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

код для вставкиСкачатьINFORMATION 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.

1/--страниц