ENGN 1969-1-F UMR1299 v.1 Vol. I Copy THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING o | Radiation Laboratory Doppler Radiation Study: Phase I Report of Contract N62269-68-C-0715 Volume I. By CHIAO-MIN CHU, SOON K. CHO and JOSEPH E. FERRIS THE UNIVERSITY OF MICHIGAN Decemberl969 ENGINEERING LIBRARY 1 1969-1-F - Volume I Each transmittal of this document outside the agencies of the U. S. Government must have prior approval of the Commander, Naval Air Development Center, Johnsville, Warminster, Pennsylvania 18974 or of the Commander, Naval Air Systems Command, Washington, D.C. 20360 Contract With: Naval Air Development Center Johnsville, Warminster, Pa. 18974 Administered through: OFFICE OF RESEARCH ADMINISTRATION. ANN ARBOR

v,/ The discussion or instructions concerning commercial products herein do not constitute an endorsement by the Government, nor do they convey or imply the license or right to use such products. SXTY Opt 1y ~ (z~~~

FORE WORD This report was prepared by The University of Michigan Radiation Laboratory, Department of Electrical Engineering. This is Volume 1 of the Final Report of Phase I under Contract N62269-68-C-0715, "Doppler Radiation Study" and covers the period 1 July 1968 through 1 July 1969. The research was carried out under the direction of Professor Ralph E. Hiatt, Head of the Radiation Laboratory and the Principal Investigator was Professor Chiao-Min Chu. The sponsor of this research is the U. S. Naval Air Development Center, Johnsville, PA., and the Technical Monitor is Mr. Edward Rickner. (i)

ABSTRACT The radiation characteristics of a doppler velocity sensor radar have been studied. A theoretical investigation has been made of the reflection of the electromagnetic radiation from an anisotropic Gaussian surface. In particular, from the known angular spectrum of ocean surfaces, the bistatic scattering cross section is derived for an open developed sea. The results thus obtained are then applied to the study of the reflected radiation from the doppler sensor equipment on an airplane. Computer programs are set up to calculate the directional distribution of the reflected radiation for a transmitting antenna of given radiation pattern. Computed results for the AN/APN-153 antenna, showing the spatial and temporal variations of the reflected radiation, are given for a wide range of relative positions of the transmitter and receiver for a few different wind speeds. Finally, the reflected radiation from an anisotropic ocean surface of Gaussian distribution is compared with models of specularly and diffusely reflecting surfaces. (ii)

TABLE OF CONTENTS Page NOME NC LATURE iv I INTRODUCTION 1 II THE MATHEMATICAL FORMULATION OF A REFLECTED RADIATION 3 2.1 Introduction 3 2.2 Coordinate System 3 2.3 Radiation Pattern 7 2.4 Reflecting Surface and Reflected Radiation 7 mI CALCULATION OF REFLECTED RADIATION 19 3.1 Introduction 19 3.2 Mathematical Formulas 19 3.3 Power Level and Directional Variation of Reflected Radiation 22 3.4 Direction of Maximum Intensity of Reflected Radiation 34 3.5 Magnitude of Maximum Reflected Radiation 56 IV CONCLUSIONS AND RECOMMENDATIONS 78 APPENDIX A: BISTATIC CROSS SECTION OF A ROUGH SURFACE 80 A. 1 Introduction 80 A.2 Scattering Cross Section 80 A. 3 Kirchhoff's Integral Formula 84 A. 4 Average Scattering Cross Section of Random Surface 97 A. 5 The Ocean Surface Wave Spectrum and the Scattering Cross Section 101 APPENDIX B: COMPUTER PROGRAMS 111 Rough Surface Program 111 REFERENCES 133 DD FORM 1473 134 (iii)

NOME NC LATURE A A dimensionless parameter: A ratio of (scale length)2 to mean-square height of a sea surface in x-direction. B A dimensionless parameter: A ratio of (scale length)2 to mean-square 0 height of a sea surface in y-direction. C A factor involved in the Neumann spectrum. dB Decibel. E1, E2 Incident and reflected electric field strengths. f Radiation frequency. F(0, ~) Normalized antenna radiation pattern. F2 Normalized radiation power density ~ F2d Normalized radiation power density per unit solid angle for a Lambert scattering in the direction of arrival of maximum radiation intensity. F2M Normalized radiation power density per unit solid angle for a sea surface in the direction of arrival of maximum radiation intensity. F2s Normalized radiation power density for specular reflection. G(r, r') Free-space Green's function. G Gain of a receiving antenna. Gt Gain of a transmitting antenna. g Acceleration of earth's gravity. A. H(i i ) Correlation function of a random surface. H1, H2 Incident and reflected magnetic field strength. J Electric surface current. e (iv)

K A function of surface slopes Am k Radiation wave number L Wavelength of the sea surface-wave with the minimum phase velocity. m A, A Correlation distances of a sea surface in x- and y-directions. x y N Index of refraction. A n Unit normal vector. Pi Incident radiation power density. P Total received power. r Pr Received power density. P Transmitted power. P2s Specularly reflected radiation power density. Sq, q q Parameters associated with the phase of the radiation in x-, y- and z-directions. R Hypothetical maximum range of detection. m R1, R11 Reflection coefficients for perpendicular and parallel field components. r Position vector. r Position vector for a transmitting antenna. r Position vector for a receiving antenna. -r t Time. U Wind speed. X Normalized x-coordinate with its origin at the transmitting antenna. x Rectangular x-coordinate. (v)

x Rectangular x-coordinate of a transmitting antenna. a x Rectangular x-coordinate of a reflection point. xr Rectangular x-coordinate of a receiving antenna. A x Unit x-vector Y Normalized y-coordinate with its origin at the transmitting antenna. y Rectangular y-coordinate. Rectangular y-coordinate of a transmitting antenna. yO Rectangular y-coordinate of a reflection point. Yr Rectangular y-coordinate of a receiving antenna. y Unit y-vector. z Rectangular z-coordinate. za Rectangular z-coordinate of a transmitting antenna. z_ Rectangular z-coordinate of a reflection point, Z r Rectangular z-coordinate of a receiving antenna. z Unit z-vector. z x, z Surface slopes in x- and y-directions. xa A parameter defined as q/qz aB A parameter defined as qy/qz r Coefficient of surface tension/water density. e Latitude angle in spherical coordinates. e1' e2 9-coordinates of incident and reflected radiations. e9-coordinate of a radiation pattern of a transmitting antenna. (vi)

02d 6-coordinate of maximum reflected radiation intensity for a Lambert surface. 02M 0-coordinate of maximum reflected radiation intensity for a sea surface. 02S 0-coordinate of specularly reflected radiation intensity. K 1' K2 Sea surface-wave numbers in x- and y-directions. X Free-space radiation wavelength. Normalized height variables. a( ~22, 1) Scattering cross section a Sea surface-wave angular frequency. T'r Correlation distances for a random surface inx- and y-directions. x y 0 Azimuth angle in spherical coordinates. 01 V02 0-coordinates of incident and reflected radiations. 0o 0-coordinate of a radiation pattern of a transmitting antenna. 02d 0-coordinate of a maximum reflected radiation intensity for a Lambert surface. 02M 0-coordinate of maximum reflected radiation intensity for a sea surface. 02S 0-coordinate of specularly reflected radiation intensity. Sea Surface-wave spectrum. Wind direction. A 21' 22 Unit vectors for the directions of incident and reflected radiations. ~2 Solid Angle. (vii)

INTRODUCTION This report summarizes the results of the continued theoretical and experimental investigation of the radiation characteristics of a doppler velocity sensor radar. In the previous work (Chu, et al, 1968) the characteristics of the direct radiation, the reflected radiation from a specularly reflecting surface, and a diffused Lambert surface were reported, along with the estimates of the detectability of such radiation. In the present investigation (Contract N62269-68-C-0715) attention is focused on the characteristics of the reflected radiation from an anisotropic Gaussian surface in the belief that such a surface poses a more realistic model for an open developed sea. Appendix A presents theoretical formulas of the reflected radiation from a random sea surface. In deriving the bistatic cross section of an anisotropic Guassian surface, through the physical optics approach, the Neumann spectrum was adopted as an angular spectrum of an open developed sea which incorporates the effect of the wind speed. The expressions thus derived were used in the various computations pertinent to the discussion of the problem. In chapter II we present the basic formulation of the problem in calculating the reflected radiation. Based on the radiation pattern of the transmitting antenna, reflected radiations are discussed in detail in terms of the angular distribution of reflected radiation. In Chapter III the various computed results for the reflected radiations from an anisotropic sea surface are presented together with the effect of the wind speed on the apparent direction of arrival, and maximum intensity per unit solid angle of the reflected radiation for various geometric conditions. These results are compared with the corresponding cases of both a specularly reflecting surface and a Lambert surface. 1

Finally, in Appendix B, the computer programs are presented for the numerical evaluation of the reflected radiation intensity for any transmitting radiation pattern, relative geometry between the transmitter and receiver and the parameters describing the sea state. The use of these computed results in the estimation of the detectibility of the reflected radiation are given in Volume II of the report (Classified Secret). Volume II also contains a description of laboratory and field tests and the results obtained.

II THE MATHEMATICAL FORMULATION OF A RE FLECTED RADIATION 2.1 Introduction In order to compute the reflected radiation from an antenna observed at any point, it is necessary to know the following: a) the radiation pattern of the antenna, b) the reflecting properties of the ground or sea surface, and c) the relative position of the antenna and the point of observation. In the previous report (Chu et al, 1968), the radiation pattern of the doppler antenna AN/APN-153 and a coordinate system specifying the orientation and relative position between the antenna and the position of observation have been given. A summary of the pertinent facts that are used in the estimation of the reflected radiation is included here for completeness. The mathematical formulations for the calculation for a specularly reflecting ground, a diffusely reflecting ground (Lambert surface) and an approximate model for an ocean surface are given. 2.2 Coordinate System Referring to Fig. 2-1, we shall choose a fixed, right-handed rectangular coordinate system with the ground as the x-y plane. The z-axis is assumed to be directed upwards, and the x-axis is chosen in the direction of the ground velocity of the airplane carrying the doppler antenna. Thus, for normal level flight, the longitudinal axis of the aircraft is parallel to the x-axis while the transverse axis is parallel to the y-axis, The position of the aircraft is represented by: ra: ECa' a

z z z T(xa Ya0 Za)Os x rvi.L0 \o 0~ | A I:Point of Observation I N (xryrzr) I['~~0 (O2 I,\ r~ / (02 2 Is Point of Reflection (xo, yo, o) Fig. 2-1: The Coordinate System

while the point of observation is represented by rr fxr, Yr, Zr Since only the relative positions between the antenna and the point of observation are essential in computing the reflected radiation, it is convenient to introduce the normalized coordinates defined by x -x X= a (2.1) z a Yr - Ya Y = (2.2) z a z and 1- r z +z z a r a a r r (2.3) z a The direction of the radiation from the antenna, observed at a point, or reflected from a point on the ground, is represented by a unit vector. Q t: X where e, 0 are the local latitude and the azimuth angles. In the vector notation, therefore, A A A Q = x sin 0 cos. + y sin 0 sin 0 + z cos 0, (2.4) where x, y, z are unit vectors in the x, y and z directions respectively. The variables used in specifying the coordinates and directions used in calculation of reflected radiation are illustrated in Fig. 2-1 and explained by the following: i) The transmitter is located at r: (xa, ya', a) ii) A ray of radiation originated from the transmitter is designated by the A direction Q: (eo, 0), iii) The ray in the direction Q would be reflected at a ground point rO: {xy,O 5

where x = x + z tan 0 cos (2.5) O a a o o Yo = Ya + Yz tan 00 sin 0o (2.6) iv) Referring to a local coordinate system at the point of reflection, the ray of incident radiation appears to come from - f1 where f1.(e1,1) =Evidently, 9 =1800~0 (2.7) 1 o and 01=o+ 18o0 * (2.8) v) The incident radiation may be reflected diffusely from the ground. The direction of the reflected radiation is represented by 2:(02' 02) vi) The reflected radiation in the direction Q2 is observed at a point rr -r (xr Yr r r where Xr xO + Zr tan 02 cos 02 (2.9) Yr =Y + Zr tan 02 sin 02' (2. 10) In general, of course, the radiation from the transmitter is distributed and specified by the radiation pattern, and the reflected radiation observed at any point may also be distributed. It appears to be coming from various directions of ~2(2, 02) For a fixed ra and rr the relation between (81 01) and (82, 02) can easily be obtained by eliminating xo, yo from (2. 5) through (2. 10). The results expressed in the normalized coordinates, are given by z (X + tn cs0)=tan 02 os ) = tan 2 (2,11) (Y+tan9 1 sin 01) = z tan2 sin02 (2.12) a

2.3 Radiation Pattern The radiation pattern of the doppler antenna of interest (AN/APN-153) has been measured and reported in the previous Final Report (Chu et al, 1968). The antenna is composed of two sets of slots alternately energized, with switching frequency 1 cps. One set of slots, energized by Feed No. 1, generates beams in the right-forward direction (Beam 1) and the left backward direction (Beam 2) while the other set, energized by Feed No. 2, generates the beams in the left-forward (Beam 3) and the right backward (Beam 4) directions. These beams are illustrated in Fig. 2-2. The reduction of measured radiation pattern for the AN/APN-153, expressed as F(e, 0 ), has been reported by Chu et al (1968). Contour plots for Feed No. 1 are illustrated in Figs. 2-3a and b. The same information may be reduced to F(e1, 01) by the change of variables given by Eqs. (2. 7) and (2. 8). The pattern of Feed No. 2 is essentially the same as Feed No. 1 except that o is replaced by 3600 - 0o In terms of F(01, 01), the power density (Poynting vector) incident at the surface z=0 from the direction - l(Ol, 01) is given by: PtGtF(8 0 1) Pi(Q21= P i.(01' 12) = (2.13) 4r z seec a 1 2.4 Reflecting Surface and Reflected Radiation When an incident radiation impinges on a surface, the energy may be specularly reflected, or diffusely scattered, as illustrated in Fig. 2-4a, b. For secular reflection, assuming that the reflecting surface is an infinitely conducting plane, the intensity (power density) of the reflected radiation is only in the direction 02 =1 (2.14) 02= 01 + 1800 (2.15) and the reflected power density is given by,r Pi. (2.16)

A z Z x \ Beam 3 Y/ / Beam 1 / l)Beam 4 Fig. 2-2: Beamp of Radiation and Ground Illumination for APN/ 153 Antenna System. 8

10 170 rq 25 tlo I Co 20 160 0 PI -:~ C'.+ ob 30 150 Or 0 5 15 ~ l e'o 25 (deg) (deg. 15 40 140 CDn,~~~~ 2 50 130 0 0o ~Z 60 120 E. C) o CO 0 I D'M 70 110 - o 80 100 100 120 140 160 0o 180 200 220 240 260 90I n 20 300 320 340 1 360/0 20 40 6b 86

01 *'qol jor'ue Jo uo;lloiap u! uoleTps. a moleq Ep ul leAel eomod st.moluoo qtoe uo oeqxmnN'tualsS eoupaooo D roTaaqdS eoo';o 0 puw O se812Ut epnIwir pue tlnmUIzv Jo su.eTL UT ~gI-NdV/NV VNNBLaiV Jo I'ON peea zo; ujoze7d uollvPBl:qe- Z'2!i CD 00 < CD v - Z FP 0t US,,..... I.! -- I CO C:) Ci3 o o oC) 00~ C o o t-h cn C~~~~~~~~ I C~~~~~~~~~~~~~~~~L ~~~~~~~~~~~E5~~~~~~~~~~~~~~~~~~,, l-.-,

" Pi(el 1 1) Pr(02, 02) Fig. 2-4a: Specular Reflection. 802=01, 2=01 +1800 Fig. 2-4b: Diffused Reflection.

On the other hand, in the case of a diffused scattering, the scattered power is distributed over the upper hemisphere. The distribution of the scattered power may be expressed in terms of a bistatic scattering cross section Af\ (per unit area), a( 22, Q1). In scattering problems, this bistatic cross section is conventionally defined as 47r times the power scattered by a unit area of the surface from an incident beam of unit intensity in the direction 2Q1, into A a unit solid angle in the direction Q2. That is, 1 AA (dd)(pr2. )1 (UdA) lPi (' 1A (scattered power (area of scat- (incident power per unit solid angle. ) tering surface) density) (2. 17) In optics work, the scattering cross section per unit area is sometimes defined in a slightly different form. That is, dP 1 r 1A d2 4r dAPios1 ( 1) (total power intercepted) with Y (Q2, 2'1) the radar cross section, related through AH a ( A22 21)? ( 2' S1)1 cos 01 This expression was used in the first quarterly report on this contract (Chu, 1968) but to avoid confusion, the cross section (Q 2' Q1) as used in Eq. (2.17) will be used henceforth. By definition, then, the scattered power density due to an area dA observed at a point at a.distance r from dA and in the direction Q2 is given by dA Pi A A dp= 2 ~2()' Q1) (2.18) 4wr 12

Now, if we sample the scattered power from the extended ground reflector at a fixed point, the scattered power appears to be coming from A various directions (different Q2). This fact is illustrated in Fig. 2-5. The directional properties of this observed radiation may be expressed in terms of the scattered power intercepted by a unit surface per unit solid angle of the beamwidth: i. e. A dpr pi( 1) ^ ^ r=- ii a(Q ~2 (2.19) dQ2 47 cos (2' 119) For a narrow beam receiver of an effective aperture area A with beamwidth dQ2 staradians, the scattered power received is then Pi(1) A Pr 4rcos 2' 1 d2 (2.20) On the other hand, for an omnidirectional receiver of an effective aperture A, the scattered power received is r/2 2r ^ Ar J Csine2 d d) (2.21) r cos=. 2 41TUQ21) (2.2 21) 0 O Other modifications of Eq. (2.18) incorporating the radiation pattern of the receiving antenna can also be deduced. However, in the absence of the detailed information on the characteristics of the receiving systems, perhaps our estimation of detectability has to be based on one or both of the above limiting cases. The evaluation of the observed scattered radiation, even with the knowledge A of a(2, 2 1), has to be resorted to the numerical computation, due to the numerical function p (Q2l) as given by Eq. (2.13), and the involved relations A i between Q1 and Q2 depending on the relative position of the transmitter and the receiver, as given by (2.11) and (2. 12). In the work reported by Chu et al AA (1968), a simple model of a(Q2 Q12) (Lambert's law of scattering) was chosen. For this simple model, 13

Point of Observation Fig. 2-5: ScattDifferentg Geometry. dA14 Fig. 2-5: Scattering Geometry. 14

A A a(Q22, Q1) = 4 cosO1 cos 62 (2.22) and the evaluation of P for this case has been reported therein (Chu et al, 1968). In the current work, a more realistic model for the bistatic cross section A A a(Q2,' 21) is considered. Due to the lack of information of this bistatic cross section for rough ground or sea surface, both theoretical and some limited experimental work has been carried out to gain some knowledge of the bistatic scattering cross section. The theoretical bistatic scattering cross section for an open developed sea (cf. Eq. A. 117) is given by A ^ x I A 2 q4 H(O, O) (qxcossink)2 + (qy cosSO - qx sin)2 (2,23) 4q H(0, 0) where H(O, 0) is the mean square height of the sea surface; %x,y are the correlation distances of the surface height parallel and xy transverse to the direction of the surface wind, respectively; 0b is the angle between the direction of the surface wind and the x axis; qZ =cos 01 + Cos 02' q = sin 01 cos 01 + sin 02 cos 02 and and = sin 01sin 0 + sin 02 sin02' A detailed derivation of Eq. (2. 23) and the approximations involved in the derivation are given in Appendix A. From the reported ocean spectrum, the dependence of H(0, 0), I and By on the surface wind velocity U are estimated and x y the results presented in Figs. A-4 and A-5, respectively. 15

In order to gain some insight into the scattering model given by Eq. (2. 23), such as the effect of the wind and aspect variations, let us introduce the following parameters: i) The parameters of the sea surface defined by A 12_/4H(, 0) and B _ 12/4H(0, 0) o y ii) The directional variables A qx sin91 cos0l + sin92 cos62 qz cos 1 + cos 02 and an y 1insin + sin + sin2 sin qz cosO1 + coso2 In terms of these normalized parameters, (2.23) may be reduced to -M )1) = / (14a +f3 ) exp -A (aCcos +Bsin) -B (cos-sin)B (cossin) (2.24) In particular, when b = 0, i. e. the incident radiation is in the direction of the wind, A A A' (1a222 2 K2 A2a B2 (P2 21) = (14a +( )2 exp -A aB (225) 2'9 1 0 0 0L o In Fig. 2-6, the normalized quantities A0, B and /A-B for the model of sea chosen in the present study is presented. Due to the large value of A and B obtained for this model it is evident that a is maximum at a = 0, 13 = 0 o (forward direction) and has a maximum value of ~VA. The cross section, O O however, decreases very rapidly away from this forward direction. For a rough estimate of the decreasing of a from the maximum, the beamwidth of the scattered power may be approximated by Ao = AO 2 0 16

For U = 1.5 m/sec, the beamwidth is approximately 100 and for U = 4 m/sec, it is approximately 15~. To discuss the effect of the direction of the wind on the scattering cross section, let us assume that the direction of incident wave and the direction of A A A /A reflected wave (i. e. O1 and f2) are fixed and compare the ratio of a (S22 1) for f/ 0 with that for b = 0. Straightforward algebraic manipulation then yields o( o0) exp 2_ Thus, as the wind direction represented by b changes, this ratio varies between the limits exp -(B -A ) a2 and exp [(B-A ) 2] corresponding to the variation of b which satisfies the relation tan 2 = 2 a 2_ 2 a -3 For those directions of incident and reflected radiations for which a and 1 are not small, the effect of the wind direction would be pronounced. This feature is not present in all isotropic models of the rough surface scattering, for which A =B O O 17

600 550 500 450 Bo 400 2 2 = x 0 Ao4H (0, 0)' 4H(0,0) 350 JA350 o = lxiy/4H(o,0), U = Wind Speed (meter/second) 300 250 200 \A \ \ Fig. 2-6: Variation of Ao, Bo and FAoBo'vs Wind Speed.,.100 50 0.O 2.0 U(m/sec) 3.0 4.0 18

III CALCULATION OF REFLECTED RADIATION 3.1 Introduction In estimating the detectability of the reflected radiation from a doppler antenna such as the AN/APN-153 by a receiver of various beamwidths, the knowledge of the angular distribution of the reflected radiation at different points of observation, relative to the transmitting antenna, appears to be of prime importance. From the general theory of the surface reflection and the radiation pattern of the antenna AN/APN-153 given in Ch. II, the computation of the reflected radiation and its distribution may be carried out. In this Chapter, the basic formulas are presented together with the numerical schemes employed and some typical results of such calculations. 3.2 Mathematical Formulas In this section we summarize the mathematical formulas and numerical schemes used in calculating the reflected radiation of the AN/APN-153 antenna for three types of the reflecting surfaces discussed in the preceding chapter. 3.2. 1 Specular Reflecting Surface For a specularly reflecting surface, the reflected radiation appears to come from a single direction, solely depending on the relative position between the transmitter and the receiver. From Eqs. (2. 11) and (2.12), it is obvious that the direction of arrival of the reflected radiation is related to the relative position z (x, Y, zr ) by z X= (1 + -r )tan 0 cos 0 (3.1) z 2s 2s a z Y=(1+,r )tanO2 sins (3.2) a 19

where Q 2 jo2s'02s3 is the direction of arrival of the reflected radiation. This radiation originates from the antenna located in a direction (00 01 )' where 0 = 02s (3.3) and 01+02+ 180. (3.4) The intensity of the reflected radiation (power/unit area) can be calculated by the method of images. The result is PtG cos 0 p2 F (el,1) (3.5) P2s 4 2 o( 10 z a (1+- ) z The calculation of P2s as a function of X, Y and Zr/Za was given in Chu et al (1968). However, the directional characteristics of the reflected radiation was not considered there. For the calculation of the angles 02s and 02s for any given X, Y and Zr/za the following relations may be used: tan 02 (3.6) tan 02s= z (3.7) z a For the estimation of the reflected power intensities and for the convenience of their comparison for three types of reflecting surfaces, we define the normalized power density as the following: 2 z cos 0 2A 2s z Fs (XMY, -) = 4 2 F(0 ). (3.8) a t t r 2 (1+ —) 2 z z a a 20

3.2.2 Diffusely Reflecting Surface For the surface whose reflecting properties are defined by a scattering cross section a(Q2 Q ), the reflected radiation observed at a point appears to be coming from various directions. The radiation which originates from an antenna located in the direction (1. 01 ) relative to the reflecting point will be distributed. To an observer stationed at a relative position X, Y and Zr/Za, the reflected power of the radiation originally in the direction (01, 01) appears to be coming from the direction (02, 02), where z _r tan 02 cos 02 = X+tan 01 cos 01 (3.9) a z r tan 02 s in02= Y + tan 01 sin01 (3.10) a The intensity of the reflected radiation, by the definition of the scattering cross section, may be expressed by dPr PtGt 2 Fo(l, 01 2 cos0 (3.11) dQ2 4 7r z 2) (3.11) a Physically, of course, this may be interpreted as the power that is intercepted by a receiver of unit aperture per unit solid angle. For the purpose of comparing the direction and magnitude, we shall define the normalized quantity, Zr AdPr GtPt F2M (X Y z' 2) dR2 2 a 2 z cos 1 A 47rcos02 Fo(01)(2' 1 (3.12) In the case of the Lambert surface, a = 4 cos eI cos0 2 (3.13) 21

Whence, F = ( cos ) (3.14) 2d Cr 1 F2d denoting the case for completely diffused scattering (Lambert scattering). 3.2.3 A Moderately Rough Sea Surface For this model, the scattering cross section is given by (2. 23). Consequently, we have 2 cos 0 F 1 1 F(2 22 2M 4-r cos 02 (0, 1 )(1++ +)2 exp -A (acosv+fBsin) -B (bcos-asinb)] (3.15) where sin 01 cos 01 + sin 02 cos 02 qx a- (3. 16) cos 01 + cos 0 q(3.16) 1 2 and 13 sin 01 sin 01 +sin 02 sin02 qy (317) cos 01 + cos e2 qz A computer program for evaluation of the normalized quantity F2M corresponding to (3. 15) is included in Appendix B. Starting from the given 0,,, Y. we first calculate Zr/za and Oz, Oz by the transformation equations (3. 9) and (3. 10). Then, for a given wind speed U(m/sec) and the direction i, the values of F2M are calculated for each set of (O1 0 1) or (I0, po). F2d and F2s can be calculated relatively easily by using (3. 14) and (3. 8), respectively. 3 3 Power Level and Directional Variation of Reflected Radiation Before presenting the computed numerical results for the angular distribution of the reflected radiation, it may prove fruitful to look at the physical situation which partially explains angular variations of the reflected radiation. Let us refer to Figs. 2-3a and b, the radiation patterns of the antenna AN/APN-153, where contours of constant F (01' 01) are shown. Each ray originating from the antenna in the direction, say, (01w 1)' would reach an observer, after diffused 22

reflection, from the direction (02, 02). The relation between (01, 01) and (082, 2)' of course, depends on the geometry between the transmitter and the receiver. For a receiver fixed at (X, Y, zr/Za), the direction of a set of (0e i1 ) along a given contour of Fo will appear to reach the receiver from different directions. This geometric effect is illustrated in Figs. 3-la - 3-1d. The contours shown in Figs. 3-la and 3-lb are the contours of Fo seen by a receiver at (X=O, Y=O. 5, Zr/za = 0. 5); the Figs. 3-1c and 3-ld at(X=O, Y=O. 5, z /z = 0. 1). As seen from these illustrations, the receiver at different locations will see different shapes of the radiation patterns, depending, of course, on its relative coordinates. Thus, as shown in Figs. 3-la and 3-lb, the receiver at (O0 0.5, 0. 5) will see the portion of the major lobe of the Beam No. 1 much more elongated, and the minor lobe somewhat contracted, while the receiver will witness the major lobe of the Beam No. 2 greatly contracted and its minor lobe very much elongated. The same antenna pattern viewed at the same X, Y coordinates, but different relative height (z /za = 0.1) is illustrated in Figs. 3-lc and 3-id. Thus, it is evident that the deformation of the beam shape, after reflection, is critically dependent on the position of the observation. It should be noted that the contours illustrated in Figs. 3-la through 3-id are not the contours of equally reflected power, due to the angular dependence of the reflecting property of the surface. For example, the normalized reflected power densities (F2M) are different even on the same contour. In Figs. 3-2a and 3-2b, on each contour of constant F0, several points are selected and indicated by the dots. The normalized reflected power density, F2M, corresponding to these points (directions of reflected radiation), are tabulated in Table 3. 1 for the down wind speed of 1.5 m/sec. It is seen that the direction in which F2M is maximum is shifted from the direction of the peak of the corresponding transmitting antenna radiation pattern. Moreover, the shape of the contour of the constant F2M, after reflection, is drastically modified. In Figs. 3-3a and 3-3b and Table 3-2, similar plots and values are given for a Lambert surface for comparison. Here the modification of the contours of the constant power (F d) is seen less drastic. In fact, the direction in which F2d is maximum appears to correspond to that of the antenna radiation pattern. 23

20 Main Lobe....~~~~~ ~~~,r~ ~ ~ ~ *... ie me.~,.. ~ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........',,.-. ~........ 0.~ ~. ". iiiI'~ ~ ~ ~ ~ ~~~~. 0 4 ~ ~ ~~~~~~ ~~~~~~~~~~~~~~~~. ~ ~ 0~~~~~~~~~~~~~~~ 0r 0 80,,, ~l..,, ~~~~~~~~~~~~::.8.~ s~~~ ~;.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~.~... ~~~~~~~~~~~~~~~~~~~~~~~~ i~~~~~~~~~~~~~~~~~~~~~~~~~~~..'..I-..'-'''''''' —~..,' 28 0 2 4 60 2040608 ~~~~~~~~~~~~~~~~~~~.~ ~~~~~3~~~...-...-.''' Side eobe F.'. CV~~~~~~~. ~ 80~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~e"' <~~~~~~~~_ 280 300 320 340 360 20 40 60 80 B2(degrees) ig 3-a: Contour of Radiation Pattern of Beam No. 1 viewed at x=0 I, y=0~ Zr/za=0.5.

20.,...eg" ~.' ~ ~.0. ~~~~~~~~ ~~~~~~~~~~~' ~,, 40..~~~~~..~.~~~~~ ~~ 40 "....'" ~.'*ee ~ e~'* ~ ~. 9'~~~~~~~~~~~~~~~~~~~ *. 80 * *~~~~~Maine Lobe ~* I ~ ~ ~ ~~ " 0~~~~~~~~~~~~ ~~2(degrees)Lob Fig. 3-ib: Contour of Radiation Pattern of Beam No. 2 viewed at x=0, y='...5. t/ZI0.5.'..e'"' k~~~~~~~~~~~~~~~ -.. ~~~~~~~~~~~~Sd Loe" L* ~~ OJ'~~~.. so~~~~~~~~~....'-.. 1Z) 2 80'.. ~2 4 6 04 08 ~'.'~~~~~2(e're Fig.~~ 3-l ~otu ~fRdainPteno emN.2vee txO 1.,z/ a0

60 70 Main Lobe g-.................. 80 ~~..............................::...".....-''.... 260 280.:...:::.300 320 340 360...:.:.. 20.. 40 02(degrees) Fig. 3-c: Contour of Radiation Pattern of Beam No. 1 viewed at x, yO. 5, z /za= 0.1...:r a.e ~'"'..........'.........$'.....0~~~ ~~~ "mbi$ O3~~~~~~ C~m,.0 ~~~~~~~ Q>.. ~.~~~~~~~~~~~~~~~~~~~~~~~ii 260 280 300 320 340 360 20 40 60 80~~~~~~~~~~~~~~~~ i' c ~~~~ ~2(eres Fi.31:Cntu fRdato atr o f Bea N. 1vee tx0 =.5, Zr/Za=0.1.

40 60 Side Lobe 0'?**.................*... b0 * Main Lobe.......... 100 _ 260 280 300 320 340 360 20 40 60 80 02(degrees) Fig. 3-ild: Contour of Radiation Pattern of Beam No. 2 viewed at x=0, y=O. 5, zr/Za=0.1

20 40~~~~~~~~~~~~~~~~~~0 *......~.. 0**.* 0 4D 0.4.. 00 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~p b~~~~~~~~~~~~~~~~~~~~~O~~~~~~~~O 80 ~~~~~~ 9.~~~~~~~~~~~. 1001II II 280 300 320 340 3D 20 40 608 02 (degrees) Fig. 3-.2a: Contour of Radiation Pattern of Beam No. 1 viewed at x=0, y=O. 5, z r/Za=0.5 Consult the Table 3. 1 for db F2 at the wind speed U=1. 5 rn/sec. the wind direction O='0 2M~~ ~ ~ ~

20 F 40 eO *.... Side Lobe 4..~~~~~O.0 C~... Io 60.0 I I * Main Lobe 0~~~ irr Consult' t Tb 3 f dF thw se U 5ne te i deoI I~~. 2M I~~~ ~ 100~~~~~ C~~ 260 280 300 320 340 360 20 40~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~ ~02 degees Fi.3-b onorofRdito Pten fBamN. iwe tx=,y=.5 zrZa O Cosl h abe31frd F2 ttewndsed 1 5mscJ h wn ieto

TABLE 3-1: Normalized Reflected Radiation Density, F2M, for Different Directions of Incidence, received at X = 0, Y = 0.5 ZZ = 0. 5. The Wind Speed U = 1.5 m/Sec, its Direction q = 0. Points and Power dB F M of Beam No. 1 dB F M of Beam No. 2 Levels of F0 2(Fig. 3-2a) 2 (Fig. 3-2b) p: peak of F0 -161.3 -678.2 -1 dB contour: read counter-clockwise, -185.3, -155.8, -130.6, -273.6 -698, -477, -620, -620 starting from A... | -5dB contour: -243.7, -214.4, -219.6, -146.7, -750, -402, -303, -360, 0 read counter clock- -114.6, -289.7, -556. 3, -554. 3, -605, -607, -607, -605,. wise, starting from B -469.1 -616 -15dB contour: -292.6, -291.0, -361.2, -214.1, -760, -450, -266, -232, read counter clockwise, -131.0, - 95.0, -235.0, -246.7, -329, -572, -576, -588, starting from C -233.9, -235.2, -558.4 -576, -542, -572 -25dB contour: -448, -402.7, -423.2, -418.7, -580, -526, -295, -206, read counter clockwise, -189.8, -113.2, - 72.3, -163.7, -232, -317, -560, -555, starting from D -158.2, -161.8, -159.2, -161.2, -559, -557, -543, -557, -145.3, -159.4, -152.3 -550 Q: peak of side lobe -141.4 -199.5 -15dB contour:, read counter clockwise -140.8, -542. 8, -542.7, -548.7 -168.0, -138.9, -263.7, -567.8 | starting from E -25dB contour: -544.2, -518.3, -335.6, -567.4, -146.1, - 90.3, -144.8, -207.7, EZ read counter clockwise, -560.1, -561.1, -564.7, -572.0, -348.6, -679.7, -683.6, -690.1, starting from F -564.7, -554.7, -555.7 -682.0, -673.0, -675 R: a point in -25dB range -557 -675.4

20 ~ ~4 *0.0 ~ ~ ~ ~ ~~~~~~~~~~*.. ~'W ~~~~~~~~~...~~~.~..~ ~,0.... ~ee ~ eeee~eeeee~e e e e " e e o ~ ~e e e 60 ~ ~ ~ ~ ~ ~ ~~~~~o MaineLobee b O %-P~~~~~~~~...80 S~~~~~~~~~~~~~.10011 1 280 300'320 340 3 20 40 60-. 02 (degrees) Fig. 3-3a: Contour of Radiation Pattern of Beam No. I viewed at x0O, y=O. 5, z z!a0e5 Consult the Table 3.2 for db F2 at the wind speed U=1. 5 rn/sec D the wind direction JP0. 2M ~ ~ ~~ ~ ~. ~~-'.e~~~ ~ ~~~~~~~~~~~~~ —.......-'"'...............' e...o ~ ~ ~r ~' - ~~" D'~~~~~ —~'- ~' ~'.-~ 60.'.' - Q)~~~~~~~-"." Main Lobe Q)~~~~~~- IrfSide Lobe Q)~~~~~~~~~~~~ ~~'' ~o~~~~~~~~~~.'~ ~'e, (N t~~~~~~~~~~~ ~ ~'. C) ~.~. -..:. 80 -:.. loo.... 280 300'320 340 360 20 40 60. 80 ~2 (degrees) Fig. 3-3a' Contour of Radiation Pattern of Beam No. 1 viewed at x=0, y=0.5, z/jza=O. 5 Consult the Table 3.2 for db F2M at the wind speed U=I. 5m/seco the wind direction ~/=0

20 F eeee ~~ ~'e * *'.... ~~~~~~40 __* ~~~ *~..-".~ e~' -Side Lobe t.. 40 ~ ~ ~ 0 9 ~~ ~~ e~ ~ l e ~o~~A.-e " ~..'-....'......'.0 ~Main Lobe k'.'-'q. ".*' -...........' b0 6 ~~- -.:.,,..""..' 6 0. ~~~P ~..e p~A... anLb.'.di'"":.'.".lP'.4.. ~ ee~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~100 360__e.2L 0 40 260 280 300 320 34036204608 02 (degrees) Fig. 3-3b: Contour of Radiation Pattern of Beam No. 2 viewed at x=O, y=O. 5, z r/za=0 5. Consult the Table 3.2'for db FM at the wind speed U =1.5 rn/sec, the wind direction ~0 2M~ ~'I 260 280 300 320 340 360 20 40 60 80 d2 (degreeS) Fig. 3-3b' Contnur of Radiation Pattern of Beam No. 2 viewed at x-0, y:0. 5, z/za:0.5. Consult the Table 3.2 ~for db F2M at the wind speed U:I. 5 m/sec 0the wind direction ~:0.

TABLE 3-2: Normalized Reflected Radiation Density, F2d, for Different Directions of Incidences, received atX = 0, Y = 0.5, Z /Z = 0.5..' r a Points and Power dB F d of Beam No. 1 dB F 2d of Beam No. 2 Levels of F0 2 (Fig. 3-3a) (Fig. 3-3b) p: peak of F0 -7. 3 -7.3 -ldB contour: read counter clockwise, - 7.5, - 7.4, - 8.7, - 9 - 7.5, - 7.4, - 8.7, - 9 starting from A 2 -5dB contour: -13, -10.5, -11.0, -10.3, -13, -10.5, -11.0, -10.3, o read counter clockwise, -11.5, -14.0, -14.7, -15.2, -11.5, -14.0, -14.7, -15.2, | starting from B -24. 3 -24. 3 -15dB contour: -24.1, -22. 7, -20.6, -19, -24. 1, -22. 7, -20.6, -19, o0 lzi |read counter clockwise, -21.5, -21.1, -26.6, -40, -21.5, -21.1, -26.6, -40, starting from C -35.1, -30. 4, -23. 7 -35.1, -30. 4, -23. 7 -25dB contour: -31, -32, -31.2, -28.2, -31, -32, -31.2, -28.2, read counter clockwise, -33.8, -32.8, -30, -40, -33. 8, -32.8, -30, -40, starting from D -37.3, -42.3, -41.6, -53.6, -37.3, -42.3, -41.6, -53.6, -28. 5, -42. 3, -32.0 -28. 5, -42. 3, -32.0 Q: peak of side lobe -20. 2 | -20. 2 -15dB contour: Q read counter clockwise, -20, -22, -23.1, -30. 3 -20, -22, -23.1, -30.3 0 starting from E a -25dB contour: -24.2, -28.2, -31.7, -39, -24.2, -28.2, -31.7, -39, 7 read counter clockwise, -33. 4, -35. 6, -40. 4, -50. 8, -33. 4, -35. 6, -40.4, -50. 8, starting from F -47. 8, -33. 8, -30. 8 -47. 8, -33. 8, -30. 8 R: A point in -25dB L -35.6 1 -35.6

3.4 Direction of Maximum Intensity of Reflected Radiation The apparent direction of arrival of the reflected radiation received at any position is taken as the direction toward which a narrow beam receiving antenna would intercept the maximum amount of radiation intensity. For a specularly reflecting surface, this apparent direction of arrival, denoted by (02s 02s), may be calculated by Eqs. (3.6) and (3.7). For a diffusely reflecting surface, such as a random sea surface, this apparent direction of arrival, denoted by (02M 02M)' is to be obtained by searching for the direction in which F (e 2 02) is maximum. Discussions on the directions of arrival are given in this section. These are illustrated in Figs. 3-4a through 3-15b inclusive. In Figs. 3-4a and 3-4b the apparent direction of arrival for an observer located at Y = 0. 5, z /z = 0.5, at the downwind speed of 1.5 m/sec is plotted against X to show the change in the apparent direction of arrival with the relative receiving position. In these curves are inserted the direction of arrival for a specularly reflecting surface for ready comparison. In Figs. 3-5a and b, similar curves are shown for the case of the cross-wind. In Figs. 3-6a and b and 3-7a and b, similar curves are shown for the receiver positioned at the relative height of O. 1. Figs. 3-8a through 3-llb show the corresponding cases at the wind speed of 4m/sec. The effects of the wind speed on the direction of arrival are shown in Figs. 3-12a and and b at Y = 0. 5, z /a = 0. 1 for the down-wind case; Figs. 3-13a and b show the similar curves for the cross-wind case. It is seen from these curves that the wind speed has little effect on the direction of arrival in the downwind, but in the cross-wind case, the direction of arrival is sensitive to the wind speed in the range I x,< 1. Figs. 3-14a and b show the effect of the wind direction on the direction of arrival at the wind speed of 1. Sm/sec at Y = 0. 5, z /z = 0. 1; r a Figs. 3-15a and b show the similar curves at the wind speed of 4m/sec. It is observed from these curves that the direction of arrival is slightly more sensitive to the change in the wind direction for the lower wind speed than in the higher wind speed, the significant effect being confined, again, more or less within the range I x.< 1. 34

For completeness, the approximate direction of arrival for the case of a Lambert surface, denoted by (02d' 02d)' are presented in Figs. 3-16a, b and Figs. 3-17a, b. In this case, based on the argument given in the last section, the maximum direction of arrival corresponds to the peak of each major lobe, and hence appears to be multi-valued. However, it should be noted that only one of the directions of arrival, corresponding to the reflecting point which is the closest to the point of observation, should have the dominant effect on the total reflected power received.

02 300 P\d 02M - 20o _O —-.. I I \G -lpo 02s -x I' I I x -3 -2 -1 0 1 2 3 Fig. 3-4a: Comparison of the Maximum Directions of Arrival (Azimuth) for the random ocean and Specular Surfaces at down-wind speed of 1.5 m/sec. The relative Receiver Height is 0. 5, Y = 0. 5. 02 80 > t o~_60 -040 -x I I I I I 2 I X -3 -2 -1 0 1 2 3 Fig. 3-4b: Comparison of the Maximum Directions of Arrival (latitude) for the random and Specular Surfaces at down-wind speed of 1. 5 m/sec. The relative Receiver Height is 0.5, Y = 0.5. 36

300 20! \ M 2.0 0 2M \ \ rl100 02S -X xI I I I I X -3 -2 -1 0 1 2 3 Fig. 3-5a: Comparison of the Maximum Direction of Arrival (Azimuth) for the random ocean and Specular Surfaces at the cross-wind speed of 1.5 m/sec. The Relative Receiver Height r = 0 5 Y=0 5 a 02 80 2M/ 60 40 / -x I I I l I x -3 -2 -1 0 1 2 3 Fig. 3-5b: Comparison of the Maximum Direction of Arrival (Latitude) for the Random Ocean and Specular Surfaces at the cross-wind speed of 1.5 m/sec. The relative Receiver Height Zr Y05 Za 37

02 300 / \ / 200 O-..-.. - -o -o -x - I I. l 1 1.x -3 -2 -1 0 1 2 3 Fig. 3-6a: Comparison of the Maximum Directions of Arrival (Azimuth) for the Random and Specular Surfaces at down-wind speed of 1. 5 m/sec. The relative Receiver Height is 0. 1, Y = 0.5. e2 100 02M 0 " — 80-O 0 - o x?~80 p I J -x' I l I, I I I Fig. 3-6b: Comparison of the Maximum Directions of Arrival (latitude) for the Random and Specular Surfaces at down-wind speed of 1.5 m/sec. The Relative Receiver Height is 0.1, Y = 0.5. 38 -3 -2 0 2 3

1 -200 x>M -— O 02 100 -X... I I I I I X -3 -2 - 1 0 1 2 3 Fig. 3-7a: Comparison of the Maximum Direction of Arrival (Azimuth) for the Random Ocean and Specular Surfaces at the cross-wind speed of 1. 5 m/sec. The Relative Receiver Height z = 0.1, Y= 0.5. a 02 \ \80 M/ \ I I \.. 160 I 2S -X - I I I x -3 -2 -1 0 1 2 3 Fig. 3-7b: Comparison of the Maximum Direction of Arrival (Latitude) for the Random Ocean and Specular Surfaces at the cross-wind speed of 1.5 m/sec. The Relative Receiver Height z = 0.1, Y = 0.5. z a 39

02 300 /0 02M 200 I'"- --' —O'~~ ~ = "~ -— > -'-0' 100 025 -_~~~~x __+ ~i~I -3 -2 -1 0 1 2 3 Fig. 3-8a: Comparison of the Maximum Direction of Arrival (Azimuth) for the Random Ocean and Specular Surfaces at the down-wind speed of 4 i/sec. The Relative Receiver Height z/z a= 0. 5, Y = 0. 5. 05 r a~~~~~~~~~ \~~~~~~~ 80 0 N\ ~'( ~~~~2S -x I I -3 -2 -1 0 1 2 3 Fig. 3-8b: Comparison of the Maximum Direction of Arrival (Latitude) for the Random Ocean and Specular Surfaces at the down-wind speed ope f 4m/sec. The Relative Receiver Heightz Z/Z = 0.5 Y05, 4~~~~~~~~~~~ r ~a 40 — 0 -X ~ X\/ -3 -2 -1 0 1 2 3 Fig. 3-8b: Comparison of the Maximum Direction of Arrival (Latitutde) for the Random Ocean and Specular Surfaces at the down-wind speed of 4 m/sec. The Relative Receiver Height Z/lZa = 0.5, Y = 0 5. 4O

02 300 200- -o -x ~ I I:I I -I I-3 -2 -1 0 1 2 3 Fig. 3-9a: Comparison of the Maximum Direction of Arrival (Azimuth) for the Random Ocean and Specular Surfaces at the cross-wind speed of 4 m/sec. The Relative Receiver Height zr/Z = 0. 5, Y = 0. 5. 2 a2M 80 /C \ -- 60 \ 40 -i X -3 -2 -1 0 1 2 3 Fig. 3-9b: Comparison of the Maximum Direction of Arrival (Latitude) for the Random Ocean and Specular Surfaces at the cross-wind speed of 4 m/sec. The Relative Reciever Height z /z = 0.5, Y = 0.5. 41

/ h P2M of - -200 — ___O 100 02S -X I I I I i X -x x -3 -2 -1 0 1 2 3 Fig. 3-10a: Comparison of the Maximum Direction of Arrival (Azimuth) for the Random Ocean and Specular Surfaces at the down-wind speed of 4 m/sec. The Relative Receiver Height z r/Z = 0. 1, Y = 0. 5. 02 2M 80 / I I I.4oJ -x - I I I -3 -2 -1 0 1 2 3 Fig. 3-lOb: Comparison of the Maximum Direction of Arrival (Latitude) for the Random Ocean and Specular Surfaces at the down-wind speed of 4 m/sec. The Relative Receiver Height Zr/z = 0. 1, Y = 0.5. 42

300! \ 2M -2~~x0...xr"'<>" 0.c -o 100 _2S -x x I -3 -2 -1 0 1 2 3 Fig. 3-1a: Comparison of the Maximum Direction of Arrival (Azimuth) for the Random Ocean and Specular Surfaces at the cross-wind speed of 4 m/sec. The Relative Receiver Height zr/z = 0.1, Y= 0.5. 02 2 O —-2- ~ -6 I \I\I -x I x 4340 I-I -3 -2 30 -- 43

2M (degrees) 90 00 3' I ~ U=1.5 m/se I -—. U=4 m/sec -3 -2 -1 0 1 2 3 Fig. 3-12a: The Effect of the Wind Speeds (U=1.5, 4 m/sec) on the Directions of the Maximum Reflected Radiation Intensities (Latitude) for the 44 44

02M 360.320 280 U U=l.5 ec...: U=4 m/s 200 el'~& 160 120 80 40 -X I I I I I -X -x l I I I x -3 -2 -1 0 1 2 3 Fig.3-12b: The Effect of the Wind Speeds (U=1. 5, 4 m/sec) on the Directions of the Maximum Reflected Radiation Intensi ties (Azimuth) for the Down-wind Case. The Relative Receiver Height is 0.1, y=0.5. 45

02M 90 80 lIl l l70 b l -I I III I I -I - -1 0 I I ( 1 I U rs I I 0 ----—:U=4.5 m/se Fig. 3-13a: The Effect of the Wind Speed (U = 1. 5, 4 mrn/sec) on the Direction of the Maximum Reflected Radiation Intensities (Latitude) for the Cross-Wind Case. The Relative Receiver Height z /z = 0. 1, Y = 0.r 5. 46

]2M 360 320 I I' I 280 I I I 10 40 -x I2004'~-'- -~,..,,/160 120 80:U=1. 5 m/sec':U=4 m/sec 40 -3 -2 -1 0 1 2 3 Fig. 3-13b: Effect of the Wind Speeds (U = 1. 5, 4 m/sec) on the Directions of the Maximum Reflected Radiation Intensities (Azimuth) for the Cross-Wind Case. The Relative Receiver Height z r/Za = 0.1, Y = 0.5. 47

82M 90 II Ii'I I I_-0 10 I I I x50 -3 - 41 0 1 2 3 i li-30 Radiation Intensity (Latitude) at Y = 0. 1 for the i! I',~1 Wind Speed of 1. 5 m/sec. r a 48

02M 360 320 I' 280 I i I' -1200 1-20 900 — 40 ~-x l I I I I I x -3 -2 -1 0 1 2 3 Fig. 3-14b: Effect of the Wind Direction on the Direction of Maximum Radiation Intensity (Azimuth) at Y = 0. 5, Zr/z = 0. 1, for the Wind Speed of 1.5 m/sec. 49

82M 90 80! _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I -xdt-i-II I I x -3 -2 -1 0 1 2 3 Wind Speed of 4 r/sec. r 50 -3 -2 - 1 0 12 3 Radiation Intensity (Latitude) at Y = 0.5, Zr/Z = 0.I for the

02M 360 320 I X 0I \240 A 160 -120'= 90~ 0 40 -x I I I I I x -3 -2 -1 0 1 2 3 Fig. 3-15b: Effect of the Wind Direction on the Direction of the Maximum Radiation Intensity (Azimuth) at Y = 0.5, zr/z = 0. 1 for the Wind Speed of 4 m/sec.

02d 200 __160 120 80 _ 40 -3 -2 -1 0 1 2 3 02d 120 __80 40 -x l l I I a I -X i i i I I I -3 -2 -1 0 1 2 3 Fig. 3-16a: Directions of Arrival for a Lambert Surface at y=0.5, z /za =0.5 for Beam No. 1. 52

02d -380 -340 300 — 220 -X I I I II -3 -2 -1 0 1 2 3 02d _40 -x I I I l I I I x -3 -2 -1 0 1 2 3 Fig. 3-16b: Directions of Arrival for a Lambert Surface at y=0.5, Zr/za0 5 for Beam No. 2. 53

02d 200 -160 — 120 _80 -40 -3 -2 -1 0 1 3 O2d 120 — 40 -x I I t I II I I x -3 -2 -1 0 1 2 3 Fig. 3-17a: Directions of Arrival for a Lambert Surface at y=0 5 Z/zaO: 1 for Beam No. 1. 54

02d 380 __340 300 260 220 -x I I - - I IIx -3 -2 -1 0 1 2 3 02d 120 40 a-x - l.. i... - -I. -.X -3 -2 -1 0 1 2 3 Fig. 3-17b: Directions of Arrival for a Lambert Surface at y-0.5, z/z -0.1 for Beam No. 2. 55

3.5 Magnitude of Maximum Reflected Radiation Due to the distributed nature of the reflected radiation from a diffused surface, we shall compare principally the intensity of radiation in the apparent direction of arrival, along which the intensity is maximum. For the case of the specularly reflecting surface, the reflected radiation observed at any point appears to come only from one direction, the magnitude of which may be represented by a normalized power density, F2s, given by Eq. (3.8). In terms of F2s' the observed power density at any point is obtained by PtG t t 2s 2 2s (watts/m ) Za For the case of the diffusely reflecting surface, the normalized intensity, F2M and F2d are given by (3.12) and (3.14), respectively. In terms of these normalized quantities, the power density per unit solid angle is expressed by dP2r PtGt dQ2= F2M (or F2d) 2 a (watts/m /steradian) In this section, the variations of Fs' F2d and F2M with relative receiving points are discussed. It should be recognized, however, that the direct comparison of F2s with F2M (or F2d) is meaningless unless the radiation pattern of the receiving antenna is taken into account. For the case of the narrow-beam antenna, the relative magnitude of F2s and F2M times the angular beamwidth of the receiving antenna (in steradians) probably could be taken for an approximate comparison of the reflected power observed. In Fig. 3-18a, we show the variation of F2M (in the apparent direction of arrival) at the relative receiver height fixed at 0. 1 and at different Y for the down-wind speed of 1. 5 m/sec; in Fig. 3-18b, similar curves at the relative receiver height fixed at 0.5 are shown. Similar curves are presented in Figs. 3-19a and b for the down-wind speed of 4m/sec. Figs. 3-20a through 3-21b deal with the corresponding cases for the cross-wind. In Figs. 3-22a and b, we show 56

the effect of the wind speed on the variation of Maximum Reflected Radiation Intensity for the down-wind case at (Y = 0. 5, Zr/a = 0. 1) and (Y = 0.5, z /z = 0. 5), respectively. Similar curves shown in Figs. 3-23a and b deal with the cross-wind case. It is noticed that the overall level of dbF2M is slightly higher in the down-wind case than in the corresponding cross-wind case, the difference being more pronounced at the lower receiver height. Figs. 3-24a and b present the effect of the wind direction at the wind speed of 1. 5m/sec at two different relative receiver heights of 0. 1 and 0. 5, respectively. Similar curves shown in Figs. 3-25a and b deal with the wind speed of 4m/sec. It is seen that, in general, the change in the wind direction has a rather minor effect on the dbF2M variation, even though the effect appears to be more significant at points at the lower receiver height. Finally, the variation of F2M for the down-wind case is compared with those of F2d and F2 in Fig. 3-26a at the relative receiver height fixed at 0. 1 and Y = 0.5. In Fig. 3-26b a similar comparison is made at the relative receiver height fixed at 0. 5. Similar comparisons are presented in Figs. 3-27a and b for the cross-wind case. The use of this information and its relation to the problem of estimating the detectability is discussed in Vol. II of this report. 57

'1'0 = z/z:w oos/ua cj'1 Jo poods pu!AM-u,o( Lot Ialitp:U8M-O'J1O ID II I II I I I I I, Ii r III I I I I II 1'1I Ij I P"9 I ii I, I II I ~~~~~, /!r ~ I I I I Ii ~~~I! Ii ~~'I:=~: —-- - ----- Ii i I O''I=~~~~~~~0I 1 1 I i _ _ _ _ _ _ I!~I I iH

dB F +10 2M -3 -2 -1 0 1 2 3 4=~ / - I.-x I I.,A~~~~~~~~~~I, i II,~ ~, I I I 11 it I I / T Il r 5 1 I 3 I I ~ ~~~~~I II I~~~~~~~~ / b~- ~ Y1. -30 / i a' Fig. 3-18b: dBF2M for Down Wind Speed of 1.5 rn/sec at Zr/Z = 0.5. 59 ~ ~ (

09'T'0 = 1Z/Iz B 3Oos/uf, Jo poaad puiM-uMoQ:oj WZagp:'66-C'Ia 08/ io, I,0 ~ ~ ~~~~~/, I \ON OL0'~~~O I I i,'*: 109 rJ% 0''Ii c\ I I \ p I Iou I I II I i II I ~~109 I! xl Ia~ O I T ~~'T = ~ ~ ~ a _(_-. -— 0I ~x,-,/I,e I~~~~~~~ ~ 1 0 1-' i I'=Xr' r J' LX E Z T O - ~ I ~~Z OT-t~~~~~ a'T~~x' I i._8

+10 dBF2M -3 -2 2-1 3 IX~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~I -x PJ/ a II ~~~~~~~~~~~~Y= i. 5 -70 cJ I~~~~~~~~4 /6~~~~~~~6 -500 ~,~~~~~~~~~~~~~~~~-b / ~~~~-7,8 Fi. -1b:dBZMfo DwnWid ped f mse a z/z = _~. 5 61\..

L( 0 to to co NT I I I II N Ii N'-4 O 0 X C') 1~~~~~~~~~~~~~~~~~~~~~~~~dG

of -)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o OR 0 0 _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~MNO4.._......_. —-- --.. |~~~~~~~-.... —-— oI — ~'' — - -I t1 _ =_L AW fmX-vw 0O -=-MN IC)t ",,-W —Mmw II Ii~~~~~~~~~~l sopo "a~~~~~~~~~~~~~I 0 (D N N~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~WN MII,,NU MAIN 0a a's

'T* 0 = Z/ z A oas/ui.o paads pulAM ssoo ioj} Z'gp:eIZ-' 2!T \ / / I A I F"I H T g'I'. ]{ X/ ~'O=X ~ _OT- Ox z | O 1- Z- ~I<Z~s I

g.O = Z/z IV oaS/Lu e Jo paead puiM ssoD3 xoJ IZgaap:qTg-E 2!i 08~~\t~~~~~ 09\ i\'I (II I/ 0I i IAYg

'*1.0 = Z/z ~'O = z/e aseC pulM UMoa aql loj &llsUaluI uo:lz!peH poolauji nuz!xseU / aow uo poadS ppulM aoq jo iooajJ:'ez-c':2!U,8f~/ I XI/ I I I:~~I g ii |1-Z II I I I - I o o - n /- -- ~fdp

9.0 = tz/2/z''0 = X 1t osco pulAX UMtoaQ o aoj ll!sualuI uoilupeui paoaoljao untum!xAIyw al uo poads pu! aqjl jo 1a;jj':qZZg-g'12I ) 08OLy \l I I t 1 - 1 \ I' -I / / 30S /xu 9\ / ~~~/

'1.0= z'9 = / 0 - X 98 D pu!M seso:I3 aql ioj,!suOUI UO.Em.pvl paqaGoo H uwntuipc a qp uo poods pulA Mao, o o looJ a oaq,:'E Z-g 2'1 \I - I I 7' \IILs8/I g -I =os- -II ~xl'~ l~ l i I I x~I; Z I I, t I I x0 1- g;- C-."EP

'9'0= z/Pz' O = 9 Ie aseD pu!Af ssoia aoT4 io; Al!sualuI uo!priptu paoaoUo LUntnlxeW aqi uo poads pu!M aoq Jo loaJJ:qZ-E'2.a 08OL09oI ~I I / I I / I / I I I'I I I I I I I 3;as/.Lr~~~ u S''I =I OIT oEs/mu 9c'T'l XI I I

dBF2M -3 -2 -1 0 1 2 3 -xl I l 0 _ I=0 _ =90~ -10 - - -20 P I I -30 I I I I I -40 I -50 I II I; I I II 7 1 - II 70 I I I I I I I i I I I ii II I Fig. 3-24a: Effect of the Wind Direction on the Maximum Reflected Radiation Intensity for the Wind Speed of 1.5 m/sec at Y = 0. 5, Zr/Za = 0.1. 70?f

dBF2M -3 -2 -10 1 2 3 -X I i Ix =00 b 1 __. - __ ~=90~ /I I A I 20 I 401 /- I I I ~ ~II I \ II L~dll i - I \ - / Fig. 3-24b: Effect of the Wind Direction on the Maximum Re ected Radiation Intensity for the Wind Speed of 1.5 m/sec at Y = 0.5, Zr/Za = 0.5.

' O = Z/az' c0O = A XI as/tu f jo peads pu!M j a oql o ZlsuauI uo!lr!peH poloaluaU tntun!xwlq aol uo uoloaoila pu!ln aql jo BoaJJa:'Z-'!ta OL- \ 0/ II I, II I I I I I - t I -, - IAIZgp

dBF2M -3 -2 -1 n 0 1 2 3 -xl I I / I 1 Ix 900 I I 0- -40 70 1 / \ -8 _0 Fig. 3-25b: Effect of the Wind Direction on the Maximum Reflected Radiation Intensity for the Wind Speed of 4 m/sec at Y = 0.5, Zr/z = 0.5. 73

AL ~ os/u7'g' T=1 =no 0o =/'t *0 = Z/Z'9 0 = a PW pur SZa!m a jo uos!BdEuODa:B9uZ-'*2!I 08-,IA11 ICI, I I Pza I oIzII ~\I I j IIj 1I ~~~~Igp~~~~~~x

oas/wi7'':=nl.zo oO:'g =P~Z cSjg iLi~'g'O = z/z'g 0 A 1 puB Lf!f;M o uosT.:tlduIoDq9g -q'2-..~ 0 I ~ ~R g j-I i0- I ~I I I3 I c*3 ~~~~~I' ~~ I I I ~ ~ J_ _ 109"I I I S~~~ I I'I \ \ r ~~~~~~~~~~I i I II I % Z I!o I t \I / I ~ —-/~or \I i ~ ~~~/ X~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~' I I III ~ X C Z' o1 0 /- C- E01+ 21b" i~I t

9L ~os/m ai'g'I = a. oJ 06 =' ZO z /' ~ = Pit, PUB SZ g atIAM Z o Jo uos0 Io03:eLZ-E -'!i I I I I III oIIg I,,I I I!I I II I ~II Io I; /'U Ij IO'I' " OI II I P:I iC x ~-;. 0 o - z- 0Ztp +

os/m tU,'.-T: =.foJ o 006 = 9,'O = z' 0 = I t aj pu~ SZ j1wA pia jo uos!iadLuo3:qLZ-E'!i II II EK.~~~~~~~~~~8 I- -109-,,-I I q I~~~~~~~~~~~~~~z \ ~ I P~'I~~~~~~I, iI / I~~~~I IPo- I Ixl~~~~ I~ I I~~~~~~ I O~- I 0 I - II \ I.-i-o- I I t ~~~I II I!, / wg' 01+ SZ~~ l\ l // IA~i 0I+

IV CONCLUSIONS AND RECOMMENDATIONS A theoretical analysis is performed to investigate the reflection properties of an anisotropic normally distributed rough surface by use of the physical optics method. Approximate formulas are derived for calculation of the reflected radiation from a moderately rough ocean surface which exhibits the Neumann spectrum. A computer program is written for this formulation, and limited numerical computations are carried out for the AN/APN-153 doppler radar antenna. The distribution of the reflected radiation is calculated for several relative geometric configurations of the transmitter and receiver. The numerical calculations are limited only to the down wind case for a few different wind speeds. The computer program, however, is valid for variations of the wind direction within the limit imposed by the Neumann theory, namely, j j < 7r/2. The cases for different wind directions will be performed in the future program. The direction of arrival and the intensity of the reflected radiation from a moderately rough ocean surface are presented and compared with those from a specularly reflecting and Lambert Surface. In most of the cases compared, it appears that, in its general trend, the reflected radiation intensity, F2M from a moderately rough ocean surface resembles that from a specularly reflecting surface, even though the direction does not. The level of the intensity sharply decreases as the receiving point moves away on the X-axis from the origin of the relative coordinate system for a low wind speed, say 1. 5 m/sec (or 2.7 knots). As the wind speed increases to 4 m/sec (or 7.4 knots), the drop in the level of the radiation intensity away from X = 0 is less drastic. The direction of arrival, both in azimuth and latitude, tends to approach the same direction each as the receiver moves away from the origin along the X-axis. Within the range IXi " 1, however, the variation of the direction of arrival is wide at points, particularly so for the case of a low receiver height. It should be noted that, all in all, the change in the wind direction has only a minor effect both in the magnitude and the direction of the maximum reflected radiation intensity. This is attributed to a small difference in A and B O O 78

The limitation of the present model and some recommendations on the future works on rough surface scattering (especially ocean surfaces) are given below. 1) The inherent inadequacy of the physical optics method somewhat limits the validity of the present calculation, particularly near the horizon. Further theoretical analysis (exact numerical solutions for some canonical problems) and experimental work (model study of scattering) which may assert the range of the validity and suggest possible refinement on the physical optics method appear to be in need for a more adequate solution to the problem of random surface reflection. 2) An alternative approach, such as the use of the concept of surface waves, in estimatingthe scattered field near the horizon should be investigated to compliment the results obtained through the physical optics method. 3) The present model is based on the Gaussian correlation for the surface height. It is possible to adapt the analysis to other kinds of statistical models for a rough surface with different correlation. However, in order to extract meaningful numerical results from such efforts, adequate statistical information on the rough surface is necessary. In particular, in the case of the ocean surface, it appears that more experiments are needed which will add a more appropriate statistical description of the ocean surface, incorporating different sea states, in estimating the reflected radiation power. 79

APPENDIX A: BISTATIC CROSS SECTION OF A ROUGH SURFACE A. 1 Introduction It has been recognized that both adequate theoretical formula and experimental data have been scarce in the case of the bistatic scattering cross section for the rough surface and ocean surface. In view of this recognition, the bistatic cross section is investigated both theoretically and experimentally by means of the physical optics approach with special reference to the ocean surface, in an effort to promote a better understanding in this area. An approximate formula, incorporating the anisotropic nature of a random ocean surface (due to wind effect) is derived for the computation of the reflected radiation from a doppler radar in the main text of this report. Recognizing the inherent shortcomings in the physical optics method, and the uncertainty in the existing sea surface wave spectra, approximations based primarily on physical grounds are introduced to simplify the result. It is realized that more experimental work and theoretical analyses based on the exact solutions of some canonical problems are necessary for a more complete understanding of the problem. It is our present feeling that the results derived in this appendix are probably inadequate for reflections near the grazing angle, for which the physical optics approach is known to be in error. A. 2 Scattering Cross Section The conventional description of the scattering properties of a rough surface is given by the bistatic cross section per unit area u( 02' 21) It is an average quantity defined as 4 r times the power scattered by a unit solid angle in the direction ~2, for an incident wave of unit intensity from the direction ~ With this definition, for an incident radiation of intensity pi(Q1) impinging on a surface of the area dA, the intensity of the reflected radiation observed at a distance r from the point of reflection in the direction ~2 is given by ^ 1 2(22' C 1) dPr Pi 1)A 2 4r (A1) r 80

Since the solid angle subtended by dA at the point of observation is dA cos 02 dQ2 2 (A.2) r the reflected radiation at a point of observation has the intensity per solid angle as given by dp2 Pi(Ql) -2 4r cos (2' 1 (A.3) dQ22 2 AA In general, the bistatic cross section o(Q1 0 2) 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 A A z x Q1 e - I (A.4) and the direction of the vertical polarization by A A A ev = x eh (A.5) - 1 1 hl Similarly, for the scattered radiation in the direction Q2, the directions of horizontal and vertical polarization are defined by A A x 2 eh2 2 ^ __ (A.6) and A A ^ 2 02 x e' (A.7) ev2 -I2xe2,81 81

A respectively, These directions are illustrated in Fig. A-1. If Q2 is the specularly reflecting direction, so that A A AA A f2 = Q1 - 2z(z ) (A. 8) then e~~~~~~h e h ~~~~~(A.9) eh ehl and e v e v-2(z 1)zxe (A. 10) V2 V1 h1 2 11 On the other hand, if f2 is the backscattering direction, so that A Q2 QR 1 pi (A. 11) then eh2 - eh (A. 12) e =- (A.1=) and e e (A. 13) V2 V1 To stress the polarization dependence of the scattering cross section, we may consider four types of scattering cross section'a, m e where I and m may stand for h or v. For example, Uchv 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 01 is limited only to nearly 900 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. 82

D~~~~~~~ Scatter Area. dA Fig. A-i: Diffraction of Incident and Reflected Waves and the Direction ofPoaiton

A theoretical model for rough surface scattering and the derivation of the cross section which is available in the literature may be classified into the following three different approaches. a) The phenomenological model. By postulation, the surface is taken as an ensemble of distributed spheres, facets and half planes. A review of this 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 solutions is feasible. 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. Most of the existing investigations, however, are limited to the case of back scattering and for the isotropic random surfaces. In the next few sections, the Kirchhoff approximation shall be used to deduce the bistatic scattering cross section for a sea surface with anisotropic wave spectra. A. 3 Kirchhoff's Integral Formula A mathematical formulation of electromagnetic scattering, which gives approximate but useful results, is the vector extension of Kirchhoff's integral formula. The Kirchhoff integral formula can be deduced from the physical concept of induced. sources as illustrated in Fig. A-2. When an electromagnetic wave is incident on this surface, the scattered radiation may be interpreted as the reradiation due to the surface electric and magnetic currents. 84

A /2 Point of Observation Incident Field r // ii /! / / y Small Part of Scattering Surface FIG. A-2: Configuration for the Application of Kirchhoff's Integral Formula.

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 given by J n'x H(r'), (A. 14) -e -s J -n' x E (r'), (A. 15) -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,) ( Hr (r r)+ VI VG(r,) due to magnetic current due to electric current (A. 16) where G = r-r (A. 17) and A a A a 8 a (A.18) V' x ~x'= + z ax, BY ay?' 8z' Thus, if the tangential field components n x E and n x H on the surface are known at all parts of the surface, the scattered fields may be obtained by adding the contributions from each part of the surface, i. e., E2(r) = dA (n' x E (r))x V' G(r, r')+ il (n x H(r )). Surface [G(rr') V'V'G(rr')+]. (A.19) 86

Similarly the scattered magnetic field is given by H2(r) = dA (n'xHs(r'))x V' G(r, r')- i We(n'xE s(r'))G(r')+ V'V'G(rr')] urface nrk- (A. 20) Equations (A. 17) and (A. 20) 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. A-2, any point of observation may be represented by A r= 02r, (A.21) and the approximation A r - r' — (r - r') -2 (r >> r'), (A. 22) is adequate. Then, A ikr kS E2(r) =ik 4 7dA 2x (nxEs(r))- Q2x 2 nxH (r)) e 47rr1 2 -s- 2L s Surface (A. 23) and ikr -ik2 H2(r) = ik, dA 1 2 2X n x Ex') +r 2X (n' xH (r) i 2 - (r)=i 4 S rsurface dA2L2 (A.24) where - / o/e. (A. 25) Equations (A. 23) and (A. 24) are the fundamental relations used in the approximate calculations of the scattered field (e.g., Hoffman (1955); Aksenov (1958)). 87

For plane wave incidence, the incident fields are given by: A ik1~ *r' El(r') = E (0) e. (A.26) ^ A l r (A. 27) -1l(_r') = i f21 x E1(0) e (A. Then, if E and H are interpreted as the surface fields due to an incident field — S -S with the electric and magnetic field respectively given by E (0) and - 21 x E (0) at each part of the surface, we may have ikr A A i 1 -2) r' E2(r) = ik 4r dA Q2x(n4'xE (r')) -.S22 x Q2x(n' xHs (r')) e 4r Surface r cL -s(A. 28) Similar equations can be obtained for H 2(r). In practice, the integral (A. 28) cannot be carried out exactly, even if we know the shape of the surface, due to the difficulties in obtaining Es and H. A commonly used approximation is the so-called local tangent plane 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, 1958) -s (A2) A (0)'I)AI' x~ _ (1+RL )(E(0)o t') A'x+' (A 29> rxn' x H = (1-R )(E (0))' x - (+R )(E(0).')' x t (A 30) where x nA t A= a —,(A. 31) I 1 x nl 88

and t' = x t32) Also 2' cR - COS sin y (A.33) cos _y+/N2 _sin2 (A. 34) 2.2.. NC os IN - Si where cos = -1 n (A.35) and N = index of refraction of the surface. In particular, if the reflecting surface is perfectly conducting, then, R[ =-1 (A.36) and R =+1. (A.37) For this case, we have, n' x E = 0 (A. 38) -s and n' xH = 2 n' xH(0), (A.39) 5 —1 which are the approximate relations used by most investigators to simplify computations. 89

By introducing Eqs. (A. 29) and (A. 30) into (A. 28), an approximate relation results between the incident electric field E1 and the scattered electric field E2 This result is given by ikr E2 =(r, 2) ik 4err A A + ( A A)( A A 2 Q2 x dA' [(1+R)(n' x)(E 1 t) + (1 - I )(n'x t )(E1 e A A - 2xdA X [(1 - Ril)(n x t)(El t - (1 +Rl)( -xt)(E * ) e) 2 2 2I (A.40) In order to clarify the polarization effect on the scattered field, let us, by using the directions of polarization defined by Equations (A. 4) through (A. 5), resolve the incident and scattered fields into the vertically and horizontally polarized components as indicated below: A ikQ1 r (A.41) E = Lhl + evl EVl ehe e -2 eh2 h2 ev2Ev2 r (A.42) From (A. 40), it is seen that the components of the incident field and the scattered field may be related by a matrix: Eh2 Shh Shv Ehl (A. 43) E v2 Svh Svv 9Ev l 2 eh Eh2 ev2

where each component of the scattering matrix Spm may be expressed by: ik ik (P1 -02)o r S m= 4f dA'e A A A -(l+RL) (e ml t) (P2xex2) 2 (n' xt) ~- (m-Rii(e lO~ (P2xe) ~1 (n' x ^ V~ -(l+Rii) (e ml t) e 2 (' xt) (A.44) 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 m (h or v), the power scattered by a surface area A, in the direction,2 v with polarization A is given by p.A 2 dp = 2 J|A (A. 45) r Therefore, if a rough surface A is relatively homogeneous, the average scattering cross section from an incident wave of polarization m to a scattered wave of polarization I is given by 4 7r ISAm > OAm - A (A. 46) where < > enclosing any quantity indicates the statistical average. 91

In order to carry out the average such as given in (A. 46), it is necessary to express SAm in terms of the configuration of the surface. To be explicit, let us assume that the surface is given by z = z (x, y) (A. 47) so that -x z -y z + z n = Y (A. 48) 1/ +z2 + z2 x y where Aaz A.z (A. 49) x ax y y (A49) Moreover, we let D [ sinl cosl0 + y sine 1inl1 + cosel (A.50) and Q2 =xsinOcos02 +ysin2 + ysin0 2 + zcos02] (A.51) Then, in terms of z, 01, 01, 02 and 02' we may, after tedious algebraic manipulation, express S m in the form ik f d k(qxxxqz f (A.52) where x = sin e1 cos 01 + sin 02cos 02 (A.53) qy = sin 01 sin 1 + sin 02 sin2 (A.54) and az Cos 1 + Cos2' (A.55) The function gr, in the case of finite index of refraction N, are complicated functions involving 1, %2, 01 and 02 and the reflection coefficients RL and RIi, 92

which in turn depend on the angle of incidence y where +sinOlcos0lz + sinOlsinolz - cose cos qy = 12' n = c (A.56) L x Anticipating that, for most cases, when N -> o, R -1 and R +1, one may arrange gem in the following form: g9m = -(l+R~)P m-(l-Rit)Qm + 2 Gpm The function P, Q and G depend on the direction of incidence, reflection and the slope of the surface. For highly conducting surfaces, the dominant contribution to the scattering matrix comes from the factors Gm, and the other terms may be treated as a perturbation. By straightforward algebraic manipulation, the explicit expressions of the dominant coefficients Giem, in terms of 01' 01 02' 02 and the slope at the surface z = az/Ox and z = az/ay are given below: Ghh= - COselCOS(02-01) +z sinO cosO2 +z sinO sin2 2 (A.57) Ghv = sin(02-01) (A.58) Gvh = csOlcos02sin(2- 01) +zx(coselSinO2sinol-co se2sinO 1sin02) +z (sine1cos02cos02-sinO2cosel cos1) (A. 59) and G = cos2cos(02- 01 -ZxCosl1 sine2 -zsin0l 2 sn (A. 60) 93

The coefficients P m and QIm, which have less effect on the scattering matrix for surfaces of large index of refraction, may be expressed as AC Phh vv= A hv vh A BC pvh Qh B (A.61) vh Qhv:a BD p -Q vw hh A where A=z2 csine cos 2sinsino2 - cos sin2 - sinOscosO 1 01c01os2] +z2 [sino cose2cosl cos2 - cos sin2 - sinO1 cosl sin sin2] -Zx z [sinlcosoe2sin(02+01) + sinOlcosls in(02+0l)] -Zx [sinOlsino2cospl - cosel1cose 2cos02 - cos 20cos02+sin2COslcos(02-)] -zy [sinO1 ssin62sin01- cosO1cosO 2 in02-cos 21sin +sin 2 O1sin0 1cos(02-01] +sin1cos (0201) [cosO1 + cosO2 (A.62) B=z2sinOl [sinscosp+cos(Scoscossin2+sinlsinsin0,) -Zy sin] [oi 2+inl(coselcoo2co +sinsino2co )] -z xzysineOl [cos(02+1)+cosOcos2o(o2+0 1)+sin 01sino 2co(ls21 -sinO 1(l+ces 1cosocos2)sin(2-01 +Z in 20 1coso cosl sin(O2-0 )-cosel(sino +cos1 cose sin0 +sine sinO sin) +zy [in2 1 cose2sin01sin(2-01) +cos01(cos2+cosO1cos82cos02+sinl sinO2cosin)] (A. 63) 94

C = -zcose cos0l -zy osl 1sinol -sine1 (A. 64) D = -zxsin0 +zycos01 (A.65) A= [+z+z +z2 3/2 2(1-sin20Cos2 )+z2(1-sin20 sin2 x y- 3 1 21 Z.j -2z z sin2e sin0lcos0l+2z sinelcoselC osol +2z sine cos1lsin0l+sin 2l] (A 66) The above expressions involved in SIm are too complicated to be of practical use. For the case of slightly rough surfaces, however, we may neglect powers involving z and z, and retaining only terms up to the linear terms. Within this approximation, we have; gj a +b z +c z, (A.67) gm = aim + bm Zx Cm Zy' (A. 67) where ahh= -2coslcos(02-01)+( 1+Rl)(cosOcoso2+C)cos(02-l0) (A. 68) (I+RI) bhh= 2sinOl cosO2 sin [(l+coso lcos co2s2) Olscos(02-0) + si1 (+coslcose2)sin1lsin(02-01)] (A.69) (1+RL) hh2 sine1 sin2 sineL (l+cos1lcose2)sin01cos(02-01 +sinl1sinlsin2- sin2 cos+l(cos l+cose 2 -ine (1+coso cose )cos sin(2-1) (A. 70) 95

ahv= 2sin(02-01))-(1- l)(l+coslcos02)sin(02-1) (A. 71) (1+R1) (1-R11) bhv = sinI (lose 1+cose2)sinl cos(02-01) + sine L(cosol+Cos2 )cosl sin(02-01) (. 2 -cose1( l+cose lcos 2)sins 2- sinlcosl sin2s (A. 72) ( 1+Rl) ( 1-R 11) Chv= s inO (~osel+c~se2)c~slc~s(02- cl)+ sini [(cosel+c~s2e)sinlsn(02-0l) +cos1( l+cos1 1cose 2)cos O2+sinO1lcos01 sin 2cos1] (A. 73) avh = 2 cos0lcos2 sin(02-1 )-(l+Rl)(l+cos1l cos02)sin( 2-01) (A. 74) (1+RL) -cose( l(+cose lcose2)sin2 - sine 1lcosle sine2sino 1 (l-RIJ) + si- (cosl+cOS2)sinelcos(s2-02) (A. 75) (1+R) Cvh= 2 [sin lcos2cos 02-o s s ino2cospl + sinO ( cosl+cose2)siinl sin(02 —01 1 1 -cos ( 1+cos0 (1cose 2)cos 2+ssin e 1 (ossin (A.78) 1 1in (0~)sc+csi2)cs1ccs(2-1 co01 avh = 2coscos(2-i)-( 1-R)(cosOl)Si(2-01) (A.274) 2 1 cosa 1 co s(02- 1)+sin81 sin82 sinoscos1-cosi(coso2o1] (A.78)

(1+Rj) cw 2 sinB sinl+ (1+coslocosO2)cos0lsin(02-0 vv 2 1 sinO1 2 (l-Rt1) + [ (l+cos8 cose 2)sinlrco9(02-0l)+msinOsinO e2sin sin1 1 -cose l(cosO1+cose2) sin02] (A.79) Here the approximate RL and Rii are evaluated through cos y = cos 1. Thus, for the case where the surface is highly conducting, which allows the approximation RI — 1, Rtll++1 it is reasonable to retain only the first terms in Eqs. (A. 68) through (A. 79) for the calculation. This approximation is introduced in the calculation of the scattering cross section, a, in this report. A. 4 Average Scattering Cross Section of Random Surface In carrying out the statistical average of the bistatic cross section for a rough surface, we assume that the surface is spatially homogeneous and temporally stationary. That is, the surface described by z=z(x, y, t) is a homogeneous, stationary random variable. In order to specify the statistical properties of this surface, we shall further assume that the random variable is normally distributed and has the correlation function given by: (z(x,y), z(x',y') >= H (T ), (A. 80) where ( T Xt - x (A.81) x T =y -y. y Based on this homogeneous approximation, it is possible to carry out the computation of the bistatic cross section. From Eqs. (A. 46) and (A. 52) through (A. 55), the formal expressions for the cross section may be given by Qm = ( dx d dx dy e x yy iq(Z-Z) (A.82) gxm(Xy )gm (xl+ oy+,-)e z(A.2) 97

where, for simplicity, we denoted z' =z(x', y). (A.83) For a spatially stationary random surface, we infer that iqz(z'-z) A <gQ m(x, Y)g, (X+T Y+T ) e K (T ), (A. 84) g, m m x y Ie x' y which is independent of x, y. Thus, we have 2 -ikqxTx -ikq T A k d i ~m ( Q2' Q1) = 4|cos IdTf dT e e K (T. T ) "I m( 12, ~21) =4 r cos I x y (A. 85) In order to estimate the statistical average K, we use the approximate I m linearized form of gQ given by (A. 67), so that ikq (Z'-Z) K ( )= ( (a, +b mz +C mzy)(a +b z +c z')e kq(z-z) Km mx y m mx em em m x cIm y (A. 86) For normally distributed surfaces whose slopes are also normally distributed, the joint probability of the five random variables U3 z (A. 87) U -- Z y can easily be written down and the statistical averages taken.= Z can easily be written down and the statistical averages taken. 98

For a simple way of evaluating K/m, we note that, for normally distributed variables < P ( j )> exp 2 Pij (A. 88) j i i where Pij= <uiu.>. (A. 89) Explicitly, we have Pl1 = 2 [H(0,0)-H(7xT y) P22 =p44 =H x(0.0) P33 =P55 =H yy(0,0).12 p14 p21 =P41 =-H x(rx' ) P13 P15 P31 = P51 = -H ('T,r) (A.90) P23 P32 =Hxy(0,0) P24 P42 - HxxI(x' T) 25 52 34 =p43 =-HX (r,r P35 P53 =-H yy(7',7y) P35 P53 yy x y P45 P54 Hyy(0, O) where, for simplicity, we denoted A aH H =-, etc. (A.91) x T x 99

It follows, therefore, that <(a+b u2+c u3)(a+b u4+c u5) [exp -i q uj]> =(a+ib + ic - )(a+ib - + ic -) exp[- 2 P.u1u 8 2 3 i 4 5 i j (A. 92) Using the above and noting that q = kq2 q2 =q3 =q4 =q5 =0 one finds that K m (Tx ) = exp {-k2 q2 H(O 0) - H(',rY) +2ambm ikq H] +2a c [ikq Hy -bm H +k q H im m m z m m z m xx z x -C2 [H +k2q2 H2] -2b c H +k q H H m.(A 93) -tm yy zHy xykz Hxy Now, it is easily recognized that fdTIr dr exp -k22 q (, O)-H(T T)] -ikq T -ikq H ikq H H2 2 ikq 22 2 2 +k q H H -kq a xy z xy H y+ 2 2 2H yy kz y -k qy * fdTxJdY exp j-k qz [H(OO)-H(T,Ty)] -ik -iky (A.94) 100

Therefore, from (A. 93) we may have the relatively simple result: 2 2 = k- 2 [amqz - b1mqx- cmqy lm 4r q e rxf d exp -k 2 [H(O, 0)- H(T, T)y -ikqxT -ikyT (A.95) Thus, based on the assumptions that i) the surface is slightly rough (neglecting the higher order terms of z, z ) x y and ii) the surface height is normally distributed, the scattering cross section may be evaluated if the surface height correlation is known. The result so obtained is probably adequate for slightly perturbed ocean surfaces. The application of the results of this section to an ocean surface is given next. A. 5 The Ocean Surface Wave Spectrum and the Scattering Cross Section In order to apply the result of the last section to a rough surface such as the sea surface, it is necessary to find the correlation of the surface height. Searching through the available literature, it has been found that such information is not readily available. However, directional spectra of ocean surface waves, based on the empirical data, have been reported (Kinsman, 1965). We, therefore, shall start from the directional spectra of the ocean surface and deduce approximately the correlation of the surface height. For small perturbations of the ocean surface, including the effect of gravity and surface tension, each component of ocean wave may be represented by the form z —aexp [-iUt+Kl +x+2] 2y (A.96) For deep water, the frequency a and the wave number / 2 2. K = 2 + K (A. 97) 101

satisfy the dispersion relation 2 3 =gK + rK, (A.98) where 2 2 g = 980 cm/sec = 9.8 m/sec and r= coefficient of surface tension/density = 74 dyn-cm /gm = 74 x 10-6 Nt-m2/kg. In Fig. A-3, the relation between a and K are sketched. For waves of small perturbations, one generally divides the ocean surface-wave spectrum into gravity waves and capillary waves. It is easy to see that the phase velocity of the wave is minimum at K = /= 364 Ra/m (A.99) corresponding to a wavelength of 27r L - = 1.73 x 10-2 meters. (A. 100) m K m nl Waves for which K> K, (L<L ) m mn are dominated by capillary effect of the sea water (surface tension) and hence are called capillary waves, while waves for which K < K (L> L) m m are dominated by gravity effect and hence are called gravity waves.* For the doppler frequency under current investigation, the wavelength is about X -3.5 x 10-2 meters, so that, roughly, components of ocean surface waves of dominant importance in the scattering process is in the upper ultra-gravity wave range. 102

IK (rad/m) 560 480 - 400 - 320 240 - 160 Gravity Waves Capillary Waves 80 0 20 40 66 80 100 120 140 1i0 rm a(rad/sec) FIG. A-3: The Wave Number vs Frequency for the Sea Surface. 103

In an open, developed sea, especially with wind blowing, the wind energy is coupled to the particular component of wave with velocity the same as that of the wind. This energy then spreads into all other wave components by the viscosity and nonlinear effects which are not accounted for by the linear perturbation theory. Due to the complicated mechanism of the coupling of energy between the wind excitation and various components of waves for an open, developed sea, we represent the sea surface as z(xyt) = fda f. y a(KcJ. c a) exp) - i at+ ricx+yr r 7 x y y J x x y -00 -oo -oo (A. 101) where a is treated as a stochastic variable. Thus, in general, we have a three dimensional correlation function H(TrrT Tt) = < z(x, y,t)z(xH, y+Tr t+t) > x jdT ( K a) exp iIi(Tt-K TKrT (A. 102) The function (Kxy, C) - Y (K,a,a), (A. 103) where K = K COS ac X K -K sina, (A. 104) is known as the three-dimensional spectrum of ocean surface waves. Actually, the measurement of the three-dimensional spectrum is not necessary, since by the dispersion relation 2 3 C = g K +rK a relation such as a(rK) or K(a) 104

exists. Therefore, we may express ~ (K, a, a) either as functions of (K, a) or, alternatively as a function of (a, a). Mathematically, therefore, we may have dKc drc dcy o4(r,K, K a)=, dKa do (Ka,a) -- K dr da da uK (K,a)6[C - r(K)] d dca dCr (o, a)6[K-K (c)] (A. 106) Substituting the above in Eq. (A. 102) and carrying out the integration over the &-functions, we have H( T T Tt)= dK (Ka) exp -i(r(K)Tt-K(T COSc+T sina)3 x y tK X y = fJd f cI (r, a) exp f-i[aTt-K(C)(T cosa+r sina)]3. (A. 107) A comparison of the two integrals above yields ( )cr(,a) d d (ma) k Ok )' (A. 108) This relation may be used in obtaining H(T, T ) to the measured directional x y spectra. According to Kinsman (1965), the directional spectra deduced from SWOP measurements is estimated to be (a,Ca) = C'f -6 exp(-2g2a,2Ua2 i 1 + t0.50+0. 82 ex(- g-4 44U2 )]cos 2a + 0.32 exp ( 2 g4a4U2)] cos 4a A (A. 109) for 7T rf 2- a <<2 105

where U is the wind velocity a is the direction measured from U and C = (2.05) m / sec On the other hand, in the calculation of the scattering cross section, we like to have the correlation function H(Tx, T )=H(,, o)=f K dKfI:(K, a)exp [iK(T COSa+T sina)]. 0 (A. 110) Thus, the information on ) (ca, a) and the relation of (A. 108), in principle, enables one to calculate H( T ) T x y Due to the uncertainty involved in the measurement of ocean surface spectra, it is felt that a complicated numerical procedure of computing H(T, T ) seems to be unjustified. For a simple approximation, we shall, by x y taking the x-axis parallel to the wind velocity, approximate H(T, y) by 2 2 T T H(T ) y H( 0O) 1 - - (A. 111) X y and neglect all the higher order terms which, in general, contribute only to the second order effect at best. With this approximation, we find that H(0, 0) = JK dK da K(K, ), (A. 112) 0 - rt2 12 -2 1|K,dd KC)Cos2a, (A. 113) x 0 - 7r/2 2100= K dK da ( in a. (A. 114) 2 2 K Y O -) 106

By using (A. 109) and (A. 110), one finds, by direct integration, H(O, 0) = 3C( )3/2 U )5 2 2g Since, however, the "correlation distances" X and I cannot be evaluated x y analytically, one has to resort to the numerical integration. The results are presented in Figs. A-4 and A-5. In Fig. A-4, the mean square surface height is plotted against wind speed, while in Fig. A-5, I and I are plotted x y against U. In this formulation, therefore, we relate approximately the effect of the wind speed to the ocean surface scattering. To include the effect of the direction of thewind, we shall now assume that the wind is blowing in a direction making an angle b with the x-axis, Then by simple rotation of coordinates, we may write F (T cos+T sino)2 ( cosb-Tx sin*) 2 H(T T )= H(O,0) L 2 x (A115) x Y By introducing this approximate correlation function into (A. 95), we obtain the following expression for the bistatic scattering cross section: iQxxz -~g —~3~P 2 (qxCos~sii)l2x I m 4q 4H(0, 0) qz-bmqx Imq xp 4q H(0,0) J z ~~~~~~~~~4z F (qyCos ~ - qxSino)22 exp 2 ~si n~) (A.116) 4qz H(0, 0) It should be noted that the correlation distances I,1 are calculated by a linearization approximation (cf eq. A-ill). The viuevs of I, I, presented in the Fig. A-5, are within the limitation, where the linearizadlonyapproximation is considered valid. For wind speeds higher than, say, 5m/sec, the nonlinear effects should be taken into account, and, as a consequence, a direct extrapolation of xa, iy for higher wind speeds is probably tenuous. 107

RMS Wave Height H'(O,0) (cam) -36'34 -32 -30 -28 -26 FIG.A- 4/'26 24 =H(OO) =4.24()2g) - 22 U = wind speed (m/sec) g = 9.8 m/sec2 20 -8H(O, 0) / 18 -16 -14 12 10 8 6 4 U (m/sec) 2 Wind Speed 1 3 4 5 6 7 108

,I (m) x y 1. 6 -1.4 1. 2 y'1.0 0. 8 x -0. 7 -0. 6'0. 5 0. 4 FIG. A-5o Variation of the correlation distances vs Wind Speed. 0. 3 0. 2 0.1 / /1 2 3 4 U(m/sec) — 109

For the purpose of carrying out the computations involved in this work, we only consider the scattering cross section under the following conditions: i) the incident radiation is horizontally polarized, ii) the conductivity of the ocean surface is assumed to be very large, sothat R~ = -1, Rl —+1, iii) the observer for the detection of the reflected radiation may receive both components (vertically and horizontally polarized) of radiation, so that o hh + vh Based on these approximations and using Eqs. (A. 68) through (A. 70) and (A. 74) through (A. 76), we find A^ I^ xy 2 (2'~ 1 q4 H( ) 1 2 exp [-"~ H(0,0) - 7(A. 117) This is the approximate relation, Eq. (2 tigation. It is recognized that a more accurage result could have been obtained by including higher order terms of (az/ax) and (Oz/ay) and also by incorporating a finite index of refraction for the sea surface. The resulting computation would be enormously cumbersome, even though such an extension is stratightforward. However, due to uncertainty involved in the sea surface spectra and also the inherent shortcomings of the physical optics approach, it is felt that Eq. (A. 117) is adequate as a first order approximation for a surface which is intermediate between a specularly reflecting surface and a surface that gives rise to a completely diffused scattering. 110

APPENDIX B COMPUTER PROGRAMS Rough Surface Program This program was used to calculate the directions of the maximum radiation received and the power level received at various points above the sea surface. The input to the program consists of the radiation pattern of the antenna, X, Y, and Zr/Za for the location of the receiving point, the wind speed and the direction. The output consists of the power levels received at various locations, coming from different directions from both the major and minor lobes of the antenna. This program took 9.5 seconds to compile and 21 seconds of CPU time to run for 13 points of the coordinate points and one case of ZRA, PSI and U The Flow Chart and the input data for the Rough Surface Program are included for reference, along with some comments on the antenna pattern. The radiation pattern of the doppler antenna AN/APN-153 employed in our present work was experimentally obtained for 360 x 90 = 32400 coordinate points covering the entire hemisphere with one degree steps in each coordinate. That is, the azimuth coordinate ranges from 10 to 3600 and the latitude coordinate from 910 to 1800. The peak intensity of the AN/APN-153 antenna pattern is 37 dB. The radiation intensity at each coordinate point was expressed in dB, multiplied by 10 and then converted into its binary form. The entire antenna pattern, then, is filed in a 360 x 90 matrix form as shown below. 111

A1,1 A1,2 A1,3 1. A,90 A1 A2 12 A22,3 A2, 90 A3, 1 A3,2 A3,3...A3, 90 A360, 1 A360 2 A360, 3 A360, 90 The element Al 1 represents the antenna radiation intensity at po=, 0 = 910; A152 50 at O = 152~, 0o = 500+ 900 = 1400 (cf. Program List No. 43). The antenna pattern is usually expressed in such a way that it is unity at its peak point, when expressed in the linear form, by normalizing the pattern intensity by its peak value (in our present case, 103. 7 ) The FORTRAN statement of the List No. 12 yields the desired antenna pattern in the linear form. It can be seen from the following. In the expression FF(I) = e 2 302585 {.1,FFI)371 (B. 1) let FF'(I) A 10 loglOF1(I) 37 - 10 log10 F2, 112

so that F(I) F 1(I) FF'(I)-37 = 10 log10 F 2 F 1(I) or 10IF = 0. 1 [FF'(I)-373. (B.2) Since Ln x = Ln 10 log10 x = 2. 302585 log10 x, F 1(I) F 1(I) Ln 10 log10 F Ln F2 e = e Fi(I) F 2 Let F1(I) -F 0\F (I) (B. 3) -2 The equation (B. 3) is the desired pattern expression. The List No. 6 is the UNFORMATED FORTRAN READ Statement. The procedure for activating this statement depends on the particular computing terminal facility through which the program is run. For convenience, the input data for; a) the selected coordinate points of (0, 0 ) b) the relative receiving coordinate points, and c) the wind speeds and their directions used in carrying out the numerical calculation are presented, along with the AN/APN-153 antenna radiation pattern F (&, 0 ) for the 57 points selected in calculating F2M. 113

FORTRAN IV G COMPILER AIN 2-11-6 10:48.35 C...... THIS PRCGRAtM COMPUTES SCATTERED C RACIATICN INTENSITTY FROM OCEAN SURFACE,... COol DIMENSION ( CNAB(2),PAA({ 100),PSIA(tC0)A,UA( 100) 0002 C U YENS ICN PO (200),TC (200 1. XO0200, t'YO(200) GOC3 DITMENSTCK F(360,90),F1(32400) FF(32400) 0004 INTFCER*2 IF(3240C) C005 F UIVALENCE (FFF), (IF(1),FI(16201) ) C.....REAL INPUT DATA CF RACIATION PATTERN...... 0006 REAr(2) IF CJC7 PI=3. 1i15' 27 0008 ACJ=37. CCC9 CC 45 I= 1,~24. C 0010 FI(I )=F(I) 0011 FF(I )=0.l *F (I) C 20312 45 FF( I )=EXP(2.3C225F 5*C.1*(FF( I)-ADJ)) __ C...... REAC INFL T CATA...... C N,......N-.-.N. CF THE ANL ES (THETACPH I) C N.('..NC. CF RECF IVFR POIN1 S(X,Y) C L.C. C. THir V-A IAT CINS FiR ZRA,.PSI AND tU CTC- [.....T _ _E__i_ C; C FPC..... PHIC C X.XC...RELATIVE X C[ CRCINATE OF RECEIVE't C.....R.. L RFLATIVF Y CC IN ATE Cf RECEIVER rC ZM *r Z.. RATIC CF HEIGHT HETWEEi N THE RECEIVER C ANCE 1F TRAN'SITTtF 1;C PS'A...,INF CIRECT ICN C L,..... INC SPEED C,') 1.:It; (, 1 C )I 015 R _ AC.,1C! )....... -J-i'.........1 1 7 FF, M a T ( I5 )........ 0C17 RLA3(5,102} ( (TC( I tP[ CI ),,I=.,N) (J'; 18 a F-A.) ( 5, 1 C ) ( ( C ( I ), C(I ) I I 1M ) U, 19_ 1G2 F:rMAIT' (2F 1 C. 5 i20C RFA t 5 1 C ) ( ( ZRD t I ), S I T.JA (I)), I =1, 0)2 1 IC0 F-CRM AT ( A T 1C.5 ) 0022 CC 9qC; K=1,L- - 0023 IRA=Z- A (K ) 0024 PSI=PSIp(KK) 0025 U=UA(KK) C C..... C.CALL S UBPREPCCRAF T-I] COCMP LTE AOF &BO..... C026 C/ LI.EFLINT(LCCKtH o027 AC=1./4./CCNA8(l) 0028 BC= 1./4. /CCNA (2) 0029 Y 3 -=SCRT ( AC*F F ) 0G) 330 0G g09 g JJ= 1, l 0031 X=XG(JJ) 0032 Y=YCtJJ) 0033 WRITE(6, 1CCC) PSI,,AC, BO 0034 - PRItE('"6,lCC1l ZR,X,'f 0035 WRITE(6, 10C2) _ 114

FORTRAN IV C COMPILER MAIN 12 1-6 1 10:: 48.35 0036 1000 FGRMAT(1H1t2X,4HPSI=F1O.5,2X,2HU=FlO.5,2*X.1 3HAOC-t1f2.5,2X,3H80=-125; ) 6C037 1001 FCRMAT (2 X,6HZR/ZA=F1.5,2X, 2X, 2HX=F 10 5, _ 1' 2 X, 2 HY= F 10o. 5//) 0038 1C02 FCRMAT 4X,4HPF,IC,2Xt,6iTHEITAC,4X,4HPHI1,2X, 1 6HTHETA1 4X,4HPHI 2,2X, 6HTHETA2, 2 I1X,2-FC, 1LX,2HF 3t2,11X,2HDKg9X 3 4FKF2, IF BKF26XBK2, 6X, 4HOBF2//) 0039 CC g9 II=1, C....... CCtPUTE FO(RACIATICN PATTERN)...... CO'q~OG PI f, i, (- C (i I} C'041 ATHC=-Tt ( [I ) 0- 04 2 - - ---- --- -- P-Pq'- - P H I-L-. 51- 1. C 0U4 3 _I 0TH0=A THC +G C51-g90. C044 FC=F(IFF I C, ITFtC) - C04', T F: l- ATi=A T FC PI / 1 C4'. 0C)0 46 PF-TC=APH I C*P I / 1. C.. CCYPUTE THEITA1 PHI 1....... U_ _ 4 l_ ATH1=I C.-A 1F t 0V04R' - THF.- TA1=ATF 1*P T/ 18C. CC;49 API- I=APH -IC+IC. c05C IF{APFI1. E. 36C.) (C TO 6G C0b l APHT I=APP I t- 3eC CU F 2 t) I.A Ph I 1 = A Pi 1s F I k 1.l. CC512 (I.) PI- 1=AhI1l*-I/1CC. C....... C tPLTE TH TA2,PH I....... Cibu3~ ARG1=X+TATN(TFETA1)*~CC(PHI]) 00)5'4 ARG2=Y-+Tih(TFFTA1 )*S'IN(PHI1) C(O55 IF( t S( AC ). LT. 1-2., aNC. PS (.ARG2) A3 __A, 1.LT.L.E-2C) CC II CJ G056 IF(AP-S(ARG1).LT 1 F-c20) GC Ti23 23 0057 I F(A,( t ARG2).LT 1 CE-20) CC TO 26 0C58 P 12=-f AN ( G2/RC 1 ) 00 59 IF:(ARGi1.T.C..T ND.KCRG2.GT.u.) GO TO 20 ~Ou~O ~IF(AkG1.LT.C..ANC.,RC22,.GT.O.) GOC T 21 O 61 IF(A. AG1.'.0. -.AND.PRC2.LT.0. ) G.n TO 21,:OE-f2 IF(A[( 1.(.T C.0.NC.tRC2.L1.O.} G)7 TO, 2 2.! 3 26 P H 1 2 = C. 0(}q 4ECC TL 20 C065 21 PF?2=PFI2+PI (00 e 6 GC TO 2C 00 t 7 22 Pt- I 2 = P - I 2 + * P2 f C0-6 G C TL 20 O6 23 PhI2=PI/2. C070 20 THPE'fA2=ATN ( ARC1/CCS (PI 2 )/ZRA) 0071 ATH2='ThE TA 2 18C./ 3 72 A PH I 2= PH I 24 1H. t/ P J C *. C.....E_ CCOPLTE CX,CY,CQZ....... 0073 Q XY=-'N-( -ET A! ) *C 5 ( PHI 1 + S I N( TXH E 2 ) TCOSI I'PH I 2 ) 0074 CY=S1N(TIETA1 )~*SIN(PF: HI1)+SIN(THETA2)*SIN(PHI2) ]075 QG Z= (S (T- ETA1)+C cS ( FETA? ) 115...~~~~ --. r

FORTRAN IV G COMPILER MAIN. _ -1 —69 108N.35. C "..... CECMPLFT THE F NCTIC-NS DKF2i C,CKF2,4EKF2 ANO DBF2...... C F2.....THE REFLECTEC RADIATIGN INTENSITY C C* J K.....TH-E FUNCTICh INVOLVED IN ESTIMATING C TFE REFLECTEG RADIATION INTENSITY FRCY AN ANISCTRPOP C OCEAN SURFACE C _DKF2...TfE APPRCXIMATE INTEGRATICN TO ESTIMATE THF LEVEL CF THE REFLECTED RADIATION C INIENSITY FPCP AN CCEAN SURFACE [ BKF2.G.DKF2 IN [E c CEPF2...F2 IN CB 00376 IF U-,S(TQZ).!E. 1.E-2C) GC'TC -99 C077 ALPHA=CX/C Z CO 7 ALPHA-= L Pt A* 2 E-ET A — 2 —B-E — -TA 2 8ETA2=BETA**2 Oje1 GAMMAI=A1 PLA*CCS (PS I)+PFT*SIN(PSI) 0,)Q2 GAMMA2 =bE TA *.C'S.(PP S I" -,A Pf1*S S- ( PS I) C3 3 PC1=GANA * _ _ A 1,* _2.....'84 PGZ=GPM A2 C** -2 0085 D- LT = 1.+AL FfP 2+BE 1 2 O") ~ 6 CFLTA 2=tELt hA*2 0C,d 7 E PS.I =A: *E G l JO' R 8 7 a U= E PQ S i 2 + f F S G 2 0089 TAU=EPSI +[:FSI2 - F ( ABS ( T AL) CE-. 174.) G-, IT — 50 COC, I F3=1./EXP(TAU) CCq2 F 1-=OELTA2TF-.CCC3 GC TC 52 CCG4 50 F3=0,0 CC95 52 CCNTINLF -- ---- W=C (1FETA1 )*CCS(TFEIAF l/CCS(THETA2) CC'7 F2-A:S (Y* FC *F 1/4./P I ) CO-8Q. IF{(+BSiF4 ).-1 T.I F-7C ) GO 10 81 CC99 FPI=ALLC( F-2 )/2. 3025 E5 Ol10 E -1c.F=i*FF2 01( 1 G TTC.C ClC? 81 L1e2=0. C104 CKI=CCS([ TF-: t I );CCS(T ETA2) 0105 OK3=1.+SIN(TFETA1 ) S.IN(THETA2)*COS( PHI1-PH12)+OK2 0106 IFi(AS(ICCSItFFETA2)).LT.,1.F-20) GC TO 99 O107 CK'+=CCS(TFFTAi)/CCSi7FETA2) C1 C K 5=ZR'C ]. 4/C C ('ht A ) C109 nK6=ZRA*ZRFA*CK4**4 0110 CK7=1o. + CK5-4 PK cl111!F(ABSiUK7).LT. I.E-20) GC TO 99 01 12 DK=ARS ( P I*DK 1**3-*K3"/0K7T /3) 0113 DKF2=AS ( CK*F2 ) 0114 [IFTABS(fKF2o.LT.1.E'-7C) GC TC 82 0O 15 4[KF2=ALCG ( fKF2 )/ 2.3 C2585 O116. EKF 2= 10.4ACKF2 01 17 GC TO 63 0118 82 [BKF2=C. C119 83 CENTINLELE 116

FORTRAN IV C CCMPILER MAIN 12-11-69 10:48.35 C *..o.. PRINT CtJT APHI-IC,ATI-O,APHIl,ATHI,APHI2, C ATH2,FOF2,CCDKF2,D8KF2 & DBF2 C1O WRITF(6,1003) APHIC-, THOAPHI1, ATH1, APH12, 1 ATH2,FO F 2,DK, K DKF2,DBKF2DBF2 0121 1,003 FORM AT16 F.8.1,4 E 13.3, F'l3,flO.3' 0122 99 CCNTINLE 0123 C9 CCNTINUE 0124 CALL SYSTEv -----------`- -— ~-~~~F~;~~ ~ —-~- - —' —~~ — -- -- 0i2 FN D TLTAL MEMCkY REQIJIFRE!'ENIS C,41714 BYTES 117

FORTRAN IV C COMPILER tSLINT 12-11-69 10:48.55 CO'l SUfBRB LIi\E CELINT(L9,CCNAB) C C.......THIS PRCGPRAM CCMPUTES CONAB(1)&CONAB(2) ~C TIN THE FUNCTICN REPRESENTING THE C SCATTFKING CRCSS SECTICN FOR AN -~C ~ ANISCTFOPIC OCEAN SURFACE....... C002 _ [r_. DI IENS IC_ AFi(630 F 0C),vF2(60C),CONAB(2) 0003 REAL K ~ ~~~-~~ ~ -~~~~- _____ ___-. ——. —----- - - -- - -- ----- -- _ 0C11O4 P I O 2= 1 -. 7 C 7 c9 C.TE INTECRATICN INTERVAL IS CIVIDED C INTC TFF TWrl~ REGIO]NS...... C C......TFE CUNULATIV. STEFS IN THE EVALUATO.N C CEF THE I.N, T G~i, T I C N B Y S I VP PS ON F CR MIUL A __......l E STEP-SIZE IN THE SIMPSON FORMULA C v...tn...H-..r...- TH~STEPS In THE INTEGRATION 0005 C=u. I 00C6 H=0.1 COC7 0(:8 101 C CNTIt: F C009 DC 2C( Ii11, 0011 K2=Ky(*K,-_JO 1 3 b Z:u,t.i 00 1 4 L 4=U2 "2 001 ] _ C 1='7.!5 +u. CC C: t?*K? C0016 C 2='?.d*R.4/(.N0CC14.*<2 6017 CS'(C2) C018 C4=1.I/ 12 tC 2 - - 9 __g_ I((C4 CE. 174.) CL TL 20 0.. C0 C = I../ p(CL) (A) ~~ L7-'j.OCL054C*Ut'C C'32 1 C7==C ": 4'i C 2 0023 IF( C7.C. 174. ) C C 30 G~02~C,t= 1../F XF ( C 7) L025 G ( T(L 21 (C027 21 CC,=6 3,7 0028 U c~- X 1 1b=K 2. * T / C f —'' _ ___76_ ____0K2 C___ (029 X22=O.41*C8 0 30 XF i=X100( 1'.";2 F + 2 C) C'_'3 I _ FO2= x1 0* fC F, 75-?C) _I C7 2 GC TE 22 C',.3 2{30 F 1=0. _ 00 4 F 2=0. 00o 22 AF1() =_I 0030 AF2( 11=F2 0037 C=Q+H 0038 2 MCO C TC- E-_ T I N L E -_ Cc~.39$...~.... C~NFUTE FUCIkCTIC SIMP(H,AF,.M)..... C04 0 8EFi-=SfF(. It ~ F F.,F1,"k) 0041 BBF2=S IPF (H,F2, F ) 118

FORTRAN IV C CCMPILER CeLINT 12-11-6< 10:48.55 0042 IF(H.EC.2.C) GC TC 103 OC43 HF=2.0 Cr044'=201 0045 BF =1BF1 0046 e F?= OC F2 00:47 GC TC lC1 0048 1C3 F FI=PFI1+F1 004c;' FF? =PF 2+9F2 005- C CNA (F 1)= IC?*PF1 _____ 0C05-1 — ( — — C —- - (2)=PI C 2*BF2 C0 52 ETUi:N CC 5 3 E N TLTAL iEP-l1CUkY kE(LIPLIONS OC1lF1 P P'VI

FORTRAN IV C COMPILFR SImP 12-11-6S 10:49.00 0001 FUNCTICN SItP(H-SftP) C.......CCM PTE FUNCTICN SIMP(HSPM~o*%*.. 0002 DIMENSICK SP(6O)j 00(3 S=SP( 1 )+SP{v )-4.*SP[2) O004 MPl=l-1 000 5 CC 10 I=4,'1,2_ 000 6 1C 0 S=S+2.*SP(I-1)+4.*SCpFI) C007 S TMP=S'HH/ 3. COC R ETUVN CC'I [N ___ TCTAL MEVC~Y RECUIPEMENTS CCC1F(- PYIFS E~XECL1IUN ~ TERWINATED ~0

The Coordinates of the Selected Locations in the Antenna Pattern. (00) ((o) > 1 151.0 140.0 > 3!55.0 1542. 0 > z! %q.n 130.0 /1 l53.4 134.0 > 5 tz1,5.n0 1.31. > 6, 14'.,6.0 > 7 155.00 164.0 > q! 5"5. OQ 172.0 > 9 15o.O0 15g.O > 1n 1I 5I1 L.!q7. > t. 41 14-4.00! 3. 9 > 9 1?133- 0 )?pf'. > 13 132. o) > lZ l4z-. Or) 37.r7 > 5 1 7.00 1 5 0.0 > 16 15o.qO! n." ] > t7 155.n0! q.O > ~ s 1 5 O * f r) 1 7. > 14 t9~.!56. 0 > 9 56.0 1 34. ~ > 21 1 46. O 120.0 > 9 1 35.00 114.n 2 3 1Itn. I 1 1.0).3 1213. 0 1 ~'. > o5!.O ~ ~ I3 4.0 q6 zo.09n m 153.n >,q7 1,::. ) 7'.,) > 7 > - q 52 15 I.5) 1 90. ( 2> 91 9 -~. O ]16 99.() > 30 16 ~.n 1 7/4.0 >:3 I1.m0 1 50r).0 32! 1,. ~2.o.) 0IP. ~ > 3!a3.qO 1 l11.0 --- 34 130.-3 11!0.) > 1 35 15 ~ ). 109 0 > 36 11-1!.0nO 106.9 37 9Z.O 106.0 > 39 I l1. OO 2.0 > s40 130.00 1.3-O~ > 14 3 R3. n q3 > A4 1 z. Or) 5 42 r 4. 1/31 903.0 > 46.50.0 237.0 > 4 R 1 62., 2 03.~ 0 > 46! 53. n qo o. n > 417 1 53- 2??. n > 50 144.0 2l3. O > S51 136.0 225. > 1P 30. 240. > 5/1 106.n 250.) 55 119.0 250.0 > 56! 33.n 2a5.0 > 57 t 1.n0 247.0 121

~' CCc Cc cL C' C Cc rc in n in in In in r In n in r C- C. C C CC C C C r Cc T- t.in.i I in.n I..n I.n.n n.%....u ConCO uoin.C w iL Cc cc cin Cc i C nCtC Ci:. C' CI Clt (7 i CnC~ Cn iC iC, in C) in CU Cl CO 0~~~~~ C''....''.... C'.....' -.4.'' C 2 a''C....''...,,C C' --,. 0: 0!,"':.'. 0C,,-, CC. cCV C un C,... V. a,, C c" Lr: C. I- F1J C" C7 c CY",- -, 03, 1 0' t,-,'-' C', Ct C, q C f \- -- 0'.t C I r I I I I I t I I I I I I C I I C i I' aP ~ ~ ~ ZY ~ (r ~''fC L6, t) C, U.) C, Lr) c U-) c- rL rl- C' U)C~ ~ ~ ~L ~ ) O h S:C C.-, 0 o9,': L/'W'- ul OQ C..-C-.C:(C 0 C,)'1 OL,.- ON'Pub. r- a-,0 o'~.,,!'-I.0' 00 C' C: et 0C: L) C" C...........C" \.',C 0.' CC 0,! e c O.r CC: 0ocCC c> cy (Y.,' ".C,'),"~-, C, 7,,,~ -,~ -.~" - -,U S cu~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,o 0 J;'p i I I III I*I II I I I I I I I I I I ~~I I I II I I I I I I I I I I I I I "s.~n cin~ ncincI~ in cl rc Incin n~~Cin~CinCini n~ 7C C~ ~i nc incI ninc LtC 1, fl s~ i n tnin C Cinc sill:11111 ggIIIIII t in ~C'~C\C \G~f~\C ~ \C C~- r- r-~ r- r~ r- r'- r~- t~ r~- Cf Cf Cf Cf Cf Cf Cf CCIX Cf C~ O~ O~ 0' O~ 0' I I I C'~

The Relative Receiver Height, Wind Direction and Speed. (Zr/Za) (d) (U) > 149. 0.0 11.5 > 15 0.1 0.03 2.N >!51 0.0! 0.0 3,0 > S..r > 153 ).I 45.0 t.5 > I5.-.5!50 2.0 > 155 0.1 45 5.!) 3 0 > 156 O 1 45.0 4.0 > i57 0. 90.0 1.5 > 159 n.1 90.0 3.n > 6t 60. 1 90.qr). > 161 0.5 0.0.5 > 162 n. o.) 2.0 > 163q 0.5 0.0 3.0 > i1 64 0.5 0-0 4.n > 1i 65 O.5 45. 1 5 > 1 ^ n.5. n 45:. n > i67 n.5 45.. 3. > 16 0.% 5. 0 4.,) > 169 0.5 90.0 1.5 > 170 0.5 9 0.0 2.0 > 1 7 1. 5 90.0 3.0 > 172 n.75 90.0 *.f -~,F",DFn iF' TI- 1 123

The Radiation Pattern of AN/APN-153. PO THETAO FO PHIO THETAO FO 250.0 1C6.C 251E-02 1 410. 2 5 1 E - 0 2 140. 0 151.0 C.851E 0C 25. 9.0 O. 314 2. 150.0 0.832E 00 245.0 133.0 0.794E-02 152.C 155.C C.776E 0 247O 0118.0 0.832E-C2 139.0 153.0 0.6C3E 00 134.0 145.0) C.724E OC 146.0 148.0 0.257E CC I 164.0 155.0 0. 372E CC 172.0 0 158.0 0.3CSE OC 158.0) 1 159.-.363E 3 —137.0 154.0 C.3CCE CC 123.0 144. j- -. 0186E 00 120.0 138.0!.2CCE CC 121.0 132.C 0 I31 bE CC 137.0 140.0 C0.257E-C 1 150.0 147.0 0.2CsE-01C 168.0 152.0 01. 2455E-01 188. 155.C 10. 3 7.372E- C I 178.0 162.0 C0.447E-C1 5. 16l2.0 C.263E-01 134.0 156.0 0.316E-01 120.) 146.) C.12C(E-C 114.C 135.C 0.8c91E-02 111.0 110.L O C.24CE-C1 126.0 128.0 O 123E-1I 134.1 t138.0. 33 It- 153.0 142. j0 O.9513E-02 172.C 148.C 0.*29E-02 190.0 153.0 0.33CE-02 199.0 156*. 0.6 1E-02 174.0 15. C O. 14 1 E- 02 150.0C 165.0 0. 182E-O 2 128.' 158.0 0.437E-02 14.C 149.0 0. 550E-C3 110.0 130.0) C.219E-02 109.0 125.3 2C.0.977E-C3 106.0 114.0 n. 214E-02 106.C 98.C C. 71~3E-C0 118.0 117.0 0.479E-01 122.0 118.0 C.178E-C2 134.0 130.0 (;.74 -14230.0 149.C. C.4Of-01 234.0 149.0 C.525E-01 216.0 155.0 0.263E-01 225.0 146.0 0 _. 269_ E-0 1 236.0 138.0 C. 7-8-02 2'37.0 150.'),! 8.82E-C 1 225.0 158.0 50.3CE-02 203.0 162.0.24 E-C2 209.0 153,C C. 513 E-3O 218.0 144.0 C.275E-C2 225.0 136.0 C.229E-02 240.0 130.0 C.1C7E-C02 242.0 117.0 0.282E-03 124

FLOW CHART FOR ROUGH SURFACE PROGRAM BEGIN READ IF I = 11, 32400 COMPUTE FF(I) READ N 45 f READ M READ L READ TO(N), PO(N) Cont'd 125

Cont'd READ xO(M), YO(M) READ ZRAA(L), PSIA(L), UA(L) KK-= 1, L ZRA = ZRAA (KK) PSI = PSIA(KK) U = UA(KK) CALL - DBLINT COMPUTE AO, BO 1I 26ECont'd 126

Cont'd x -- XO(J J) ~ = YO(JJ) WRITE PSI, U, AO, BO WRITE ZRA, X, Y /<~ ~ WRITE PHIO, THETAO, PHI1, TEHTA1, PHI2, THETA2, FO, F2, DK, DKF2, DBKF2, DBF2 127/ I II=1, N 127

Cont'd APHIO = PO(II) ATHO = TO(II) COMPUTE FO COMPUTE THETAl., PHIO COMPUTE THETAO, PHIO COMPUTE THETA1., PHI1 COMPUTE ARG1, ARG2 Cont'd 128

Cont'd Gll 2 1T0 ~.- GO TO g21 <( 10-20 99 F SET T SET PHI2 = PI/ 2 AR110PHi2-= - COMPUTE - FR 0 PHI2 AR11 F F A I AR2 > q IARG21)0 LARG21100 ARG2K / \IRG21< T T SET SET PHI2= PHI2 + PI PHI2=PHI2 + 2PI _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ COMPUTE THETA2 COMPUTE ATH2, APHI2 129

Cont'd COMPUTE QX, QY, QZ T ~ GO TO F COMPUTE ALPHA, BETA, ALPHA2, BETA2, GAMMA1, GAMMA2, BG1, BG2, DELTA, DELTA2, EPSI1, EPSI2, TAU ITAU 10-2 T GOTO F COMPUTE F3, F1 COMPUTE COMPUTE F2 Cont'd 130

Con't F COMPUTE DBF2 COMPUTE DK1, DK2, DK3 oGO TO 1eF3 COMPUTE DK4, DK5o, DK6, DK7 GO TO Cont'd -9 131

Cont'd COMPUTE DK COMPUTE COMPUTE DBKF2 WRITE APHIO; ATHO; APHI1, ATH1, APHI2, ATH2, FO, F2, DK, DKF2, DBKF2, DBF 132

REFERENCE S Aksenov, V.I. (1958), "The Scattering of Electromagnetic Waves by Sinusoidal and Trochoidal Surfaces with Finite Conductivity, " Radioteknika i Elektronika, 4, 459-466. Beckmann, P. and A. Spizzichino (1963), "The Scattering of Electromagnetic Waves from Rough Surfaces, " The MacMillan Co., New York. Chu, C. M. et al (1968), "Doppler Radiation Study, "The University of Michigan Radiation Laboratory Report 1082-1-F, SECRET. 140 pp. Hoffman, W. C. (1955), "Scattering of Electromagnetic Waves from a Random Surface, " J. Appl. Math. XIII, 3, 291-304. Kinsman, B. (1965), Wind Waves, Prentice-Hall, New York. Pidgeon, V.W. (1966) "Bistatic Cross Section of the Sea, " IEEE Trans., AP-14, 3, 405. Senior, T. B.A. and Hunter, I. M. (1966), "Experimental Studies of Sea-Surface Effects on Low-Angle Radar, " Proc. IEE, 113, 11, November, 1966. 133

UNCLASSIFIED Security classification DOCUMENT CONTROL DATA - R & D (Security classification of title, body of abstract nd tl idexlin iannotation m.ust be entered when tile overall report is classified) I. ORIGINA TING ACTIVITY (Corporate author) 2a. 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: Phase 1 Report of Contract N62269-68-C-0715 Volume I. 4. DESCRIPTIVE NOTES (Type of report and Inclusive dates) Interim Report July 1968 - July 1969 5. AUTHOR(S) (First name, middle initial, last name) Chiao-Min Chu, Soon K. Cho and Joseph E. Ferris 6. REPORT DATE 7... TOTAL NO. OF PAGES 7b. NO. OF REFS December 1969 133 7 8a. CONTRACT OR GRANT NO. 9u. ORIGINATOR'S REPORT NUMBER(S) N62269-68-C-07 15 b. PROJECT NO. 1969-1-F, Volume I c.,h. OTHER REPORT NO(S) (Any Ither nulmher thoat miy boe ssl.ned this report) d. 10. DISTRIBUTION STATEMENT'ransmitt al outside agencies ot U.. Government must nave prior approval of NAVAIRDEVCEN or NAVAIRSYSCOM 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Air Development Center Johnsville, Warminster, PA. 18974 13. ABSTRACT The radiation characteristics of a doppler velocity sensor radar have been studied. A theoretical investigation has been made of the reflection of the electromagnetic radiation from an anisotropic Gaussian surface. In particular, from the known angular spectrum of ocean surfaces, the bistatic scattering cross section is derived for an open developed sea. The results thus obtained are then applied to the study of the reflected radiation from the doppler sensor equipment on an airplane. Computer programs are set up to calculate the directional distribution of the reflected radiation for a transmitting antenna of given radiation pattern. Computed results forthe AN/APN-153 antenna, showing the spatial and temporal variations of the reflected radiation, are given for a wide range of relative positions of the transmitter and receiver for a few different wind speeds. Finally, the reflected radiation from an anisotropic ocean surface of Gaussian distribution is compared with models of specularly and diffusely reflecting surfaces. D 1D, NOV 6M5 ~1i473 UNCLASSIFIED Sic IrilV ('I,.Sli jtic.ti....

UNCLASSIFIED Security Classsification 14. LINK A LINK LINK C KEY WORDSN ROLE WT ROLE WT ROLEI WT Doppler Radar Velocity Radar Detectability Radiation Characteristics Specular Scatter Diffuse Scatter S.argtm'U LA SSIFIE, ~iclrri~ (lirG~jifjed~ii

//1//1/1/UNIVERSITY OF MICHIGAN 3 9015 02827 4663 THE UNIVERSITY OF MICHIGAN DATE DUE ~I/?

DISTRIBUTION LIST: Air Task No. A3605337/202B/F08-232-602 Work Unit A53373A-3 NAVAIRSYSCOM, AIR-604 (2 for retention) (1 for AIR-533) (1 for AIR-5337) (1 for AIR-360E) 5 cys DDC 20 cys NAVAIRDEVCEN, Johnsville, Warminster, Pa (3 for ADL) (6 for AMD) (3 for AMX) (1 for AMXI) (2 for AMXA) 15 cys