INFORMATION TO USERS This reproduction was made from a copy of a manuscript sent to us for publication and microfilming. While the most advanced technology has been used to pho tograph and reproduce this manuscript, the quality of the reproduction is heavily dependent upon the quality of the material submitted. Pages in any manuscript may have indistinct print. In all cases the best available copy has been filmed. The following explanation of techniques is provided to help clarify notations which may appear on this reproduction. 1. Manuscripts may not always be complete. When it is not possible to obtain missing pages, a note appears to indicate this. 2. When copyrighted materials are removed from the manuscript, a note ap pears to indicate this. 3. Oversize materials (maps, drawings, and charts) are photographed by sec tioning the original, beginning at the upper left hand comer and continu ing from left to right In equal sections with small overlaps. Each oversize page is also filmed as one exposure and is available, for an additional charge, as a standard 35mm slide or in black and white paper format. * 4. Most photographs reproduce acceptably on positive microfilm or micro fiche but lack clarity on xerographic copies made from the microfilm. For an additional charge, all photographs are available in black and white standard 35mm slide format.* *For more Information about black and white slides or enlarged paper reproductions, please contact the Dissertations Customer Services Department. T T-1K /T -T Dissertation U 1V11 Information Service University Microfilms International A Bell & Howell Inform ation C o m p an y 300 N. Z e e b R oad, Ann Arbor, M ichigan 48106 8619741 Durden, S tep h en Loren MICROW AVE SC A TTER IN G FR O M THE OCEAN S U R F A C E P h .D . S ta n fo rd U n iv e rs ity University Microfilms International 300 N. Zeeb Road, Ann Arbor, Ml 48106 1986 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this docum ent have been identified herewith a ch eck mark V . 1. Glossy photographs or p a g e s_____ 2. Colored illustrations, paper or print______ 3. Photographs with dark background_____ 4. Illustrations are poor copy______ 5. Pages with black marks, not original copy______ 6. Print shows through as there is text on both sides of p a g e _______ 7. Indistinct, broken or small print on several pages 8. Print exceeds margin requirem ents______ 9. Tightly bound copy with print lost in spine_______ 10. Computer printout pages with indistinct print______ 11. Page(s)___________ lacking when material received, an d not available from school or author. 12. Page(s)___________ seem to be missing in numbering only as text follows. 13. Two pages num bered 14. Curling and wrinkled p a g es______ 15. Dissertation contains pages with print at aslant, filmed a s received_________ 16. Other_______________________ .__________________________________ . Text follows. ___________ University Microfilm s International M IC R O W A V E S C A T T E R IN G F R O M T H E O C E A N S U R F A C E A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY By S te p h en Loren D u rd en J u n e 1986 I certify th a t I have read this thesis and th a t in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (Principal Adviser) I certify th a t I have read this thesis and th a t in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify th a t I have read this thesis and th a t in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ul Approved for the University Com m ittee o n ^ i^ d u a te Studies: Dean of G raduate Studi idiefe ABSTRACT Knowledge of microwave scattering from th e ocean surface is necessary for rem ote sensing of ocean surface w inds, waves, and other ocean surface phenom ena. Em pirical models of microwave sc atter from the ocean have been used successfully, b u t are lim ited to the ocean conditions and param eters observed. Here, we investigate the physics of microwave scattering from the ocean for its own sake a n d to understand th e lim itations of em pirical models. T he physics of scattering from rough surfaces such as the ocean has been analyzed using th e sm all perturbation, physical optics, and two-scale ap proxim ations, although th eir ranges of valid application have not been well established. The b est previous calculations of scattering from the ocean were achieved using th e two-scale approxim ation and an ocean surface described by a modified Pierson-M oskowitz spectrum a t sm all wavenumbers an d a spec tru m found from wave ta n k observations a t large wavenumbers. Agreem ent of calculated and observed cross sections is good, b u t surface slope variances calculated from the sp ectru m are much sm aller th a n independently m easured slope variances. In this work we exam ine the validity of th e aforementioned scattering approxim ations by com paring their predictions w ith calculations done using th e m ethod of m om ents, which is an exact num erical m ethod. N ext, a new large wavenumber spectrum for the ocean is proposed. The spectrum has a physical interpretation, and slope variances calculated using it are in good agreem ent w ith observations. Our m om ent m ethod calculations show th a t th e two-scale approxim ation should be accurate for an ocean described by this spectrum , and, in fact, cross section calculations do com pare well w ith observations. Swell is included in the m odel, an d we find th a t its effect on ra d a r m easurem ent of surface winds is m inim ized by use of large incidence angles, high frequencies, and vertical polarization. Finally, th e theory of cross section m odulation by long ocean waves is extended to th e case of two-scale scattering. Calculations of th e m odulation transfer function show th a t th is extension to two-scale scattering is prim arily im p o rtan t for small incidence angles. ACKNOW LEDGM ENTS For by Him were all things created, th a t are in heaven, and th a t are in earth ... C olossians 1:16 K J V T his work is dedicated to the Lord Jesus C hrist, who created the subject m atter of this work and who has provided the tim e and resources for carrying it out. T hanks to Him especially for His guidance, comfort, and fellowship during my tim e a t Stanford. My than k s also to my parents. T heir support and encouragem ent during the various difficulties I encountered in my graduate studies have been of enorm ous help. T his work was carried out a t the Stanford C enter for R adar Astronom y under th e guidance of Prof. John Vesecky. M y thanks to him for his advice on research m atters and on preparation of th is dissertation. I appreciate very m uch his friendship throughout my stay at Stanford. I w ould also like to th an k Profs. Allen Peterson and Joseph Goodm an for their critical reading of this dissertation and Prof. Len Tyler for his helpful com m ents and criticism s during the course of this work. T hanks to Ja n M artin, Dave Hinson, Dick Simpson, Paul Rosen, D onna G resh, Tom Spilker, and Essam M arouf for their comments, questions, and suggestions, particularly as provided a t our W ednesday afternoon m eetings. Finally, I would like to acknowledge the Office of Naval Research for their financial support in the form of an ONR G raduate Fellowship. v Contents 1 In tr o d u c tio n 1.1 1.2 2 3 M otivation 1 ........................................................................................ 1 O v e r v ie w ............................................................................................ 3 A p p r o x im a te A n a ly tic a l S o lu tio n s for R o u g h S u rface S c a t te r in g 5 2.1 In tro d u c tio n ........................................................................................ 5 2.2 Far-Field Equations ........................................................................ 6 2.3 Random S u rfa c e s .............................................................................. 8 2.4 Small P erturbation T h e o r y .......................................................... 10 2.5 Physical O p tic s .................................................................................. 15 2.6 Two-Scale M o d e l.............................................................................. 24 2.7 S u m m a r y ............................................................................................ 28 N u m e r ic a l S o lu tio n fo r R o u g h Su rface S c a tte r in g 30 3.1 In tro d u c tio n ........................................................................................ 30 3.2 M ethod of M o m e n ts ........................................................................ 30 3.3 Scattering from Sinusoidal S u rfaces.............................................. 38 3.4 Scattering from Random S urfaces.................................................. 42 3.5 S u m m a r y .................................................................................... 46 vi 4 V a lid ity o f A n a ly tic a l M e th o d s f o r R o u g h S u rfa c e S c a t t e r in g 5 6 48 4.1 In tro d u c tio n ........................................................................................ 48 4.2 Results for Horizontal P o la riz a tio n ............................................. 49 4.3 Results for Vertical P o larizatio n ................................................... 68 4.4 Application to Two-Dimensional Dielectric S u r f a c e s ............ 84 4.5 S u m m a r y ........................................................................................... 88 W in d S p e e d D e p e n d e n c e o f th e R a d a r C ro ss S e c tio n 90 5.1 In tro d u c tio n ........................................................................................ 90 5.2 Ocean W avenumber S p e c tru m ....................................................... 91 5.3 Two-Scale Scattering from the O c e a n ..............................................103 5.4 D eterm ination of Spectrum P a r a m e t e r s ....................................... 107 5.5 Comparisons w ith D a t a .....................................................................114 5.6 Effect of Atm ospheric S t a b i l i t y ........................................................125 5.7 Effect of S w ell........................................................................................ 127 5.8 S u m m a r y ............................................................................................... 128 C ro ss S e c tio n M o d u la tio n b y L o n g O c e a n W aves 132 6.1 In tro d u c tio n ................................................................................. 132 6.2 Scattering Model for Imaging R a d a r s .............................................. 133 6.3 T ilt M o d u la tio n ..................................................................................... 135 6.4 Hydrodynamic M o d u l a ti o n 6.5 Total M odulation Transfer F u n c t i o n ..............................................147 6 .6 Comparison of Linear and Nonlinear M o d u la tio n ....................... 153 6.7 S u m m a r y ............................................................................................... 157 . 139 7 C on clu sion s and R ecom m en d ation s for F u tu re R esearch 159 7.1 C o n clu sio n s........................................................................................... 159 7.2 Recommendations for Future R ese arch ......................... 162 L ist o f T ab les 3.1 Comparison of scattered field predicted by MOM w ith Zaki and Neureuther’s method for a sinusoidal surface...................... 40 3.2 Efficiencies for scattering from sinusoidal surfaces having an amplitude of 0 .1 A............................................................................... 42 4.1 Comparison of theoretical and calculated ratios of coherent reflected power a t normal incidence to power reflected from a flat surface for horizontal polarization.......................................... 67 4.2 Comparison of theoretical and calculated ratios of coherent reflected power at normal incidence to power reflected from a flat surface for vertical polarization............................................... 5.1 Standard deviation of the RADSCAT wind speed exponents. 82 110 5.2 Comparison of slope variances for several wind speeds................. 110 5.3 Comparison of RADSCAT upwind wind speed exponents with calculated wind speed exponents........................................................I l l 5.4 Comparison of calculated upwind/downwind and up wind/cross wind cross section ratios with those from RADSCAT for a wind speed of 7 m /s ........................................................................................113 5.5 Comparison of calculated upw ind/dow nw ind and upw ind/crossw ind cross section ratio s w ith those from RADSCAT for a wind speed of 13 m /s ........................................................................................114 5.6 Comparison of calculated upwind w ind speed exponents w ith those from NRL aircraft observations................................................ 119 5.7 Comparison of calculated upwind w ind speed exponents w ith those from SKYLAB spaceborne observations a t K u-han& .. . 120 5.8 Comparison of calculated upwind/ downwind and upw ind/crossw ind cross sections w ith RRL observations.................................................125 5.9 Comparison of calculated upw ind/dow nw ind and upw ind/ crosswind cross section ratio s w ith those from NRL observations.................126 x L ist o f F ig u res 2.1 Geometry for the scattering problem............................................ 7 3.1 Effect of windowing on the calculated scattering pattern. . . . 37 3.2 Effect of surface length on scattering p a tte rn ............................. 39 3.3 Estim ated power spectrum for the K ~ 3 ensemble. 44 4 .1 Effect of spectrum enhancement for horizontal polarization for ................ surfaces with 0 jw 0.1........................................................................ 4.2 R adar cross section as a function of angle of incidence for horizontal polarization for surfaces with /? sa 0.1 4.3 40................................ 56 R adar cross section as a function of incidence angle for hori zontal polarization for surfaces with 0 s=s 160.............................. 4.7 54 R adar cross section as a function of incidence angle for hori zontal polarization for surfaces with 0 4.6 53 R adar cross section as a function of incidence angle for hori zontal polarization for surfaces with 0 « 10................................ 4.5 52 R adar cross section as a function of incidence angle for hori zontal polarization for surfaces with /? w 1.................................. 4.4 50 57 RMS difference between the method of moments and two-scale model as a function of K& for horizontal polarization............... xi 58 4.8 Small scale 0 and large scale k for various am plitudes and transition wavenum bers............................................................ 4.9 59 Illustration of error in cross section when 0 is too small. . . . 4.10 RMS error for the two-scale model as a function of 0 an d for horizontal polarization for the K~* surface.................. k 63 4.11 RMS error for the two-scale model as a function of 0 and for horizontal polarization for th e K ~ s surface.................. k 64 4.12 RMS error for the two-scale model as a function of 0 an d for horizontal polarization for the K ~ 2 surface.................. 61 k 65 4.13 Exam ple of need for two-scale division in quasispecular scat tering. . ............................................................................................ 66 4.14 Effect of spectrum enhancem ent for vertical polarization for surfaces w ith 0 « 0 .1 ................................................................ 4.15 R ad ar cross section as a function of incidence angle for 0 f» 69 0 .1 for vertical polarization...................................................................... 70 4.16 R adar cross section as a function of incidence angle for 0 » 1 for vertical polarization...................................................................... 71 4.17 R adar cross section as a function of incidence angle for 0 « 1 for vertical polarization........................................................ 4.18 R adar cross section as a function of incidence angle for 0 « 73 10 for vertical polarization...................................................................... 75 4.19 R adar cross section as a function of incidence angle for 0 as 40 for vertical polarization...................................................................... 76 4.20 R adar cross section as a function of incidence angle for 0 ta 160 for vertical polarization...................................................................... xii 77 4 .2 1 RMS difference between the m ethod of m om ents and two-scale m odel as a function of K& for vertical polarization.................... 78 4.22 RMS error in th e two-scale cross section (TSM) as a function of 0 and re for vertical polarization for the K ~ 4 surface. . . . 79 4.23 RMS error in the two-scale cross section (TSM) as a function of 0 and re for vertical polarization for the K ~ 3 surface. . . . 80 4.24 RMS error in the two-scale cross section (TSM) as a function of 0 and re for vertical polarization for the K ~ 2 surface. . . . 81 4.25 Exam ple of need for two-scale division in quasispecular scat tering for vertical polarization.......................................................... 83 4.26 Calculations of backscattering from a cylinder for horizontal polarization........................................................................................... 85 4.27 Calculations of backscattering from a cylinder for vertical po larization................................................................................................ 86 4.28 Calculations of backscattering from a sphere............................... 87 5.1 Ocean spectrum a t 5 m /s and 5.2 Com parison of calculated cross sections w ith RADSCAT ob 20 m /s ...............................................109 servations................................................................................................... 112 5.3 Variation of observed and calculated cross sections w ith inci dence angle a t L -band ............................................................................116 5.4 V ariation of observed and calculated cross sections w ith inci dence angle at C -band ........................................................................... 117 5.5 V ariation of observed and calculated cross sections w ith inci dence angle a t X -b a n d ................................................................ xiii 118 5.6 Variation of the calculated and observed cross sections with wind speed at normal incidence and X -band....................... 5.7 121 Variation of the calculated and observed cross sections w ith wind speed a t normal incidence and i f u-band................................ 1 2 2 5.8 Variation of the calculated and observed cross sections with wind speed a t 25° and i f u-band......................................................... 123 5.9 Variation of the calculated and observed cross sections with wind speed a t 50° and Jfu-band......................................................... 124 5.10 Effect of swell on wind speed dependence of the cross section at L -band................................................................................................ 129 5.11 Effect of swell on wind speed dependence of the cross section a t i f u-band..............................................................................................130 6 .1 6 .2 Geometry for scattering from a tilted surface................................ 137 Tilt m odulation transfer function for various scattering ap proxim ations...........................................................................................140 6.3 Hydrodynamic m odulation transfer function for various scat tering approxim ations...........................................................................146 6.4 Hydrodynamic M TF as a function of incidence angle for sev eral directions of long wave p ro p ag atio n .........................................148 6.5 Total M T F as a function of incidence angle for several direc tions of long wave propagation............................................. 149 6 .6 Comparison of calculated and measured M TF m ag n itu d es.. . 151 6.7 Comparison of calculated and measured M TF phase..................152 6 .8 Comparison of m odulation transfer function with actual cal culated m odulation for a long wave with slope 0.01....................... 154 xiv 6.9 Comparison of m odulation transfer function w ith actual cal culated m odulation for a long wave w ith slope 0.05. 6 .1 0 , . . . . 155 Comparison of m odulation transfer function w ith actual cal culated m odulation for a long wave with slope xv 0 .1 156 L ist o f S y m b o ls a wavenumber of periodic surface; coefficent in amplitude p a rt of waveheight spectrum An + k cos 6B n b coefficient in am plitude p a rt of waveheight spectrum 6 (n) z component in Bpace harmonic expansion B coefficient in am plitude p a rt of waveheight spectrum Bn m agnitude of n th component in space-harmonic expansion c coefficient in angular p art of waveheight spectrum C (x ,t) autocorrelation function for surface height C2 variance of second derivative of surface Cn nth component of com puter generated surface Cp drag coefficient D ratio of integral of K 2S ( K ) exp(—s K 2) to integral of K 2S { K ) c base of th e natural logarithm E m agnitude of total electric field vector E ( K ,w ) space-time energy spectrum E total electric field vector E* m agnitude of incident electric field E1 incident electric field vector E* m agnitude of scattered electric field E* scattered electric field vector g acceleration due to gravity g* g+7K2 G scalar Green’s function G dyadic Green’s function h surface height relative to the x-y plane h, Fourier transform of the surface height hxtp slope of surface at a specular point hx first derivative of surface w ith respect to x ha second derivative of surface with respect to x H m agnitude of total magnetic field; standard deviation of surface height h H2 height variance of surface Hn n th component of filter transfer function for generating random rough surfaces E total magnetic field vector Hi m agnitude of incident magnetic field H* incident magnetic field vector H* m agnitude of scattered magnetic field H* scattered magnetic field vector Hankel function of the second kind of order Hankel function of the second kind of order one I(p) physical optics integral In coefficient in pulse function expansion of the surface current / square root of negative 1 xvii J surface surrent J o(0 zeroth order Bessel function k m agnitude of k k electrom agnetic wavevector; wavevector of sub-resolution ocean waves kx x component of k kv y com ponent of k K m agnitude of K K wavevector of surface wavenum ber component Kc low wavenum ber cutoff for wavenum ber spectrum Kd transition wavenum ber separating large and sm all scale surfaces of th e two-scale model K{ wavenumber of large scale ocean wave K Ttt wavenumber of sm allest com ponent th a t can be resolved by radar Kt wavenumber w here wind drift layer begins to influence wave breaking Kx x com ponent of K K xm wavenumber in x direction of swell peak Kv y component of K Kym wavenumber in y direction of swell peak Ko sample interval in wavenumber dom ain for com puter generated surfaces L period of periodic surface; rad a r resolution cell size; M onin-Obukhov length m [ K ,K i ) m odulation transfer function between long wave slope and xviii sm all scale spectrum nc index of low wavenumber spectral cutoff K c h un it vector norm al to surface of scatterer Nt num ber of com puter generated surfaces used in scattering calculations p probability density function P(n-) m agnitude of Fourier component of periodic surfaces Pn n th pulse function q speed of flow in wind drift layer Q to ta l energy input/dissipation Qd energy dissipation due to wave breaking Qi energy input to waves through wind action Qn energy input to waves through nonlinear resonant interactions r m agnitude of r r position vector (in three dimensions) of observation point f unit vector in direction of r r' m agnitude of r* r* position vector (in three dimensions) of point on surface of scatterer R m agnitude of r —r*; ratio of cross wind to upwind slope variances; norm al incidence Fresnel reflection coefficient R (K ) m odulation transfer function between long wave height and radar cross section; s coefficient in am plitude p a rt of waveheight spectrum 3 Fourier transform of surface slope S am plitude spectrum ; stan d ard deviation of slope S2 to ta l slope variance Sx stan d ard deviation of slope in the x direction Sv sta n d ard deviation of slope in the y direction T ( K , Ki) m odulation transfer function between long wave height a n d sub-resolution spectrum ' Ta air tem perature Tt sea-surface tem perature u, friction velocity U (x, y, z ) w ind velocity U (x) velocity of ocean surface current w ind speed a t 19.5 m above the ocean surface Wn window function for surface current in num erical calculations x position vector in the x-y plane x u n it vector in the x direction Xn real p a rt of n th com ponent of com puter generated w hite G aussian noise y u n it vector in th e y direction Yn im aginary p a rt of n th com ponent of com puter generated white G aussian noise z height above x-y plane z u n it vector in th e z direction za [ 2 defined by erf z a / 2 = Z ( uj) frequency spectrum Z0 roughness length in planetary boundary layer model (1 —ot)/2 xx a coefficient in small perturbation theory cross section; coefficient of power law representation of cross section /3 4 k 2{h2) 7 ratio of surface tension to w ater density; phase of m odulation transfer function 6 delta function; tilt angle of resolution cell out of the plane of incidence A {hxtp h * )//in Ax spatial sampling interval for com puter generated surfaces e dielectric constant f, param eter in dielectric constant model eo perm ittivity of free space rjo impedance of free space 6 angle of incidence t param eter in dielectric constant model k A2 C 2 A electromagnetic wavelength A wavelength of periodic surface p. coefficient in first order expansion of the wind source/dissipation term Q fio perm eability of free space v wind speed exponent in power law representation of cross section xxi p m agnitude of p p position vector (in two dimensions) of observation point p unit vector in direction of p p' m agnitude of p* pf position vector (in two dimensions) of point on surface of scatterer a radar cross section or0 radar cross section per unit area okx w idth of swell spectrum in x direction crKv w idth of swell spectrum in y direction r param eter in dielectric constant model <f> electric or magnetic field; direction of small scale wave component 4>' incident electric or magnetic field <pB scattered electric or magnetic field $ angle of long wave propagation relative to radar look direction $(<£) directional p a rt of waveheight spectrum ip tilt angle of resolution cell in the plane of incidence; phase of integrand in physical optics integral; stability param eter in boundary layer model ^ (K ) wavenumber sp ectrum x(-, -) joint characteristic function for surface height h lo radian frequency of electric field ft radian frequency of long ocean wave xxii statistical average m agnitude of vector or complex quantitycomplex conjugate xxiii Chapter 1 Introduction 1.1 Motivation An understanding of microwave scattering from th e ocean surface is needed for several areas of application. One of the first applications was rad ar de tection of targets at sea, where rad ar backscatter from th e ocean surface is tre a te d as clutter. M ore recently, emphasis has been on use of ocean backscatter as a tool for sensing of ocean surface winds and waves, as well as other ocean surface phenom ena [l]-[5]. Several future spacecraft systems, notably ER S - 1 (European Space Agency - 1990), NROSS (U.S. Navy - 1991), RADARS AT (Canada - 1990’s), and SIR-B’, C, and D (NASA - late 1980’s), will use rad a rs to remotely sense various ocean surface properties for both research and commercial purposes. Surface properties can also be observed by passive rem ote sensing techniques, in which we observe the natural ra diation em itted by the surface, rath e r th an ra d a r backscatter [3]. Because backscattering and emission from a surface are related, th e theory of mi crowave scattering from the ocean surface also applies to th e case of passive rem ote sensing. In addition to to rem ote sensing of ocean surface properties, there are 1 other applications of microwave scattering from the ocean and rough surfaces. In particular, th is same scattering theory is used in th e design of m arine communication and radar systems [7]. In these sytems th e desired mode of operation is line-of-sight or over-the-horizon propagation via th e troposphere. The presence of the ocean surface creates m ultipath, and in order to properly model m ultipath, an accurate theory of scattering from the ocean surface is needed. Finally, the theory of microwave scattering from rough surfaces can be applied to rad ar remote sensing of terrain [8 ]. Over the past several decades, theoretical research on microwave scatter ing from the ocean surface has focused on electromagnetic scattering from random rough surfaces and wind generation of ocean waves. Although it is not possible to obtain exact analytical solutions to the scattering problem , asymptotic approxim ate solutions have been obtained [1 ,2 ]. However, the limits of validity of these solutions have not been well established. Thus, the accuracy of these approximations for the ocean surface has not been known. As in the scattering problem, there is no exact analytical solution to the problem of ocean wave generation. In fact, there are not even any approxi m ate analytical solutions, so th a t the only existing descriptions of the ocean surface are derived from observations or dimensional analysis [2]. Problems w ith existing descriptions of the ocean surface include lack of obvious phys ical interpretation and disagreement of predicted ocean surface slopes w ith independently observed slopes [54]. Microwave scattering from the ocean is affected by surface winds and by the presence of long ocean waves. These long ocean waves can be detected by high resolution radars, such as synthetic aperture ra d a r [3,4,5]. Although wave-like pattern s are visible in th e images produced by these radars, their 2 exact relation to th e surface waves is not well understood. It is thought th a t the long waves m odulate the radar cross section both by tilting the sub-resolution waves and by changing their height. Theories describing this m odulation have been developed but are limited to incidence angles greater th an about 20° [4]. 1.2 Overview The primary goal of this work is the prediction of the rad ar backscatter cross section for the ocean surface a t microwave frequencies as a function of var ious rad ar and ocean surface param eters. In Chapter 2 we formulate the scattering problem. We then examine asym ptotic approxim ate methods for cross section calculation and use these m ethods to derive the cross section of a surface which is rough in one dimension. We also review the existing cross section calculations for surfaces rough in two dimensions. In Chapter 3 we describe a numerical m ethod for cross section calculation which does not have the restrictions of the asym ptotic approxim ate theories and apply this m ethod to scattering from a surface rough in one dimension. In Chap ter 4 calculations made using th e numerical m ethod for random surfaces are compared w ith calculations m ade using the approxim ate m ethods. We then make conclusions concerning the conditions under which th e asym ptotic ap proximate theories are correct and the conditions under which they break down. W ith the validity of the approxim ate theories determ ined, a specific ap proximate theory, the two-scale model, is used to predict the variation of the cross section with surface wind speed in Chapter 5. We propose a new 3 spectrum to describe waves with large wavenumbers and compare predictions of wind speed dependence of the cross section w ith observations. Finally, the m odulation of the cross section by long ocean waves is examined in Chap ter 6. Here, we extend the previously derived theory to include two-scale scattering. The contributions of this author to research in microwave scattering from the ocean surface are: • Numerical determination of the limits of validity of physical optics and small perturbation theory in the scattering of electromagnetic waves from random rough surfaces. • Numerical determination of the error in the two-scale scattering ap proxim ation as a function of the large scale curvature and small scale height variances. • Development of a comprehensive new model for rad ar backscatter from a wind-driven sea, including proposal of a new spectral form for large wavenumber ocean waves. • Calculation of the effect of swell on windspeed measurements. • Extension of existing long wave m odulation theory to include two-scale scattering. 4 Chapter 2 Approximate Analytical Solutions for Rough Surface Scattering 2.1 Introduction In this chapter we form ally specify the problem of scattering from an arbi tra ry surface. Although exact analytical solutions do not exist, several ap proxim ate m ethods are available for this problem . The two sta n d ard asymp totic approxim ations are physical optics, which is a high frequency m ethod, and small pertu rb atio n theory, which is a low frequency m ethod. In addition, th ere is the two-scale approxim ation, which is a combination of th e physical optics and small p erturbation approxim ations. We use these m ethods to de rive th e rad a r cross section of a perfectly conducting one-dim ensionally rough random surface for comparison w ith the num erical m ethod to be described in C hapter 3. We also review previously derived results for a two-dimensional dielectric surface. These results will be used in Chapters 5 and 6 to analyze th e effects of wind speed variation and long ocean waves on th e rad a r cross section of the ocean surface. 5 2.2 Far-Field Equations The scattering problem may be formulated in term s of the fields (or currents) on the surface of th e scatterer. Once these fields are known, th e scattered fields can be calculated from a vector version of Huygens’ Principle. This relates the fields a t a point r outside a closed surface S to the fields on S and is given in [9] for an eJC"f dependence: ^ ( * 0 = f s j ^ 0W { r ^ ) ^ { r l x H { f ,) ) d S , + f s V x G ( r , f ,) • { h x E { f ) ) d S , ( 2 . 1) J?T(r) = - j ju>eo&(rt ? ) • (» x £ ( f ,))dS' - f V x ^ [ r . f 1) • (n x ^(f'JJcfS ' ( 2 .2 ) where E and H are the electric and magnetic fields on the surface, respec tively, n is the un it norm al to the surface, r* is the position vector of a point on the surface S , u/ is the radian frequency of the fields, Ha is the free space permeability, e0is the free space perm ittivity, G — (I + l/fe2VV)C? is the dyadic Green’sfunction, G = exp{—j k R }/47ri2 is the free space Green’s function, and R is (r —? \, the magnitude of r —r \ The overbars denote vec to r quantities. T he geometry of the problem is illustrated in Fig. 2.1. The surface may have height variation in the x-direction (shown in the figure) and in the y-direction (not shown). In most scattering problems we are interested in the far-field region. For surfaces with linear dimensions X and Y y th e far-field is defined as th at region which is a t a distance R 3> X Y j X [l]. W hen the observation point is in the far-field, the following approximations are valid [9j: G (r,f0 » eikf r e~ikr/ 4 n r (2.3) V G ( r ,r ') « j l t r G ( r ,r ') (2.4) 6 z X Figure 2.1: Geom etry for the scattering problem . T he x -z plane is the plane of incidence. T he y-axis points into the plane of the page. % , ^ ) « ( 7 - r r ) ( ? ( r , r ') (2.5) where r is a u n it vector in the direction of r, r is |r|, and k is the wavevector in the scattering direction. W hen we substitute (2.3)-{2.5) into (2.1) and (2.2), we obtain th e following far-field integrals for the scattered fields [10]: £T(r) = ~ J *C % 4 7 IT x f {-rjof x { h x H ) + h x £)e>WF'<fS' JS W { r ) = ~ J *e ,kT f x (— f x (n x E ) + h x H ) e ’kf ¥'dS' Anr Js t)a (2.6) (2.7) where r)o is th e impedance of free space. For a surface th a t is rough in one dimension only, the above equations simplify considerably. The incident and scattered fields have the sam e po larization, which allows us to trea t the fields as scalar quantities. T he scalar version of (2.1) and (2.2) is: (2-8) 7 where <f>is the electric field for horizontal polarization and the magnetic field for vertical polarization, G(p,J?) — ( j / 4 ) H ^ ( k \ p —p'\) is the two-dimensional G reen’s function, H q2\ x) is the Hankel function of the second kind of order zero, p is the position vector of the observation point in a cylindrical coordi n ate system, and p1 is th e position vector of a point on the surface, n is the outw ard pointing surface norm al defined by: A= S )/(l + (2.9) We have replaced the element of differential area dS' by th e element of dif ferential arc dV.Although the integral is in fact over the entire surface, the integrand has no variation in the y-direction, so th a t the integration becomes one-dimensional. In the far field approxim ation the Green’s function is: G (p.p') « J yOTTKp e-(j3*/<+jMej W (2.10) W hen (2.10) is substituted into (2.8) we arrive a t the following integral for the scattered field in the far-field region: & » • * * - M i* * '* (2 ' n ) Using (2.6) and (2.7) for a surface rough in two dimensions and (2.11) for a surface rough in one dimension we can calculate the scattered fields given the total fields on the surface. The approxim ate m ethods are devoted to approxim ate determ ination of these surface fields. 2.3 Random Surfaces We model the random rough surface as a two-dimensional Gaussian random process h(a:,y), where h is the deviation of the surface from its m ean height. T h e autocorrelation function is defined by C (x, y) = (fc(x\ y’)h(x* + x, y' + y)) (2.12) T h e wavenum ber spectrum for height fiuctuations (also called the waveheight spectrum in th e case of the ocean) is defined by: V ( K x t K v) = - ^ i f f " C { x yy ) e - ^ K^ +K^ d x d y (2.13) If th e surface is rough in only one dim ension, then the above definitions are replaced by: C(x) = (h(x')A(x' + x)) *(«■) = ^ /_” C ( x ) e - * * d x (2.14) (2.15) Various statistics of the surface can be determ ined from the wavenumber spectrum . T he height variance (for the two-dimensional case) is: 9 { K x, K v) d K t d K v / £ (2.16) T h e variances of th e x and y derivatives are: ( { 0 ) } = f (217) C and ( ( P ) ) =I SZK «"HKi'K')iKidK" (218) In th e case of a surface rough in one dim ension the variances of the surface height and its derivatives are: = H K 2nV { K ) d K J —QQ (2.19) T h e existence of these expressions for th e variances in term s of th e power spectrum m eans th a t for th e purpose of scattering, the ocean surface is com pletely specified by its power spectrum . 9 2.4 Small Perturbation Theory The small perturbation theory (SPT) {11]-[16] applies to surfaces which are slightly rough relative to the electromagnetic wavelength. In this section we apply this theory to the determ ination of the radar cross section of a perfectly conducting surface which is rough in one dimension. We begin by considering a horizontally polarized plane wave impinging on a finite length L (in the x-direction) of the rough surface. The mean of the rough surface is the x-y plane, and the plane of incidence is the x-z plane. After finding the scattered field and cross section, we will let L —►oo. Since we are dealing w ith a finite length L of h(x), we can assume th a t A(x) is periodic and expand it in a Fourier series: M*) = £ P{n)e’anx n—^oo (2.20) where a = 2itfL. W hen L —* oo, this sum is replaced by a Fourier-Stieltjes integral. The P ( n ) ’s are related to th e wavenumber spectrum SP(iif) by the following expression: 7 <P(m )P(ny> (»)*> H = { I„ :'f(a n ) m = n m ^n [2.21) where * indicates complex conjugate. The incident electric field (of unit magnitude) is represented by: Ei = (2.22) where k T — k sin &>kt = k sin 0, and 9 is th e angle of incidence. The geometry of the problem is the same as in Fig. 2.1, except th at there is now no height variation of the surface in the y-direction. The observation point r is also the 10 source, since we are analyzing the backscattering case. T he reflected wave (the com ponent in the specular direction) is then represented by: E r = ~ e~i{klX+kMM) (2.23) For a periodic surface the scattered field divided by the incident field is also periodic. We can thus represent th e scattered field by a space harm onic expansion. E* - £ B ne~i{kl+an)x^ih[n)z n=—oo (2.24) where 6(n) = (k 2 — (kx 4* a n )2)1/ 2. T he total electric field is then: E { x , z) = s in k t z + ]T (2 .25) Since the surface is a perfect conductor, the tangential electric field a t the surface m ust be zero. Thus, 2 j e - ’k*x sin k xh{x) + £ = 0 (2.26) This equation cannot in general be solved for the Bn s. Instead, we expand th e B n ’s in a perturbation expansion where th e expansion param eter is the height of the surface: = + + - (2.27) We su b stitu te th is expansion into (2.26) and keep only the first order term s in the surface height h[x). This requires th a t the surface be slightly rough. The result is th e following first order boundary condition: e~’k' x £ B ^ e ’™1 = 2je~ik*‘kxh{x) rt=—OO 11 (2.28) Substituting (2.20) into (2.28) and equating coefficients yields: B™ = 2j k xP[n) (2.29) T he harm onic expansion of E * expresses the scattered field in a discrete set of directions. In fact, we are interested in the power radiated into an infinitessimal element of arc dO. We use the above expansion to find the field on th e surface z = 0. We then use this in (2.11) to find th e scattered field as a continuous function of 6 when L —*■oo. For the surface 2 = 0, (2.11) reduces to the following when 4>— E*: e -j(3 )r f i + k p ) ™ f L/2 { Q E * , \ = e (230) S ubstituting (2.24) into this equation yields: Ea= >/87Tkp s~?<x>^ n= f L/2 e~mi+an)x'dx' J-Lf2 (2.31.) where A n = (I7n6(n) + k cos 8B n) T his simplifies to: j e -i(3*/4+M E ~ ~ Jtokfi sra((2fc* + a n ) L /2 ) AnL {2kc + a n ) L / 2 . (2>32) T he rad a r cross section per unit surface area is defined by: o° = lim L-oo (2 . 3 3 ) L T he average of the field m agnitude squared is: sin((2fcg + a m ) L f2 ) sin((2fca + an )Lf2 ) (2kx + a m ) L / 2 (2 fc*■+• a n ) L f 2 (2.34) Using (2.21) and (2,29) we find th a t: 87r/:3(cos3 0 / £ ) ^ (am ) {AmA'n) = - m = n *(&J(n ) + 2fc cos 8bin ) + k2 cos* 9) 0 (2.35) m ^ n 12 Using (2.34) and (2.35) in (2.33) we obtain th e following expression for the cross section: 2 tt o° — lim fc cos2 6 Y . m jL) L-*°° m L , i ^ sin s((2fc* + am )L /2 ) . ( 6 » + 2fcco3 Ob{n) + k cos t>)L ((2^ + a m )£ /2 )i . . (2 -36) As Xr —►oo we replace th e sum m ation by an integral and 27rm f L = am by the continuous variable p. Using the identity [14]: we find th a t: o° — 2nk cos2 0 f J —DO 'Jf(p)(62(n) + 2fccos06(n.) + k 2 cos2 Q)6(2kx + p)dp = 27r/:cos2 0\P(—2fcx)4/:2 cos2 0 (2.38) T hus, we have the final result th at: (8) = 87rA3 cos4 8^i (—2fc sin 0) (2.39) This is the cross section p er u n it area (norm alized rad ar cross section) of a perfectly conducting surface th a t is rough in one dimension for horizontal polarization. T he derivation of the vertical polarization cross section is very sim ilar to the horizontal case except th a t now we deal w ith the m agnetic field rather th a n the electric field. T he incident m agnetic field is given by: TV = (2.40) T he to tal m agnetic field is the sum of th e incident, reflected, an d scattered fields: H { x yz) *= 2e~ik*x cos k . z + ]T n 13 (2.41) T h e to tal magnetic field on the surface m ust satisfy: H - 0 (2.42) We follow the sam e procedure as in th e horizontal case and su b stitu te the su m of the incident, reflected, and scattered fields into (2.42). We then use th e perturbation th eo ry to find the coefficients of the harm onic expansion. We find th at: ~ ak*n )P (n ) B ”l) = (2*43) Substitution of th e harm onic expansion w ith these coefficients into the farfield equation (2.11) and averaging yields th e following cross section: Oy = 8?r/:3( l + sin2 8)2^ { - 2 k sin $) (2*44) T his is the cross section per unit area for a perfectly conducting random rough surface rough in one dimension for vertical polarization. W hile the cross sections for one-dim ensionally rough surfaces given in (2.39) and (2.44) h a d no t been derived in th e literature before, derivations of the cross sections for dielectric surfaces rough in two dimensions have been presented several tim es (e.g., [14]). Thus, we m erely state them here w ithout derivation. The cross section for P-poIarized send and P-poIarized receive is: o%p = 16n-A:4cos4 5|o:pp|2lJf(—2A:sin0,O) (2.45) For horizontal polarization (P = H): aHH (cos 6 + (e —sin3 0)1/ 2)2 (2.46) For vertical polarization (P = V): „ ^ = ( e ~ l ) ( e s in 2g + ( e - s i n 2 fl)) (ecos 6 + (e - sin2 0)1/ 2)2 14 * . ' T he p aram eter e is th e dielectric constant. For a perfect conductor e = —joo and a h as the following values: OtRR ~ 1 (1 -t- sin2 6)2 a v v --------T hus, in th e case of a perfect conductor, the two-dimensional cross sections have th e sam e angular dependence as the one-dimensional cross sections and can be obtained by m ultiplying th e one-dimensional cross sections by 2k. In b o th th e one-dim ensional and two-dimensional cases the small perturbation theory results in a cross section which depends on th e wavenum ber spectrum evaluated a t a single wavenum ber. This wavenumber is th e wavenum ber for which p a th s from adjacent crests have a phase difference of 2tt. This is the Bragg condition (in analogy w ith Bragg diffraction of x-rays from crystals), and th e scattering m echanism is called Bragg scattering. As m entioned a t the beginning of this section, SPT should be valid only for surfaces th a t are slightly rough com pared to th e electromagnetic w avelength. To m easure the surface roughness, we define a param eter (3 = 4 k2(h2). T h e sm all p ertu rb atio n theory is then rigorously valid only when f3 approaches zero. 2.5 Physical Optics In physical optics th e unknown fields at a point on th e surface are assumed to be th e fields which would exist if the surface were replaced by an infinite plane tan g en t to th e original surface at th a t point [14,17]. This m ethod is also called th e tangent-plane or Kirchhoff m ethod. For a one-dimensional 15 surface and horizontal polarization the electric field on the surface is zero, so (2.11) reduces to: e-j(s*j4+kP) r E ‘M — j s z r f * i * * ' de {2Aa) T he physical optics approxim ation replaces the derivative of th e electric field by: H « 2/Arp ■t ie ’** ? on (2.49) Substitution of (2.49) into (2.48) yields: « •- w For vertical polarization (2.11) reduces to: ■ike~i(s*f4+kri r .. . 1 * > • « * * '* <2-51> T he physical optics approxim ation is: H - 2JT = 2e}k>? (2.52) S ubstitution of (2.52) into (2.51) yields: H ' = / > '» ’«"»» * We see from (2.50) and (2.53) th a t for backscattering physical optics predicts identical fields for horizontal and vertical polarizations. As in the p ertu rb atio n theory, we apply (2.50) to th e case of a surface of finite length L. We th en let L —*■oo to get the cross section p er u n it area of an infinite surface. For a surface of length X, the ensemble average of the m agnitude squared of th e scattered electric field is: le r fL/2 I X{2k-' 16 (2.54) w here x(2fc*, —2kx) = (e~i2k^ x^~h^x"^) is th e joint characteristic function for h { x , y). For a Gaussian random surface x(2kz,2fc*) = exp{—4fcf.ff2(l — C (x ' —x"))} where C(x) is the autocorrelation function and H 2 is th e height variance {h 2). Letting u = x 1—x", we find th a t (E ' E '*) = ------——r— f L {L - lu |)c -4fc;ffa(1- c M>ei2t‘udu 27 iy»cos* u J - l (2.55) L etting L —* oo, as in (2.33), the ra d a r backscattering cross section per unit area is: o°(0) = J L - H (2.56) COS v J —oo This is a general expression for the backscattering cross section using the physical optics approxim ation. T here axe two cases in which we can obtain closed form expressions for the cross section. For a very rough surface {Ak\H2 l) the integral in (2.56) is dom inated by th e region around u = 0. In this case we may approxim ate C (u) by its Taylor expansion ab o u t the origin: 62u 2 C(u) ~ 1 ----- — — |- ■■• z (2.57) where 62 = —C"(0). Substituting th is expression into (2.56) we get: or0 = COS3 9 J -o o e-*k‘H7b7tl^ 2ei2k^ d u (2.58) ' T his integral can be evaluated exactly to yield: <7° = . , ^ 1 V 2 Hbcos3 0 «-■»»■/«■»• (2.59) v 1 We note th a t th e slope variance S 2 for th e surface is —H 2C"(Q), which is equal to H 2b2. Using this fact in (2.59), we get: T his is called the quasispecular cross section because it can be physically interpreted as the cross section due to reflection from p arts o f th e surface norm al to the incident wavevector. T he quasispecular cross section has been derived for two-dimensional surface roughness by first evaluating the re tu rn from the specular points and then averaging over all possible realizations of the surface [18,19,20]. We now use this procedure for a surface rough in one dimension. In this case th e scattered field predicted by physical optics is given in (2.50) o r (2.53). We identify the integral in these equations as J(p). Thus, I{p) = r F {x)eik^ d x J —OO (2.61) where ^ (x ) = 2(x sin 0 —h(x)cosfl) ^ '(x ) = 2sinfl — 2hI (x)cos0 F (x ) = p • n '( l + M x ) 2)1/ 2 We are prim arily interested in values of this integral for large k. To estim ate I{p) as k —*■oo we use th e m ethod of statio n ary phase [21]. C ontributions to I(p) from the vicinity of stationary phase points (x) = 0) a re 0 { k ~ 1^2) while contributions from intervals which have no stationary phase points are 0(A;“l ). T hus, as k —* oo contributions from stationary p h ase points dom inate. Around th e stationary phase points, the phase can b e represented by its Taylor expansion, and the result can be integrated analytically. We find th a t ( 2 -62) 18 and 2ike~i(rl A+kp} E ’ = ^ m r m (2 6 3 ) as k —> cx>. T h e x / s are the points of stationary phase. At these points h x(xj) = ta n 0 , so th a t stationary phase points are also specular points. For a surface w ith Ak2 x H 2 !S> 1, the p a th difference between specular points is random , and th e phase difference will be a uniform random variable which takes on values betw een —n and tt. In this case, th e retu rn s from th e specular points add incoherently. The cross section due to a single specular point is ° = = (2-64) where h xx(xip) is th e second derivative of the surface a t th e specular point. T h e total cross section per un it area is the average of N tim es this cross section: = cos3 0 ( | / i „ ( i „ ) | } (2,6S) where N is the num ber of specular points per u n it length. To find N , we find the probability of a specular point in the interval (xo,xo + dx): P ( h , { x ) - h xtp{x) = 0) for some x in (x0,xo + dx). We expand h xtp in a Taylor expansion about x0: h x{x0 + A) = h x(x0) ■+■has(x0)A(xo) ----R earranging th is expression gives: Aw = ^ r 1 Thus, we desire th e probability th a t A is in (0, dx). (2-66) This is simply the probability density function of A integrated from 0 to dx. 19 If we define h xtp{x) — h x(x) as the random variable u(x), then A is the quotient of the two random variables u and hXI, and its probability density function is [22]: / oo h xx)dhxx (^np (2.67) -00 Assuming th a t the surface height and its derivatives are stationary processes, the density function given in (2.67) is independent of xo* The probability of a specular point in the interval (zo, £o + dx) is the integral of (2.67) from 0 to dx. Since this is a small interval, we approxim ate the integral by the integrand a t A = 0 multiplied by dx. Thus, P{hx h-ztp = 0) w d x f |^ix|phahn(^s»p) hxx)dhxx J “ OO (2.68) But, it is also known th at P ( h x — h xsp = 0) « d x { N ) t so (W) ~ f /-OO l^xi |PAjh„ { h x t p i h z z ) d h xx (2.69) The joint density can be re-w ritten as p(Ax,/iie) = p(h*)p(/ijX|/is) Thus, the num ber of specular points per unit length is ■W— [^**|Pj», (h'up) (2.70) We substitute this expression into (2.65) and write the expectation as an integral over d hxx: ff poo ° ~ coss0 J For a Gaussian surface th e slope density function is: where S is the slope sta n d ard deviation. Since h Xip — ta n 6, th e cross section for a Gaussian random surface (in one dimension) is: a°(B) = \ / ^ - ^ T 7 e~ tan3<VaS3 w V 2 5 cos5 9 (2.72) v ' which is identical to the cross section derived by averaging th e physical optics integral and th en evaluating it, viz. (2.60). We now retu rn to (2.56). It can also be evaluated in th e case of a slightly rough surface, and it is interesting to compare this result w ith th e results of the sm all perturbation theory. When 4k%H2 -C 1 (2.56) can be w ritten as: c r° = “ COS — — COS2 9 J—oo ° ° t1 + 0 J1—oo t *k*n*<Hu)J2kmUdu + • ■■) = — P e,SAlUdu + A k \ H 2 H C(xi)ei2k*udu COS2 0 7-co * J-oo v J (2.73) v ' T he first term in the last line represents the coherent (zero order) return from the surface and is non-zero only in the forward direction. In the back w ard direction it is zero an d will be dropped. The second term is the Fourier transform of the correlation function, which is ju st th e w avenum ber spec tru m . Thus, th e cross section predicted by physical optics for a slightly rough surface is: o° = 87rfc3'4,(—2fcsin0) (2*74) T his cross section is identical to those predicted by S P T , except th a t the angular dependence factors related to polarization are missing. This simi larity has caused some controversy over the years concerning th e relation of physical optics to the sm all perturbation theory [23,24]. In order to investigate the SPT and physical optics approxim ations for a slightly rough surface, we compare the surface currents predicted by each. F irst, we consider horizontal polarization. In this case we have been dealing w ith the electric field; to find the surface current, we need th e m agnetic field. This can be found from the first Maxwell equation: H = V x E (2.75) For physical optics we have from (2.9), (2.20), (2.22), and (2.75): w 2e jfci*(l -i- jfc*/t(aj))(/t*(a;) sinfl + cos0) 0)P (n)cJ'anrj (2.76) For the small p ertu rb atio n theory, using (2.9), (2.20), (2.25), (2.29), and (2.75), we have: 2 jk ze~jk‘x n B nb{n)eianic - i k' T > :' x ^cosfl + T j b ( n ) cos 0P(n)e,ani^ (2.77) The currents given by (2.76) and (2.77) are equal only when all term s b u t the n = 0 term are zero. This corresponds to a flat surface. We also com pare the surface currents for vertical polarization. From (2.20) and (2.40), physical optics predicts: J = 2H* » 2e_jfclI( l + j k Mk(x)) ( 2 . 78 ) 22 For th e small p ertu rb atio n theory, using (2.20), (2.40), and (2.43), we have: J = 8 = 2 cos (kt k{z))e~ikxX + ^ 2 B ^e~j{'kl+an)X^ b{nWx) n « 2e-jfc,s(l + 1/2 = 2e~lk*x ( l + J E ^ rt B„e,'°“ ) ~ fc(!n )t)P(ri)'e,°nZ) (2-79) C om paring (2.78) and (2.79), we see th a t in the vertical polarization case th e physical optics an d SP T surface currents are equal only w hen all term s except n = 0 are zero. T hus, for b o th polarizations physical optics and th e small p erturbation theory predict identical surface currents only in the case of a perfectly flat surface. W hen any roughness, is present the two approxim ations predict different currents. In C hapter 4 we will examine the accuracy of these tw o approxim ations for slightly rough surfaces. Physical optics can also be used to derive the rad a r cross section for a dielectric surface rough in two dimensions. T he two-dimensional equivalent of (2.56) is [17]: = fcg^ f r 7TCOS* 0 J J-oo e~4k‘H2^1~c (x’v^e?2klXdxdy (2.80) where R is the norm al incidence Fresnel reflection coefficient. For the case of a very rough surface (4k*H2 1) th is integral can be evaluated exactly to yield quasispecular scattering: _o _ |-R|2sec4 0 2 SXS„ 8f7Sa where S z and S v are th e slope standard deviations in the x and y directions, respectively. For an isotropic surface (2.80) and (2.81) can be re-w ritten as: °° = [ e~ik’H3^ ~ c ^ J 0{2kxp)pdp cos3 0 Jo 23 (2.82) where Jo is the zeroth order Bessel function, and CTo _ \R\2sec4 0 e_ tan3fl/53 ^ g3j where S 2 is the to tal slope variance of the surface ( S 2 + S^). In order th a t physical optics apply, the surface m ust be gently undulating. As a m easure of undulation, we define k = A2C 2, where A is th e electrom ag netic wavelength and C 2 = (h2x) is the curvature variance. This applies only to surfaces th a t axe rough in one dimension. For surfaces rough in both dimensions, the curvature variance may be greatest in a direction other than th e x-direction. In th is case C 2 = (hi,), where s is the coordinate in the x-y plane in the direction in which C 2 is m axim um . For surfaces rough in one dimension, s is sim ply x, since the x direction is the only direction in which there is curvature. Physical optics is rigorously valid only as k approaches zero. 2.6 Two-Scale Model Few naturally occurring surfaces can be considered to be either slightly rough or gently undulating. For these surfaces the two-scale approxim ation was devised [25]-[32]. In this approxim ation th e surface is divided into a large scale surface height fluctuation hi and a sm all scale surface h eight fluctuation h g. T he to tal height fluctuation is equal to h t + h t . We assum e th a t the two surfaces are independent, allowing th em to be separated in wavenumber space. Thus, can be represented by: V { K ) = tf|(JO + 24 (2.84) where 'Pi and W, are the large and small scale spectra, respectively. They are defined by: * ,( * ) = * ( * ) K < K t * .(K ) = <V(K) K > K d ' where Kd is th e transition wavenumber, separating the large and small scale surfaces. T he total cross section is found by combining quasispecular scat tering from the large scale surface and Bragg scattering from the small scale surface. To do this, we m ust determine the effect of the large scale surface on Bragg scattering and the effect of the sm all scale surface on quasispecular scattering. We have seen th at the quasispecular cross section is simply the sum of the cross sections due to norm al reflection from the specular points on the surface. Now, we wish to calculate the effect of small scale roughness on norm al incidence reflection, in order to understand its effect on reflection from specular points. We have, in fact, already done this using physical optics (Section 2,5). We repeat (2.56) here for convenience: a°{$) = COS3 r e- « ; f f 3(i-c(u»eyfc,udu 8 J - 00 (2.86) This can be re-written as: COS3 W hen 4k \ H 2 8 f°° e4k‘H3cMei2k*udu J-co (2.87) 1, we can proceed as in (2.73). Now, however, we are interested in the coherent component rath er than th e incoherent. This is the first term of (2.87): o°(0) = e"4fc^ 3 f ° ° e*“ -udu J —OO (2-88) 25 where — Ak2H 2. Thus, the effect of the small scale roughness is to m ultiply the coherent component by exp{—/?}. Since the quasispecular cross section is simply the sum of the cross sections for specular points, the quasispecular cross section is m ultiplied by exp{—/J}: a 0( $ ) = e ^ a ° QS{d) (2.89) when small scale roughness is present. Next, we calculate the efffect of the large surface on Bragg scattering. The prim ary effect of this surface is to tilt the Bragg scattering components. The average cross section is ju st the cross section for a tilted slightly rough surface averaged over all possible large scale tilts: — f J -<X> f f B i ^ ^ ) p { h x ) d h t (2.90) where p[hx) is the slope probability density function. This form ulation ne glects the effect of the large scale curvature on small scale scattering. Also, it assumes th a t th e return from th e various portions of the surface can be added incoherently. This should be a good assum ption, since, according to the sm all perturbation theory, fields a t different angles of incidence are proportional to different Fourier components of the surface. Since these components are independent, the fields are independent and can be added incoherently. Because the large and small scale surfaces are independent, the quasispecular scattering and Bragg scattering are independent, and the quasispecular and Bragg cross sections can be added. The two-scale cross section for a one-dimensional perfectly conducting surface is given by the following: o°(0) = e~pOQS (8) + J p(tan i/>) rf(tan ^ ) 26 (2.91) where Oq S is the quasispecular scattering cross section for the large scale surface, is the Bragg cross section for the sm all scale surface, and ip is the tilt of the small scale surface due to the presence of the large scale surface. In the case of two-dimensional roughness we again divided into a large scale, gently undulating surface height fluctuation hi and a small scale surface height fluctuation h t . The total height fluctuation (about the mean) is equal to hi + hB. In the wavenumber domain: = % { K xyK v) + V t ( K X)K v) (2.92) where ^ ( iC * ,! ^ ) = V { K s, K v) K <Kd ^ t {KXiK„) = * (J T „ K ,) K > Kd (2 931 K = (K \ + K*) 1/ 2 and K d is the transition wavenumber. The rad ar cross section due to the large scale surface is found by physical optics, which leads to quasispecular scattering. The cross section for the small scale surface is found using the sm all perturbation theory. Because we m ust average this cross section over large scale tilting, we need th e cross section of a slightly rough surface tilted by an angle ip in the x-direction and <5 in the y-direction. This was derived by Valenzuela [27] and is given by: <rs(®)V'J^) = 167rfc4cot40i|o:(0,^,6)| *'®rJ (2isin(0 + ^),2fccos(0 + ip) sin 5) (2.94) where 0 is the angle of incidence relative to th e x-y plane, 0, is th e local angle of incidence defined by cos 9i — cos(0 + ip) cos 5, k is the electrom ag netic wavenumber, and a is a complex-valued function of the local angle of incidence, the polarization, and the dielectric constant e. 27 T he two-scale cross section for a surface rough in both dimensions is: a°(0) = e x p (-/J)o £ s (0) -f f J p f ta n ip, ta n g )d (ta n ^ )d (ta n 6) cos ip cos o (2.95) where p (ta n ^ ,ta n f i) is the slope probability density function for the large scale surface. T his expression has been derived from physical argum ents [28] is identical to the expression for a com posite surface derived by B ahar [31] when th e suface consists of a gently undulating portion and a slightly rough portion. In order th a t the two-scale cross section be accurate, th e two-scale di vision m ust be done so th a t quasispecular scattering from th e large scale surface and Bragg scattering from th e small scale surface are correct. It has been suggested in the literature th a t the separation wavenum ber Kd be chosen to satisfy large scale criteria [29] and sm all scale criteria [30]. For the two-scale m odel, we require th a t Kd be chosen correctly. 2.7 Summary We have seen th a t scattering by an arbitrary surface can be form ally specified by th e far-field equations, which relate the scattered field to th e fields (or currents) on th e surface. In th e case of a surface rough In only one dimension, there is no depolarization, and the vector far-field equations reduce to a scalar equation. T he sm all perturbation theory was used to find the cross section for a perfectly conducting surface, rough in one dimension. T he results for the two-dimensional dielectric case were stated. In b oth cases th e cross section depended on the wavenumber spectrum evaluated at the Bragg wavenumber 2fcsin0. 28 Physical optics was then applied to the one-dimensional case and cross sections for both very rough and slightly rough surfaces were derived. For very rough surfaces, the quasispecular cross section was derived and was shown to correspond to scattering from specular points on the surface. For slightly rough surfaces, physical optics yields a Bragg scattering cross sec tion. However, this cross section lacks the angular dependence factors (and thus polarization difference) found in the small perturbation theory results. Furtherm ore, the surface currents predicted by the physical optics and small perturbation theories agree only in the case of a perfectly flat surface. Lim its on the validity of these approximations are not known, although a small curvature seems necessary for physical optics, and a small height variance for the small perturbation theory. Finally, we examined the two-scale approxim ation, in which quasispecular scattering from th e large scale portion of the surface and Bragg scattering from the small scale portion of the surface are combined. In the wavenumber dom ain the transition point between the large and small scale surfaces is Kj. T he two-scale model should be valid when Kd is chosen properly. 29 Chapter 3 Numerical Solution for Rough Surface Scattering 3.1 Introduction An alternative to the approxim ate analytical m ethods discussed in th e pre vious chapter is numerical com putation of the rad ar cross section using the m ethod of m oments. This m ethod has the advantage of not requiring as sum ptions about the surface such as small roughness or gentle undulation. The price paid for this advantage, however, is a large com putational require m ent. Although numerical solution is theoretically feasible for a surface rough in two dimensions, we present only one-dimensional calculations be cause of com putational lim itations. 3.2 Method of Moments In this section we summarize the m ethod of moments for calculating the radar cross section of an arbitrary perfectly conducting surface which is rough in one dimension only. We will follow the treatm ent of Lentz [33], In the case 30 of horizontal polarization, the incident electric field is represented by: E \ x ) = e->(**iv0 -*cose) (3 .1.) The geometry is the same as th a t used in Chapter 2 (Fig. 2.1), so the electric field is pointing in the y-direction. From (2.8) the scattered electric field is given by: E- = - t f /4 ) ( X ' W p - p 'I ) ^ ' (3.2) where d E j d h is th e normal derivative of the total electric field evaluated at th e surface and S is the surface profile. From Maxwell’s equations d E j d h = —jkrioh x H . Since we are dealing w ith a perfectly conducting surface, the boundary condition requires th a t h x H be equal to the surface current J. Thus, E‘ = (3 .3 ) On the surface E* + E* = 0. This results in the following integral equation for J(p ): E 1= - J c J & ) H ™ ( k \ p - Tf)dt (3.4) We can re-write this integral equation in term s of x rather th an the position vector along the surface profile: £ ’(*) = 4 J —L j 2 J W H ^ i k R X l + ( M z ') ) 2) ^ (3.5) where R = {(x —x')2 + (h(x) —h(x'))2}1/ 2 is the distance between the obser vation point and a point on the surface. There is no exact analytical solution to (3.5). However, it can be solved numerically using the m ethod of moments. The first step is to expand the unknown, J (x ), in a set of pulse functions: J(x) = j t . I » p (x - z » ) n=l 31 (3-6) where x n = n A x and P (x ) is the unit pulse function of width Ax: - {0 M ile rw i^ 2 We substitute this expansion into (3.5): £*(*) = ^ X ',!» [ B P W K 1 + ( A .M ) !)dx' 4 n—1 J**n (3.8) The integration interval A xn is centered at xn and has w idth Ax. This is now one equation in N unknowns. We generate N equations by enforcing (3.8) at the xn’s. This converts (3.8) to a m atrix equation of the form [Z][I\ = [K] (3.9) The elements of [Z] are given by: z mn= ^ f H ^ ( k R m) - ( i + (hI {x‘) y y i 2dx' 4 J A*„ (s.io ) where Rm = {(xm - xf)2 + (A(xm) - /t(x'))2}1/2. [I] and [V] are given by I n = J(x „ ) and Vm = E* (xm) . Once [/] is found, the backscattered field can be found by substituting (3.6) into the far-field expression found in Chapter 2: r~i(kp+3r/*) = kri°—~ ^ S7rkp + r€3H^tin«+h(xn)coB»)A x (3 For vertical polarization it is m ore convenient to work with the magnetic field. In this case we replace (3.1) - (3.11) by th e following: JP (x ) = J(x ) (3 . 1 2 ) j k f L/* -L/2 n H[2){kR) R \h[x) —h(x') — h x(x')(x — x ’)]dx' 32 (3.13) m —n 1 /2 = < a S l^ ln W _ fc(l.) (3.14) —h e(x')(xm — x')]dx' m ^n p—Hkp+ZrH) H*(p,6) = fe ^ - ^ - - ^ J ^ c o s fta n " 1 hx{xn) + # )(l -f- (M®**))2) 1 . eji(*n»in«+M*«))Aa; (3.15) In the integral equation and in the integral in the expression for Zmn> the neighborhood around the point x' = x is deleted. C om puter codes which implement these equations for a sinusoidal surface were w ritten by Lentz [33]. Starting w ith these codes, we developed codes for arb itrary surfaces specified by the surface height a t points separated by A x. We created two versions of th e code for each polarization. In the first version the integrals required in th e calculation of the Z m atrix elements are approxim ated by assum ing th a t the integrand is constant over th e interval A x. T hus, in the case of horizontal polarization (called transverse magnetic or TM in [33]) th e m atrix coefficients axe approxim ated by: Zm = + ( M * » ) n ,/2A * (3.16) where Rm„ = {(xm — x„)2 + (hfa:,*) — ^(^r*))2}1^2- W hen m = n a small argum ent approxim ation is m ade to the Hankel function. In this case = k~ f A , H P ( * £ ) (3.17) w here e is the base of the natural logarithm . For vertical polarization (called transverse electric or T E in [33]) the m atrix coefficients are approxim ated 33 When the integrals are evaluated in this approximate m anner, the height of the surface is needed only at th e center of the pulse functions. These are the surface heights specified as input to the programs. We will refer to these two codes as T M l and TE1. T he second set of codes do not use the above approxim ation. Instead, the integrals are evaluated by Gaussian Q uadrature. Since this requires an analytical description of the surface, we use linear interpolation to approx im ate the height of the surface between the specified points. In this case we consider th e specified points to represent the endpoints of the segments. Thus, the height of the surface is given by: - * ) (3.19) h{x) ~ i ( j _ Xn) h (* „) + x > Xn The codes which use this piecewise linear representation of the surface will be refered to as TM2 and TE2. T he codes we use do not solve (3.9) directly. Instead, to reduce com puta tional tim e, the surface current is calculated only at every other point along the surface. The continuity of th e surface current allows the current at the in-between points to be calculated by interpolation. Given I n- i and I n+i we calculate I n from: (3.20) In order to find J„_i and I n+i we substitute (3.20) into (3.9). We assume th a t the original system in (3.9) was N X N with N odd. The new system is now of order ( N — l) /2 = M . T he new m atrix elements are: Z mn = + £ ^(2m-l),{2n) for th e interior columns (m — 1 ,2 , ...M and n = 2 ,3 , Z ml = Z (2m-l),l + ^ z (2m-i),2 (3.21) M — 1), (3 .2 2 ) for the first colum n and Z m,M - (3.23) for th e last colum n. T he incident field vector has elements V ^ = V2 m- i , m = T hus, we solve the system [^'][7'] = [V'] for [/']. T hen we find [/] from: h i -1 = / ; for j ~ 1,2, (3.24) and hi = \ ( I ' i + I'i+ l) (3-25) for j = 1 ,2 ,..., M —1. Because interpolation reduces the m atrix order by onehalf, it greatly decreases com putation tim e, and is, therefore, used exclusively in our calculations. T he last num erical code which we will describe is based on the physical optics approxim ation and is refered to as PO . In C hapter 2 we used physical optics to find an analytical expression for the rad ar cross section for two asym ptotic cases. W hen we do not m ake these asym ptotic approxim ations, no analytical solutions exist for even the physical optics approxim ation, so we evaluate th e resulting integral numerically. The PO code has a structure th a t is sim ilar to the m ethod of m om ents codes. However, instead of solving 35 an integral equation for the surface current, it finds the surface current from the physical optics approximation: 7 - 2h x W (3.26) Once the surface current is calculated in this m anner, PO proceeds in exactly the same m anner as the m oment m ethod codes. The problem of estim ating the cross section of an infinite surface from a finite length sam ple of the surface is analogous to the problem of power spec tru m estim ation for a function when only a finite length sample is available. It is also sim ilar to the problem of sidelobe supression in antenna arrays. In b oth cases a window or weighting function other th an the rectangle function is norm ally used [34]. This has the advantage of reducing sidelobe levels, at the expense of broadening the mainlobe. In our case we desire to window the surface current in order to get a better estim ate of the cross section for an in finite surface. This reduces sldelobes from specular component, which m ight obscure the much weaker backscatter component. For all codes m entioned above, the surface current from —L f 2 to L j 2 is m ultiplied by a Hanning window: Wn = sin2((n — l)7r/JV) (3.27) for n — 1,2, ...,1V. This window has sidelobes of —32 dB as compared w ith the rectangle which has sidelobes of —13 dB [35], Figure 3,1 shows a compar ison of the scattering patterns for a flat surface using the rectangular window and the Hanning window. In order to get the best estim ate of the scattered field from an infinite surface, a finite surface which is as long as possible should be used. However, because th e com putational cost of m atrix factorization grows as N z (N is 36 o tI I I, It O I | i I t L '* h i \ • H < ? ' , I* ? F i > M v" * * I o ep_|----------------,----------------1--------------- 1--------------- 1--------------- 1--------------60 BO 10 0 120 140 160 1B0 SCATTERING ANGLE (DEG.) Figure 3.1: Scattered field for a flat surface when the current is m ultiplied by a rectangular window (—) an d when the current is m ultiplied by a Hanning window (-------- ). 37 th e m atrix order), a surface which is as sh o rt as possible should be used. Fig. 3.2 shows the field scattered from a flat surface for various surface lengths. Increasing th e length of the surface from 12A to 24A significantly decreases the m ainlobe w idth. However, a further increase from 24A to 36A shows a sm aller decrease. Because of the rapid increase in com putation tim e with surface length, we use a surface length of 24A in all our com putations. This is approxim ately the same as th a t used in [36]. 3.3 Scattering from Sinusoidal Surfaces There has been m uch work over the p a st few decades on scattering from si nusoidal surfaces. T he p erturbation an d physical optics m ethods developed in C hapter 2 have been used to solve this problem [16,17], and several nu m erical m ethods have been developed [37]-[40], In this section we present calculations for sinusoidal surfaces as a check on the accuracy of th e MOM codes we have described. We consider scattering from the surface h[x) — H sin ax, w here a — 27t/A and A is the period. We recall from Section 2.4 th a t the field scattered from a periodic surface divided by the incident field m ust be periodic in the xdirection. T his allows us to represent th e scattered field as a space harmonic expansion: < £ '(x ,a)= OO £ (3.28) n = —oo where 6(n) = (ft3 — (kx + a n )2)1/ 3. T his expansion is a set of plane waves travelling in directions satisfying the grating equation: sin = sin 0, + 38 (3.29) O o- SCATTERED FIELD (dB) CM t» CD 90 100 110 120 130 140 150 SCATTERING ANGLE (DEG.) Figure 3.2: Scattered field for a fiat surface of length 12A (— ), of length 24A (-------- ), and of length 36A (—• —). 39 TM1 (Ax = .2) (Ax = .1) (Ax = .075) TM2 (A x = .2) (Ax = .1) (Ax = .075) TE1 (Ax = .2) (Ax = .1) (Ax = .075) TE2 (Ax = .2) (Ax = .1) (Ax = .075) 0.17 0.62 0.70 3.17 0.79 0.98 4.11 0.81 0.37 8.53 1.23 0.02 Table 3.1: Difference in dB between scattered field (n = —1 mode) predicted by MOM and Zaki and Neureuther’s m ethod [37,38] for a sinusoidal surface with height of 0.25A, wavelength of 1.5A, and normal incidence. The n — 0 mode is the specular component, and the grating equation reduces to the law of reflection. Table 3.1 shows the differences in dB between calculations using the MOM codes described in Section 3.2 and th e calculations Zaki and Neureuther [37,38] for a sinusoidal surface with a height of 0.25 A and a wavelength of 1.5A. For this surface, only the n = 0 and n = ± 1 term s in (3.28) are radiat ing; all other term s are evanescent. The values in the table are the differences in the n = —1 components (normalized by the n — 0 components) calculated by the two m ethods. In general, the difference decreases as A x is decreased, w ith the change in difference being rath er large in going from A x = 0 . 2 to Ax = 0.1 and rather small in going from A x = 0.1 to A x = 0.075. In the TM1 case the difference actually increases as A x is decreased. We sus pect th a t the error for Ax = 0.2 nearly cancels other errors either in our calculations or in Zaki and Neureuther’s. As A x is decreased, there is less 40 cancellation and the overall difference increases. T he errors present in the TM2, T E 1, and TE2 cases for A x = 0.2 are too large for our purposes; however, th e errors for sm aller A x are quite acceptable, since'm any radars are accurate to w ithin no b e tte r th an 2 dB. Since the com putational tim e in using A x = 0.075 is three tim es th a t for Ax = 0.1, we choose A x = 0.1A for this work. This is identical to th e value used in [36]. A way of testing the codes which does not depend on the accuracy of other m ethods is to calculate the scattered power and com pare it to the incident power. Since power conservation is a necessary b u t n o t sufficient condition for th e accuracy of the scattered field, this test does not prove th e accuracy of the results. However, since th e correct answer m ust conserve power, finding th a t th e scattered power is reasonably close to th e incident power would increase o u r confidence in th e results. Table 3.2 shows the efficiency, which is the scattered power norm alized by the incident power, for a surface w ith H = 0.1A and A — 8.0A and a surface with H — 0.1A,A — .8A, and 8{ = 45°. We see th a t the TM codes b o th produce results which conserve power w ithin 1%. T h e T E codes are less accurate, w ith power being conserved to w ithin 10% for b o th TE2 cases and one of the TE1 cases. For calculations using the rough surfaces to be described in the next section, we found it necessary to use T E 2 for vertical polarization, while TM1 was quite satisfactory for horizontal polarization. T his choice of codes leads to power being conserved w ithin ab o u t 15%. 41 Code TM1 TM 2 TE1 TE2 Surface Wavelength A = 0.8A A = 8.0 A 1.00 1.00 1.00 1.00 0.90 0.87 0.91 0.90 Table 3.2: Efficiencies for scattering from sinusoidal surfaces having an am plitude of O.lA. 3.4 Scattering from Random Surfaces T he num erical m ethod described above calculates the rad ar cross section of an a rb itrary deterministic rough surface. However, we desire to estim ate th e average cross section of a random rough surface. To do this, we create a finite num ber of com puter-generated random surfaces. We th en use m om ent m ethod scattering calculations for these com puter-generated surfaces to es tim ate the average rad ar cross section of th e corresponding random rough surface. A single realization of the random surface can be represented by a Fourier integral. However, since com puter-generated surfaces m ust be of a discrete n atu re, we represent a single realization by an inverse Discrete Fourier T rans form: N- 1 h(z.) = £ CnesnK°Xi (3.30) n=0 where Ko ~ N A x and t ranges from 0 to N — 1. In our case we have N = 256 and A x = 0.1, so th a t Ko = .25. T he transform (3.30) is im plem ented using a 256 point F F T . Because the ocean surface and m any other n a tu ra l surfaces have spectra which fall of as a power law in I f , our com puter-generated surfaces should also have a power law spectrum . T hus, we let the spectrum 42 have the following form: * < « - { r - £ 5 £ . -<“ > where K n = nifoj K e is the low wavenum ber cutoff, and a is a positive constant. T he stan d ard m ethod for generating a random process w ith a desired spectrum is to filter w hite Gaussian noise. We generate the noise and apply the filter in th e wavenum ber domain. T hus, th e C„ s in (3.30) are given by: Cn = ( X „ + ir „ ) iT „ (3.32) where H n is th e filter transfer function, an d X n and Yn are independent G aussian random variables w ith zero mean. T heir variance is chosen so th at th e surface found by using (3.30) has an RMS height of IX. T he height spectrum of the surface is proportional to the m agnitude squared of Cn. For th e spectrum to have the form given in (3.31), H n m ust have th e following form: = { V ' where n c — K c/ K q. Fig. 3.3 shows an estim ate of the spectrum for the ensemble of surfaces w ith a Jif-3 fall off. A lthough the filter transfer function specifies a sharp low wavenum ber cutoff, th e estim ated spectrum resembles a Pierson-M oskowitz ocean spectrum . T he prim ary quantity of interest in rad ar backscattering is th e ra d a r cross section per un it area. For the tth m em ber of th e ensemble, th e cross section for horizontal polarization is: o° = ? f \ E r 43 (3.34) LOG OF SPECTRUM o- co_ -1.5 1 0 -0 .5 0 .5 1 LOG (K) Figure 3.3: Estim ated power spectrum for th e K ~ s ensemble. 44 1.5 For vertical polarization: a° = (3.35) We estim ate the ensemble averaged cross section <t° by the sample average: (3.36) 1=1 where N , is the num ber of surfaces in the ensemble. The param eters which need to be chosen in the numerical technique are A x, L , and N t . As mentioned previously, L is 24A and A x is O.lA. The remaining param eter is the num ber of surfaces N ,. N t was chosen by starting out with N t = 10 and increasing it by 5 until the RMS change in cross section was less than 1 dB. The RMS change in o° is defined by: \/lZ (^ + 5 “ ff° )7 8 where a ° is th e sample averaged cross section using n surfaces. o°n+5 and a° are calculated in the range 0° to 70° in 10° increments. We found th at — 40 should provide an accurate estim ate of the cross section. This is confirmed by calculating confidence intervals for our estimate of the mean cross section. Our estim ate a0 is itself a random variable with m ean a0. Since it is the sum of a large number of independent, identically distributed random variables (the cr,*’s), it is approximately Gaussian (by the Central Limit Theorem). The approxim ate 100(1 — a) confidence interval for the actual m ean a 0 is found from: P { ~ z a/z < (ff° - a ° ) 0 ^ / s d ( a ° ) < zaf2} — 1 - a (3.37) where erf za/ 2 — ( l —a )/2 and sd(o°) is the standard deviation of the o f’s. The confidence interval is thus (o° - zajj s d (a ? )/0 v 7 ,o ° + zaj 2 sA{ai)/ (3.38) Since is approximately exponentially distributed [14], its mean and stan dard deviation are equal. Replacing the standard deviation by the sample average, we obtain: p * (l - ^ (1 + ^ /a /0 v i)) (3.39) The 90% confidence interval is: (o °(l - 1 .6 4 5 /^ /w j), ct°(1 + 1.645 / iJ n ",)) (3.40) For N t = 40 the confidence interval (on a dB scale) is: {a3 - 1.32 dB, a3 + 1.00 dB) (3.41) Thus, the 90% confidence interval is approxim ately 1 dB on either side of our estim ated cross section. This analysis is correct only when no coherent component is present in the return. For normal incidence there will be a coherent p art if th e surface is sm ooth enough. In this case o° has a mean greater than the standard deviation, causing the confidence interval to be smaller than the one given in (3.41). The confidence interval in (3.41) is thus a bound on the confidence interval a t norm al incidence. 3.5 Summary We have examined the use of moment methods in the calculation of scatter ing from arbitrary surfaces and have described two moment m ethod codes for each polarization, as well as a numerical physical optics code. Scattering predicted by the m ethod of moments for sinusoidal surfaces was compared w ith independent calculations and was found to be accurate for a spacing of 46 0.1 A between points on the surface. In th e case of a random surface, a tech nique for estim ating the average backscattering cross section was described. This technique is used in Chapter 4 to calculate the cross section of random surfaces of different amplitudes and spectral slopes. 47 Chapter 4 Validity of Analytical Methods for Rough Surface Scattering 4.1 Introduction As sta te d in Chapter 2 th e asymptotic approxim ate techniques are rigor ously valid only if certain criteria are m et. The validity of th e physical op tics a n d perturbation approximations for one-dimensional determ inistic sinu soidal surfaces has been investigated using th e m ethod of moments [33,37,38] an d using a Bpace harm onic representation [39]. T heir validity has also been investigated for linear-correlated and Gaussian-correlated one-dimensional ran d om surfaces using the m ethod of moments [36,41,42,43,15,44]. In this chapter we present results of numerical experiments used to study th e physical optics, perturbation, and two-scale approximations for surfaces w ith power law spectra. This work is new both in the use of surfaces with power law spectra and in the investigation of the two-scale model. To perform these numerical experiments, we estim ate cross sections using the moment m ethod described in C hapter 3 and compare w ith predictions of th e approx im ate analytical theories which were described in Chapter 2. In this work we do not consider scattering at grazing incidence (incidence angles greater 48 th an 70°) since shadowing and other effects not included in the analytical models may become im portant [13]. 4.2 Results for Horizontal Polarization In this section we describe the results of numerical experiments for horizon tal polarization. We begin by looking a t the validity of Bragg scattering, which is the scattering mechanism predicted by the small perturbation the ory and physical optics for slightly rough surfaces. We used the m ethod of moments to calculate <7° for the J f “3, and K~^ surfaces with /? « 0.1 (RMS height of 0.025A). As stated in Section 2.4, /? = 4k*(h2), so /? « 0.1 represents a very sm ooth surface. We then enhanced these spectra by 10 dB a t 2fcsinS0°, which is the Bragg wavenumber for an incidence angle of 50°. This enhancement was performed by multiplying the spectrum by U(Jf)> which is: K fK ) = / 0.9K b < K < 1.1K b ( 1 + 1 0 si"S \ 1 , , otherwise where K b is the Bragg wavenumber 2A:sin500, Thus, the spectrum is en hanced w ithin ±10% of the Bragg wavenumber. Bragg scattering theory predicts th a t a 0 should be increased by 10 dB at 50°, and it can be seen in Fig. 4.1 th a t this is exactly w hat happened in the m ethod of moments cal culations. Furtherm ore, Bragg theory predicts no change in the cross section for incidence angles corresponding to Bragg wavenumbers w ith no spectral enhancement (0 less th an about 40°,9 greater than about 60°). Again, it can be seen in Fig. 4.1 th a t this is exactly w hat happened in the MOM calcula tions. We also performed experiments in which the spectrum was enhanced a t both half the Bragg wavenumber and twice the Bragg wavenumber and 49 oM _l CO 3 o- 3 o- z O oI N K O LU - t±i CO CO • CO CO CO ' CO o g. oc ? o goc ? CJ c_> 00 00 0 20 40 0 60 INC. ANG. (DEG.) 20 40 60 INC. ANG. (DEG.) (a) (b) CM 00 3 O- CM - 00 0 20 40 60 INC. ANG. (DEG.) (c) Figure 4.1: R ad ar cross section o° as a function of incidence angle 6 for horizontal polarization for surfaces w ith 0 « 0.1 (RMS height of 0.025A). (— ) no ehancem ent of spectrum and (------- ) spectrum enhanced by 10 dB at 2k sin 50°. (a) K ~ A surface, (b) K ~ 3 surface, (c) K ~ 2 surface. Calculations are done a t 10° intervals. 50 found no change in o° a t 50°. This experim ent strongly supports the validity of the B ragg scattering m echanism for slightly rough surfaces. Next, we compare cross sections calculated by th e num erical an d ana lytical m ethods. We use the m ethod of moments (MOMJ to calculate cross sections for the com puter-generated surfaces described in Section 3.4. We then calculate cross sections for these same surfaces w ith the approxim ate m ethods —■sm all perturbation theory (SPT), physical optics w ith num erical evaluation of the far-field integral (PO ), quasispecular theory (QS), and the two-scale theory (TS). Fig. 4.2 shows the cross section as a function of the incidence angle 0 for the various surfaces when 0 & 0.1 (RMS surface height of 0.025A). A t 9 = 0° th e cross section is strongly dom inated by the specular (zeroth order) component. T he S P T cross section at 9 = 0° is zero and is not shown. For $ > 10° SP T and M OM are w ithin about 1 dB at all angles of incidence. P O , on the other hand, is close to M O M only for 9 < 20°, For large angles of incidence, it predicts values which are too large by up to 10 dB. This indicates th a t physical optics is valid only near the specular (in this case vertical) direction, as suggested in [45]. Fig. 4.3 shows M OM and SPT calculations for surfaces w ith 0 « 1 (RMS height of 0.08A). As in the 13 fa 0.1 case, b oth m ethods give very similar results for $ > 10°, This shows th a t S P T is valid for 0 up to 1, in agreement w ith results reported in [15]. Fig. 4.4 shows SPT , T S, and MOM for surfaces w ith 0 « 10 (RMS height of 0.25A). At sm aller incidence angles (less th a n about 20°), there is some deviation between S P T an d MOM (up to 5 dB). TS, however, is in close agreem ent w ith MOM at all incidence angles. Fig. 4.5 shows the MOM, S P T , and TS results for 0 « 40 (RMS height of 0.5A). In this case SPT an d M OM disagree at alm ost all incidence angles (by up to 10 dB). TS, 51 i ---------- r 20 40 INC. ANG. (DEG.) INC. ANG. (DEG.) (b) (a) CO o _ CO O o EC ? CJ o 000 20 40 60 INC. ANG. (DEG.) 00 Figure 4.2: R ad ar cross section a0 as a function of angle of incidence & for horizontal polarization for surfaces w ith (3 ss 0.1 (RMS am plitude of 0.025). (— ) MOM, (--------) S P T , and (------ ) PO . (a) K ~ A surface, (b) K ~ 3 surface. (c) K ~ 2 surface. Calculations are done a t 10° intervals. 52 o_, CM CD OQ S o- 3 oz z o O CM o hO sLU LU <0 ' CO CO ' CO O g- O o o O CO oc ® DC ? o CD C O - 0 0 60 40 20 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (a) (b) CO o- CO CO ' CO O ee g< o co 0 20 40 60 INC. ANG. (DEG.) (c) Figure 4.3: R adar cross section a0 as a function of angle of incidence 0 for horizontal polarization for surfaces w ith ft « 1 (RMS height of 0.08A). (—) MOM and (--------) SPT. (a) K ~ 4 surface, (b) K ~ 3 surface, (c) K ~ 2 surface. Calculations are done at 10° intervals. 53 CM CQ 3 oO oCM - z z o o o LU CO CO 1 CO LU to ' to o O o °*d - gc ■ DC ? u U CO CO 0 40 20 I 60 0 INC. ANG. (DEG.) (a) 20 40 60 INC. ANG. (DEG.) (b) m S o- CO 0 20 40 60 INC. ANG. (DEG.) (c ) Figure 4.4: R adar cross section a 0 as a function of incidence angle 0 for horizontal polarization for surfaces w ith « 10 (RMS height of 0.25A). (—) MOM, (-------- ) SPT , and (------- ) T S. (a) K ~ A surface, (b) iiT-3 surface, (c) K ~ 2 surface. Calculations done a t 10° intervals. 54 however, agrees quite well w ith M OM indicating th e soundness of th e twoscale approach for surfaces th a t cannot be described by SP T alone. W hen p « 160 (RMS height of 1.0A), even TS can differ from MOM by several dB, as seen in Fig. 4.6. We also notice th a t the TS approxim ation becom es worse as th e spectral slope of the surface decreases. T his is expected because both sm all height variance of th e sm all scale surface and small curvature variance of th e large scale surface are required by the T S approxim ation. Because the K ~ 2 surface has a m uch greater curvature variance th a n the K ~4 surface for the sam e height variance, TS is a b e tter approxim ation for the K ~ 4 surface. In order for the TS approxim ation to be accurate, the transition wavenum b er wavenum ber K j , which separates th e large and small scale surfaces (Sec tion 2.6), m ust be chosen properly. In order to b e tter understand th e effect of varying Kd on the cross section, the RM S difference between M OM and TS was calculated for a variety of conditions. T he RMS difference is defined as the square root of the sum of differences squared divided by th e num ber of incidence angles (eight in this case) and is taken to be the overall error incurred in using th e two-scale model. Fig. 4.7 shows the the RMS error as a function of Kd for each surface for four different surface am plitudes. Fig. 4.8 shows th e surface conditions in term s of the sm all scale /? and the large scale k for th e various am plitudes and choices of Kd- In general, sm aller errors te n d to occur for m oderate values of K&. W hen Kd is very large or very sm all, th e errors can be large. W hen Kd is sm all, /? is large, and errors are due to the failure of SP T for the sm all scale portion of th e surface. As Kd is increased, fi decreases and from k k increases. Large errors can then result being so large th a t the large scale portion of the surface is not gently 55 o CM .—. CD *o CO S ©- 2 2 O O o 1— *?" O LU _ CO g . CO ’ CO O g_ cc *? o 1— o LU CO CO CO o CE o o o _ |----------1------------------- i 1----------i i 1— J>A • 0 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (a) 0>) oCM CO 3 ©- 2 O o F «?- o LU _ co 5_ co ' CO o g_ DC ? u co 0 20 40 60 INC. ANG. (DEG.) (c) Figure 4.5: R ad ar cross section a 0 as a function of incidence angle 0 for horizontal polarization for surfaces w ith /? « 40 (RMS height of 0.5A). (— ) MOM, (-------- ) S P T , and (------ ) TS. (a)K ~ * surface, (b) i f ”3 surface, (c) K ~ 2 surface. Calculations done a t 10° increm ents. 56 CO o- o- LU _ CO § CO ' CO O gQC ® to - CO CO 0 20 60 40 0 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (a) (b) CO Q oZ o j— o LU CO CO CO O DC u - ■ o to • oco _ 0 20 40 60 INC. ANG. (DEG.) (<0 Figure 4.6: R adar cross section a0 as a function incidence angle 0 for hor izontal polarization for surfaces w ith 0 ps 160 (RMS height of 1.0A). (—) MOM and (-------- ) TS. (a) K ~ A surface, (b) i f -3 surface, (c) K ~ 2 surface. Calculations done at 10° intervals. 57 10 10 CM - CD CD oc oc *o o r LU O - CO LU (0 - co Kd Kd (a) (b) 10 00 T3 CO CO to - CO nr « - 1 3 5 7 9 Kd (c) Figure 4.7: RMS difference between M OM and TS as a function of K j (in A-1) for horizontal polarization. (—) RMS height of 0.25A, (---------) RMS height of 0.50A, (------- ) RMS height of 1.0A, (--------- ) RMS height of 2.0A. (a) K ~ * surface, (b) K ~ 5 surface, (c) K ~ 3 surface. 58 n o- g o a. < u- T O CD O _i CM _ 4 3 -2 -1 0 I 1 2 -3 0 1 1 LOG O F BETA LOG O F BETA (a) (b) 2 co Q- < u. ° O CD O -i CM _ CO -2 1 0 1 2 3 LOG O F BETA (c) Figure 4.8: Small scale 0 and large scale k for various am plitudes as Kd ranged from 1 to 9* (— ) RMS height of 0.25A, (--------) RMS height of 0.50A, (------ ) RMS height of 1.0A, (--------- ) RMS height of 2.0A. (a) K ~4 surface. (b) K ~ s surface, (c) K ~ 2 surface. 59 undulating. Large errors can also occur even if the small and large scale scattering is accurately described by Bragg and quasispecular.scattering, re spectively. As Kd increases, the smallest incidence angle for which Bragg scattering occurs also increases. If this cutoff angle is too large, quasispec ular scattering may reach very small values before Bragg scattering takes over, resulting in a dip in th e cross section as a function of incidence angle. This is illustrated in Fig. 4.9 for the K ~4 0 rs 10 surface. When this occurs, both scattering mechanisms are quite accurate, b u t the way in which the surface has been divided is incorrect. This is one of the problems in using a two-scale approxim ation for a surface w ith a continuous, monotonically decreasing spectrum . In order to avoid this type of behavior, 0 should be chosen as large as possible, bu t less than about one. As the surface am plitude is increased or as the spectral slope is decreased, the RMS error increases. It is interesting th a t even in cases w ith both large 0 and large rc, there are choices of Kd which yield small errors (e.g., less than 2 dB), However, in these cases the error is a very rapidly varying function of Kd. For smaller 0 and k the error is much less sensitive to variations of Kd. Because Kd is a m athem atical construct and not a physical param eter, the TS model is useful only in cases for which there is a fairly wide range of Kd over which the error remains nearly constant. The d ata shown in Figs. 4.7 and 4.8 were combined to generate RMS error surfaces as a function of 0 and k . Figs. 4.10-4.12 show contour plots of the error for each spectral slope. For the K ~ A surface the largest errors occur for 0 too large or 0 too small. For the K ~ s surface we again see large errors for very small and very large 0. In addition, there are large errors when k takes on very large values. The problem of choosing Kd so th a t 0 is 60 Oo CD X - c/> CD - 00 0 10 20 30 40 60 60 70 INC. ANG. (DEG.) Figure 4.9: Cross section as a function of incidence angle for the surface w ith J3 « 10 for horizontal polarization and Kd = 7. Calculations done a t 10° intervals. Erroneous m inim um a t 30° is th e result of Kd being chosen so large th a t neither quasispecular nor Bragg scattering contribute strongly enough a t 6 « 30°. 61 to o small does not occur for the K ~ 2 surface. In this case the largest errors are for large /? an d large re. The results in these contour plots are in contrast w ith Brow n’s suggestion th a t the choice of Kd be based only on /? [30]. In fact, it appears th a t criteria sim ilar to those of b oth Brown and T yler [29] m ust be considered in choosing K d - In general, Kd should be chosen so th a t (3 and re lie in a region of the contour plot w ith sm all error. So far, we have examined quasispecular scattering only in the context of the two-scale model. Near norm al incidence it is often assumed th a t scattering is dom inated by the quasispecular m echanism [19], Even in this case the two-scale division should be perform ed. If slope variances for the entire surface are used instead of slope variances for th e large scale portion of th e surface, th e QS cross section m ay be in error. This is seen in Fig. 4.13. QS has the wrong level at $ = 0° and the wrong angular dependence for all B, On the other h and, if we perform the two-scale division, b u t neglect the Bragg term in th e two-scale cross section, then th e calculated cross section (QSTS) is w ithin 3 dB of MOM for B less th a n about 30°. For larger 0 Bragg scattering begins to dom inate and QSTS fails. For scattering near norm al incidence, the two-scale division should be performed in th e sam e m anner as the full two-scale m odel, and b oth re and 13 should lie in a region of the TS contour plots w ith sm all error. The need for small re is obvious, since re determ ines th e probability th a t the radius of curvature at a specular point is less th a n some constant [29]. However, th e need for sm all (3 in the case of quasispecular scattering is no t so obvious. T he reason th a t /? m ust also be small is th a t the quasispecular term m ust dom inate the Bragg term in the two-scale cross section. If 62 is too large, 1.0 - CO - 3.0 - 2.0 LOG OF KAPPA O" 4.0 3.0 2.0 1.0 0.0 1.0 2.0 LOG OF BETA Figure 4.10: RMS error for th e two-scale model as a function of 0 and /c for horizontal polarization for th e K ~ 4 surface. 63 -X g CL < # U_ O . ^ tt- .m O 3 CM “ CO 3.0 0.0 1.0 2.0 LOG OF BETA F igure 4.11: RMS error for the two-scale model as a function of 0 and /c for horizontal polarization for the K ~ z surface. 64 iC V : 0 .0 1.0 LOG OF BETA Figure 4.12: RMS error for the two-scale model as a function of j3 and horizontal polarization for the i f - 2 surface. 65 k for CQ T> O o - u LU CO CO CO CC o 0 20 10 30 40 INC. ANG. (DEG.) Figure 4.13: Cross section er° dependence on angle of incidence 6 for QS (—• —), QSTS ( ) t and MOM (—) for th e K ~ 2 surface w ith P « 10. Calculations done a t 1 0 ° intervals. 66 Theoretical (exp(—/?)) 0 K ~4 0.91 0.35 4.13 x 10 “ 3 0.90 0.37 4.5 x 10“ 5 0 .1 1 .0 1 0 .0 MOM K~* 0.91 0.36 4.85 x 10- 3 K ~2 0.89 0.27 4.95 x 10 “ 3 Table 4.1: Com parison of theoretical and calculated ratios of coherent re flected power a t norm al incidence to power reflected from a flat surface for horizontal polarization. Theoretical value is exp(—j9) from physical optics and calculated values are from m ethod of m om ents. th en this will not be th e case, since the e~& factor, which m ultiplies the qua sispecular term , will be near zero (Section 2 .6 ). Physically, this m eans th a t th e specular points m ust be sm ooth enough to have a coherently scattered com ponent. If specular points are too rough, then all retu rn from them is diffuse. In this case specular points no longer have any significance. To test the correctness of the e~& factor, we calculated th e ratio of the coherently reflected power a t norm al incidence to power reflected by a flat surface as a function of 0 . Table 4.1 shows these results. T he theoretical and calculated values generally m atch quite well for /3’s of 0 . 1 and 1.0. For 0 of 10 there is disagreement by two orders of m agnitude. However, b oth the theoretical and calculated values are very m uch less th a n one. T hus, when the sm all scale 0 has a value of 10 or more, there is essentially no quasispecular scattering. In perform ing the two-scale division for quasispecular scattering, th e division m ust be done so th a t the physical optics approxim ation holds an d so th a t there is coherent retu rn from the specular points. 67 4.3 Results for Vertical Polarization In this section we repeat the work in Section 4.2 for the case of vertical polar ization. We begin by showing results of spectrum enhancem ent experiments in Fig. 4.14 . This figure shows th a t enhancement of the spectrum leads to increased cross sections only a t incidence angles w ith Bragg wavenumbers in the enhanced portion of the spectrum (40° < 6 < 60°). This is the same result as in the horizontal polarization case (Fig. 4.1), and confirms th a t Bragg scattering is the physical mechanism for scattering from slightly rough surfaces a t vertical polarization. Next, we compare cross sections calculated by the num erical and analyt ical m ethods. Fig. 4.15 shows S P T , PO , and MOM for vertical polarization for 0 « 0.1. The PO plot, which is identical to the plot for horizontal po larization is close to MOM, differing by about 3 dB a t 70°. For large 6 SPT predicts cross sections that are m uch larger than M OM (by up to 7 dB a t 70°). This is also the case for j 9 w l , as shown in Fig. 4.16. This sam e type of discrepancy has shown up in the comparison of labora tory measurements with SPT. Using a radar looking a t waves in a wind-wave tank, Lee [46] found that the m easured polarization ratio {oyfo%) was sev eral dB sm aller than the ratio calculated by SPT. T his is consistent with our calculations. In an experiment using a radar looking a t an aluminum plate th a t h ad been roughened by a shotgun blast, Axline [41] found th at the vertical polarization SPT cross section was 5 dB too large a t 70°, while the horizontal polarization SPT cross section was very close to measurements. At optical frequencies, measurements of backscatter from metallic surfaces have indicated very little dependence on polarization, in contrast to the large 68 o CM L 3 o - I\ 00 z z o 00 Oo F= V o LU CO g CO ' CO T3 \ \ \ p V u LU CO CO CO V og_ oc ? o o oc o 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (b) (a) CM CO 3 oOo *=?o LU w2<D00 0 20 40 60 INC. ANG. (DEG.) (<0 Figure 4.14: R a d a r cross section a 0 as a function of incidence angle 6 for ver tical polarization for surfaces w ith 0 « 0 . 1 . (—) no ehancem ent of spectrum and (--------) sp ectru m enhanced by 10 dB a t 2 fcsin 50°. (a) K~* surface, (b) K ~ 3 surface, (c) K - '1 surface. C alculations are done a t 10° intervals. 69 i ---------- r 20 40 INC. ANG. (DEG.) INC. ANG. (DEG.) (b) (a) oN z o i— u LU CO nr CO ' CO O £_ oc ? O o 20 40 60 INC. ANG. (DEG.) (c) Figure 4.15: P lot of th e cross section cr° as a function of th e incidence angle $ for vertical polarization for surfaces w ith /? « 0.1. (—) M OM , (-------- ) S P T , and (— • —) P O . (a) K ~ 4 surface, (b) K ~ s surface, (c) K ~ 2 surface. Calculations done a t 10° intervals. 70 O o O o r— n - ®CO 0 20 60 40 0 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (a) (b) CM “ S o- LU co 2 co O g . CO 0 20 40 60 INC. ANG. (DEG.) (c) Figure 4.16: Plot of the cross section o° as a function of the incidence angle 8 for vertical polarization for surfaces w ith « 1 . (—) MOM and (-------- ) SPT . (a) K ~ 4 surface, (b) K ~ 5 surface, (c) K ~ 7 surface. Calculations done at 1 0 ° intervals. 71 polarization difference predicted by S P T [47]. A lthough th e spectrum en hancem ent experim ents described above dem onstrate th e correctness of the Bragg scattering mechanism, th e m echanism predicted by SP T , these exper iments and our calculations indicate th a t SPT predicts an incorrect angular dependence of the cross section for vertical polarization. Because we could find no source of error in the p ertu rb atio n theory, its correctness should be further investigated by laboratory m easurem ent and calculation w ith differ ent num erical codes. We did find th a t we can bring th e SP T cross section into close agreem ent w ith MOM by m ultiplying it by th e em pirically derived factor exp{—0/47.2°}. This factor was found by exam ining calculations for th e and K ~ 2 surfaces for 0 « 0.1. Fig. 4.17 shows the empirically modified SPT (SPTM ) com pared w ith M OM for 0 r# 1 . For all three surfaces there is good agreem ent in b o th an gular dependence and cross section m agnitude. Fig. 4.18 shows th e SPTM , TSM (TS using the modified S P T ), and MOM for 0 a# 10 . As in the hori zontal polarization case, the pertu rb atio n theory (SPTM ) begins to deviate from MOM, although the deviation is somewhat sm aller th a n in the horizon ta l polarization case. Fig, 4.19 shows SPTM , TSM , and MOM for 0 as 40. In this case there is significant disagreem ent between S P T M and MOM (up to 10 dB), particularly a t smaller incidence angles and sm aller spectral slopes. At large incidence angles there is still good agreement, indicating th a t Bragg scattering for vertical polarization is less sensitive to large scale tilting th an for horizontal polarization. This m eans th a t even for fairly rough surfaces, the cross section at large incidence angles is proportional to the surface spec tru m evaluated a t the Bragg wavenum ber, making vertical polarization more suitable for rem ote sensing applications. When 0 « 160 (Fig. 4.20), SPTM 72 o_, wl CO CQ S o- 2 z z o I— Io co co co ' CO o g ee . u co ' CO o- O o(M- O oCM“ LU LU o £_ cc ? o CO 0 60 40 20 0 INC. ANG. (DEG.) 20 40 60 INC. ANG. (DEG.) (a) (b) oCM“ <D~ C0 0 40 20 60 INC. ANG. (DEG.) (c) Figure 4.17: P lo t of the cross section er° as a function of th e incidence angle 6 for vertical polarization for surface w ith 1 . (—) M O M and (-------- ) SPTM . (a) K ~ 4 surface, (b) K ~ s surface, (c) K ~ 2 surface. Calculations done at 1 0 ° intervals. 73 an d MOM disagree at all incidence angles, so th a t averaging over large scale tilts is required for very rough surfaces, such as this one. It can be seen th a t the TSM agrees w ith M OM for p « 160. As in th e case of horizontal polarization (Section 4.2), we calculated the RMS error for the two-scale model for vertical polarization (TSM ). Fig. 4.21 shows these results. T he behavior of th e error for vertical polarization is sim ilar to th a t of horizontal polarization with one exception: the error is n o t large for large K j when the surface am plitude is large. The reason for th is is the lack of sensitivity of Bragg scattering for vertical polarization to large scale tilting in the two-scale m odel. Thus, in m any regions in which the horizontal polarization TS had large error, the TSM for vertical polarization has much sm aller error. Figs. 4.22-4.24 show contour plots of the RMS error for the three sur faces. In all three cases the largest errors are associated w ith very large ft and very small /?. E rrors for very large k are relatively sm all, as a result of the insensitivity of Bragg scattering for vertical polarization to large scale tilting. As m entioned above, this insensitivity makes vertical polarization m ore suitable for rem ote sensing since the cross section is proportional to the waveheight spectrum a t a single Bragg wavenumber. Thus, a set of cross section m easurem ents m ade a t different frequencies or wavenumbers can be directly converted to the ocean waveheight spectrum . In the horizontal po larization case this is not true, since m any wavenum ber com ponents m ay contribute to scattering at a p articular incidence angle through large scale tilting. However, until the disagreem ent between MOM an d SPT for vertical polarization is resolved, it seems best to use horizontal polarization in rem ote sensing applications where analytical scattering model calculations are used 74 o_, CM CO *o o - O o LU 0 V) I CO CO o g- f f l - oco co 0 80 40 20 0 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) 00 (b) CO 0 20 40 60 INC. ANG. (DEG.) 00 Figure 4.18: Plot of the cross section a 0 as a function of the incidence angle 9 for vertical polarization for surfaces w ith (3 « 1 0 . (—) MOM, (-------- ) SPT M , and (------) TSM . (a) K ~ 4 surface, (b) K ~ z surface, (c) K ~ 2 surface. Calculations done a t 10° intervals. 75 o_ CM CM O o hO UJ CO ■<frCO ‘ CO CO o <o- u o CO - 0 g DC ? 20 40 60 o co0 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (a) (b) INC. ANG. (DEG.) (c) Figure 4.19: P lo t of th e cross section a 0 as a function of th e incidence angle 0 for vertical polarization for surfaces w ith j3 « 40). (— ) M OM , (---------) SPTM , and (------ ) TSM . (a) K ~ 4 surface, (b) i f - 3 surface, (c) K ~ 2 surface. Calculations done at 10° intervals. 76 S O- LU L ii <o - CD - V CO 0 60 40 20 0 20 40 60 INC. ANG. (DEG.) INC. ANG. (DEG.) (a) (b) CM LU _ CO 0 _ CO CO 1 Q g . CO 0 20 40 60 INC. ANG. (DEG.) (c) Figure 4.20: Plot of the cross section a 0 as a function of the incidence angle 8 for vertical polarization for surfaces with f3 « 160. (—) MOM, (-------- ) SPTM , and (------) TSM. (a) K ~ 4 surface, (b) K ~ 3 surface, (c) K ~ z surface. Calculations done at 10° intervals. 77 Ifl m *o CO cc _ O ® cc cc O CM_ •o cc QC 111 to - tli CO C/) CO 2 2 cc « CC « - 1 3 7 5 9 5 3 t Kd Kd (a) (b ) 7 9 in CO T3 g CC CC LU co CO 2 o 1 3 6 7 9 Kd 00 Figure 4.21: RMS difference between M OM and TS as a function of Kd for vertical polarization. (—) /? » 10, (-------- ) /? « 40, (------- ) ft ta 160, (----— ) (3 « 640. (a) K ~ 4 surface, (b) K ~ 3 surface, (c) K ~ 3 surface. 78 o - 2.0 - 1.0 LOG OF BETA Figure 4.22: RMS error in the two-scale cross section (TSM) as a function of /? and k for vertical polarization for the K ~ 4 surface. 79 - 1.0 “ - 3.0 - 2.0 LOG OF KAPPA 0 - 3.0 - 2.0 1.0 0.0 1.0 2.0 LOG OF BETA Figure 4.23: RMS error in the two-scale cross section (TSM) as a function of /3 and k for vertical polarization for the K ~ s surface. 80 0 .0 1.0 LOG OF BETA Figure 4.24: RMS error in the two-scale cross section (TSM) as a function of (3 and k for vertical polarization for the K ~ 2 surface. 81 0 0 .1 1 .0 1 0 .0 Theoretical (exp(~/5)) 0.90 0.37 4.5 x 10"B K~* 0.91 0.37 1.70 x 10- 3 MOM K ~3 0.90 0.35 4.85 X 10" 3 K~2 0 .8 8 0.26 4.95 X 10~ 3 Table 4.2: Comparison of theoretical and calculated ratios of coherent re flected power a t normal incidence to power reflected from a flat surface for vertical polarization. Theoretical value is exp(—/?) from physical optics and calculated values are from m ethod of moments. to interpret observational data. O ur conclusions concerning the use of the quasispecular theory are the same for both vertical and horizontal polarization. Fig. 4.25 illustrates the necessity of using the two-scale division even for scattering at incidence angles where the Bragg component may be neglected. Table 4.2 compares the ratio of coherent reflected power at normal incidence to power reflected from a flat surface as calculated by MOM and theory (e~^). The results are very similar to the horizontal polarization case (Table 4.1). We conclude th a t for both horizontal and vertical polarization, scattering near norm al incidence can be well approxim ated by the quasispecular theory. However, in order to correctly calculate slopes for use in the quasispecular theory, a two-scale division of the surface m ust be performed, and this two-scale division should be done in the same m anner as in the full two-scale model (both quasispecular and Bragg scattering). 82 m oUJ C/3 c/> 0 20 10 30 40 INC. ANG. (DEG.) Figure 4.25: Cross section o° dependence on angle of incidence 6 for QS (—■—), QSTS (-------- ), and M OM (—) for the K ~ 2 surface with « 10 for vertical polarization. 83 4.4 Application to Two-Dimensional Dielec tric Surfaces All results derived in this chapter have been for perfectly conducting sur faces rough in one dimension only. In general we are interested in calculat ing the cross section of surfaces rough in b oth dimensions having arbitrary dielectric constant. We thus need to determine how the results presented in Sections 4.2 and 4.3 relate to problems of interest (such as the ocean surface). The criteria for the validity of the analytical theories th a t were found in the numerical experiments were in term s of the height and curvature statistics of the surface. Since these param eters relate only to the geometry of the surface and not the m aterial properties, we would expect th a t our criteria for validity should be independent of the dielectric constant. Thus, th e use of a perfectly conducting surface in our experiments should not be a problem. The use of surfaces rough in one dimension only is a potentially more se rious problem. Since we cannot do numerical calculations for surfaces rough in two dimensions, our only source of guidance concerning the extension of th e results to two-dimensional roughness is the application of asym ptotic m ethods to deterministic objects. In particular, we find th a t low frequency (Rayleigh) and high frequency (geometrical optics) approximations are valid for b oth spheres and cylinders of similar radius [48]. This is illustrated in Figs. 4.26-4.28. For both the cylinders and the sphere the Rayleigh ap proximation begins to break down as the wavenumber-radius product, k R , becomes greater than about 0.5. As k R becomes greater than about 5.0, geometrical optics becomes valid for both the cylinder and sphere. Thus, 84 H O rtCO CO O cc o O CM - U J IM _l < 2 cc o z 0.1 kR Figure 4.26: Cross section norm alized by geometrical cross section for a cylinder as a function of the product of wavenum ber k and radius R for transverse electric polarization [48]. (■—) E xact solution. (---------) Rayleigh approxim ation. (— • —) Geometrical optics approxim ation. 85 LL) CO n - CO CO CC O Q NLU N CC 0.1 kR Figure 4.27: Cross section normalized by geometrical cross section for a cylinder as a function of the product of wavenum ber k and radius R for transverse m agnetic polarization [48]. (—) E xact solution. (--------) Rayleigh approxim ation. (—* —) Geometrical optics approxim ation. 86 co- LLI CO ® “ W a LU *- -1 oi- 0.1 kR Figure 4.28: Cross section normalized by geometrical cross section for a sphere as a function of the product of wavenumber k and radius R [48], (—) Exact solution. (------- ) Rayleigh approximation. (------ ) Geometrical optics approxim ation. 87 th e addition of curvature in a second dimension (changing from cylinder to sphere) does not have m uch effect on th e region of validity for the low frequency and high frequency asym ptotic approxim ations. T his leads us to believe th a t changing from roughness in one dimension to roughness in two dimensions does not substantially affect the results of Sections 4.2 and 4.3. As stated in Section 2.5, for a surface rough in b oth dim ensions, we define k as th e curvature variance in the direction of maximum curvature, i.e. k = X 2C2 (4.2) w here C 2 is the variance of the second derivative of the surface taken in the direction of m axim um curvature. T he param eter /? has no direction associated w ith it and is the same as in th e one-dimensional case, i.e. 0 = 4 k 2 (k 2) 4.5 (4.3) Summary- Using one-dimensional m ethod of moments calculations as a standard, we have exam ined the validity of th e physical optics approxim ation (quasispec ular scattering), the sm all perturbation approxim ation (Bragg scattering), and th e two-scale m odel. We found th a t th e small p ertu rb atio n theory for horizontal polarization is accurate for (3 < 1 for all three surfaces. For 0 larger th an one, the accuracy decreases. For vertical polarization there is a large difference between th e m ethod of m om ents and the sm all perturbation theory even for very sm all /?- W hen this discrepancy is em pirically corrected, the sm all p erturbation theory is accurate for {3 < 1 for all three surfaces, as in th e horizontal polarization case. We also noted th a t for {3 larger than 88 one the empirically modified sm all perturbation theory for vertical polar ization rem ains in b etter agreem ent w ith the m ethod of m om ents th a n the p ertu rbation theory for horizontal polarization because Bragg scattering is less sensitive to large scale tilting in the vertical polarization case. C ontour plots of th e RMS error were derived as a function of (3 for the small scale portion of the surface and K for the large scale portion of the surface, and it was found th a t the validity of the two-scale m odel depends on both /? and and k k. The transition wavenum ber Kd should be chosen so th a t /? lie in a region of these contour plots where th e RMS error is sm all and where th e RM S error is slowly varying w ith respect to 0 and k. We also found th a t quasispecular scattering should be treated as a special case of the twoscale model. Even when quasispecular scattering is th e dom inant scattering mechanism , a two-scale division of the surface should be perform ed. Finally, we discussed the use of the results of this chapter for surfaces th a t are rough in two dimensions and th a t have a rb itrary dielectric constant. We noted th a t th e use of perfectly conducting surfaces should not change our results since they concern only surface geom etry and not m aterial properties. We also presented evidence, derived from determ inistic scatterers, th a t our results should hold for two-dimensional surfaces, provided th a t we define the param eter k to be taken in the direction of m axim um surface curvature. 89 Chapter 5 Wind Speed Dependence of the Radar Cross Section 5.1 Introduction Over the past two decades considerable effort has been devoted to the un derstanding of how the rad ar cross section is related to ocean surface winds. T he two-scale m odel has been assum ed to describe scattering from the ocean surface and has been used in various models [50] - [53] to predict the varia tion of rad a r cross section w ith wind speed and direction. A significant step in this effort was th e work of Fung an d Lee [54]. T heir m odel can account for variations in cross section w ith rad a r frequency, polarization, angle of incidence, and w ind velocity. However, their model is restricted to angles of incidence greater th a n 20°. Also, the slopes predicted by th eir spectrum are significantly sm aller than empirically determ ined slopes (Section 5.4), requiring use of em pirical slopes in the two-scale scattering m odel. Finally, the large wavenumber spectrum used in their model is purely em pirical and has no obvious physical interpretation. In th is chapter we propose a new spectrum for th e w ind-driven ocean surface. This spectrum is based on b o th empirical and theoretical 90 considerations an d does have a physical interpretation. Slope variances cal culated from it are in agreement w ith measured slope variances, allowing use of surface slopes derived from the spectrum rather th a n ad hoe outside measurements. We calculate the rad ar cross section of a surface described by this spectrum using the two-scale scattering model, including quasispec ular scattering so th a t our calculations are valid for the range of incidence angles 0° — 70°. We then compare our model predictions w ith L -t C7-, X-, and i f u-band observations so th a t its validity can be determined. Finally, we use our model to examine the effect of atmospheric stability and swell on the radar cross section. Our objective here is not to create a better empirical model. Instead, our objective is to p u t as much physics as possible into the model so th at areas of disagreement w ith observations can point to areas where physics needs to be better understood and new effects included. 5.2 Ocean Wavenumber Spectrum The ocean surface is a time-varying random rough surface. In the space-time domain, it can be described by a correlation function: C (x ,t) - (h(x’, t ,)h(x, + x ,t>+ 1 )) (5.1) The space-time power spectrum is defined by: E ( K , u ) = {2 n)-* J l ° ° C { x ,t)e -3^ - ^ d x d t (5.2) This spectrum contains much more information than can be measured with current technology, either m situ or remote sensing. 91 The most easily m easured quantity is the frequency spectrum , which is defined by: Z {u ) = / / : E i K ^ d K ' (5.3) From the space-time spectrum we can also calculate the wavenumber spec trum , which is needed for the scattering theories described in Chapter 2 . i-oo E(K,w)dio (5.4) This spectrum is more difficult to measure th an the frequency spectrum , since it requires m easurement of the instantaneous surface height as a function of position, asopposed to local surface height as a function of tim e [55], If there cj is a dispersion relation between frequency and wavenumber, = w(iif), then it is possible to convert the frequency spectrum to the wavenumber spectrum . We assume th a t the wavenumber spectrum can be separated into an am plitude spectrum and a directional spectrum : *(«■) = (5.5) S { K ) is now related to the frequency spectrum by: S(K) = (5.6) For surface water waves (in deep water) th e dispersion relation is [56,57]: w5 = g , K where g+ = g + 7 K 2 and g is the gravitational constant. surface tension to w ater density and has a value of 7.25 7 (5.7) 7 is the ratio of x 10- 6 m 3 /s 3.When i f 2 ;» <7, the dom inant restoring force for the wave is surface tension, and the waves are called capillary waves. For smaller wavenumbers g, ~ g and 92 the dispersion relation is ws = g K . In this case gravity is the dominant restoring force, and the waves are called surface gravity waves. The transition between capillary and gravity waves lies at a wavelength of about 1 cm. Unfortunately, the theoretical expression for the dispersion relation (5.7) is not likely to be very accurate in a wind-driven sea at large wavenumhers. Its derivation assumes th at only a single gravity-capillary wave is present. In fact, a wind-driven sea has m any wavenumber components, including very long waves. The orbital velocity of these long waves is large enough to severely distort the dispersion relation by Doppler shifting. The new relation [58] is: = y/gJC{l + y / K K l h f a t ) - K M r , t ) ) (5.8) where K \ is the wavenumber of the long wave and k\ is the surface height due to th e long wave. Since there is a spectrum of random long waves, it is not possible to accurately determine the dispersion relation for a wind-driven sea a t large wavenumbers. Hence, wavenumber spectra for the gravity-capillary range which have been derived from frequency spectra m ust be used with caution. T h e wavenumber spectrum satisfies the following transport equation [59]: + ; - v * ( l r ) = <?, + <?„ + «,, (5.9) where Qi is the input due to surface winds, Qn is the exchange in energy between various wavenumbers due to nonlinear resonant interactions, and Q* is th e dissipation due to factors such as wave breaking. This equation is valid only in the absence of currents. If currents are present, then th e more general equation for action density transport is needed [60]: =<?•+«-•+<?<> 93 (s i °) where N (x, K , t) is the action density spectrum , which is the wavenumber spectrum divided by the intrinsic frequency u>'(K). This is the frequency in a coordinate system moving with the current. In a fixed coordi nate system the frequency is w (if,x ,i) = u '(K ) + K * t/(x ,t), where U ( z ,£) is the velocity of a current which may have spatial and tem poral variations. The tim e derivatives of x and K are given by [61]: dx du) = -= dt dK dK 5w ■5— S (5.11) K * . . (5'12) If the current velocity U is zero, (5.10) reduces to (5.9). There has been considerable theoretical and experimental work done con cerning the wind input term Q i. The pioneering work in this area was done by Phillips [62] and Miles [63,64] and provides a qualitative understanding of how waves are generated. W hen wind blows across a sm ooth water surface, waves are first generated by pressure variations due to atmospheric tu rb u lence. This corresponds to a constant Qi, which causes linear growth of the spectrum w ith time. However, as soon as small waves have been gener ated, the air flow is modified, resulting in a feedback mechanism. W hen this mechanism dominates, Qi becomes proportional to the spectrum , causing an exponential growth in the spectrum with time. In this case Qi — 0 [ K ) 9 ( K ) where is the growth rate. T he growth rate was calculated by Miles [63,64], but was found to be in poor agreement with experiment. However, he ex amined only linear instabilities in laminar flow w ith shear only in the air flow. Valenzuela [65] extended this theory to coupled shear flow (shear in both air and w ater). The shear flow in the w ater is in the form of a very thin layer of flow, called the wind drift layer, w ith velocity q relative to the 94 wave orbital velocity. This is Bimply a boundary layer ju s t b eneath the w ater surface resulting from the tangential stress of the wind on th e w ater surface. Deeper w ithin th e water, the velocity is ju s t th a t due to wave orbital motions. Valenzuela found th a t th e presence of th e wind drift layer does indeed affect wave growth, and his calculated values of th e growth ra te for short gravity and gravity-capillary waves agree well w ith experim ent in m any cases [6 6 ]. Further work in this area has included turbulence [67] and nonlinear effects [6 8 ]. The m ost reliable estim ates of w ind input still seem to be ocean and wave-tank m easurem ents. As a result of examining d a ta from a variety of sources, P lan t [69] found the following expression for th e w ind input: Qi = 0.04 cos <f>ulK*/2 g ; 1 / 2 V ( K ) (5.13) where u* is th e friction velocity, defined as the square ro o t of th e ratio of surface stress to air density. From this equation it can be seen th a t the wind input is a function of u ,, y,, and K. The term due to nonlinear resonant interactions, Q„, has been found for b o th gravity-capillary and gravity waves. Nonlinear interactions occur when several wavenum ber components combine to form a new com ponent. In this way energy is transfered between com ponents of different wavenumbers. For p ure gravity waves these interactions occur only to th ird order, i.e., three waves combine to form a fourth wave. T h e resonance condition for this case is: i f j + i f 2 — i f s 4* if* (5*14) Uq -J- Wz = (jJ3 -}- U4 (5.15) and where the i f ’s are the wavenumbers an d the w’s are th e frequencies. The 95 interaction term for this case has been derived by Hasselm ann [59] and is an integration over three wavevectors. In th is case Qn is a function of the wavenum ber K . For gravity-capillary waves resonance occurs at second or der, so th e resonance condition is: Ks = Ki + K 2 (5.16) W3 — Cdj + U>2 (5.17) and Q n for gravity-capillary waves has been derived in [70] is a function of both K an d gt . T h e dissipation te rm Q i is the least understood of th e three term s. It is prim arily the result of wave breaking, although for large wavenumbers (capillary range) viscosity is im portant. Originally, it was th o u g h t th a t wave breaking is not a function of the wind speed. However, it was found by B anner and Phillips [71] th a t the presence of the wind drift layer ju st below th e surface can significantly reduce the height at which waves break. In the presence of the wind drift layer, the breaking height is: h-max — ~ ~ ( C ?) (5.18) w here c is the phase speed (c = \J g ,(K ) and q is the drift speed. Wu [72] found th a t q ta 0.55u. using wave tank m easurem ents. It can be seen from (5.18) th a t the height a t which a wave breaks is not affected by the wind d rift layer only if th e condition Ku^jg* < 1 is satisfied (i.e., 5 « c). If it is n o t satisfied, the height at which a wave breaks is reduced. T he height a t w hich a wave breaks is thus a function of u 4, a n d K . Since Qd is a function of the breaking height, Qi also depends on u ., <7 ., a n d K . 96 To date, no spectra covering all wavenumbers have been obtained as a result of solving (5.9), so we m ust use observations and guidance from dimensional considerations. A spectrum derived from observations is the P ierson-Moskowitz spectrum [73]: Z {u ) = ^ where a = 0.008, exp (-0.74 (we/w )4) (5.19) = g{U\v&, and t/ig.s is the wind speed at 19.5 m above the ocean surface. The Pierson-Moskowitz spectrum assumes th a t the ocean is fully developed (infinite fetch and duration), which is usually a good ap proxim ation over the open ocean. The observations from which this spectrum was found contained waves w ith frequency up to about oj2 1 Hz. In this range = g K is valid and (5.19) can be transform ed to a wavenumber spectrum using (5.6): S ( K ) = B K - i & c p i - O M i K '/ K ] 1) (5.20) where B — 0.004 and K c = gfU j 9 B. For K K c the Pierson-Moskowitz spectrum is not wind speed depen dent and is not useful for modeling wind speed dependence of radar backseatter. R ad ar observations in the microwave region show a strong dependence of th e cross section on th e wavenumber on wind speed [54]. Because th e crosssection depends spectrum at large wavenumbers, the spectrum a t these wavenumbers m ust be wind speed dependent, indicating th a t th e PiersonMoskowitz spectrum m ust be modified at these large wavenumbers. Also, recent observations of the frequency spectrum for w >■ u c show deviation from th e w-B form of the Pierson-Moskowitz spectrum [74,75,76], A large wavenumber correction to the Pierson-Moskowitz spectrum has been proposed in [77] and used after modification by Fung and Lee [54] for 97 rad ar cross section calculation. As mentioned in Section 5.1, the model of Fung and Lee predicts a cross section which agrees w ith observations to w ithin a few dB for incidence angles in the 20° — 70° range. The large wavenumber spectrum used in their work, however, has no physical basis bu t was simply suggested as the form for d ata taken in a wind-wave tank. Also, as will be seen in Section 5.4, slopes predicted by the spectrum used in their model are considerably smaller than slopes m easured by Cox and M unk using sun-glitter techniques [78]. A spectrum w ith physical basis can be developed from dimensional con siderations. Assuming equilibrium between wind input, dissipation, and non linear interactions, the transport equation reduces to Q i + Q n + Qd — 0 [79]. In this case, th e spectrum depends only on the modified gravitational con stan t <7*, th e ocean wavenumber K , and the friction velocity u«, which is defined as the square root of the ratio of surface stress to air density. Di mensional considerations require the spectrum to be of the following form [56]: S {K ) = K ~ * f( K u l/g ,) (5.21) where / is an unknown function which approaches a constant as the argu m ent approaches zero. If K u \ j g <£ 1, this form reduces to the form of th e Pierson-Moskowitz spectrum . However, if this condition is not satisfied, the spectrum deviates from the K ~ 4 form and depends on the surface wind velocity through the friction velocity. The condition K v.\fg C 1 is also the condition under which the wind drift layer has no effect on the height a t which a wave breaks, according to (5.18}. For a surface wave with wave length greater th an 1.7 cm, an increase in wavenumber decreases the phase velocity, reducing the height a t which the wave breaks. This indicates th a t 98 the deviation from the K ~4 spectrum at large wavenumhers is a result of the presence of the wind drift layer. For wavelengths less th an 1.7 cm, the phase velocity increases w ith wavenumber, so a change in the spectral form may occur. A more im portant factor for these very short waves is viscosity, which is the prim ary dissipation mechanism for capillary waves. T hus, our analysis is strictly valid only for gravity and gravity-capillary waves. In order to proceed further, we need an explicit form for / . Because frequency and wavenumber spectra do seem to be power laws (e.g., Phillips [79]), we replace / by a power law and arrive a t the following form for the wavenumber spectrum: S { K ) = B K -* { b K u 2J g , ) rW and r ( i f ) is an unknown exponent which depends on K and (5.22) 6 is a constant. The exact form of r(Jf) is not known. However, the frequency variation of w ind speed dependence [54,80] suggests th a t r may have a logarithmic dependence on K . Hence, we take r = a log( K / K t) y where a is a constant and K t is the wavenumber where K u \ / g 1 is no longer satisfied. Depending on u« this condition probably becomes invalid for K between 1 and 10 . In lieu of more exact information, we choose K t = 2 m -1. T he other function to be chosen is $(<£). We follow Fung and Lee [54] and choose $(<£) = l + c ( l - e - ,,f 3 )c o s 2 ^ (5.23) where s and c are constants. Fung and Lee included the dependence on K to model the observed increase in cross section dependence on wind direction w ith radar frequency. The constant s is taken to be 1.5 x 10-4 m 2. The constant c m ust be found in term s of one or m ore measureable quantities. 99 T he to ta l, upwind, and cross w ind slope variances are: roo S 3 = f°° S ( K ) K 3d K to (5,24) and sl = f " V {K , 4>)K3c p[4>)dKdt where Cp(<f>) = cos3 4>for th e upw ind slope variance £ 3 (5.25) and Cp[<f>) = sin 3 <j>for the cross w ind slope variance S 3. Realizing th a t ’I' (K , <f>) = (27t)_15 ( jK‘)$(^6), Sp can be re-w ritten as: S l = \ f “ S ( K ) (1 ± | (1 - *-**)) Jl * d K (5.26) where th e + is used for S 3 and th e — for S 3. We can define the cross wind to upw ind slope variance ratio as R . Using (5.26), we find th at: ‘ = (1 ~ \\-D ) ^ ^ where R = ^ (5.28) u and „ JS° K i S W e x p i - a K ^ d K Jo” K 2S { K ) i K l J The param eter D can be calculated from (5.22). R y however, m ust be m ea sured. We use the Cox and M unk m easured ratio for a clean surface [78]: .003 +192X 1 0 - ^ ,,, 3.16 x 10“ 3 v 1 T he com plete wavenumber spectrum (in m 4) is given by the following ex pression: ' c-.74(jre/jr)a •(1 -I- c (l —e-,Jf3)) cos 2 ^) K < 2 (5.31) jqlogfjf/a) * (1 + c (l —e- *^3)) cos 2^) 100 K >2 We now note th a t the spectrum is completely specified except for the con sta n ts a and ft. Once these param eters are determ ined, the spectrum is known. The rem aining inform ation needed for our ocean surface model is a de scription of the planetary boundary layer so th a t we m ay relate the wind a t observation level to the friction velocity. T he ocean is a rough surface w ith tu rb u lent air flowing over it. The w ind profile is the well-known logarithm ic form: = ^5^ 0nY a ~ ^ ) m/B ^5'32^ Zo is the roughness length and corresponds to the size of roughness elements on the surface. Since the surface shape itself depends on th e friction ve locity, Z0 also depends on the friction velocity u«. T he following empirical expression has been given for Z q [77]: Zo = .00684/u. + .428u? - .000443 m (5.33) T he function ip depends on atm ospheric stability, which is determ ined by th e air-sea tem p eratu re difference. W hen air tem perature is less th an sea tem perature, there is a heat flux from sea to air, increasing buoyancy and enhancing atm ospheric turbulence. T his is an unstable condition and ip is negative. The condition of equal air and sea tem peratures is the neutral stability condition, and ip — 0. For air tem perature greater th an sea tem p eratu re (stable conditions), ip is positive. An em pirical expression for ip is given in [81]: ’ - 5 z/L ip{z(L) = ^ z/L > 0 ln [(l + x)/2) „ + ln [ ( l + i* ) /2 ] — 2 ta n ” 1 a: + ?r/2 (5*34) 2 101 z { L < 0 where x = (1 - 16z /L ) 1' 4 z j L = 9 .2 77ffzA 0(l/ro + 1.72 x 1O-6 :ToA Q/A0)/U 2 (z) A Q - 6.404 x 108[0.98exp(—5107/T.) - 0.75 e x p (-5 1 0 7 /ro)] T0 = Ta + 826T l exp(-5 1 0 7 /T o ) A 0 = T , - T a - 0.01 2 Ta is the air tem perature in K and T, is the sea tem p eratu re in K. L is the M onin-Obukhov length from turbulence theory. Using th is expression, along w ith Zo and u«, th e wind velocity a t an a rb itrary level m ay be calculated. If we are given th e velocity at a particular level and desire to calculate u„, one approach is to use Newton’s m ethod on (5.32). However, this is some w hat cum bersom e when tft ^ 0. A second m ethod is to introduce another p aram eter called th e drag coefficient, which relates U (z) to it. directly: u? = CDU2 (5.35) T he friction velocity is th en given by [82]: VC ^U (5.36) where C o n is th e drag coefficient for a neutral atm osphere (0 = 0). An em pirical expression for at z = 10 m was given in [81]: 1.14 4 < 17 < 10 m /s 0.49 + 0.06517 10 < 17 < 26 m /s For other heights the drag coefficient is: 0.16Cpw(10) T his boundary layer model allows us to do all calculations w ith I/ io.e. 102 (5.38) 5.3 Two-Scale Scattering from the Ocean In the two-scale model the quasispecular cross section and the Bragg cross section are combined to yield an approxim ate cross section for the total surface, as described in Section 2.6. We repeat the two-scale cross section here, for convenience: o°( 0 ) = e x p ( - £ ) o js (0 ) + f aBraff( ^p ( t a n ^ ,ta n 6 )d (ta n ^ )d (ta n £ ) J cos cos o (5.39) where p(tan Vs tan 6 ) is the slope probability density function for the large scale surface. Given the surface m aterial properties and height spectrum , the cross section can be calculated. There is some controversy concerning the probability density function for the slope of the ocean surface. Using sun-glitter m easurements, Cox and Munk [78] found a slope density th at is skewed toward the upwind direction. This slope probability density function is used in the model of Fung and Lee, causing the upwind cross section to be greater th an the downwind cross section, in agreement with radar observations. On the other hand, th e recent laser-optical sensor measurements of Tang and Shemdtn [84], show a zeromean Gaussian slope density function. A Gaussian slope density function was also found in the laboratory measurements of Wu [85]. Following these results, we take the surface height, slope, and curvature to be zero-mean Gaussian processes. The slope density function is: n p(tan'(''tan*) = 2 / 1 tan 2^ ta n 2 5^ ^ “ P (— 2 S f - - 2 S f ) . I5'40) where Sx and Sv are the slope standard deviations. These variances can be 103 found from th e ocean spectrum using the following expression: S l = f 2’ f K* V [ K t <f>)KsC{<f>)dKd$ Jo Jo where ' (5.41) — cos2 <f>for S j and C{<f) — sin 2 4>for If the slope is a zero-mean Gaussian process, th en p(tan ta n 5) for the large scale surface is the sam e in the upwind and downwind directions. This m eans th a t th e difference in th e upwind and downwind cross sections is re lated to the sm all scale spectrum . In particular, th e sm all scale spectrum m ust be m odulated by th e presence of the large scale surface, and th a t m od ulation m ust increase the sm all scale spectrum for large scale tilting toward the downwind direction and decrease the sm all scale spectrum for tilting tow ard th e upw ind direction. One m odulation m echanism is hydrodynam ic m odulation of th e small scale waves by the large Beale waves. Wave tank observations [8 6 ] have Bhown th a t the m axim um m odulation occurs on the downwind face of the large scale waves. This w ould produce a larger cross section in th e upw ind direction th a n the downwind, so we include this effect in our model. P ropagation of short waves in th e presence of long waves can be modeled by propagation of waves in a periodic current whose velocity is th e long wave orbital velocity. As the sm all scale waves propagate through th e large scale, the small scale spectrum is shifted from equilibrium and m odulation occurs. In order to find perturbations from the sm all scale equilibrium spectrum , we use (5.10). We represent the action density spectrum by No + S N , where No is the equilibrium spectrum and 6 N is the p ertu rb atio n from th e equilibrium caused by th e large scale waves. Also, we assum e th a t the source/dissipation function Q = Q i + Q n + Qd has the form —f i ( K , x ) 6 N to first order. Because 104 of the lack of theory for the dissipation function Qd, ft is considered unknown. Substitution of these expressions for N and Q into (5.10) yields the following equation for 6 N [60]: <542> We define a transfer function T ( K „ K i) which gives the m odulation of N ( K t ) due to a long wave w ith wavevector K \\ 6 N = N 0h (T e ’(Rl* - nT) + c.c.) (5.43) when th e surface height is given by h(x) = h(e, tffjl*~n^ + c.c.) (5.44) In b o th of these expressions c.c. represents the complex conjugate. We can find T by substituting (5.43) into (5.42): - mrl&F ■* > ( £ * ■ ■ it) <s -45> This expression can be re-w ritten in term s of the w avenum ber spectrum : •*«> (£*« ’W " t ) (546) We can define another transfer function which relates th e change in the action density spectrum to the large wave slope rath er th a n height. This function, which we call m ( K , K i ) , is equal to T/jTTj. Because 6 N f N 0 — S'b/'S'a, we have: ¥ ( i f , x , i ) = $ 0( ^ ) ( l + / + c.c.) . (5.47) where 3 (if|) is the Fourier transform of a realization of th e large scale slope. This expression gives the small scale spectrum in term s o f the large scale slope. 105 T here axe two difficulties in using (5.47) in the two-scale scattering model. F irst, th e param eter fi is not known w ith any degree of certainty since it depends on th e dissipation term Qd in th e tran sp o rt equation. Since fi de term ines the phase of th e transfer function, it is rath er critical in calculation at x of th e upw ind/dow nw ind difference in th e cross section. Second, depends on the slope a t ail points on th e ocean surface. In o th er words, is a functional of the slope, as opposed to a function of slope. In the two-scale m odel th e sm all scale cross section and hence th e spectrum depend only on th e local slope. At present it is not known how to p u t (5.47) into a form suitable for use in th e two-scale m odel, and we m ust settle for use of the following empirical expression [87): * ( K ) = *oCK')(l + “ *„) (5.48) w here S u is the large scale upwind slope stan d ard deviation. T hus, in us-, ing (5.39), the equilibrium spectrum is replaced by (5.48). As mentioned above, this models th e observation th a t th e m axim um m odulation is shifted downwind from the large scale wave crests. We have now specified all values needed in (5.39) to calculate the radar cross section, except for K& and the dielectric constant e. Because th e ocean is rough in b oth dimensions, we use th e definitions for fi and k given in Section 4.4. From Sections 4.2 and 4.3 we know th a t fi should be as large as possible b u t less th a n about one for th e two-scale m odel to be accurate. To satisfy this condition, we choose K& so th a t a t a high w ind speed (20 m /s) fi has a value of 0.5. For lower w ind speeds fi is, of course, less than 0.5. In Section 5.4 we will examine the validity of the two-scale m odel for an ocean described by our new spectrum . T his m ethod of choosing K& means 106 th a t b oth Ka and th e large scale slope variances will change w ith th e rad ar frequency, as expected physically. This effect is no t included in th e m odel of Fung and Lee [54], since they use empirically derived slope variances. T he dielectric constant m ay be found in [8 8 ]. It is of the form: e , „ , e, - 4.9 ' l + j'27r/r . t 7r/e 0 where to = 8.854 X 10~ 12 is th e perm ittivity of free space, and t, r , and e, are functions of th e salinity and sea tem perature, and have been found by a least squares fit to experim ent. T he salinity was chosen to be 0.035, which is a typical value for m ost of the w orld’s oceans [89]. T hus, using (5.49), the dielectric constant can be calculated as a function of ocean tem perature and rad a r frequency. The only rem aining param eters in th e ocean-scattering model are th e spectral constants a and 6 , which we now determ ine from radar m easurem ents. 5.4 Determination of Spectrum Parameters We use the spectrum of (5.31) w ith (5.39) and related equations to calcu late the ra d a r cross section of the ocean surface. T he spectrum param eters a and b are determ ined by a least squares fit of cross section calculations to RADS CAT GHz d a ta for horizontal-send, horizontal-receive (HH) polar ization looking in the upw ind direction [90]. This d a ta was taken from an aircraft ra d a r operating a t .ffu-band (13.9 GHz). Horizontal polarization is used because th e results presented in Section 4.3 indicate a possible problem in the sm all pertu rb atio n cross section for vertical polarization. T he wind presented w ith th e RADSCAT d a ta is the neutral stability wind a t 19.5 m. The actual wind was m easured a t flight level (100 m ), converted to the 107 friction velocity using air-sea tem perature difference inform ation, and then converted to th e 19.5 m level assum ing a neutral stability atm osphere. Thus, we assum e neutral conditions in our fit of the m odel to this d ata. Since the calculated wind speed dependence is prim arily determ ined by a, we varied a until the m ean square error between calculated and observed w ind speed exponents was minimized. T h e wind speed exponent u was determ ined by fitting the power law a0 = ctU(9 B to calculations and observations. We found th a t a = 0.25. We then adjusted b until the m ean square error in th e cross section was minimized. We found th a t b — 2.25. T hus, the final spectrum is given by (5.31) w ith a = 0.25 and 6 = 2.25. Fig. 5.1 shows the spectrum for C/1 9 .E = 5 m /s an d for Uiq,s — 20 m /s. A t large wavenumbers ( K > 2 m -1) the slope of th e spectrum is a function of the w ind speed. T his agrees w ith the findings of Lawner and M oore [91] who m easured the ocean spectrum using tower-based radar. The errors in a and b can be estim ated from errors in the m easured radar cross section. In doing so, we assume th a t errors in the cross section are random , since the reported cross sections have been corrected for system atic biases. T he error in a is given by [92]: ( 5 -5 0 ) where s„ is th e error (stand ard deviation) in a and is the error in the m easured w ind speed exponent due to errors in th e cross section for th e tth incidence angle. The errors in the i c a n be found from the following: <-£(£)* (5-«) Table 5.1 shows s„ as a function of 8. Using these values in (5.50), we find th a t sa « 0.05. This is an error of about 2 0 %, since a = 0.25. T he derivatives y• o - CO h - Tf_ O * LU 0 . tO u. W- -3 2 1 0 1 2 3 LOG OF K (RAD/M ) Figure 5.1: Log of ocean spectrum (in m4) as a function of log of wavenumber for a wind speed of 5 m /s (---- ) and 20 m /s (-------- ). For K < 2 m - 1 the spectrum is the Pierson-Moskowitz spectrum , and for i f > 2 m - 1 the spectrum is our new spectrum . 109 0° Incidence Angle 10° 20° 30° 40° 50° 0,23 0.33 0.21 0.33 0.25 0.26 Table 5,1: Standard deviation of the RADSCAT wind speed exponents. Source Observed [78] Calculated {K& ~ 10 m -1) Fung and Lee [54] W ind Speed (m /s) 5 10 15 20 1.5 2.2 2.8 3.4 1.1 1.7 2.2 2.3 0.5 0.7 0.8 0.9 Table 5.2: Comparison of observed slope variances (xlO 2) [78] with calcula tions using our spectrum and values from th e spectrum used by Fung and Lee [54]. of a w ith respect to v are estim ated by finite difference approximation. The error in b can be estim ated in a similar m anner since 6 cross sections for a wind speed of 15 m /s. The error in is a function of the 6 is estim ated from the equation: T he standard deviations .siojo,. can be calculated directly from the d ata given in [90]. Using them , we find th a t the standard deviation of b is sj « 0.4. T his is an error of ju st under 20%, since b — 2.25. Table 5.2 compares large scale slope variances for several wind speeds. The observed slope variances [78] are for an ocean surface covered with a surface film which dam ped out waves w ith wavenumber greater than about 10 m -1. Thus, in calculating slope variances from spectra, we used K <t — 110 Source 0° Calculated HH pol. Observed HH pol.[90] C alculated W pol. Observed W pol.[90] -0.80 -0.36 -0.80 -0.46 Angle of Incidence 10° 40° 30° 20° 0.06 1.43 1.69 1.69 0 1 .0 0 1.65 1.98 0.07 1.38 1.55 1.53 0 1.05 1 . 6 8 1.77 '50° 1.75 1.93 1.56 1 .6 6 Table 5.3: Com parison of RADSCAT upwind w ind speed exponents w ith calculated w ind speed exponents. 10 m ” 1. T he slope variances calculated from our new spectrum are reason ably close to observations a t all wind speeds. T h e spectrum used in the m odel of Fung and Lee, however, predicts slope variances which are consid erably sm aller th an observations. As mentioned in Section 5.3 Fung and Lee use th e em pirical slopes from [78] rather th an slopes calculated from their spectrum . Fig. 5.2 shows the calculated and RADSCAT observed cross sections at a w ind speed of 15 m /s as a function of angle of incidence. We em phasize again th a t only the RADSCAT HH observations at 13.9 GHz were used in the em pirical determ ination of the spectrum param eters a and 6 . B oth the HH and VV (vertical-send, vertical receive) calculated values are w ithin 3 dB of th e observed values, although the predicted polarization ratio is about 5 dB too large a t 50°. Table 5.3 compares th e calculated wind speed exponent u w ith th e ob served values given in [90]. Although only HH values were used in finding a, th e calculated and observed values for b o th polarizations are in good agreem ent. T he largest difference was 0.4 a t 20°. T his is com parable to the stan d ard deviation in the exponents used in th e determ ination of a an d 6 , as 111 o_ h O LLI </) CO to g g - OC t o co- 0 10 20 30 40 50 60 70 INCIDENCE ANGLE (DEG.) Figure 5 .2 : V ariation of the rad a r cross section w ith angle of incidence at 13.9 GHz and 15 m /s . (---- ) C alculated HH pol. (---------) Calculated W pol. Values were calculated a t 10° intervals. The A ’s are observed values for HH polarization, and the -f's are observed values for W polarization. Observations are aircraft data from RADSCAT [90]. 112 Source Calculated HH pol. Observed HH po l.[90] Calculated W pol. Observed W pol. [90] U /D 20° 0° 0 1.5 1.4 0.5 0 1.3 1.7 0.5 50° 1.3 3.5 0.5 2 .0 U /C 20° 50°0° 4.5 7.6 0 0 .0 1 .0 5.7 0 4.7 7.2 0.4 0.5 6.3 Table 5.4: Com parison of calculated upw ind/dow nw ind (U /D ) an d upwind/crossw ind (U /C ) cross section ratios (in dB) w ith those from RAD SCAT 13.9 GHz observations for a wind speed of 7 m /s and incidence angles of 0°, 20°, and 50°. seen in Table 5.1. Tables 5.4 and 5.5 show the upw ind/dow nw ind and upw ind/crossw ind cross section ratios from calculations and from th e RADSCAT d a ta for wind speeds of 7 and 13 m /s , respectively. T he calculations show a qualitative be havior th a t is sim ilar to the observed values. However, there are several cases in which the calculations and observations differ by 3 or more dB. In par ticular, upwind/dow nw ind ratios for 13 m /s and 50° and upw ind/crossw ind ratios for 7 m /s and 20° are in error. In using the two-scale model, we have chosen Kd so th a t the m axim um value of/? is 0.5, as described in Section 5.3. For our spectrum w ith a — 0.25 and b = 2.25, we find th a t th e large scale k rem ains below 0.002 and the small scale /? is always greater th an 0.01 and less th a n 0.5. Contour plots showing th e RMS difference between two-scale and m ethod of m om ent calculations (Sections 4.2 and 4.3) show th a t for these values of (3 and model should have an RMS error of less th a n 2 k, th e two-scale dB. This indicates th a t the two-scale model is a good approxim ation for calculation of scattering from th e ocean surface. This result is supported by th e absence of second order 113 Source ° 0 -0.4 0 -0.4 0 Calculated HH pol. Observed HH pol.[90] Calculated W pol. Observed W pol.[90] U/D 2 0 ° 50° 1 .2 1,5 0 .6 5.1 1 .1 0 .6 0 .1 3.7 U /C 20° 0 3.3 - 0 . 1 4.4 0 3.6 -0.3 4.3 0 ° 50? 7.6 8 .2 7.3 7.2 Table 5.5: Comparison of calculated upwind/downwind (U/D) and up wind/crosswind (U/C) cross section ratios (in dB) w ith those from RAD SCAT 13.9 GHz observations for a wind speed of 13 m /s and incidence angles of 0°, 20°, and 50°. peaks in Doppler rad ar measurements of the ocean, as pointed out by W right et at. [1 1 0 ], 5.5 Comparisons with Data In this section we compare predictions of our model w ith independent ob servations. Several useful sets of observations have been published. For Land C-bands we use the aircraft observations taken in th e North A tlantic by the Naval Research Laboratory (NRL) in the 1960’s [93,94]. X -band aircraft d ata has been taken by Raytheon [50] and by the Radio Research L abora tories of Jap an (RRL) [97]. Finally, observations from space at i f u-band have been obtained by the SKYLAB scatterom eter and by the SEAS AT-A Satellite Scatterom eter (SASS) [54,95,96]. We begin by examining the dependence of the cross section on incidence angle. Figs. 5.3 and 5.4 show calculated and NRL upwind cross sections a t Zr-band (1.2 GHz) and C-band (4.5 GHz). For C -band upwind values were available only for vertical polarization. Fig. 5.5 shows the calculated and the RRL cross sections for a wind speed of 9 m /s a t X -band (10.0 GHz). For the 114 NRL 1.2 GHz d ata the HH calculations are w ithin 1 dB of the observations for 0 > 30°, while the W values are up to 4 dB greater th an the observed values. For 6 < 30° the calculated values for both polarizations are up to 10 dB greater than the observed values. At C -band the model predicts cross sections which are about 4 dB greater th an the observed NRL values for vertical polarization. This result and the results for NRL L-band d ata and RADSCAT d ata are in agreement w ith the numerical results of Section 4.3, where it was found th a t the small perturbation theory predicts values up to 7 dB larger th an the moment m ethod. The RRL d a ta a t X -band for horizontal polarization is within 3 dB of calculations for all incidence angles except 15°, 60°, and 70°. In general the model correctly predicts the dependence of the rad ar cross section on incidence angle. There are, however, cases in which the errors are as large as 10 dB. As mentioned at the end of the last section, errors caused by use of the two-scale scattering model should be less th an about 2 dB, so we expect th a t large differences between the predicted and observed cross sections are caused by measurement errors or by effects not included in our ocean calculations. In our calculations we assume neutral stability conditions and absence of both swell and surface films, since these param eters are usu ally not known. If these assumptions are incorrect, then differences between calculations and observations would occur. M easurement errors include in correct radar calibration and errors in m easurement of surface conditions. The presence of measurement errors and dependence of the cross sec tion on environmental param eters other than wind velocity causes a large variation in m easured radar cross sections for the sam e reported wind veloc ity. R adar cross section measurements made by the SEAS AT scatterom eter 115 GO “O LU CO CO CM co 0 10 20 30 40 50 60 70 INCIDENCE ANGLE (DEG.) Figure 5.3: V ariation of the upw ind rad a r cross section w ith incidence angle a t L -band (1.2 GHz) and 18 m /s. (-----) Calculated HH pol. (---------) Cal culated W pol. Calculations done at 10° intervals. T h e A ’s are observed values for HH pol., and the + ’s are observed values for W pol. Observed values are NRL aircraft d a ta [93]. 116 o04 m *D o z o H o o LU • CO CO CO oS C °M' CC I CJ o n - \ i o “T" 10 " 1“ 20 T “ 30 "T " 40 —T“ 60 60 —t 70 INCIDENCE ANGLE (DEG.) Figure 5.4: V ariation of th e ra d a r upwind cross section w ith incidence angle at C -band (4.5 GHz) and 15 m /s for W pol. (---------) Calculated cross section. Calculations done at 10° intervals. T h e + ’s are observed NRL aircraft d a ta [94]. 117 CM O g_ LU CO • CO CO o ° = CM - cc < o 20 30 40 50 60 INCIDENCE ANGLE (DEG.) Figure 5.5: Variation of the upwind radar cross section w ith incidence angle a t X -band (10.0 GHz) and 9 m /s for HH polarization. (---- ) Calculated cross section. Calculations done at 1 0 ° intervals. The A ’s are observed values from RRL aircraft d ata [97]. 118 Source Calculated HH pol. NRL HH pol. [80] C alculated W pol. NRL W pol. [80] L 0.87 1 .1 2 0.78 1.47 so 6 C X 1,23 1.47 1.30 1 . 2 0 1.14 1.36 1 .2 0 1 .1 0 60fi C 1.41 1.59 1.25 1.30 X 1.65 1.45 1.45 1.35 Table 5.6: Com parison of calculated upwind w ind speed exponent u w ith those from NRL aircraft observations for incidence angles of 30° and 60° at L -band ( 1 . 2 GHz), C -band (4.5 GHz), and X -b an d (8.9 GHz). (SASS), for example, vary by up to 10 dB for th e sam e wind velocity [96]. Thus, th e errors of im portance in these com parisons of calculations a n d ob servations are consistent and system atic differences, present in com parisons w ith several different d a ta sets. T he errors in our comparisons of calculated and observed cross sections as a function of incidence angle are n o t sys tem atic for horizontal polarization, so we believe th a t the model accurately predicts th e variation of the cross section w ith incidence angle for horizontal polarization. For vertical polarization the calculated cross section is a few dB greater th a n observed cross sections for large incidence angles. T his sam e problem was observed in Section 4.3 in our com parison w ith m om ent m ethod calculations. To continue our com parison of calculations an d observations, we exam ine the dependence of the rad ar cross section on w indspeed. Table 5.6 com pares NRL exponents at £ -, C-, and X -bands [80] w ith calculated values a n d in dicates agreem ent w ithin 15% in m ost cases. One exception is a t 30° and L-band (1.2 GHz) for vertical polarization. T here, we calculated 0.78, and th e observed value was 1.47. Table 5,7 is a com parison of calculated wind speed exponents w ith exponents from SKYLAB d a ta a t i f u-band (13.9 GHz) 119 Source Calculated HH pol. SKYLAB HH pol. [54] Calculated W pol. SKYLAB W pol. [54] Incidence Angle 30° 40° 50° 1.69 1.69 1.75 1.32 1.89 1.81 1.65 1.31 1.15 1.55 1.53 1.56 1,39 1.89 1.69 1.61 1.36 1.32 Table 5.7: Comparison of calculated upwind wind speed exponents with those from SKYLAB spaceborne observations at i f u-band (13.9 GHz). taken on two different missions [54]. As in the case of th e NRL exponents, there is good agreement between observations and calculations, with calcu lations generally falling between the two observed values. Fig. 5.6 shows the calculated and observed values as a function of wind speed for X-band (9.0 GHz) a t norm al incidence [50]. Because of the scatter in the data, it is difficult to compare the overall wind speed dependences. However, we do note th a t the level of th e observed d ata is close to the calculated values. Figs. 5.7-S.9 show plots of the upwind cross section versus wind speed for the SEASAT scatterom eter (SASS) d ata for various angles of incidence. For normal incidence the calculated cross section is several dB larger than ob served values, although the slopes of the curves are very similar. For both 25° and 50° th e calculated and observed cross sections are within a few dB. We conclude th a t the new spectrum combined with the two-scale scattering theory can predict the wind speed dependence of microwave backscatter at L-, C-, X - , and i f u-bands over the range 0° —70°. Finally, we examine th e dependence of the cross section on th e wind di rection. Table 5.8 compares calculated values of the upwind/downwind and 120 c> cmH CD S Io 2 " LU CO CO CO o DC O CO- IO 4 10 W IND SPEED (M /S ) Figure 5.6: Variation of the calculated and observed cross sections with wind speed at norm al incidence and X -band (9.0 GHz). A ’s are observed Raytheon aircraft d a ta [50]. 121 ca 2 I*c5 LU CO CO CO o GC u 4 10 W IN D SPEED (M /S) Figure 5.7: Variation of the calculated and observed cross sections with wind speed a t norm al incidence and i f u-band (14.6 GHz). (---- ) Calculated. The A ’s are observed values from SEAS AT scatterom eter (SASS) [96]. 122 CD “O CO _ in WIND SPEED (M/S) Figure 5.8: V ariation of the calculated and observed cross sections w ith wind speed a t 25° and i f u-band (14.6 GHz). (---- ) Calculated HH polarization. (--------) C alculated W polarization. The A ’s are HH polarization SEASAT scatterom eter (SASS) observations. + ’s are W polarization SEASAT ob servations [96]. 123 in 00 *£L in O T- cc in cm - co WIND SPEED (M/S) Figure 5.9: Same, as Fig. 5.8, except incidence angle is 50°. 124 Source Calculated HH pol. Observed HH pol.[97] Calculated W pol. Observed W pol.[97] U /D 9.3 m /s 14.5 m /s 1.3 1.4 3 1 0.5 0.5 1 0 U /C 9.3 m j s 14.5- m /s 7.5 7.9 5 5 7.6 7.2 6 4 Table 5.8: Comparison of calculated upwind/downwind (U/D) and up wind/crosswind (U /C ) cross section ratios (in dB) w ith those from RRL X-band (10.0 GHz) aircraft observations. upwind/crosswind cross section ratios with those from RRL aircraft obser vations a t X -b an d (10.0 GHz) and incidence angle of 52° [97]. Table 5.9 compares calculated upwind/downwind and upw ind/crossw ind ratios w ith NRL aircraft observations for vertical polarization a t a wind speed of 19 m /s and incidence angle of 60° [94]. In b o th the NRL and RRL comparisons and the RADSCAT comparison in Section 5.4, the calculated upwind/downwind ratio is always smaller than the observed ratio, indicating a problem in our model. Because the upwind/downwind difference in th e cross section results from m odulation of the small scale spectrum by the large scale surface in the two-scale model, we must conclude th at either (5.48) is an incorrect de scription of hydrodynamic m odulation, or some other effect is present. The modulation of short waves by long waves will be discussed in more detail in Chapter 6 . 5.6 Effect of Atmospheric Stability We noted in Section 5.5 th at m easured radar cross sections for the same wind velocity can vary due to m easurem ent errors and dependence of the 125 Source Calculated (1.2 GHz) Observed C alculated (4.5 GHz) Observed C alculated (8.9 GHz) Observed U /D 0.3 3 0.4 U /C 1.5 4 6 .6 1 6 0.5 6 .8 2 4 Table 5.9: Com parison of calculated upwind/dow nw ind (U /D ) and upw ind/crossw ind (U /C ) cross section, ratios (in dB) w ith those from NRL aircraft observations a t 19 m /s for vertical polarization a n d incidence angle of 60° [94]. cross section on environm ental param eters other th an w ind velocity. These p aram eters include atm ospheric stability (as m easured by air-sea tem pera tu re difference) and presence of swell, as mentioned in Section 5.5. D a ta concerning the effect of atm ospheric stability is given in Keller et of.[82]. It was taken w ith an X -band (9.375 GHz) ra d a r operating at ver tical polarization an d an incidence angle of 45°. T heir d a ta indicate a 3 dB increase in cross section for a change from stable to u nstable conditions (a change in air-sea tem perature difference from + 5 K to -5 K ). O ur model, however, indicates an increase of only 0.4 dB. A large increase in cross section has also been noted by W ismann et of. [83]. There are a t least two possible explanations for th e difference between th e observed and calculated effects of atm ospheric stability. One possibility is th a t the boundary layer model sim ply underestim ates th e change in the friction velocity for un stab le conditions (air cooler th a n th e sea). A second possibility is th a t u n d er unstable condi tions, th e form of th e spectrum is changed, so th a t the sp ectru m for a given friction velocity differs for stable and unstable conditions. Regardless of the 126 cause, th e dependence of the radar cross section on atmospheric stability has serious implications for rem ote sensing, since information ab o u t‘.the air-sea tem perature difference is needed for proper inversion of measurements. In light of our comparison w ith the measurement of Keller et of., our model is best used under stability conditions th a t are near neutral. 5.7 Effect of Swell Swell refers to long ocean waves which have propagated away from the region where they were generated. If the presence of swell changes the cross section, then rad ar estim ates of wind speed will be incorrect, since swell is not related to the local wind. We model th e swell as a narrow band Gaussian process w ith the following spectrum [98]: 2nO Kx<7Kv where {h2) is the variance due to swell, K xm and K vtn are the wavenumbers of the spectral peak, and Ckx and ctkv are the spectrum widths in the x- and j/-directions, respectively. A Gaussian shaped spectrum is used since swell is narrow band b u t not monochromatic. To include swell in our model for th e ocean radar cross section, we add the swell spectrum (5.53) to windwave spectrum (5.31). This will increase the large scale slope variance and contribute to quasispecular scattering. Bragg scattering will also be affected by tilting and hydrodynam ic interaction. To examine the effect of swell as a function of frequency and incidence an gle, we choose swell w ith a wavelength of 300 m and Ojcx = c Kv — 0.0025 m -1. 127 T his is typical of swell param eters obtained from synthetic ap ertu re rad ar images [99]. To em phasize the effect of swell, we take the; RMS height to be 4 m , which is large am plitude swell, and we take th e direction of propagation to be th e rad ar look direction. Thus, the calculations presented here tend tow ard a worst case analysis. Fig, 5.10 shows th e effect of swell on th e ra d a r cross section a t L -band ( 1 . 2 GHz), and Fig. 5.11 shows the effect at i f u-b an d (13.9 GHz). T he m axim um effect is a t Z-band and 6 = 20° for horizontal polarization. In this case swell increases the cross section by approxim ately 6 dB at 5 m /s and by 3 dB at 20 m /s . This would cause errors of several m /s when attem pting to rem otely sense surface wind speed using L-band radars such as the SEASAT and th e SIR-A and -B synthetic aperture rad ars. As th e frequency or incidence angle is increased, th e effect of swell is decreased, until a t i f u-band and 50°, the change in cross section is less th a n 2 dB a t all wind speeds. For vertical polarization th e effect of swell is sm aller, since Bragg scattering at vertical polarization is less sensitive to tilting th a n a t horizontal polarization. From our model we conclude th a t error in w ind m easurem ent due to swell is m inim ized by using high frequencies, large incidence angles, an d vertical polarization. 5.8 Summary A new spectrum for large wavenumber ocean waves has been derived from dim ensional analysis a n d empirical considerations. This new spectrum has been combined w ith th e Pierson-Moskowitz spectrum for sm aller wavenum b e r waves and w ith th e two-scale scattering m odel to form a m odel relating 128 CO r- CO r- I I I II 10 T-r-T 10 W IND SPEED (M /S) WIND SPEED (M /S) (a) ' - ] 4 I I I 1---- 1 1------------------- (b) * 10 | I 4 WIND SPEED (M /S) I » 1 1 i I 10 W IND SPEED (M /S) (c) (d) Figure 5.10: Variation of the cross section at w ith w indspeed a t L -band ( 1 . 2 GHz) w hen swell is absent (---- ) and when swell is present (-------- ). Swell height (RMS) is 4 m. Swell wavelength is 300 m. (a) Incidence angle is 2 0 °, HH polarization, (b)Incidence angle is 20°, W polarization, (c) Incidence angle is 50°, HH polarization, (d) Incidence angle is 50°, W polarization. 129 a? S <M- z o H O ' ■ / CO CO 2 / CM - *+ yy + y + y > y + y + y + y z o h O / LLl CO s ' ' s ** J, + y *+ yy + + yy + y + y 0 y * y / / LLl CO V / CO CO O / _ / o co C O o u o 1 4 10 t o 1 i 4 W IND SPEED (M /S) i \ * i i i 10 W IND SPEED (M /S) (b) 00 CO 2 - l 4 l r™ m n T 10 I l i "T 10 W IND SPEED (M /S) (c) W IND SPEED (M /S) (d) Figure 5.11: Variation of the cross section w ith wind speed a t i f u-band (13.9 GHz) when swell is absent (---- ) and when swell is present (---------). Swell height (RMS) is 4 m. Swell wavelength is 300 m . (a) Incidence angle is 20°, HH polarization. (b)Incidence angle is 20°, W polarization, (c) Incidence angle is 50°, HH polarization, (d) Incidence angle is 50°, W polarization. 130 the ocean rad ar cross section to wind velocity and radar param eters. This model explains most of the observed dependence of the radar cross section on wind speed, wind direction, frequency, angle of incidence, and polarization for incidence angles in the 0° —70° range. We did find th a t the calculated ver tical polarization cross section is always several dB large than observations at large incidence angles. Also, the model does not properly predict the upwind/downwind difference in th e cross section, b u t instead consistently underestim ates it. We noted th a t in addition to wind velocity, the cross section depends on other environm ental param eters, meaning th a t there is a range of possible cross sections for each wind speed and direction. We have examined the effect of atm ospheric stability using our model and have found th a t the predicted dependence of the cross section on atmospheric stability is much weaker th an the dependence seen in observations. This indicates th a t our model is accu rate only under neutral or stable conditions. We have also used the model to examine the effect of swell on wind speed measurements and have found significant effects a t low frequencies (X-band) and small angles of incidence. Use of high frequencies (e.g., JTu-band), large incidence angles (e.g., 50°), and vertical polarization nearly eliminates error in wind speed m easurement due to swell. A sum m ary of the research described in this chapter has been published in IE E E J. Oceanic Eng, under the title “A physical rad ar cross section model for a wind-driven sea with swell” [1 0 0 ]. 131 i Chapter 6 Cross Section Modulation by Long Ocean Waves 6.1 Introduction In the last chapter, we dealt with the dependence of the rad ar cross section on wind speed. In this chapter we examine variations in the cross section due to the presence of ocean surface waves with a wavelength larger than the radar resolution cell. Using real aperture radar (RAR) or synthetic aperture radar (SAR), it is possible to image these ocean waves [99], Theoretical analysis of this imaging process is normally carried out in term s of a m odulation transfer function (M TF), which specifies the change in radar cross section due to a long ocean wave. Existing theories for the M TF [102]-[105] have considered only Bragg resonant scattering from a slightly rough surface. As we saw in Chapter 4, the two-scale cross section and the Bragg cross section may differ considerably when the surface is not slightly rough. At microwave frequencies the ocean is not slightly rough but can be accurately analyzed by the two-scale model, as shown in Chapter 5. Thus, the existing theory needs to be extended to the case of two-scale scattering. In this chapter we describe extension of previous work to the case of 132 two-scale scattering — quasispecular scattering plus Bragg scattering weighted by th e large scale slope density. After presenting numerical results showing the effect of two-scale scattering on the m odulation transfer function, we compare calculated and m easured m odulation transfer functions. Finally, we compare m odulations predicted by our m odulation transfer functions with calculations from a nonlinear model. 6.2 Scattering Model for Imaging Radars In previous chapters it has been tacitly assumed th a t the resolution cell size of the rad ar is very large compared to all ocean waves. This is normally the case for scatterom eters, which m ay have resolution cell sizes up to 50 km. Imaging radars, however, have much smaller resolution cells. The SEAS AT SAR, for example, had a resolution cell size of approxim ately 25 m. Because a fully-developed ocean may have waves with wavelength much longer than 25 m , we m ust modify our two-scale model. We now divide ocean waves into three different wavenumber regions: K < K ret K re» < K < Kd Kd < K Long waves resolved by the rad ar Subresolution waves of large height - contribute to qua sispecular scattering Subresolution waves of small height - contribute to Bragg scattering where K ret is the wavenumber of the smallest wavelength wave th a t can be resolved by the radar. If the rad ar has a resolution cell of length L, then by th e Nyquist sampling theorem, the shortest wave th a t can be imaged m ust have a wavelength of 2 L %since we m ust have at least two resolution cells per wavelength for proper sampling. A wavelength of 2L corresponds to a 133 wavenumber of wjL> so K ret — tt/ L . Thus, we have a random rough surface which consists of sub-resolution waves (K > The cross section of this random surface is modulated by deterministic waves w ith wavelength greater th an 2 L ( K < K rtt), Modulation of the cross section by long ocean waves is initially assumed to be a linear process. We will examine the validity of this assum ption in Section 6 .6 function, . A linear system can be represented by a m odulation transfer 12 (If): Sa°fa° = |I2|h c o sp ? ■x - tit + 7 ) (6.1) where 6a° is the change in cross section due to the presence of the long wave, a0 is the m ean cross section (when no long waves are present), and 7 is the phase of R . This expression gives the m odulation of the cross section resulting from surface height variation due to a monochromatic surface wave: h(x) — h cos{K ■x — fit) (6.2) If we have a spectrum of long waves, then we m ust use Fourier representations of the surface and the m odulation. In this case the height variation due to long waves is given by: k{x) = J (&(JT)e^-*“ n,>+ e .ejd tf (6.3) The limits of integration in this expression are ± K re§»since waves with larger wavenumber cannot be resolved by the radar, h is th e Fourier transform of the surface height fluctuation due to th e presence of the long waves (i.e., the surface height w ith sub-resolution waves removed). The m odulation transfer function is now defined by: tfo% 7° = J {R (K )h (K )e i (**“<«> + c.c)dK 134 (6.4) R can be w ritten as: R = R tilt 4" R h yd ro Rtut represents the m odulation due to tilting of the rad ar resolution cell by long ocean waves, and Rhydro represents the m odulation due to hydrodynam ic interactions between the long waves and th e sub-resolution waves. We neglect m otion effects since these do no t affect th e actual cross section (e.g., as observed by a scatterom eter), b u t only th e cross section as m easured by a ra d a r which is sensitive to Doppler shifts, such as a SAR. 6.3 Tilt Modulation To calculate th e change in the rad a r cross section due to tilting of the resolu tion cell by long ocean waves, we m ust find th e cross section of a tilted rough surface. In the case of Bragg scattering, we now have two sources of tilting: ran d om tilting by the subresolution waves and tilting by th e longer (resolved) waves. We use the sam e theoretical form ulation as th a t in Section 2 .6 . Now, however, we separate the tilting angles into two components: tjt = tfri + f a (in plane of incidence) 6 = Si + S2 (out of plane of incidence) (6.5) (6 .6 ) where rpi and 6i are the tilts resulting from waves w ith K greater th an K re„, an d and ^2 are the tilts due to th e longer waves. We can then sta te the B ragg scattering cross section of the resolution cell as a function of the long wave tilts, rj)2 and S2 [103]: 2 t 62) = J J a%(di)p(tZLn ^ i,ta n 5 i) d ( ta n ^ i) d ( ta n ^ i) 135 (6.7) where 0 ,- is th e local incidence angle given by cos 0 ,- = cos 0 cos ( 0 + 1/>) and a% is the Bragg cross section for a tilted surface given in [27]. For sm all to moderate angles of incidence, quasispecular scattering is significant. To include this mechanism, we need the quasispecular cross section for a tilted rough surface. In this case the local angle of incidence in the tilted resolution cell is defined by: cos 0 ,* = cos Si cos( 0 + ^ 2 ) (6 -8 ) The quasispecular cross section for a flat surface is (from Chapter 2): _ \R \ 2 sec4 0c. tan*,/as; 2 S XS V (6.9) The effect of tilt ip2 in the plane of incidence is included by replacing 0 by 0 4- tp2 , and we now examine the effect on (6.9) of tilt out of the plane of 62 incidence. Fig. 6 .1 shows the primed coordinate system, in which the surface is flat, and th e unprim ed coordinate system, in which the surface is tilted by 62 . We consider th e coordinates (—ro,0,Zp) in the prim ed coordinate system. Then, in the unprim ed system, the coordinates of this point are given by [1 0 1 ]: x 1 0 y 0 z 0 cos 02 —sin 02 sin 02 cos 02 —Xq 0 (6.10) 0 . 4 . The slopes of the specular points in the primed system are: _ , ^ z ip — hytp — f (6.11) 0 (6 .12) In the unprim ed system, the slopes are: Xq t n * ,P " Z& CO Sfc 136 (6 . 13) z V Figure 6 .1 : Geometry for scattering from a surface tilted out of the plane of incidence. The x -z plane is the plane of incidence. The x-axis points out of the plane of the page. (6.14) — tan ^2 Letting Xq = rsinfl and = rcosfl we have: h xtp — tan 0 sec S2 (6.15) hytp — tan 6 2 (6.16) Using (6 .8 ) and (6.16), (6.9) becomes cr°(0 ,tf2,£2) = •exp 2SxStt cos4 8 2 cos4 ( 6 + ip3) ta n 2 ( 8 + rf)2) sec2 8 2 2 SI {■ ta n 2 8 2 2S 2 (6.17) ) The sum of (6.7) and (6.17) is th e total radar cross section of a random rough surface which is tilted by ip2 and 8 2. Henceforth, we replace these tilt angles by tp and 6 . We find the M TF by expanding the change in cross section in a Taylor expansion and keeping only the first (linear) term s: o° _ 1 da°[d, ip,0 ) a0 a0 dip n j;\ 8 *=0 137 96 8 5=0 (6.18) Since ip and 6 are assum ed to be sm all, they can be replaced by the x and y height derivatives, respectively: 6o° 3h 3<7°(fl,0,5) 36 a0 $=o &v dh dx 1 do°{0,ip,0) dip 1 (6.19) T he surface height and its derivatives are given by th e following expressions: k (x ) = h(e*(K'x (6 . 20 ) + c.c.) + c.c.) (6 .21 ) = Aynf.c'!*-*-*") + c.c.) (6.22) ox We su b stitu te (6 .2 1 ) and (6 .2 2 ) into (6.19): So0 = h f 1 do0 [ o 5Up 3tr° o° 36 1 jK x + V>-o jK v I + c.c. (6.23) 5=o Using the representation for surface height in (6.20), th e M T F is defined by: 6o° o° (6.24) On com paring (6.24) w ith (6.23), we find th a t the M T F for tilt m odulation is the following: drf) J _ dcP_ <7° d ip , 1 d °° 3 x + o° 36 V>=o 3Kv 5=0 TCCOS* J f, , JK + ^1 - g9o° f *=0 j K sin $ 5=0 Because o°[6>ip,6) is a sm ooth and even function in 5, 3 o ° / 3 6 |y,=o = 0. T hus, th e tilt M TF is: 3a° o° dip 1 Run - i K cos $ V-=o 138 (6.25) w here <& is th e angle of long wave propagation relative to th e radar look direction (projected onto the m ean surface). This m eans th a t waves propa gating perpendicular to th e rad a r look direction are not im aged by the tilt m echanism . We now evaluate (6.25) for an 1 .2 GHz rad ar w ith 25 m resolution (SEASAT SAR param eters). At 1.2 GHz the Pierson-M oskowitz K~* spec tru m w ith isotropic angular dependence is a good approxim ation for M TF calculations, even though its wind speed dependence is not correct. Fig. 6.2 shows the M T F norm alized by j K when a 0 is the cross section due to Bragg scattering w ith large scale tilting ignored. These plots are essentially iden tical to those given in [104]. Also shown in this figure is th e M T F when the full two-scale m odel is used. The M T F using two-scale scattering is greatest for a 10° incidence angle. This is in contrast to the Bragg scattering M T F, which continues to increase as the origin is approached. A t large incidence angles th ere is only a small change in the M TF due to use of the two-scale model, and for these angles the M T F is well approxim ated by the Bragg scattering M T F. 6.4 Hydrodynamic Modulation T he m odulation of the cross section by hydrodynam ic interaction of long waves w ith sub-resolution waves is analyzed using the transfer function be tween th e long wave height and the action density spectrum derived in Sec tion 5.3. We repeat it here for convenience: 1 — 9*o - ~ K • —~ *o dk 139 K *k k2 ( 6 .26 ) o_ (O U_ O- O0 40 20 60 INC. ANG. (DEG.) M o_ (M LU h2 O- 0 20 40 60 INC. ANG. (DEG.) 0>) Figure 6.2: T ilt M T F { R in tfjK ) as a function of the angle of incidence 6 for B ragg scattering when th e large scale tilting is neglected (---------) and for two-scale scattering (—). (a) Horizontal polarization, (b) Vertical polariza tion. Calculations done a t 1 0 ° intervals. 140 Note th a t we have m ade a Blight change in notation. Previously, we have used K to denote th e wavenumber for all ocean waves and k to denote th e elec trom agnetic wavenumber. Since th e electrom agnetic wavenum ber does not directly enter th is discussion, k now denotes the wavenum ber in the spectrum of sub-resolution waves and K denotes the w avenum ber of long (m odulating) waves. As m entioned in C hapter 5, the increm ental source/dissipation term fi is not well understood theoretically, prim arily because th e dissipation term in the tra n sp o rt equation is not known. If fi is not too m uch greater th an 0 , the factor fl(fl —j n ) / ( t l 2+ fi2) will have an am plitude of approxim ately 1 , so th a t its p rim ary function will be to shift the transfer function phase. Hence, we sim ply replace this factor by e*'7, where 7 is the phase shift. Wave tan k m easurem ents [1 0 2 ] indicate th a t th e hydrodynam ic m odulation has a phase shift of approxim ately 45° ahead of the long wave crest, so we take 7 = 45°. Once we know the m odulation of the spectrum , we can find the m od ulation in cross section. The sim plest case, analyzed in [60], is the case of Bragg scattering where the tilting due to the large scale sub-resolution waves is ignored. In th is case S N /N o = S a 0/tT°i so R h yd ro = T { k Bx , K ) (6.27) where k& is th e Bragg wavenumber. For the Pierson-Moskowitz K~* spec tru m , Rhydro for Bragg scattering is given by: Rhydro = 4.5K e 3'1cos2 $ (6.28) where $ is th e angle between th e ra d a r look direction and the direction of th e long wave. N ote th a t this function is independent of the angle of incidence. Also, like Run, it is zero for waves travelling in the azim uth direction. 141 The above derivation of the hydrodynamic M TF assumed th a t long waves m odulate only the small scale sub-resolution waves. In fact, it has been shown th at this type of m odulation occurs for waves up to one-fourth the wavelength of the long waves [107]. Thus, a 100 m long wave m odulates both large and small scale sub-resolution waves for m any imaging radars (e.g., SEAS AT SAR), and should cause m odulation of both Bragg and quasis pecular scattering. The Georgia Strait experiment supports this hypothesis [106]. In this experiment quasispecular scattering from the ocean was found to occur in bursts th a t were correlated w ith increases in slope variance due to the presence of long waves. This indicates th a t long ocean waves m odulate quasispecular scattering through hydrodynam ic interactions. To find the M T F for the case of two-scale scattering, we begin with the two-scale cross section: 0 ° = Oq s + J J o % p ( h ^ h v) d h x d h v (6.29) The m odulation in c° is, to first order: So0 = dffgg + J J 6 a % p [ h ^ h y) d k xd h v + jj a % 6 p ( h Xi h s ) d h x d k v (6.30) The first term on the right is the change in the cross section due to quasis pecular scattering. The second term is the change in the Bragg scattering cross section due to m odulation of the sub-resolution small scale spectrum , and the third term is the change in the Bragg scattering cross section due to m odulation of the sub-resolution large scale slopes. The quasispecular cross section can only be m odulated through th e change in slope variances. Thus, where S*0 and are the slope variances in th e absence of long ocean waves. The changes in slope m odulation can be found from the change in the spec trum : 6S i = J J 6V {k)k ld k xdkv = h (eft**-***) J J y 0[k )T (k ,K ) cos2 tftk*dkd<f> + c.c.) (6.32) The change in slope variance in the y-direction is: 6S i = h J J 4ro(fc)T(fc, JC) sin 2 tfj^dkd^ + c.c.) (6.33) We substitute (6.32) and (6,33) into (6.31). After some algebraic manipula tion, the result can be p u t into the form Sa° = hex0 (Rase’^ - W + c.c.) Upon doing this, we find the following expression for R q s - l] / / **T (k,K ) cos’ ik ’d k d t J J V 0T{k, K ) Sin2 <f>k3dkd<t> - (6.34) This is a general expression for the M TF for quasispecular scattering due to hydrodynam ic m odulation. Next, we analyze the second term in (6.30). The change in the local Bragg cross section can be found directly from T ( k ,l ( ) . J 6asp{kSihv)dhxdhv — k\^J J a%(8i ^>,^)T(fc,iiL)p(tan ^ ,ta n 5 ) d(tan ^>)d(tan + c.c.] 143 (6.35) where k — 2 fcsin(d + ifi)x + 2fccos(0 + t/>) sintfy. The third term in (6.30) is sim ilar to the first term in th at m odulation comes through the.slope vari ances: «P(A., h . ) = J § l f S l + (6.36) The th ird term is then: J e^Sp ih zth Jd h xd h g = h ~ •d (tan ^ )d (tan fi) j J tf 0 (E )r(fc,tf}cos 2 # sd fc # n ^ ,ta n f i) + “ x) j ^ ) p ( t a n t a n 5)d(tan $ ) d(tan d) J J ^ o { k ) T (k t K ) sin2 $ k 3dkd$J (0« ,ei(Kx-nt) + c c ^ (6.37) T he sum of the second and third term s in (6,30) is the total change in the cross section due to Bragg scattering. Comparison of (6.37) w ith the following Scr% = ha\j + c.c.) yields R b (K )= + 2h [y j ^ I I a °(0 ,0,5)r(A :,K ')p (tan 0,tan fi)d (tan ^ )d (tan 6) x)a£ ( ^ ^ % ( t a n ^ t a n f i ) ■d(tan^f)d(tanS) J j ’J'o(£)T(£, K ) cos2 4>kzdkd<i> + •a£ (0 , J J ^~p T ~ " <5)p(tan tft, tan S)d(tani/>)d(tan 6) J J 9Q(k)T{kt K ) b,\t? <j>kzdkd<j> (6.38) ThiB is a general expression for the M TF for Bragg scattering from a twoscale surface when m odulation is due to hydrodynamic interaction. 144 To get the total M T F for hydrodynam ic m odulation, we m ust combine th e M T F ’s for quasispecular and Bragg scattering. The to ta l change in cross section is: 6 = h [( o%Rb + <t%s R q S) -f- c.c] (6.39) From this expression, th e total M T F is: R h yd ro = ^ { C q s R q S + <?B^b) (6.40) This expression is a general expression for the total M T F where R b R qs and are given by (6.34) and (6.38). As in the case of tilt m odulation we evaluate (6.40) for a Pierson-Moskowitz K ~ 4 spectrum for th e SEASAT SAR param eters. Substitution of a K ~4 spectrum into (6.34) and (6.38) yields: R q s = 1.125Ke?'* (1 + 0 .5 cos 20) - (1 - 0.5 cos 20) (6.41) and R b = 4.5.KV 7 cos 2 0 + J ( 1 + 0.5 cos 2 0 ) j J ^-2t^ - 1j <rfiO( 0 ,0 ,6 )p (ta n 0 ,ta n 6 ) d (ta n 0 )d (ta n 5) + ( l —0.5 cos 20) I / — ^ ffflo P (ta n 0 ,ta n 5 )(f(ta n 0 )d (ta n 5 )| (6.42) T he to tal hydrodynam ic M T F is found from (6.40). Fig. 6.3 shows the hydrodynam ic M T F [Rhydro f K ) calculated using Bragg scattering w ith large scale slopes ignored and using the two-scale model. As in th e case of tilt m odulation, use of two-scale scattering changes th e angular dependence at sm all incidence angles. Also, as in th e tilt m odulation case, the two-scale M T F a t larger incidence angles is well approxim ated by the Bragg scattering 145 o_ 10- U_ O- ID 0 40 20 60 INC. ANG. (DEG.) 00 lo ll. f— 2 ID 0 20 40 60 INC. ANG. (DEG.) (b) F igure 6.3: Hydrodynam ic M TF (RhydrofK) as a function of th e angle of incidence 8 for Bragg scattering when th e large scale tilting is neglected (-------- ) and for two-scale scattering (—). (a) Horizontal polarization, (b) Vertical polarization. Calculations done a t 10° intervals. 146 M T F. Since both the tilt and hydrodynamic M T F ’s for two-scale scattering are close to the M T F ’s for Bragg scattering for large incidence;angles, we conclude th a t derivation of the M TF using only Bragg scattering is a good approxim ation for large incidence angles. Fig. 6.4 shows the hydrodynam ic M T F (using the two-scale scattering theory) for a long wave propagating in several different directions. An in teresting result is the fact th a t the M TF is small but non-zero at small 0 for waves propagating norm al to the rad ar look direction. This was not true when th e effects of the large scale sub-resolution waves axe neglected. How ever, because of the smallness of the M T F, it is likely th at these waves would not be detectable in practice. 6.5 Total Modulation Transfer Function T he to ta l modulation transfer function is sim ply the sum of the tilt and hy drodynam ic M TF’s, Rtm + Rhydro, since the use of an M TF assumes linearity. This is a complex addition, since the hydrodynamic M TF has a phase of 45° and the tilt MTF has a phase of 90°. Fig. 6.5 shows the total M TF for a long wave propagating in several directions relative to the rad ar look direc tion. For a wave propagating in the radar look direction and 45° to the look direction, the tilt m echanism is dominant. In the case of propagation normal to th e look direction, m odulation is caused by hydrodynamic m odulation of quasispecular scattering. A t present, very few measurements of the M T F are available for compar ison w ith theory. One possible method is to measure the long wave spectrum in a region that is being imaged. Then, knowing the characteristics of the 147 IOU_ O- ID 0 20 40 60 INC. ANG. (DEG.) M ID - LL. O- to 0 20 40 60 INC. ANG. (DEG.) 0>) Figure 6.4: H y d ro d y n a m ic M T F {Rhydro/K) f°r two-scale scattering plotted as a function of the incidence angle 6 for a long wave travelling in th e radar look direction (—), 45° to the rad ar look direction (---------), and 90° to the rad ar look direction (—• —)• (a) Horizontal polarization, (b) Vertical polarization. Calculations done at 1 0 ° intervals. 148 n o _ CM LL t— 2 O- o0 40 20 60 INC. ANG. (DEG.) (a) CO 0 CM _ u_ o0 20 40 60 INC. ANG. (DEG.) (b) Figure 6.5: Total M T F {Rtot/K) for two-scale scattering plotted as a function of th e incidence angle Q for a long wave travelling in th e rad a r look direction (— ), 45° to the rad ar look direction (--------- ), and 90° to the rad ar look direction (— • —)• (a) Horizontal polarization, (b) Vertical polarization. Calculations done a t 10° intervals. 149 rad ar system , we can. find th e m odulation transfer function. Because of uncertainties in the ra d a r’s param eters and in the m easured long wave spec tru m , it is difficult to get a quantitatively accurate m easurem ent of th e M T F. However, it is possible to obtain an estim ate of th e dependence of th e M T F on incidence angle over the range of incidence angles for which a particular rad ar operates. This has been done for the SEASAT SAR, and a decrease in the estim ated M T F w ith incidence angle is found for incidence angles in the 23° —25° range [108]. A lthough this agrees w ith the calculations shown in Fig. 6.5, it is over m uch too small a range of incidence angles to be con clusive. U nfortunately, all other ocean imaging rad ars have also operated over a sm all range of incidence angles. Because of calibration and long wave spectrum m easurem ent problem s, it is not possible to learn anything about the dependence of the M T F on incidence angle from combining d a ta from different rad ars operated a t different incidence angles. T he m ost promising m ethod for m easurem ent of the M T F is th e twoscale probe [109], Briefly, this probe is a CW rad a r which looks a t a small area on th e ocean surface. T he am plitude m odulation of the retu rn signal is directly related to the change in cross section caused by long waves, and the frequency m odulation of th e retu rn is related to orbital motions of th e long waves. By doing am plitude and frequency dem odulation, inform ation on th e cross section m odulation and th e long wave o rbital velocity can be found. This allows a m odulation transfer function relating the long wave orbital velocity to th e cross section m odulation to be determ ined. In theory, this m easured M T F should be equal to R divided by K . However, this is not the case in practice. The theoretical M TF is a m easure of the change in th e cross section relative to the case w hen no waves are present. The m easured M T F, 150 u. o_ to O- 0 40 20 60 INC. ANG. (DEG.) Figure 6 .6 : Comparison of calculated and m easured M T F m agnitudes for a wave travelling in the rad ar look direction for vertical polarization. (—) Calculations. A ’s are measurements from [110], and + ’s are measurents from [ ill] . Calculations done at 10° intervals. however, measures the change in cross section relative to the tim e averaged cross section when long waves are present. Thus, m easured M T F ’s depend on long wave properties and will be expected to differ somewhat from the theoretical M TF. Fig. 6 .6 shows the m agnitude of the total M T F for a wave travelling in the radar look direction for a rad ar operating a t 1.5 GHz with a resolution of 5 m. Also shown are measurements of the M TF taken under stable or near neutral conditions [110,111]. It appears th a t th e theory predicts values th a t are generally smaller than measurements, particularly at larger angles of incidence. However, as noted before, the theoretical and m easured M T F ’s differ somewhat in their definitions, so exact agreement is not expected. Fig. 6.7 compares the M T F phase from theory and observation. The observations have a smaller phase than predicted by theory, so th a t the 151 CO LU o_ CD U. LU S CD Q_ 0 60 40 20 INC. ANG. (DEG.) Figure 6.7: Comparison of calculated and measured M TF phase for a wave travelling in the radar look direction for vertical polarization. (—) Calcula tions. A ’s are measurements from [1 1 0 ], and + ’s are m easurents from [111]. Calculations done at 10° intervals. maximum m odulation is closer to the long wave crest than predicted by th e ory. Several explanations are possible. First, the laboratory data according to which we set the phase 7 to 45° (Section 6.4) may not hold true in the open ocean. Second, another type of m odulation may be im portant. A possible mechanism is m odulation of th e small scale spectrum through m odulation of airflow by the long wave [1 1 2 ]. The need suggested here for another m odulation mechanism is in agree m ent w ith the results of Section 5.5. There, we found th at an em piri cal expression for hydrodynamic m odulation predicts an upwind/downwind cross section difference th a t is too small. We concluded th a t either the empirical expression describing th e modulation Is inadequate or th a t some other effect is present. The presence of another m odulation mechanism, such as m odulation of airflow, m ight explain the smallness of the calculated 152 upwind/downwind cross section difference said the largeness of the phase of the calculated M TF (Fig. 6.7). 6.6 Comparison of Linear and Nonlinear Mod ulation In order to test the assum ption of linear m odulation, we calculate the actual 6o®joQfor various long wave amplitudes and wavenumbers. We use the first order hydrodynam ic modulations of the spectrum since the full nonlinear theory is not tractable. Once the new spectrum is found in this m anner, we calculate the change in cross section using the full two-scale scattering theory rath er than a Taylor expansion. Figs. 6 .8 - 6 .1 0 show comparisons of R totf K (linear m odulation assumed) w ith 6oDj h K o fi (linear m odulation not assumed) for various choices of k and K . The actual change in cross section Sa° was com puted at the point along the long wave where the M TF was a maximum. For a slope of 0.01 the linear and nonlinear calculations are in good agreement. In the 0.05 case the nonlinear m odulation is about 50% greater th an the linear m odulation at an incidence angle of 20°. Finally, for a slope of 0.1 the nonlinear modu lation is several times greater th an the linear for 8 — 2 0 °. The reason for the failure of the linear theory for large slopes is the presence of quasispecular scattering, which is highly nonlinear in the long wave tilt. In practice, this nonlinearity is probably not a problem, since for m oderate wind speeds (e.g. 10 m /s) the RMS slope for resolvable waves is approxim ately 0.02 (according to the Pierson-Moskowitz spectrum ). Slopes of 0.05 and greater occur only under gale or hurricane conditions, and under these high wind conditions it is likely th a t quasispecular scattering is reduced by the presence of foam 153 m Z Q o_ U < —J Ol 3 Q o„ O *" 5 0 20 40 60 INC. ANG. (DEG.) W CO Z — o _ _1 0 20 40 60 INC. ANG. (DEG.) 00 Figure 6 .8 : Com parison of linear calculation, J t / K , (—) w ith nonlinear cal culation, S a °/h K o °, (--------) for a long wave w ith slope of 0 . 0 1 (h = 1 . 0 and K = 0.01). (a) Horizontal polarization, (b) Vertical polarization. Calcula tions done a t 1 0 ° intervals. 154 O 40 20 GO INC. ANG. (DEG.) « O©0 40 20 60 INC. ANG. (DEG.) 0 >) Figure 6.9: Com parison of linear calculation, R / K , (—) w ith nonlinear cal culation, 6 a ° fh K o 0, (--------) for a long wave w ith slope of 0.05 (h = 1.0 and K — ,05). (a) Horizontal polarization, (b) Vertical polarization. Calcula tions done a t 1 0 ° intervals. 155 CO o s- O o _ *5 rg O0 20 40 60 INC. ANG. (DEG.) W o m _J o . 0 40 20 60 INC. ANG. (DEG.) (b) Figure 6.10: Comparison of linear calculation, R /K , (—) w ith nonlinear cal culation, 6 o °jh K o a, (--------) for a long wave with slope of 0.1 [h = 1.0 and K — 0.1). (a) Horizontal polarization, (b) Vertical Polarization. Calcula tions done at 1 0 ° intervals. 156 and breaking waves. Furtherm ore, spectral analysis of SEAS AT S AR images of ocean waves shows no evidence of harm onic distortion, which would he expected if this type of nonlinearity were present [113,4]. Since the SEASAT SAR o p erated at an incidence angle of 20°, which is th e angle of m axim um nonlinearity according to our calculations, it appears th a t the nonlinearity discussed here is not a problem . As m entioned a t th e beginning of this sec tion nonlinear hydrodynam ic effects have been neglected in our calculations. However, th e lack of identifiable distortion in the SEASAT images suggests th a t th eir effect is small. We conclude th a t imaging of ocean waves should be approxim ately linear for all angles of incidence, so far as cross section m od ulation effects are concerned. Nonlinearities relevant to SAR (introduced by surface m otion effects) are discussed by Alpers [114]. 6.7 Summary The existing theory of cross section m odulation due to long waves has been extended to the case of two-scale scattering. It was found th a t the inclusion of quasispecular scattering makes a significant difference in the m odulation transfer function at small incidence angles. However, a t larger incidence angles, where Bragg scattering dom inates, we found only a sm all change in the m odulation transfer function. Thus, for incidence angles greater than about 30°, th e M TF derived using Bragg instead of two-scale scattering is a good approxim ation. W hen two-scale scattering is used, the m odulation transfer function has a m axim um at about 10°. We found th a t nonlinear effects are greatest a t 2 0 °, b u t under m ost conditions on a real ocean these effects should be small, so th a t for wave observation, imaging radars are best 157 operated a t sm all incidence angles. Com parison w ith d a ta showed th a t the am plitude of th e theoretical M T F is som ewhat sm aller th an m easurem ents and th a t th e phase is considerably larger than m easurem ents, indicating th a t laboratory m easurem ents of th e hydrodynam ic m odulation phase differ from open ocean values or th a t smother m odulation m echanism is present. 158 Chapter 7 Conclusions and Recommendations for Future Research 7.1 Conclusions As a result of this research we have gained better understanding of scattering from rough surfaces. In Chapter 2 we used small perturbation theory and physical optics to calculate the cross section of random rough surfaces. Both the perturbation theory and physical optics predict Bragg scattering as the mechanism for scattering from slightly rough surfaces. However, the angular dependences of the two cross sections are different and exam ination of the surface currents showed th at the two m ethods are equivalent only in the case of a flat surface. Physical optics is also used for very rough surfaces and, in this case, predicts quasispecular scattering. In Chapter 3 a numerical approach to rough surface scattering based on the m ethod of moments was described and applied to surfaces rough in one dimension. This numerical approach was used in Chapter 4 to study the validity of th e approxim ate analytical m ethods. The following results were obtained: 159 • T he Bragg scattering m echanism for slightly rough surfaces was veri fied. • The sm all perturbation theory cross section for vertical polarization was found to be larger th a n the m oment m ethod calculations at large incidence angles, and an empirical correction to the perturbation result was obtained. • A criterion for validity of the small pertu rb atio n theory was found in term s of the surface height variance. • The two-scale model was verified, and an error analysis was given in term s of th e large scale curvature and sm all scale height variances. • Q uasispecular scattering (as derived from physical optics) was found to be a special case of two-scale scattering. This work should allow the approxim ate analytical m ethods to be applied m ore knowledgeably in the future. In C hapter 5 a new high frequency spectrum for the ocean was proposed. The results of C hapter 4 indicate th a t the two-scale model is a good ap proxim ation for microwave scattering from an ocean described by this new spectrum . We combined the two-scale scattering model w ith our spectrum to create a m odel for the ra d a r cross section of th e ocean surface as a func tion of wind velocity and rad a r param eters. Com parison of calculations with observations revealed the following: • Calculated and observed slope variances are in good agreem ent. • Calculated and observed cross sections agree in their dependence on w ind speed, rad ar frequency, and incidence angle. 160 • Calculated cross sections for vertical polarization are always a few dB larger than observed values for large incidence angles. • Calculated upwind/downwind cross section differences are consistently smaller th an observations. • The calculated cross section has a much weaker dependence on atm o spheric stability (as measured by the air-sea tem perature difference) than the observed. After completing our comparison with observations, we used the model to investigate the effect of swell on wind speed measurements and found that: • Large errors in wind speed estimates can occur for radars operating a t lower frequencies (e.g., L-band) and smaller incidence angles (e.g., 20°). • Errors can be nearly eliminated by using high frequencies (e.g, K uband), large incidence angles (e.g, 50°), and vertical polarization. This work will allow wind measurement radars to be designed so th a t the effect of swell is minimized. In Chapter 6 the existing theory of cross section m odulation due to long waves was extended to include two-scale scattering. The following results were obtained: • At small incidence angles the m odulation transfer function derived us ing two-scale scattering is very different from the one derived using only Bragg scattering. 161 • At larger incidence angles there is little difference between the two theories, indicating th a t use of the two-scale model is n o t necessary for this case. • The greatest m odulation of two-scale scattering occurs for a radar op erating at an incidence angle of about 1 0 °. • Nonlinear effects upon wave scattering are greatest a t an incidence angle of 2 0 °, b u t are not significant under m ost real ocean conditions. T hese results indicate th a t for ocean wave observation, imaging radars are best operated a t small incidence angles. Com parison w ith d a ta showed th at; • The am plitude of the calculated m odulation transfer function is some w hat sm aller th an m easurem ents. • The phase of th e calculated m odulation transfer function is consider ably larger th an m easurem ents. Possible explanations for the discrepancies between calculations and observa tions include a difference between laboratory and open ocean m easurem ents of m odulation phase and presence of other m odulation m echanisms. 7.2 Recommendations for Future Research In scattering theory, priority should be given to explaining an d correcting the discrepancy between the sm all perturbation theory a n d the m ethod of m om ents for vertical polarization. Work should also be done on the exten sion of num erical m ethods to surfaces rough in two dimensions. A m om ent m ethod solution for arbitrary surfaces which divides th e surface into tria n gular patches has been developed [115]. However, as m entioned before, it 162 is not currently feasible to use this code for random rough surface calcula tions because of com putational time. If param eters similar to those used in the one-dimensional case are used, a 10000 x 10000 m atrix would need to be solved. This is too large to be handled by most com puters, and it is expected th a t some type of iterative solution would be required [116]. The m ost promising m ethods for this problem are the conjugate gradient method [117,118] and spectral methods [119]. There has been a m oderate am ount of effort over the last ten years to produce an analytical scattering theory which does not have th e limitations of the two-scale model. The diagram m ethod [1 2 0 ] uses Feynman diagram techniques to derive an infinite set of one-dimensional integral equations for the various statistical moments of the scattered field. This technique has been used to obtain a more accurate coherent intensity th an available from the physical*optics approximation. However, few other quantitative results have been derived. In the stochastic Fourier transform m ethod [121,122,123] the scattering problem for random surfaces has been reduced to a single finite dimensional integral equation. Numerical and asymptotic evaluation of this equation may provide some interesting results in the future. The full wave approach of B ahar [124] has been applied to surfaces not treatable by the two-scale m ethod. Unlike the two m ethods ju st mentioned, m any numerical results have been obtained. However, since multiple scattering is neglected in the full-wave theory, its validity needs to be further investigated. In the case of air-sea interaction, m uch work remains. The processes responsible for wave growth and dissipation need to be b etter understood. In particular, a theory of dissipation due to wave breaking and viscosity needs to be developed and included in th e transport equation. This step is 163 basic to all further theoretical progress. Secondly, th e effect of atm ospheric stability should be further investigated. If its only effect is to change the friction velocity for a given windspeed, then a new boundary layer model should be developed. If it affects the spectrum a t a given friction velocity, th en this effect m ust be understood and quantified. Finally, the m odulation of sh o rt waves by long waves needs to be b e tter understood so th a t theories of th e m odulation transfer function and the upw ind/dow nw ind difference in the cross section can be improved. As m entioned above, it appears th a t a m echanism in addition to hydrodynam ic m odulation, such as m odulation of airflow by large scale waves, is im portant. If this is true, this m echanism needs to be identified and incorporated into th e two-scale model a n d into m odulation transfer function theories. In su m m ary, work should be done in the following areas: • correction of the discrepancy between sm all perturbation and m om ent m ethod calculations for vertical polarization, • extension of the num erical m ethod to surfaces rough in two dimensions, • development of new analytical scattering theories, • development of a theory for dissipation by breaking waves and viscosity, • explanation of the effect of atm ospheric stability on th e ra d a r cross section, and • development of a m ore com plete theory of short wave m odulation by long waves. 164 Bibliography [1] G. R . Valenzuela, “Theories for the interaction of electrom agnetic and ocean waves - a review,” Boundary Layer Meteorol.,vol. 13, pp. 61-85, 1978. [2] R. K. Moore and A. K . Fung, “R adar determ ination of w inds a t sea,” Proe. IE E E , vol. 67, no. 11, pp. 1504-1521, Nov. 1979. [3] K. Tomiyasu, “T utorial review of synthetic aperture ra d a r (SAR) with applications to imaging of the ocean surface,” Proc. IE E E , vol. 66 , no. 5, pp. 563-583, M ay 1978. [4] J. F . Vesecky and R. H. Stewart, “T h e observation of ocean sur face phenom ena using imagery from th e SEASAT synthetic aperture rad ar: an assessment,” J. Geophys. Res., vol. 87, no. c5, pp. 33973430, April 1982. [5] C. Elachi, T. Bicknell, R. L. Jordan, an d C. Wu, “ Spaceborne syn thetic aperture im aging radars: applications, techniques, and tech nology,” Proe. IE E E , vol. 70, no. 10, pp. 1174-1209, O ct. 1982. [6 ] E. G. Njoku, “Passive microwave rem ote sensing of th e E a rth from space - a review,” Proe. IEEE, vol. 70, no. 7, pp. 728-750, July 1982. 165 [7] H. V. Hitney, J. H. Richter, R. A. P ap p e rt, K. D. Anderson, and G. B. B aum gartner, “ Tropospheric radio propagation assessm ent,” Proe. IE E E , vol. 73, no. 2, pp. 265-283, Feb. 1985. [8 ] F . T . Ulaby, “R adar signatures of terrain: useful m onitors of renew able resources,” Proe. IE E E , vol. 70, no. 12, pp. 1410-1428, Dec. 1982. [9] C .-T . Tai, Dyadic Green’s Functions in Electromagnetic Theory. Scran ton: Intext Educational Publishers, 1971. [10] A. K . Fung and H.-L. Chan, “Backscattering of waves by composite rough surfaces,” IE E E Trans. A ntennas Propagate vol. AP-17, no. 5, pp. 590-597, Sept. 1969. [11] S. O. Rice, “Reflection of electrom agnetic waves from slightly rough surfaces,” Commun. Pure Appt. M ath., vol. 4, pp. 351-378,1951. [12] G. R . Valenzuela, “Depolarization of electromagnetic waves by slightly rough surfaces,” IE E E Trans. Antennas and Propagat., vol. AP-15, no. 4, pp. 552-557, July 1967. [13] F . G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces. New York: Pergam on Press, 1979. [14] A. Ishim aru, Wave Propagation and Scattering in Random Media. New York: Academic, 1978. [15] A. K . Fung, “Review of random surface scatter m odels,” Applica tions o f mathematics in modern optics: Proceedings of the conference, SP IE , San Diego, CA, August 1982. 166 [16] J . R. W ait, “P erturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci., vol. 6 , no. 3, pp. 387-391, M ar. 1971. [17] P. Beckm ann and A. Spizzichino, The Scattering o f Electromagnetic Waves from Rough Surfaces. New York: McMillan, 1963. [18] T. Hagfors, “Scattering and transm ission of electrom agnetic waves a t a statistically rough boundary between two dielectric media,” in Electromagnetic Wave Theory II, J. Brown, Ed. New York: Pergamon Press, 1966, pp. 997-1012. [19] D. E. Baxrick, “Rough surface scattering based on the specular point theory,” IE E E Trans. A ntennas P r o p a g a tvol. AP-16, no. 4, pp. 449-454, July 1968. [20] — , “Relationship between slope probability density function and the physical optics integral in rough surface surface scattering,” Proe. IE E E , vol. 58, no. 10, pp. 1728-1729, 1970. [21] C. A. Bender and S. A. Orszag, Advanced M athematical Methods for Scientists and Engineers. New York: McGraw-Hill, 1978. [22] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1965. [23] J . C. Leader, “T he relationship between the Kirchhoff approach and sm all pertu rb atio n analysis in rough surface scattering theory,” IE E E Trans. A ntennas Propagat., vol. AP-20, pp. 786-787, Nov. 1971. [24] G. R . Valenzuela, J . W. W right, and J . C. Leader, “Com ments on the relationship between the Kirchhoff approach and sm all perturbation 167 analysis in rough surface scattering theory,” IE E E Trans. A ntennas Propagat., vol. AP-20, no. 4, pp. 536-539, July, 1972. [25] J. W. W right, “A new model for sea clutter,” IE E E Trans. A ntennas Propagat., vol. AP-16, no. 2, pp. 217-223, M ar. 1968. [26] F . G. B ass, I. M. Fuks, A. I. Kalmykov, I. E. Ostrovsky, and A. D. Rosenberg, “Very high frequency radiowave scattering by a disturbed sea surface,” IE E E A ntennas Propagat., vol. AP-16, no. 5, pp. 554568, Sept. 1968. [27] G. R. Valenzuela, “Scattering of electrom agnetic waves from a tilted slightly rough surface,” Radio Set., vol. 3, no. 11, pp. 1057-1066, Nov. 1968. [28] G. R. Valenzuela, M. B. Laing, and J. C. Daley, “Ocean spectra for the high frequency waves as determ ined from airborne ra d a r m easurm ents,” J. M arine Res., vol. 29, no. 2 , pp. 69-84, M ay 1971. [29] G. L. Tyler, “Wavelength dependence in radio-wave scattering and specular-point theory,” Radio Set., vol. 11, no. 2, pp. 83-91, Feb. 1976. [30] G. S. Brown, “Backscattering from a G aussian-distributed perfectly conducting rough surface,” IE E E Trans. A ntennas P r o p a g a t vol. AP-26, no. 3, pp. 472-482, M ay 1978. [31] E. B ahar, “Scattering cross sections for com posite random surfaces: full wave analysis,” Radio Set., vol. 16, no. Dee. 1981. 168 6 , pp. 1327-1335, Nov.- [32] E. Bahar, D. E. Barrick, and M. A. Fitzw ater, “C om putations of scat tering cross sections for composite surfaces and the specification of the wavenumber where spectral splitting occurs ,’1 IE E E Trans. Antennas Propagat., vol. AP-31, no. 5, pp. 698-709, Sept. 1983. [33] R . R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surface,” Radio Sei.t vol. 9, no. 12, pp. 1139-1146, Dec. 1974. [34] R. B. Blackman and J . W. Tukey, The Measurement of Power Spectra. New York: Dover, 1958. [35] F . J. Harris, “On th e use of windows for harmonic analysis with the discrete Fourier transform ,” Proe. IE E E , vol. 66 , no. 1, pp. 51-83, Jan . 1978. [36] H. L. Chan and A. K. Fung, “A num erical study of th e Kirchhoff approximation in horizontally polarized backscattering from a random rough surface,” Radio Sei.t vol. 13, no. 5, pp. 811-818, Sept.-Oct. 1978. [37] K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly con ducting surface w ith a sinusoidal height profile: TE polarization,” IE E E TVons. A ntennas Propagat., vol. AP-19, no. 2 , pp. 208-214, M arch 1971. [38] — and —, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TM polarization,” IE EE 3Vons. Antennas Propagat., vol. AP-19, no. 6 , pp. 747-751, Nov. 1971. 169 [39] A. K. Jordan and R. H. Lang, “Electromagnetic scattering patterns from sinusoidal s u r f a c e s Radio Set,, vol. 14, no. 6 , pp. .1077-1088, Nov.-Dee. 1979. [40] S.-L. Chuang and J. A. Kong, “Scattering of waves from periodic surfaces,” Proe. IE E E , vol. 69, no. 9, pp. 1132-1144, Sept. 1981. [41] R. M. Axline, Ph.D. Dissertation, University of Kansas, 1975. [42] R. M. Axline and A. K. Fung, “Numerical com putation of scattering from a perfectly conducting rough surface,” IE E E Trans. Antennas and Propagat., vol. AP-26, no. 3, pp. 482-488, May 1978. [43] A. K. Fung and H. J. Eom, “Note on the Kirchhoff rough surface solution in backscattering,” Radio Sci., vol. 16, no. 3, pp. 299-302, M ay-June 1981. [44] A. K. Fung and M. F. Chen, “The effect of wavelength filtering in rough surface scattering,” Proceedings of IG A R S S ’84 Symposium, Strasbourg, August 27-30, 1984. [45] E. F. K nott, “A progression of high-frequency RCS prediction tech niques,” Proe. IE E E , vol. 73, no. 2, pp. 252-264, Feb. 1985. [46] P. H. Y. Lee, “Laboratory m easurements of polarization ratios of wind wave surfaces,” IE E E A ntennas Propagat., vol. AP-26, no. 2, pp. 302-306, M ar. 1978. [47] J. Renau, P. K. Cheo, and H. G. Cooper, “Depolarization of linearly polarized em waves backscattered from rough m etals and 170 inhomogeneous dielectrics,” J. Opt. Soc. A m er., vol. 57, no. 4, pp. 459-566, April 1967. [48] G. T . Ruck, D. E. Barrick, W. D. S tuart, and C. K. Krichbaum, Radar Cross Section Handbook. New York: Plenum , 1970. [49] F. J. Wentz, “A two-scale model for foam-free sea microwave bright ness tem peratures,” J. Geophys. Res., vol. 80, no. 24, pp. 3441-3446, Aug. 1975. [50] D. E. Barrick, “W ind dependence of quasispecular microwave sea scat ter,” IE E E Trans. Antennas Propagat., vol. AP-22, no. 1, pp. 135136, Jan. 1974. [51] H.-L. Chan and A. K. Fung, “A theory of sea scatter a t large incidence angles,” J. Geophys. Res., vol. 82, no. 24, pp. 3439-3444, Aug. 1977. [52] M. A. Donelan and W. J. Pierson, “Bragg-scattering and equilibrium ranges in wind-generated waves - w ith application to scatterom etry,” subm itted to J. Geophys. Res., 1985. [53] G. S. Brown, “Estim ation of surface wind speeds using satellite-borne radar m easurm ents at norm al incidence,” J. Geophys. Res., vol. 84, no. B 8 , pp. 3974-3978., July 1979. [54] A. K. Fung and K. K. Lee, “A semi-empirical sea-spectrum model for scattering coefficient estim ation,” IE E E J. Oceanic Eng., v ol. OE-7, no. 4, pp. 166-176, Oct. 1982. [55] M. A. Donelan, J. Hamilton, and W. H. Hui, “Directional spectra of wind-generated waves," Phil. Trans. R. Soc. Lond. A , vol. 315, pp. 171 509-562, 1985. [56] O. M. Phillips, The D ynam ics o f the Upper Ocean. New York: Cam bridge University Press, 1977. [57] A. S. M onin, V. M. Kamenkovich, an d V. G. K ort, Variability of the Oceans. New York: Wiley, 1977. [58] Yu. A. Sinitsyn, I. A. Leykin, and A. D. Rozenberg, “T he space-time characteristics of ripple in the presence of long waves,” Izv., Atmos. Oceanic Phys., vol. 9, no. 5, pp. 511-519, May 1973. [59] K. Hasselm ann, “On th e nonlinear energy transfer in a gravity-wave spectrum ,” J. Fluid Mech., vol. 12 , pp. 481-500, 1961. [60] W. R . Alpers and K. Hasselm ann, “T he two-frequency microwave technique for m easuring ocean-wave spectra from an airplane or satel lite,” Boundary Layer Meteorol.,vol. 13, pp. 215-230, 1978. [61] P. H. LeBlond and L. A. M ysak, Waves in the Ocean. Am sterdam : Elsevier, 1978. [62] O. M. Phillips, “On th e generation of waves by tu rb u len t wind,” J. Fluid Mech., vol. 2 , pp. 417-445, 1957. [63] J . W . Miles, “On th e generation of surface waves by shear flows,”J. Fluid Mech., vol. 3, pp. 185-204, 1957. [64] — , “On the generation of surface waves by shear flows. P a rt 4.,” J. Fluid Mech., vol. 13, pp. 433-448, 1962. 172 [65] G. R . Valenzuela, “T he growth of gravity-capillary waves in a coupled shear flow,” J. Fluid Mech., vol. 76, p a rt 2, pp. 229-250, July 1976. [6 6 ] W . J. P lan t and J . W. W right, “Grow th and equilibrium of short gravity waves in a wind-wave tank,” J. Fluid Mech., vol. 82, p a rt 4, pp. 767-793, Oct. 1977. [67] P. R. Gent and P. A. Taylor, “A num erical model of the air flow above w ater waves,” J . Fluid Mech., vol. 77, p a rt 1, pp. 105-128, Sept. 1976. [6 8 ] T. R. Akylas, “A note on the generation of surface waves by inviscid shear flows,” Wave M otion, vol. 6 , pp. 141-148, 1984. [69] W . J . P lant, “A relationship between wind stress and wave slope,” J. Geophys. Res., vol. 87, no. c3, pp. 1961-1967, M arch 1982. [70] G. R. Valenzuela and M. B. Laing, “Nonlinear energy transfer in gravity-capillary spectra, w ith application,” -7. Fluid Mech., vol. 54, p a rt 3, pp. 507-520, Aug. 1972. [71] M. L. Banner and O. M. Phillips, “On th e incipient breaking of small scale waves,” J. Fluid Mech., vol. 65, pp. 647-656,1974, [72] J. Wu, “W ind-induced drift currents,” J. Fluid Mech., vol. 1 68 , p a rt , pp. 49-70, M arch 1975. [73] W . J . Pierson and L. Moskowitz, “A proposed spectral form for fully developed wind seas based on the sim ilarity theory of S. A. Kitaigorodskii,” J. Geophys. Res., vol. 69, pp. 5181-5190,1964. 173 [74] H. M itsuyasu, “M easurem ents of the high-frequency spectrum of ocean surface waves,” J. Phys. Oceanogr., vol. 7, no. 6 , pp. 883-891, Nov. 1977. [75] K. K. K ahm a, “A study of the growth of the wave spectrum w ith fetch,” J. Phys. Oceanogr., vol. 11, no. 12, pp. 1503-1515, Nov. 1981. [76] I, A. Leykin and A. D. Rosenberg, “Sea-tower m easurm ents of windwave spectra in the Caspian Sea,” J . Phys. Oceanogr., vol. 14, no. 1, pp. 168-176, Jan . 1984. [77] W. J. Pierson, “The theory and applications of ocean wave m easuring systems a t and below the sea surface, on the land, from aircraft, and from spacecraft,” NASA C ontractor Rep., CR-2646, N 76-17775,1976. [78] C. Cox an d W . M unk, “Statistics of the sea surface derived from sun glitter," J. M arine Res., vol. 13, pp. 198-227, 1954. [79] O. M. Phillips, “Spectral and statistical properties of th e equilibrium range in w ind-generated gravity waves,” J. Fluid M eek., vol. 156, pp. 505-531, 1985. [80] W. L. Jones and L. C. Schroeder, “R adar backscatter from the ocean: dependence on surface friction velocity,” Boundary Layer Meteorol., vol. 13, pp. 133-149, 1978. [81] W . G. Large and S. Pond, “Open ocean m om entum flux m easure m ents in m oderate to strong w inds,” J. Phys. Oceanogr., vol. 3, pp. 324-336, M arch 1981. 174 11, no. [82] W. C. Keller, W . J. P lant, and D, E. Weissman, “The dependence of X -band microwave sea return on atmospheric stability an sea sta te ,” J. Geophys. Res., vol. 90, no. c l, pp. 1019-1029, Jan. 1985. [83] V. W ism ann, W. C. Keller, and F. Feindt, “The dependence of the X - and C -band rad ar backscatter cross section on air-sea tem pera ture difference m easured a t the N orth Sea Research Platform ,” URSI Symposium, Amherst, MA, 1985. [84] S. Tang and O. H. Shemdin, “Measurement of high frequency waves using a wave follower,” J. Geophys. Res., vol. 88 , no. cl4 , pp. 9832- 9840, November 1983. [85] J. Wu, “Slope and curvature distribution of wind-disturbed water surface,” J. Opt. Soc. A m ., vol. 61, pp. 852-858,1971. [8 6 ] J. W . W right, “Detection of ocean waves by microwave radar: the m odulation of short gravity-capillary waves,” Boundary Layer M eteorol., vol.13, pp. 87-105, 1978. [87] W. L. Jones, Frank J. Wentz, and Lyle C. Schroeder, “Algorithm for inferring wind stress from SEASAT-A,” J. Spacecraft and Rockets, vol. 15, nol 6 , pp. 368-374, Nov.-Dee. 1978. [8 8 ] L. A. Klein and C. T. Swift, “An improved model for the dielectric constant of sea w ater at microwave frequencies,” IE E E Trans. A n ten nas Propagat., vol. AP-25, no. 1, pp. 104-111, Jan . 1977. [89] A. E. Gill, Atmosphere-Ocean Dynamics. New York: Academic, 1982. 175 [90] W . L. Jones, L. C. Schroeder, and J. L. Mitchell, “A ircraft mea surem ents of the microwave scattering signature of the ocean,” IE EE Trans. Antennas P r o p a g a tvol. AP-25, no. 1, pp. 52-61, Jan. 1977. [91] R. T . Lawner and R. K. Moore, “Short gravity and capillary wave spectra from tower-based rad ar,” IE E E J. Oceanic Eng., vol. OE-9, no. 5, pp. 317-324, Dec. 1984. [92] Lyman G. P a rra tt, Probability and Experimental Errors in Science. New York: Wiley, 1961, pp. 114-115. [93] N. W. Guinard, J. T. Ransone, Jr., and J. C. Daley, “Variation of the NRCS of the sea with increasing roughness,” J. Geophys. Res., vol. 76, no. 6 , p p .1525-1538,1971. [94] J. C. Daley, “W ind dependence of rad ar sea return,” J . Geophys. Res., vol. 78, no. 33, pp. 7823-7833, Nov. 1973. [95] J. D. Young and R. K. Moore, “Active microwave m easurem ent from space of sea-surface winds,” IE E E J . Oceanic Eng., vol. OE-2, no. 4, pp. 309-317, Oct. 1977. [96] L. C. Schroeder, W. L. G rantham , J. L. Mitchell, and J. L. Sweet, “SASS measurements of the K u-band radar signature of the ocean,” J. Oceanic Eng., vol. OE-7, no. 1, pp. 3-14, Jan. 1982. [97] H. Masuko and S. Niwa, “M easurements of microwave backscattering signatures of the ocean surface using X -band and K a-band airborne scatterom eter/radiom eter system,” Proc. Aug. 1984. IG A R S S ’84, Strasbourg, [98] D. L. Johnstone, “Second-order electromagnetic and hydrodynamic effects in high-frequency radio-wave scattering from the sea,” Ph.D . dissertation, Stanford University, Stanford, March 1975. [99] P. L. Jackson and R. A. Shuchman, “High-resolution spectral esti m ation of synthetic aperture radar ocean wave imagery,” J. Geophys. Res., vol. 88, no. c4, pp. 2593-2600, M ar. 1983. [100] S. L. Durden and J. F. Vesecky, “A physical rad ar cross section model for a wind-driven sea with swell,” IE E E J . Oceanic Eng.y vol. OE- 1 0 , no. 4, pp.445-451, Oct. 1985. [101] E. Bahar, C. L. Rufenach, D. E. Barrick, and M. A. Fitzw ater, “Scattering cross section m odulation for arbitrarily oriented compos ite rough surfaces,” Radio Sci., vol. 18, pp. 675-690, 1983. [1 0 2 ] W. C. Keller and J. W. W right, “Microwave scattering and the strain ing of wind generated waves,” Radio Sci.y vol. 10, no. 2, pp. 139-147, Feb. 1975. [103] C* Elachi and W. E. Brown, Jr., “Models of radar imaging of ocean surface waves,” IE E E Trans. Antennas Propagat.,vol. AP-25, pp. 84-95, 1977. [104] W. R. Alpers, D.B. Ross, and C. L. Rufenach, “On th e detectabil ity of ocean surface waves by real and synthetic aperture radar,” J. Geophys. Res.,vol. 86 , pp. 6481-6498, 1981. [105] O. M. Phillips, “The structure of short gravity waves on the ocean surface,” in Spaceborne Synthetic Aperture Radar for Oceanography, 177 R. C. Beal, P. DeLeonibus, and I. K atz, eds. Baltimore: Johns Hop kins Press, 1981, pp. 24-31. [106] E. Caponi, D. Kwoh, B. Lake, H. Rungaldier, and H. Yuen, “Real A perture R adar Backscatter M odulations,” in The Georgia Strait Ex periment, Chapter 4. [107] O. M. Phillips, “The dispersion of short wavelets in the presence of a dom inant long wave,” J. Fluid Meek., vol. 107, pp. 465-485, June 1981. [108] J. F . Vesecky, S. L. Durden, M. P. Smith, and D. J. Napolitano, “Synthetic aperture radar images of ocean waves: theories of imaging physics and experimental tests,” in Frontiers of Remote Sensing of the Oceans and Troposphere from A ir and Space Platforms, NASA Conference Publication 2303. Washington, DC: NASA, 1984. [109] W. J. Plant, W . C. Keller, and J. W. Wright, “Modulation of coherent microwave backscatter by shoaling waves,” J. Geophys. Res. vol. 83, no. c3, pp. 1347-1352, M arch 1978. [1 1 0 ] J. W. Wright, W . J. Plant, W. C. Keller, and W. L. Jones, “Ocean wave-radax m odulation transfer functions from the West Coast Ex perim ent,” J. Geophys. Res., vol. 85, no. c9, pp. 4957-4966, Sept. 1980. [111] W. J. Plant, W. C. Keller, and A. Cross, “Param etric dependence of ocean wave-radar m odulation transfer functions,” J. Geophys. Res., vol. 88 , no. cl4, pp. 9747-9756, Nov, 1983, 178 [112] J. Smith, “Short surface waves with growth and dissipation,” J. Geo phys. Res., vol. 91, no. c2 , pp. 2616-2632, Feb. 1986. [113] J. F . Vesecky, H. M. Assal, and R. H. Stewart, “Remote sensing of the ocean waveheight spectrum using synthetic aperture radar images,” in Oceanography From Space, J. F . R. Gower, ed. New York: Plenum, 1981, pp. 449-457. [114] W. Alpers, “M onte Carlo simulations for studying the relationship between ocean wave and synthetic aperture radar image spectra ” J. Geophys. Res., vol. 88 , no. c3, pp. 1745-1759, 1983. [115] S. M. Rao, D. R. W ilton, and A. W. Glisson, “Electromagnetic scat tering by surfaces of arbitrary shape,” IE E E Trans. Antennas Prop agate vol. AP-30, no. 3, pp. 409-418, May 1982. [116] T . K. Sarkar, K. R . Siarkiewicz, and R. F. Stratton, “Survey of nu merical m ethods for solution of large systems of linear equations for electromagnetic field problems,” IE E E Trans. Antennas Propagate vol. AP-29, no. 6 , pp. 847-856, Nov. 1981. [117] — and S, M. Rao, “The application of the conjugate gradient m ethod for the solution of electromagnetic scattering from arbitrarily oriented wire antennas,” IE E E Trans. Antennas Propagat., vol. AP-32, no. 4, pp. 398-403, April 1984. [118] — and E. Arvas, “On a class of finite step iterative methods (con jugate directions) for the solution of an operator equation arising in electromagnetics,” IE E E Trans. Antennas Propagat., vol. AP-33, no. 10 , pp. 1058-1066, Oct. 1985. [119] R . K astner an d R . M ittra , “A sp ectral-iteratio n technique for analyz ing scattering from a rb itra ry bodies, p a rt I: cylindrical sc attere rs w ith E-wa.ve incidence,” IE E E Trans. A ntennas Propagat., vol. A P-31, no. 3, pp. 499-506, M ay 1983. [1 2 0 ] J . A. D eSanto, "G reen’s functions for electrom agnetic sc atterin g from a ran d o m surface,” J. M ath. Phys., vol. 15, p p . 283-288, 1974. [1 2 1 ] G. S. Brown, “A stochastic Fourier tra n sfo rm approach to scattering from a ran d o m surface," IE E E Trans, A n te n n a s Propagat., vol. AP30, no. 11 , pp. 1135-1144, Nov. 1982. [1 2 2 ] — , “A pplication of th e integral equation m eth o d of sm oothing to ran d o m surface scatterin g ,” IE E E Trans. A n ten n a s Propagat., vol. A P-32, no. 12 , pp. 1308-1312, Dec. 1984. [123] — , "Sim plifications in th e stochastic F o u rier tran sfo rm approach to ran d o m surface scatterin g ,” IE E E Trans. A n ten n a s Propagat., vol. AP-33, no. 1 , pp. 48-55, Ja n . 1985. [124] E . B ah ar and D . E. B arrick, “Scattering cross sections for com posite surfaces th a t can n o t be trea te d as p e rtu rb e d physical optics prob lems,” Radio S c i., vol. 18, pp. 129-137, 1983. 180

1/--страниц