 0
continue
230
c
c generate inverse transform of the frequency data
do 240 n = 1, fftsize
datai(n) = -datai(n)
! conjugate data
240
continue
call fft(datar, datai, fftsize, powertwo)
do 250 n = 1, fftsize
datai(n) = -datai(n)
! conjugate data
250
continue
c
c do time-domain multiplication
if (centertime .gt. 0) then
j = fftsize + 1 - int(fftsize * centertime / maxtimerange)
else
j = -int(fftsize * centertime / maxtimerange) + 1
end if
if (j .gt. fftsize) j = 1
do 260 n = 1, fftsize
datar(n) = datar(n) * gater(j)
datai(n) = datai(n) * gater(j)
j = j + 1
if (j .gt. fftsize) j = 1
260
continue
c
c take transform of the resulting data
call fft(datar, datai, fftsize, powertwo)
do 270 n = 1, fftsize
datar(n) = datar(n) / dble(fftsize)
! scale data
datai(n) = datai(n) / dble(fftsize)
270
continue
c
c un-wrap the resulting gated frequency domain
tempr = datar(centerpoint)
tenpi = datai(centerpoint)
do 280 n = (fftsize - centerpoint + 2), fftsize
datar(n - numzeros - 1) = datar(n)
datai(n - numzeros - 1) = datai(n)
280
continue
do 290 n = 1, (centerpoint - 1)
tenp = datar(n)
datar(n) = datar(n + centerpoint - 1)
datar(n + centerpoint - 1) = tenp
temp = datai(n)
datai(n) = datai(n + centerpoint - 1)
datai (n + centerpoint - 1) = tenp
290
continue
datar(totnunpoints) = tenpr
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
183
datai (totnunpoints) = tenpi
c
c
un-window the resulting data
if (windowtype .eq. 2) then
do 300 n = 1, (centerpoint - 1)
wvalue = 0.5 * (1.0 - dcos(tpi * n / totnunpoints))
datar(n) = datar(n) / wvalue
datai(n) = datai(n) / wvalue
300
continue
do 310 n = (centerpoint + 1), totnunpoints
wvalue = 0.5 * (1.0 - dcos(tpi * (totnunpoints + 1 - n)
*
/ totnunpoints))
datar(n) = datar(n) / wvalue
datai(n) = datai(n) / wvalue
310
continue
else if (windowtype .eq. 3) then
do 320 n = 1, totnunpoints
wvalue = 0.44959 - 0.49364 * dcos(tpi * (n - 1) /
*
(totnunpoints - 1)) + 0.05677 * dcos(2.0 * tpi *
*
(n - 1) / (totnunpoints - 1))
datar(n) = datar(n) / wvalue
datai(n) = datai(n) / wvalue
320
continue
else if (windowtype .eq. 4) then
do 330 n = 1, totnunpoints
wvalue = 0.35875 - 0.48829 * dcos(tpi * (n - 1) /
*
(totnunpoints - 1)) + 0.14128 * dcos(2.0 * tpi *
*
(n - 1) / (totnunpoints - 1)) - 0.01168 *
*
dcos(3.0 * tpi * (n - 1) / (totnunpoints - 1))
datar(n) = datar(n) / wvalue
datai(n) = datai(n) / wvalue
330
continue
else if (windowtype .eq. 5) then
do 340 n = 1, totnunpoints
wvalue - de wvalue
datai(n) = datai(n) / wvalue
340
continue
end if
C********WRITE GATED OUTPUT DATA TO DISK**********
open (3, file=gatefile, status='new',
*
carriagecontrol=1list1)
c
do 101,n=(numfut+1), (numfut+numfreqpts)
write(3,9999)datar(n),datai(n)
101
continue
40 continue
close (2)
close (3)
c
c ********b e g i n s m o o t h i n g p o r t i o n o f p r o g r a m **********
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
77
open (3, file=gatefile, status='old')
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
184
open (4,file=smoothfile,status=1new1)
c
5
do 5 N = 1,numfreqpts
avgdata(N) = 0 . 0
smagsqavgdata(N) = 0 . 0
avgpowersq(N) =0.0
continue
c
scounter = 0
freqincr = (stopfreq - startfreq)/float(numfreqpts-1)
bandsmooth = 2.0*freqincr*(float(smoothnum))
if(outflag.eq.O)numprint = 9
if(outflag.eq.l)numprint = 17
if(outflag.eq.2)numprint = 101
if(outflag.eq.3)numprint = 401
c
write(4,*)'input filename =>',infile
write(4,*)'bandwith of smoothing =
bandsmooth,1 GHz'
write(4,*)'number of samples in smoothing = ',2*smoothnum+l
write(4,*)'number of frequencies printed to output file =>'
write(4,*)numprint
write(4,*)' '
write(4, *)'freq.(GHz) avg.(dBsm)
smoothavg.(dBsm)'
c
do 400 1 = 1 , numsweeps
c
o o
15
do 15,N= 1,numfreqpts
read(3,*)fdatar(N),fdatai (N)
continue
n
rselect = float(1-1)/float(fileincr)
iselect = (I—1)/fileincr
flag = rselect - float(iselect)
o n
if(flag.eq.O)then
#### take magnitude sq of data sweep for power value
c
call magsq (fdatar, fdatai, numfreqpts, magsqsweep)
c
c
c
#### add data sweep into avgdata array
call arrayadd(avgdata, magsqsweep,numfreqpts)
c
c
c
55
c
c
#### sq the power value which is the sq of the sq of the mag.
do 55 N=l,numfreqpts
powersq(N) = magsqsweep(N)*magsqsweep (N)
continue
#### add the power sq term to the avgpowersq array
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
185
call arrayadd (avgpowersq, powersq, numfreqpts)
c
c
c
#### smooth the data sweep
call smoothdata (magsqsweep, numfreqpts, magsqsmooth, smoothnum)
c
c
c
#### add smoothed data sweep to smagsqavgdata array
call arrayadd (smagsqavgdata, magsqsmooth, numfreqpts)
c
scounter = scounter + 1
c
endif
c
400
c
c
c
c
continue
#### divide data by the number of sweeps
stacknum = dble(scounter)
call arraydivide(avgdata,numfreqpts, stacknum)
call arraydivide (smagsqavgdata, numfreqpts, stacknum)
call arraydivide (avgpowersq, numfreqpts, stacknum)
c
c
c
c
#### smooth avgdata array to form savgdata array
call smoothdata (avgdata, numfreqpts, savgdata, smoothnum)
c
c
c
#### smooth the avgpowersq array
call smoothdata (avgpowersq, numfreqpts, savgpowersq, smoothnum)
oc
c
c
85
c
c
c
c
#### calculate the std dev values
do 85 N=l,numfreqpts
stdev(N) = dsqrt (avgpowersq (N) - avgdata (N) *avgdata (N))
tenp = smagsqavgdata (N) *smagsqavgdata (N)
smstdev(N) = dsqrt (savgpowersq (N) - tenp)
continue
#### calculate the confidence intervals for the res
if (conflag.ne.0. and.conflag.ne.1. and.conflag.n e .2) then
write(*,*)1*** confidence interval incorrected specified'
write(*,*)1*** defaulting to the 95% interval1
confvalue = 1.96d0
else
if(conflag.eq.O)confvalue = 1.645d0
if(conflag.eq. 1)confvalue = 1.96d0
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
186
if(conflag.eq.2)confvalue = 2.576d0
endif
c
95
c
c
c
do 95 N=l,numfreqpts
tenp - confvalue * smstdev(N)/dsqrt (stacknum)
lower(N) = smagsqavgdata(N) - tenp
upper (N) = smagsqavgdata (N) + tenp
continue
#### convert values to dBsm
call
call
call
call
call
c
c
c
c
701
800
dbconv(avgdata, numfreqpts)
dbconv(savgdata,numfreqpts)
dbconv(smagsqavgdata,numfreqpts)
dbconv(upper,numfreqpts)
dbconv(lower,numfreqpts)
#### write output to disk
if(outflag.ne.0.and.outflag.ne.1.and.outflag.n e .2
.and.outflag.n e .3)then
write(4,*)'Invalid output flag specified'
else
if(outflag.eq.O)outstep=50
i f (outflag.eq.1)outstep=25
if(outflag.eq.2)outstep=4
if(outflag.eq.3)outstep=l
do 701 N=l,numfreqpts, outstep
freq = startfreq + float(N-l)*freqincr
write(4,9997)freq,avgdata(N),savgdata(N)
continue
write(4,*)'
'
write(4,*)' freq
smstdev
lower
write (4,*)' (GHz)
(sq.m)
(dBsm)
------------write (4,*)'----do 800 N=l, numfreqpts,outstep
freq = startfreq + float(N-l)*freqincr
write(4, 9998)freq,smstdev(N),lower(N),upper(N)
continue
endif
upper'
(dBsm) '
c
79
format (la)
format (80a)
9999
format (lx, fl6.9, lx, f16.9)
9998
format (2x,f5.2,3x,f9.6,5x, f9.3,5x, f9.3)
9997
format(2x,f5.2,3x,f9.3,6x, f9.3)
1000
c
close(3)
close(4)
c
stop
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
187
end
c
0 *******************SUBROUTINES**********************************
q* * * * * *** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* *
SUBROUTINE fft(x, y, n, m)
c
c
c
c
c
c
this is a standard fast fourier transform routine
the real input array is in the x array and the imaginary
input array is in the y array
the real and imaginary output data is placed in these same
two arrays
implicit none
c
real*8 x(l), y(l), E, c, cl, s, si, xt, yt, t
c
integer i, j, k, 1, m, n, nl, n2
c
30
20
10
n2 = n
do 10 k = 1, m
nl = n2
n2 = n2 / 2
E = 6.283185307179586 / dble(nl)
c = 1.0
s = 0.0
cl = dcos(E)
si = dsin(E)
do 20 j = 1, n2
do 30 i = j, n, nl
1 = i + n2
xt
= x (i)
- x (1)
x(i) = x (i)
+ x(l)
yt
= y(i)
- y(l)
y(i) = y(i)
+ y(l)
x(l) = c * x t + s * y t
y(l) = c * y t - s * x t
continue
t = c
c=c*c l - s * s l
s=t*sl+ s* cl
continue
continue
j = 1
nl = n - 1
do 40 i = 1, nl
if (i .It. j) then
xt
=
=
x(i) =
yt =
y(j) =
y(i) =
X(j)
x(j)
x(i)
xt
y(j)
y (i)
yt
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
188
60
50
40
end if
k= n / 2
if (k .ge. j) goto 50
j= j - k
k = k / 2
goto 60
j= j + k
continue
return
end
c
c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SUBROUTINE memcoef (data, n, m, cof)
c
c
real*8 data(l), cof(l), wkl(1601), wk2(1601), wkm(150)
real*8 neum, denom
c
integer i, j, k, n, m
c
10
30
40
50
60
20
wkl(l) = data(l)
wk2(n - 1) = data(n)
do 10 j = 2, (n - 1)
wkl(j) = data(j)
wk2(j - 1) = data(j)
continue
do 20 k = 1, m
neum = 0 . 0
denom = 0 . 0
do 30 j = 1, (n - k)
neum = neum + wkl(j) * wk2(j)
denom = denom + wkl(j)**2 + wk2(j)**2
continue
cof(k) = 2 . 0 * neum / denom
if (k .ne. 1) then
do 40 i = 1, (k - 1)
cof(i) = wkm(i) - cof(k) * wkm(k - i)
continue
end if
if (k .eq. m) return
do 50 i = 1, k
wkm(i) = cof(i)
continue
do 60 j = 1, (n - k - 1)
wkl(j) = wkl(j) - wkm(k) * wk2 (j)
wk2(j) = wk2(j + 1) - wkm(k) * wkl(j + 1)
continue
continue
pause 'never get here'
end
c
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
189
SUBROUTINE predictor(data, ndata, d, npoles, future, nfut)
c
implicit none
c
real*8 data(l), d(l), future(1)
real*8 reg(75), sum
c
integer ndata, npoles, nfut, j, k
c
10
30
40
o o
20
do 10 j = 1, npoles
reg(j) = data(ndata + 1 - j)
continue
do 20 j = 1, nfut
sum = 0.0
do 30 k = 1, npoles
sum = sum + d(k) * reg(k)
continue
do 40 k = npoles, 2, -1
reg(k) = reg(k - 1)
continue
reg(l) = sum
future(j) = sum
continue
return
end
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
o
SUBROUTINE smoothdata (mdata,NF,smoothout,smoothnum)
o o o
real*8 mdata(1),smoothout(1)
integer NF,smoothnum
#### local variables ####
real*8 tempmag
integer N,J
c
c
c
c
c
c
c
c
calculate smoothed data
average 2M + 1 points where M = smoothnum
unless in the first and end sections of the sweep
####
20
30
smooth first section of sweep
do 30 N = 1, smoothnum
terrpmag = 0.0
do 20 J = 1, (N + smoothnum)
tempmag = tempmag + mdata(J)
continue
smoothout (N) = tempmag / dble(N + smoothnum)
continue
c
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
190
c
c
####
40
50
c
c
c
do 50 N = (smoothnum + 1), (NF - smoothnum)
tempmag = 0 . 0
do 40 J = (N - smoothnum), (N + smoothnum)
tempmag = tempmag + mdata (J)
continue
smoothout(N) = tempmag / dble(2 * smoothnum + 1)
continue
####
60
70
c
smooth center section with (2*smoothnum + 1) points
smooth end section of sweep
do 70 N = (NF - smoothnum + 1),NF
tempmag = 0 . 0
do 60 J = (N - smoothnum),NF
tempmag = tempmag + mdata(J)
continue
smoothout(N) = tempmag / dble(NF - N + smoothnum + 1)
continue
return
end
c
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SUBROUTINE magsq(datai,data2,NF,magsqout)
c
real*8 datai(1),data2(1),magsqout(1)
integer NF
c
c
c
#### local variables ####
integer N
c
c
c
20
c
#### take magnitude
of each complexarray
do 20 N = 1,NF
magsqout(N)= datai(N)*datal(N)
continue
pair
+ data2(N)*data2(N)
return
end
c
c
SUBROUTINE dbconv(data, NF)
c
real*8 data(l)
integer NF
c
c
c
#### local variables
integer J
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
191
c
n n
10
do 10 J=1,NF
if(data(J).GT.0)then
data(J) = 10. * dloglO(data(J))
else
data(J) = -99.
endif
continue
return
end
c
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SUBROUTINE arrayadd(datai, data2,NF)
c
real*8 datai(1) ,data2(1)
integer NF
c
c
#### local variables
integer J
c
10
do 10 J=1,NF
datai(J) = datai(J) + data2(J)
continue
c
return
end
c
c
c
c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SUBROUTINE arraydivide (data, NF, d)
c
real*8 data(l),d
integer NF
c
c
c
#### local variables
integer J
c
10
do 10 J=1,NF
data(J) = data(J)/d
continue
c
return
end
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
APPENDIX G
SURFACE AND VOLUME
SCATTERING PROGRAM
192
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
193
(* This program calculates backscattering from
the top surface, bottom surface, and volume.*)
(* Written by:
Date:
Last Modified:
Ron Porco
December 10, 1990
March 30, 1994
*)
Off[General::spelll];
Module[
{
(*GENERAL INPUT PARAMETERS*)
Freq=10.75,
(*GHz*)
eLayr=3.35-I 0.22,
eBot=6,
(*TOP SURFACE INPUT PARAMETERS*)
CorLenl=2.0
(*cm*)
sigmal=0.25
(*cm*)
(*VOLUME INPUT PARAMETERS*)
albedo=0.018,
opdepth=12,
(* RadiusScat=0.2,
VolFrac=0.1,
eScat=3.2-I 0.003,
d=12,*)
(*BOTTOM SURFACE INPUT PARAMETERS*)
CorLen2=0.87,
sigma2=0.087,},
{
theta=N[thetadeg Pi/180];
s=N[Sin[theta]];
c=N [Cos [theta] ];
CL12=€orLenl*CorLenl;
CL22=CorLen2 *CorLen2;
eBotr=eBot/eLayr;
kAir=N[2 Pi Freq/30];
kLayr=N[kAir Sqrt[Re[eLayr]]];
ksigmal=kAir sigmal;
kLl=kAir CorLenl;
ml=sigmal/CorLenl;
ksigma2=kLayr sigma2;
kL2=kLayr CorLen2;
m2=sigma2/CorLen2;
Cl=N[Sqrt[1-(s~2)/eLayr] ];
Eta0=l.;
Etal=N[1/Sqrt[eLayr]];
Eta2=N[l/Sqrt[eBot]];
kLayrI=Abs[Im[kAir*Sqrt[eLayr]]]//N;
eLayrR=Re[eLayr]//N;
(*eScatI=Abs[Im[eScat]]//N;*)
(*l/cm*)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
194
(*
Compute the field reflection coefficients *)
R01H=(Etal*c-EtaO*Cl)/ (Etal*c+EtaO*Cl);
R01V= (EtaO*c-Etal*Cl)/ (EtaO*c+Etal*Cl);
(*
R01V=(eLayr-Sqrt[eLayr])/(eLayr+Sqrt[eLayr]);
R01H=(1-Sqrt[eLayr])/(1+Sqrt[eLayr]); *)
(Compute the power reflection coefficients*)
PR01H=N[Abs [R01H] A2] ;
PR01V=N [Abs [R01V] A2] ;
(CONTRIBUTION FROM THE TOP LAYER*)
(*********************************)
ul=2 kAir s;
kzl=kAir c;
Yl=N[((kAirA2)/2) Exp[(-2)*(kzl sigmal)A2]];
(Compute the Kirchhoff and copmplementary terns*)
(*if kL>4, a Gaussian surface will appear to be
planar for Kirchoff terms, i.e., angle of incidence
appears to be 0 degrees*)
If[kLl<4.0,
fwl=2. ROlV/c;
fhhl=-2. ROlH/c,
fwl=2 (eLayr-Sqrt [eLayr])/ (c eLayr+c Sqrt [eLayr])//N;
fbhl=-2 (1-Sqrt[eLayr])/(c+c Sqrt[eLayr])//N];
Fwl = 2 sA2 (1+R01V) A2 (eLayr cA2 (eLayr-1)
+eLayr-(sA2)-eLayr cA2)/ (eLayr*eLayr*cA3);
Fhhl=-2 sA2 ((1+R01H)A2) (eLayr-1)/ (cA3) ;
(* Calculate the I to the nth power terms *)
Inwl=N[((2 kzl)An f w l Exp[-((sigmal kzl)A2)]) +((kzl)An Fwl)/2];
Inhhl=N[((2 kzl)An fhhl Exp[-((sigmal kzl)A2)]) +((kzl)An Fhhl)/2];
(*Choose Surface Roughness*)
(*VJnexpl=n CL12/(n*n+(CorLenl ul)A2)A1.5; *)
(* Exponential *)
Wnexpl=N[ (0.5 CL12/n) Exp [-((CorLenl ul)A2)/(4 n) ]];
(Caussian*)
Zlv=N[sigmalA (2 n) Abs[Inwl]A2 Wnexpl/n!];
Zlh=N[sigmalA (2 n) Abs[Inhhl]A2 Wnexpl/n!];
Sumwl=N[Sum[Zlv, {n, 1,6} ]];
Sumhhl=N[Stim[Zlh, {n, 1,6}]] ;
BkscatwTOP=Yl Sumwl;
BkscathhTOP=Yl Sumhhl;
(* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * )
(*CALCULATE THE REAL ANGLE OF TRANSMISSION*)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
195
(*Fran Ulaby,et. al.,Vol.l, ECN.(2.69)*)
eLI=Im[Sqrt[eLayr]]//N;
eLR=Re [Sqrt [eLayr] ]//N;
p=2*eLI*eLR;
q=eLR*eLR-eLI*eLI-s*s;
thetat=ArcTan [Sqrt [2*s*s/ (q+Sqrt [p*pfq*q]) ]]//N;
st=Sin[thetat]//N;
ct=Cos[thetat]//N;
Ctl=Sqrt[l-eLayr*st*st]//N;
Ct2=Sqrt[l-st*eLayr/eBot]//N;
(*COMPUTE THE FIEID REFLECTION COEFICIENTS*)
(*Note that RIO = -R01*)
R10H=(EtaO *ct-Etal*Ct1)/(EtaO*ct+Etal*Ctl);
R10V= (Etal*ct-EtaO*Ctl)/ (Etal*ct+EtaO*Ctl);
R12H=(Eta2*ct-Etal*Ct2)/ (Eta2*ct+Etal*Ct2);
R12V=(Etal*ct-Eta2*Ct2)/ (Etal*ct+Eta2*Ct2);
(*COMPUTE THE POWER REFLECTION COEFFICIENTS*)
PR10H=Abs[R10H]"2//N;
PR10V=fibs [R10V] "2//N;
PR12H=Abs[R12H]"2//N;
PR12V=Abs [R12V] A2//N;
(* CONTRIBUTION OF VOLUME SCATTERING*)
(* absorption coeff Ka from Fung, p.122 *)
(*
NumScat=N[3 VolFrac/(4 Pi RadiusScatA3)];
Ks=N[(8/3) Pi NumScat kLayrA4 RadiusScatA6
Abs[(eScat-eLayr)/ (eScat+2 eLayr)]A2];
Kal=2 (1-VolFrac) kLayrl;
Ka2=N [VolFrac*kLayr* (eScatl/eLayrR) *
Abs[3*eLayr/(eScat+2*eLayr)]A2];
Ka=Kal+Ka2;
Ke=Ka+Ks;
(* extinction coeff Ke *)
albedo=Ks/Ke;
opdepth=Ke*d;
Kes=Ke/c;
*)
(* Calculate commonly used terms
*)
Loss=N[Exp[-2 opdepth/ct]];
XH=(1-PR10H PR12H Loss)A2;
XV= (1-PR10V PR12V Loss) A2;
(* Conputation of Rayleigh Phase Terms
PH=1.5;
PV1=1.5-6*(stA2)* (ctA2);
*)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
196
FV2=1.5;
T10H=1-PR10H;
T10V=1-PR10V;
(* Calculate first-order contrib of back scatter coeff *)
BkscathhVOL= ((albedo T10H T10H PH)/(2 XH)) ((1-Loss) c (1+PR12HA2 Loss)
+(4 opdepth PR12H Loss c/ct));
BkscatwVOL= ((albedo T10V T10V)/(2 XV)) ((1-Loss) (PV2 c) (1+PR12VA2 Loss)
+(4 opdepth PV1 PR12H Loss c/ct));
(* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * )
(CONTRIBUTION FRCM BOTTCM LAYER*)
kz2=kLayr ct;
u2=2 kLayr st;
Y2=N[kLayrA2/2 Exp[-2 (kz2 sigma2)A2]];
(* Compute the Kirchhoff and complementary terms *)
(*note kL2=4 is switching point for Gaussian surface*)
If[kL2<4.0,
fw2=2 R12V/ct;
fhh2=-2 R12H/ct,
fw2=2 (eBotr-Sqrt [eBotr])/ (ct eBotr+ct Sqrt [eBotr])//N;
fhh2=-2 (1-Sqrt[eBotr])/(ct+ct Sqrt[eBotr])//N];
Fw 2 = 2 stA2 (1+R12V) A2 (eBotr ctA2 (eBotr-1)
+eBotr-stA2-eBotr ctA2)/ (eBotr*eBotr*ctA3);
Fhh2=-2 stA2 (1+R12H)A2 (eBotr-1)/ (ctA3) ;
(* Calculate the I to the nth power terms *)
Inw2=N [((2 kz2) An2 f w 2 Exp[- ((sigma2 kz2) A2) ]) + ((kz2) An2 Fw2) /2] ;
Inhh2=N[((2 kz2)An2 fhh2 Exp[-((sigma2 kz2)A2)]) +((kz2)An2 Fhh2)/2];
(*Choose Surface Roughness*)
(*Wnexp2=n2 CL22/(n2*n2+(CorLen2 u2)A2)A1.5;*)
(* Exponential *)
Wnexp2=N[(0.5 CL22/n2) Exp[-((CorLen2 u2)A2)/(4 n2)]]/
(*Gaussian*)
Z2v=N[sigma2A (2 n2) Abs[Inw2]A2 Wnexp2/n2!];
Z2h=N[sigma2A (2 n2) Abs[Inhh2]A2 Wnexp2/n2!];
Sumw2=N[Sum[Z2v, {n2,1,6} ]];
Sumhh2=N[Sum[Z2h, {n2,1,6}]];
BkscatwBOT=N[T10V T10V Loss Y2 Sumw2 (c/ct)/Sqrt [XV] ];
BkscathhBOT=N[T10H T10H Loss Y2 Sumhh2 (c/ct) /Sqrt [XH] ];
^'k'k'k'k 'k'k 'k'k 'k'k 'k'k 'k'k ic 'k'k 'kic k 'k'k 'k'k 'k'k 'k'k 'k'k 'k'k 'kic k 'k'k -k 'k'k 'k'k 'k'k 'k'k 'kie 'k 'k 'k 'k -k ic k ^
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
197
BkscathhTOT=BkscathhTOP+BkscathhBOT-HBkscathhVOL;
BkscatwTOT=BkscatwTOP+BkscatwBOT+BkscatwVOL;
BkscathhTOTdb=N[10 Log[10,BkscathhTOT]];
BkscatwTOTdb=N[10 Log[10,BkscatwTOT] ];
BkscathhVOLdb=N[10
BkscatwVOLdb=N [10
BkscathhBOTdb=N[10
BkscatwBOTdb=N [10
BkscathhTOPdb=N[10
BkscatwTOPdb=N [10
Log[10,BkscathhVOL]];
Log [10, BkscatvWOL] ];
Log[10,BkscathhBOT]];
Log [10, BkscatwBOT ]];
Log[10,BkscathhTOP]] ;
Log [10, BkscatwTOP ]];
Do[Print[thetadeg,"\t",BkscathhTOPdb,"\t",
BkscathhVOLdb,"\t",BkscathhTOTdb], {thetadeg, 10,50,10}]}]
10
20
30
40
50
-2.81771
-8.94237
-16.5821
-24.7241
-33.8125
-19.578
-19.8716
-20.3994
-21.2339
-22.5158
-2.72709
-8.60517
-15.0739
-19.6271
-22.205
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
APPENDIX H
ANTENNA PATTERNS
198
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
199
-26
2 GHz
-28
-30
4>
I
-32
'I
i
I
-34
-36
-38
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
-110
2 GHz
-120
S? -130
u
& -140
« -150
o- -160
-170
-180
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B .l: 2 GHz Horizontal polarization, near-field horizontal
scan of lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
200
-28
3 GHz
Magnitude (dB)
-30
-32
-34
-36
-38
-40
-42
-44
-23 -17.25-11.5 -5.75
0
5.75 11.5 17.25 23
Distance from Center (inches)
(a) Magnitude
180
Phase (Degrees)
90
0
-90
-180
-23 -17.25-11.5 -5.75
0
5.75 11.5 17.25 23
Distance from Center (inches)
(b) Phase
Figure B.2: 3 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
201
-32
Magnitude (dB)
-34
-38
-42
-44
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
Phase (Degrees)
4 IGHz
-90
-180
■23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.3: 4 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
202
-30
GHz
Magnitude (dB)
-35
-45
-50
-55
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
jGHz
Phase (Degrees)
135
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.4: 5 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
203
-3 6
6: GHz
-38
-40
i
0
2
T3
1
-42
. .
-44
-46
-50
-52
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
45
6 GHz
22.5
0
-22.5
-45
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.5: 6 G Hz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
204
-38
GHz
-40
-42
0
1
I
-44
‘46
-48
-50
-52
-54
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
-60
7 &Hz
-80
?-100
g.
&
-120
sP -140
-160
-180
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.6: 7 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
205
-35
-40
-45
u
3
-50
U
I
-55
-60
-65
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
-40
<iHz
-60
-80
-100
-160
-180
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.7: 8 GHz Horizontal polarization, near-field vertical
scan of lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
206
-38
-40
-42
u
"O
3
I
-44
i f
-46
-48
-50
-52
-54
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
120
GHz
100
-20
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.8: 9 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
207
-42
10 GHz
-44
-48
-54
-56
23 -17.25-11.5 -5.75
0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
GHzi
-150
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.9: 10 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
208
-4 4
11 GHz
-46
-48
3 -50
•o
S -52
-56
-58
-60
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
Phase (Degrees)
11 GHz
-90
-180
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.10: 11 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
209
-4 4
12 GHz
Magnitude (dB)
-46
-48
-50
-52
-54
-56
-58
-60
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
Phase (Degrees)
12 GHz
-90
-180
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B .l 1: 12 GHz Horizontal polarization, near-field horizontal
scan of lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
210
-40
13 GHz
-45
«
T3
-50
3
&
CC -55
s
-60
-65
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
100
13 GHz
C/3
4)
-100
-150
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.12: 13 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
211
-4 6
14 GHz
Magnitude (dB)
-48
-52
-56
-58
-60
-62
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
Phase (Degrees)
14 GHz
-90
-180
■23 -17.25-11.5 -5.75
0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.13: 14 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
212
-4 6
15 GHz
-48
_
-50
3
«
-52
1
‘54
I
-56
S'
-58
-60
-62
-23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
15 GHz
-90
-180
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.14: 15 GHz Horizontal polarization, near-field horizontal
scan of lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
213
-50
16 GHz
Magnitude (dB)
-52
-54
-56
-58
-60
-62
-64
-66
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
16 GHz
Phase (Degrees)
135
-45
-90
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.15: 16 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
214
-5 0
17 GHz
-55
u
3
•M
§>
C3
£
-60
-65
-70
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
17 GHz
120
oC/5
j-=
CL,
-60
-120
-180
23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.16: 17 GHz Horizontal polarization, near-field horizontal
scan of lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
215
-5 0
18 GHz
-55
i
u
3
-60
Ic3
S
-65
-70
•23 -17.25-11.5 -5.75 0
5.75 11.5 17.25
Distance from Center (inches)
23
(a) Magnitude
180
18 GHz
120
-120
-180
23 -17.25-11.5 -5.75
0
5.75 11.5 17.25
Distance from Center (inches)
23
(b) Phase
Figure B.17: 18 GHz Horizontal polarization, near-field horizontal
scan o f lm offset-fed reflector antenna.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
REFEREN CES
[1] Y. Kuga, "Laser Light Propagation and Scattering in a Dense Distribution o f Spherical
Particles," Ph.D. Dissertation, The University o f Washington, Seattle, 1983.
[2] L Tsang and J. Kong, T h e o ry o f M icrow ave R em ote S ensing, John W iley and
Sons, Inc. New York, 1985.
[3] F. Carsey, ed., M icrow ave R em ote S ensing o f Sea Ice, Am erican Geophysical
Union, 1992.
[4] S. Tjuatja, "Theoretical Scatter and Emission Models for Inhomogeneous Layers with
A pplication to Snow and Sea Ice," Ph.D . D issertation, The University o f Texas at
Arlington, 1992.
[5] "Earth System Science: Report on Earth System Sciences Committee NASA Advisory
Council", National Aeronautics and Space Administration, W ashington, D.C., pp. 61-62,
82-83, January 1988.
[6] J. H oughton, G. Jenkins, and J. Ephraum s, ed., C lim a te C h an g e, T h e IP C C
S cien tific A ssessm ent, Cam bridge The University Press, New York, 1990.
[7] R. Porco and J. Bredow, "Robotic Aided Dense M edium Target Fabrication," IEEE
Transactions on Geoscience and Remote Sensing, vol. 32, no. 1, pp. 217-219, 1994.
[8] R. Porco, "Radar Cross Section M easurements of Volcanic Ash Particles," Master's
Thesis, The University o f Texas at Arlington, 1990.
[9] S. Nadimi, "Extinction o f Solid Dense Random M edia at M icrowave Frequencies,"
M aster's Thesis, The University o f Texas at Arlington, 1992.
[10] A. Ishim aru and Y. Kuga, "Attenuation Constant o f a Coherent Field in a Dense
Distribution o f Particles," J. Opt. Soc. Am., vol. 72, pp. 1317-1320, 1982.
[11] D. Gibbs and A. Fung, "Measurement o f Optical Transmission and Backscatter from
a Dense Distribution o f Particles," IGARSS '90 Proceedings , pp. 1029-1032.
[12] G. Koh, "Investigation o f M illim eter Wave Propagation and Scattering in Random
Scattering M edia," PIERS '91 Proceedings, p. 538.
[13] G. Koh, "Experim ental Study o f Electrom agnetic W ave Propagation in Dense
Random Media," Waves in Random Media, vol. 2, pp. 39-48, 1992.
[14] C. Nance, "Scattering and Image Analysis o f Conducting Rough Surfaces," Ph.D.
Dissertation, The University o f Texas at Arlington, 1992.
216
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
217
[15] S. C handrasekhar, Radiative Transfer, D over Publications, Inc., New York,
1960.
[16] F. Ulaby, R. M oore, and A. Fung, Microwave Remote Sensing, vol. 3, Artech
H ouse, Norwood, MA, 1986.
[17] J. Leader, "Polarization Dependence in EM Scattering from Rayleigh Scatterers
Em bedded in a Dielectric Slab," Journal o f Applied Physics, vol. 46, no. 10, pp. 43714385, 1975.
[18] P. Beckm ann and A. Spizzichino, The Scattering of Electromagnetic Waves
from Rough Surfaces, The MacMillan Co., New York, 1963.
[19] A . Fung, "On Depolarization o f Electromagnetic Waves Backscattered from a Rough
Surface," Planetary Space Sciences, vol. 14, pp. 563-568, 1966.
[20] M. Sancer, "Shadow-corrected Scattering Electrom agnetic Scattering from a
Randomly Rough Surface," IEEE Transactions on Antennas and Propagation, AP-17, pp.
577-585, 1969.
[21] S. Rice, "Reflection o f Electromagnetic W aves from Slightly Rough Surfaces,"
Communications in Pure and Applied Mathematics, no. 4, pp. 361-378,1951.
[22] G. Valenzuela, "Depolarization of EM Waves by Slightly Rough Surfaces," IEEE
Transactions o f Antennas and Propagation, vol. AP-15, no. 4, pp. 552-557, July 1967.
[23] F. Ulaby, R. M oore, and A. Fur g, Microwave Remote Sensing, vol. 2, Artech
House, Norwood, MA, 1986.
[24] A. Fung and M. Chen, "Numerical Sim ulation o f Scattering from Simple and
Com posite Random Surfaces," Journal o f the Optical Society o f America, vol. 2, no. 12,
pp. 2274-2284, 1985.
[25] M. Chen and A. Fung, "A Numerical Study of the R egions o f Validity o f the
K irchhoff and Small-Perturbation Rough Surface Scattering Models," Radio Science, vol.
23, no. 2, pp. 163-170, 1988.
[26] P. Beckmann, "Scattering by Composite Rough Surfaces," Proceedings o f the IEEE,
vol. 53, pp. 1012-1015, 1965.
[27] I. F uks, "Theory o f Radio-wave Scattering at a Rough Surface," S o v ie t
R adiophysics, vol. 9, pp. 513-519, 1966.
[28] G. Valenzuela, "Scattering o f Electromagnetic Waves from a Tilted Slightly Rough
Surface," R adio Science, vol. 3, no. 11, 1968.
[29] A. Fung and G. Pan, "A Scattering M odel for Perfectly Conducting Random
Surfaces Part I: Model Development," International Journal o f Remote Sensing, vol. 8,
no. 11, pp. 1579-1593, 1987.
[30] Z. Li and A. Fung, "Scattering from a Finitely Conducting Random Surface," PIERS
Conference, 1989.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
218
[31] K. Chen, "Numerical and Theoretical Study o f Rough Surface Scattering," Ph.D.
Dissertation, The University o f Texas at Arlington, 1990.
[32] A. Fung, M ic ro w a v e S c a tte r in g a n d E m issio n
A pp licatio n s, Artech House, Norwood, MA, 1994.
M o d els
and
T h e ir
[33] M. Vant, R. Ramsier, and V. Makios, "The Complex Dielectric Constant o f Sea Ice at
Frequencies in the Range 0.1-40 GHz," Journal o f A pplied Physics, vol. 49, pp. 12641280, 1978.
[34] L. Tsang and J. Kong, "Scattering o f Electromagnetic W aves from a H alf Space o f
Densely Distributed Dielectric Scatterers," Radio Science, vol. 18, no. 6, pp. 1260-1272,
1983.
[35] B. W en, L. Tsang, D. W inebrenner, and A. Ishimaru, "Dense M edium Radiative
Transfer Theory: Com parison with Experiment and Application to M icrowave Remote
Sensing and Polarim etry," IEEE Transactions on Geoscience and Remote Sensing, vol.
28, no. 1, 1990.
[36] R. W est, L. Tsang, and D. W inebrenner, "Dense M edium Radiative Transfer Theory
for T w o Scattering Layers with a Rayleigh D istribution o f Particle Sizes," I E E E
Transactions on Geoscience and Remote Sensing, vol. 31, no. 2, pp. 426-437, 1993.
[37] R. Olsen and M. Kharadly, "Experim ental Investigation o f the Scattering of
Electromagnetic W aves from a Model Random M edium o f Discrete Scatterers," Radio
Science, vol. 11, no. 1, pp. 39-48, 1976.
[38] C. M andt, Y. Kuga, L. Tsang, and A. Ishim aru, "M icrowave Propagation and
Scattering in a Dense Distribution o f Non-tenuous spheres: Experiment and Theory,"
Waves in Random Media, vol. 2, pp. 225-234, 1992.
[39] V. Tw ersky, "A coustic Bulk Param eters in D istributions o f Pair-correlated
Scatterers," Journal o f the Acoustical Society o f America, vol. 64, no. 6, pp. 1710-1719,
1978.
[40] V. Twersky, "Propagation in Pair-correlated D istributions o f Small-spaced Lossy
Scatterers," Journal o f the Optical Society o f America," vol. 69, no. 11, pp. 1567-1572,
1979.
[41] T. W allace and J. Kratohvill, "Comments on the Com parison of Scattering o f
Coherent and Incoherent Light by Polydispersed Spheres with M ie Theory," A pplied
Optics, vol. 8, pp. 824-826, 1969.
[42] H. Nelson, "Radiative Scattering Cross Sections: Comparison o f Experim ent and
Theory," A pplied Optics, vol. 20, pp. 500-504, 1981.
[43] J. M entzer, S c a tte rin g a n d D iffractio n o f R ad io W aves, Pergam on Press,
New York, 1955.
[44] D. Atlas, "Advances in Radar Meteorology," Advances in Geophysics, vol. 10, pp.
317-479, 1964.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
219
[45] D. H arris, R. Roe, W. Rose, and M. Thom pson, "Radar O bservations o f Ash
Eruptions," US Geological Survey: Professional Paper, vol. 1250, pp. 323-333, 1981.
[46] F. Ulaby, R. M oore, and A. Fung, Microwave Remote Sensing, vol. 1, Artech
House, Norwood, MA, 1981.
[47] J. Kong, Electromagnetic Theory, John W iley and Sons, New York, 1986.
[48] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation,
Academic Press, New York, 1969.
[49] A. Ishim aru, Wave Propagation and Scattering in Random Media, Volume
I, Academic Press, New York, 1978.
[50] L. Bayvel and A. Jones, Electromagnetic Scattering and Its Applications,
Applied Science Publishers, Englewood, NJ, 1981.
[51] A. Fung and H. Eom, "A Study of Backscattering and Emission from Closely Packed
Inhomogeneous M edia," IEEE Transactions on Geoscience and Rem ote Sensing, vol. 23,
no. 5, pp. 761-767, 1985.
[52] G. G oedecke, "Radiative Transfer in Closely Packed Media," GOSA, vol. 67, no.
10, pp. 1339-1348, 1977.
[53] I. G radshteyn and I. R yzhik, Table of Integrals, Series, and Products,
Academic Press, Inc., New York, 1980.
[54] J. Bredow, R. Porco, M. Dawson, and C. Betty, "A M ultifrequency Laboratory
Investigation o f Attenuation and Scattering from Volcanic Ash Clouds," IEEE Transactions
on Geoscience and Remote Sensing, accepted for publication.
[55] R. Porco, J. Bredow , and S. Nadim i, "C onstruction and M easurem ents o f
Robotically Fabricated D ense M edia Targets," Progress in Electromagnetics Research
Symposium Proceedings, pp. 519, 1993.
[56] R. Porco, J. Bredow , and A. Fung, "Synthetic D ense M edia Fabrication with
Application to Sea Ice," IGARSS '93 Proceedings.
[57] J. Bendat and A. Piersol, Random Data, John W iley and Sons, Inc., New York,
1986.
[58] The Adept One Manipulator Handbook.
[59] J. Percus and G. Yevick, "Analysis o f Classical Statistical Mechanics by Means of
Collective Coordinates," Physical Review, vol. 110, pp. 1-13, 1958.
[60] J. Rochier, A. Blanchard, and M. Chen, "The Generation o f Surface Targets with
Specified Surface Statistics," International Journal o f Remote Sensing, vol. 10, no. 7, pp.
1155-1174, 1989.
[61] H P 8510 Network Analyzer Operating and Programming Manual, Hewlett-Packard
Company, Santa Rosa, CA, 1985.
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
220
[62] B. Jersak, "Time Dom ain Analysis o f Measured Frequency Domain R adar Cross
Section Data," Master's Thesis, The University o f Texas at Arlington, 1988.
[63] T. G riffen, "R adar Cross Section M easurem ents o f D ense Vertically Structured
Dielectric Cylinders," Master's Thesis, The University of Texas at Arlington, 1987.
[64] J. E aves and E. Reedy, P rin cip les o f M o d ern R a d a r, Van Nostrand Reinhold,
New York, 1987.
[65] J. Bredow , K. X ie, R. Porco, and M. Shah, "An Experimental Study on the Use of
M ultistatic Imaging for Investigating Electromagnetic Wave-Object Interaction," Journal o f
Electromagnetic Waves and Applications, vol. 7, no. 6, pp. 811-831, 1993.
[66] C. Balanis, A n ten n a T h eo ry , Harper and Row, New York, 1982.
[67] K. Jezek, P. G ogineni, L. Peters, J. Young, S. Beaven, E. Nasser, and I. Zabel,
"M icrowave Scattering from Saline Ice Using Plane W ave Illum ination," IG A R SS '94
Proceedings.
[68] W. W iesbeck and D. Kahny, "Single Reference, Three Target Calibration and Error
Correction for M onostatic, Polarimetric Free Space M easurements," Proceedings o f the
IE E E , vol. 79, no. 10, 1991.
[69] B. Jersak, "Bistatic, Fully Polarimetric Radar Cross-Section Calibration Techniques
and M easurem ent E rro r A nalysis," Ph.D. D issertation, The U niversity o f Texas at
Arlington, 1993.
[70] Personal com m unication with Elias Nassar, ElectroScience Laboratory, The Ohio
State University, June 1994.
[71] J. B redow and S. G ogineni, "Com parison o f M easurem ents and Theory for
Backscatter from Bare and Snow-covered Saline Ice," IEEE Transactions on Geoscience
and Rem ote Sensing, vol. 28, no. 4, pp. 456-463, 1990.
[72] E. H echt and A. Zajac, O ptics, Addison-W esley Publishing Company, M enlo Park,
CA, 1974.
[73] H. Chuah, S. Tjuatja, A. Fung, R. Porco, and J. Bredow, "Phase Correction Factor
for a D ensely Packed Random M edium," to be subm itted to IE E E Transactions on
Geoscience an d Rem ote Sensing.
[74] R. W alpole and R. M yers, P ro b a b ility a n d S ta tis tic s fo r E n g in e e rs a n d
S cientists, M acm illan Publishing Company, New York, 1985.
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
7 083 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа