J ---~1 1969-2-Q 1969-2-Q = RL-2039 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGWINEING Radiation Laboratory DOPPLER RADIATION STUDY Interim Report No. 2 1 October 1968 - 1 January 1969 By: C-M Chu and S-K Cho Contract N62269-68-C-0715 March 1969 Prepared for: Naval Air Development Center Johnsville, Warminster, PA 18974 Ann Arbor, Michigan

1969-2-Q FOREWORD This report (1969-2-Q) was prepared by The University of Michigan Radiation Laboratory, Department of Electrical Engineering. The report was written user Contract N62269-68-C-0715 "Doppler Radiation Study" and covers the period 1 October 1968 - 1 January 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

I 'j6'4-2-Q ABSTRACT Approximate formulae for the bistatic cross section of a homogeneous, stationary surface are derived4 The formulae are based on physical optics. These formulae give the estimates of the aspect and polarization dependence of the cross section based on a single funmctionaf the incident and reflected direction. This results from an integral involving the correlation function of surface height. It is suggested that a study of this "universal" function based on some realistic choice of correlation function of surface hetg, should be carried out for a reasonable estimate of the aspect dependence of the scattering from a rough surface. ii

1969-2-Q TABLE OF CONTENTS I INTRODUCTION 1 n THE SCATTERING MATRIX 3 III THE SCATTERING CROSS SECTION 9 IV ASPECT DEPENDENCE OF CROSS SECTION 12 V CONCLUSIONS AND RECOMMENDATIONS 16 REFERENCES 17 APPENDIX A: STATISTICAL DESCRIPTION OF ROUGH SURFACE 18 APPENDIX B: EVALUATION OF STATISTICAL AVERAGES 22 DD FORM 1473 29 iii

1969-2-Q INTRODUCTION The investigation of the scattering properties of rough surfaces, particularly that of the sea is continued in this period. Based on the Kirchhoff integral formula and tangent plane approximations, integral formulae for the scattering matrices were given in the last quarterly report (1969-1-JQ Dec. 68). In terms of these scattering matrices, the conventionally used bitatic cross section al, akw, etc may be considered as the statistical average of some funetional forms of the scattering matrix. The primary concern in estimating the cross sections, therefore, is the choice of a reasonable statistical description of the rough surface. After an extensive survey of the experimental and theoretical works concerning the spectrum of the sea surface, we limit ourselves to rough surfaces that are described statistically as staionary and homogeneous, corresponding to that of an aged open sea. By assuming a height correlation function H(Tx, ry) for the rough surface, it is shown in Appendices A and B that the integral involved in calculating the average bistatic cross sections can be expressed in relatively simple form. In Chapter III, the explicit forms of the various components of the bistatic cross section showing the aspect and polarization dependence of such rough surfaces are given. All these formal results depend on a "universal" function F which may be expressed as an integral.of the height correlation of the surface. A realistic choice of the correlation function is somewhat uncertain. It is suggested that some simple models based on either measured results on the angular variation of the sea spectrum, or conventional anisotropic Gaussian correlation might be used. This phase of the study shall be continued in the next research period. 1

1969-2-Q In order to check the theoretical formulae derived, and/or to obtain the constants that might be involved in the formulae, an experimental system has been designed and constructed to study the bistatic scattering cross section of surfaces including water surfaces. Preliminary measurements involvisg flat surfaces of finite and finite conducttt have been carried out to test the transmittin receiving and the recording system. It is felt that this system functions very well. Experimeatal work involving the reflection from rough surfaces of known characteristics (corrugated surface), ground surfacean agitated salt water surface are currently under consideration. 2

1969-2-Q THE SCATTERING MATRIX The expressions for the components of the scattering matrix, specifying the scattering properties of a rough surface, derived from the Kirchhoff integral formula and tangent plane approximatin for the boundary conditions, have been given in the First Interim Report (1969-1-Q, December 1968). The essential results, expressed in Sowms that are more convenient for later use, are summarized in this section. Referring to Fig. 2-1, a plane wave incident from direction f lis scattered by a rough surface. We are interested in finding the field scattered in the direction f2. To be specific, let the rough surface be described by the function z(x, y) of which only some partial statistic behavior is known. The incident direction is given by S-1i 9 9taeoos +^ nsin n osej. (2.1) The directions of polarisation associated with the incident field are given by the following i) the direction of horizontal polarizeaion ~xt eh = l sin-4cosl (2.2) and, ii) the direction of vertical polarization = A IA Asli4 V1 Sx eh = -x cose coe 1 -y cos sinl +z sLn0l. (2.3) Similarly, any direction of reflection is given by A A l2=x sin2cos24syn2 s inO s a cos2. (2.4) The directions of polarization associated with the reflected field are, direction of horizontal polarization, 3

z A a 1 y A "2 A z Scattering Area, dA FIG. 2-1: DIFFRACTIONS OF INCIDENT AND REFLECTED WAVES AND THE DIRECTIONS OF POLARIZATION.

1969-2-Q Xl I ' = -X 2 + y O 2 (2.5) and, ii) vertical polarization, e2 = -x co cos2 P2-y oY 2 1n2+ sin2(2.6) If the incident field is represented in terms of polarized components by ikI - r 1 [^Eh+i v e (2. 7) _A the far zone reflected field, in the direction n2,is also resolved into the two dire -s of polarization, and is expressed as ikr L2= vv]r * + e22 r The relation between the components of the scattered field and those of the incident field are related formally by a scattering matrix. This relation is given by Ek S ShvF Eh. EV] [ S 2 R J (2.9) By using Kirchhoff's integral formulation and the tangent plane approximation for the boundary conditions, the elements of the scattering matrix may be approximately represented by the integrals; r r -ikq x'+q Iq zz) S W Ir: J Jm (2.10) Im $'im where qx=sin o s100 +sin2cos2 ' (2.11) q^=sinO8in l"+sn2Bi2n, (2.12) qyen181n2l1nO2att2 5

1'6-)-2-Q and q cos01+ co2. (2.13) The functions g, in the case of finite index of refraction (N), are complicated tm funotions involving the conventionally used reflection coefficient, R.L= coe 2 (2.14) CoO~+,/,2-,n2' and __........2 + (2. 156) N2coSy - /N2-sin2 i(2.15) where y is the angle of incident wave with the local unit normal and varies with xfu, Q, ardng to the relation -sineghly co surfaces the domin sin Ox 11 1... com Y=ay 2=,.16 1+ Anticipating thatfor most cases, when N -- 0, Rj -1, and R1 r +1, one may express gm in the following form. m = -(l+Ri.)P -(l-Ri//)Q +2 Gm The function, P, Q, and G depend on the direction of inoidence, reflection and the slope of the surface. For highly conducting surfaces, the dominant contribution to the scattering matrix oemi trm the ba(etw G and cthe othr terms may be treated as a pertutatlon. By atraigtjbrwmad algebraic manipulation the explicit expressions of the dominant coefficients, in te s I - 1 1' 02' 82 ' and the slope at the surface zza az/ax and z = az/oy are given below: Ghh z sin o s^ +z sin l - (2.17) Gh1= *u|2t) (2. 18) 6

1969-2-Q G v = z.(costG IsinG sli40-coS02in0,sin2)4zy (sine cosO2 c0st2 -sinO 2ocososI080) +-cosO cost02sin(o -01 (2. 1 9) G = -z cs0 sin02-zsn sin02cIno( -1).(2. 20) The coefficients PI andVW QQ which have leeis effect on the scattering mati fr surfaces of large index of rfatomay be rexpsew as P AC hh vv A AD ~hv vh A Pih BC BD where A =z 2 jIJnscos02Sino114-o0sn~~l6cs1o~os2 X 1 2. ls'$2-coo~1 2 1 1o~~oslo +Z2 BO 00 2c~1~ioeo2 -oos6 8130 +2-s'lnOoos0 lsinoin -Xz y Wi81n2coBe 218coeG0co0 2iole2wmvcs2n O4o26n10oe~~ 22 B= z ( in I2ioL+Oo1+0o86~ os~1 O+snl0sinG2 sin'~ -Z sinG 7)"0+ 0"10m20(2#)+nl'o "0 +ZX in2,)1clse~oeolmi(02-0*'COBuinG200 ( l00os21o29oaG2sin(2-s1)] +Z [i o00ise 1sn 001)c1(si142oSG8oo02si14+sino sine slz41) 1si2 ooO l'4Sin(02~ G(14+co02 cs2os2 4in12co 7

1969-2-Q C = -Z CosO lCos -ZyCOos ilnt1 -sinO1 D = -,zsinl y 1 +Zy oos8 1 2 2 2 2 2 2 2 2 2 A Z= [l(+z+I (l-sin,91ooe ) +zy(1-sin98Isin ) l i X Jl A jr 1 y -2z z sin2 01sinloosl+2z sin2OlmOlool +2z sin61coeO sinl +sOin2 J. 8

1969-2-Q m THE SCATTERING CROSS SECTION In terms of the scattering matrix introduced in the previous chapter, the conventional scattering cross section can easily be deduced. For example, if the incident field is horizontally polarized, ik 21' r Eh EIhe (3.1) 1 = ehl Eh. (3. 1) thee the component of the scattered field polarized in the orizontal direction is given by ikr E =Is Ehi (3.2) If the scattering area iAsut Ity A, then the scattering cross section is easily shown to be ahh(S, 2) 4w o co.l A (3.3) In terms of the integral representation of Sh we have shSh WI ^k2r dy. Ir' Ih' hy) bh(x'y') 5 -ikqfh(x-x) -i -y') -iya)i7 h e e,e where for simplicity we denote '/ zAxy ) '= z (x',y') (3.4) (3. 5) In oint rouh uurfaotes, only some pretial statistical knowledge of the sura Is sws eu t. m el el the 0gr d4s\t (3.4) is not feasible. From the statistical properties of the surface, however, one may find the expected value, or statistical averages of thae expressions;. 9

1969-2-Q Denote x' = x + and y' = y +. The expeted vale of Eq. (3.4) may be x y pressed as PShhS >5 2 dy I X.eQ ~9 16w J ikq ('-z) <h(x, y)g(x+1- y ry)e > (3.6) For a homogeneous random surface, for which the statictioal properties are invriant under the translation of ooordinate (i. e. the scatterin area is uniform), we infer that iCQZ(~'- i) A <ghhlY)g;*(Vxr+y7 >= yh-l(or (37) which is independent of the ooordinat, x, y. With this homogeneity assumption, Eq. (3.6) may be reduced approximately to 1 ( ikrX ik;7Y (ShhS>h 2 A drj dre e K Y(Tr, < (3.8) 16, 2 w y y~ Thus, we have ah( 1)= o 01f f e e K X x ( y). (3.9) In general, therefore, for a rough surface that may be statitioally represented by a stationary random variable, the scattering cross sections may be estimated by evaluating the expected values K (r, )= e< M(X Y)g* (x'+r, y+r ) ' > (3. 10) Tho scattering ro sectlyd by the The scattering cross eection conventionally used can then be evaluated by the 10

1969-2-Q expression A k2 ik&Xx ikqyy am ) -k2 oose Jf dTf e Km(, ) (3.11) Am (4' r xx Y ' y Due to the oomplicated integrals Involved in the procedures in calculating the cross section and th uncertainties in tbe statistical description of a rough surfae such a that of the s, estimato of am for a rough surface presents an extremely difficult task. In the present case, some simple models for the rough surface were chosen, and approximate formulas for evaluatin the scattering cross sections based on these models are derived. 11

1969-2-Q IV ASPECT DEPENDENCE OF CROSS SECTION In order to evaluate the statistical averages of the scattering cross sections given in Eq. (3.3) of the last section, some partial statistical knowledge of the surface must be known. Due to the complexity of physical processes involved in causing the disturbanoe over a surface such as the sea, a single analytical description for the rough surface is of course impossible. Kinsman (1965) gives an Itensive review of the theoretical and experimental work on the wind waves. For a fully aroused sea, it seems that teh following two statemens concerning the sea waves might be approximately true: i) The surface is spatially homogeneous and temporally stationary. ii) The surface height is approximately Gaussian-distributed, although the Gram-Charlier distribution fits the epprimental data better; the difference between these two distributions are very small. iii) Due to surface wind, the sea spectrum is anisotrppic; the directional dependence of the sea spectrum has been measured experimentally. Based on these approximate statements, we shall assume the rough surface to be anisotropic, stationary, homogeneous random surface. The surface is statistically specified by a correlation function H(rx, /y=< Z ( y)Z(x It yI-y)> = H('r,c) (4.1) where T: T cos a (4.2) r =r sina. (4.3) y If the x-direction is chosen as the direction of the wind, some of these directional spectra have been experimentally determined. As illustrated in Appendix A and B for such a model the statistical average of the scattering 12

1969-2 -Q cross sections may be evaluated. In this section, the analytical expressions for the scattering cross sections., knowin the function H(Tx,., is developed. To develop the formal expressions for the scattering cross section,, we shall follow the approximate procedure developed in Appendix B. First the coefficient 91 M defined in (~WtinriI are expanded into series for ms to tefirst order approximation in z an zY Xx y Explicitly, we have =-2cs 0oos(20 1) +( '+RI) (cos 01+csO )os8(02 -01) a. 2si0cos 00(2 - (ics1cos2)cs1~s~+cosO1sin0lsin02-cols6(cosO1o+cos02) cosJ + [(l+cose cose)sinnlsin( -0iJ sinG11 2 1 2 = 2cos01si sin (R) (l+cosG coso )sin~1os2-1 cm COS Iiin2 11 sinG(0 +sinolsinG 1sinG 2 -s inO2 cos01(cos01+cosG2) (l-R11) - (1-$cosO G S os)cosOsin(O -01) sinO1 1 2 'l'2' =2sin(02 -0, ('-RWO(1. O61ck gsn(02 -01 bh=. (cosG +cosG )sin-/ CS cs hv sin1 1 2 sinG1 j02os11+cos2)cosOIsin(02-01) _Co0,90(1+cosG Icos02)sinO2 -sinolcos91 sin02sinO2] Ch -sn (cos0 +cose )coo Cos(Cse+-8 )i)+sn lR) 1 2 1 2 01) sine h snO 1 21 20-1 +CosO0 1(l+eosO IooG2) oo502+sinO 1cosG IsinO2Coso1 13

1969-2-Q ah= 2cos~lC0802si( ~~1R)1+ICOS0]Cor 0)sin(~2-1 bhz2[s 01s 22s0211-O12) bv-2oO1 sin slno 18Jif ~Osino1 1 2 2 vh 2 L)css+oq)OOsn (l+RcoL)(IcosOicose 2) sl nO2in~2 sine I 1 2 1sinG1 +snecos01 (1+csnO 1C080 2s inO (coo iI0COS02 Sif02C08(0-0] c - in (cost 0,1+cos0 ie -- s c~ fd i a + co IS 01Gca(8-1)os( 2-1) (cos028el~co88)cs(~si - 02COSO1 (+R4)e co0 1 2 =-sn sinG1 ()~sl+Cosoi 2 +oG av cs cs(20 (-,,(oo~oe2O8G1(o201 cG) c8 cv =-2 sin02sino1+ (i 1 +cose 1cos02 )cSOIin(O 2-01) I 1 1 2 -COs 0 (Cos 0 +cosG )Sln HereJ the approximate R// and RIare evaluated through coo -Y cos 0. Then., following Appendix B,, we choose correlation function H(Tr,8 T and compute the integral 14

1969-2-Q dT V A J x J dr yexp [o0) -H -ik qzC -00 -CD (4. 5) and then the integrals involving the evaluation of the cross sections may be expresed in terms of F. From Eq. (3. 11) we immediately deduce A A 4ik cosO fim <S2' = -2-) [b mb ^y] * 2. (4. 6) z It is evident that from any analysis carried out here the choice of the correlation function H(Tr, y) and subsequent evaluation of F(qx, qy, qz) completely determines the aspect dependence of the average value of the scattering cross sections. The choice of some reasonable expressions for H(T, T ) seems to be the major problem confronting us at present, coupled by some approximate method of carrying out the integration for F(q, qz qz). We expect that, after careful study of the experimental data as mentioned in Kinsman (1965), some reasonable functional form of H(Tx, T ) may be deduced. If this proves to be impossible, then, perhaps, following the commonly used technique in dealing with random variables, assuming scale lengths I and I we will postulate x y 2 2 H(Tx T y)=H(O 0) exp - - (4.7) x y This jaase of work shall be continued in the next research period. 15

1969-2-Q V CONCLUSIONS AND KOO ENDATIONS A first order theory for the bistatic scattering cross section of a rough surface incorporating the effect of the dielectric constant of the surface, has been developed. The results depend on the choice of a model of surface height oorrelation. Due to the approximations involved in the derivation, and uncertainty in the choice of a reasonable model of the height correlation, the usefllness of the results derived cannot be certain. Hlwever, it extends the present theories of rough surface scattering which are mainly limited to backscattering, to arbitray direction, and within the same degree of approximation. Therefore, it seems to be worthwhile to choose some reasonable form of height correlation and investigate the implications of the aspect variations of scattering cross section as predicted by these first order formula. If possible, the results of the theory should be cheked against the limited experimental data available in open literature. An experimental system for the measurement of bistatic cross section that is designed in conjunction with this work should also prove to be useful in asserting the validity of the theoretical results and suggest possible modifications. 16

1969-2-Q REFERENCES Kinaman, B. (1965), Wind Waves. Prents e-HallU New Jersey Moyal, J. E. (1949), "Stochastic Processes and Statistical Physics, " J. Royal. Stat. Soc.. Ser. B., A, id.2, pp. 150-210. 17

1969-2-Q APPENDIX A STATITICAL DESCRIPTION OF ROUGH SURFACE In carrying out the sttitical average of SmS to obtain V statistical xm Im *m' preperties of the surface are required. Descriptions of a random surface and the approximate distribution fimctions involving the surface height and slopes are given n this ppendix. We consider the urface height given by z = x. y) (A I) as a random function. By proper choice of the reference height, we let the average value of z be zero, i. e. Zt(x y)> =0 (A. 2) A statistical description of this random function is the correlation of the height between two positions on the surface defined as H(x, y; x', y') = (x, y), z'(x, y)>. (A. 3) For a homogeneous (spatial)surface, the correlation function should be invariant with respect to any coordinate translation. Thus, if we denote x' = x- (A. 4) x y'=y-y. (A. 5) then H(x, y;x' y) = <z, z'> = H(T, ) (A. 6) where, for simplicity, we denote z'z(x', y')=z(x — y-'ry). (W 7) If we let T = cT Oo (A. 8) T = T in, (A. 9) Y y then H(Tx T )=H(T. ). (A. 10) 18

1969-2-Q hi practical measurements involving sea surfaces, some knowledge of the fuwtion H(r, 0) is known. The mean square value of the surface height, from (A. 6), is given by m2 = 42 H(OO) 0). (A. 11) o For mort physical problems, H(rT, y ) must be analytical functions of the variabl and is svidntly.Y i a r.. Therets-, 3HA ^H(0o,0) '0 (A. 12) x 7x=O Ty=0 and H (O, 0) = 0. (A. 13) Since, for this class of random functions, the pato f d ia n and taking the average commute (Moyal, 1949 the correlation between slopes and heights can be easily obtained. The results are, explicitly, <z,z') -H (T, ) (A.14) zz x>= 0 (A. 15) <z x')> =H (T T ) (A. 16) < X ' -H( Ty) (A. 17) 4kk^ > 3-H (0, 0) (A. 18) <z, = -Hy(, y) (A. 19) <4, z > =0 (A. 20) O Y'> = H (r ^T ) (A. 21) O l> = -H (TrT) (A.22).4d,Z > =-H (0, 0). (A. 23) In evaluating the aversge values of the scattering cross sections given in Chapter I, we are interested in obtaining the averap of quantities, 19

1969-2-Q -ikq (z-z') 4 m(Za zy)g zm( ' z)e For evaluation of these quantities, it is necessary to know the higher order correlations uch as z y;Z; '; Z ^Z > X 'y x y Knowledge of these higher correlation functions for rough surfaces or sea surfaces is practically unliown and too difficult to measure. By rt heavily on the physical reasoning that a random surface may be considered as a summation of isbtad ly many small perturbations, we may infer that the random surface is apprximately Gaussian so that the joint probability distributions of the quntites z, z z' ' and (z-a) may be expressed from the correlation of each e yo ix' y pair of the variables. To simplify notations, let us denote A A A A A u = = ' a't u zx U1 z-z, u, 'X' u3zyy U4 u5^ y and the coreelation Pij <uiuj ~ For a random surfaoe of given H (', r ), then it is easily seen that x y P 11= 2 H(0, 0)-H(T, r] P22 P44 = -Hx(0~ 0) P33 55 -H (0, 0) P12 =14=P21=P41 -H(7x 7y) P13=P15 P31=P51=-H y(x, y) yxy P23 =P32 -Hx (0' 0) P34 =42 -Hxx(TX Ty) P5 P52 P34= P43=- H( xy (T y) P35=P53-" yytxr y) P45 P54=-Hyy(0 ).* 20

1969-2-Q In terms of those orrelations, the joint probability distribution can be writen apprimately as 1 m uu f(u,u2,u3,u 4 u ) = exp 1 B 3' 4P 5 y2 pu lot 2|p| where I pl is the determinant formed by Pij. and mi are the cofactors of Pi in the determinant. Therefore, for the average of any function of the five variables, we may use the probability distribution and evaluate J> du du2 du duJ J(u, u2, U3 u4 u5) L00 <<l -<x> -CD f(Ul"'U U'3U4U) 5 Thus, in principle, using the Gaussian approximation, if we know the correlation function H(7r., of thi surface, the scattering matric can be computed. The procedure is, however, extremely ompliated and some extra approximations may be introduced in order to obtain manageable results. 21

1969-2-Q APPENDIX B EVALUATION OF STATISTICAL AVERAGES Based on th development of the approximate joint probability of u1 z-z, u2z u3 z, 4 and u = z', the statistical averages involved 1 X y in the callations of scattering cross sections can be, in principle, carried out. However, due to the complicated funtional forms of gm as shown in Ch. IV, xact evaluations of the cross sections seem to be very impractical. From the expression gm= -(l+RL)Pm -(1-R//) Qlm+2 G1m1 we see that GI. the dominant term,is a linear combination of constant z and z. To simplify our oomputtion, therefore, we shall approximate the other Y terms also by a combination of constant termz and z. This approximation x y would give exact results for the case of a perfectly conducting surface and is expected to give good results when the reflective surface if fairly smooth (i.e., z is small). With this approximation, we may write g m aim+ bm u2+ olm u (B.1) and 'I* R +bt u4+cI u5 (B.2) I+m m m 4 m 5 gimgim lm+ almblm(U2+U4)+~ml m(U3Us5)+m Im(U2U5s+3U4) so tham m (u2u4)+camY us). (B. 3) The statistical averages to be carried out, therefore, are basically three types. These are evaluated below. a) For the constant term, we use the distribution 22

1969-2-Q ul -L2pi, _ This yields (B. 4) -ikqzu1 (C ~~2,pI fD r 2 duex L"'i -j]=ep. ~ k2q2 (B. 5) b) For the terms fut Uj) = 1 lierin u,, we useth distribution j (B. 6) where A. I = I p11 Pij P13 = 2 (B. 7) ImL "I <uje 1> = 1 OD OD Juiduif -00 -00 du 1 exp -ikq u1. 2 Aj (B. 8) The above integral can be carried out analytically by re-arranging the exponential term isuch as 2 A2 22 2 A32 2-j- [U1+ikcp 1qj 2 (B. 9) - I- ii - or - 0 24

1969- 2-Q Straightforward successive integration yields -ikq U 2 u c) for the quadratic termsexp -we ue the ditribution c) for the quadratic terms, we use the distribution (B. 10) I! exp where Pll A= P21 P31 (B. 11) (B. 12) Pl2 P13 P22 P23 P32 P33 and M is the cofactor of the element iJ easy to verify that -i u -I Mijuiuj = - -ikqzU- 1 2 A zli p in A. By rearranging terms it is LM-,-,u M12u2+tMl3u3+ikqzA- 2 M 1 2M11 1 P33 [r P23u3ikqzM12] 2 V3 3k 2 M J L 3 zJ -3 2 (B. 13) Using this result, straightforward integration one at a time yields 0 00o 00 f du1 udu2 f u3du3f(ul, u2, u3)ex [-ikqzu1 -0o -co) -o) {p23+k q2 [ k2 ] ] 1 ^ ^ ij ^;'"'2 > 24

1969-2-Q In general, therefore, we have the relation -uiujeikqzul > ij-kqplip exp q2 ZP1(B.14) With these results,we have the following approximate formula useful in the evaluation of the average values of the scattering cross sections: <(gmglm = exp [ k2 q Pl { mikqzalmblm [14+ Pl] -ikqal mclml3+Pl +b mctm [P25+P3J 2 2 22 +b p2 p-k qZ bLmCm (P]2Pl5+p13P14 +bmP24+ClnP35 c m ( 1513 14) - DZtP 12P14+c mP13 ] (B. 15) Using these relations, we can deduce the following relation which is useful in the evaluation of cross sections: <g (z, y* i ', e-ikq(z-z) pr I 2 2 21 gm(zx, y mx z)e-q( ' e xp - k qz a2 +2aL b (-ikqzP2)+2al m (-ikqzP13)+ m(Pa24 - 2 2 2 222 22 +c2(P3 -k2qzP13)+2bm c m (P2 z qz P12P13) (B. 16) In evaluating the cross sections we m t oatry eut the Intgral 00 r(0.... -lafz-z^ "11 f dx yglm(z Zygl( ze q( Zy e -a), 00 Now, from the following relations in Appendix A, 25

1969-2-Q P112 [H(0. 0) -H(X. TX] P 12 -HX(7X TY) P13 Hy( ) p -Hx T) 5 y(T7, T) P13 Y= Y 35- YY X Y P25 -HXY(TX.7Y) P24=-H )rx, we see that Eq. (B. 16) may be simplified completely. To deduce the simplified relation, let us define a function yF(qx q r x dq dT [2 ep k qzP lexp ikqTx i- ] i -00 -00 (B. 17) By Fourier inversion we have exp {-k2q [H(0, oO)-H(T7X T)] x 2 -dqyF(q, qz) exp +ikq kq k2 x (B. 18) By differentiating (B. 18) with respect to x we have k2 qH,( T eXP { -kq e A (0, O0)-H(r T)] | 2 r rx k q -ik 12ep t-"- 2Z P11 (B. 20 26

1969-2-Q Fourier inversion of (B. 20) yields I dx d-y (-ikqzpl ) exp 2- P = — F(qx qy,). (B. 21) Similarly, with Ty, cOD 00 r k2 2 2 1 d,x f dry (-ikqzpl3) exp [- 11= - q F(qy, ). (B. 22) Twioe differentiation of the above with respect to x and 7y yields H +k2q2 H2 xx z x k2q2 H H1Hy 1 exp [k2q [H(0, 0) -H<TX, JT 22 2 2 H+k q. Hy k2 (2?) -a) _ ~ (B. 23) Identifying the respective values of terms, etc., i. corresponding correlation functions, the above relation may be expressed as 27

1969-2-Q 2 2 2 222 P24-k qzP12 2 2 P25-k2qP12Pl3 P35-2 13 _ 7I expF -kp+ = —.. l x Xp - 2 zP, 7)2 h -I - (2 I) -aO qz dqx dqy-F(qx, qz) -O0 1 (B. 24) The Fourier inversion of the above becomes -O - dT y exp [ Pklli q2 =-2 Fy,, ) yQy.(B. 25) Substituting these relations in (B. 16) and carrying out tbe integration, we have a relatively simple relation for the calculation of the scattering cross section. O0 J dT -a:)D *r, e-q(z -z t) d (Zx Zg (ZXz ' ) > C ikqx-tk..... - - F(qx'qy qz) amqzblmqx-c 2. (B.26) q2 z28 28

UNCLASSIFIED Cl.rrirt vrl (I Q fic tion DOCUMENT CONTROL DATA E R & D (.fuflty rl ifmlllraton onf tl to. hnl), otf nhiafnrl stl fthlextng rtnhotflfln noun# be eolered when tho overall re port ml I An l lfleif0) _ I I G1N9 A T I N o A C t ITV I T (Cotpo tO nthor^o) as. RFO"t iC Ccu.. C. S.S Fc A TION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, ib. anouAnn Arbor, Michigan 48108 0 %F — - DOPPLER RADIATION STUDY 4. OESCRlPTIVE NOTEc (7ype of report ar d Incrluelv d&tea) Interim Report No. 2 (1 October 1968 - 1 January 1969). A TO ~ rt. Au.4 ( m.d4Om * *t nt lwm ) Cleao-Ma Soondk K. Cho POT OATI.. TOTAL NO. OF PAO ". NO. r CS March 1969 28 2 Cs. CONTrACT Or GcANT NO. *. ORIO IN ATOR nt RPOT NUMS ERtI - N62269-68-C-0715 1969-2-Q 6. PROJIC T NO c. h. OTHER REPORT NO l0S (Any othoe ntmotuf ee ft may be eslliged this report) d. I0. OISTrIUTION STATEMINT II *SUPLMNITARY NOTItS 12 iSPONSORINQ MILITARY ACTIVITY U. S. Naval Air Development Center Johnsville, Warminster, PA 18974 Approimate ome r l r the bttic cross section of a homogeneous, stationary surface are derived. The formulae are based on physical optics. These formulae give the estimates of the aspect and polarization dependence of the cross section based on a single function of the incident and reflected direction. This results from an integral involving the correlation function of surfce height. It is suggested that a study of this universal function based on some realistic choice of correlation function of surface height should be arried out for a reasonable estimate of the aspect dependence of the scattering from a rough surface. 11r F IA07I Lop Lopt too 6 I + I ' UNCLASSIFIED -- SIevurlivC~*~i~rcr 29

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