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Backscattering of E.M. waves for rough surface models 1971

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BACKSCATTERING OF E.M. WAVES FOR ROUGH SURFACE MODELS by PATRICK DONALD 0'KELLY B.A.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1965 A "THESIS SUBMITTED IN PARTIAL'FULFILMENT'OF THE REQUIREMENTS FOR THE DEGREE OF > DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the req u i r e d standard Research Supervisor Members of the Committee Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA November, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date f-cl^/TX- i i ABSTRACT Baclcscattering from c e r t a i n models of rough surfaces i s studied by a p p l i c a t i o n of a Monte-Carlo technique and by experiments on a physi c a l model. The models considered are l o s s l e s s arrays of hemicylinders and of hemispheres on a l o s s l e s s ground plane. For the Monte-Carlo simulation, the incident r a d i a t i o n i s considered to be a c y l i n d r i c a l or s p h e r i c a l wave with f i n i t e beamwidth. The shape of the beam i s chosen to be the same as the f a r f i e l d r a d i a t i o n pattern of an open waveguide. M u l t i p l e s c a t t e r i n g e f f e c t s are investigated f o r a per i o d i c array of hemicylinders and found to be s i g n i f i c a n t for object diameters greater than one wavelength, and den s i t i e s greater than 30%. It i s assumed that these r e s u l t s are also approximately v a l i d for random arrays. The s i n g l e s c a t t e r approximation i s used f o r a l l studies of the random case with these l i m i t a t i o n s i n mind. A s p e c i a l surface d i s t r i b u t i o n function i s developed and tested which includes the constraint of f i n i t e s c a t t e r e r s i z e i n a phys i c a l surface model. I t i s used to generate random coordinates from which a set of ph y s i c a l surfaces are formed out of die-stamped aluminum. These surfaces are scanned with 35 GHz. r a d i a t i o n from a pyramidal horn. Samples of the backscattered f i e l d are converted to d i g i t a l information and numerically analysed to determine the scattered f i e l d s t a t i s t i c s . These s t a t i s t i c s are compared to those obtained from the simulation. The means (coherent i n t e n s i t y ) are found to agree to within 2.5% while the variance (incoherent i n t e n s i t y ) obtained experimentally i s higher by a fa c t o r of about 15. This discrepancy i s a t t r i b u t e d to s i g n i f i c a n t phase measuring errors introduced by the present scanning system. i i i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v i i i LIST OF TABLES x LIST OF SYMBOLS x i ACKNOWLEDGEMENTS ..... xv 1. INTRODUCTION 1 1.1 Existing Methods of Solution 1 1.1.1 Continuous Surface Model 2 1.1.2 Discrete Scatterer Model 3 1.1.3 Experimental Studies 4 1.2 Aims of this Study 4 1.2.1 Surface Model 5 1.2.2 Incident Beam Model 5 1.2.3 Monte-Carlo Method 6 1.2.H Experiment 7 1.3 Restrictions , 8 2. THE SCATTERED FIELD FROM AN ARBITRARY CONFIGURATION OF DISCRETE SCATTERERS ON A PERFECTLY CONDUCTING PLANE 9 2.1 Introduction 10 2.2 Scattering from a Configuration of Hemicylinders .... 10 2.2.1 Cylindrical Incident Beam 10 2.2.2 Scattering Coefficients for Grating 14 2.2.3 Image Method for Transformation to Surface Problem ; 16 2.2.h Backscattered Field 17 2.2.5 Normalization of Field Equations 18 iv Page 2.3 Scattering from a Configuration of Hemispheres 20 2. 3.1 Spherical Incident Beam 20 2.3.2 Single Scattered F i e l d 24 2.3.3 Far-Zone Backscatter 25 2. 3.h Normalization of F i e l d Equations 28 2.h Antenna Model 30 "2.5 R e s t r i c t i o n to Narrow Beamwidth 34 2. 5.1 Two Dimensional Case 34 2.5.2 Three Dimensional Case 37 2.6 Importance and Range of Parameters 41 3. PERIODIC ARRAYS OF HEMICYLINDERS 43 3.1 Solutions f o r Various Orders of Accuracy 43 3.1.1 Exact Solution 43 3.1.2 Nearest Neighbour Approximation 46 3.1.3 F i r s t Order Nearest Neighbour Approximation .... 46 3.1.U Single Scatter Approximation 47 3.2 Numerical Calculations of the Scattered F i e l d 47 3.2.1 Determination of Active Scattering Area 48 3.2.2 E f f e c t of the Orders Of Scattering Approximations 48 3.3 Scattered F i e l d using Single Scatter Approximation 52 k. RANDOM ARRAYS OF HEMIC YLINDERS AND HEMISPHERES 54 h.l Generation of Scatterer Coordinates . . .' 55 U. 1.1 Uniform D i s t r i b u t i o n 55 4.1.2 Non-Uniform D i s t r i b u t i o n 55 k.2 C a l c u l a t i o n of F i e l d S t a t i s t i c s 59 v Page 4 . 3 Determination of Active Scattering Area 61 4 .4 Comparison of Results 64 4 . 4 . 1 Coherent F i e l d 65 4 . 4 . 2 Incoherent Intensity . 68 4 . 5 Summary 71 5 . CONSTRUCTION OF A LABORATORY SURFACE AND THE MEASUREMENT OF ITS SCATTERED FIELD 72 5 . 1 Design of the .Experimental System 72 5 . 1 . 1 Parameter Values 73 5 . 1 . 2 Method of Measurement 75 5 . 1 . 3 Construction of the Surface 78 5 .2 Simulation Test of the Surface D i s t r i b u t i o n 79 o n 5 . 2 . 1 Determination of Minimum Grid Subdivision 5 . 2 . 2 E f f e c t of Minimum Separation 8 0 5 . 2 . 3 P r o b a b i l i t y Density of Coordinates 83 5 . 2 . 4 D i s t r i b u t i o n of the Number of Scatterers per Independent Sample 92 5 . 3 I n i t i a l Testing of the Experimental Surface 92 5 . 4 Experimental Results Compared with Simulation Results 94 5 . 4 . 1 Preliminary Processing of Experimental Data 94 5 . 4 . 2 C o r r e l a t i o n Distance for the Scattered F i e l d 98 5 . 4 . 3 S t a t i s t i c s of the Scattered F i e l d 100 5 . 5 Summary 101 6. CONCLUSIONS 102 6 . 1 Surface Model 102 6 .2 Simulations.* l y ^ 6 . 3 Experiment 104 6 .4 Numerical Results 105 6 . 5 General Recommendations 106 v i Page APPENDIX A DETERMINATION OF UNKNOWN SCATTERING COEFFICIENTS 1 08 APPENDIX B RELATION BETWEEN REAL AND IMAGE FUNCTIONS m APPENDIX C FIRST ORDER NEAREST NEIGHBOUR APPROXIMATION 113 APPENDIX D. SINGLE OBJECT SINGLE SCATTER STATISTICAL CALCULATIONS 115 APPENDIX E INTERGRATION OF SINGLE SCATTER COHERENT AND INCOHERENT INTENSITY FOR THE CONTINUOUS UNIFORM DISTRIBUTION 118 E. 1 Two Dimensional Problem 118 E . l . l Coherent Field 118 E.1.2 Incoherent Intensity 121 E.2 Three Dimensional Problem 126 E.2.1 Coherent Field 126 E.2.2 Incoherent Intensity 129 REFERENCES 130 v i i LIST OF ILLUSTRATIONS Figure Page 2.1 Scattering Geometry for Two Dimensional Problems H 2.2 Real and Image Fields Incident on a Cylinder 16 2.3 Image Source 19 2.4 Geometry of Three Dimensional Scattering Problem 20 2.5 Radiation Pattern for Horn Antenna and for Approximate Beam Functions 32 2.6 Effective Surface Width. 34 3.1 Periodic Two Dimensional Surface 44 3.2 Magnitude and Phase of Scattered Field as a Function of the Ratio . of Surface Width to Beamwidth 49 3.3 Error Introduced by the Single Scatter Approximation as a Function of Hemicylinder Density for Various Values of Hemicylinder Radius. 50 3.4 Error Introduced by the Fi r s t Order Nearest Neighbour Approximation as a Function of Hemicylinder Density for Various Values of Hemicylinder Radius 51 4.1 Discrete Distribution of Scatterers 57 4.2 Modified Discrete Distribution of Scatterers 58 4.3 The Determination of Active Beamwidth - Coherent Field 62 4.4 The Determination of Active Beamwidth - Incoherent Field 63 4.5 A Comparison of Three Methods for Calculating the Single Scattered Coherent Field for an Array of Hemicylinders with p = 2% and a = 0.2X 66 4.6 A Comparison of Three Methods for Calculating the Single Scattered Coherent Field for an Array of Hemicylinders With p = 2% and a = 0.05 A 67 4.7 A Comparison of Three Methods of Calculating the Single Scattered Incoherent Intensity for an Array of Hemicylinders 69 5.1 Experimental System 76 5.2 Reference Plate for Normalization 77 5.3 Forming a Hemispherical Scatterer in Aluminum Sheet 78 v i i i Figure P a 8 e 5.4 Computer Generated Hemisphere Coordinates 79 5.5 Effect of Distribution Parameters 7 9 5.6 Scattered Field Statistics as a Function of N g for1 Various Densities of Hemispheres 81 5.7 Scattered Field Statistics vs. Density of Hemispheres for Various Separations > 82 5.8 Probability Density of Hemisphere Coordinates for an Object Density of 5% and 500 Sample Surfaces, N * 1 84 s 5.9 Probability Density of Hemisphere Coordinates for an Object Density of 5% and 500 Sample Surfaces, N g = 2 85 5.10 Probability Density of Hemisphere Coordinates for an Object Density of 5% and 500 Sample Surfaces, N g = 3 86 5.11 Probability Density of Hemisphere Coordinates for an Object Density of 10% and 500 Sample Suffaces, N g * 1 87 5.12 Probability Density of Hemisphere Coordinates for an Object Density of 10% and 500 Sample Surfaces, N =2 88 s 5.13 Probability Density of Hemisphere Coordinates for an Object Density of 10% and 500 Sample Surfaces, N =3 89 s 5.14 Variance of Scatterer Coordinates as a Function of N 90 s 5.15 Probability Density of the Number of Spheres in the Illuminated Area for N = 1,2,3 and Densities of 5% and 10% 91 s ' 5.16 Results from Test Surface 93 5.17 Data from a Typical Sheet 96 5.18 Statistics of Simulated and Experimental Data 99 ix LIST OF TABLES Table Page 2.1 The Parameters of the Scattering Problem 42 5.1 Statistics of Simulation and Experimental Data 100 x LIST OF SYMBOLS <(E)> = ensemble average of function E Re(E) = real part of complex function E Im(E) = imaginary part of complex function E |E| = magnitude of complex function E Arg(E) = phase of complex function E E* = complex conjugate of E A = position of antenna An,A^ = scattering coefficient for a single cylinder a = scatterer radius a n,a^ = scattering coefficient for a single sphere B n s = multiple scattering coefficient of the s^ n cylinder B = a real constant b,be,b^, = real constants C = column vector of incident wave coefficients o C = coherent intensity C a = antenna beamwidth factor Cs = distance from antenna to s^h scatterer C n(x) = any cylinder function of order n, real argument x D = width of antenna aperture Ds = amplitude factor of plane wave at s* n cylinder d =: separation of periodic scatterers djjĵ fj = minimum separation of scatterers E s c a t = total f i e l d scattered from an array of cylinders or spheres Eplane = total f i e l d scattered from a ground plane •̂ norm = ^ e l ^ normalization factor x i E o l 3 j = total f i e l d scattered from an array of hemicylinders Eyg = total scattered electric f i e l d Ep = electric f i e l d amplitude at some polarization angle E i j = polarization components of the electric f i e l d E T = Total el e c t r i c f i e l d E i = incident electric f i e l d E s = f i e l d scattered from the s^*1 cylinder E 0 = complex constant f i e l d amplitude factor E = normalized total scattered el e c t r i c f i e l d fb' fp' fMC = average scattering amplitudes f(9)> f(0 >0) = antenna beam functions *nm>̂ nm'̂ nm = s e r :"- e s expansion coefficients g( u)j sC^v) = antenna beam functions g n(u) = antenna beam function with variable sidelobe level H = matrix of multiple scattering amplitudes H n(x) = Hankel function of the f i r s t kind of order n, argument x h n(x) = spherical Hankel function of the f i r s t kind = a linear combination of H m_ n(x) and H m + n(x) i , j , k = cartesian unit vectors (j?) = incoherent intensity J n(x) = Bessel function of order n, argument x . j n(x) = spherical Bessel function of order n, argument x K = ratio of effective surface width to width of main antenna beam at the surface Kr, Ks, K Q = experimental system overall channel gains k = free space wave number L = a lower triangular matrix L = antenna to surface distance x i i N Q = number of scatterers i n the act i v e s c a t t e r i n g area Ng = number of subcells of the modified d i s c r e t e uniform d i s t r i b u t i o n P = point of observation P^(x) = Legendre f u n c t i o n of order n, argument x P™, Q,™ = functions of legendre functions R, 0 = polar coordinates r e l a t i v e to centre of antenna Ra, 0 a > 0 a = s p h e r i c a l coordinates r e l a t i v e to centre of antenna Rs, C£)g = polar coordinates of s**1 s c a t t e r e r R > S ,(p = reference l e v e l , s i g n a l l e v e l , r e l a t i v e phase l e v e l P P P rs> 6s>0s = coordinates r e l a t i v e to the l o c a t i o n of the s ^ 1 s c a t t e r e r s, t = indices s p e c i f y i n g the and t^* 1 s c a t t e r e r U = a -wave fu n c t i o n U = an upper t r i a n g u l a r matrix W = column vector of s c a t t e r i n g c o e f f i c i e n t s W = act i v e surface width W = width of antenna main beam at surface o X, Y, Z = rectangular coordinates r e l a t i v e to the centre of the acti v e s c a t t e r i n g area Xg, Yg = rectangular coordinates of the s**1 s c a t t e r e r X n s ' ^ns = s c a t t e r i n g c o e f f i c i e n t s f o r hemicylinders x s, y s , z s = rectangular coordinates r e l a t i v e to the centre of the s*" s c a t t e r e r va> z a = rectangular coordinates r e l a t i v e to the centre of the antenna (X = angle of incidence ^3 s = angle to antenna from the s sc a t t e r e r (5 = p o l a r i z a t i o n angle F = Neumann fa c t o r x i i i s p h e r i c a l unit vectors free space wavelength area density of objects c o r r e l a t i o n c o e f f i c i e n t complex c o r r e l a t i o n c o e f f i c i e n t normalized r a d i a l coordinate of the s s c a t t e r e r normalized c a r t e s i a n coordinates of the s s c a t t e r e r variance of the random v a r i a b l e x standard deviation of the random v a r i a b l e x covariance of the random va r i a b l e s x and y x i v ACKNOWLEDGEMENTS I wish to thank my research supervisor Dr. M. Kharadly for his expert guidance throughout this project. Acknowledgement i s given to the National Research Council of Canada for postgraduate scholarships from 1965 to 1969, and for support under grant A-3344. I wish also to thank Mr. D. Daines of the machine shop staff for his invaluable assistance with the construction of the experimental surface model. Finally, I thank Miss J. Murie for typing the manuscript, and Mr. K. McRitchie and Dr. D. Corr for their proficient proofreadin xv 1 1. INTRODUCTION The a n a l y s i s of electromagnetic or acoustic wave s c a t t e r i n g by rough surfaces requires the s o l u t i o n of the wave equation\7^U = - subject to the appropriate boundary conditions on the "rough" boundary. By "rough", one should understand that the boundary i s s o l e l y described by i t s s t a t - i s t i c a l properties. Examples of t h i s type of problem include radar returns from t e r r a i n or sea surface, radar exploration of the surfaces of the moon and planets, sonar returns from the ocean f l o o r , o p t i c a l r e f l e c t i o n s used i n d u s t r i a l l y to determine the q u a l i t y of machined surfaces, and many others. 1.1 E x i s t i n g Methods of Solution B a s i c a l l y , one must f i r s t solve the wave equation f o r a general boundary and then integrate the f i e l d over a l l possible configurations of t h i s boundary to obtain the required s t a t i s t i c s of the scattered f i e l d . Of course, a s o l u t i o n t h i s general may not be obtainable by the methods presently a v a i l a b l e . Therefore, s u f f i c i e n t r e s t r i c t i o n s must be placed upon the d e f i n i t i o n of the problem to allow one to obtain u s e f u l r e s u l t s . The following r e s t r i c t i o n s have been imposed, not n e c e s s a r i l y a l l at once, by previous i n v e s t i g a t o r s : 1. Harmonic time dependence. 2. I n f i n i t e plane wave incidence. 3. Boundary surface p e r f e c t l y conducting or pure d i e l e c t r i c i n t e r f a c e . h. Only s c a l a r s o l u t i o n obtained. 5. One dimensional roughness only. 6. Surface i s l o c a l l y plane and/or 2 7. Scattering elements are much smaller than the wavelength. 8. Active scattering areas are either very dense or very sparse. 9 . A particular restricted model i s chosen for the surface. A l l treatments of the problem have been subjected to limitation S. Generally speaking, there are two main methods for representing the scattering boundary which divides the methods for solving the problem into two distinct classes. The f i r s t method i s to define the boundary as some continuous random variable z = f(x,y) which is ultimately defined by i t s s t a t i s t i c a l moments (mean, variance, etc.) and subject to appropriate restrictions. The second method is to represent the surface by a collect- ion of discrete scatterers which are, in general, random in size, shape, and position. However, the only random variable considered so far has been the position of each object. 1 . 1 . 1 Continuous Surface Model Consider f i r s t the cbntinuous surface model. This was f i r s t invest- igated by Lord Rayleigh in 1 8 9 6 . He assumed a sinusoidal perfectly conducting surface with normal incidence. His approach required a Fourier 2 series expansion of the incident wave and the surface profile. S.0. Rice generalized the Rayleigh approach to a slig h t l y rough surface described by i t s small deviation from a mean plane. Rice's reflection coefficient was 5 verified experimentally with reflections from blacktop roadway . W.C. Hoffman used a similar expansion of the surface, but applied i t to the Stratton - Chu integral for the scattered f i e l d . Hoffman's results were no more general than Rice's, but his derivation had the merit of being 6 mathematically rigorous. T.B.A. Senior assumed that the roughness was a perturbation of a mean plane, and by using a Taylor series expansion of the 3 f i e l d at the boundary he developed a surface impedance tensor f o r the rough surface. Again, t h i s method i s r e s t r i c t e d to s l i g h t l y rough surfaces. 7 Beckmann used a p h y s i c a l optics method to obtain the required r e f l e c t i o n 8 c o e f f i c i e n t . More recently, Middleton attacked the problem from the point of view of pure communications theory with l i t t l e consideration Q given to the electromagnetic aspect of the problem. Also, Bass et a l appl i e d a perturbation method to sea r e f l e c t i o n s . I t should be noted that a l l the above methods require surfaces that are only s l i g h t l y rough. That i s , the surfaces have e i t h e r small deviations i n height from a mean -Diane, or they have a long c o r r e l a t i o n distance. 1.1.2 Discrete Scatterer Model !0 11 The main i n v e s t i g a t i o n s have been c a r r i e d out by Ament , Biot , Spetner 3"^, and Twersky''"^'"'*^'1^'1^. Twersky"^' has a l s o considered the s t a t i s t i c a l problems which are connected with t h i s method when higher order s t a t i s t i c a l moments of the f i e l d are to be considered. Most i n v e s t i g a t o r s have only considered the problem of the mean scattered f i e l d , or at best, the mean and variance. Ament modelled his surface as a c o l l e c t i o n of randomly spaaed h a l f - planes. Biot considered a uniform d i s t r i b u t i o n of hemispheres on a per- f e c t l y conducting ground plane, but these objects were required to be so small that very l i t t l e s c a t t e r i n g could take place. Spetner assumed a d i s t r i b u t i o n of point s c a t t e r e r s , which y i e l d e d r e l a t i v e l y simple r e s u l t s . Twersky has contributed the l a r g e s t amount of information towards the s o l - u t i o n of these problems. He considered e i t h e r hemispheres or hemicylinders on a p e r f e c t l y conducting ground plane. The main advantage of the Twersky approach i s that use i s made of our extensive knowledge of the s c a t t e r i n g properties of a member of the d i s t r i b u t i o n alone. Here again, the random 4 variable has been l i m i t e d to be the position of the objects. Under s u i t - able approximations the results are simple, at least f o r the mean of the scattered f i e l d . 1.1.3 Experimental Studies Some workers have inve s t i g a t e d t h i s problem experimentally. In these studies, r e s t r i c t i o n #2 i s removed by the use of actual receiving and transmitting antennas with t h e i r associated f i n i t e beamwidth and non-plane wavefront. The remaining r e s t r i c t i o n s , except f o r 8, may also be removed. The choice of the surface model divides the experimental studies i n t o two sections corresponding to the choice of s p e c i f i c natural surfaces or of laboratory constructed models. Natural surface data has 5 been analysed by such investigators as W.H. Peake who considered r e f l e c t i o n s 19 from blacktop and lawn grass, and M. Katzin ̂  who considered radar sea on c l u t t e r . For the laboratory studies, R.H. Clarke and G.O. Hendry worked with a water surface agitated by a controlled a i r flow; Hiat t , Senior, and 21 Weston considered a surface produced by rough casting of metal; B.E0 22 Parkins constructed two surfaces, one by denting sheet metal with a hammer, and the other by flowing grout over sand. In a l l the above experi- ments, a method had to be devised f o r measuring the s t a t i s t i c s of the models as the s t a t i s t i c s could only be roughly controlled. 1.2 Aims of t h i s Study B a s i c a l l y , the aims of t h i s investigation are the following. 1. To choose a feasibl e and controllable surface model. 2. To include the effe c t of the f i n i t e beamwidth and non- plane character of incident radiation from an antenna. 23 3. To develop a Monte-Carlo technique f o r c a l c u l a t i n g the scattered f i e l d s t a t i s t i c s ; the main problem here i s to 5 d e v i s e a s u i t a b l e n u m e r i c a l m e t h o d f o r d e t e r m i n i n g t h e d i s t r i b u t i o n o f s c a t t e r e r s . k. To c a r r y o u t a s e r i e s o f e x p e r i m e n t s u p o n a n a r t i f i c i a l s u r f a c e f o r w h i c h t h e p r o p e r t i e s c a n be a c c u r a t e l y c o n t r o l l e d . 1.2.1 S u r f a c e M o d e l F o r t h i s w o r k t h e T w e r s k y m o d e l o f a r o u g h s u r f a c e was c h o s e n . The r e a s o n f o r t h i s i s t h r e e f o l d . F i r s t , i t was f e l t t h a t a more e x a c t s o l u t i o n s h o u l d b e o b t a i n a b l e f o r h i g h e r d e g r e e s o f r o u g h n e s s b e c a u s e t h e s c a t t e r i n g c h a r a c t e r i s t i c s o f t h e i n d i v i d u a l o b j e c t s w e r e known e x a c t l y . S e c o n d , a M o n t e - C a r l o m e t h o d was p r o p o s e d f o r t h e c a l c u l a t i o n o f t h e s c a t t e r e d f i e l d s t a t i s t i c s w h i c h r e q u i r e s t h e g e n e r a t i o n o f t h o u s a n d s o f d i f f e r e n t s u r f a c e p r o f i l e s . T h i r d , e x p e r i m e n t a l m o d e l s c o u l d be more e x a c t l y c o n s t r u c t e d , c a n d w o u l d h a v e d e s i r e d r e p e a t a b i l i t y . The g e n e r a l a n a l y s i s o f t h e p a r t i c u l a r s u r f a c e s c o m p o s e d o f a r r a y s o f c i r c u l a r hemicylinders or hemispheres on a l o s s l e s s ground plane i s c o n s i d e r e d i n C h a p t e r 2. I t i n c l u d e s t h e e f f e c t o f f i n i t e n o n - p l a n e wave i n c i d e n c e . M u l t i p l e s c a t t e r i n g i s c o n s i d e r e d i n C h a p t e r 3 . 1.2.2 I n c i d e n t Beam M o d e l T h e r e a r e two r e a s o n s f o r t h e i n c l u s i o n o f f i n i t e b e a m w i d t h n o n - p l a n e i n c i d e n t r a d i a t i o n . The m a i n one i s t o d e t e r m i n e t h e b e h a v i o u r o f t h e f i e l d i n a more r e a l i s t i c s i t u a t i o n . H o w e v e r , i t s h o u l d a l s o be m e n t i o n e d t h a t t h e i n f i n i t e b e a m w i d t h c a s e ca.nnot b e t r e a t e d b y t h e M o n t e - C a r l o a p p r o a c h b e c a u s e t h i s w o u l d r e q u i r e t h e s t o r a g e o f a n i n f i n i t e a r r a y o f numbers o r an i n f i n i t e amount o f c a l c u l a t i o n t i m e . The s p e c i f i c m o d e l p r o p o s e d i s t h a t o f t h e f a r - f i e l d r a d i a t i o n f r o m k a n a p e r t u r e w i t h a n a p p r o p r i a t e i l l u m i n a t i o n f u n c t i o n . T h i s m o d e l was c h o s e n f o r t h e f o l l o w i n g r e a s o n s : 1. It can be made to resemble very closely the radiation pattern of a laboratory antenna. 2. • It is simple enough mathematically to bs easily included in the field equations. 3. The beamwidth, sidelobe level, and sphericity of the wavefront can be easily controlled. This problem of the incident beam is discussed in detail at the begin- ning of Chapter 2. 1.2.3 Monte-Carlo Method The Monte-Carlo method is the most direct method for calculating the scattered field statistics and i t has known accuracy. Thus i t is suitable for obtaining numerical results and for evaluating theoretical approaches. Analytic solutions are usually obtained by the following sequence of opera- tions: 1. Calculate the scattered field due to one object. 2. Use the above to calculate the total field scattered by an a r r a y o f the above o b j e c t s . 3. Integrate the field over a l l possible configurations using some particular distribution function for the locations of the objects to obtain a l l the desired statistical moments. The drawback of the analytic method is the inherent connection between the calculation of the field, and the calculation of its statistics. On the other hand, the Monte-Carlo technique proceeds as follows: 1. As above. 2. As above. 3. Determine a single configuration according to some numerical random or pseudo-random process. k. Calculate the field due to this configuration by 2. 5. Keep a running mean, variance, etc. 7 6. Repeat 3»4 and 5 u n t i l a s u f f i c i e n t number of surface configurations have been used t o approximate the desired s t a t i s t i c a l moments. It thus eliminates the drawback i n the a n a l y t i c method as i t w i l l work on any sequence of random functions, provided only that the elements of the sequence themselves may be numerically calculated. The implementation of the Monte-Carlo method i s therefore d i v i d e d i n t o three d i s t i n c t p a r t s : 1. Obtaining an a n a l y t i c expression f o r the s c a t t e r e d f i e l d from a f i x e d configuration. 2. Numerical determination of the coordinates of each configuration. 3. Numerical c a l c u l a t i o n of the f i e l d using 2 and the c a l c u l a t i o n of the s t a t i s t i c a l moments. The a p p l i c a t i o n of the Monte-Carlo method to the surface models studied i n Chapters 2 and 3 i s given i n Chapter h, where the problem of s u i t a b l e methods of determining the d i s t r i b u t i o n of s c a t t e r e r p o s i t i o n s i s a l s o discussed. 1.2.H Experiment The main purposes f o r performing an experimental study were t o compare a c t u a l r e s u l t s with theory and t o give another method which could be used where the approximate theory became i n c o r r e c t . These aims r e s t r i c t e d the choice of the experimental surface t o be of the c o n t r o l l e d l a b o r a t o r y type. The experimental surface used was, i n f a c t , a d u p l i c a t i o n of the one used f o r the t h e o r e t i c a l simulation of the problem: p e r f e c t l y conducting metal hemispheres on a n e a r l y f l a t metal ground plane. The l o c a t i o n s of the hemispheres could be determined by the same method as that used i n the simulation. Thus, unlike previous experiments the d i s t r i b u t i o n and density of the experimental surface could be s t r i c t l y c o n t r o l l e d . In p r i o r 8 experiments these f a c t o r s had t o he estimated a f t e r the surface was constructed. The implementation of the experiment i s described i n Chapter 5.* 1 . 3 R e s t r i c t i o n s In summary the approach taken i n t h i s t h e s i s required that the previous l i s t of r e s t r i c t i o n s on the problem be modified t o the f o l l o w i n g : * 1 . Harmonic time dependence. 2. F i n i t e beam incidence with non-plane wavefront. 3 . Boundary surface same. h. Vector s o l u t i o n . 5. Surface two or three dimensional. 6. Surface need not be l o c a l l y plane. 7 . Object dimensions up t o a wavelength (simulation) or greater (experiments). 8. Object density from sparse to about 50$. 9. Surface model c i r c u l a r hemicylinders or hemispheres on a p e r f e c t l y conducting ground plane. 9 2 . THE SCATTERED FIELD FROM AN ARBITRARY CONFIGURATION OF DISCRETE SCATTERERS ON A PERFECTLY CONDUCTING PLANE 2.1 Introduction In t h i s chapter the s c a t t e r i n g of a f i n i t e non-plane electromagnetic beam by a sing l e configuration of objects i s analysed. A simple far-zone r a d i a t i o n pattern i s assumed f o r the i n c i d e n t wave used i n the d e r i v a t i o n of the f i e l d scattered from an array of p e r f e c t l y conducting hemicylinders on a p e r f e c t l y conducting ground plane i n section 2 .2 , and from an array of p e r f e c t l y conducting hemispheres on a p e r f e c t l y conducting ground plane i n section 2 . 3 • The array of hemicylinders i s formulated as a purely two dimensional problem by considering i n f i n i t e l y long cylinders and the plane of incidence perpendicular to the axis of the cylinders as shown i n f i g u r e 2 . 1 . Hence 2k the r e s u l t s may be derived from the s c a l a r wave equation . Of course, t h i s model of a rough surface i s a very r e s t r i c t e d one but the r e s u l t s could be applied to any p h y s i c a l problem which i s e s s e n t i a l l y (or l o c a l l y ) two dimensional such as radar returns from water waves. 13 The d e r i v a t i o n e s s e n t i a l l y follows Twersky . F i r s t , a gra t i n g of objects i s analysed by the separation of va r i a b l e s method, and then the 13. Rayleigh Image technique i s ap p l i e d t o obtain the s o l u t i o n to the surface problem. Because of the r e l a t i v e s i m p l i c i t y of the c a l c u l a t i o n s and i n a n t i c i p a t i o n of the study i n the next chapter of the p e r i o d i c array, the c a l c u l a t i o n s are given i n d e t a i l f o r the two dimensional case. For convenience backscattering only i s considered. For the present purpose i t i s s u f f i c i e n t , to consider only the s i n g l e s c a t t e r approximation f o r normal incidence i n the three dimensional case. The reasons f o r the 10 normal incidence r e s t r i c t i o n i s given i n section 2.3.*+. Beckmann states that the geometry of the problem i s perhaps the most dominant f a c t o r i n rough surface scatter. Under t h i s assumption then even the approximate solutions considered here should prove u s e f u l . The experimental r e s u l t s are given i n Chapter 5 The development generally follows the treatment of hemicylinders, but fu r t h e r complications are introduced by the vector nature of the problem. The f i e l d scattered from a sing l e sphere cannot i n general be derived from the two s c a l a r equations. These vector solutions are discussed f u l l y by S t r a t t o n ^ and. by Morse and Feshbach^, but the form used here i s essent- i a l l y that given by Twersky . In section 2.h a s p e c i f i c model i s proposed f o r the antenna beam used i n a l l subsequent studies. The model i s chosen to be representative of a laboratory horn antenna. F i n a l l y , i n section 2 . 6 the relevant parameters of the problem are discussed and t h e i r role i n the problem and range of values are decided upon. 2.2 Scattering from a Configuration of Hemicylinders 2 . 2 . 1 C y l i n d r i c a l Incident Beam conditions on the surfaces of the cylind e r s . Although the s o l u t i o n may be obtained as e a s i l y f o r d i e l e c t r i c c y l i n d e r s , only the p e r f e c t l y conducting case s h a l l be considered i n the following problems f o r computational s i m p l i c i t y . Let - i o t Assume harmonic time dependence, e Hence the reduced s c a l a r must be solved with the correct boundary ( 2 . 1 ) 6 = 1 where i s the t o t a l e l e c t r i c f i e l d i s the incident beam, at angle Ct; E g i s the complete multiple scattered f i e l d from the s object N 0 i s the t o t a l number of objects At t h i s point, consider the form of Ej_. The usual treatment would TT be to put E± = e i k * ? = e i k r c o s ^ + 2~̂  and r e f e r t h i s to the coordinate of the s * h object. That i s , one would assume plane wave incidence. However, Ej_ may be considered t o be the f a r - f i e l d r a d i a t i o n from some d i r e c t i o n a l but two dimensional source such as a long narrow s l o t . The incident wave must be considered i n terms of a s e r i e s of elementary c y l i n d r i c a l waves i n order t o match boundary conditions. Now, l e t = F(R,0), where R and0are r e f e r r e d t o the centre of the radiator. In p a r t i c u l a r , since only the far-zone f i e l d of the r a d i a t o r need be considered, l e t i k r E. = E _e f(0) ( 2 . 2 ) which i s the form of a c y l i n d r i c a l wave m u l t i p l i e d by the space f a c t o r , f(0). The scattered f i e l d i s derived using the geometry given i n f i g u r e 2 . 1 . Figure 2 . 1 Scattering Geometry f o r Two Dimensional Problem 12 In the above diagram, A = p o s i t i o n of antenna P = point of observation ( X g , 0 ) = p o s i t i o n of the s^*1 c y l i n d e r Ct = angle of incidence L = distance from antenna t o surface Assume that L i s large, as the far-zone f i e l d of the antenna i s to be considered. Also, assume that r g i s small, which i s possible because 4- V* r s w i l l be f i n a l l y placed at the surface of the s object to s a t i s f y boundary conditions. Thus, using f i g u r e 2.1, R 2 = ( L sinCi + X_ + r„ cos0 g )d + ( L cosQt - r s s ina ) 2 (2.3) l e t C 2 = ( L cosa) 2 + ( X s + L s i n a ) 2 (2.k) t 8 n & = Xo + L s ina ( 2 ' 5 ) and, since r L S s R 2 * ; C 2 + 2 r g c g c o s ( 0 s +/3g) ( 2 . 6 ) As C i s a c t u a l l y the distance from the centre of the antenna to the s th s object (see f i g u r e 2.1), and r w i l l be l i m i t e d t o the v i c i n i t y of s the surface of the object, i t i s a l s o true that r « C f o r any s, s N N s provided that the antenna i s s u f f i c i e n t l y f a r from the surface and that the i l l u m i n a t e d area of the surface i s s u f f i c i e n t l y l i m i t e d i n extent. Of course, the extent of the surface considered w i l l be l i m i t e d by the beam 13 width of the antenna. Therefore, R ~ C + r cos(0 + Q) (2.7) s s s A s i m i l a r approximation i s applied to the angular antenna coordinate. From f i g u r e 2.1 r r - i f L c o s Ct- r s s i n Q 0 = K - a - tan 1 - L\ (2. 8) * L s i n a + X +r cos 0 ^ S S s Now, make the assumption that the amplitude, f(0), i s constant and equal to the value i t a t t a i n s at the centre across the e n t i r e surface of the s c a t t e r i n g object. This approximation i s reasonable provided that r « L cos a s (2.9) • r « X + L sinCX s s which implies that the objects are small compared to the distance from the antenna to the surface. Hence, equation (2.8) becomes _. , L cos Ct 6* jj--a- t a n ' X (2.10) ^ X + L sinCt s t h Therefore the incident wave a r r i v i n g at the s object can be assumed to be approximately plane i n the v i c i n i t y of that object, that i s , E . * E f i f i e i k r s c O S ( e s -Ps)f{E_a_p) ( 2 > 1 1 ) F o r s i m p l i c i t y , l e t ikC — 14 so that E. = E 0 D s e i k r s C ° s ( ^ + P s ) (2.13) Also, note that for convenience f(0) = 1 and that E c is merely a complex constant amplitude factor which may later be adjusted to give a normalized solution. 2.2.2 Scattering Coefficients for Grating The plane wave (2.13) may now be expanded in a series of cylindrical waves2** referred to the axis of the s**1 cylinder. DO % = E 0 D s V ^ J n ( k r s ) ( i ) n e i n ^ s + PS) (2.lk) n oo +v» The wave scattered from the s object may be expanded in a similar series of outgoing cylindrical waves. oo E s - ^ B n s H n ( k r s ) e i n 0 s (2.15) n - - o° where H^Z) shall always refer to H^ 1\z), the Hankel function of the fir s t kind. Thus, from (2.1), the total field is given by E T * E i Y~\ sH n(kr s)e i n e £ 3 <2.l6) s = i n--°o for a particular set of positions X^, Xg, ... , and the unknowns B n g are determined from the boundary conditions. The calculation is outlined in Appendix A and the results are N 0 co 3 - i f 3ns *n where E 0 D s ( i ) n e l r ^ s + XjWW kl*t " X s ! ) f r ,st nm t-l m. - co (2.17) An = -HnTkaT ( 2' l 8 ) £ = - Jn{ t e) (2 va) S t f 1 ' * > s nm / \ m+n (-1) , t<s ( 2 . 2 0 ) At t h i s point, one i s faced with the formidable task of s o l v i n g the i n f i n i t e set of simultaneous l i n e a r a l g e b r a i c equations, ( 2 . 1 7 ) , f o r the i n - f i n i t e number of unknowns, B . F i r s t , reduce (2 .17) to a f i n i t e number of ns equations i n a f i n i t e number of unknowns by assuming that the A n become n e g l i g i b l e a f t e r some n = N, say. This i s reasonable e s p e c i a l l y i f ka i s small ( i . e . small s c a t t e r e r s ) . Second, introduce the change of v a r i a b l e s , ns (B + (-1) B n s ) 2E -" s o n = 1 ,2 , B X os os 2E (2 .21) ( i ) n ns (B 2E ~ n s o ns' (2.22) Substitu t i o n of ( 2 . 2 l ) and ( 2 . 2 2 ) i n t o (2 . 17) y i e l d s r No N ns = ^ 1 D gcos n/̂ , + , .n-m + , • M X m t V ^ I X t Y n s » A -n £=•/ tjG mro No _N - i D . s i n nfi + ) ) ( i ) Y , H (k X. s <"~s / , [_ ; mt mnv 1 ^ t-/1.?s w-o i % st X s l ) f n m , v st I X ) f .} si ' nm( ( 2 . 2 3 ) (2.24) where H* (s) = H (x) ± ( - l ) n H m + n ( x ) mn m-nv ' v ' Tn+n̂  > (2.25) ( 2 . 2 6 ) 16 Use was made of the f a c t that C (s) = (-l) C (z) f o r any c y l i n d e r function -n n 27 C . Note that the systems of eauations to be solved, ( 2 . 2 3 ) and (2.2k), n are each h a l f the s i z e df the o r i g i n a l system (2 .17) but the operation must be performed twice. This i s , of course, a great saving of work i f an exact numerical s o l u t i o n i s required. Before attempting t h i s , however, 1 ,7 ,13 i t i s advantageous to apply the Rayleigh image technique t o obtain the proper form f o r the s o l u t i o n to the surface problem. Figure 2 .2 Real and Image F i e l d s Incident on a Cylinder 2 . 2 . 3 Image Method f o r Transformation to Surface Problem The image method i s as follows. For a wave incident at angle QL as shown i n f i g u r e 2 . 2 , the scattered f i e l d from the h a l f c y l i n d e r w i l l be composed of the f i e l d produced by a f i e l d incident at angle Ct and i t s image at angleJT-CUncident upon a whole cyl i n d e r . The s o l u t i o n w i l l be correct i n the region Y)>0 only. The v a l i d i t y of t h i s may be r e a d i l y checked by t e s t i n g t o see i f the boundary conditions are s a t i s f i e d . I f E r j , i s now defined to be the t o t a l f i e l d f o r the surface problem, E^(CQ = E .(a) ± E^TT-CL) + = E . + E . + E , . mc plane obj E s c a t ^ ± E s c a t ^ - a ) (2.27) ( 2 . 2 8 ) There i s considerable s i m p l i f i c a t i o n of the problem a f t e r a p p l i c a t i o n of the above equations due to the symmetry of the r e a l and image functions. Now, the E Q t ) j i n equation (2.28) i s composed of the f i e l d s scattered by the i n d i v i d u a l objects: E o b j » E s c a t ( a > * E s c a t ( ^ - a ) (2.29) E f d ) ± E s ( 7 7 - a ) (2.30) ^ (2-31) I t i s i n t e r e s t i n g t o note that only one set of the c o e f f i c i e n t s Xns or Y n g need to be c a l c u l a t e d f o r each p o l a r i z a t i o n f o r the surface problem (see Appendix B). The r e s u l t i n g f i e l d s scattered by the s t h c y l i n d e r are: ^ E s = ^ Eo Xj^Xs 0 0 8 n 0 s H n ( k r s ) (2-32) N E s = - , + i E o X ^ ( i ) n Y n s S i n n W k r s ) ( 2'33) 2.2.U Backscattered F i e l d From t h i s point onward, only the scattered f i e l d i n the backscatter d i r e c t i o n w i l l be considered. The analysis f o r other d i r e c t i o n s i s no d i f f e r e n t i n p r i n c i p l e . Backscatter i s a reasonable choice as i t repre- sents the monostatic radar problem. From a consideration of the geometry of the problem (see figu r e 2.1), i . e . p u t t i n g point P at point A ° S (2.3*0 (2.3k) holds a l s o f o r forward s c a t t e r i f C(CL) and R (P) are replaced s / s with C (-0) and,G s(-a). 18 H n(kC sW (-i) n-7= F-t/EST ( 2- 3 5> Since C i s assumed t o be large, HL(kC„) may be approximated by the s n s 2 7 f i r s t term of i t s large argument asymptotic expansion t n ( l ^ Q e ^ C s ( 2 . 3 2 ) and ( 2 . 3 3 ) when combined with ( 2 . 3 ^ ) and ( 2 . 3 5 ) give the f o l l o w i n g approximate expression f o r the far-zone backscattered f i e l d from the array of hemicylinders: N ° ikC N v Ji 5=1 ^ ^S n - Q . .\ ^ ikC N E 1 = hiV-ZjdE V %=^V ( - D V a i n nO ( 2 . 3 7 ) 2 . 2 . 5 Normalization of F i e l d Equations F i n a l l y , the constant E q must be selected so as t o y i e l d a normal- i z e d r e f l e c t e d f i e l d . This normalization i s such that the t o t a l r e f l e c t e d f i e l d i s unity when the the surface i s f l a t . From equation ( 2 . 2 8 ) E = E, + E , + E ( 2 . 3 8 ) 1> inc plane obj = E . + \ a ( 2 . 3 9 ) mc l b where E i s the incident f i e l d , E i s the f i e l d r e f l e c t e d by the inc plane plane and E i s the t o t a l scattered f i e l d . The required normalized f i e l d Tb i s then E = E t s = 1 + E ° b j (2 .U0) E E norm plane Now, E ^ a r e i s simply the image of the incident wave i n the plane (see fi g u r e 2 . 3 ) . Therefore, from equation ( 2 . 2 ) , the corresponding image source becomes 1 9 "plane ~ "~ "o/kR E H ^ _ _ = ± E , 5 ~ f (9) (2.41) where R and Q are determined from the configuration shown i n fi g u r e .2. 3. s>- x Figure 2.3 Image Source For backscatter, i| j_ ei2kL cosQ E p l a n e = ± EoJKL~7^am) (2.42) while f o r s c a t t e r i n g i n the forward d i r e c t i o n ,i2kL E ' = ± E 7ff= plane o/2kL (2.43) Equations (2.36), (2.37), (2.4o) and (2.42) may now be combined t o y i e l d the f i n a l form of the normalized backscattered f i e l d . E ' ' = 1 + i | ( 1 -jQvtekL cos a e rrr m) i k ( C s - 2L cos a) s= 1 No ( - l ) n X cos nO J ns ik(C„ - 2L cos a) r\-o N (2.44) S= 1 (2.45) 20 Now, i f the unknown c o e f f i c i e n t s X and Y can be determined from ' ns ns equations (2.23) and (2.2.4) then the normalized backscattered e l e c t r i c f i e l d may be determined from equations (2.44) and (2.45). In summary, the problem has thus f a r had the f o l l o w i n g r e s t r i c t i o n s placed upon i t : 1. Expressions have been e x p l i c i t l y given only f o r the backscattered f i e l d . 2. The surface must be i l l u m i n a t e d by a f i n i t e width beam located a large distance from the surface. 3. The s c a t t e r i n g objects must be small compared to the e f f e c t i v e width of the beam at the surface. 2.3 S c a t t e r i n g from a Configuration of Hemispheres 2.3-1 Spherical Incident Beam The geometry of the problem i s shown i n f i g u r e 2.4. Note that only normal incidence i s considered from the outset. Figure 2.4 Geometry of Three Dimensional Scattering Problem In keeping with the two dimensional problem, consider a s i m p l i f i e d f a r - f i e l d r a d i a t i o n pattern of spherrcal waves, Y 21 e i k R a M s in(0 +BK + cos((7V e i k R a ^ ° kR a (2.46) where 8 i s the polarization angle: 6=0 for || polarization ^ = If f o r I polarization and 6" , are unit vectors in the antenna coordinate system. & P NOT?, by standard vector analysis, <Ee = c o s 0 a c o s 0 a i a + c o s 0 a s i n 0 a j a - sin0 a " k ^ - s i n 0 a r + eos0aj; ( 2 . 4 7 ) (2.48) i s the relation between the spherical and rectangular unit vectors. Therefore, at the antenna, E p = (s in(0 a +6)cos 0 a c t>s0 a - cos(0 a + t))sin(7j) a ^i a + (sin(0 f t +5)cos 0 a s i n 0 a + cos((4-+6)008 0 ^ 3 ^ ( 2 . 4 9 ) - - s i n(0 a +6) sin 0 ^ Refer the incident f i e l d to the x s , y g , z g axes at the s object instead of to the antenna coordinate axes by the two translations along L and V y + Y„ (2.50) z a = L and the unit vectors become 22 consequently, J a = J 8 (2 .51) R ! = r s + R ! + ^ + 2 ( x s x s + y * Y s " z s L ) ( 2 ' 5 2 ) where, X s = R s c o s * s Y s = R B s i n c ^ s (2 .53) are the coordinates of the s^*1 scatterer. To f u r t h e r s i m p l i f y the problem, rotate the x s , y s , z g coordinate axes through the angle db„ about the z„ axis. That i s , x s = x s c o s < ^ s - v s s i n < P s ' y g = x^sin<£ s + y scos $>g (2.5*+) z = z' s s Equation (2.52) becomes, upon the s u b s t i t u t i o n of equation ( 2 . 5 ^ R a = r s 2 + C s + 2 ( R s x s " L z s > < 2 ' 5 5 ) where C 2 = R2 + L 2 (2 .56) s s From f i g u r e '2. U6 s i n 9 = R./C C S S (2 .57) cos G c = L/CS Hence, R a = r s 2 + Cl + 2 C s ( x s s i n e c ~ z s c o s 0 C ) ( 2 > 5 8 ) And, since r « C i n the v i c i n i t y of the s t h object, C g + x s s i n 0 c - z s c o s 9 c (2.59) I t i s reasonable to assume, as i n the two dimensional case, that the amplitude functions f and e may be considered to be constant across the P surface of the s scatterer. This assumption w i l l be v a l i d f o r small objects. Therefore, from f i g u r e 2. k, (2.60) and from equations (2. 51) and (2.5*+), i = - cos cb i ' + s i n dp j / a - s s u s'Js J a = sindbgi^ + cos<$>sJ/ (2.61) Thus, by equations (2.6o) and (2.6l), equation (2.^9) becomes: 7 ~ sin(dps - d)(cos 0 c T s + s i n Q*'B) + cos(£ s - 6)T S (2.62) F i n a l l y , then, the incident wave may be approximately represented as ikC E. « E Q - - S e i k ( x s s i n 6c - z scos 8 c ) f ( 0 ,77- $ ) 1 k C s ' c s (2.63) sin(cp - 6 ) ( c o s Q i ' + s i n A k') + cos(d> - 5)V / \ U c s ^ C S S / U S i n the x',y/)z' coordinate system. This corresponds t o an incident wave, s ^ t h which i s plane across the s scatterer, with angle of incidence and angle 24 of polarization and a = - 6 c 6' = <5-<£> (2.64) r e s p e c t i v e l y . 2.3.2 Single Scattered F i e l d As the d e r i v a t i o n of the two dimensional scattered f i e l d has been considered i n d e t a i l , the f o l l o w i n g e x p r e s s i o n 1 ^ f o r the three dimensional f i e l d scattered from a single hemisphere on a ground plane with the incident plane wave of equation (2.63) i s presented without f u r t h e r comment: h F - - ^ V ^ - C ^ i ) , h , J E s ' " ^ s ~ / _ , ( i ) n W s ) n = x CO P™ cos m^sinp'-mP^sin m(7̂ cos(j)' * sTnTJ m - o m + n o d d Ac ^ k r s ^ P™ cos m^sinS-m?^ sin m^cosO^ m + i a n h n ( k r s ) Y>-\+n odd sintt nTF^cos n d ) sinS+mPl? sin m.ct>cos6> T ^ 1 - sin0/sina sin9. n = t +2iE4^ (OVtll) P kW k r^V" [m P " C ° S ^ P ° ^ < s i n WB?1"̂ sin@ ssina sinQs, • i a n h n ( k r / ) m - o m +n o d d —| j m + r> even P™ cos mC^coso^+mP^sin wtpsinff , ifnCX 3 1 JJ where, (2.65) J n ( k a > a n " " h n(ka) / • j k a j ^ k a ] ' a n " " [ t a h j k a j ' (2.66) and ,ra -- (n-m). jaf Q ' N m, . = 6 7 ;;—P (cos H )P„(cos(X) n ^m(n+m).' n v u s ' i k C s E ' = E _ — — f ( e c ^ ) o kC (2.67) (2.68) m The subscripts (X and Q of P denote d i f f e r e n t i a t i o n with respect to CC or Q m ra of the appropriate Legendre function P ( c o s d ) or P (cos@ ) i n equation n n s ( 2 . 6 7 ) . 2 . 3 . 3 Far-zone Backscatter As i n the c y l i n d r i c a l case, only the far-zone backscattered f i e l d w i l l be considered. Therefore, r ' = C s s es7 =ec 01= 7 7 ( 2 . 6 9 ) m . . . Consider the expressions f o r P given by equation ( 2 . 6 7 ) and i t s n . der i v a t i v e s . Substitute the va r i a b l e s from equations ( 2 . 6 4 ) and ( 2 . 6 9 ) . Then, n ni. (n+«j).' P m(cos e ) n v ^c ( 2 . 7 0 ) P n = ^ m T n 7 ^ P n ( C 0 S e c ) p n ( c ° s 0 c ) s i n 0 C (2-7D = -P_ n, But 28 pjTcos Q ) = . 2 nK w c sin y -n cos @ cP m(cos 0 C) + (n+m)P^_ 1(cos Qc) CL_ ( 2 . 7 2 ) Let m - n cos 6 - f n + m 1 P n - l ( c o s 0 c ) ^ ( c o T B X (2 .73) Then, m m m m °n P = -p^ = p n e n « n s i n Second, consider the function h (kr ) and i t s d e r i v a t i v e s . Since r = C i s large, h (kr ) may be approximated by the f i r s t term of i t s s s n s 27 asymptotic expansion . ( 2 . 7 5 ) . ikC -n r -,/ ikC k r 7 s th Equation (2.65) f o r the backscattered f i e l d from the s hemisphere s i m p l i f i e s under the above approximations and s u b s t i t u t i o n s to the follow- ing expression. E « -2iE'-s' o kC s n n m m m2 \ \ m 2 m> \ \ vn, v f \ v , , m V  m > ( - D . ( ^ l ) a n ) - ( - 1 ) ^ 0 + a ) (-D m P n 4 ' n( n+1) C L i • 2A nL / J m+n odd m+n e v e n sin(<5 - & ) € a ( 2 . 7 6 ) / N 1 1 , v / m 2 m \ \ m m m27| _ . — a n ) (-D J i l n " a n ) F n Q n f o s ( 6 " € 0 n . i n ( n + l ) ^7p s i n ^ S i n 2 0 c ^ m+n odd - m+n even The scattered f i e l d given by the above expressions i s resolved i n t o s p h e r i c a l components i n the x 7, y', z ' coordinate system. s s s The f i n a l form of the f i e l d equations i s most conveniently given as components i n the antenna coordinate system. In t h i s case, the component: can be recognized as the d i r e c t and cross-polarized components. Let ikC ! EQ ssin(5 - 's E = ~ 2 i E y e 2? s' EA sin(5 - cj> )C a + cos(<5 -&)€, u& s V / r s s cp, (2 .77) Wow, resolve these components i n t o the corresponding rectangular components. That i s , as i n equation ( 2 . 4 8 ) , and using ( 2 . 6 9 ) , E =-.-2iE7 e ikC, s' o E ^ Sin(6-O s)cosei s / + E 0 qcos(5-^) ; j q / s s' + E& s i n (5 ~ Cp ) s i n g k s' - c s' (2 .78) From equation ( 2 . 6 l ) , i = - coscbi + sincpj s' *'s a g a j s,= s i n c p i + coscpj a (2 .79) k - - k s' a So that at the antenna, — ikC E = 21E e -E sin(5 - <$> )cos0 coscb + E, cos(£ - CD )sincp y s s c "s >"s ~s s E„ sin(5 - (f) )cos0 sincp + Ei cos(($ - cp )coscp - E Q sin((5 - cp )sin@ k ( 2 . 8 0 ) I t i s s u f f i c i e n t , however, to consider only the two p o l a r i z a t i o n s f o r 5= §-77 and 5= 0 respectively. That i s , _ 2 ikCL E = 2 i E s o L . k C S - . -Eg cos <£> cos@ + E^ sin CD s s c rs s L E e cos0 + E 0 sin<$>coscJ> j ~s s E (2.81) -E cosc£> sin0 k \ es s c a J and TikC " 2 i E 1 f i ( e C , 7 T - ^ S ) I E e s c o s 6 c + E$s£ sinCD coscbi ~s T3 a 2- r ^ 2 ~ - E e sin Cp cosQ + E^ cos Cp 'a ( 2 . 8 2 ) +EQ s i neb sin0 k 1 s - s c ai 2 . 3 . 4 formalization of F ie ld Equations Again, the normalized to ta l backscattered f i e l d i s the actual quantity which should be considered. Since the plane reference surface w i l l not depolarize the incident radiation, the direct and depolarized components must be considered separately. That i s , • E V 11,-L , o b ^ E = 1 + ~ E plane ( 2 . 8 3 ) as i n equation (2.ko) for the direct components. For the depolarized 29 components however, E .1 I ob,j plane E , i s , as before, the image of the incident f i e l d i n the plane. The plane geometry i s similar to that shown in figure 2.3 with CL = 0. II J_ e i 2 k L E = + E (2.85) plane 0 2kL It must be noted that the to ta l f i e l d scattered by the ensemble of objects w i l l be given simply by the vector sum of the field, scattered by each object, since the single scatter approximation i s assumed. The normalization of the to ta l f i e l d given by the sum over s of equations ( 2 . 8 l ) and ( 2 . 8 2 ) by equations (2.84) and ( 2 . 8 5 ) may now be performed. F i r s t , the various polarization components of equation ( 2 . 8 3 ) must be recognized. That i s , the f i e l d i s i n the form ^ = V a + E l 2 Ja + E i k a E ^ E 2 1 i a + E 2 2 j a + E 2 k a ( 2 . 8 6 ) where E i s the direct component due to the || pol . incident wave E^g i s the depolarized component due to the 11 pol . incident wave E i s the depolarized component due to the | pol . incident wave E 2 2 ^ S "k*16 c& r e c t component due to theJ_pol. incident wave These four components may now be correctly normalized. The other components E and E along the z axis are not considered here for two reasons. F i r s t , 1 2 . a ' they w i l l not be detected by an antenna i n the same or cross-polarized 30 orientation as the transmitting antenna, which i s physically the most r e a l i s t i c configuration. Second, they w i l l tend to cancel out when averaged over the ensemble of the objects i f the objects are distributed according to a continuous uniform distribution. This point w i l l be further discussed in Chapter h. The normalized backscattered f i e l d from an array of perfectly conducting hemispheres on a perfectly conducting ground plane, illuminated by a narrow beam antenna at normal incidence, by the single scatter approx- imation can now be cast into the form: E = 1 + 4 i M J e i k ( C s ~ L ) l f (0 , TT- cb) 1 1 kC„ • Ea cos CD cosh+E, sinCp es s c ^s s E. 12 l+ikLJe i k (°s" L ) | f (0c,7T-cT>s) kC sincb coscb s ŝ (2.87) E21 = ^ i k L ( e i k ( C s " L ) ) f $ ) kC E A cos0 + E M sincfacoscb 0s c 9s\ s f l E = 1 + 22 . ( ik(C -L)' 1+ikL e f (e c,77-$ B) -E„ sirKJ) cosQ + E^ cos^Cp s s c ^s s 2.k Antenna Model It i s now possible to choose a specific function for the antenna space factors f ( 0 ) and f (0',0) respectively. There are several choices which could be made. The simplest one is the rectangular 31 k 16,1 >e0 (2.88) This simple case was dismissed because the function has no sidelobes; only the beamwidth can be co n t r o l l e d . A s l i g h t l y more complex function was tested and found to be a reasonable representation of a narrow beam antenna with r e a d i l y c o n t r o l l a b l e beamwidth and sidelobe l e v e l . The model chosen i s that of an aperture with an appropriate n 1 i l l u m i n a t i n g function cos (yJTx) The form of the space f a c t o r f o r the r a d i a t o r i s a s l i g h t l y modified version of that given by S i l v e r . n c o s ^ n±Z 'XX * ( u ) - (n-l ) / 2 r _ / 2 > ? 9 o ( n + 2 ) u N 2 g » - k=0 _ o _J n/2 \TTA ovfr- J k=0 L X 7 T A 2k0o n odd n even (2.89) where f(0) = g n ( s i n e ) (2.90) and 0 i s the angular distance to the f i r s t n u l l of the beam i n degrees, o The integer n reduces the sidelobe l e v e l i n di s c r e t e steps as n i s i n - creased. Note that n = 0 and n = 1 reduce equation (2.89) to the approx- imate r a d i a t i o n patterns of an open waveguide i n the E-plane and H-plane respectively. THETR [DEGREES] Figure 2.5 Radiation Patterns f o r Horn Antenna (-£ &•) and for Approximate .Beam Functions. A comparison of (2 .89) with the a c t u a l r a d i a t i o n pattern of a horn antenna i s shown i n fi g u r e 2.5. Here i t can be seen that the simple forms given with n = 0 and n = 1 are s u f f i c i e n t to quite accurately describe t h i s p a r t i c u l a r antenna. Therefore, the functions g. (u) = sinj Q u) 180 ej /180 1 u g (u) = c o s ^ ; 0 6 (2 .91) ,270 \ 1 . ^ 2 / 2 7 0 u \ 2 w i l l be used almost e x c l u s i v e l y f o r the study cf the hemicylinder problem. For the three dimensional problem i t i s s u f f i c i e n t to consider the product of the above two functions with s u i t a b l e arguments: L80 (2.92) s i n tee7 c o s ( \ 6 h u / = g(u,v) S i m i l a r l y , g ^ U j v ) = g(v,u) (2.93) where f ( 0 , 0 ) = g ( s i n 0 c o s 0 , s i n 0 s i n 0 ) (2.9k) The subscripts e and h on the beamwidths r e f e r to the beamwidths f o r the pattern i n the E-plane and H-plane of the antenna. 2.5 R e s t r i c t i o n to Harrow Beamwidth As has already been mentioned, i t was decided to use the beam f a c t o r s ( 2 . 9 l ) and ( 2 . 9 2 ) because of t h e i r s i m i l a r i t y to a laboratory horn antenna. Therefore, i t i s also reasonable to assume that the beamwidths to be considered w i l l be of the same order as that of an a c t u a l antenna . An e f f e c t i v e surface area may now be introduced which i s a function of the beamwidth,, and suitable approximations may be made which s i m p l i f y the f i n a l form of the f i e l d equations. 2 . 5 . 1 Two Dimensional Case Let 0 be the angular distance to the f i r s t minimum of the r a d i a t i o n Figure 2 .6 E f f e c t i v e Surface Width o pattern. Then from fi g u r e 2 . 6 W 2 L tan 0 o (2 .99) o W L s i n 6 + L s i n 6 \ cos + (X) ( $ . 9 6 ) cos (0 1 - a) 35 Define a constant K such that K tan Q = t a n 0 1 :(2.97) and le t C = 2K tan Q a ( 2 . 9 8 ) Further assume that the coordinate, X , may be derived from some normalized s distr ibution, p , -\SL. p — h Th 6 1 1 Let Then - L sin @ L sin 0 1. cos - (X) I ( O + i cos(6, - a) cos(6, +a) ! 2pscosCC + |casin(X = t L C a cos a- ( I c ^ ^ s i n ^ n' i. 2PHcosCX+ |cRsinCX 5 = "2" cos2CC- (ic a ) 2 s i n 2a X = LC n ' s aH s . ( 2 . 9 9 ) (2;100) (2.101). (2.102) (2.103) Note that the condition a <|TT- e x (2.101+) must be f u l f i l l e d in order that the surface width, W, may be kept f i n i t e . This stipulation rules out the case of grazing incidence. For this part of the problem a different method must be devised for determining the width of the effective scattering area, and w i l l not be discussed here. Equations (2.k) and (2.103) may now be combined to give / 2 2 C = L/(C n / + s i n a) + cos a ( 2 . 1 0 5 ) a J a.1 S L C ' (2.106) s The constant K of equation (2.97) can be determined by increasing i t s value u n t i l computed f i e l d values become constant. This w i l l be investigated thoroughly i n Chapters 3 and k f o r the p e r i o d i c and random cases, respec- t i v e l y . At t h i s point, the discussion of the two dimensional case w i l l a lso be r e s t r i c t e d to normal incidence only. I t seems p o i n t l e s s to keep the more general case as the three dimensional problem has already been re- s t r i c t e d i n t h i s manner. . I t should however be pointed out that unlike the three dimensional case, a l l but grazing incidence can e a s i l y be con- sidered i n the two dimensional problem i f i t i s desired. Since C O << 1 (2.107) f o r a narrow beam, the following r e l a t i o n s h i p s are applicable. W = LC (2.108) a P/ = P B (2.109) 2 2 C g X L ( l + (2.110) s i n / i * 1 - Ic^Og (2.111) c°s/3S*CaPs ( 2 . H 2 ) These approximations may now be introduced i n t o the f i e l d equations (2.23), (2.24),. (2.25) and (2.26) along with the antenna function (2.9l) to give the f i n a l form of the scattered f i e l d due to the array of hemi- c y l i n d e r s : E Nl0 N II ^ = 1 + C e - i k L ^ u | ^ ( - l ) n X n s c o s nCs Mo N ( 2- 1 1 3) E X = 1 + i C e ' i k L ^ U ^ ( - l ) n Y n s s i n n ^ where X = A"j4SuV(C 0 ) c o s nO y~V) n" mX .H + (k|X_-X | )F S t> ns n JKL S B x a^s <Ms / , /_ , mt mnv 1 ^ s i ' : t = l m=o i k L i v <> Y „ e = A l - i W ^ C D ) s i n nfj +) ) ( i ) n " m Y H _ (klx -X I ) F 4*5 and 2 2 U g = e i K W a A . ( 1 . l ^ 2 , (2.115) 2.5-2 Three Dimensional Case These same approximations may be applied t o the three dimensional problem. I f the act i v e s c a t t e r i n g surface i s made l a r g e r than necessary, obviously the r e s u l t s must remain unaffected. Therefore, assume f o r s i m p l i c i t y that the e f f e c t i v e area i s a c i r c l e i n s t e a d of an e l l i p s e , where the radius i s determined by the widest of the two beamwidths of the antenna. Thus, the diameter of the act i v e area w i l l be given by W = C L (2.117) which i s the equivalent of equation (2 . 1 0 8 ) . S i m i l a r l y i f p x and p are some-normalized d i s t r i b u t i o n of coordinates over the c i r c l e of unit diameter, then 5 s v s and therefore, 38 s a M s (2.119) tancj> - ^ s •s PX s Furthermore, (2.111) s t i l l holds so that equations (2.56) and (2.57) become c ~ L ( I + |c2n2) (2 .120) s a s 2 2 cos0„ « 1 - §C O (2.121) c ^ a' s s i n A C 0 (2.122) u c aMst as f o r the two dimensional case. The above approximations considerably s i m p l i f y the expressions (2.87) f o r the scattered f i e l d , e s p e c i a l l y the functions E e and E^ . These functions may now be expanded as a power s s series i n the small quantity CgPs. These complex functions of s p h e r i c a l Bessel functions and Legendre functions have already been s i m p l i f i e d by the r e s t r i c t i o n to normal incidence. In f a c t , i f non-normal incidence i s considered the fol l o w i n g approximations do not y i e l d a more usable form of the Legendre functions. From equation (2.76) i t can be seen that the functions 2 m m m / = k ^ L I ^ (2 .123) ™ s i n 2 9 c and m 2 m 8 (-1) m P n Bnm = sin y c are to be determined. The fol l o w i n g approximation may be applied f o r 39 small Q m, •, ( - l ) m Q \2m' (n+m)! P ( c o s A ) - i ~ (1 - cos0 ) ; \ n c 22* m:(n-m)J (2.125) « i4La(C aa)" (n+m)l 2 ml (n-m) I m Note that for non-normal incidence, cos0 « 1 and hence the above expression becomes a much more complicated function of C fl. Now, QJS < f « m - | n M n a s . 2m p™~ f ; ( n + m ^ •' (C a*°f) • . n ~ m(n~m)I (m'.)d 2^ (2.126) so that "nm nm m 2(m-l) ( - D e m ( n + m ) : (C aO s) (ml) (n-m) 2̂m~ m m 2(m-l) (-1) m(n +m)' (C aP s) 2 2 , 2 ^ ^'aPs + * n V s (2.127) (m«) 2 (n-m)' 2 2 m combine equations (2.127) and ( 2 . 7 7 ) , keeping only terms to the order of 2 (CgPs) • This operation yields ( - l ) n (2n+l ) n - i n n \ * S V v s -a ' ) f + a ) g n m n / i nm n / m = o m= o m+n odd m+n even (2.128) « | b ( l +b f lC&f) 40 where even \ N a n ' n b = ) (2n+l) n L—/ a , n odd V n' n - l (2.129) CO \ = b 2 _ J 2 n + 1 ) < n = l -na' + v(n+2)(n-l)a n even n + |(n+2)(n-l) a^, n odd (2 .130) Similarly, E 0 S « -|b (1 + ^ C 2 J S ) (2.131) where (2n+l) a , n even n' n - i _2(n+l) + Kn+2)(n-l) -na n + £(n+2)(n-l) a£, n odd (2.132) The f i n a l simplified form of the backscattered f i e l d by an array of hemispheres on a perfectly conducting ground plane for narrow beamwidth and normal incidence therefore becomes': 2 2 2ib i2kLC„Dc, E l l = 1 " " k L 6 s( C£Ps c o s ^s' C aOsSinCP s ) E 1 2 = ^ ^ ^ W 0 ^ ' W v - ^ i2kLC?Ps E 2 1 - kL 6 (2.133) g(C^ ssincT s , CaP s cosc^) | ( l^- | ) -b^ C ^ f sinc^cosc^ 2 2 E 2 2 = 1 - ^ e ^ ^ g C G ^ s i r ^ . C ^ c o ^ ) • 1 + k -1) s i n ^ + t y cos%-1] 41 The above equations w i l l be used f o r a l l the s t a t i s t i c a l studies on the three dimensional surface i n Chapter k. S i m i l a r two dimensional equations are given at the end of the next chapter, a f t e r a more d e t a i l e d i n v e s t i g a t i o n of t h e i r more exact counterparts ha.s been made. 2.6 Importance and Range of Parameters Before any r e s u l t s ( s t a t i s t i c a l or otherwise) are obtained from equations (2.11*0 or (2.133) i t i s advantageous to summarize the r e s t r i c t - ions and proposed treatment of a l l the parameters of the problem. I t i s i n s t r u c t i v e at t h i s time to decide upon suitable values f o r these parameters which n a t u r a l l y divide i n t o two sections, those of the beam, and those of the surface, as l i s t e d i n Table 2 . 1 . The reasons f o r the choices shown are the following. The angle of incidence i s chosen mainly f o r convenience, n and 0 O are chosen to agree with the laboratory antenna, while L i s chosen to put the surface just i n t o k the f a r f i e l d of t h i s antenna using the c r i t e r i o n 2 R 2 > y (2.13*+) The upper l i m i t s of the size of the objects and object density are d i r e c t l y r e l a t e d to multiple scatter, and w i l l be i n v e s t i g a t e d i n the next chanter. The f a c t o r K w i l l be determined f o r the p e r i o d i c and random problems, and the r e s u l t s f o r both cases, w i l l be compared. With these ideas i n mind, the p e r i o d i c configuration of scatterers may now be investigated. Parameter Description Treatment Value as fixed parameter Antenna Parameters a angle of incidence set to normal incidence, Ct= 0, for convenience a= o n sidelobe level i s a function of n vary to find effect of sidelobes n" = o n1- = 1 L distance from antenna to surface vary to fi n d effect of antenna position L = 8oA e c width of main beam vary to find effect of beamwidth keeping QQ < 30 ee = 8° 6 , - 8 ° Surface Parameters a radius of objects variable, but of lesser importance use a few values .̂0.5 a = 0.2A, 0.5A. K proportionality between width of main beam and surface width vary to find the smallest surface width required small as possible P area density of objects the main independent variable accuracy decreases with increasingp P<0.25 Table 2.1 The parameters of the Scattering Problem 43 3. PERIODIC ARRAYS OF HEMICYLINDERS One of the simplest rough surface problems i s that of the two dimensional periodic array of hemicylinders. As has already been mentioned, there are three main reasons for choosing this simple configuration. F i r s t , exact numerical values may be obtained for the X and Y of equations (2.111+) J ns ns v ' so that the single scatter approximation (and other higher order approxi- mations) may be compared with the exact solution. Second, the periodic results can be indicative of the behaviour of the random problem. In fact, i t i s hoped that any estimates of parameter l imi ts w i l l be even better for the random case due to the averaging processes. Third, a l l computation times are much shorter because only one configuration, not thousands, need be considered for each set of parameters. The three dimensional surface of course, w i l l not be considered here due to i t s extreme complexity. 3.1 Solutions for Various Orders Of Accuracy 3 . 1 . 1 Exact Solution In principle, equations (2.Ilk) can be solved. The solution i s obtained merely by inverting the matrix of coefficients of the X ^ and Y ̂ . However, i n a case where N = 25, which i s quite reasonable, and N = 5, which o corresponds to objects about 1 . 2 ^ i n diameter, one i s faced with solving a set of 125 simultaneous l inear algebraic equations i n 125 unknowns - a formidable problem. Therefore, the parameters in the cases that w i l l be considered w i l l be selected such that a solution i s feasible. For s implici ty, locate the origin of the coordinate system at the centre of the middle hemicylinder for an odd number of hemicylinders and halfway between the two centre hemicylinders for an even number of hemi- cylinders as in figure 3 . 1 . Ar ih- -w- Figure 3 .1 P e r i o d i c Two Dimensional Surface Put A. =< | ( 2 s - l ) ( ^ ) / N D even ( 3 . 1 ) s =< -KN 0 -1), -KN 0+1), . . - 1 , 0 , 1 , . - . |(NQ-1) , NQ odd -k®0, -Kv 1)'-- •• •••^No ' N 0 e v e n ( 3 . 2 ) and hence X = d(t - s) s ( 3 . 3 ) Equation (3-3) vrhen applied to the arguments of H~ i n equations ( 2 . 2 3 ) mn and (2.2k) immediately makes a l l the elements of every diagonal above and every diagonal below the main diagonal the same. Hence, the minor problem of merely c a l c u l a t i n g the elements of the c o e f f i c i e n t matrix i s now withi n the area of computational f e a s i b i l i t y . A quick glance at the number of unknowns i n equations (2.111+) 45 immediately rules out any hope of obtaining a s o l u t i o n by d i r e c t numerical i n v e r s i o n of the matrix (e.g. Gauss e l i m i n a t i o n ) . The d i g i t a l computer a v a i l a b l e (IBM 360/67) d i d not have a large enough memory, and i f a u x i l i a r y memory such as magnetic tape were used, the computation time and roundoff e r r o r would become p r o h i b i t i v e . Therefore, an i t e r a t i v e method, the only a l t e r n a t i v e , must be r e l i e d upon, but t h i s does not n e c e s s a r i l y converge i n 29 a l l cases. Twersky has a c t u a l l y w r i t t e n down the se r i e s s o l u t i o n which one 30 obtains by applying the method of d i r e c t i t e r a t i o n to equations s i m i l a r to (2 .23) and ( 2 .2k). In a form s u i t a b l e f o r i t e r a t i o n these can be written as M0 -N + > ) K :ns * / J Vst^t or, more compactly ti-s (3.4) , W = HW + C ( 3 . 5 ) where the main .diagonal of the .matrix H i s zero and H i s symmetric. Then the i t e r a t i o n follows the scheme —k+1 —k. — ° W = HW + C, W = 0 ( 3 . 6 ) "til f o r the k i t e r a t i o n . Conditions f o r the convergence of ( 3 . 6 ) t o the correct s o l u t i o n (the s o l u t i o n i s correct i f convergence i s obtained at a l l ) are 30 given by Faddeeva . In f a c t , the matrix H - I, I being the u n i t matrix, must have a dominant main diagonal. Note that Twersky claims that (3 -6) always converges f o r the s c a t t e r i n g problem. 30,31 A b e t t e r method i s the s o - c a l l e d Gauss-Seidel method . Let H = L + U ( 3 . 7 ) where L i s a lower t r i a n g u l a r and U an upper t r i a n g u l a r matrix. Then, perform the i t e r a t i o n according to the scheme -k+1 —k+1 -k _o , W = LW + UW + C, W = 0 ( 3 . 8 ) 46 That i s , the new values are used immediately as they are cal c u l a t e d . Faddeeva' states that t h i s method w i l l converge whenever (3.6) converges, w i l l probably converge f a s t e r , and w i l l converge i n other cases as well. Therefore t h i s method, ( 3 . 8 ) , was used i n a l l subsequent c a l c u l a t i o n s . 3.1.2 Nearest Neighbour Approximation For the nearest neighbour (N-N) approximation, one s t i p u l a t e s that the multiple s c a t t e r i s produced only by i n t e r a c t i o n s between those cylinders on e i t h e r side of the c y l i n d e r i n question. That i s , 30 r + H (kd),• t = s+1 mnv '' — H . = < mnst ( 3 . 9 ) o, t £ s+1 i n equation (3.*0« The system matrix i s now a t r i - d i a g o n a l matrix which g r e a t l y reduces the amount of computation required i n equation ( 3 . 8 ) . 3.1.3 F i r s t Order Nearest Neighbour Approximation The f i r s t order nearest neighbour approximation (N-N-l) i s obtained by f i r s t assuming that there i s a minimum separation of s c a t t e r e r s such that k X X « 1 s i (3.10) which implies a r e l a t i v e l y low density. Then, i t must be f u r t h e r assumed that the m ultiple s c a t t e r e f f e c t s are small compared t o the primary scattered f i e l d due to an i s o l a t e d c y l i n d e r (e.g. low density and f a i r l y small objects). The r e s u l t i n g N-N-l s c a t t e r i n g c o e f f i c i e n t s N Y ^ -iA" ns l e J T T 2 V - L i . nykL t o r ns .^2C a i k d s+1/ . m m> s+l X ~ ns i k L ] 1 ,ikd U . ii+n I II (-1) A h . v . m m, s-1 N s - l l_, s+1/ , m m,s+i (3.11) 4 7 where are derived i n Appendix C. Note that these are the ac t u a l c o e f f i c i e n t s ; no i t e r a t i o n i s required. 3.1.4 Single Scatter Approximation The s i n g l e s c a t t e r approximation i s obtained by assuming that Hmnst = 0 (3.13) f o r a l l s and t, or eq u i v a l e n t l y that the i t e r a t i o n scheme (3.6) converges s u f f i c i e n t l y with no i t e r a t i o n at a l l . Therefore, only the f i r s t terms are kept i n equation ( 3 - l l ) with the r e s u l t that . a . e i k L \ x Y « - i A 7=-U^h ns n x/kL s ns ll e i k L \ || X n s "~ \ / k L U s h n s (3.14) f o r the sing l e s c a t t e r s c a t t e r i n g c o e f f i c i e n t s . 3.2 Numerical Calculations of the Scattered F i e l d The preceding four methods f o r determining the X n s arid Y n s were pro- grammed on the IBM 360/67 d i g i t a l computer along with equation (2.117) f o r the scattered f i e l d produced by these c o e f f i c i e n t s . Up to 30 objects and an N of 10 were allowed f o r . Any increase i n these values would tend to use too much memory f o r the e f f i c i e n t running of the program under the MTS time- shared system. 48 3.2.1 Determination of Active S c a t t e r i n g Area The f i r s t problem i s to determine the optimum value of the r a t i o between the beamwidth at the surface and the necessary surface area. This f a c t o r (K i n equation (2.98)) can be determined by i n c r e a s i n g i t s value u n t i l the f i e l d i s v i r t u a l l y i n v a r i a n t to any f u r t h e r increase. The r e s u l t s are shown i n f i g u r e 3-2. The c y l i n d e r s i n t h i s case are 1.0 wavelength i n diameter and the spacing i s 5.0 wavelengths which gives a density of 20%. For these parameter values i t can be seen that the e r r o r due t o the s i n g l e s c a t t e r approximation i s appreciable. From f i g u r e (3.2) the smallest value of K that can be used appears t o be about 1.5 and t h i s value w i l l be used f o r a l l f u r t h e r studies of the p e r i o d i c surface. As K = 1 gives a surface width equal to the width of the main beam, t h i s value (K = 1.5) implies that at l e a s t the f i r s t sidelobes make some contr i b u t i o n to the scattered f i e l d . Cylinder diameters of O.U and 0 . 6 wavelengths were a l s o t r i e d , but with s i m i l a r r e s u l t s . The only d i f f e r e n c e to be noticed was the degree of multiple s c a t t e r . 3.2.2 E f f e c t of the Orders of Scat t e r i n g Approximations The r e l a t i v e errors of the magnitude and phase of the scattered f i e l d f o r the single s c a t t e r and f i r s t order nearest neighbour approximations com- pared to the exact s o l u t i o n are shown i n f i g u r e 3-3 and 3.k. Here, the independent v a r i a b l e i s the area density of the objects, p=_^o_ (2a) (100%) (3.19) A density greater than 35% has not been shown because i n several cases the exact s o l u t i o n d i d not converge. S i m i l a r l y , a c y l i n d e r diameter greater 29 than 1.0 wavelength was not used. This f a c t i s i n disagreement with Twersky who claims that the s e r i e s should always converge on p h y s i c a l grounds, although an even b e t t e r converging seri e s (3-8) was used here. As a consequence. Figure 3.2 Magnitude and Phase of Scattered F i e l d as a Function of the Ratio of Surface Width to Beamwidth. DENSITY (PERCENTJ DENSITY (PERCENT) Figure 3.3' Error Introduced by the Single Scatter Approximation as a Function of Hemicylinder Density f o r Various Values of Hemicylinder Radius. DENSITY (PERCENT) DENSITY (PERCENT) Figure 3.4 Error Introduced by the F i r s t Order Nearest Neighbour Approximation as a Function of Hemicylinder Density f o r Various Values of Hemi- cyli n d e r Radius. 52 the usefulness of the s e r i e s s o l u t i o n obtained by i t e r a t i o n i s severely l i m i t e d . In f a c t , i t can be seen from ( 2 . I l k ) that the diagonal elements of the system matrix increase as the s i z e of the objects i s decreased, and the off-diagonal (multiple scattering) elements decrease as the spacing i s increased Therefore, the diagonal i s dominant f o r small c y l i n d e r s and large separation, and a point can be reached by increasing the r e l a t i v e amount of multiple s c a t t e r i n g so that the diagonal i s no longer dominant. This point corresponds to a f a i r l y small amount of multiple s c a t t e r f o r the parameters considered. Now, the a c t u a l problem to be dealt with i s that of the random case. As the errors are seen to f l u c t u a t e as a function of object spacing, i t i s reasonable to expect that the average errors f o r the random array w i l l be somewhat l e s s than the maximum errors shown i n f i g u r e 3 . 3 and 3.*+. Therefore, with these points i n mind, the f o l l o w i n g r e s t r i c t i o n s w i l l be imposed; 0.2^Xa<( 0 . 5 A 5 and average density <^20 .̂ Of course, very small value of a/A^ 0 . 0 5 w i l l again be acceptable. These l i m i t s have been selected on the basis of using only the s i n g l e s c a t t e r .approximation. The higher order approximations would give smaller errors, but use too much computation time to be considered f o r the Monte-Carlo simulation. 3 . 3 Scattered F i e l d Using Single Scatter Approximation As the e r r o r i n using the s i n g l e s c a t t e r approximation has been ascertained, the f i e l d equations f o r the two dimensional problem equivalent to ( 2 . 1 3 3 ) f o r the three dimensional problem may now be determined from (2.11^) and ( 3 . 1 7 ) . The r e s u l t s are v a l i d under the f o l l o w i n g r e s t r i c t i o n s . o 1. Narrow beamwidth approximately 15 2. Average separation about 2A or l a r g e r 3. Object diameter about l A or smaller k. Normal incidence backscatter 53 Ho N E " =1 + / i E v l , ( caPs> L (-D V 0 8 Ho N! (3 . 2 l ) The above equations w i l l be used for a l l s t a t i s t i c a l studies of the two dimensional rough surface. 54 h. RANDOM ARRAYS OF HEMCYLINDERS AND HEMISPHERES The equations (2.91) and ( 3 . 2 l ) derived in Chapters 2 and 3 may now be put to the use for which they were intended, namely the calculation of certain average properties of the electric f i e l d scattered from a rough surface. The so-called rough surface in this particular case consists of perfectly conducting hemicylinders or hemispheres situated on a perfectly conducting ground plane. The "roughness" i s determined by the choice of the distribution of object coordinates , p g . In Chapter 3, p was chosen s such that the objects were located periodically. This situation i s , i n one sense rough because the surface so created i s not f l a t . However, by rough, i t i s meant in this context, that at least one s t a t i s t i c a l moment other than the mean must be non-zero. Therefore the f i e l d s t a t i s t i c s must be calculated by integration or by a Monte-Carlo method. The la t t e r has been chosen for reasons already stated in the Introduction. The essence of the particular Monte-Carlo technique to be used has already been outlined but the method i s given here i n more detail as follows. 1. Generate a set of coordinates, p„ . 2. Calculate the electric f i e l d from equation (2.91) or (3.21) as desired. 3. Keep a running total of s t a t i s t i c a l sums of the results of 2. • 4. Repeat 1, 2, and 3 u n t i l enough configurations have been included to be reasonably representative of a l l configurations. 5. Take the f i n a l results from 3 and calculate the required s t a t i s t i c s . These st a t i s t i c s may be used to determine when "enough" has been reached in k. If not, 1, 2 and 3 may be repeated a number of additional times, and the results rechecked. The usefulness of this method has already been discussed so now consider the implementation of the above sequence of events. Note that step 2 has already been studied at length i n the preceding chapters. The f i r s t problem i s that of determining the coordinate distr ibution ĵ jĈ J. A special distr ibution i s developed here for f in i t e scatterer separation with particular reference to the experimental problem. With a suitable distribution chosen for the coordinates, steps h and 5 may be executed to f ind the f i e l d s ta t i s t ics . The remainder of this chapter i s devoted to an investigation of the effects of the various parameters under the restrictions which have been imposed as a result of the study of the periodic problem. J+.l Generation of Scatterer Coordinates k.1.1 Uniform Distribution The simplest method of obtaining the object locations i s to use the continuous uniform distribution where a l l values of D in the unit interval ' s (-§-,|) are equally probable. This distribution w i l l be.used in most cal- culations for three reasons. F i r s t , this distr ibution may be obtained approximately from a d ig i t a l computer in the form of a pseudo-random sequence. 32 The s t a t i s t i c a l accuracy of such a sequence i s discussed by Olsen . Second, most theoretical calculations of the f i e l d s ta t i s t ics by the approximate integration method use this distr ibution. Third, i t i s a reasonable approx- imation to many pract ical problems. In this model the separation of the scatterers may take any value including those which make the objects overlap. The qualitative results may not be greatly affected by this , but i t certainly does not completely represent a physical situation. Like the single scatter approximation i t i s useful in i t s s implici ty provided that the density of scatterers i s f a i r l y low. ( i . e . probability of overlaps i s low). k.1.2 Non-Uniform Distributions There are situations which cannot be considered using the uniform 56 d i s t r i b u t i o n , e. g. when attempting to construct a p h y s i c a l rough surface from a t a b l e of the coordinates. I t i s t h i s problem, the construction of a lab- oratory surface f o r experiments, v h i c h prompts the development of a non- 32 uniform d i s t r i b u t i o n . Olsen has developed two methods f o r computing d i s t r i b u t i o n functions -which allow f o r f i n i t e dimensions of the s c a t t e r e r . One d i s t r i b u t i o n of t h i s type i s formed by generating a set of coordinates using the continuous uniform random number generator RAND which i s supplied with the IBM 360/67 computer. The distances between each p a i r of objects are c a l c u l a t e d ; any objects closer than a given value are rejected; then new co- ordinates f o r the r e j e c t e d ones are c a l c u l a t e d u n t i l a l l the minimum separation c r i t e r i a are s a t i s f i e d . These c a l c u l a t i o n s are straight-forward when a p p l i e d to a one dimensional array such as the coordinates f o r the hemicylinder problem and should give an accurate representation of a p h y s i c a l c o l l e c t i o n of f i n i t e s c a t t e r e r s . Note that a many-body d i s t r i b u t i o n f u n c t i o n of t h i s type has only been determined numerically but never a n a l y t i c a l l y . There are three drawbacks inherent i n the above method. The f i r s t i s t h a t many c a l c u l a t i o n s and hence a r e l a t i v e l y l a r g e c a l c u l a t i o n time i s required to produce a set of coordinates. Also, the time increases sharply as the s c a t t e r e r s become more dense. The second i s that the program i t s e l f becomes very complex when generating p a i r s of coordinates i n a plane. A l s o too many numbers have t o be remembered simultaneously. The t h i r d drawback i s not an inherent problem, i t i s merely caused by the p a r t i c u l a r algorithm used by Olsen t o c a l c u l a t e the d i s t r i b u t i o n , and could therefore be remedied. Using t h i s algorithm the Olsen d i s t r i b u t i o n tends toward p e r i o d i c i t y as the density i s increased or the width of the empty regions i s increased. Each s i n g l e configuration must become p e r i o d i c , of course, but each succeeding one should be d i f f e r e n t w i t h i n the width of the empty region. Therefore, the Olsen d i s t r i b u t i o n w i l l give a smaller variance than one would expect from a t r u l y 57 random surface. 32 The second distribution considered by Olsen has removed the comput- ational problems noted above. This distr ibution provides the necessary minimum separation but cannot exactly represent a physical situation because the coordinates are allowed to attain only specific values. This method (see figure 4.1) consists of dividing the surface into square boxes, the width of which i s the minimum allowed separation of the scatterers. The output of X X X X X X X X Figure 4.1 Discrete Distribution of Scatterers RAND i s truncated to yie ld an integer between 1 and the number of boxes in a row (7 in figure 4.1) . A pair of these integers i s then used to place an object into the corresponding box. I f there i s already an object there, a new set of coordinates i s calculated u n t i l an empty box i s found. This distribution should be reasonably good for a low ratio of number of objects to number of boxes ( 25% shown in figure 4.1) . A distribution function that l i e s between the discrete and continuous methods outlined above and removes both the computational and theoretical disadvantages i s now proposed. By increasing the number of discrete - positions, it ' should be possible to approximate the desired continuous function as nearly as i s required. The one problem associated with this method is that the condition (full or empty) of each cell must be remembered. Therefore, as the number of cells is increased, an increasingly large amount of computer storage is required. Hopefully, then, a large number of cells will not be required. This method is similar to the discrete Olsen method only now each cell is divided into a number of subcells, Ng, which gives a larger number of cells (and hence a larger number of discrete positions) per row (compare figure k.2 where N = 3 with figure k.l). In this case, a pair of cell s coordinates are generated by RAND and then the required number of cells around the chosen one and the chosen cell itself are checked for the :. presence of an object. If there is no object detected, a l l these cells are set to f u l l . The process is repeated until a l l the objects have a position. For this distribution there is the added problem of determining Ns, which obviously must be kept as small as possible. Ng, like the antenna factor K, must be determined experimentally by increasing its value until the change in the field statistics is negligible. x X X X X r X r Figure k.2 Modified Discrete Distribution of Scatterers k.2 Calculation of Field Statistics The remaining part of the Monte-Carlo technique i s the actual calcu- lation of the statis t i c s from the collection of f i e l d values Ĵ EJ . F i r s t of a l l , i t must be decided upon which particular s t a t i s t i c a l functions should be considered. A thorough discussion of the properties of a l l 17 the related s t a t i s t i c a l functions i s given by Twersky . The results presented here w i l l primarily be concerned (for simplicity) with the f i r s t and second s t a t i s t i c a l moments only. Of course, higher order moments may be calculated as readily by this method, with only a small sacrifice in computational time and storage requirements. This i s the great advantage of the Monte-Carlo method. Analytical methods become so complex that usually only the mean may be determined; even the, calculation of the variance i s extremely d i f f i c u l t . A brief outline of the s t a t i s t i c a l parameters which w i l l be used i n conjunction with this particular problem may now be considered. Assume that N values of the complex elec t r i c f i e l d E have been calculated for n N different configurations of the objects according to the desired dis- tribution of coordinates. The functions required for the analysis of the mean and variance of the elec t r i c f i e l d for a rough surface are then: <Ex> = n = 4 ¥E I m ( E n } o 1 N /• >|2 < Ex>" ¥E{ R e ( E n } ) n=4L ^ J <4>-TE{Im(E">}2 M < w -¥ 2_ yRe(E n)lm(E n) n-1. 60 The set of numbers calculated according to equations (k.l) are the five basic outputs of the Monte-Carlo portion of the analysis. From these numbers, the desired statistical results may be derived. Note that there will be additional sums of higher order products i f higher statistical moments are required. The means of the real and imaginary parts of the field are useful as given above in the fi r s t two equations of (k.l). However, the usual functions which are considered are the so called "coherent intensity" and "coherent phase". p 2 2 C =<E X> + <EY> (*.2) - l <Ey> CL = tan <Ex> The variance of the real and imaginary parts of the field may be calculated from VlE{Re^)-<Ex>2} = <E2>-<E X > 2 (U.3) cr 2 = <E 2> - <E > 2 y N y yy Here, the usual function to be considered is the " incoherent intensity". Finally, the co-variance and correlation coefficient may be determined from (>•. 5) P o- xa y 61 32 The accuracy of the means may readily be calculated. These estimates should give a sufficient indication when enough samples have been used in the Monte-Carlo calculation. That i s , the results may he checked after a certain number of samples have been processed, and then more samples may be taken. The size of the error w i l l also give a good estimate of how many more samples are required. These errors of the means are: 10? A<E v> = 2 p l _ N-1 ( 4 . 6 ) A<E > = 2 y 2 N-1 The above calculations w i l l give the desired means and variances for any complex function E of the random variables X1,X2, .. X^ for any distribution of the X . These calculations must be adhered to unless the n "til f i e l d equations and distribution function for the s object are functions of the s^ n coordinate only (e.g. single scatter approximation with con- tinuous uniform distribution). For this case the means may be calculated exactly and the variances calculated approximately with a considerable saving of computational effort. These special calculations are given i n Appendix D. The main advantage of this method i s that the density,p, appears after the Monte-Carlo operations have been completed. The disadvantages are i t s limited application and limited results. This method is only used as a comparison with the Monte-Carlo simulation. k.3 Determination of Active Scattering Area The value of K = 1.5 for the ratio of the effective surface width to the width of the area illuminated by the main beam of the antenna (see equation 2.97) was checked by performing the same experiment as i n section 3.2.1 for the periodic surface, upon the random surface. Z9 63 64 In this case, the beam sharpness factor, n, was varied also. As a finite number of objects are being considered, the density must take on only discrete values. So that a l l the results could be referred to one fixed density, two sets of data were calculated, one above and one below P= 0.25. The results were then linearly interpolated to p = 0.25 and are shown as functions of K in figures 4.3 and 4.4. From figure 4.3 i t can be clearly seen that the value K = 1 is sufficient, even for the case of large sidelobes (n = 0 ) , to determine the coherent field. However, from figure 4.4 i t appears that K = 1. 5 would be preferable for calculation of the incoherent intensity. On closer inspection of the curves of figure 4.4 i t can be-seen that the curves are functions of K for 1.0 < K< 1.5 but not of n. This fact may be inter- preted as indicating that the effect of the sidelobes is negligible. Perhaps this variation with K appears because the main beam is seeing only a portion of the distribution of objects end hence slightly different surface statistics. Thus, in cases where i t is necessary to restrict the number of objects for computational reasons, K = 1.0 will be used. In most cases, how- ever, the value K - 1. 5 (which agrees with the periodic array results) will be retained. 4.4 Comparison of Results Two sets of functions are available with which to compare the Monte-Carlo 15 simulation. The first of these is the Twersky results for the coherent and incoherent field scattered by the hemicylinder problem for a continuous uniform distribution and infinite plane wave incidence. The second functions are given in Appendix E and are obtained by approximate integration of the field equations derived in Chapter 2. These functions are for the continuous uniform distrib- ution but include finite beam incidence. A l l methods are based upon the single scatter approximation. 65 k.k.l C o h e r e n t F i e l d 15 The Twersky c o h e r e n t f i e l d i s g i v e n b y <E p> = 1 +pfj (4.7) where 11,1 _ ATE 11,1. n n = o and 2a' . , 1 o w (4.8) (4.9) i s t h e number o f a c t i v e s c a t t e r s p e r w a v e l e n g t h . The f u n c t i o n m o d i f i e d f o r f i n i t e beamwidth g i v e n i n A p p e n d i x E i s <E b>. = 1 + p'f b (4.10) where f = - 2 IB kL a n d C a/2kL II 1350 (1 - K & e 1 * * 1 + k L C a ) ~1 B ~ e B ~ 1.526.8 (4.n) (4.12) As t h e above f u n c t i o n s , (4.7) and (4.10) a r e e x p l i c i t f u n c t i o n s o f t h e d e n s i t y , p y , and a r e b o t h i n t h e same f o r m , i t i s e a s i e s t t o compare o n l y f ^ a,nd f i n d e p e n d e n t l y o f pf However, f o r t h e M o n t e - C a r l o r e s u l t s t h e f u n c t i o n P 0.2 60 90 100 L / X 130 300 300 no 300 r~ 300 330 _ T _ 320 L / X 330 320 L A 3I0" 320 L / X I 330 320 L / X 330 I 330 330 330 Figure 4.5 A Comparison of Three Methods for C a l c u l a t i n g the Single Scattered Coherent F i e l d for an Array of Hemicylinders with p = 2% and a = 0.2X: (a) Twersky (b) Monte Carlo — A & - (c) Approximate Integration 0.06 67 •130-1 L A L/X Figure 4.6 A Comparison of Three Methods for C a l c u l a t i n g the Single Scattered Coherent F i e l d f or an Array of Hemicylinders with p=2% and a=0.05^. (a) Twersky (b) Monte Carlo h— (c) Approximate Integration >*~ 68 f M C = ^ { < E > - ^ 1 3 > must be calculated for every value of p ' . f, , f , and fx,n are shown in figure 4. 5 as functions of the antenna to b p Mt • surface distance l>/\ for a density, p , of 2% and object radius, a, of 0.2/\. The same functions are given in figure 4. 6 for a = 0.05A- From figure 4 . 5 the following results are evident: 1. The Monte-Carlo method is the best method for calculating the coherent intensity. At least, for the parameter values chosen, the analytic solution is too much in error. 2. The analytic method gives quali tat ively the form of the curves, but their variation i s too large. 3. The approximate analytic solution becomes better as the antenna to surface distance i s increased. 4. The results for f in i te beam incidence are f a i r l y close to the plane wave results even for the narrow beamwidth used here. From figures 4 . 5 and 4 . 6 i t can be seen that the accuracy of the methods i s independent of the object radius. This fact verifies the result found i n Appendix E that i t is permissible to s e t = \TJ i n the f i e l d equations. Thus, the use of this assumption (see equation ( 2 . 8 0 ) ) for the three dimen- sional case i s indeed reasonable. 4.4.2 Incoherent Intensity 15 The Twersky plane wave incoherent intensity is given by P P (4.15) P' where <I*> = -2Re(f ) (4 .16) which i s derived by assuming that a l l the incoherent power i s i n the backscatter direction only. 0.06-1 0.05- A <\i o 0.04- 69 <Ift> =0.578 °TW a = 0.2X - T & & - ~& A &— 4S fe —H- 4S H & hi fes- 4? & U D .03—|—i—i—i—i—j—i—i—i—i—1—i—i—r 90 100 80 0.4-. L/A 1 j 1 i I 1 1 1 1 1 ! j 1 1 1 1 j 110 300 310 320 330 L/A -4 A. « o 0 . 3 4 <i:> =i. i i 7 . a = 0.2X O.i- 4s n n & a a -& & —& &— 0.0015- _ 0.0010- • A. (4 o V T—i—i—i—i—i—i—i—i—|—i—i—i—r 80 90 100 L/A —I |—i—i—i—i—j—i—i—!—i—j—i—i—i—r—j 110 300 310 320 330 L/A & A <I„> =0.0063 °TW a - Q.05X 0.0005-4 -a *s & K & » ?s K - ~& & 43- N — • it 80 o.D045-j 1 1 1 1 P—I i i 1 j 1 1 ! 1 1 90 100 L/A I — I — I — 1 — I — j — I — I — I — I — j — I — I — I — I — I U0 300 310 320 330 L/A & A . o l O V D.0030' 0.0015- -0.0000 -5i ft >£- 4 * ~ — a H- <I > =0.0126 TW a = 0.05X 80 T — i — i — i — i — i — i — i — i — j — i — i — i — i — I | — i — i — i — i — j — r 90 100 110 300 310 L/A T—i—i—j—i—i—r 320 L/A Figure 4.7 A Comparison of Three Methods of Cal c u l a t i n g the Single Scattered Inco- herent Intensity f o r an Array of Hemicylinders. (a) Monte Carlo: p=2% A — p=10% —-A- -A — (b) Approximate Integration —# K- (c) Twersky r e s u l t s are constants as indicated on each graph. The values are too large to p l o t . 70 The f in i te beamwidth modified function i s derived in Appendix E as < V • P <4 > -P' (^.17) where 2 II 2 1 II 2ft8e 2 f" 1 p 90 JtbCa 2 n 2 6 h 270 Again, the Monte-Carlo results may be put into the same form MC = P' <XMC> -P' MC (4.19) so that "Co P' <EMC> (4.20) The behaviour of f, , f , and f „ has already been discussed so now i t i s b' p' MC 2 2 2 necessary only to compare <I > , <I. > , and <L.- > to determine the behaviour L o o M O o of the incoherent intensity. These functions are shown in Figure 4.7 for the same parameter values as the coherent intensity. From 4.7 i t i s clear that: 1. The Twersky model gives far too large an incoherent intensity, and therefore the assumption that most power i s scattered into the backscatter direction i s not va l id at least for this range of parameters. 2. The Twersky results are better for smaller diameter objects. 3. The analytic method ignores the fine variations due to changes in L/\, but gives a reasonably good result, especially for low densities. 4. The analytic and Monte-Carlo results agree more closely for a lower number of objects. k. 5 Summary In summary, then, i t has been found that the Monte-Carlo method as implemented in this chapter i s a useful method for calculating the mean and variance of the f i e l d scattered from the discrete scatterer model of a rough surface. The Monte-Carlo method also allows the use of the same distributions of object coordinates to be used for both computer simulations and physical experiments. The actual distribution function to be used has been developed, and w i l l be studied i n detai l in the next chapter. The analytic solution has the advantage of being simple to calculate numerically but gives .large errors in some cases. In fact, the plane wave solution i s better than the analytic solution for the coherent intensity while the opposite i s true for the incoherent intensity. Thus, a good estimate of the behaviour of the scattered f i e l d may be obtained by using a combination of the plane wave and analytic beam solutions. That i s , with F ' given by equation (E. kk). For the three dimensional problem, the Monte-Carlo method remains the only useful means of numerical analysis. 2 / 2 a « arg(l + p f p ) (4.21) 72 5. CONSTRUCTION OF A LABORATORY SURFACE AND THE MEASUREMENT OF ITS SCATTERED FIELD Perhaps the most accurate method f o r determining the f i e l d s t a t i s t i c s i s the d i r e c t measurement of the f i e l d scattered from a s u f f i c i e n t l y large number of independent ensembles of scatterers which are i l l u m i n a t e d by a r e a l antenna. This experimental i n v e s t i g a t i o n i s a t t r a c t i v e because of the inherent freedom from mathematical approximations (such as the single scatter approximation). On the other hand, there are two main disadvantages to the experimental approach. The f i r s t a r i s e s from the d i f f i c u l t y of ad j u s t i n g the system parameters to exact values, that i s the system must have close mechanical tolerances and s t a b i l i t y . The second i s the inherent i n f l e x i b i l i t y of the system. The v a r i a t i o n of most parameters requires the construction of a new part of the system. These disadvantages are outweighed, however, by two features which are d i f f i c u l t , i f not impossible, to implement i n the computer simulation. F i r s t , object shapes other than hemicylinders or hemispheres may be considered with l i t t l e extra e f f o r t . Second, a d i s t r i b u t i o n of sizes as w e l l as locati o n s of objects may be used. Although these features w i l l not be u t i l i z e d i n t h i s study, t h e i r a p p l i c a t i o n i s immediate and therefore provides impetus f o r the development of a workable experimental system. 5.1 Design of the Experimental System For the design of the experiment, four major problems must be considered. The f i r s t i s the choice of suitable parameters so that the greatest amount of information can be obtained f o r the simplest and l e a s t number of changes i n the experimental set-up. The second i s the method of construction of a s p e c i f i c rough surface. The t h i r d i s the method of measurement and data c o l l e c t i o n . The f i n a l problem i s the method of analysis of the data i t s e l f . 5.1 .1 Parameter Values For the experimental problem, the s e l e c t i o n of s u i t a b l e parameter values i s l i m i t e d mainly by the dimensional constraints of the experimental set-up and the computation time l i m i t a t i o n s of the corresponding Monte-Carlo simulation. The wavelength was chosen i n the 8 millimeter range to keep a l l the p h y s i c a l dimensions of the experiment reasonably small. For example, at .3 • cm. wavelengths, the dimensions of a sin g l e surface would be too large, while at h mm. wavelengths, the mechanical tolerances (to a f r a c t i o n of a wavelength) could become a problem. The simulation time i s a function of the number of objects so i t i s necessary to work with as few scatterers as possible. This implies that t h e i r size/wavelength should be large, t h e i r density low, the antenna close to the surface, and the beamwidth narrow. In terms of the approx- imations used i n the simulation, the narrow beamwidth and low density are desirable while the large s i z e of scatterers and small antenna to surface distance are undesirable. The upper bound of the object s i z e , then, i s l i m i t e d by the sin g l e s c a t t e r approximation, while the lower bound i s determined from the l i m i t e d number of objects. These considerations give a u s e f u l range f o r the object radius of 0 . 3 < - ^ - < 0 . 5 (5 .1 ) The upper l i m i t was chosen f o r t h i s experiment to y i e l d as large an amount 74 of incoherent s c a t t e r i n g as possible. Therefore, at 8 mm. , a =X/2 (5.2) = .169 i n . The antenna-surface distance i s l i m i t e d by the requirement that the k surface l i e i n the f a r f i e l d of the antenna. That i s D 2 L > . - ~ (5.3) where D i s the width of the r a d i a t i n g aperture. For the horn used i n figu r e 2.5 and at a frequency of 35-0 GHz., K ,7.62 2 L 2 ( 7 B 5 I ) >79A (5-4) (5.5) Therefore, choose L = 80A. * 27 i n . k Actually, twice the above value i s to be p r e f e r r e d to be absolutely c e r t a i n that the surface i s i n the f a r f i e l d of the antenna, but (5-5) must be used to keep the i l l u m i n a t e d surface area as small as possible. The laboratory antenna has a f i x e d beamwidth of 6 = 8 ° (5.6) o which w i l l also be used f o r the simulation. Some parameters i n t h i s experiment can be varied without too much e f f o r t . The most r e a d i l y v a r i a b l e parameter i s the frequency, which may be varied from 32 to 38 GHz. f o r the p a r t i c u l a r generators used. The 75 other e a s i l y v a r i a b l e parameter i s the object density. As the density i s to be kept low, a good s t a r t i n g value i s chosen to be P~% (5-7) 5.1.2 Method Of Measurement The problem of experimentally measuring the s t a t i s t i c s of the f i e l d scattered from a rough surface proceeds as follows: 1. Construct a rough surface s u f f i c i e n t l y l a r g e r than the area i l l u m i n a t e d by the main beam of the antenna. 2. Place the surface i n the beam at a c e r t a i n p o s i t i o n . 3. Measure the scattered f i e l d from t h i s surface. 4. Measure the scattered f i e l d from a f l a t surface at the same p o s i t i o n . 5. formalize the f i e l d (3) by the f i e l d (4). 6. Repeat steps ( l ) to (5) keeping a l l parameters f i x e d f o r a large number of'independent surfaces ( l ) . As such a large number of independent surfaces are required, i t was decided to reduce the manual labour by constructing a surface which was several beamwidths i n area. By t h i s method, several independent samples may be obtained from one surface. This surface may then be moved contin- uously past the antenna to give a continuous scattered f i e l d as a function of p o s i t i o n . The maximum number of independent samples may then be selected by some method from t h i s continuous set of data. This method was implemented using the transverse p o s i t i o n e r and anechoic chamber 32 developed by Olsen f o r h i s s c a t t e r i n g i n v e s t i g a t i o n s . This scanning method has one disadvantage, however. I t i s d i f f i c u l t to construct a surface where the ground plane portion i s f l a t over as large an area as i s required f o r a large number of samples. I t i s also hard to adjust the p o s i t i o n i n g of the surface on the scanner such that the 76 ground plane i s always the same distance from, and i n the same o r i e n t a t i o n with respect to the antenna. These errors can e a s i l y introduce phase errors of the order of 2 radians at 8 mm. wavelengths. The s o l u t i o n to t h i s problem was to devise a system which tends t o cancel the phase errors. The system used f o r t h i s experiment i s shown i n f i g u r e 5.1. The d i r e c t i o n of motion of the surface i s normal to the page. Here, 35Ghz. klystron isolator mixers X I mi rcvr. <3 REF* <3 SIC. tape rec. 4 chart. scanner Figure 5-1 Experimental System the surface i s covered with the d i s t r i b u t i o n of hemispheres i n the lower h a l f , while the upper h a l f remains f l a t . A reference s i g n a l i s obtained from the upper antenna, while the a c t u a l scattered s i g n a l from the rough surface i s obtained from the lower antenna. The d i v i s i o n of the rough surface s i g n a l by the f l a t surface s i g n a l should then give the normal- i z e d scattered f i e l d . The remaining errors are now due only to t i l t i n g of the surface as i t i s moved. Every e f f o r t was made to mount the surface so that these errors were kept to a minimum. The reference s i g n a l i s then f e d to the reference channel of the 77 Scientif ic Atlantic Model 1751, #8 phase-lock receiver while the random signal i s sent to the signal channel. The three outputs (signal phase - reference phase, reference magnitude, and phase magnitude) are recorded on an Ampex Model SP - 300 F.M. tape recorder for later analysis. Before any measurements can be taken, an absolute reference must be established since the gain and phase shifts w i l l not necessarily be the same over the signal channel and the reference channel,This reference is obtained by placing a f la t metal plate against the rough surface and para l le l to i t as in figure 5.2. The slight variation i n position between the f la t surface and the actual one under i t should not noticeably affect the magnitude, and the phase shift w i l l cancel due to the measurement method. The system gain controls and phase shifts could now be set to give a Figure 5-2 Reference Plate for Normalization reading of 1 (0 db) on the magnitude channels and 0 on the phase channel. It i s more convenient, however, to merely record the actual readings on the three channels, and then record the subsequent data from the rough surface without altering the controls. The data can then be normalized later at the time of processing (see section 5 . 4 . 1 ) . 78 5.1.3 Construction of the Surface The material chosen f o r the surface was .05 i n . t h i c k s o f t annealed aluminum. Ordinary aluminum and so f t copper sheet were t r i e d , but these materials were not d u c t i l e enough. The hemispheres were then formed i n the sheet at the appropriate l o c a t i o n s using a punch and die as shown i n figur e 5-5- Figure 5-3 Forming a Hemispherical Scatterer i n Aluminum Sheet The die was machined to be a perfect hemisphere of the desired radius, but the shape of the punch had to be c a r e f u l l y determined by t r i a l and error. The main problems encountered were tearing of the metal and pointed "hemispheres". The d i s t r i b u t i o n of s c a t t e r e r positions.was generated by the algorithm developed i n Chapter h f o r the discrete d i s t r i b u t i o n of f i n i t e - s e p a r a t i o n scatterers. As the method was programmed on the IBM 360/67 d i g i t a l computer, i t was advantageous to use the Calcomp d i g i t a l p l o t t e r to obtain a d i r e c t p l o t of the object p o s i t i o n s f o r each surface. Part of a scaled down output i s shown i n f i g u r e J.k. The coordinates were p l o t t e d i n r e a l size so that the computer output could be fastened d i r e c t l y to the aluminum sheet. The object l o c a t i o n s were then t r a n s f e r r e d to the metal surface with an automatic centre punch. X X X X X X X X X X X X X X K K X X X X X X X X X X X X X X X X X X K X X X X Figure 5.4 Computer Generated Hemisphere Coordinates 5.2 Simulation Test of the Surface Distribution Recall at this point (see section 4.1.2) that there are two main para meters which characterize the distribution function developed for use in the experimental study. These parameters are the minimum separation, dmin* a n d t h e n u m b e r o f subdivisions of the basic c e l l , N g . The l imit ing effect of these parameters i s shown i n figure 5.5- Obviously, K s should be as large as possible and dn]j_n as small as possible to allow continuous distribution ^ i n - * - 0 < discrete distribution continuous ' uniform distribution -4- i n Figure 5.5 Effect of Distribution Parameters 80 comparison to be made with the continuous uniform distr ibution (ideal case). 5.2.1 Determination of Minimum Grid Subdivision The coordinate distribution function of section 4.1.2 may now be investigated by the Monte-Carlo simulation to determine the minimum value of N , the number of c e l l subdivisions required to approximate a continuous s distribution. The minimum value i s used, of course, to minimize the comp- utation time. As the same parameter values should be used as for the experiment, this problem was not discussed earl ier in Chapter k. 2 Figure 5.6 shows the coherent intensity, C , the coherent phase, (X, and the incoherent intensity, <̂ I ) as a function of N for densities of 2. 5$, 5$, and 10$. The minimum separation has been conveniently chosen to be 0.5 in . or 1.48X at 35-0 GHz. This i s about the minimum separation that can be mechanically formed because of the f in i te dimensions of the walls of the die. These curves (figure 5.6) indicate that N = 3 i s S- sufficient for densities of 10i, while N = 2 i s sufficient for lower densities s In the interest of minimizing computation time, the value of N = 2 was s selected as sufficiently large for the purposes of this experiment. Note that the coherent phase i s the quantity most sensitive to changes in N and shows some variation even for a density as low as 2. 5$. s This increase of the coherent phase with increasing N g shows that the surface actually becomes "rougher" even though i t can be seen that the incoherent intensity i s • decreasing. The reason for this effect i s that the increased roughness causes more power to be scattered into other directions instead of increasing the fluctuations in the specular direction. 5.2.2 Effect of Minimum Separation The other distribution parameter, d [ n ^ n may now be investigated, although the value of d m i n = 1. has already been selected. Figure 5-7 shows the coherent intensity, coherent phase, and incoherent intensity as 1.0 2.0 3.0 CELL DIVISIONS 1.0 2.0 3.0 CELL DIV IS IONS -e- 1.0 2.0 3.0 CELL DIVISIONS 10% - Q 5% 2.5% 4.0 Figure 5.6 Scattered F i e l d S t a t i s t i c s as a Function of N s for Various Densities of Hemi- spheres . Figure 5.7 Scattered F i e l d S t a t i s t i c s vs. Density of Hemispheres Various Separations. f o r a function of density for a minimum separation of 0, 1, and 1. U8 wavelengths for Ns=2. For a density of 5% (the value chosen for the experiment) the variation over the range of d ^ can be seen to be relatively slight. In this case, the surface is actually becoming more periodic as d is increased. This fact is illustrated by the decrease in min coherent intensity. Obviously, the coherent f i e l d w i l l approach some limiting value <1 since even the periodic surface is rough to the extent that i t w i l l scatter power into directions other than the specular one. 5.2.3 Probability Density of Coordinates A more direct check upon the behaviour of the distribution function may be obtained by numerically calculating the probability density of the normalised polar coordinates of the scatterers as a histogram. Polar coordinates are used to minimize problems in calculating the histograms due to an accidental correlation between the discrete divisions of the d i s t r i - bution function and the discrete "boxes" of the histogram. The radial coordinate i s that given by equation (2.118), that i s R + < ( 5 - 8 ) while the angular coordinate is given by 1 P ^ 0 = — t a n " 1 p (5.9) The individual coordinates are sorted into 25 discrete ranges for 500 independent samples of the surface. On the average there are about 15 scatterers per surface giving about- 7500 coordinate values per graph. Figures 5-8 - 5-10 show the probability density histograms for an object density of 5% with. N = 1, 2, and 3 respectively. Figures 5.11 - 5-13 show the same functions for an object density of 10%. The continuous straight lines on the graphs indicate an ideal continuous uniform distribution. CO 0.0 0.1 0.2 0.3 0.4 0.5 RRDIflL C00RDINRTE 03 <° Q 2 A A A I T 1 •—4 CD C L CO O Q-o' -1.0 -0.6 -0.2 0.2 0.6 1.0 RNGULRR COORDINATE figure 5.8 Probability Density of Hemisphere Coordinates for an Object Density of 5$ and 500 Sample Surfaces, N = 1. s o CO A 0.0 0.1 0.2 0.3 0.4 0.5 RRDIflL COORDINATE 00 o o >—to CO UJ Q t—HQ CQ <X CO o *& * & A I I ! r i 1.0 -0.6 -0.2 0.2 0-6 1.0 RNGULRR COORDINATE Figure 5-9 P r o b a b i l i t y Density of Hemisphere Coordinates f o r an Object Density of 5% and 500 Sample Surfaces. N = 2. Figure 5-10 P r o b a b i l i t y Density of Hemisphere Coordinates f o r an Object Density of 5% a n d 500 Sample Surfaces, N = 3- ID 0.0 0.1 0.2 0.3 0.4 0. RADIAL COORDINATE Q3 >-cn I— •• c n UJ a A A A A A A A A A A A CD CC CD O 1 1 1 - 0.6 -0.2 0.2 ANGULAR COORDINATE -1.0 0.6 Figure 5 « H P r o b a b i l i t y Density of Hemisphere Coordinates f o r an Object Density of 10$ and 500 Sample Surfaces, N = 1. Figure 5-12 P r o b a b i l i t y Density of Hemisphere Coordinates f o r an Object Density of 10% and 500 Sample Surfaces, N = 2 . Figure 5.13 P r o b a b i l i t y Density of Hemisphere Coordinates f o r an Object Density of 10% and 500 Sample Surfaces, N = 3 - 90 The following important characteristics of the surface distr ibution are i l lus t ra ted by these graphs: 1. The distr ibution for Ng=2 (h subcells) i s more uniform than for N s=l (no subcells). For Ns=3> however, the improvement i s not so pronounced. 2. As N g i s increased, the density function i t s e l f becomes more random. A moderate periodic effect can be seen for N =1. s 3. For N g=l, the probability density functions are almost identical for an object density of either 5$ or 10$ while for larger values of N , the higher object density shows the greatest amount of improvement. Characteristics #1 and #2 show that the distribution function behaves as desired, and point out the poor characteristics of the simple discrete distribution (N. = l ) . Thus, the use of such a distr ibution i s jus t i f ied and #3 shows that i t i s even more important to use this distr ibution as the object density i s increased. #2 also strengthens, the previous choice of N : s = 2 for the experimental study. The variance of the coordinate distr ibution i s shown in figure 5- iM as a function of N g . These curves also show that increasing N g causes the distribution of coordinates to behave more l ike a continuous uniform distribution. Of course, (see figure 5.5) the minimum separation must also 0.034 0.34-1 0.032 N, Figure 5 .ik Variance of Scatterer Coordinates as a Function of N : (a) P= % ' S (b) p= 10$ (c) Continuous Uniform Distribution 0.150 0.120 >— 0O.OSDH 5 a g 0.060 - 0.030- cu -0.000 0 0.150-1 0.120 £ 0.090 H a a a. 0.060- 0.030- -0.000 0.150-1 0.120 >«. 3 0.030 H 5 a re a. 0.060 0.030-1 -0.000 ft OF SPHERES ft OF SPHERES * OF SPHERES 0.150 n 0.120 H <n O.OSO a -0.000 30 O.lSD-i 0.120 H 3 0.030 30 O.lSO-i 0.120 H <£O.DS0 5 a -0.000 30 91 10 20 30 40 4* OF SPHERES 20 30 40 # OF SPHERES 50 10 20 30 40 # OF SPHERES Figure 5.15 P r o b a b i l i t y Density of the Number of Spheres i n the Illuminated Area for N g = 1, 2, 3 and densi t i e s of 5% and 10%. 92 go to zero before the continuous uniform distribution i s reached exactly. Notice that for the particular parameters chosen for this surface, the higher density behaves more like the continuous uniform distribution. Such behaviour is not general, however, as can be seen by the following example. Take N =.:.l and a density high enough so that a l l boxes are f i l l e d . For s this case,the distribution has become periodic for a large density, not more random. The means of the coordinates have not been illustrated because they are quite insensitive to changes in the distribution parameters and hence remain essentially constant over the range of variables studied here. 5.2.k Distribution of the Number of Scatterers per Independent Sample As a f i n a l check upon the surface distribution, the probability density of the number of spheres per independent illuminated area is shown in figure 5.15- The solid curves are Normal (Gaussian) distributions calculated using the mean and variance which were determined from the actual distributions. In a l l cases i t can be seen that the number of scatterers i s distributed almost normally. The correspondence becomes better for increasing N , further i l l u s t r a t i n g the improvement obtained using this distribution function. The fact that the distributions are almost normal also indicates that the illuminated areas are chosen from a sufficiently larger area. That i s , edge effects and odd effects due to the fact that the density must be actually fixed over a limited area (for computational expediency) are negligible. 5.3 I n i t i a l Testing of the Experimental Surface For an i n i t i a l test to determine the magnitude of the edge effects, separation distance of the antennas, and the best widths for the f l a t and rough sections of the experimental surface, a special test surface was used. This surface consisted of a 2 f t . square aluminum sheet with hemispheres formed only in the lower right hand quarter. The surface was scanned by the system shown i n f i g u r e 5.1 with the an- tennas a r b i t r a r i l y separated by 4.75 i n . centre to centre. The r e s u l t i n g output from the receiver i s shown i n f i g u r e 5.l6. The distances shown are a c t u a l l y the average of several t e s t s performed with the surface at d i f f e r e n t elevations. The v a r i a t i o n between these t e s t s was s l i g h t . From f i g u r e 5«l6 i t can be seen that both antennas should be about • k i n . from the edge of the surface and about 2 i n . from the t r a n s i t i o n l i n e . These f i g u r e s give a smallest surface that i s 12 i n . wide with random scatterers over the lower 6 i n . t e s t surface magnitude phase 6 f — 81 distance Figure 5.16 Results from the Test Surface The "length" of the surface i s determined from the f a c t s that there must be k i n . extra at each end to take care of edge e f f e c t s while the chamber i t s e l f i s 96 in- wide. These dimensions give a surface width of 52 i n . with a 94 scanning distance of hk i n . I f the apparent figure of h i n . i s assumed to give the distance between independent samples, then each surface should give at least 13 independent samples of the scattered f i e l d . 5.4 Experimental Results Compared With Simulation Results A set of ten experimental surfaces were bu i l t according to the preceding specifications. These surfaces were scanned with the measuring and recording apparatus of figure 5-1 using the method of section 5.1.2. Frequencies of 35 to 39 GHz. in steps of 0.5 GHz. were used. A Monte-Carlo simulation was also performed, using the equations (2.133) to calculate the f i e ld . The object coordinates were generated by the same program as used for the experiment and the relative coordinates were calculated for the antenna at 100 positions across the surface to simulate the experimental scanning as closely as possible. 5.4.1 Preliminary Processing of the Experimental Data The analog signals from the tape recorder were converted to a set of d ig i t a l samples using the DEC PDP-9 d ig i t a l computer and i t s associated multiplexed analog to d ig i t a l converter. Use of the multiplexer allowed essentially simultaneous sampling of the three signal channels plus a fourth control channel, which was generated automatically by the scanner, to indicate the beginning and end of data on the other three channels. The digi ta l ized samples were then transferred from the PDP-9 memory to punched paper tape. This paper tape was then used as input data for the much larger IBM 360/67 d ig i t a l computer, which i s a superior machine for performing the many accurate numerical operations required for the s t a t i s t i c a l analysis of large amounts of data. The raw data as read from the paper tape i s unnormalized and consists of the following: ( l ) the level on the phase channel (0) for 0° 95 phase shift on the receiver phase meter, (2) The reference plate reading (R , S , d) ), and (3) The signal from the surface (R , S , 0J. P P p . s s 0 The next part of the data processing i s concerned with the calculation of the actual normalized samples of the scattered f i e l d . F i r s t , the 0° phase reference i s averaged over a l l samples to minimize effects of channel noise and frequency d r i f t . Second, the reference plate signals are averaged over a l l samples (there are added fluctuations here due to deviations from a perfectly f l a t reflector). For the normalized fields, R / = K <R > p r N p' p S N V ( 5 . 1 0 ) where KR, KS, and K0 are the overall channel gains, and RZ, S', and (£/ are the true normalized f i e l d value; Thus = 1 P 0 ' = o c (5.11) th so that the k sample of the normalized f i e l d scattered from the rough surface i s given by S s < V R s <SP> a ^ ( k ) ) = {<0P>-0s}f (5 .12) These two quantities were calculated, for each sheet (approximately 100 samples per sheet per frequency) and stored on d i g i t a l magnetic tape for the subsequent s t a t i s t i c a l analysis which comprises the third and f i n a l stage 0 . 9 4 0 .8 - 0 4 8 F I L E # 30 T 12 IB 20 24 28 32 SCANNING D3STRNCE C1NCHE5J (a) 1 24 r 36 40 44 SHEET 4t 6 1.2- l . l - D i .0 0 . 9 4 0 . 8 r 45.0 h 22 .5 0 4 *T" B i 12 F I L E & 12 I I 1 I T 16 20 • 24 28 32 SCPNN1NG D1STRNCE (INCHES) (b) UJ a 1—22.5 « -45.0 36 40 44 SHEET * 6 1 . 2 - 1 r 45 .0 T 4 F I L E % 37 12 15 20 24 28 32 SCANNING DISTRNCE (INCHES) (c) Arg(E) 35 40 44 SHEET * 5 (r Figure 5.17 (a) Experimental Data from a T y p i c a l Sheet (b) Simulation Data from a T y p i c a l Sheet (c) Experimental Data Averaged over the Frequency Range 97 of the processing of the experimental data. The corresponding samples generated by the Monte-Carlo simulation were also stored on the same magnetic tape for ease of comparison. These sets of normalized data were plotted against the scanning position for a preliminary inspection before any further processing was carried out. Data for a typical sheet i s shown in figure 5-17 (a) and (b). I t was noticed that some of the experimental curves showed an appreciable l inear offset from the normalized values, as well as the expected random variations. These offsets were completely different from sheet to sheet and for different frequencies, indicating that some measurement errors due to improper normalization remain in spite of the precautions which have been taken (see section 5.1-2). The Monte-Carlo simulation results, however, do not exhibit this characteristic and furthermore, they show very l i t t l e variation over the range of frequencies used (due. t o the fact that the f i e ld i s normalized for each frequenc5r). The absence of this variation with frequency was fortunate, and suggested one f i n a l method for reducing measurement errors. Assume that the offset errors are completely random for each scan, and that the actual variation with frequency i s essentially negligible as indicated by the simulation. Then, average the data over the various frequencies to obtain one set of frequency averaged data. The data so obtained w i l l then be approximately that obtained by performing the ident ical experiment nine times and using the average of the nine readings. This process w i l l also reduce errors due to measurement of the parameters of the system such as the antenna to surface distance. Typical frequency averaged data i s shown in figure 5.17 (c). Notice that the offset for this sheet has been considerably reduced. 5.4.2 Correlation Distance for the Scattered F i - l d To determine the maximum number of independent 'samples which can be obtained from each surface, the magnitude of the complex autocorrelation coefficient, was calculated both for the experimental data and for the simulation data. to yie ld the curves shown i n figure 5«l8 (a). For comparison, i t can be seen from figure 2.5 that for the experimental antenna to surface distance of 27 inches, the 3 db width of the beam at the surface i s approximately 3.6 i n . while the width of the entire main beam i s about 7.9 i n . The simulation curve gives an uncorrelated distance of about 1.75 i n . which i s considerably less than the half-power width, while the experimental results y ie ld an uncorrelated distance s l ight ly larger than the half-power wi dth ( similar results were obtained in the preliminary testing of section 5.3). This discrepancy may be explained by considering the opposing errors for the two methods. For the Monte-Carlo simulation, the main source of error is caused by the single scatter approximation. The absence of M.S. w i l l cause the auto- correlation function to be narrower, and have a sharper transition from correlated to uncorrelated samples because interactions from adjacent but unilluminated areas are not considered. A secondary source of error, that of considering the antenna beam amplitude to be zero beyond a certain angle w i l l also contribute to this effect. On the other hand, the main experimental errors w i l l tend to widen the correlated area. These errors are caused by improper normalization and surface P was calculated separately for each sheet and then averaged over a l l sheets RRGtE).IDEGREES) u ; RRG(E3 [DEGREES] Figure 5.18 S t a t i s t i c s of Simulated ( l e f t ) and Experimental (right) Data. (a) Autocorrelation of the E l e c t r i c F i e l d as a function of the distance between samples. (b) P r o b a b i l i t y Density of the Magnitude of the F i e l d . (c) P r o b a b i l i t y Density of the Phase of the F i e l d . 100 waviness due to s l i g h t bending of the aluminum sheet. Both of these e f f e c t s cover areas -wider than the beamwidth, and hence w i l l increase the c o r r e l a t i o n over a f a i r l y long distance. In view of the above r e s u l t s , i t was decided to keep samples that were the half-power width ( i . e . 3 .6 in.) apart. Thus, 13 independent samples per sheet were obtained. 5 . 4 . 3 S t a t i s t i c s of the Scattered F i e l d For the simulation and f o r the experiment, the coherent and incoherent f i e l d s were ca l c u l a t e d along with t h e i r p r o b a b i l i t y density functions. The p r o b a b i l i t y d e n s i t i e s are shown i n f i g u r e s 5 - l 8 (b) and ( c ) . The s o l i d curves are normal de n s i t i e s c a l c u l a t e d using the mean and variance of the respective data. The s t a t i s t i c a l moments are given i n table 5-1. c 2 a I 2 <|Ef> <afg E> simulation • 9 5 4 l .3469° .00221 .9835 .595 .0009 2.79 experiment • 9755 o 2.335 .03265 1.0025 2.17 .0052 9 8 . 0 Table 5.1 S t a t i s t i c s of Simulation and Experimental Data. From f i g u r e 5 . l 8 (b) and table 5-1 i t can be seen that the d i s t r i b u t i o n s of the f i e l d magnitude f o r the experiment and f o r the simulation are reason- ably s i m i l a r . The means are very close, while the standard deviation f o r the experiment i s increased by a f a c t o r of about 2 . 4 . A reasonable agreement between these figures i s expected because the measurement of the magnitude of the f i e l d i s not nearly as s e n s i t i v e to experimental e r r o r as i s the phase measurement. Thus, i t i s reasonable to conclude that a high percentage of the deviation i s due to multiple s c a t t e r i n g e f f e c t s . The s e n s i t i v i t y of the phase measurements i s i l l u s t r a t e d i n f i g u r e 5>l8 (c) where the phase 101 distributions are shown. Here there i s an appreciable difference between the experiment and the simulation. In fact, the standard deviation for the experimental data i s now almost 6 times larger. 5.5 Summary The experimental approach as investigated in this chapter can be considered to be a qualified success. The method i t s e l f i s an excellent one, but the use of the existing scanner introduced some large errors which made i t d i f f i cu l t to assess quantitatively the comparative behaviour of the simulation and the experiment. Suggestions for improving this situation are given in sections 6. 3 and 6.h. The design of the surface model and i t s associated distr ibution function, though, was found to be experimentally practical and a good representation of a random surface. The actual distribution used for the experiment was based upon a division of the or iginal |- i n . ce l l s into four i n . subcells. The coordinate distribution generated for this choice of N g was found to give an acceptable trade-off between computation time and the required characteristics of the distribution. A detailed study of the behaviour of the model would require that many more surfaces be bu i l t for several different parameter values. This extension of the work, was considered to be beyond the range of this thesis. 102 6. CONCLUSIONS 6.1 Surface model The r e s u l t s of section 5.1.3 and 5.2 i n d i c a t e that the surface model developed i n t h i s t h e s i s has a l l the c h a r a c t e r i s t i c s that were or i g i n a l l y - required. B a s i c a l l y , these requirements were 1. That the s t a t i s t i c s could be c o n t r o l l e d as desired. 2. That the model be usable both f o r computer simulation and f o r a c t u a l experiments. The f i r s t c r i t e r i o n was s a t i s f i e d by using an array of d i s c r e t e scatterers (as opposed to a continuous rough surface) where the randomness was determined by using a computer generated pseudo-random sequence. In the case studied here, the randomness was introduced by the s c a t t e r e r coordinates although other v a r i a b l e s such as s c a t t e r e r s i z e and/or shape could have been used. In the simplest case, the simulation of the single scattered f i e l d , no s p e c i a l problems a r i s e . That i s , a continuous uniform d i s t r i b u t i o n may be used d i r e c t l y to generate the coordinates because overlapping scatterers do not matter. For any simulation which includes multiple s c a t t e r (and hence sca t t e r e r separation) or to s a t i s f y the second c r i t e r i o n above, a more soph i s t i c a t e d coordinate d i s t r i b u t i o n f u n c t i o n must be u t i l i z e d . Such a d i s t r i b u t i o n function was developed i n section 4.1 and f u l l y tested i n section 5-2. The t e s t r e s u l t s , which are i l l u s t r a t e d i n f i g u r e s 5.6 to 5 . l 6 , show that t h i s d i s t r i b u t i o n i s an acceptable function to use f o r the study of any d i s c r e t e s c a t t e r e r problem. I t i s a good approximation to a continuous uniform d i s t r i b u t i o n , and i t i s simple and quick to numerically c a l c u l a t e f o r one, two, or three dimensional p o s i t i o n vectors. 103 The second c r i t e r i o n f u r t h e r l i m i t e d the model t o be composed of s p e c i f i c shapes (hemicylinders and hemispheres) so that e i t h e r the scattered f i e l d could be represented mathematically with ease, or that a metal surface model could be machined i n t o the same shape. The mathematical s c a t t e r i n g functions are derived i n Chapter 2 while the method of surface construction i s o u t l i n e d i n section 5 « 1 . 3 . The accuracy of data obtained using t h i s surface model depends upon the degree of approximation which i s applied to the mathematical s c a t t e r i n g functions and the degree of p r e c i s i o n with which the surface i s machined. For t h i s study, the s i n g l e s c a t t e r approximation was mainly used f o r the simulation, and the experimental surface was formed with a tolerance of b e t t e r than +5$ f o r object s i z e and shape. S l i g h t d i s t o r t i o n of the surface between scatterers was i n e v i t a b l e . The surface model developed i n t h i s t h e s i s f o r the study of rough surface s c a t t e r i n g was, therefore, found to be e n t i r e l y s a t i s f a c t o r y . This model i s v e r s a t i l e , easy to use, and lends i t s e l f to g e n e r a l i z a t i o n to higher orders of randomness. 6.2 Simulati on The aims of the computer simulation of the rough surface s c a t t e r i n g problem were e s s e n t i a l l y : 1. To f i n d a, good mathematical method f o r the f i e l d c a l c u l a t i o n s based upon the p r i o r choice of surface model. 2. To use f i n i t e non-plane incident r a d i a t i o n , representative of a p h y s i c a l antenna. 3. To provide comparison with experimental studies. As only a numerical method f o r determining a surface d i s t r i b u t i o n was developed, the average f i e l d c a l c u l a t i o n s were based on a Monte-Carlo method. A single c a l c u l a t i o n w i l l therefore involve only the instantaneous f i e l d 104 value due to a f i x e d configuration, while the a c t u a l s t a t i s t i c s are c a l c u l a t e d from a set of such values. I t was found i m p r a c t i c a l to use any other than the s i n g l e s c a t t e r or f i r s t order nearest neighbour approximation f o r the f i e l d due to a single configuration, but only the s i n g l e s c a t t e r method was generally used. Higher order multiple s c a t t e r i n g e f f e c t s were inve s t i g a t e d however, f o r the r e s t r i c t e d case of a p e r i o d i c array of hemicylinders. The re s u l t s i n d i c a t e d that f o r de n s i t i e s of greater than 30% and object s i z e greater than §-,\ i n radius the multiple s c a t t e r i n g e f f e c t i s d e f i n i t e l y not n e g l i g i b l e . The Monte-Carlo method requires that only a f i n i t e number of scatterers be considered. This i s not a serious constraint because p h y s i c a l problems are nearly always composed of f i n i t e areas of surface. In f a c t , the second aim of t h i s part of the study automatically r e s t r i c t s the number of scatt e r e r s per saciple by r e q u i r i n g only a f i n i t e beam of i l l u m i n a t i o n . The p a r t i c u l a r form of the incident r a d i a t i o n i s developed i n section 2.k and i l l u s t r a t e d i n f i g u r e 2.5. The functions were chosen to reasonably approximate a horn antenna r a d i a t i o n pattern, have independently v a r i a b l e beamwidth and sidelobe l e v e l , and be simple to calculate numerically. The values of the vari a b l e parameters of t h i s model were subsequently set to approximate the horn antenna used f o r the experiment. The one problem which i s encountered with the f i n i t e beam i s that i t i s d i f f i c u l t to d i r e c t l y compare the r e s u l t s with other methods of c a l c u l a t i o n . This was attempted i n se c t i o n h.k, where some a n a l y t i c a l solutions are given f o r the coherent and incoherent f i e l d s . I t was concluded that the Monte-Carlo simulation was the best means of c a l c u l a t i o n . 6 . 3 Experiment The aims of the experimental study were 105 1. To demonstrate that a su i t a b l e experiment could be performed. 2. To compare with numerical methods. 3. To extend the i n v e s t i g a t i o n to areas where numerical methods are p r o h i b i t i v e . The f i r s t goal above was amply demonstrated. The technique of scanning a formed metal surface was shown to be a p r a c t i c a l method. The forming of the sheet according to the pre-calculated d i s t r i b u t i o n of coordinates was quick and very simple once the proper punch and die had been constructed. The scanning system that was used though, was less than perfe c t , g i v i n g r i s e to r e l a t i v e l y large phase measurement errors. Some methods of preventing t h i s problem i n future experiments are discussed i n the next section. The experimental r e s u l t s compare favorably with the simulation, but the aforementioned phase errors make i t d i f f i c u l t to in t e r p r e t the deviations i n terms of s p e c i f i c mechanisms. A set of experimental data for various den s i t i e s would most l i k e l y help to analyse the various e r r o r s , but t h i s i s beyond the scope of t h i s work. The t h i r d goal has been attained i n theory, but was judged also to be beyond the present scope. The formed metal surface can d i r e c t l y accomodate high densities of sc a t t e r e r s , large s i z e s of s c a t t e r e r s , and odd shaped s c a t t e r e r s . With a s l i g h t m odification to the d i s t r i b u t i o n function random siz e s of scatterers could also be included. 6.4 Numerical Results The main r e s u l t of t h i s study has been to develop c e r t a i n v a l i d methods for i n v e s t i g a t i n g the behaviour of rough surface s c a t t e r i n g . There are a large number of v a r i a b l e parameters, and even f o r a f i x e d set of parameters, a large number of random v a r i a b l e s inherent i n t h i s problem. Because of t h i s , a d e t a i l e d numerical study of the scattered f i e l d was not 106 undertaken. Instead, the behaviour of parts of the sc a t t e r i n g were investigated f o r s u i t a b i l i t y and for the best parameter values to use for subsequent c a l c u l a t i o n s . For the per i o d i c model of hemicylinders i t was found that 1. The width of the act i v e s c a t t e r i n g area should be at lea s t comparable to the width of the main beam of the antenna at the surface. 2. The r e l a t i v e error from neglecting multiple scatter i s l e s s than 20% f o r a £. 0.5X and p $ 35% except for some s p e c i a l values of a. 3. The f i r s t order nearest neighbour approximation reduces these errors by a factor of 1/2. 4. Computation time i s p r o h i b i t i v e f o r higher order approximations. For the random s i n g l e s c a t t e r model of hemicylinders with a continuous uniform coordinate d i s t r i b u t i o n , the coherent f i e l d and incoherent i n t e n s i t y x^ere determined a n a l y t i c a l l y . Comparison with the Monte-Carlo simulation showed that the c a l c u l a t i o n of the coherent f i e l d i s r e l a t i v e l y inaccurate while the c a l c u l a t i o n of the incoherent i n t e n s i t y gives excellent r e s u l t s . The following r e s u l t s were noticed for the coordinate d i s t r i b u t i o n : 1. The improvement attained by using t h i s d i s t r i b u t i o n increases with object density. 2. D i v i s i o n of the simple model into only four subcells i s s u f f i c i e n t to produce a reasonably uniform d i s t r i b u t i o n without excessive computation time. 3. The f i n i t e separation of the scatt e r e r s necessitated by the experiment causes a n e g l i g i b l e e f f e c t . 6.5 General Recommendations The main problem involved i n t h i s i n v e s t i g a t i o n i s the need f o r con t r o l of the various errors so that t h e i r e f f e c t can be evaluated. This was done 107 to a l i m i t e d extent by studying a s p e c i a l case - the p e r i o d i c two dimensional array. It would thus be very i n s t r u c t i v e to extend the accuracy i n a s p e c i f i c manner of the three dimensional problem f o r the simulation and for the experiment. The accuracy of the simulation could be improved by d e r i v i n g the f i r s t order nearest neighbour approximation for the array of hemispheres. The comparison of these r e s u l t s with the s i n g l e s c a t t e r r e s u l t s would give a better i n d i c a t i o n of the amount of m u l t i p l e sca t t e r i n t h i s case. The experimental r e s u l t s could be greatly improved, and the actual s e t t i n g up of the experiment ( i t took a long time to properly mount the sheets and a l i g n the scanner for each data set) made easier by redesigning the scanner i t s e l f . The scanner which was used for the experiment was o r i g i n a l l y designed as an antenna p o s i t i o n e r for pattern measurements, and therefore moved i n a v e r t i c a l plane. The problems encountered were due to the surface to antenna distance changing due to improper alignment. These problems would be eliminated i f the surface was l y i n g i n a h o r i z o n t a l plane, and supported upon a f l a t backing plate which was i n turn supported on r o l l e r s . The antennas could then be dir e c t e d s t r a i g h t downward from some point above the device (e.g. suspended from the c e i l i n g ) while the surface was gently p u l l e d across beneath by means of a wire winding on a drum. Mounting of the surface upon the scanner would also be f a c i l i t a t e d because gravity would hold i t i n p o s i t i o n , unlike the v e r t i c a l mounting system that was used. 108 APPENDIX A DETERMINATION OF UNKNOWN SCATTERING COEFFICIENTS The to ta l e lectr ic f i e l d , using the equations (2.1), (2. lU), and (2.15) of Chapter 2, may be written as: Ho E_ = E. + ) E T mc / s sTt' Mo = E i nc + E s + / E t t*-s (A. l ) E D \ J n (k r o s / Xi> s B n a H n ( k r J e i n 9 s ns n v s' t*S 27 Now, Graf's addition theorem for cylinder functions i s 00 C (w)e ina C , (u)<J. (v)e n+kv ' k v ' ik (A. 2) where jv[ <|u| and u,v,w,(2, and-̂ Q are related by the following geometry, and C n(x) i s any cylinder function. Apply this theorem to equation (A. l ) for t <s: >? P From the above diagram, P<3 . ^ H m + n ( k ( X s - X t ) ) j n ( 1 c r s ) e 1 " ( 7 7 - 6.') (A.3) 109 or, by using the fact that cn(z) = (-Dnc_n(z) (A. h) replacing n by -n and summing in the opposite direction obtain the equiv- alent expression Now, for t > s, Again, H (kr )e m t X ] V m ( k ( V V ) J n K ) i n a H (kr )e t =) H (k(X,-X ) )J (kr )e mv t / , n-m\ t 5') n v s ; (A. 6) (A. 7) The combination of equations (A.5) and (A.7) may be written CO V k r t ) e i m 9 t = HHn-mU!Xs-Xtl ) J n C k r s ^ i n 9 s C ^ 8 ) f\--ao with F st nm * m+n (-1) } t<s 3 t>s (A.9) Therefore, the to ta l e lectr ic f i e l d becomes E y E D f j (kr ) ( i ) n e i n ( 0 S ^ s ) + T"̂  B H (kr ) e i n 9 s T l — j o s {—> n s L - J n s n s 5:1 I— w~—c& + E E v r W * i v\i )^(^)-in0s c i s mt inO g ^ t nm (A:10) 110 The following boundary conditions must be satisfied on the surfaces of the cylinders, i . e . at r = a, s E T = 0 , _L polarization • -• = 0 , j j polarization Or (A.11) in*9 Since (A.10) must hold for a l l 0 , the coefficients of each e may be equated to zero i n equation (A.10) or i t s r derivative, which yields the s equation for the unknown coefficients, B : ^ ' ns where ns n No £*0 EDD ( i ) n e i n A ; +Y~Ny\tH (k X -X, )pst ° s / , [_ f M-t n-m s t nm (A.12) A =s - ^ H (ka) A" => - n j'(ka) (A.13) n n <(ka) I l l APPENDIX B RELATIONS BETWEEN REAL AND IMAGE FUNCTIONS The problem Is to simplify the expression for the scattered f i e l d t h caused by the s hemisphere as given by equation (2.30) of Chapter 2, that i s E U A = E r + E i (B. l ) g s - s where the superscripts r and i refer to the real and image functions respectively. Therefore, from equation (2.15), CO E l ' L = y"(B r + B 1 )H ( k r s ) e i n Q s (B.2) s / , s ns - ns' n v s v ' XS---K) And, from equation (2.17), B r = A,, ns " , mt nmst t-1 rn - - t (B.3) Note that G ^ i s not a function of(2. To find B ^ g , Qmust be replaced by TT- QL. F i r s t , i t i s obvious that the distance C g must be the same for the real and image functions. Second, the angle B i s given by i L cos(77- a) tan/3g X + L sin(7T-0t) s -L cos a (B. h) X + L sinCt s -tan ̂3 Therefore, B 1 = - B r (B. 5) s s 112 Third, the antenna radiation pattern i s given by f 1 = fiU-jj+a+jQ = f(-J +a +/£) (B.6) But, the antenna pattern may be assumed to be an even symmetric function, therefore 1 r f = f (B.7) The above may be applied to equation (B.5) 31 ns I T = A n E N„ Bmt^nmst (B.8) By a comparison of (B.3) and (B.8) i t can be seen that B 1 = ( - l ) V -ns v ns (B.9) Hence, after dropping the superscript r, E,U=V^(B ± ( - l A n s ) H (kr ) e l n 6 s s /_ f ns x ' -ns' n v s' n - - oo ( B _ n s + ( - l ) n B n s ) ( - l ) n H j k r J e - i n e s nN s' + <Bns ± ( - l ) n B - n s ) H n ( k r s ) e i n 0 s + ( B Q S ± B o s ) H 0 ( k r g ) (B.10) " ( - l ) n H n ( k r s ) e i n 6 s ( B _ n s ± ( ~ l ) \ s ) ( e ' 1 * s ± e i n s) *> -1 + ( B o s ± B 0 S ) H 0 ( k r g ) oo f n = -1 C * n s c o s n%) . f 2 E o X D s H o ( k r s ) ] Y n s s i n n0 e 113 APPENDIX C FIRST ORDER NEAREST NEIGHBOUR APPROXIMATION For a low density of objects, the average separations w i l l be large. Thus, any terms i n v o l v i n g the object separation may be approximated by t h e i r 27 large argument asymptotic 'expansion. In p a r t i c u l a r , the asymptotic expansion , N t~o~ i Z , . x n - i-jrK f i(4n - l ) 1 . . vz^]4e (-i)e (l+-Sz— ••• (c-l} may be u t i l i z e d to c a l c u l a t e the c o e f f i c i e n t s 4 = ( H m - n ( k d ) i ( - 1 ) n H m + n ( k d ^ i ) n " m f n m <C-2> i n equation ( 3 . 9 ) with the r e s u l t that i n equation (3 .k) ( l - i ) e i ^ d f 1 "] T-TtfP 2 ,-LpoO Hmns,s+1~ ^ " | k T ^ - l ) f f i + n J ^ + +n , || p o l . J (C3) For large kd, the c o e f f i c i e n t s f o r the perpendicular p o l a r i z a t i o n tend to zero f a s t e r than those f o r p a r a l l e l p o l a r i z a t i o n . Thus i t i s expected that most c a l c u l a t i o n s w i l l be more accurate f o r the perpendicularly p o l a r i z e d case. As a large separation has been assumed, i t i s a l s o reasonable to assume that multiple s c a t t e r i n g e f f e c t s are small and hence only one i t e r a t i o n i s required to obtain convergence of equation ( 3 . 6 ) . This means p h y s i c a l l y th that the multiple scattered wave from the s c y l i n d e r i s caused by the s i n g l y / \th . .th scattered waves from the (s - 1 ) and (s+1) cylinder. That i s , (o .M f o r t = s+1 only. S u b s t i t u t i o n of equations (C. 3 ) , ( C . 4 ) , and ( 3 - 9 ) i n t o equation ( 3 . 4 ) gives the N-N-l approximation to the s c a t t e r i n g c o e f f i c i e n t s : 114 Y « ns -iA x„ikL ns <U2h' s ns ikd I S v U 2 n ) (-1) mA h" s-1/ , — lYlzl H A n J k L <U h ,ikd s ns ^ C ( k d ) ^ : IF s-1 1 m, s-1 •HI* > A'h" s+lZ , ra ra,s+l^ (C .5 ) with cos (C . 6 ) Equation (C .5 ) for the parallel polarized f i e l d was derived under the additional _1 - 2 assumption that the (kd)~ 2 term sufficiently dominates the (kd) a term i n equation (C.3). 115 APPENDIX D SINGLE OBJECT SINGLE SCATTER STATISTICAL CALCULATIONS This s i m p l i f i e d method consists of vising the s t a t i s t i c s of a sing l e object averaged over a l l p o s s i b l e p o s i t i o n s to c a l c u l a t e the exact means and the approximate variances f o r the given ensemble of objects. Higher order moments may not be c a l c u l a t e d with any accuracy at a l l . The p o t e n t i a l saving here i s very large, because the values c a l c u l a t e d by the Mohte-Carlo section of the ana l y s i s are independent of the density of the objects, thus e l i m i n a t i n g one of the many v a r i a b l e parameters of the problem. For the single s c a t t e r uniform d i s t r i b u t i o n , E = 1 + Therefore, <E> = <E x> + i<Ey> 5 = 1 = 1 + <E Es> •5-1 But, as a l l the objects are i d e n t i c a l and independent, C v = E< Eo ( xo» = N <E S o N o / (*3) Here, the subscript o means that any one of the N Q objects i s to be used t o c a l c u l a t e the average. Therefore <V - 1 + "c<V ( D A ) 32 To calculate the variance, an approximation must be made . -<(1+ L\) >-<v ( B-5 ) = N 0<V - No<v2 + £ E<\v It may be assumed that, for large numbers of objects in conjunction with a fairly low density, the last term of equation (D.5) may be neglected. If this is indeed possible, then (D.5) reduces to ° Z * N o « \ > " Ko< V > 2 2 2 Similarly, Q = N « E > - N <E > ) (D.6) " >y Q 0 ^ 0 Now, from the area density function used in Chapter ks N =p((̂ -)i4P i P(C kL) P o ( C 7 ) where 1 for hemicylinders P = 2 for hemispheres (D.8) 117 Therefore, the s t a t i s t i c a l functions become • <E >- 1 + p ( C o k L ) P < E x > <E > =p(C kL) P<E > y ° y 0 G* ~ P(C M / f o f > -p(C ckL) P<E >^ (3).9) o \ xQ xo Oy* P(C 0tt) P(<By o>-p(C 0KL) P<E y o>^ 0 x y * P ( C o k L ) P ( < E X o E y o > -P(C 0kL) P<E X o><E y o>) These functions are valid only for a uniform distribution with the single scatter approximation. 118 APPENDIX E INTEGRATION OF SINGLE SCATTER COHERENT AND INCOHERENT INTENSITY FOR THE CONTINUOUS UNIFORM DISTRIBUTION The comparison of the mean scattered f i e l d calculated by the Monte- Carlo method f o r the f i n i t e beamwidth problem and the Twersky i n f i n i t e plane xrave problem i n Chapter h indicates that the differences are small, and hence amenable to approximation. E.1 Two Dimensional Problem E . l . l Coherent F i e l d The average f i e l d scattered by a single object i n the active scattering area i s (B.D l e t be the continuous counterpart of the coordinate d i s t r i b u t i o n Q • Then 0 . 5 (E.3) Therefore, using equation (3.2l), 0 . 5 2 0,1 (cos n/3) i 2 -}dQ n (sin n/J 0 7 7 JkL ikL(C a p) ( l - | C a p )g ( C ^ ) ^ ( - l A 1 1 9 33 2 The method o f s t e e p e s t d e s c e n t s i n d i c a t e s t h a t , p r o v i d e d kLC i s l a r g e , a t h e m a j o r c o n t r i b u t i o n t o t h e i n t e g r a l o c c u r s f o r Q n e a r z e r o . T h i s f a c t means t h a t £3 ( f r o m e q u a t i o n 2 .5) may be a p p r o x i m a t e d b y Hence, ,11,1. r- -i#T£77 <E > % V2e — Q-5 2 2 i21cLCap J k L P v 0 . 5 0 . 5 2 2 i k L C ^ ( i - J c ^ ) g " ' t c D ) a p • 0 . 5 A v e r y i m p o r t a n t p o i n t t o n o t e h e r e i s t h a t i t was n e c e s s a r y t o assume t h e e q u i v a l e n t o f = §-77 i n t h e t h r e e d i m e n s i o n a l c a s e i n o r d e r t o o b t a i n a u s e f u l s o l u t i o n . I t was m e n t i o n e d a t t h a t t i m e t h a t t h i s w o u l d be v a l i d o n l y f o r a v e r y n a r r o w beam. However, i t i s i n d i c a t e d b y e q u a t i o n L 2 (E .U ) t h a t o n l y t h e p r o d u c t x C a must be l a r g e . Thus, i t w i l l be r e a s o n a b l e t o e x p e c t good r e s u l t s even f o r f a i r l y w i d e beams p r o v i d e d t h a t t h e a n t e n n a i s a f e w hu n d r e d w a v e l e n g t h s away f r o m t h e s u r f a c e . T h i s w i l l c e r t a i n l y be t h e case i n most p h y s i c a l s i t u a t i o n s . To p e r f o r m t h e i n t e g r a t i o n o f (E.k), l e t t = - i k L C ^ p 2 2 a/" ( E . 6 ) and, s i n c e p i s s m a l l , expand t h e a n t e n n a space f a c t o r , g, i n a T a y l o r s e r i e s a b out p = 0 . From e q u a t i o n ( 2 . 9 5 ) (E .7) 120 or more simply, r (u) * 1 - b e j h u ( E . 8 ) where 5kO0 b e = 9 e 2 36450^ ( 8 N (E.9) Then, equation (E.5) becomes x o ^ pyjf N l k L e - l 2 2\ 4) <* + b e , h ^ C a P j d P (B.10). Further, l e t 1 + 2b B = e,h ( E . l l ) Thus, -i-frkLC kLC _ i - t 2 t 2e dt + BC a i K L C L I - t t 2 e dt fo (H.12) But e _ t t b d t ^ F ( b + l ) - z b e ' z 0 (E.13) i s the asymptotic expansion of the incomplete gamma function f o r large 27 | z| and arg(z) < 37T/2 . Ap p l i c a t i o n of equation (E.13) to (E.12) y i e l d s the f o l l o w i n g expression f o r the single object average: 2TTf r 2iB 1 - kLC kL C jKkL aN 2 S i ^ ( l + — -(1 - BC a)e 7 1 (E.lU) and hence the coherent f i e l d i s given by, using equation (D . 9 ) . <E> = 1 H - ^ f 1 2a I B i p i ^ C l + ^ C )• + . ( l - B C )e - r T " kL C aJjTkL (E .15) E. 1.2 Incoherent I n t e n s i t y For the second s t a t i s t i c a l moment of the scattered f i e l d , under the assumption of a continuous uniform d i s t r i b u t i o n , the si n g l e object function i s given by 0 . 5 <E > = E(wp)E (Wp)dp (E.16) - 0 . 5 Again, from equation ( 3 « 2 l ) with p = |-7T, 0 . 5 Z l 2 14-7T 1 , t. <- c c . 0 . 5 For the above i n t e g r a l , i t i s not s u f f i c i e n t to use the Taylor ser i e s exp- ansion f o r g(C ) because the method of.steepest descents i s not applicable. Therefore, the function g must be included i n i t s e n t i r e t y . Thus, 2 II 2J£[(i- c a p 2 ) s i ^ f d p kL, (E.17) 0 122 , 2.1 o 28x kL -z (1 0 2 2 c o s CP) 2 U . J - » I - ^ ( b C a P ) 2~2dP (E . 1 8 ) where II h b 180 0e 270 0h (E . 1 9 ) Let (E . 2 0 ) with a= hc_ (E . 2 1 ) Then, , 2 I' <E*> o 8ft f +2, s i n 2 t t 2 dt '0 2 J-<E > o kLa sin(t+|jt) sin( t-|-ft)j t - ! -|jt dt Now l e t '0 i 1(a) i2(a) i 3(a) 2 s i n t •dt '0 ha s i n ^ t dt '0 r- , 2 Jo s i n dt (E . 2 2 ) (E . 2 3 ) (E . 2 4 ) ik(a) t s i n t dt Therefore, 2 II ,2 f l 8H~ -, - p - ! — r (a) --tWa) kLQ 1 b 3 (E . 2 5 ) For the perpendicularly p o l a r i z e d case, s u b s t i t u t e u = t + \tt (E . 2 6 ) i n the appropriate i n t e g r a l s to obtain a f t e r s i m p l i f i c a t i o n . .2 ? , | f _ l 2J . 2 . J . r p <E > = it + 1 y i 2 ( a + i t ) - i (a-n) "3ic 2_n 2 b 2 " Jt i3(a+3t) +1 u-n) ik(a+ri - ifi-n) — - 2 ~ t rtb 2 - — _ * (E . 2 7 ) Evaluate the i n t e g r a l s of equations (E.2k): l a 2 s i n t . . C T l ( CZ) = J ~ t 2 0 'dt 2 . . 2rt r s i n • g s i n ^ + j dt (E . 2 8 ) = - | s i n 2 | + Sia 27 where S i d i s the w e l l known sine i n t e g r a l f u n c t i o n . let . 2 s i n t I2(a) = I dt • = |(y+ ma- c i a ) (E . 2 9 ) Here, CiCZis the cosine i n t e g r a l f u n c t i o n and y i s Euler's constant. 124 M I (CO = I s i n 2 t d t 3 4 = iKCX - sinQ) (E .30) F i n a l l y , ret I (a) = | t s i n ^ t d t = - asina + i - cos a) (E.31) I t i s now pos s i b l e to s i m p l i f y (E.28) and (E . 2 9 ) by using the asymptotic expansions f o r the sine and cosine i n t e g r a l s . This procedure i s v a l i d since ( Z « l . In f a c t , a= be (180) i % 7 o k ) K ^ ^ o (E . 3 2 ) >10 o 27 even f o r Q as small as 8 . Therefore , o s i (CO Ci(C() Tt cosq sinCX 2 " a ~ a2 s i n a c o s a a ~cT2 (E . 3 3 ) Thus, the sing l e object incoherent i n t e n s i t y f o r the p a r a l l e l p o l a r i z a t i o n becomes 2 I to o W^bC \2r WoCc d 2 - 1 + - r + T ^ r ( l - -=)sin bC 4 b<V 1/ a (E . 3 ] + ) For the perpendicularly polarized case, lAOL + rt) +1, (a - rc) ~ it + Icosa +-2 (̂i[2cosa + sina) i i u a 2 (E. 1 ( a + n ) - i ( a - TC) = c i (a+rc) - c i ( a - n ) •+ m - g 2 2 £ + 1 -2.-1 (E. but 27 In & + 1 2Jt a (E. Therefore, i 2 (a + n) i 2 ( a - n ) ~ f sina a (E. Without approximation, and i 3 ( c x + j t ) + i (cx-at) = K a + sina) (E. i ,(a + K ) - i . (a -n) =^(a+sina) (E. 4 4 Ij. so that the perpendicular component becomes 4 ^ ^ f t - i - o kLbC a I jtbCj ca/rt? b \ U —(h - C 2) sin bC bcn a a (E. Now, from equation (D.10). . 2 ^ . „C^kLC a 2a * 2Jt /E 2\ A ^ ^ E S (E. T ~ — 2 If terms to the order of (-=-) 2 .or less and (bC &) or less are neglected the approximate incoherent intensity i s given by 126 (E.U3) where b JtbCc (E .UU) E.2 Three Dimensional Problem E.2.1 Coherent F i e l d For a continuous uniform distribution on a plane, <E Q> =j- E o(x,y)dxdy A 27T-|w 'v^f f Eo(R,^)RdRd<|) ( E > U 5 ) 0 0 2TTh = | J E Q ( W P ^ ) ) O C P * I > 0 0 Consider the angular integral f i r s t . Then, i t can be immediately seen that the cross polarized components have the following form (see equation (2.133)) for i r j 27T <E^> = k Jf(c|>)sin&os$ac|) 0 (E.'46) = 0 127 since f i s an even function of Note, as was mentioned i n Chapter 2, the same reasoning may be applied t o the z-axis component which has already,been neglected. Thus, the cross p o l a r i z e d components are zero to the degree of approximation that has been used. P h y s i c a l l y then, one would expect very l i t t l e d e p o l a r i z a t i o n of the backscattered f i e l d at normal incidence f o r any surface which does not e x h i b i t an appreciable amount of multiple s c a t t e r i n g . That i s , equation (E.17) w i l l be zero f o r any p h y s i c a l d i s t r i b u t i o n . The d i r e c t components of the coherent f i e l d may now be calculated. Again, from equations (2.133), E. - . ^ i ^ a P 2 ! ^ 0 ^ 1 0 * ' 0 ^ 0 8 * ^ ! - C 2 0 2 ) g ( C ^ o s < £ , C l i n c h . ) ! i i 0 kL 2 2 1 - ' 2) cos (£> 2* sin (p (E.47) As before, the antenna r a d i a t i o n pattern, g, may be expanded i n a Taylor seri e s about p = 0. 2 2 g(u,v) as 1 - ( b h u + b g v ) (E.l*8) where and b^ are given i n equation (E.8). Substitute equations (E.22) and (E.23) i n t o (E.20) and keep only terms t o the order of pc I 2J7 2 1+ib < E i i >* * 5 e i k L C a P 2^2/ i - c a p ( i + < 0 0 ^osf^+lcosC b sinfp+b cosCJp+§sinC Carry out thec^)-integration f i r s t . Then, )d$p<*P (E.1+9) <E > = <E > = <E > N i r N 22^ x ° (E.50) and, \ 2 2 8 i b f ikLC p 2 2 < E o > ! : f " T T I6 S ( l - B C a p ) p a p ( E . 5 1 ) with 2h + b„ + b. B = 1 * (E. 52) Let 2 2 t = ikLC (E. 53) Then, i£kLC 2 2 2 I 2 fiB H^LC . iB ! (E. 54) IkLC J IkL kL H a v. a' v Therefore, the approximate coherent f i e l d i s given by In this (three dimensional) case, the expression for the average scattered f i e l d i s not an asymptotic evaluation. In fact, i t i s to be expected that the results w i l l be accurate only for small surface areas. This implies that (E .55) w i l l only be v a l i d i f the product (L) (C ) remains sufficiently small. In fact, i t can be seen that (E .55) becomes proportional to e for large L. Clearly, this cannot be va l i d unless 1 2 C —> 0 simultaneously to remove the indeterminate phase factor e u k^^a. 129 E.2.2 Incoherent I n t e n s i t y From equations (2.133) pcpdcp (E .56) Unfortunately, when the form of g has been s u b s t i t u t e d from equations (2.96), the in t e g r a t i o n s above become impossible to perform a n a l y t i c a l l y . The problems a r i s e because i n t e g r a l s of the form vhere P 1 and P are polynomials and a and b are functions of Q are encountered. Whichever i n t e g r a t i o n i s performed f i r s t i s immaterial - the second i n t e g r a t i o n becomes impossible because the asymptotic expansions of the sine and cosine i n t e g r a l s cannot be applied a f t e r the f i r s t i n t e g r a t i o n alone. Therefore, the incoherent i n t e n s i t y f o r the three dimensional case must be c a l c u l a t e d by the MonterCarlo technique i n a l l cases i n v o l v i n g the antenna model chosen f o r t h i s study. An approximate expression could be derived by assuming a square beam (see equation (2.92)), but t h i s i s not considered here. The same reasoning i s appli c a b l e t o the c r o s s - p o l a r i z e d components. a b 2 130 REFERENCES 1. Lord Rayleigh. Theory of Sound. Dover, New York, 1945. 2. Rice, S.O. "Reflection of Electromagnetic Waves from S l i g h t l y Rough Surfaces", The Theory of Electromagnetic Waves. Dover, New York, 1951. 3 . Hoffman, W 0C 0 "Scattering of E.M. Waves from a Random Surface", Quarterly of Applied Mathematics, 13, # 3 , 1955, P« 291 4. S i l v e r , S. Microwave Antenna Theory and Design. Dover, New York, 1965 5. Peake, W.H. "Theory 0 f Radar Return from Te r r a i n " , IRE National Convention Records, 7 , Part I, 1959, p. 27. 6. Senior, T„B.A. "Impedance Boundary Conditions f o r S l i g h t l y Rough Surfaces", Applied S c i e n t i f i c Research B (Netherlands), 8, #5-6, I960, P . T 3 7 . 7. Beckmann, P. 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