8( THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory STUDY OF ANGLE OF ARRIVAL ERRORS DUE TO MULTIPATH PROPAGATION EFFECTS (U) Chiao-Min Chu and John J. LaRue May 1967 Final Report No. 8003-1-F 1^ Contract No. N123(60530)56078A 8003-1-F = RL-2173 003-1-F opy Contract With: U. S. Naval Ordnance Test Station China Lake, California 93555 Thls documont cont na ir vrmnaon *fftctln# the erAtor0;T; dftense of t ritad l.tat*s withnl th* mnAnln: of '.he 'apic, rt. Laws. (Title 18 U. I. C.i tsecions 703 anJ 7S9,/::.-, ns mlsalon or the r4ev elitlon of Its conten, in *n innnerto *n iunut!h ori4e parson i prhlod I h rt - b Adminiftered through: I I iGROUP DOWNGRADED T EAR INTERVALS,.DECLASSIFIE TER 12 YEARS OFFICE OF RESEARCH ADMINISTRATION. ANN AR BOR Co '. TdeIa U''TCArSSIFIED

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN Study of Angle of Arrival Errors Due to Multipath Propagation Effects (U) Chiao-Min Chu and John J. LaRue May 1967 Final Report No. 8003-1-F May 1966 - May 1967 Contract N123(60P30)56078A Prepared for U. S. Naval Ordnance Test Station China Lake, California 93555 CONFIDENTIAL CNIDETA

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F ABSTRACT (U) A unified approach using the generalized concept of angular spectra for the study of the radiation received at any point in space from a transmitter due to multipath propagation effects is formulated. The various mechanisms contributing to the multipath effects, such as scattering by discrete objects, and by extended objects such as ground, are formulated in general. Although the formulation is based on CW transmitter and stationary receiver, the results may also be applied to other transmitter signals, scanning and moving receivers (and transmitters) with slight modification. iii I CONFIDENTIAL

THE UNIVE'RSITY OF MICHIGAN 8003-1-F TABLE OF CONTENTS (Unclassified) ABSTRACT iii I INTRODUCTION AND SUMMARY 1 II CHARACTERIZATION OF THE RADIATION FIELD 5 2.1 Introduction 5 2.2 The Angular Spectrum 5 2. 3 Direct Signal 9 2. 4 Scattering Matrix 11 III SCATTERING FORMULATIONS 15 3.1 Introduction 15 3.2 The Sphere 16 3. 3 Dipole Scattering 22 3. 4 Physical Optics Approximation 24 IV GROUND REFLECTION 29 4. 1 Introduction 29 4.2 Angular Spectra of the Reflected Radiation 29 4. 3 Ground Reflection Matrix 33 4.4 Reflection Matrix for Plane Ground 40 V REFLECTION FROM RANDOM ROUGH SURFACES 44 5.1 Statistical Averages 44 5.2 Slightly Rough Surfaces 47 5. 3 The Shadow Effect 60 APPENDIX A: NOTES ON STATISTICAL AVERAGES 70 APPENDIX B: ALTERNATE FORMULATION OF GROUND REFLECTION 84 REFERENCES 87 DD 1473 iv

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F INTRODUCTION AND SUMMARY (U)The objective of this research is to carry out a general study of the multipath propagation effects on the radiation received at any point in space from a transmitting source. Anticipating the fact that the polarization and distributional characteristics of received signal are important in utilizing the received signal to estimate the range or position of the source, a unified approach using the generalized concept of -angular spectra of fields is suggested here. (U) In Chapter II the characterization of radiation by angular spectra is introduced. The electric field strength at any point r associated with radiation coming from a small solid angle d Q in the direction Q may be expressed as dE(r)= l(r, + ( )2. (1. 1) 1 1 2 2( where e and e are directions of horizontal and vertical polarization and 6 and are the two components of angular spectra of radiation. From the far zone approximation, the directed radiation from a transmitter located at Et may be expressed as,. % f F1(Q) ikr-Itl I- ^ dr) (1.2) 2 F (Q) direct where ( dr Ir-rj -t F1 and F2 are related to the antenna pattern and gain. (U) In Chapter II the scattering of the directed signal by discrete objects is discussed. In terms of a scattering matrix, the contribution of the angular spectra due to a scatterer at r. may be characterized by a scattering matrix I 1 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F [s (QKQdi) where Ai li-, (1.3) di Jr.-rj is the direction of the incident radiation seen by the scatterer. The scattered field at any point in space can then be represented by the angular spectra A F1(2"di) ilr.-rI ilr-rJ &e ] A j iIr-r.Ii 6(Xa^) 1 4) scattered where = r-r. (1.5) i fr-r.I 1 _ 1 is the apparent direction of the scattered radiation seen at any point. Theoretical models for the approximate calculations of the scattering matrix are also presented in Chapter III (U) The reflection due to a rough ground is investigated in Chapter IV. Using the geometric optics approach, the contribution of the angular spectra due to ground reflection may be expressed in terms of an integral such as given by Eq. (4. 50). If the transmitter is far from the ground, then this angular spectra may be expressed in terms of ground reflection matrix [I such that ]e e.r:R] ___ e (1.6) J F(O) t reflected direct where ro is some chosen center of the illuminated region of the ground, and A _r -: =.... (1.7) 2 -CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F is the apparent direction of direct radiation relative to the center. (U) A study of the ground reflection matrix for a slightly rough ground, using the geometric optics approximation is carried out in Chapter IV. For a slightly rough, random ground, the formulas for the statistical average, and correlations between the elements of the reflection matrix, including the effect of finite index of refraction of the ground are investigated. In Chapter V, a preliminary study on the effect of shadowing on the ground reflection matrix is carried out. (U) In principle, the radiation at any point in space can then be obtained by adding the direct signal, scattered signal, and reflected signal given by A =+ + 18 d(r 0) < 2 ci] (1.8 2 direct lscattered -ireflected For a receiver at any point with any receiving pattern, the received signal can then be obtained by integrating over the angular space. (U) Due to the uncertainties involved in the problem, especially the ground reflection for which various statistical modbels of ground can be chosen, no specific calculations were made on the theoretical model proposed in this work. It is felt that before meaningful numerical analysis can be made, some experimental results characterizing the statistics of ground reflection are necessary. (U) It is to be noted that although the present formulation is presented on the basis of CW transmitter and stationary observer, the extension of the formulation to FM and moving observer is relatively uncomplicated. By assuming the transmitted signal to be of the form -iuW t f(t)e ~ where w is the carrier frequency, the angular spectra may be expressed in the form C D3 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F Il t) -iwot.. e so that the time variation of the power spectra of the radiation may also be included in the formulation. (U) For a moving detector, the point of observation changes with time, so that r=r(t), describing the trajectory of the detector may be used in the angular spectra. The angular spectra then takes the form 1(, r(t)t) -iwot, (-, r(t) Thus the effect of a moving receiver may also be investigated using this formulation. (U) Other corrections, including moving and scanning of the transmitter, the tropospheric or meteorological effects may also be incorporated in this formulation. (U) In summary, a unified approach sditable in the investigation of the radiation at any point from a transmitter due to multipath propagation effects is formulated. This approach, incorporated with experimental results that may yield a reasonable statistical model or models of the ground may be employed to obtain a detailed characterization of the received radiation by a stationary, as well as moving, detector. CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F CHARACTERIZATION OF THE RADIATION FIELD 2. 1 Introduction (U) The radiation observed at any point from a transmitting source generally consists of several components due to the multipath propagation effect. This multipath effect is illustrated in Fig. 2-1. If the transmitter is a low angle radar such that the ionospheric reflection may be neglected, the received radiation may be roughly classified into three categories: i) the direct signal whenever the point of observation is in the illuminated region of the transmitter (main beam or any side lobe); ii) the scattered signal, due to the presence of obstacles in the illuminated region, and iii) the ground reflected signal. (U) In order to infer from the received signal at any point, the possible configuration of the transmitting system and its surroundings, it is necessary to have a unified, detailed characterization of the radiation at any point. (U) The basic characteristics of the radiation received at any point that may be subject to detection and analysis are; a) the temporal variation of the signal, b) the angular distribution of the signal, and c) the polarization of the signal. (U) For CW transmission, the temporal variation may be accounted for by the phase variation of the signal. To incorporate both the information of angular distribution and the polarization of the radiation, which is necessary when ground reflection is important, it seems most natural to employ and generalize the notion of angular spectrum to characterize the radiation. The idea of angular spectrum has been successfully used in the study of ionospheric reflections (Booker, Ratcliffe and Shinn, 1950) and are therefore adapted and generalized in this report for the characterization of the radiation. 2.2 The Angular Spectrum (U) For a simple introduction of angular spectrum and the notations involved in this work, we shall choose a fixed coordinate system, with the average level of ground taken as the z-plane, as illustrated in Fig. 2-2. Any point in space is then CONFIDENTIAL

CONFI DENT IAL THE UNIVERSITY OF 8003-1-F /4 I / MICHIGAN /I /I /I / zI 9 H, p4 E-4, cJ & C.) / / / Q / / / / I / / / i/ */1 /// i ) C.) 6

CONFIDENTIAL THE UNIVERSITY OF 8033-1-F MICHIGAN -- z I I I I I I I y I / / / A e1 / / 0 -/0 I z r e2 z x-y Ground Plane x FIG. 2-2: COORDINATE SYSTEM FOR ANGULAR SPECTRUM 7 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F represented by a vector A A A r = xx+yy+zz (2.1) A and any direction may be denoted by a unit vector Q. In terms of the latitude angle a and azimuth angle,3 the unit vector Q is = sin acos+ysinasin+3+zcosa. (2.2) (U) Since the radiation reaching r may be distributed in all directions, one may define the electric field of the radiation reaching r from a small solid angle dQ A in the direction Q by dE= (r, 2)dQ (2.3) where _(r, Q) may be called the angular spectrum of the electric field. For CW transmission, the angular spectrum may be expressed in terms of a complex amplitude and phase, so that (r A (, )eikd() (2.4) where d(r) is the total path travelled by the wave from the transmitter to the point of observation, and A(r, 2) is the complex vector amplitude. (U) For the radiation (far zone) field, the electric vector must be normal to the direction of propagation so that A must be a two-dimensional vector normal to. Following Green and Wolf (1953), one may define two mutually perpendicular unit vectors '1 and e2 both normal to Q by el= [ = x sin ( -y cos (2.5) and e2=x el-xcosacosH3+ycosasin-z sina. (2.6) These vectors are illustrated in Fig. 2-2. The complex vector amplitude A can AA then be decomposed in the directions e and e2, such as CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F A(r, A A e(r, l+A2(r, )e2 (2.7) In other words, the radiation reaching any point may be expressed in terms of ikd(r-) two scalar amplitudes (A1 and A2) and a phase delay (e (-) by the angular spectrum _(r ) =A (r )e eikd(r+A2( Se eikd(). (2.8) Evidently, A1 is the electric field of the horizontally polarized component while A2 is the electric field of the vertically polarized component of the radiation. (U) In most problems, the phase factor due to time delay is relatively easy to determine. Thus, apart from the phase factor, one may describe the radiation field by two scalars, or a two-dimensional vector such as... r_^ L&Q ) 1. (2.9) (represented by) A2(Q, r) Representation of the incident field from the transmitter in the components A and A2, and the study of change of A1 and A2 due to various scattering processes shall be of prime importance in the present investigation. 2. 3 Direct Signal (U)The direct signal seen at any point in the far zone approximation is generally represented locally by plane waves. For a plane wave travelling in a direction f2, the electric field is given by A ikQ * r E= el-E22e e. (2.10) Since the radiation for a plane wave appears to come from one direction only, we may represent its spectrum by = [ikQ ~ r 1 C = L +E ]e ~ (- ) (2.11) 0 where 6(Q-0 ) is the Kronecker delta function in the angular space. CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) For a transmitting antenna, the field seen at any point may also be represented by the form of (2.11), but the amplitudes and phase factor,of course,vary with direction and distance. To express the direct signal in terms of the radiation pattern, gain, and the transmitted power of the transmitting antenna, let us consider a horizontally polarized antenna with gain G and the power pattern P(Q). If the total power radiated is Wt, then the Poyntingvector in any direction n is given by WtG P(2) P (2.12) 47rr where r is the distance measured from the antenna. The magnitude of the electric field is then E =2pr =0 rWt (2.13) Now let the transmitting antenna be located at position r, then the radiation received at any point appears to be from a direction r-r * (2. 14) The magnitude of the electric field is given by _E___l v~ (2.15) For a horizontally polarized antenna and if the phase of signal at the antenna is assumed to be zero, we have E=160G e (2.16) Thus, for a horizontally polarized antenna, we may represent the angular spectrum Thus, for a horizontally polarized antenna, we may represent the angular spectrum of the radiation by ' 1... 10 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F ikJ r-r \ (F e V 6(A QA (2.17) where F1() = 60GP()W. (2.18) 1 (U) In general, the direct signal may be expressed as ~r A A ikir-jr AA t (L 6): [F1l(~)+2F2() irrtLIQ...... (2. l9) (rF ) 6(,) (2.19) in order to specify the angular and spatial variation of the radiation. (U) It is obvious that; i) for a horizontally polarized antenna, F2=0, ii) for a vertically polarized antenna, F = 0, and iii) for a circularly polarized antenna, F= i F2 (U) Elliptically polarized antennas can be expressed in different combinations of the two complex factors, F and F2. For mathematical simplicity, we may represent the direct signal by the two-dimensional vector; d () )ikir-4 ^ d 1(- ) e-1-) A (2.20) F2(7) j (represented by) 2.4 Scattering Matrix Any obstacle in the beam of the transmitter scatters the incident radiation into different directions. As illustrated in Fig. 2-3, the incident radiation may be assumed coming from the direction o with the electric field E, while the scattered radiation is distributed over all the directions 2 and with different electric fields s E. The relation between E and E,due to the linearity of the Maxwell's equation, -S -S o may be expressed in terms of a scattering matrix for plane wave scattering (Saxon, 1955). Mathematically, one may deduce, from Maxwell's equation, that the scat tered field in the far zone approximation may be expressed by ii i, 11 CONFIDENTIAL

CONFIDENTIAL - THE UNIVER SITY OF 8003-1-F MICHIGAN s Incident Scattered E Direction of Polarization FIG. 2-3: GEOMETRY FOR THE SCATTERED SIGNAL 12 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F ikr A e E )= - (P, ), E ) (2. 21) -S s s 0 -o 0 r where A A s(Q, o) is.known as the scattering matrix. (U) Explicitly, if one represents the incident field by E ( =A( A) e (2. 22) -D o L1 1 2 and the scattered field in any direction by A F (s) A (5)A ek ie o Es = s e +A2 ee (2.23) then the scattering matrix may be represented by ) A nA A 11 s 0 12 0 0 A (A 1 (2. 24) 21 s o 0 22 s'o such that 1 [11 2 1 21 (2.25) S O s21 s222 2 2 A (U) If the incident field appears to come from direction 2 with phase angle o, and if the scatterer is located at r., then the scattered radiation seen at any point r appears to come from a direction A n r-r. S2 = 02, A (2.26) s 1 I - r-r.I -1 In terms of the scattered amplitudes, one may easily represent the scattered signal by the delta function distribution iklr-r.I r -,I, riI A Sr)= A(,+A2 r e 6(QQi). (2.27) 1.A2 "J.I13... CONFIDENTIAL

* CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) In principle, if the scattering properties of various objects, including ground are known, the composite signal that is seen at any point originating from a transmitter can be obtained by summing over all the components. The discussion on the scattering matrices of discrete objects and the ground are given in Chapter III of this report. 14 *CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F III SCATTERING FORMULATIONS 3. 1 Introduction (U) In Chapter H, it was shown that if the radiation field is represented by two scalars or a two-dimensional vector associated with the angular spectrum, the scattered field may be expressed in terms of a scattering matrix. Thq description of the scattering properties by a scattering matrix is the natural extension of the ordinary concept of scattering cross section. For example, from Eq. (2.25), for a scatterer located at the origin, the scattered field is given by E1 eikr s 11 S12 E1 1 i- S 12 E2_ (3.1) 2 21 22 2 Hence, the conventional bistatic scattering cross section for a horizontally polarized incident wave is given by 2 2 E r + E a =lim 47rr2 = 4r S +S 21. (3.2) Thus, the knowledge of the scattering matrix contains all the information about the conventional cross section. The inverse, however, is not true. Therefore, it is necessary in the present work to discuss the means of evaluating the scattering matrix. (U) Just as in the case of calculating the scattering cross section, the exact solutions of the problem are only possible in a very few cases. In most cases, approximate methods developed in the evaluation of scattering cross section such as physical optics, Rayleigh approximation, etc., must be used. In this chapter only a general formulation of these approaches are given. For a detailed appli cation of these approximate methods in the calculation of radar cross sections the work of Crispin, Goodrich and Siegel (1959) may be referred to. Extension of __ 15 _ CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F their results in the form of scattering matrices seems to be straightforward but in most cases, very detailed. 3.2 The Sphere (U)Perhaps the only possible shape of finite scatterer whose scattering matrix may be formally written down in relatively simple exact form is the sphere. To illustrate the derivation of scattering matrix from the exact solution of Maxwell's equation, the scattering matrix for a sphere is derived here. (U)The standard problem for the scattering of a plane wave by a sphere is summarized in detail by Stratton (1941). Refer to Fig. 3-1, a plane wave whose electric field is polarized in the x-direction is impinging on a sphere of radius a and dielectric constant -N' (the permeability is assumed to be 'uo). Since ikz E =E xe (3.3) x the scattered field may be expressed in spherical wave functions as -=E in (2n+l) a M: -lb ' (3)7 E E am n -Ol bn ie (3.4) -s o n(n+l) n-01n eln( where the spherical wave functions are expressed in terms of spherical Hankel functions, associated Legendre functions, etc., by (3) F cos) (1) -1 FdP'cos3) (1) 5) M). osO h (kr sin h (kr) (3.5) (()1) eln P ((cosn0) n cos [kr ((k sin sin kr h(. (3.6) The scattering coefficients in (3.4) are expressed in terms of the spherical Bessel functions and the normalized radius of the sphere a = ka (3. 7) ____________________ 16 ______. -,,, CONFIDENTIAL

CONFIDENTIAL THTE UN I 7 E R I TY OF 8003-I-F M{ ICHI GAN - z A Sphere Ho Eo Incident Field x FIG. 3-1: GEOMETRY FOR SCATTERING BY A SPHERE 17 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F by ji(Na) Ej n(aj '-,j(a) [Na j(Nat ' a n (3. 8) jn(Na W -ha() (a a hn (No -n. n 1n n and j n() [N j(Na -N2jnN a)aj (a ' b 2f f(3.,9) hn (a) Naj(Na -N2jn(Na) [a (ac In the literature, the scattering coefficients have been calculated for various sphere sizes and indices of refraction. (U) For far zone fields, the asymptotic form of the Hankel function ikr h((kr)( i)n+l e (8.10) n kr may be used. This yields M(3)n(-1)-kr n+1 A_ s- ]c__[ _]_ (3. 11) | (3) nin e 1n -ic N E { cos S1(n ) cos+ sin0 S2() (3. 13) e.n k, de _n r sine a A whr1e ( (1) (1) OD (2n+) P (cos)3 dP (cosO) i s__n)=-i +r n (3.14) an(n+) n sin n de and ________ 18 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F ) 2 n(n1) (dos) OD, 2 dP~ ( cos))(3. 15) S(0)= i (2n+l)' a +b. (3. 15) n=l Both S (0) and S2(0) can be calculated for any sphere from the corresponding scattering coefficients. (U) From Eq. (3. 13), it is easy to infer by symmetry that the field scattered by a sphere from an incident wave polarized in the y-direction is given by ikr E =E y r sin0 S1()- 'cos0 S(0)o. (3.16) Thus, if the incident field is expressed in terms of the amplitudes E E. x (3.17) E y while the scattered field is expressed in terms of the amplitudes E SE,- (3. 18) one may have. E0 eikr cos0S1(0) sin0Sl(0) E1 s = 1 (3.19) E k sin0S2(0) -cso0S2(0) E (U) In order to adapt (3.19) to the present convention for the direction of polarization, one shall assume that the incident field comes from a direction Ao(ao, fo) instead of the z-direction. The direction of polarization of the incident radiation can then be assumed to be, respectively A A Q xz el =f = sin o-y cos (3. 20) 10: Io 2I o- o 0 CONF_9 ETIAL CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F and e20o 2xe10:xcosa cos 13+ycos a0sin 30-zsina0.(3.21) Moreover, the scattered signal in any direction S (as, fs) should be expressed in terms of the components in the two directions A A xz s sn =X sinP -ycos 3 (3.22) and e2 =n xels=xcosa cos +y coasa sina -z sin (3.23) instead of the 0 and p components. (U) Mathematically, this means that one must re-express A sAS E = E +0 E (3.24),-s E in the form A S A els E + e2s (3.25) -s is 1 2s 2s by a rotation of reference coordinates. It is easy to verify from (3. 24) and (3. 25) that the vector components are transformed according to E] e[ i ) (e2 0)] (3.26) E2 2s' () e2s' E Now, ife, 20, o are identified, respectively, with the x, y, z directions used in the 0 2l0' o derivation of (3. 19), one can rewrite (3. 19) in the form 1 eikr ) (e. ) l[cossl(0) sin0sl(0) " El1 E J L( s) (2s(ein0s2(s) -cos0sl2( EJ CONFIDENTIA_______ 20 _______ CONFIDENTIAL

CONFIDENTIAL ----- THE UNIVERSITY OF MICHIGAN 8003-1-F L! (U) To express the matrix relation between the scattered and incident fields in terms of Q Q and the derived directions eO, e20, e and e2, one notes that _ o s A A A A (3.28) A A A A A A^ A (Q')-Q 0=0 X = -.0 s and e sin cos 0+e 20sinO cos0 + It follows, therefore, A A Q Q = cos e s o I X AI = sinO o0 S sin0cos0 Q A 10 s sinos0= 20*0 - s e 4=-: e els sinO 2s and ^ A A o 0 2s A ^ e2s sin o = els (U) Using the above relations in (3. 27) may be expressed explicitly by (3.29) (3. 30) (3.31) (3. 32) (3. 33) (3. 34) (3. 35) (3.36) the components of the scattering matrix a A A A AAA -(Qo 2s)(Qe )s(.O) 21 CONFIDENTIAL (3. 37)

.CONFIDENTIAL -- THE UNIVERSITY OF MICHIGAN 8003-1-F A 1h (4. (AA. A 12"(,s' jo):k0 is2) ^2) A A A A A +(Q e )(Q e )(n (Qe ) (3. 38) o 2s s 10 2 s o A A A A A s21( 0,[1-& * 'fi l[k] | o 2s s 10(s S +(' els)(QS e20)s2(Q ) (3. 39) and AA 1 AA AA A A s22(Qs, Qo): (Q''e M.e )s( (Q'o' 22(. [ )2[k 2s s 20)s s A A A A J.) -( el)(Q e20)s2(Q S )} * (3. 40) o Is s 20 2 s o 3. 3 Dipole Scattering (U) For obstacles with dimensions much less than the wavelength the Rayleigh approximation may be used in calculating the scattering matrix. Roughly, when a static electric field is applied to an isotropic body, the-body is polarized. In the low frequency approximation (wavelength large in respect to the dimension of the body), the dominant terms of the scattered field may be approximated by the field radiated from the induced oscillating dipole. From the solutions of electrostatic problems involving spheres, spheroids, etc., one generally recognizes that a small body has three mutually perpendicular principal axes of polarization. In referring to a fixed coordinate system, these directions are denoted by n, n2 and n3, then the induced polarization caused by any incident field are given by P (E ni)aini (3.41) i=1 where ci are known as polarizability of the body (vande Hulst, 1957). (U) If E = E 1 + e (3.42) o El0 el E20'2 ______ 22 _ CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF 8003-1-F MICHIGAN then 3 3 -' A A + 7 A A A - p= a.n. ( ni. e)E + an ( )E -;ini( i e10) 10 I 1 i 20 20 (3. 43) (U) Since the far zone field due to an oscillating dipole located at the origin is given by (Stratton 1941), k2 E= - 47rC 0 1 ikr - e r Qs X(s xp) (3. 44) The components of the scattered field are, therefore, given by A E = E- e1 Is - Is k2 4= O47 0 ikr e A r p. els (3. 45) and A k2 E2s=E- e2 47r e *, ikr e - e2 r 2s (3.46) Substitution of (3. 43) in the above, yields k2 1 ikr E = -e is 47 E r 0O (e 10' ni)(ni e1)E10 and k2 E =2s 4r e 0 1 r 3 A A A A 3 e i(e10 o' ni )i e2s)E 10 3 -1 A+ ^i(Ai e2) JLO] (3.47) (3.48) The scattering matrix for small bodies are therefore given by ij( S o) 4 c o 1 r 3 ak(Ais I n)- eo) = 1I 1 I J (3.49).- - - I I I I I I I i23 i CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) For the special case of a sphere, a 1=a2 = a=a, one has SA eikr k2 lo e (e el) s o 47T f A A. (3.5 010 2s 20 2s (U) For most regular bodies such as ellipsoids, spheroids, cylinders, etc., values of polarizability are known, so that (3. 50) may be used to calculate the scattering matrix. For bodies of other irregular shapes, approximate values of a may be obtained by using the Born approximation which yields Va - VC6 (C-1) (3.51) where V is the volume of the body and 5 is the dielectric tensor. The limitations of such an approximation have beendiscussed by van de Hulst (1957). 3. 4 Physical Otics Approximation (U) When an obstacle of infinite conductivity is illuminated by an incident wave, currents are induced on the surface of the body. The scattered field can then be interpreted as the fields radiated from the surface currents. In general, the surface currents are not known unless one can solve the scattering problem involving the obstacle exactly. In the physical optics formulation, the following two physically plausible approximations are made regarding the surface currents (see Fig. 3-2). (U) a) For a body of finite size, a part of the surface is in the shadow region, where the field is small. Therefore, for a first order approximation one assumes that the surface current is zero in the shadow region. (U) b) To calculate the current on the illuminated surface of the obstacle, one assumes the local radius of curvature of the obstacle to be much larger than the wavelength. Under this approximation, the surface currents may be approximated everywhere by the currents that would be induced on a plane tangent to the surface. This approximate current may be expressed in terms of the incident mag netic field strength by ___ 24 I______........ CONFIDENTIAL

CONFIDENTIAL q I - THE' UNIVERS ITY OF 8003-1-F MICHIGAN \\\ \ \\ \\ \ //I da \ r r ShadowRegion Illuminated Region FIG. 3-2: GEOMETRY FOR PHYSICAL OPTICS APPROXIMATION 25 CONFIDENTIAL I

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F K= +2n xH amp/m2 (3.52) where ns is the normal to the surface. (U) Using this approximate value of surface currents, the scattered magnetic field due to any surface element is given by iklr-r I dH (= + r- n xH.(r x V e da (-r) 2. 7r — I- -— I~ 3.5 If Ir-_rlis large, ikIr-r ikir-r_ e __ e A Vs _ -ik..s (3. 54)* -r -rs I's A where Q is the direction of the scattered field. Thus, S iklr-r I -- -S H ik (3.55) dH, 2r s x H( x Qs da * (3.55) A (U) If the incident field intercepted by the area comes from the direction o, then (3. 55) may be expressed in terms of the electric field by the relations, dE (r)=dH r)x% - (3.56) Eo and -(rs) = x E. (rso (3.57) Using these relations in (3. 55) one obtains ikir-r I - I (ns Eo)Qs s -(ns ) oE)-EO da. (3.58) In terms of the horizontally and vertically polarized component, dE (re +e -dE dE )=esls2sdE2s (359) _______ 26 CONFIDFNTIA I

CONFIDENTIAL THE UNIVERSITY OF 8003-1-F and Eo(-) elOE10(r2 e0 20 s) MICHIGAN (3. 60) one finds that dEIs ike-r s else2 =-ik e d E 2j n.(Ax e20) dEs I ns (e2sx e20) A A A -ns (elsx e10) A A A2 -ns (e 2sx e10) E0 (3.61) E 2 A ^ A (U) For any arbitrary incident field and an extended surface, e10, e20, els e2 are not constant vectors, so that the integration of {3. 61) over the illuminated region of the obstacle is somewhat cumbersome. In most calculations, one assumes A A 4 that the incident field is a plane wave, thus the vectors fo e, e are constant 0 1 a 20a and = e (3. 62) E20(rs) E20J 20-s 20 Moreover, if the far zone approximation is introduced into the scattered field, then A 2 =r, (3.63) ikr-r I A l-s ikr -ik i2 r e r ~e s - e (3. 64) and the vectors e and e are constant. Thus, Ils 2s Els()A A 4 A AikrA -k ikr ns (elx e20) -ns (esx e10) da Ix 2rA 4 A A A E2s lt ns' (e 2x e20) -n s (e^2s x e10 region Eo10 ik. (Aos) 0El xe J -20 27 (3. 65) _1 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F Therefore the scattering matrix of the obstacle is given by A A 4ik S(mS, Q.)=- 2i S' s.2k ~-o'):^ ikr - (o da e A A 4 -n s* (e sxelo ns I(es e10) A A A -n.(eg xe n) (3.66) lit - v region (U) As an example, consider the scattering matrix of a plane defined by.a a b b -<x< a, -b<y<2 2 -2 2 -_ and oriented in the z-direction. The scattering matrix is easily calculated to be ik) A sinc [ S(a 43 a 43 - A sinc L(sina cos l-sina cos s s O o 2r 2 0 S sinc I (sina sing -sina sin 3)] X cosac cs(s -jO) -sin(jo-') ) cosaocosa sin(O -1 ),4cosa cos(f3 -3,, 0 S 0 S S 0 S (3. 67) where A is the area of thb plate and sinc y siny y 28 CONFIDENTIAL

CONFIDENTIAL ----- THE UNIVERSITY OF MICHIGAN ----- 8003-1-F IV GROUND REFLECTION 4. 1 Introduction (U) The reflection of waves by a rough surface such as the ground has been a subject of investigation by many authors. The approaches used by various investigators and their results have been summarized by Beckmann and Spizzichino (1963). In general, most formulations deal with the scattering of scalar waves using the Kirchhoff approximation. In the case of reflection of electromagnetic waves, the scalar formulation has been applied individually to the vertically and horizontally polarized components of the electromagnetic wave. A practical, explicit formulation which considers the inter-polarization coupling ( depolarization effect due to the non-planar nature of the reflection) has yet to be developed. (U) Only recently (Fung, 1966) the vector reflection problem has been formulated in terms of the vector form of the Kirchhoff-Huygen principle. In this work, however, a different formulation of the problem using the concept of angular spectra is given in order to study the polarization as well as the angular distribution characteristics of the ground reflected wave. Approximate boundary conditions using geometric optics are then used to deduce the reflection matrix of the ground. 4.2 Angular Spectra of the Reflected Radiation (U) The ground reflected wave may be considered as the radiation due to 'induced sources' on the ground as a result of interaction of incident fields with the ground. Thus, the reflected field satisfies the source-free Maxwell equations everywhere above the ground. (U) It is well known that at a single frequency o = 2 ir f (or each Fourier component of a time varying field), the solutions of the homogeneous Maxwell equation may be represented by two scalar functions. Those functions, 0(j) and ) (U) satisfy v2 j =0 (4.1) __ 29 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F where k =. (4.2) c X (U)To conform with the geometry appropriate to the problem of ground reflection, it is convenient to choose some reference plane z = 0 near the ground, and describe the ground profile by a function z =z (x,y). (4.3) s The reflected fields, therefore, exist in the region of space defined by z <Z <. (4.4) sIn terms of the two scalar functions, one may express the field in the following formE A 1 E = Vx (z0)+ Vx Vx(zVI) (4.5) and i H= - VxE= 1 v r Vxz0)+kVx(z )]. (4.6) - i/~o L (U) By taking the spatial Fourier transform of (4. 1) with respect to x and y coordinates, one finds that (r) and p(r) may be represented by ik x iky i/k2-kk2 z (U)=rdkXdk~ (kky)e x e e x (4.7) and i 2 r p ikx ikx y i/k2 — z (r)= dk (k, ky)e e e (4.8) respectively. In the above, we choose - Im I k 2 > O (4.9) in order to satisfy the radiation condition as z oo. Although, mathematically, the integrations in (4. 7) and (4.8) extend from -ao to +ao for both kx and ky physically meaningful solutions for the field far (several wavelengths) from the ground plane may be obtained by carrying out the integration over the range of kx and ky such ___ 30 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF 8003-1-F that Jkh-k-k is real. Thus, we may denote kx=ksina cos 3 k =ksinasin. y MICHIGAN (4.10) (4. 11) (4. 12) and -2-k2 = k cos a A where a and 3 are real angles. Physically, if we denote by Q the unit vector in the direction defined by the latitude angle a and the azimuth angle '(, then (4. 7) and (4. 8) may be reduced to more meaningful terms, ik~ ' df3cosa 4(a,!)e e r ) o, ik. r dpcos al(a, 3)e - (4.13) (4.14) (4. 15) (U) Substituting these equations in (4. 5) and (4. 6) yields 7 /2 2 E(r)=ik3 sin2acosada de I (a, i)j+ ie2 e (a, ])J e 2 c~~ ikaI r 0 and r/2 2r H(r)=7? ik3 sin2acosa dJda 0 0 A ikQ r di2 (a;, )-ie1 (, el]e (4. 16) where n - 1i20 0o V u 120 7r 0 (4.17) These expressions for E(r) and H(r) may be interpreted as the angular spectra representation of the radiation. It is easy to see that C1(a, ) = ik3 ( (a, )sin a cos a (4. 18) and '2(a', = -k3 3 (a, )sin a cos a (4.19) 31 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F are, respectively, the components of vertically and horizontally polarized electric fields of the radiation coming to a point r from a small solid angle d Q in the direcA tion n. Mathematically, one may rewrite (4. 15) and (4. 16) as A dE()= (a, el ei e - d (4.20) and dH(![) j(a, ()e (, 13 e( ' A d;D. (4.21) From (4.20) and (4. 21), it is easy to see the advantage of using angular spectrum in the characterization of the radiation. In the calculation of ground reflected radiation, the angular spectrum expressed in the form of (4.20) and (4.21) are functions of direction only and are independent space coordinates. (U) To evaluate the angular spectrum in terms of the boundary condition, let the ground be defined by A 4 r =xx yy +zz (4.22) -'S S S A and the z-component of the reflected field oh the ground is given by E (rZ) and HZs). Then, integrating the z-component of (4.21) and (4.22) yields A Ez(r)= — sincda dlsina e t (, 3) 12 a ' ikx ikyys ik —k z( dkysinma os e e e (4. 23) -~1 and ) Equations (4. 23) and (4. 24) may be considered a two-dimensional Fourier (U) Equations (4.23) and (4.24) may be considered a two-dimensional Fourier transform involving the functions G and G2. For slightly rough ground, where 1 2 Zg does not differ greatly from zero, one may argue that the inverse transforms.....,......, 32 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F are approximately given by.2 _ /" (" H(r) -ik r ( -k2 ost Hr - (4.25) (2ir) 1 and A 2 f C -ik 0 - r. ( -k cosa id E(r) (4.26) (27r)2 sina s s Thus, approximately, the knowledge of the z-components of the electric and magnetic field strength of the radiation determines the angular spectra completely. 4. 3 Ground Reflection Matrix (U) From the results of the last section, it is seen that a study of the ground reflected wave may begin with the knowledge of the z-component of the reflected electric and magnetic field at the ground, or at some reference plane. In general, these two components are not known, so that approximate evaluations, or even direct postulations concerning these components (in the case of randomly rough surfaces) must be used. Three possible approaches to the estimation of Ez(rs) and Hz(rs) are given below. a) The Layer Approach (U) Borrowing the idea from the random screen approach for wave propagation through the ionosphere, we may postulate directly the phase, amplitude and polarization variation due to a plane wave reflected from the surface layer of the ground. Such a model has been used successfully in ionospheric diffraction, but a great deal of measurement is necessary to determine the parameters involved in such a model. b) The Multiple Scattering Approach (U) Assuming that the surface of the ground is composed of a random distribution of scatters of appropriate properties, the reflected waves due to some plane wave incident on the ground may be calculated by the method of multiple scattering. Such a model has been used successfully for the transmission of solar radiation through the atmosphere. However, the correct model of the scatterers, and meaningful solutions for engineering use are both difficult to obtain. 33 CONFIDENTIAL I

I CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F c) The Reflection Approach (U) Assuming that the ground is locally smooth so that the law of plane wave reflection is applicable locally, then for any ground surface defined by zs(Xs, Ys), the reflected waves may be obtained. A postulated statistics of the surface (and the electric properties of the ground) would then suffice to specify the problem. (U) In the present work we shall follow approach c) due to its successful application in scalar wave reflection. (U) Referring to Fig. 4-1, the radiation from a source located at r (xo, yo, 0) is reflected by the ground surface defined by Zs=Zs(xs, y). (4.27) At any point of reflection as indicated in the figure, the radiation appears to come A A from a direction fo. This Qf is given by l 3(Xs o)+y(ys-yO)+z(zr-zo) -xsin cos 0+y s~inaOsinJ0+zcOsaO (4.28) where CosaO S S S 0 (4. 29) and x -x cos~ s _ (4.30) o r-r sina - -0 0 The incident radiation, except for a phase factor, may be represented by A 4 ES(r )=E e1+E2 e0 (4.31) where A A e x z A e 0 =x sin -ycos (4. 32) 0o-= sina oo 0 = f x e =x +ycosoo +yoa sin -zsino (4.33) o 10 o (3) and E1 and E20 are the complex amplitudes of the vertically and horizontally polarized components, respectively, of the incident radiation. 34 ___________........... CONFIDENTIAL I

CONFIDENTIAL THE UNIVERSITY OF 8003-1-F MICHIGAN Source r(X(, Y, zo) A rP n Tangent Plane Point of-Reflection rS [X8., Y s(X8, ys)]. FIG. 4-1: GEOMETRY FOR REFLECTION APPROACH - I [[ I [[ IIIll ] [ [] [[ --- 35 CONFIDENTIAL I II l I I II I I I I I I I I Iiiiii i iln -j

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) At the point of reflection, r, the normal to the surface is given by a Qzz az therefore, if Go l< 0, 01 the incident field is reflected. The reflected field is now assumed to be that reflected by a local tangent plane at rs. The direction of the reflected wave is therefore A A AAA r = n2-2n1(npi). a(4. 35) (U) In order to express the reflected radiation in terms of the local reflection coefficient for plane waves, one may resolve the incident field in the direction of perpendicular and parallel components. Represent these directions associated with the incident radiation by A e X 1 - oX1l (4.36) o1 )txfll sin l A A e/ = o xe (4. 37) otl 0 OrL where cos y=- -n =-[osa Cos +sina sina cos(P -1 (4.38) By simple rotation of coordinates, one may have E = (eo e1 )E O(e eO) E2eo 01) oIl 20 O e (4. 39) Similarly, the reflected field at the point of reflection may be resolved in the parallel and perpendicular directions of polarization defined below/. x n A r 1 A er 1 e (4.40) r.L x_1 0 r I CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F A A A +4 A e 1Oxe =e +2nxe cos y r// r ri. oI 1 o.L (4. 41) The fields of the reflected radiation at the point of reflection are therefore given by ) 1 L )Eo+(e e202 -RLe+2nlxeoL cos -/ (eo/o elO)E10 /; e20)E21O (4.42) and AlB AO ^ A [ 1A Al A eH(r)= l ~ 'e10)E 10+(eoj e20)E2J e 1+2n xecoy =LB ol 0 1 20 0/11 ~ An A o A,a +nT Of,, eoLeo elo)E1+(e0)eo OJ l f/ 20 2 (4.43) where Rj and R// are the plane wave reflection coefficients, given by / 2. 2 Rj= cos y+JN2-sin (4.44) and ( E cos -) N- sin i R N2i =.N2.2, where N is the index of refraction of the ground. (U) From Eqs. (4. 42) and (4. 43) it is easily seen that (4.45) -H (r ) z-s rlo 1 sina sin2 O' RI ac+R//b2 R//ab-R//bc R bc-R// ab R b2+R//ac, w Elo(rs) (4.46) E20(! -E (r ) z -5 J where a = cosa l+cos 7 cos a b = sin a sina1 sin(31-' ) O1 0 37 - CONFIDENTIAL (4.47) (4.48) - i~~~~~~ iiii i i iiiii!i

CONFIDENTIAL ----- TH.E UNIVERSITY OF MICHIGAN 8003-1-F and c = -cosy cosa +cos a (1-2cos2. (4.49) Introducing (4. 46) into (4. 48) and (4. 49), one finds the angular spectra of the ground reflected radiation to be related to the incident radiation (approximately) by the following, (a,l 2 a r f -ikQ_ r 1'1 k2 cos a fd e (2f (Qa ~ =(27r)2 0sin sin2a 2 R ac+Rb2 Rj.bc-R//ab E (r) (4.50) Rj.ab-R//bc Rj b2-R//ac E20 1s)) (U) If the incident field is a plane wave, then E (r E A 01s] 01 ikr (4.51) e (4.51) E02(rs) 02 where E and E02 are respectively the complex amplitudes of the electric fields the ground by a ground reflection matrix such that A R] (4.52) (U) This reflection matrix is then given by 38 8 CONFIDENTIAL CONFIDENTIAL

CONFIDENTIAL - THE UNIVERSITY OF MICHIGAN -- 8003-1-F.2 F- l k Cosa ( 2 sina sina (27r) o ikr ' (% -Q) dx y e Is s 2 J sinm Rlac+R// b2 R.L bc-R//ab - Rjab-R//bc Rj b2+R/I ac (4.53) (U) For any given definitive ground profile, this reflection matrix, in principle, may be evaluated by integration. In general, however, due to the uncertainties in the exact ground profile configuration, one has to use statistical approaches, and obtain the statistical description of [R]. The statistical study of the reflection matrix LR] shall be treated in Chapter V. -- 39 _____ CONFIDENTIAL -

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN --- 8003-1-F 4. 4 Reflection. Matrix for Plane Ground (U)The approximate form of the angular spectra for the ground reflected wave and the reflection matrix discussed in the previous section is basically deduced from the geometric optics approximation. In order to investigate the accuracy of such approximations, the limiting cases of reflection of a plane wave by a plane ground are carried out using this approximate formulation for comparison with the known results. (U)For the case of a plane ground, cos a1 = 1 hence cos y = -cos a.2 a = c = sin a 0 (4.54) (4.55) (4. 56) (4.57) b =. e Equation (4.53 ) then becomes A A 2 cos a sina ikr (o-fC) R1= 2 dx dy e (27r) 2 sma s s - RI/L. R//-j (4.58) For an infinite plane, one,notes that A A 1 /oo ikr ' (o0-2) -2 f"cs fd y s (27r) 2- -00 01 O s/ ikxs (sino-cos ~-sincos s -2 dxs f dys e (27r) 2 - O ikys (sina sinn -sin sin).. - n - e = 6 Lksinocospo-Kx 6 LksIosinpo-kyJ 1 (or-74 )6(3p-Bo) k sinacosa (4.59) ----— 40 - CONFIDENTIAL I IIII

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F Thus sina [R] = 2 6(a-7r+)6(P-f). (4. 60) sin a This indicates that the reflected wave is indeed a plane wave given by A A -refi. (r) =R e10E10 e E20 ik2e (4.e61) A where 2r is given by the angles (nr-a), B (which of course is the specular direction of reflection). Therefore it is seen that the present formulation reduces to the known form of specular reflection in the case of an infinite plane ground. (U) In general, if a transmitter illuminates a part of ground defined by an area AT, then the contribution of the transmitted radiation to the angular spectra of the reflected radiation within the geometric optics approximation for a plane ground.can be expressed in terms of the reflection matrix A A k2 cosasina' RL 0 i C -] 2 sina 0 0 se AId AT osasi F( ]. (4.62) sina R//J For the special case that AT is defined by the region a a b b one finds that (=A L kasin ^ F(, )=sinc (sinacos,-sinacos) sinc (sinacosp-sinacos) (4. 63) (U) In terms of this reflection matrix, the ground reflected wave is given by,. l-. 41 CONFIDFNTIAI

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F k2 ~/A A i2K. (27r) 0 0 1 0 El0 L OR 0 (4.64) ~ R/ E20 In the far zone approximation, if a receiver is far from the illuminated region, and whose coordinate is given by fQsr, then one may use the asymptotic expression A -ikr ikr2/v 27ri (A A e 27ri A e e - k '6(+ ) — k 6(n-2s). (4.65) Thus, the scattered field observed at a point far from the reflection area is given by ikr sina cosa ^ A ik e AT s F(s o)elsRE10+R/E (4 66) refl. 27r. r sina +eRE (U) From (4. 66) it is easily seen that the scattering matrix for a plane ground, within geometric optics, is given by S.-ik sina cosa Ri O Es] A - F(Q ) (4.67). LO R//J In the case of a planar rectangle, this becomes =ik kb [s] = 2A sinc [2(s!inaocoso~-sinascos-s sinc[L-(sinaosinO-sin0 sin -cos)] so1innc (4.68) cosa sinm 0 ~0 0~ sina R// Lo - ---— 42, CONFIDFNTIAI

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) This equation does not agree exactly with the result of physical optics as given by Eq. (3. 67). The primary reason for the discrepancy is due to the fact that the surface current assumed in the physical optics approach is inconsistent with the result predicted by the geometric optics approach, using local tangent plane reflection even in the case of infinitely conducting flat planes, as pointed out by Fung (1966). Insofar that both approaches are approximate, it is difficult to argue precisely which is the more correct one. In the present work, therefore the geometric optics approach and the resulting ground reflection matrix given in the previous section shall be used throughout because of its relative simplicity. *,^ It is to be noted that the scattering matrix given by (4. 68) contains singularities in the normal direction as=0. For distributed radiation, where one has to carry the integration of angular spectra over all directions, this singularity does not cause any trouble in the integration. For the calculation of scattering matrix, this singularity, caused by the simple formulation of using z-components of the fields, may be avoided by an alternate formulation\using the tangential components of the electric field. This alternate formulation is given in Appendix B. 4 -,,43 CONFIDENTIAL

I CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F V REFLECTION FROM RANDOM ROUGH SURFACES 5. 1 Statistical Averages (U) In general, most surfaces such as terrain, sea surfaces, etc., are either too complicated to be characterized by a simple contour, or are time varying such that characterizing by a single contour is inadequate. For such surfaces, it is common practice to consider them as random rough surfaces specified by their stochastic properties. For a random rough surface, the contour of the surface may be represented by a random function in two dimensions, such as Zs= (sys). (5.1) Since each element of the reflecting matrix is a function of e and its derivatives, it is therefore necessary to consider the reflecting matrix as a random quantity. In most practical cases, based on physical grounds, one may assume the function e to be second order stationary. In such case, a statistical investigation of the reflection matrix may be carried out by evaluating some of the statistical average quantities. (U) To carry out formally some of the, statistical averages, consider each element of the reflection matrix given by (4. 53). These may be written as k Cos a ikx (sinaocossina1 cos 1) 1i (27)2 sinxa sina iky (sinao sin3 -sinx sinC3 ) dys C.. (5.2) where { i,iJ = 1,2 and the coefficients ik(cosa -sinc) c1 si e c [Rac+Rl, b (5.3) s4i 7 2, ---II —44 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F 1 ik(cosa -sina)g c 2= 2 e [~ Rlbc-R//ab] (5.4) sin 7 1 ik(cosao -sina)cRl = -e Rab-R//bc (5.5) sin ' ik(cos a -sina)_ C22= 1 e 0 [Rlab+R,,b2] (5. 6) sin are functions of the random variable e. Explicitly, if one denotes the partial derivatives of e with respect to xS by ~ x and the partial derivative of e with respect to ys by ~y, then the quantities r -1/2 Cos_ =i [+ + ]y (5.8) sinaisinI31=-[2+g21 by (5.10) contained in C.. are all random variables. Functionally, therefore, i j (xs (x y] (5. 11) (U) The mean reflected field and the correlation of the field components are therefore dependent on the expected values [R.i.] and&[R. '7. Here, we use the symbol [x] to denote the expected value of the random variable x instead of the conventional E to avoid confusion with the electric field. Formally, one has k2 cos a ik(sinacos -sinacosx & [Rinj = )2 sina sir es I ik(sina cosf -sincsind3)y e 0 0 dy C. (5.12) 45

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F and 42 r ik(sinr cosf0 -sin(cosfj)(x,-Xs) ckos a 1 r 0 0 s (2r) sin a sina 0 f iik(sina s sin3-sinasin)(ys-ys') dy dysI e f Cij( x' y)Ci( ' ' (5.13) where, for simplicity, the prime is used to indicate that the function, etc., are evaluated at x, Y'. For example, ' s ' (x, y). (U) Equations (5. 12) and (5. 13) can be somewhat simplifed from the assumptions that e is a second order stationary random variable. In (5.12), since[ C..i is not a function of x and y (due to stationary) one may carry out the integrals involving xs and ys separated. Just as in the case of plain, smooth ground, one may carry out the integral over the area of illumination, and denote k2 cosa ( f ik [sir cos -sicos0 1 2 ' ' dx dy [eiJ 2 sina sin r s s (2er) 0 ik [sin sinJ -sinsin drea of F ] illumination e -A F (,). (5.14) where A is the area of integration. Thus, one may express E [Ri= AF (A), [C.J] (5.15) (U) In (5.13) if one denotes that y x -x8 =, ys-yS = 'r (5.15) sN y ___________________ 46 __ CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F due to stationary assumption, functionally, one may denote 95.19( W ('rx'r5).'I (5.17) ij( x y)Ci } x y ij, ij' (5 x Y) Thus (5.13) may be reduced to 4 2 Ri..'1- k cos a dx dy 4 '2.2 (2ir) sin a sin a s s o r iik(sina cos3 -sircosT3)r dT e x l ik(sina sing -sinasinM) e 0 0 ('x ) y eij, ij' x y (5.18) (U) The evaluation of the integrals involved in the average values of reflection matrices for any arbitrary stationary random surface is extremely complicated. Reduction of these equations for a relatively simple case of a slightly rough surface with gaussian statistics is described in the following section. 5.2 Slightly Rough Surfaces (U) The expressions for the average values of the reflection matrix is not useful in practice unless some approximations are introduced so that one may have a scheme to evaluate these integrals numerically. Some approximations introduced in reducing these equations are given in the steps below. (U) A. The surface is slightly rough, so that in all the expressions, it is only necessary to retain terms up to an including the second power of x and. Thus, from 2 1/2 ni- X+y 2 [^X(- y- ] (5.19) one may have approximately n [-Z+ 4y-X -y Z -( ) (5.20) 1, j 2 x y Within the same order of approximation one has -4 ---,7 ____.___..... 47. CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F + cos a -sin a cos 2 +cos2 a -sin a sin -2 sin aocosjosinjl3xy. (5.22) and v2 c sin a -3cos sina o ox-sc2 osa osi osin sm + O[5 cos — 2s cos [ co - -2sin aCOsin -4sin C+sina cosj3Sx~y ^~ (5. 25) (U) B. The index 0f refraction of the surface is high, i. e. yNI > 1. Then ~.2 one may have 2 2 cos2 2 R-s a cosac -cs +siasin o gy 2- (526) 0 2 r o o X R ( Ncsa nsinacooo-3Csa sir sin(527) the incident wave if the reflecting surface is smooth, i. e. I = 0. (U) With these approximations, one has, + R-cos ac - +sina cos a - -2sin2 ( 2 o NL 2 x o oy 2 -4Rm( oosa L cos ox 2y (5.27) e B. The index 6f refraction of the surface is high, i.e. IN=0 >>. Then on e may have U)With these approximations., one has,.____________________ 48 CONFIDENTIAL

H,Al i4 cos%-cos 2z R sino)asin a c2s..2. 2 (a0)sa -sina (sin a 2c aIcosp3M L 0O N o o ox 2sina 4an csco~ F. 2 2~ CDoi22 -L 2iacosoa s (sina -2cos% )Js14a + 4sinCos~aosjC F 2os.2y Nco% Los(2sin a+1)-iJ CA a0sbf I l sin ina 0 2 2 2 o 2 0Z 0N — 2cos a +1 +(2sin a +1)cos A - I 0 L2 o 0 0 2 2cos a + na+ Lsinc2a 2a+1~+(2sin a +1)*2n + os 2 0 N 2 0 0 o JJy Cos a (5.28)0 z

- a m 0C k 2 2 12 2 Z | =e [2sina +-Rtana (1+Cos a )sing L -+2smna + tana (1+cos a0) cosj 21 p - (1+c +- tan sin tana s o+ tNan2%0(cos2O-sin23)y cos 0 x 2 2 cos tan a sin o cos /3 ~ + 0 o 0o 'y z C,) Z P 0 m r — ~o FT to Z - r (5. 29) 4 P-4 0

. H rm zM o ik cosao -Cos] 12 Ncos a 0 0I r 2sina cosa (cos2a L~ + N cos a jC0SI3 cU1 2 2 21 (sinla -Cos a1 )(Cos _3 sin A3) i 00 0 Oj~ y 2 2 2- 2 2 22 2 22 — tan ao(sin a -cos a )sin13 cosj3 9 +-tan a (sin a -cos a )sn( cosf3P N 0 0 0 0 o 0x N 0 0 0 0 0oy ni z CIO) 00 - 0c I0. 0 (5.30) z

CNIDENTIAL IVERSITY OF MICHIGANI THE UN 8003-1-F x 0 0 Ca eq~ 0 C)4 eq 0 a0 U) C ) 0,U) 0 CT) 0 0 eq U) 0 LU' LO' 0 a t 0 0 C) C) 0 0 U) 0 (a 0 U)s 0 eq t 0 a 0tj +0 L+J LO' eta, C)Q 0 0 0 elq 0a 0 0 0 0 0 C) cmJ 0 C) 0 C.) 0 eta, eq "4 U) eq0 0 C)4 0 cm z r") CY) 52 CNIDEN)TIA

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F To simplify notation, one may write c 1 expik(cosao-cosaQ] [R (aO)sin2ao+dlcosPo X ty+d smoy i+d2 x y+d4] (5. 32) 1 oy 2xy 12 exp ik(osa0- -cos [clsinB g -c cos3 (5. 33) +C2xgy+C3e2+ 42] (. c exp[ik(cosacos)] rblsinoSx-blcosPy 2 3 + (5.34) c2 exp ik(cosa +cos)] [R,(a )sin a +acoso 22 0L 0 I 0 X +alsinj3 gy+a 2gxy+a3 x+a4gy (5.35) to show the explicit dependence of Cij on the random variable g. The coefficients ai, b., etc., of course can easily be identified from (5. 9) through (5. 12). (U) C. In order to carry out the expected values of Cij and their products it is necessary to use the joint probability distribution densities of 9, I,I etc. In order that all these high order joint probability distribution functions can be written down with relative' ease, one shall assume that the surface is isotropic and obeys gaussian statistics. For such surfaces, the statistical description can be expre ssed in terms of one correlation function defined by B(r)=<f [ (s+r)?(s)] (5. 36) where s, s-Hr are distances measured along any straight line. For such surfaces the statistical average for the variables involving,, e and theirproducts can, in x y principle, be carried out explicitly. In Appendix A explicit formulas necessary for evaluating the expected values of Cij and their products are deduced. (U) Using these approximations and the formulas in Appendix A, the statistical averages of the reflection matrix can be written down explicitly. 53 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) For the expectation value of the reflection matrix, one has R R12 2 11 12 ^ k2(cos a -cosa) B(O) 21 R J=AF( Qv, — ) exp-2 R21 R22 0 R. (~B)- B(0) (d +d ) sin2 o3 4 B"(0) (b +b ) sin2a3o B"(0) - (c +C ) sin2a (5.37) (U) For the products involving C.. Cij, if one represents 1j 1jI y -y = T sin T, (5. 38) then their expectation values may be expressed in terms of T and 0. Some explicit results are given in Eqs. (5. 39) to (5. 42) inclusive. Due to the systematic representations of C.. given by (5. 32) to (5. 35) it is easy to write down the expected value for the products'between any other two C.. 's. For example,s [cl1c* I 1] 11 1.J can be obtained from e C *c by replacing R,,( ) and a. \with R (a) and d. R22 - ) 1\. 0 i 54 CONFIDENTIAL I.

-n z-I 'U> sin a R (a) R (a )sin 2a -BT (O) (a) (a +a )+ R (a )(a + a)] } 2 * 1 + ik(cos - cos a)B'(T) sin a cos ^ cos IR (a) a* + RR (aa) a ] + ik(cosa-cos)3l'(T)sin in s an0 R() a + R (a)al + sin 0 cos 0 k (Cos cosa)2 Bt (T) [R (a )sin a a2 + R (a )sin a a + 2a a sing cosIpl + 2a a* sin cos F - ( T)} +COS k(cos - cos2 a B() ()([Ri sina a + R<)sin aa+a aa cos)(] z 0O CA z- ~ H z - a a* cos2, "(T) - a a sin2, (7)} 2 2 2 --- 2 + sin k (cos o -- cos a) B () T O [R 11(ao)sin a 4+ R 1 (o)sin aa4+a1 a sin2 0] w - ala sin2 B" ( 7 ) - a1 a; cos2 } -1 1 0 0 (5. 39)

[c2c] eP{k(Cos a ' cs)2[B(O) -B1(Tr)] 2 * * {- ()R1%)inab3+ b) H +I ik(cos a -~ Cos a)B' I ( )snaCos Cose 0 R (ca 0 0 0 T 11r 0 I' zI) zT ~ 2 +k k(cos a -4 COS a)Bt3(T)sin a~ sin3 sine Rll(a) b* 0 0 0 T 10 1 sin0CS Jk2 (COS a- COSa)2 B'T FR (a )sin a11 2abCO i sin cos 0 1 0 2 + ab 0o 0 i z ~iQ O ~ W 0 m ~II C) 1 1 0 s3 2-2 r 2 * * + Cos20{k2(Cosa Cosa) Br IT) LR (a )sin a + a b~ coszj -a b* COS T " r b b* sin23 T) II 0*i 073 2 2 2 2 r2 * 4 21 + sin 07k (Cosa -.Cosa) $' (r) [R1 (a sin ab +a bsi 11 0 0 4()} 0 z (5. 40)

4 [ c 22c11 exp -k (cos a cos a)2 r(o -B(T)]} x {sinR2 t (a ) R* (a sin2 B() [R )d* + d* )+ R (o)(a3 + s! 2, 2 * * 0 0 1 4 +ik(cosa - cos)B'( 'T) sin2a sine sin 0 l(a)d R (a) a.0 0T, + sin0 (COS- COS)2 B() Rll(a)sin2 d2* + R(O) Sin a - 2a d* sinOS cos B + 2a- d1 * sin3 cos ( - L 2 fk,,2 sine + osi 0 ck2( osa - cosa)) (R' () ) sinad4+ Ra sin a4 + ad sin + o ) o) sin2 2 + - 2a d sin,8 B" (T) - al dl sin c o - () o o 22 2 '(7)2 H C) 1rT1 - -1 loZ oH "-I m-4 o CI Z 2L )m

CNIDENTIA I-THE UNIVER SITY OF MICHIGANI 8003-1-FI m 020% I. s001 02?I m 0 LJ 1~. 0 op4 02 kC) ~41 0 C)L 0 02 I 0 C)C 02 0 Ca ~0 + 02 C)L 0 02 eq 02O 0 C) tO; i 58 CNIDETA

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F (U) In general, to carry out the integrals involved in (5. 13) one may represent the function g...,= C. gij, ix' y)Cij' ( ' x' y by the form gij ij,(T,T y)=exp -k (cosa -cosa)2 [B(O)-B(T x [Ao(T)+A (T) Ccs +A2(T)sin0 +A3() sinO cos0 1 T 2 T T T +A4(T)cos20 +A5(T)sin2O] (5.43) where for each set of ij, ij', the functions Ai(T) can be obtained from (5. 39) - (5.42). Using this representation the correlation between the elements of the reflection matrix may be represented in the integral form ' =R k4 2cos, 0 R._R_ COS A TdT G jkjk 3 A TdT j1(2 r)3sin asin J 0 JO(T) 2A (T)+A (T)+A (T7 -2iJ (kr)A () cos 0 o 4 5 1 1 0 -2iJ1(kTc)A2(T)sin0O-J2(kr)A3(T)sin 0o -J2(kT) [A4(T)+A(T] exp -k (osa-cos) B(O) -B( (5.44) where A 2 2 k= kin a +sin a-2dinasina cos(P3-fo) sinacosp-sina cospl cose = sin a +sin -2sinasinc cos(P3-fo),0 0 0 and Jn 's are Bessel functions. (U) Thus, within the approximation of slightly rough, high index of refraction, and guassian statistical description of the surface, the statistics of the field reflected from the surface may be obtained by carrying out a set of integrations. If a 59CONFIDENTIAL CONFI DENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F suitable model for B(r) is chosen, such integrations can be carried out at least by numerical methods. 5. 3 The Shadow Effect (U) The statistical description of the reflection matrix given in the previous section neglects the effect of shadowing, i. e. a part of the ground that is being shadowed by other parts. These parts therefore have negligible contribution to the reflected radiation. To the first order approximate solution of the shadow effect, Beckmann (1965) introduced the notion of shadow fraction, i. e. the average fraction of the nominally illuminated part of the ground that is actually being illuminated by the incident wave. In terms of this shadow fraction, the reflection matrix such as derived in the previous section can then be corrected appropriately. For a more sophisticated treatment, the statistical averages given by Section 5. 2 should be carried only over the illuminated part of the ground. Hence a study of the statistics of the illuminated part of the ground becomes necessary. (U) Due to the complicated nature of the statistical problem involved in the shadow problem, even the estimation of the shadow fraction becomes controversal. Beckmann (1965) derived an expression for the estimation of the shadow fraction. Due to some oversight in his statistical analysis, his result does not seem to be correct. Brockelman and Hagfors (1966) computed the shadow fraction by computer simulation and found that their results do not check with Beckmann's formulation. In this section, a general study of the problem of shadowing is carried out, but due to lack of time only some preliminary results are reported. (U) In the last stages of this contract, the report by Wagner (1966) was made available to us. In that report some preliminary results on the theoretical investigation of the shadow effect are given. Our approach seems to be different from that work, but it is interesting to note that both approaches yield the same first order correction to the shadow factor. (U) A study of the shadow effect can be carried out by generalizing the classi cal approach of investigating the zero crossing of random variables (Rice, 1954). Referring to Fig. 5-1, consider the cross section of a random surface given by a g —r 60 - CONFIDENTIAL

CNI DETAL. I THE UNIVERSITY OF MICHIGAN 8003-1-F '-4 '-4 '-4 t PZ4 Fz4 0 H 0 Fx4 ~H 0 '-4 & I 0 411 0 0 t uA - mmmmm 61 CNI DN A

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F curve. If a ray inclined at an angle 0 to the vertical (g-axis) reaches the point (Toj O) i. e. the point (To, 0o) is not shadowed, this ray must not intersect the 9-r curve at finite T. For simplicity, this ray shall be hereafter referred to as the ray a. Mathematically, for a ray not to be shadowed in the range (T1, rT+dr1) (T1) = 1 < go+(T 1- To)t o (5. 45) This condition may be written as g1,la <90-T a o(5. 46) A where for simplicity, we denote a = cot a. If one introduces a new variable y _A (T)-Ta. (5. 47) then (5. 46) may be reduced to Y1<Yo - (5.48) In terms of the random variable y,the fundamental problem of shadowing therefore may be reducedto: giveny=yo at T=TO=0, find the probability that y in the range of T given by (T1, T1+dT1) crosses the level yo. For the crossing to occur, one must have for small dT1, Y1=Yo (5.49) y; > 0 (5.50) T=T and dyl=yj dT1. (5. 51) If one denotes the conditional probability that given y=Yo at T=To=0, Y1 y(T 1) < Yl+dyl and y dy < T <y + d 'by. by f(y1,y)dyldyd II________.62 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F then the conditional probability that given y=yo (or g=go) at T=To=0, the ray a would cross the level go in the range T1 < T < T+dTl is given by (. 1 / 5oo)d =f1 f(o', Y)Y dy1 dT (5. 52) *0 which of course is the familiar result of the crossing problem. (U) If 9 is a second order normally distributed random function, then joint probability density distributions such as fogo, g ), f1(g, go, T)......... ofn(go',go, ~l,l... nJnl T2 *. In), ete for en <(Tn) < Vn+dn, n< ' (Tn)< n +dn-, T1> T2 >T3.... >T can be formally written down. Then the conditional probability density that given go, 'o at 7-=o' 1K+dg1 >(T71)>1' e1 +dE1 > (71)> e is I O l(go.$0 go J 1 v 1) fl/fo (5. 53) since by definition, Y1= 'i-Ta, y1 = 1 -a, it is easily seen from (5. 52) that given o, gO the conditional probability that the ray a is shadowed in the range T1 < T < T+dTl is gl(Tl', a, /o.,O )dTl 0 dT1 f1 (goT 1a). (Y), Ty Y~dy (U) Similarly, given 0, Sg the conditional probability density that (rT1) is in the range ( 1, 1l+dr1), (T2) is in the range (2;, 2+dT2)...., etc., is fn(o' go'. 1 * * n ' n.T2 T) fn/fo * (5. 55) 63 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F Following arguments similar to those used in deriving (5.55), it is easily seen that given go, C,' the event that the raya would be multiply shadowed at(T,rT +dr) ),(T2+dT2).... (Tn+dTn) has a probability (n 1T 2l... n' )00 dT.d2 dT where gn(T 2** ' Tna/o' o )= / yldyl Y2Y2 * *y yndy 0O n 00 0 no '(o +~0 )r (Yl+a). o+dT ), (yIan T'12, T2 n*T1 (5 56) no 0 n n (5.56) (U) Using (5. 55) and (5. 56) and following Lonquet-Higgens (1962), it is then obvious that given 9, ~ the event that a ray from g is being shadowed the first 0o0 0 time in the range (T1, T +dT1) has a probability given by the following infinite series g(Tr, a%,l o,)dT g1 ' a/ )d T1 - fdT2g2(T1'?2' a/o' o)d 1 1 T2 2 23 3 020 3 1 0 0 +........ (5.57) (U) In terms of g, one may deduce that given, the conditional probability that the ray a is being shadowed at all is 0 P (shadow/(, eo)=| g(T,/(Oo)d 1 * (5.58) JO It follows therefore, that given g, the ray a illuminates the point g is 00 0 ____N__ 64 _ CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F Pi(illuminated/So, o)= -P (shadow/9,? ). (5. 59) 1 0 0 S 0 0 The probability that any point on the surface is being illuminated by the ray a, therefore, is given by s(a)= ( d ( d' Pi(illuminated/o,. )f(,o) (5. 60) -ao 0-a which should be the correct form of the 'shadow fraction'. (U) Equation (5. 59) also yields some statistics of the illuminated region. Since P.(illuminated/gO, 9o)fo0(o, o) is the joint distribution density of height (go), slope ( o) and the event being illuminated by ray a the height and slope distribution for the illuminated region is evidently fo (illuminated) o/ill. )= (/) (5. 61) (U) Owing to the complicated multiple integrals and series involved in the expression (5. 57), the rigorous, formal solution of the shadowing problem, is difficult to apply. In order to obtain some acceptable numerical results, approximations simplifying the numerical process seem to be necessary. The simplest types of approximation involve the use of approximate joint distribution functions f or f. For a first approximation, f f (oE ')fo('1g, ) fn(n (5.62) n 0o000 1 nn n which neglects the correlation between height and slope distribution at different values of To. For a second approximation, one may take into account the correlation between two adjacent values of T, so that '- f (o 1 ' l ) f(l (5 63) I — f —' -o 1.- '- 2 n n - l ' n - - 9 n5 n) f= (5. 63) _____ 65 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN -- 8003-1-F Such processes may be continued indefinitely. The advantage of such an approximation lies in the fact that the distribution densities are now approximated by aproduct involving factors with less variables, and thereby simplifying the integrations involved. (U) As an example for the first approximation, one notes that n f f k=1 n r 2 2' T 1 k _k _exp(5. 64) n= I 2r B'(0)B[ 2B(O) 2( 21B'(BI (5. k 1 It is easy to verify that / gn '.y... r 2 2 _n0 / dy1 yk exp L 2B"(0)I = IB"(0)exp [ 2 (O1 -!B"(0X ^ erfc [T a ] (5.66) a /2 IB"(0)f so that g may be expressed as n (2)"n B) e - V erfc V e k(5.67) (27n ~ - o2 k=1 66 - CONFIDENTIAL I

CONFIDENTIAL THE UNIVERSITY OF 8003-1-F MICHIGAN - Therefore, from (5. 57), (5. 58) and (5. 59) one finds that P.(illu. /~o, )= 00 GzO 1+n=l n=l f)00 Tr1 I n-l (-1)n dT1 dT2... '0 "07 1* dTn gn (5. 68) (U) To find Pi, one has to evaluate the integral o ln T J1 drl Tn-1 TT -(o+k) / 2B(0) dT dTn k= (5. 69)n k (5.69) If one denotes 'T -(oh)2/2B(0) F(T) = e /#0 d' (5. 70) so that - (9o+r)2/2B(0) F'(T)= e (5. 71) Equation (5. 69) may be evaluated easily. The result is JI o n-0 F'(7-)dT1 -j O 1nF'(7n)dTn n - n - F (a))] Zn (5. 72) Now F(oo) '0 -(o+ra) /2B(O) e d = B(0 erfc 2 a 2B(0)] (5.73) Hence, one has I IIIII I II II III I I II I I II I II L II.....N67... CONFIDENTIAL.I I I I I I I I I I II II

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F P. (illu. /90.9 g') 1 0 = 1+ I(-1)n / n"- =L _ _7 -v2 { 2/2Ta' U -/; erfc n erf /2B() -exp eV () (e-exP 2 2r a = exp -A erfc ~ 1 t /2B(0).J -/'FV erfc V)erf:o0 /2B(0) (5.74) (5. 75) where V2 A e -Jf V erfc V A= 4/'F V Using (5. 74) and (5. 60) one may express the shadow fraction as I1 OD OD s(a)= d9 I d9 0 - 0 OF -.a OD exF 2 e- 2B(0) - 2' 21B"(0)I -A erfc 2 7 r V9 (- U) A -BI 1-(-O -) -F (5.76) Equation (5. 76) can easily be integrated by using these two relations deo 27 IB"(0)l exp U. IOD 1 - Pr v -x2 e dx= [1 + erf V] 68 CONFIDENTIAL

CONFIDENTIAL THE UNIVERSITY OF MICHIGAN 8003-1-F -and / OD ~ oD /d p 2B(0) -A erf L exp-x2 -A erfx dx] -o * ( 2B<) f2B( J -oo =J -/expE-Aerfx] [erfix] dx = 1[-e-2]. Using these relations, one finds that s(a) = [ + erf V]l - e. (5. 77) (U) This expression has been derived by Wagner (1966) and was shown to check closely with the simulated computer results of Brockelman and Hagfors (1966). In this report, this result is obtained as a first order approximation of the rigorous result, expressed in the form (5. 57 - 5. 60). Higher order approximations to the shadow fraction within the present formulation, in principle, seem to be possible. Due to lack of time, these are not completely carried out. 69 CONFIDENTIAL

THE UNIVERSITY OF MICHIGAN 8003-1-F APPENDIX A NOTES ON STATISTICAL AVERAGES (This Appendix is Unclassified) The reflecting surface defined by the random function g(x, y) will be assumed to have the following properties. a. The mean value is zero '(5 ) =0 (A. 1) b. The surface is isotropic, i. e. the statistics of e are invariant to a rotation of coordinates about the z-axis. Thus, the statistics of e may be specified by b(s) where s is a straight line in the xy-plane. c. The function is stationary and the correlation function given by B(T)=b De(8+s)g(s)] (A. 2) is known. In most physical problems we may assume that B(T) is an analytic function of T. It is obvious that B(T) is an even function of T to that B'(0) =0 (A. 3) B"(0)< 0 (A 4) and ' = B"(0). (A. 5) T=0 d. e is a Guassian random variable and the probability density function is AOf()= 2B1 1(A. 6) F2 rB(o)o The statistical averages evaluated in Chapter V, Section 5. 3, are (Cij) (A. 7) and (Cij, Cij ) (A. 8) 70

THE UNIVERSITY OF MICHIGAN 8003-1-F Ci may bewritten as - o So c eY nA. 9) where a k(cosatcosa0) (A. 10) {F ax (A. 11) x ex - (A. 12) y ay and the C are given in Eqs. (5. 32) through (5. 35) inclusive. iy _ For the conditions discussed inChapter V of this report, only the terms up to the second order in xand t need be considered in (A.7) and (. 8). In order to evaluate 6 (C1) the joint probability density fauctionf(, x, ) is required, It is easy to see that 4() "(x "~(y) "0 (A. 13) ~ (g2)= B(O) (A. 14) <5 (x) =!(S) = <5(6Y') =0 (A. 15) and < 4(c) <~y ) = -B"(0). (A. 16) Therefore, the joint probability density is 1 2 x _ f(Ag p - g i 1,_ g X. ^y I Y (2)W -B(o)2B(() o2B 2 2 ) (A. 17) From (A. 17) at is clear that (ri[ a2B(o)] (A18) 71

THE UNIVERSITY OP MICHIGAN 8003-1-F Tho statistiao l averaps Inoessery to evaluate (A.?) Oaa be -btaaied- from (A. 17) and (A. 18) and are 6 texp(-iagx9 = (A. 19) 4 exp(-iag)E = 0 (A.20) &, exp(-iagA1 = 0 (A. 21)' an ep(-iag)gJ Cep(-iaj)g ] -B"(O)ep(- aB(O) (A 2) In order to evaluate 4CtJ sip) we consider a general term of the form exp 9 g-'j (gx~m(g ) F m(E P (A. 23) where ' g y-'r. ) andy and g are the partial derivktlres of evaluated y 9x y at x —'r y-r respectively. To simplify the valuation we introduce the x Y 'r r 0os 0 (A. 24) X 7' ~7 = sin D (A. 25) y T The joint probabltf density f(-', -g ~x, g ' ) can be obtained from the y correlation matrix. The elements of the correlation matrix are determined by the following averages which are obtained by straightforward differentiation: B= Ic =:BI(r)0o T 9 G( )= - )=-B'(')sin 90 yT 6 -B"(T)cos2 sin 7' 7 7' 27 BI(T 28 4( a )-B"(7)oas B Cos2 yy Tr T O a B(()= B"(T Sin Cos 9 7'UX L 7' xy The correlaton marxM Is of order five with elements P~given by 'i - - r7 C)

THE UNIVERSITY OF MICHIGAN 8003-1-F p1 -2 B(O) - (7) 2 33 p44 - B(O 12 14 21 p41 B( Ioo p13 - 15 - 31 - p1 - ''rsn -2 P'32 - '45 - P54 -0 p24 - p42 - Bl('r)oos2 BT(in 2T B'(T P25 W.'P52 - -3 5 P43 - T B"(T) sinGT08 oos p35 - p53 - 3 537' 7' 7. 73,

THE UNIVERSITY OF MICHIGAN 8003-1-F Although in principle, we may always evaluate the expected values from the joint probability density of these five variables, the average values for the first few terms of the expansion when any one of the m, m', n, n' is zero can be evaluated in a less complicated manner. In Chapter V, when we take the powers of expansion in x etc to the second power, some special cases for the average canbe deduced as follows: (a) If m = n = mI' n' = 0, we need only the distribution density for u1 h, 2 f_ _ _ '(0) exp t.2L u ( ] 2I-m 2 [(0 -iau I j$ f(u1)e I du1 exp a2 [B(0) - ()]] (A 26) -100 (b) if any of the powers m, n', n, n' is non-zero, we may consider only the correlation matrix M between ul and u2, where u2 is the variable (2) (9 o ', B,' c). Denote the determinants of this matrix by M1( and its x x ~ y1-\ Y\2 cofactors by:~ M then iju f(u) exp ( 2 (2). (2) 2 2 (l + 12 1U2 22 2 1 '22 (A. 27) The integral a bdu1 jl2 f (u1 u2)e e -00 -00D can be evaluated as follows: If we write the exponent in (A. 27) as

THE UNIVERSITY 8003-1-F OF MICHIGAN 1 2M2 (2) 22) LM 11 Ui + 2M12 u1u2+ (2)U 2] 22 2 - tau1 s. - 2 M() 11 1 + 2 a 2M (u2-iaM12))2 - a2 (o) - (T)] The integration over U yields 1 [14 -s m l] 2 u2 du e 2.ma (0) - AM 0 For n " 1, we have * a (2) 12 exp [-a2 [(0) - A(T) ] (A. 28) and for n = 2, we have bI- [a M > l - a2 M1(2) exp [-a [-(0) - i(] ] (A. 29) using these expressions, it is straight forward to obtain the following Integrals that are used in the statistical averages, Co Co dh f(h, C ) e x 75

THE UNIVERSITY OF 8003-1-F M.ICHIGAN Co co '1xd C dhl h. 1) -a x I*ian' (T) COS 0 7..exp{-a' [LB(O) -S T OD SI co 9 d dhf (h. 9)e - ~ y.MOD co dh f(hs 9' e ia y miapt(T) sine 7' 2 exp ma. (0) - 19 (.r) 00 00 2.9~ xx OD dh f(h, 9 ) eiah x -CO x x OD de' dh f(ho, ia x K a 1 1 ) 2-...2 2 ~ R(Tr) Cos OPT exp{-a.1 l) < ( o]}.Co OS 92d y y y y S0 S0 dhf~h ' )-iah dh f(hp El 1)*ia y -COD

THE UNIVERSITY OF MICHIGAN 8003-1-F = [-B"(o) - a2 ) sin2 exp {-a2 [J(0) () (c) If any two of the exponents m, n, m', n' are non-zero, we have to evaluate the expected values by using the correlation matrix of the three variables, u = h, u1 and u2 and u3 respectively for the two values of x f x ' y 1 2 6x3 y' y whose exponents are non-vanishing. Denote the elements of the matrix by (3) p and, the determinants by M, and cofactors by. M then, ii ijI. 1 u 33 3 1k1 f(u, u ) U -la _ _ _ - - --- 3/2 2 M2 / (27r). (A.30) To evaluate the integral c0 00 00 If I du ' du2 c I U2 du2 m - iah ua du3f(ulo u2,3) e -00 -Co we expand the exponents in A. 30 in form, -(3) 1 fM 3 -u + iaul 2M jl l ^ ^ 2M 3) 1 N(2). 2N (3) (2) 2 12 2 13 3 (3) u+iaM 1 1 N2 u3 + 22 3 2 yAC2a. 2Z22 77

THE UNIVERSITY OF MICHIGAN 8003-1-F 2 2P22,-1 2] + 2 P], (A.31) where (2) 22 N 32 32.23 P33 (A. 32) If we substitute,. *f22 +P22 P23 P22 2 'i ap13 U2 2 Y2 + laP12 then after integrating over ul we obtain 9.. 2 c. exp a Pll f — ip11 00 -OD P23 2P22 Y2- iaP13 dy3 00 -00 <j. 2 2 yiap Y2 Y3' Y2 P12 dY2 e These integrals can be carried out explicitly, the results are: o00 00 00 du1J u2du2 -0o -00 -a -iau u3du3 f(ul, u2, u3) e *

THE UNIVERSITY OF MICHIGAN 8003-1-F p2 e [ 21] 1 ap11 3LP23 l2 -- 12 L 2 (A. 33) -) -o0 00 -00 -00 -iau1 u3 du f (U1 U' u3) e 3 3.1 2p 3 3 2 11 I i P13 -aP22 P13 + 2ap P23 ep 2- 2 J 3~ e x [ (A. 34) 00 -MD du - -2 2 -lau1 U3du3f(ul' U2, U3)e (2) 2 2 2 4 2 2 [N +3 23 [ 3 P22 32+ 2 33 4P 13P23 Pa 12 P13 - s *exp - 2 p11 ~ exp (A. 35) d, if any three of the exponents are non-zero, we use the correlation matrix t between 4 variables ul, u2 u3, and u4. Denote determinant of this matrix by M, we have, f(Ul, U2, u3 4)= e To evaluate the integral 4 (4) (A. 36) 7Q

THE UNIVERSITY OF 8003-1-F MICHIGAN aS 00 -ia1u I 1"I j (D U2;d2 1 — 00 u3-du3 -,D u4du4 f (ul, u, 3, U4) a we expand the exponential in (A. 36) ja1 4 (4) k jk ujuk + iau1 kc k j 1 2M(4) u4+ 11 u1 (4)u + (4) u + (4)u + iaM:12 2..13 3 14 4 N / +4) 11 2 1 2N(3) + (2) v33 U4+ (3) (3) - (4) 13 + 23 + 3 _ ia 14 v133 2 (2) 22 (2) (3) 2 N12 - iaM 13 u + - - 3' ^22 + 1 2 2 P22 [u2-iap12] 2 2 + 2+ P11 (A. 37) where N(3) is the determinant cf

THE UNIVERSITY OF 8003-1-F MICHIGAN N(3) N P22 P32 P42 P23 P24 P33 P34 P43 P44 (A. 38) By change of variables, U2 = 2Y2 + laP12 u3 a + P23 2 P22 Y2 + iap13 U = 4 ^s, - aI P22;i (3) 23 3) Y3 33 P24) P 22. - iap14 and after the uI integral is carried out, Id 2?.-. dY2 -.00 S a0 a 2 2 2 Y - Y2 Y3 dy3 e n i1 U2 U3 -00 For n I I3 -i au 00 00 du1 e 2 U 2 2J u3 du3 - - -00 u4 du4 f (ul' 2 3 u4) 4 4 1' 2# I3' U4

THE UNIVERSITY OF MICHIGAN 8003-1-F 2 {la P12 13 14 1234 P13P24 14P } 22 (A. 39) For n a 2 we have, O -00 -00 -OD {[22 + 2p3P24 + a2[1214 22+ 2p12p14p23 - 12 13 24 2 P2 4] + a4 P12P3P4} exp [ 11 (A.40) (e) For the expectated values of the functions Involving all the powers, we need the complete correlation matrix between the five variables. Denote the determinant of (5) ' the matrix by M ', and its nth order sub-matrix involving the second n rows and columns by N, the integral co OD poo coo -iau J du J u du du u44 u5du5 ( 112 3 41 5 - -00 -00 -c00 -co -o00 can be simplified by substitutions similar to those given previously. If we substitute, U2 iap12 QO

THE UNIVERSITY OF MICHIGAN 8003-1 —F 3 -Ip23 y +2 3 a 3 22 p22 (3) U4 y4-2 N23 33 N(4) N34 (4) y3 22i~ p24 -. y2 + lap 14 ~22 U 5 y5 - y4 + 2) 1 '22 p53 - 52 p23) N33 +4/i K- 2+Ia 15.22 the integral, after carrying out the u1 integration is reduced to Ia2 1 D00 rco2O1 e e -p 2 dy2 dy3 d dy4 d y5 ex Ly 2 Y3 Y4 Y j "l'~2'3 -OD -00 -co -00 =exp 2 J0 - { P34P~25 + p24 P3 + p23 p45 -2[14 15 23+1 1 24 12 l5 P43 + 14p13p52 +1P4 12 53+1125 + a p12p13p1p15 l (A. 41) 83

THE UNIVERSITY OF MICHIGAN 8003-1-F APPENDIX B ALTERNATE FORMULATION OF GROUND REFLECTION (This Appendix is Unclassified) In the derivation of the angular spectra of the ground reflected radiation given in Chapter IV, the z-components of the incident field at the ground were used in the interests of simplicity. However, the employment of the z-components of the field causes some difficulties in the limiting case of normal incidence, since in this case the values of Ez and Hz are zero, although the angular spectra is finite by taking limiting values. In order to consider such cases, an alternate derivation of the angular spectra, based on the tangential components of electric and magnetic fields, are given here. Starting from dE_(r)= [l(a, + 3)+(a, ) e - (B. 1) A ^ 4 el=x sin 3 -y cos 3 (B. 2) and e = xcos a cos 3 ycos a sin3-z sina (B. 3) one finds that Eo(r)= tj h 3)sin/3+2(a, )cosacos 3] e > - dQ (B. 4) E (,=] [-e(,P)cos3+ 2(,a)cosasin e -- dQ2. (B. 5)..2 Let r * s, and consider the above approximately as a two dimensional Fourier transform, then one finds that 1 ^(a, 0sinf34f (a, 13)cos c cos d P =2 Cios f' a r EJ (r )e e - r (B. 6) (2-r)2 s s 84

THE UNIVERSITY OF MICHIGAN 8003-1-F and E- C(', 3 cosoi-Be(a, l3)os sin3Co0S 2 A d E (r)e e (B. 7) (2r)2 s Ys y From the above it follows 1( = (2= 2 dx rdy E (r )sin)3-E (r)coss ] cosae-i s 8 (27r) s x (B. 8) and k1 -ikQ- r A PC'd, (r +E (r9) 2k2 I22 f dyEx,,~ os3Eys, s in e -ik( B r9, Now, for an incident field approaching the ground in a direction 2o, the incident electric field may be expressed as A A A A +ik o r E (r)=IE o+E e e (B.10) -o- 1010 2020 (B. 10) where E10 and E20 are respectively the amplitude of the perpendicular and parallel polarized components of the incident field referring to a plane ground z = 0. The amplitude of the reflected field on the ground is therefore given by A ikQ -r ' L(B. 11) E(r-s) [EOR sin3o+E 20Rlcosacos e (B. 11) and A iky2 (B E (r E10RcosO o+E R cosain (B. 12) -s 10 o 20/ 2i0e (B.12 85

THE UNIVERSITY OF MICHIGAN 8003-1-F Thus A - - ik Qo rs Ex (_)cos3+Eysin3 = E10R sin(o3- )+E20 RcosaocO (B-3) e (B. 13) and [Ex(rs)sinj-Ey(rs) cosa cosr A [E R cosacos( -j3)-E2o cosacosa sin(-e (B.14) Introducing the above to (B. 8) and (B. 9), one obtains the following matrix form for the reflected angular spectra of the reflected radiation. 1 k2 r ik(-)k= 2 / dx dy e C (a, C (2r)2 sJ s R1 cosacos(O-J3) -R//cosacosaosin(3o- ) E1 LRLsin(o-3) i R1/cosaocos(o-) j E2 (B. 15) This result seems to be closer, but not identical to the physical optics results, but the singularities in the scattering matrix are avoided. 86

THE UNIVERSITY OF MICHIGAN 8003-1-F REFERENCES (Unclassified) Beckmann, P. and A. Spizzichino (1963), The Scattering of Electromagnetic Waves From Rough Surfaces, John Wiley and Sons, Inc., New York. Booker, H. F., J. A. Ratcliffe and 0. H. Shinn (1950), "Diffraction from an Irregular Screen with Application, to Ionospheric Problems, " Phil. Trans. Roy. Soc., A242, pp. 579. Crispin, J. W., R. F. Goodrich and K. M. Siegel (1959), "A Theoretical Method for the Calculation of the Radar Cross Sections of Aircraft and Missiles," The University of Michigan Radiation Laboratory Report 2591-1-H, AD 227695 (423 pp). Fung, A. K. (1966), "Scattering and Depolarization of El ectromagnetic Waves from a Rough Surface, " Proc. IEEE 54 p. 395. Green, H. S. and E. Wolf (1953), "A Scalar Representation of Electromagnetic Fields, Proc. Phys. Soc., A66, p. 1129. Lonquet-Higgens, M. S. (1962), "The Statistical Geometry of Random Surfaces," Proc. Symposia in Applied Mathematics, Vol. XIII, p. 105 Rice, S. O. (1944), "Mathematical Analysis of Random Noise," B. S. T. J., 23 p. 282 and (1945), B. S. T. J., 24 p. 46. Saxon, D. S. (1955), "Tensor Scattering Matrix for the Electromagnetic Field, " Phys. Rev., 100, p. 1771. Sekera, Z. (1966), "Scattering Matrices and Reciprocity Relationships for Various Representations of the State of Polarization, " Rand Corporation Report RM4924-PR. Stratton, J.A. (1941), Electromagnetic Theory, McGraw-Hill Book Company, New York. vande Hulst (1957), Light Scattering by Small Particles, John Wiley and Sons, Inc., New York. Wagner, R. J. (1966), "Shadowing of Randomly Rough Surfaces, " TRW Systems, Inc., Report 7401-6012-R0000. 87

CONFIDENTIAL Security Classification DOCUMENT CONTROL DATA R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATlON The University of Michigan Radiation Laboratory CONFIDENTIAL Department of Electrical Engineering 26 CROUP Ann Arbor, Michigan 48108 1 3. REPORT TITLE Study of Angle of Arrival Errors Due to Multipath Propagation Effects (U) 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Final Report: May 1966 through May 1967 5. AUTHOR(S) (Lost name. first name, initial) Chu, Chiao-Min 6. REPO RT OAT 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS May 1967 87 12 8a. CONTRACT OR GRANT NO. 4. ORIGINATOR'S REPORT NUMBER(S) N123(60530)56078A 8003-1-F b. PROJECT NO. c. 9b. OTuHFER RPORT NO(S) (A nY other nmbera thatr may b*e as4ined this report) d. 10. AVA IL ABILITY/l.jMITATION NOTICES 11. SUPPI EMAENTARY NOTES.1a. SPONSORING MI4lITARY ACT IVITY U. S. Naval Ordnance Test Station China Lake, California 93555 13. ABSTRACT (U) A unified approach, using the generalized concept of angular spectra, for the study of the radiation received at any point in space from a transmitter due to multipath propagation effects is formulated. The various mechanisms contributing to the multipath effects, such as scattering by discrete objects, and by extended objects such as ground are formulated in general. Although the formulation is based on CW transmitter and stationary receiver, the results may also be used for other transmitter signals, scanning and moving receivers (and transmitters) with slight modification. I- - - —.I - --- -T - - I -- - -. --- - - -- - - -- - -- m DD J^N6.4 1473 CONFIDENTIAL Security Classification

I CONFIDENTIAL Security Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE! WT 3 -Radiation Multipath Propagation Scattering Rough Surfaces INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Di~rective 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. '3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified, If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first namre, middle initial. If military, show rank and branch of service. The name of the principal.au.thor is an absolute minimum requirement. 6. REPORT DATE- Enter the date of the report as day, month, year; or month, year. If more than one date appears on the report, use date of publication. 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information. 7b. NUMBER OF REFERENCES: Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report. 9b. OTHER REPORT NUMBER(S): If the report has been assigned any other report fumbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those imposed by security classification, using standard statements such as: (1) "Qualified requesters may obtain copies of this report from DDC." (2) "Foreign announcement and dissemination of this report by DDC is not authorized." (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through t, (4) "U. S. military agencies may obtain copies of this report directly from PDC Other qualified users shall request through (5) "All distribution of this report is controlled. Qualified DDC users shall request through 0, b \ If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 11. SUPPLEMENTARY NOTES: Use for additional explanatory notes. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S), (C), or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. I I _ CONFIDENTIAL Security Classification I