1969-1-Q 1969-1-Q = RL-2038 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory DOPPLER RADIATION STUDY Interim Report No. 1 1 July - 1 October 1968 By C-M Chu Contract N62269-68-C-0715 December 1968 Contract With: Naval Air Development Center Johnsville, Warminster, Pennsylvania 18974 Administered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

DOPPLER RADIATION STUDY Interim Report No. 1 1969-1-Q (1 July - 1 October 1968) By Chiao-Min Chu Contract No. N62269-68-C-0715 Prepared for U.S. Naval Air Development Center Johnsville Warminster, Pennsylvania 18974

1969-1-Q FOREWORD This report (1969-1-Q) was prepared by the University of Michigan Radiation Laboratory, Department of Electrical Engineering. The report was written under Contract N62269-68-C-0715 "Doppler Radiation Study" and covers the period 1 July - 1 October 1968. The research was carried out under the direction of Professor Ralph E. Hiatt, Head of the Radiation Laboratory, and the Principal Investigator was Professor ChiaoMin Chu. The sponsor of this research is the U. S. Naval Air Development Center, Johns ville, Pennsylvania and the Technical Monitor was Mr. Edward Rickner i

1969- 1 -Q ABSTRACT In order to arrive at a model for the bistatic cross-section of a rough surface, which is adequate for the estimation of the reflected radiation from a doppler radar, a survey of the literature on the experimental work and theoretical models, concerning rough surface scattering, has been carried out. As a result of this survey, the dominant characteristics of radiation reflected from rough surfaces, and the basic approaches in arriving at various theoretical models are summarized. It is suggested that for the present work, a theoretical model based on the statistics of the slope of the surface, incorporating the effect of wind is probably adequate. An outline of the approach in arriving at the expressions for the bistatic cross-section based on the slope characteristics is given. ii

1969-1-Q TABLE OF CONTENTS ABSTRACT ii I INTRODUCTION 1 II REFLECTIVE PROPERTIES OF ROUGH SURFACES 3 2.1 Introduction 3 2.2 The Scattering Cross-Section 3 2.3 Experimental Determination of a (6, ~) 7 2.4 Survey of Experimental Results 9 2.4.1 Aspect Variation 9 2.4.2 Frequency Dependence 10 2.4.3 Polarization Effects 11 2.4.4 Effect of Wind and Sea State 11 III THEORETICAL MODELS FOR ROUGH SURFACE 13 3.1 Introduction 13 3.2 The Method of Series Expansion 14 3.3 Kirchhoff's Integral Formulation 16 3.4 Model of Specular Reflecting Regions 24 3.5 The Kirchoff Integral and Statistical Average (Physical Optics) 32 IV CONCLUSIONS AND RECOMMENDATIONS 36 LIST OF ILLUSTRATIONS 37 BIBLIOGRAPHY 38 DD FORM 1473 iii

1969-1-Q I INTRODUCTION This is the first quarterly report on Contract N6229-68-C-0715, "Doppler Radiation Study, " and covers the period 1 July through 1 October 1968. Contract N6229-68-C-0715 is a continuation of Contract N62269-67-C-0545, in the study of the characteristics of the radiation from airborne doppler navigational radar systems. The conclusions derived from the numerical computations of the radiation from a doppler antenna such as AN/ADN-153 seem to indicate that the diffusely reflected radiation from the ground is appreciable and should be investigated more thoroughly. Since our previous calculations were based on the simple model of a Lambert surface, the principal effort of this contracting period is to investigate the realistic model for the reflective properties of a rough surface (particularly of the sea surface) that are appropriate in the calculation of the diffusely reflected radiation. An extensive literature survey on the theoretical and experimental results concerning the reflection from rough surfaces was carried out. It has been found that various phenomenological and/or statistical models have been proposed to characterize the reflective properties of rough surfaces. Some of these models have been experimentally checked under specific experimental conditions, such as a given frequency, fixed angle of incidence or reflection, or both, and therefore it is difficult to appraise their validity in other situations. Some of these models are theoretically plausible and allow the "scalng" of the reflective properties in wavelength and/or some characterisitic roughness of the surface, but have not been checked experimentally. In Chapter I of this report, the experimental results on the basic characteristics of the reflected radiation from rough surfaces (particularly the sea surface), which are reported in the open literature available, are summarized. In Chapter m, various theoretical and phenomenological models concerning the reflection of waves from rough surfaces are summarized, and in some cases extended. 1

1969-1-Q It is hoped that a detailed correlation study between the experimental results and various proposed theoretical models will enable us to choose, or propose a practical model for the reflective properties of a rough surface (particularly the sea), which is appropriate for the present work. It is also expected that the proposed model may involve some numerical constants that can be determined experimentally, and suggest some simple, scaled experiments for their determination. This phase of the work has been started in this research period and shall be the primary objective for the next research period. 2

1969-1-Q II REFLECTIVE PROPERTIES OF ROUGH SURFACES 2.1 Introduction The study of the diffused reflection from a rough surface has been conducted experimentally and theoretically during the past few decades in connection with radar echoes from rough surfaces. An extensive survey of the theoretical and experimental results has been given in the book by Bechmann and Spizzichino (1963). Since then, additional theoretical and experimental investigations have been carried out by various investigators in the attempt to clarify the effect of polarization, angle of incidence and surface roughness, on the magnitude and angular distribution of the power scattered by a rough surface. It is the object of this section to summarize the results of these studies, including those summarized by Bechmann and Spizzichino, in order to choose a realistic model of surface reflection for the study of doppler radiation. For the present, it appears that the rough terrain model is the most appropriate one in order to investigate the reflected electromagnetic field. The discussion which follows is based on this model. 2.2 The Scattering Cross-Section The average quantity which is used to characterize the reflective properties A A of a rough surface is the bistatic cross-section o(Q2, Q2), per unit area. It is conventionally defined as 4ir times the ratio of the average power scattered per unit area of the surface, per unit solid angle in the direction 2* to the incident A 2 power from direction n1 per unit area of the surface. As illustrated in Fig. 2-1, the mean level of the scattering surface may be taken as the z-plane, the incident direction Q1 may be expressed in terms of two angles 01 and p by S = -[x sin 01cos 01+y sin 0 sin 01+ z cos 1j. (2.1) 3

1969-1-Q A Direction Q2 may be expressed in terms of two angles 02, 02 by A A A 2 = (x sin2 cos 2 + ysin 2 sin 2 + z os2 ) (2.2) A Then, for an incident power P1 in the direction /1, the scattered power interA cepted by an area dA2 normal to 22 and at a distance r2 from dA is given by 1 dA cos0 ^ P2 2 2 (1' 2 (23) 47 r2 A A In general, the bistatic cross-section a({ol Q2) depends on the polarization of incident and reflected radiation. To fix the direction of polarization, we shall define the direction of the horizontal polarization of the incident radiation by Y^ A zxQ2 eh (2.4) and the diri of the *vrtioal pilarization by A A e 21 xe (2.5) 1 h1 A Similarly, for the scattered radiation in the direction 2, the directions of horizontal and vertical polarization are defined by A A zx2 (2.6) eh2= iXn ~2 and A A A e =Q xeh (2.7) 2 2 h2 A respectively. These directions are illustrated in Fig. 2-1. If Q2 is the specularly reflecting direction, so that Q = 1 -22 (z 1 (2.8) 2 1 1 4

z scattering area dA FIG. 2-1: DIRECTIONS OF INCIDENT AND REFLECTED WAVES AND THE DIRECTIONS OF POLARIZATION. N. 5

1969-1-Q then A A ete =eh (2.9) N h! and A A e = - e. (2.10) v2 v1 A On the other hand, if Q2 is the backscattering direction, so that A. A, 02 1, (2.11) then A A e -he (2.12) 2 h and A A e =e. (2.13) V2 1 To stress the polarization dependence of the scattering cross-section, we may consider four types of scattering cross-section CAB, where A and B may stand for h or v. For example, ah is the scattering cross-section corresponding to horizontally polarized scattered radiation when the Incident wave is vertically polarized. Experimental data concerning the bistatic cross-section are very scarce. In the work of Hunter and Senior (1966), Pidgeon (1966) and others, where the bistatic cross-section is measured, the incidence angle 91 is limited only to nearly 900 degrees and the reflection direction is either nearly specular or nearly in the backscattering direction. The detailed information on the directional distribution of the scattered power from a rough surface in the microwave range has not been so far reported in experimental work. On the other hand, experimental work on the backscattering cross-section has been reported extensively. In the subsequent sections of this chapter, we shall therefore confine our attention to the backscattering cross-section. In the 6

1969-1-Q backscattering direction, since 2 la 0, 02 = 0 (2.14) 9 we may represent a(1, 2) a (6e, ) (2.15) 2.3 Experimental Determination of a(6, 0) In the experimental work, the backscattering cross-section is generally correlated with the backscattered power. For a c.w. system, it can be easily seen that the backscattered power is 22 PtGt 2 r (4r) 3R f dOd cos2 6f4 (o.0)a(.) (2.16) where Pt = transmitted power Gt= L gain of the transmitting antenna x R = operating wavelength = distance from the transmitter to the scattering area and f (0, 0) = the field pattern of the transmitting antenna. Therefore, for a wide beam transmitting antenna, only some spatial average of the scattering cross-section is measured. For a narrow beam, we assume f(eo) = 1 = 0 o-te 2i otherwise, (2.17) 7

1969-1-Q where {, and () are the angular beam width of the transmitting antenna, and 9 0 o is the direction of the major lobe of the transmitting beam, then PtG2X2. cos a(, ) (2.18) r (47r)3R2 ~ ~ ~ Alternately, the backscattering cross-section can be determined from the average power returned by using wide beam transmitter and short pulsed signal. In that case, the returned power corresponding to each pulse spreads in time, and may be identified for any Interval of time as the scattered power from each portion of the surface, where 0 and 0o may be different. Such a scheme has been used by Edison et al (1960). For a transmitter with its major lobe incident on the surface, the relation a (, 0) may be expressed in the following form 2 P (t) tr- 0- - (2.19) Pt+ T t(d-t) G2(t) dt 2 at 3 64r2 (h+ 2) where h - height of the antenna t = time measured from the starting of received pulse c = velocity of light d = time delay from the starting of the transmitted pulse and the,.received pulse T= pulse width and cos = - (t If one assumes that the scattering surface is isotropic, so that a(6 0,0 ) is independent of o0, then 8

1969-1-Q Pt(r) 0 2 t+T P (d-t)G2(t)dt (2.20) 647r t (h+ )3 which is the formula used by Edison, et al. The limitation here, besides the assumption of isotropic surface, is that the result is correct only for a small range of 8. Most experimental results reported in the literature, based their calculation of a(O, 0) from correlation formulas such as (2. 18) or (2.20) or some variations of these equations. Thus, the results of experimental data should be understood as some spatial average, and therefore independent of 0. It is felt that for rough surfaces such as the sea, the effect of the wind should be illustrated in the spatial variation of a(8, 0) with 0. But such an explicit relation for such dependence does not seem to be available in the literature. 2.4 Survey of Experimental Results The dominant characteristics of the back scattering cross-section from rough surfaces have been reported by experimental investigators, in numerous articles. A brief summary of these dominant features is outlined in this section. Since there exist numerous publications which discuss different aspects of this problem, only some representative ones are referenced here. 2.4.1 Aspect Variation Wiltse, et al (1957) reported that at 9.6, 24, 38, and 48.7 GHz, for a moderate sea, a (0, 0) is a decreasing function of 8, with the slope being greatest at 8 e 30~. The change of a(0, 0) from = = 0 to = 300, even for a calm sea, may vary by a factor as large as 5000. The results of Grant and Yaplee (1957), at wavelengths of 8.6, 1.25, and 3.2 cm of various types of terrain seem to indicate similar variation. Edison, et al (1960) reported that the measured average values of a(8, 0) at 3.8 GHz and 415 MHz over terrains, forests, deserts, etc., in the range of 8 from 00 to 250 seem to correlate fairly well with the relation 9

1969-1-Q 2c8C exp a2 tan 0 (2.21) 44 2 4 where = the mean square height deviation of the rough surface and a = the horizontal correlation distance of height variation. Finally Renau and Collinson (1965) performed back scattering measurements for the angle of incidence varying from 0 to 89 when a laser beam is incident upon manufactured rough surfaces whose rms surface slope is denoted by h/I. They determined that at 0 angle of incidence the cross-section varies as the square of h/I and that it is independent of wavelength. As the angle of incidence increases the cross section increases with h/l and h/X with an upper limit closely agreeing with what Lambert's scattering law would predict. In this case h represents the the rms surface height and h/I is the rms surface slope. 2,4.2 Frequency Dependence Goldstein (1946) reported that a(0, () varies with the zeroth power of X -4 for a rough sea and varies as X for a calm sea. Katzin (1957), based on the results of measurements at 3. 2, 9. 1, and 24 cm wavelengths Indicates that ca(0,) -1 varies as X for a sea surface. Grant, et al (1957) reported that for various types of terrain, a decreases with X. WHtse, et al (1957) reported that a VV(0, ) seems to be relatively independent of frequency, especially for small values of 0. Boring, et al (1957) made an extensive experimental study of sea returns at frequencies of 6.3 GHz and 35 GHz. Their results indicated that a(0, 0) for any polarization at 35 GHz is about 3 to 4 dB higher than that of the corresponding value at 6.3 GHz. Roughly, there is a 1/X dependence for aV, aHH and l;/an dependence for aVH or a HV. VH HV' 10

1969-1-Q Long (1965), made a further analysis of the Boring results, and reported that aHH and aVV are independent of frequency at low frequencies, and have 1/X dependence for larger frequencies. Moreover, the ratios aHH/aHV and aVV/aHV are relatively insensitive to frequency. Chia, et al (1965) report the reflection coefficient of forest, desert and water terrain, for measurements taken at 9 MHz from an altitude of 1000 km. They also present tables of data taken by other groups for various terrains at several frequencies and from different heights above the earth. Finally Renau and Collinson (1965) determined that the cross section differs for the two polarizations. At grazing incidence, the cross-section is larger for the parallel polarization. 2.4.3 Polarization Effects Wiltse, et al (1957) reported that for a moderate sea, aHH(0,0) is usually less than aV by several dB for large values of 0; while for small 9, aHH and aVV are nearly the same. aV (aHV) is on the average 12 dB less than aV for large O and 20 dB less at O =0. The results reported by Boring, et al (1957) seem to imply that for a calm sea, aVV >> aHH, but as the wind speed increases, aHH approaches closer to aVV. In some cases, aHH may exceed aVV for very rough sea conditions. Hunter, et al (1966), reported that at 9.375 GHz, with right circularly polarized transmitter and horizontal and vertically polarized receivers, aRV < aRH for cross wind and aRV > aRH for upwind or downwind. aRV, the cross-section for right circularly polarized transmission and vertically polarized receivers, is easily shown to be equal to aVV + VH aRV = 2 2.4.4 Effect of Wind and Sea State Wiltse, et al (1957) reported that for 9 > 20~, rough sea results in large values of a(9, 0), but for near vertical incidence, the trend is usually reversed. 11

1969-1-Q The work of Boring, et al (1957) reported the results of a statistical analysis of a large number of experiments of sea return at frequencies of 6.3 GHz and 35 GHz. It is found that the variation of a(0, 0) may be empirically represented by the expression. C2 Ca 3 1C4H + C5cs a(, ) =C W (90 ~-) 1 where H is the height of waves in feet 0 is so chosen that 0 = 0 is the upwind direction W is the wind speed in knots. This expression was obtained from a statistical analysis of a large number of experimental results. The constants C1, C2, C3, C4 and C5 differ for different states of polarization and for the two frequencies measured. From their data, it seems that in most cases, aAB increases with wind speed. Clarke and Hendry (1964) studied the power reflected in the specular direction, when the incident wave is horizontally polarized and at an angle of incidence of 450, for various states of roughness of the reflecting surface. Beard (1967) examined the forward scattered field for artificially created water waves by various methods. The water samples varied from distilled water to ocean water and the frequency used was 100 GHz. His results indicate that in any theoretical approach for the study of scattering from rough surfaces the sphericity of the wave fronts must be taken into account. Finally, Renau and Callinson (1965) report that for a constant rms height h the cross section increases with higher frequency and large angles of incidence. 12

1969-1 -Q III THEORETICAL MODELS FOR ROUGH SURFACE 3. 1 Introduction In order to explain the various aspects of the return from a rough surface, several theoretical models have been proposed to account for the dominant features of the reflected signal. Due to the lack of detailed information on the structure of the rough surface, and the dielectric properties of the surface, only the average properties based on some statistical description of the surface can be described. Basically, if one specifies a rough surface by some random function z = z(x,y) (3.1) with zero mean value, and a postulated statistical property, then together with the dielectric constant of the surface, the formal solution of the scattered field from the surface may be obtained approximately, and a statistical average of the return signal can be taken. Most of the theoretical work is based on a gaussian surface. Recently (Valenzuela, 1967) the use of the actual wave spectrum of the sea was attempted. It is felt that, introducing the actual statistical description of the sea surface, including the effect of the wind, one should obtain more accurate prediction of the scattered radiation for the sea surface. Mathematically, the theoretical model for rough surface scattering may be classified into three different classes. These are: (t) The phenomenological model. By postulation the surface is taken as an ensemble of distributed spheres, facets and half planes. A review of this phenomenological model has been given by Beckmann, et al (1963). (b) The use of series expansion. In this approach, the incident and scattered fields are expanded in series and boundary conditions are used in determining the coefficients of expansion. Due to the complicated procedure in determining the coefficients, only the first and second order approximate solutions have been reported so far. It is doubtful that any attempt to obtain the more detailed, higher order 13

1969-1-Q solutions is feasible. A brief description of the results from this approach is given in Section 3.2. (c) The use of the Kirchhoff approximation. The use of the Kirchhoff integral representation of the scattered field offers a convenient means of finding the approximate expressions for the scattered field from a rough surface. This has been used by various investigators in obtaining theoretical models for rough surface scattering. In the high frequency limit, this offers some basis for the phenomenological models introduced. It is felt that the extension of Kirchhoff's formulation, incorporating the dielectric properties of the surface and some realistic statistical description of the surface, may lead to a useful model appropriate for the present work. In the remainder of this chapter, the Kirchhoff approximation and possible extensions are discussed in some detail. 3.2 The Method of the Series Expansion The method of approximate solutions, for the fields scattered by a slightly rough surface, by a series expansion has been introduced by Rice (1951), used by Peake (1959) and recently extended by Valenzuela (1967). The results of these investigations are summarized in this section. The surface described by the random function z =z(x,y) (3.1) is assumed to have the following statistical properties: (a) The expected value of z is zero, i.e., <z>=0. (3.2) (b) The expected value of z is given by 2 > A -23 <z > z. (3.3) 14

1969-1-Q (c) The correlation function for the random function z is given by <z(x,y), z(x+, y+n)> A p(.,n), (3.4) and (d) The surface is slightly rough, i.e., a8z z |z < (3.5) Based on these assumptions, the first order results of Peake (1959) may be generalized to the following expressions for the bistatic cross section: aHH(81' 01' 02' 02) 2 = 4irk cos1 cos02cos2-1) 2.(Cl).2 X (cos 81+ -sinO ) X w[k(sin81 cos 0+ sine2cos 2) k(sin61 sin 0 + sin 2 cos 2)] (3.6) and aVV('1 01'2 2 ( -1) (1+ sin208)-sin2 2 = 47k cos 0lcos 2 cos (2 -1) 2 1 2 co2s' 2,cos0 + e-sin X w [k (sin ossi +sinO2cos 02) k(sinl cosl +sin92cos 2)] (3. 7) 15

1969-1-Q where E is the relative complex dielectric constant of the scattering surface, and w (a,, 4) is the Fourier transform of z p (, n). For back scattering and an isotropic rough surface, i.e., p(,) P n,), (3.8) the above reduces to the back scattering cross-section given by Peake (1959). A second order solution, for a perfectly conducting rough surface has been given by Valenzuela. The results were applied to cross polarization effect of sea returns. In terms of the sea spectrum A (v), Valenzuela obtained the following result for the scattering cross-section: 2 2 2 4 a (e) = H(O) a k cos4e T VH ( HV - k2 VH X 0 v5A2 (v- k sinO)A (v +ksin8) dv (3 8) 0.12 2 2 2 yk -v + Yk -v Using the Neumann spectrum for a fully developed sea, Valenzuela presented the value of aVH for several values of wind speed for X = 24 cm. Application of this results for different ranges of wavelength, involves lengthy calculation&so that whether this model agrees with the experimental features of wavelength dependence cannot be assessed at thi present time. 3.3 Kirchhff' s Integral Formulation A mathematical formulation of electromagnetic scattering, which gives approximate but useful results, is the vector extension of Kirchhoff's integral formula. The Kirchhoff's integral formula can be deduced from the physical concept of induced sources as illustrated in Fig. 3-1. When an electromagnetic wave is incident on this surface, the scattered radiation may be interpreted as the re-radiation due to the surface electric and magnetic currents. 16i.

1969-1-Q A z / Point of Observatioi \,// Incident / / Field r / / / /y 1n' /x Small Part of Scattering Surface FIG. 3-1: CONFIGURATION FOR THE APPLICATION OF KIRCHHOFF'S INTEGRAL FORMULA 1 17

1969-1-Q Consider for example an elementary area dA', with normal n', located at r'. If the total (incident and scattered) field at this surface is given by E (r') and H (r'), then the surface electric and magnetic currents per unit area at r' are 5Sgiven by J = xH (r'), (3. 10) -e - J = ' xE (r') (3.11) -m -s respectively. The electric field due to these induced sources, observed at any point r, is given by dE(r) = (n x E (r'))x V' G(r,r' )+ i (n'x H(r') fG(r, r+ V' V'G(r r' k due to magnetic current due to electric current (3.13) where ikir - r'j e - - G 41-r - r (3.13) and V' x + + + (3.14) ax, 8 ' azI A A Thus, if the tangential field components nx E and n x H on the surface are -- -8 known at all paitts of surface, the scattered fields may be obtained by adding the contributions from each part of the surface, i.e., E() dA{('xE (r'))-x VG(r,r')+ iop (nxH(r))*[G(r,r')+V'V'G(r,r'}. Surface (3.15) 18

1969-1-Q Similarly the scattered magnetic field is given by _ ( Surface = k durf H adA{(XHs(r))x v' G(r, r')-iWEO('xE (r))[G(r,r')+-1V'V'G(r, ' 1t (3.16) Equations (3.13) and (3. 16), are exact, provided that the exact surface fields are used in the integration. In most practical cases, the point of observation is high above the ground, so that the far zone approximation may be introduced. Referring to Fig. 3-1, any point of observation may be represented by A r = 2r, (3.17) and the approximation | r - r ' | (- ')2 (r> r), (3.18) is adequate. Then, Eq. (3. 15), and Eq. (3. 16) may be reduced to A iwMo eikr su -ik| r' and H2 (r)=-~ - dxAQ 2nx 'xE (r'- x 2('xH (r)' ~2- n ^J Surface { {2 x - - Sur4face (3.20) Equation (3.19) and Eq. (3220), are the fundamental relations used in the approximate calculations of the scattered field, [for example, see Hoffman 1955, Aksenov, 1961]. 19

1969-1-Q For plane wave incidence, the incident fields are given by: A ikQn r' -1- -1 El(r') = (0)e (3. 21) 1 ik2 * r' H1(r ') (3.22) -1 — ~ x1 (0) then, if E and H are interpreted as the surface fields due to an incident field -8 -S with the electric and magnetic field respectively given by E1(0) and - 1 x E (0) at each part of the surface, we may have E (r) ~ — dA x(ni'xE (r))- x x(n'xH (r e Similar equations, can be obtained for 2(r). In practice, the integral given by Eq. (3. 23) cannot be carried out exactly, even if we know the shape of the surface, due to the difficulties in obtaining E -s and H. A commonly used approximation is the so-called local tangent plane '-s approximation. Assume that the local radius of curvature of the scattering surface is much larger than the wave length, so that locally, the reflected fields at any point may be considered the same as those reflected by a plane tangent to the reflecting surface at that point. With this approximation, it may be easily shown that, (Aksenov, 1961) x E = (1 +R )(E (0). n'x + (1-R)(E (0) t') nxt (3.24) 1 ) Axt - (+R ( A) ( A A2 r7'xH = (1-R)(E (0)) n'x'- (I+R )(E (0)) nxt (3.25) -s I -1 )U -1 20

1969-1-Q where Q xn' t= (3.26) and A A A t' = l xt (3.27) 1 R =co N - -sin 2, (3.28) cosy + lN2 sin2 2 2 2 R = N. ' (3.3029) 'o N = index of refraction of the surface. In particular, if the reflecting surface is perfectly conducting, then, R/ = -1 (3.31) and R = +1. (3.32) RIi=+ 21

1969-1-Q For this case, we have, 'xE = 0,(3.33) and r f' xH = 2 'xH (0) (3.34) which are the approximate relations used by most investigators to simplify computations. By introducing Eqs. (3. 24) and (3.25) into Eq. (3. 23) an approximate relation results between the incident electric field E and the scattered electric field E2. This result is given by A i i ikr E2 (r, (2) - r 2 rik 1 5 x [dA' AAl+R|)(n'Ax(E* t) + (1-R) (anxt') (E t')J e 1d (3.35) A A A A ik(i-42 -.r' -42x^2xdA' 1-R)(n'xt)(E1 t)-(l+R n)('xt)(E 1 2 (3.36) In order to clarify the polarization effect on the scattered field, let us, by using the directions of polarization defined by Eq. (2 4) through Eq. (2. 5), resolve the incident and scattered fields into the vertically and horizontally polarized components as indicated below: ikl * r E1 =ehl Ehl + ev E 1e 1 (3.37) -1 L hl ehl VEvlj e 22

1969-1-Q and E2 h2 Eh2 ikr + E v2 v2j r (3.38) From Eq. (3. 36), it is seen that the components of the incident field and the scattered field may be related by a matrix: Eh2~ = hh Shv hl39 E = S S E v2 L vh vv vl 1 where each component of the scattering matrix Sab may be expressed by: iwHp S = i-o IdA' ab 47rr J - (I+R) - (l-R1) + (1-R1) - (1+R 1) ik (6 1- 2), rt (be it) (2xe a2) (,'xt) (e t) e ( x t' ) ( bl. ) a2 (xt' } (e b t, ) ea2' (nI (3.40) The relations between the elements of the scattering matrix and the scattering cross-section may be easily deduced from the definition of the scattering cross section. For example, for an incident wave with polarization direction a, (h or v), the power scattered by a surface of A, per unit solid angle with polarization b is given by dP dP2 2 d | ISbaI P1 23

1969-1-Q Thus, if the surface A is relatively uniform, the scattering cross-section per unit area is given by b Acos (3.41) where < > enclosing any quantity indicates the statistical average of that quantity. In general, the evaluation of Sab and the subsequent deduction of aab is complicated, and depends on the postulation of the statistics of the surface. Most theoretically models in the literature may be interpreted as the results of carrying out this integration and averaging process approximately. Some of these approximate procedures and subsequent results are discussed in the next few sections. 3.4 Mdel of specular Reflecting Reions The first order approximation of the Kirchhoff integral, is obtained by means of a stationary phase integration and yields the geometric optics result for the reflected field. Although this is a generally known fact in classical field theory for a long time [MacDonald, 1912], specific application of this concept to rough surface scattering was not used until 1966 when Kodis applied t with the scatteer being an infinitely conducting rough surface. Extension of the ratUtaofKbdis torough surfaces with a finite index of refraction, and some qualitative results of this approach are investigated in this section. For the approximate evaluation of the integral (3.40), by mens of a stationary phase integration, we assume that the exponential factor e -, due to its rapid oscillatory behavior as r' varies, is the dominating factor, and the rest of the integrand is slowly varying. Thus, the contribution to the value of the integral is primarily due to regions in the neighborhood of some r', such that ( 1 -2) r = constant. (3.42) 1 2 - 24

1969-1-Q There may be several values of r', say r1, r2... r.... which satisfy -— 3 this condition. These points are known as stationary points or specularly reflecting points, because at these points the geometry is such that waves may be A A considered as specularly reflecting from the direction 1 to Q2. To illustrate this, let us denote A A A A A q = (1 2) xq + yqy + qz (3.43) and any point on the rough surface by A A r' x' +yy' + z' (x',y') (3.44) where z' = z' (xy') is the equation for the rough surface. Then, Eq. (3.42) means that qxx' + qay' + qz' (x; y') = constant at a stationary point. Therefore, zx + z 0 (3.45) and qy+q z, -O 0(3.46) at a stationary point. For a physical interpretation of the stationary points which satisfy these equations, we note that the normal to the surface is given by A az'. az ^A n' ax, a (3.47) a'z2 8' z,2+1 x' ay' 25

1969-1 -Q Thus, it is easy to see that the condition satisfied by a stationary point may be represented by the vector relation n'xI = n' x(l- 2) = O. (3.48) As illustrated in Fig. 3-2, at any stationary point r', Eq. (3.49) indicates A A A 1 that n' bisects the angle between ~1 and Q2 as indicated, and waves are A A reflected from 21 to ~ 2 specularly at r',where j = 1,2, 3,.. Thus, with the geometrical optics approximation, the first order scattered field from an incident direction 61 to any direction 2 may be considered as the diffraction field due to regions near all the stationary points of the surface. This mathematical observation indicates that the phenomenol i, —l models consider the reflecting surface as a raddom distribution of hemispheres, paraboloids, or other bumps on the surface and-tbey mr be heonalditrad ~the same claes of ajproximta:.bn a:se for apeoulari eflWfbtg regoi. Equivalent qualitative results should be obtained if the statistical descriptions of the surface are the same. To carry out the mathematical approximation of geometric optics further, A the direction of incidence ~1 is fixed and the reflected field is observed at a fixed direction S2, then, at any reflection point r' the vectors a, t, t' are fixed (independent of the position r' ), hence may be taken out of the integral sign in Eq. (3.40). The results are, S F(~2- F k)1 ab 4r F 1, 2' k) - (1+RL) (ebl. t)(22x ea2) (a'xt) (- RIt)( t t )( Xea2)' ('x') + (1+-R})(1 ) 1ea2 (Ext') - (1+RI)(e t)ea xt)] (3.49) 26

1969-1-Q A "1 A 11 A " 2 -/. Tangent Plane at r' N../ 7 —K / r'. 01 -"2 FIG. 3-2: WAVE REFLECTION FROM A SPECULAR (STATIONARY) POINT. 27

1969-1-Q where A ^ F(n Rik k) ( '1 - l)' r Fj Area Near r' (3.50) is the factor denoting the geometrical properties of the surface near each specular point r' -j' Explicitly, if I1 = - (sinl 1 coso +ysin 1 sinfl + z cos ) (3. 51) f = (x sin e2 cos ~2 + y sine 2 sin~2 + i cos 2) (3.52) then, by straight forward algebraic manipulation, we have, hh = r- [C1 R C2 RlI F ( 1, 2'- k) (3.53) i r.54) iSh = 41 — R +C Rk) (3.55) S = Cr C 4 I 2 vv 47r -CR2R C1RI F(Q1, /2, k) (3.56) The constants C1, C2, C3, C4 depend on the incident and reflected directions, and may be expressed as 28

1969-1-Q C1 C2 -2PQ S 2sin e1 sin 2 sin2(2 -1) 1 2, (3.57) (3.58) 2 sin 02 P sin (2 -1 ) 3 = (3.59) -2sin 1Q sin (2 - 1) S C4 (3.60) and P = cos 1 sin 2 - sin 0 cos 2os 2- 1) Q = cos 2 sin 1 - sin2 cos 01cos (2 - 01) (3.61) (3.62) S =[1 -cos1 cose2 - sine1 sin e2 cos (2 - 1)[cos e1 os2] (3.63) The expressions for aab as indicated by the expression (3.41 ), depend on the expected value of the following relation F( A Ak A <F(f1 2n2, k) F (21, i2, k)> = ik (f -2 ).(r;-rt) ' d 'd dxdye i. (3.64) The estimation of the value of the above expression depends greatly on the statistical description of the rough surface. For high frequency scattering, Kodis (1966)

1969-1-Q showed that an asymptotic result for the integral may be given by <F( 2,,k) F*(1, f2, k) > r A A = - - 2 <(R><N (Q1',Q2)> A k cose 1+ cose2] where < K > is the average gaussian curvature at the reflection point, and A A < N ( 21, ~2) > is the average specular reflecting points per unit area. Both < i > and < N ( 1, u2) > depend on the particular statistical model used for the rough surface. By using different postulates, it is shown by Spetner, et al (1960) that the wavelength dependence for the back scattering cross-section may be 1/X, 1/\, 1/X over different frequency ranges. These models, however, do not seem to agree with the experimental evidence of 1 /X (or less) dependence on the wavelength. Recently Barrick (1968) relates the integrals involved in Eq. (3.64) to the probability distribution of the slope of the surface. His results may be expressed as: A A A A <F(Q, Q2 Tk)F (QI12' k)> 4Ar2 1 rk(sine1 cos1+sin92cos2) 2 (cos e cos (cos+ +cos0 ) k (sin 0 sin 1 + sin e sin 2) X (cos + cos e2) where P (u, v) is the probability densily distribution for the slopes of the surface 30

1969-1-Q az ax az a = V For back scattering and a Gaussian distribution of slopes, e1 = 2 = e, ~1 2 = 2 ' U +1v 1 2k 2 P(u, v) = 2 e and we have, <F( 1s 22, 2'k) F (^21 '2 sk)> k2 cot2 e SrA 1 2S -- e 2k2cos20 s This exponential form of variation of back scattering cross-section with 0 occurrs in almost all the theoretical models. It is felt that by uaing more realistic forms of the slope distribution of the sea surface, such as the slope distribution of the sea surface in the presence of wind measured by Schooley (1962), would yield a more realistic result for the aspect dependence of the scattering cross-section. It is to be noted that the approximate theoretical model obtained by using the geometric optics approach does not reveal the cross polarization effect in the back scattering direction, as being reported by various experimental works. Refer to Eq. (3. 53) through (3.63), for the back scattering direction, one finds that in the limit 31

1969-1-Q C = C =o 1 3 4 and C - 2 2 cos O So that this model predicts that aHV = VH 0 For a theoretical model which indicates the effect of cross polarization a more accurate integration of the Kirchhoff's integral which corresponds to the physical optics approach should be used. This is outlined in the next section. 3.5 The Kirchoff Integral and Statistical Average (Physical Optics) The formal evaluation of the expected value of the bistatic cross-section from Kirchhoff's integral is very complicated for surfaces of finite conductivity. Following the formulation given by Eq. (3.49), we may formally write, m 47 dy' e ^~ WT m 4- r7 n z - (1+Rl) AB m -(1-Rli) CDm +(1-R1)MlBm -(I+Ril) NiD } (3.65) where the function Al, Cl, etc. are given by A I( ^xe ) nxt (3.66) C (Q xe) nxt' (3.67) 2 32

1969-1-Q ^ A A Ml = e2 *n x t' (3.68) 2 A A A N = e * n x t (3.69) 2 A B =e *t (3.70) m m and A A D =e t'. (3.71) m m1 Due to the complicated dependence of R and R 1 on the slope of the surface through the factor z sin 01 cos 01+z sinOsine cos 0 cos= - = -- "s 2 1 (3.72) +z +z x y the evaluation of the expected value of < S. S* > for the general case has not been investigated. Approximations for the expressions of Sr by assuming that the reflection coefficients R I and R are nearly constant, or by assuming that the rough surface is infinitely conducting can simplify the mathematical results considerably. For example, if the reflecting surface is infinitely conducting, so that R -1 R- -1 RK| = l, then Eq. (3.65) may be simplified into iuO r r ik(T-4 )r' Sm 2r dxj dy e Ge m (3.73) where =- [MB - ND ]. (3.74) m n-z m 33

1969-1-Q Explicitly, if AA A l xssinecosl+ ysin8 l0n + zCos (3.75) A FA Al 2 LXsin 02CO02+ Ysin%928i. 02+ z Cos e 2j (3.76) and A A +A -xz -yz +z A 2 (3.77) n2 = Vi+z +z x y It can be easily shown that, A A A hh h2' nx eV1 = sln(01-062) (3.78) A A A Gvh =ev2 nx e 2 = sin6 1cos [zx sinL- z0 os)2] 0 -sLnO8cosO z sin1 - zCos1 - - cos9 1Cos602sin (O1- 02) (3.79) -4 A A Ghv =-eh2 nx hl cOS61 Cos (01- 02)- sin6[ zoos02+ z sin 02 (3.80) and A.A A G e2 nx eh COS % Cos(01 - 2) - sinejZ zcos 02 + zsin 021 (3.81) The expected values of <S SA k > for the case 'of the surface height z = z (x,y) Is a Gaussian distributed random variable whiohs been curried omt 4Ibyffmrai( 1955) 34

1969-1-Q and extended in detail by Fung (1964). The results are quite complicated, and because the assumption of Gaussian distributed surface appears to be an unrealistic model for the sea surface with wind effects, they are not quoted here. It is felt that for the case of a sea surface with wind effects, a more realistic approach is to start with the statistical description of the slope of the surface. Experimental values of such distribution have been reported by Schooley (1962). But analytical expressions for such a distribution and the correlation of such distribution with the theoretical spectrum of sea waves, (Wright,1967) has yet to be postulated. If a realistic statistical model of the slope given by the distribution function F (z x, z ) can be postulated, the expected values of <S mS* > can be carried out, for high frequency scattering by jfm fcm extending the approach given by Barrick (1968). In principle, each of the integrals S1 may be represented by the form A. ^ lm (* r ik (Q2 -slm du' dy' K (zx. y Im x I where K is a function of the slope, different for different polarizations. Thus, if the statistical description of the slope of the surface is known, the expected values of the cross-section may be carried out. The theoretical analysis involved in this approach has started, and it is expected to be finished in the next quarterly report. 35

1969-1-Q IV CONCLUSIONS AND RECOMMENDATIONS In order to find a realistic model for the reflection properties of a rough surface to be used in calculating the reflected radiation, a survey of existing theoretical models and experimental results for rough surface scattering has been carried out. It is concluded that a simple, realistic, yet adequate model for the bistatic cross-section incorporating the roghnsas and wind, for surfaces such as the sea is lacking. From the review of the basic approaches for arriving at a theoretical model, it is concluded that two basic approaches may be employed to postulate a theoretical model. The use of geometric optics approach yields a simple model for the bistatic cross-section. This model, however, does not agree with the experimental results in the cross polarization effects. The approach of physical optics using a realistic statistical description of the slope of the surface seems to be very promising in arriving at a model for the purpose of this investigation. Derivation of expressions for the expected values of bistatic cross-section for several postulated slope statistics may be carried out by extending the approach of Barrick (currently under study and expected to be completed in the near future). It is planned that a controlled experiment using a water tank to simulate the sea s surface will be carried out during the next research period in order to verify the validity of the theoretical expressions and estimate the parameters (such as mean value of slope, correlation distance, etc.) involved. 36

1969-1-Q LIST OF ILLUSTRATIONS Fig. 2-1: Directions of Incident and Reflected Waves and the Directions of Polarization 3-1: Configuration for the Application of KirchhoffI's Integral Formula 3-2: Wave Reflection From a Specular (Stationary) Point 37

1969-1-Q BIBLIOGRAPHY Aksenov, V.I. (1958) "The Scattering of Electromagnetic Waves by Sinusoidal and Trochoidal Surface with Finite Conductivity", Radiotekhnika i Electronika, 4, 459-466. Beard, C.I. (1961) "Coherent and Incoherent Scattering of Microwaves From the Ocean'', IEEE Trans., AP-9, 5, pp. 470-483. Beard, C.I. (1967) "Behavior of Non-Rayleigh Statistics of Microwave Forward Scatter From Random Water Surface", IEEE Trans., AP-15, 5, pp. 649-657. Beckmann, P. and A. Spizzichino (1963) The Scattering of Electromagnetic Waves From Rough Surfaces, The MacMillan Co., New York. Boring, J.G. et al (1957) "Sea Return Study", Final Report, Project 157-96, Georgia Institute of Technology, Atlanta, Georgia. Chia, C., A.K. Fung and R.K. Moore (1965) "High Frequency Backscatter From the Earth Measured at 1000 Km Altitude", Radio Science, NBS., Vol. 69, 4, pp. 641. Clarke, R.H. (1963')"Theoretical Characteristics of Radiation Reflected Obliquely From Rough Conducting Surface", Proc. IEE, 110, pp. 91. Clarke, R.H. and G.O. Hendry (1964) "Prediction and Measurement of the Coherent and Incoherent Power Reflection From a Rough Surface", IEEE Trans., AP-12, 3, pp. 353-363. Edison, A.R., R.K. Moore and B.D. Warner (1960) "Radar Terrain Return Measured at Near Vertical Incidence", IEEE Trans., A P-8 3, pp. 246. Goldstein, H. (1946)"Frequency Dependence of the Properties of Sea Echo", Phys. Rev., 70, pp. 938. Grant, C.R. and B. S. Yaplee (1957) "Backscattering From Water and Land at Centimeter and Millimeter Wavelengths", Proc. IRE, pp. 976. Hoffman, W. C. (1955) "Scattering of Electromagnetic Waves From a Random Surface", J. Applied Math., XI, 3, pp. 291-304. Hunter, I. M. and T. B.A. Senior (1966) "Experimental Studies of Sea-Surface Effects on Low-Angle Radars", Proc. IEE, 113, No. 11, pp. 731. Katzin, M. (1957) "On the Mechanism of Sea Clutter", Proc. IRE, 45, 1, pp. 44. Kodis, R.D. (1966) "A Note on the Theory of Scattering From an Irregular Surface", IEEE Trans., AP-14, 1, pp. 77. Long, M. W. (1965) "On the Polarization and the Wavelength Dependence of Sea Echo", IEEE Trans., AP-13, 5, pp. 749. 38

1969-1-Q Bibliographies (continued) Peake, W.H. (1959) "Theory of Radar Return From Terrain", IRE Convention Record, Part pp. 27. Pidgeon, V. W. (1966) "Bistatic Cross Section of the Sea", IEEE Trans., AP-14, 3, pp. 405. Renau, J. and J.A. Collinson (1965) "Measurements of Electromagnetic Backscattering From Known Rough Surfaces", Bell Systems Tech. J., 44, 2203-2226. Rice, C.O. (1951) "Reflection of Electromagnetic Waves From Slightly Rough Surfaces", Comm. Pure and Applied Math., 4, pp. 351. Schooley, A. H. (1962) "Upwind-Downwind Ratio of Radar Return Calculated From Facet Size Statistics of a Wind-Disturbed Water Surface", Proc. IRE, pp. 456. Spetner, L. M. and I. Katz (1960) "Two Statistical Models for Radar Terrain Return", IRE Trans., AP-8, 3, 242-246. Valenzuala, G. R. (1967) "Depolarization of E. M. Waves by Slightly Rough Surfaces", IEEE Trans., AP-15, 4, pp. 552. Wiltse, J.C., S. P. Schlesinger and C. M. Johnson (1957) Back-Scattering Characteristics of the Sea in the Region From 10-50 Kmc"', Proc. IRE, 4 pp.220. 39

UNC LASSTFTFI DSecurity Classification lmR DOCUMENT CONTROL DATA R & D,.,...turtv lasification onl Itlo. hItot, l of nbsrtticl nitl Itit.lin nniotnfloln mon be entereod when tl overall report Is clnanliled) 1. ORIGINATING ACTIVITY (Corporate author) 2B. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 - 3. REPORT TITLE DOPPLER RADIATION STUDY 4. DESCRIPTIVE NOTES (Typ of repott nnd Inclusive dater) Interim Report No. 1 1 July - 1 October 1968 a. AUTHOR(S) (First name, middle Initial, lIst name) Chiao-Min Chu 6. REPORT DATE 7.' TOTAL NO. OF PAGES 7b. NO. OF REFS December 1968.39 24 88. CONTRACT OR GRANT NO. 90. ORIGINATOR'S REPORT NUMBER(IS N62269-68-C-0715 b. PROJECT NO. 1969-1-Q c. '!h. OTHER REPORT NO(S) (Any other tlnumers Ihot many ha oeaaitned this report) d. 10. DISTRIBUTION STATEMENT Unlimited distribution II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Naval Air Development Center Johnsville, Warminster, PA 18974...... ~ - I...... 13. ABSlTRACT In order to arrive at a model for the bistatic cross-section of a rough surface, which is adequate for the estimation of the reflected radiation from a doppler radar, a survey of the literature on the experimental work and theoretical models, concerning rough surface scattering, has been carried out. As a result of this survey, the dominant characteristics of radiation reflected from rough surfaces, and the basic approaches in arriving at various theoretical models are summarized. It is suggested that for the present work, a theoretical model based on the statistics of the slope of the surface, incorporating the effect of wind is probably adequate. An outline of the approach in arriving at the expressions for the bistatic cross-section based on the slope characteristics is given. v FOR d A -Y UL. NOV.1 4 /j UNCLASSIFIED Se.curit (C'lssiiii. iitao