015239-1 -F RADAR SYSTEM ANALYSES - I Final Report by Chiao-Min Chu Radiation Laboratory Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan 48109 dork Performed Under 8398-6 (AF Prime F30602-75-C-0082) 1 February 1977 - 30 September 1977 Submitted To Purdue University RADC Post-Doctoral Program (C-E Systens) Wlest Lafayette, Indiana 47397 15239-1-F = RL-2281 February 17, 1978

015239-1-F TABLE OF CONTENTS SECTION Page No. 1 INTRODUCTION AND CONCLUSION............. 1 2 A SIMPLE SAR SYSTEM................. 3 2.1 Seometry of a Simple SAR Systen......... 3 2.2 Transnitted and Returned Signal......... 6 2.3 The Ideal Resolution.............. 7 3 4AVE PROPASATION IN INHOMOGENEOJS VrEDIA........ 11 3.1 Introduction.................... 11 3.2 Stratified edia................. 12 3.3 lodels of Turbulent Atnosphere......... 15 3.4 Wave Propagation in leak Turbulence....... 17 4 ATIOSPFERIC EFFECTS ON SAR OPERATION.......24 4.1 Introduction.................. 24 4.2 Atmospheric Stratification........... 24 4.3 Effect Due to Weak Turbulence......... 27 4.4 Effects of Strong Turbulence and Discrete Scatters................... 35 5 REFERENCES.......................39 i

l15239-1-F 1. INTRODUCTION AND CONCLUSION In this report, the results of a study concerning the effect of atnosphere on the operation of the synthetic aperture radar (SAR) are summarized. It is well known that the success of SAR depends on the oroper choice of the weighting function in the signal processing to account for the phase difference of the radar signals returned from different parts of the ground surface. [Cutrona, et al (1961), Brown (1967)]. So far, in iost analysis concerning SAR, the weighting functions are constructed on the basis of free space propagation of signals. In reality, of course, the atrosphere is inhonogeneous spatially and temporally. The effect of atmospheric refraction and scattering causes the deviations and uncertainties on the amplitude, phase, and direction of the signal. In this report, based on the present state-of-the-art in the solutions of wave propagation in inhonogeneous (and/or randorr) nedia, several models ol atmosphere are chosen, and the effect of these atrospheric models on the operation of SAR are discussed. In Section 2 we describe a sirple SAR system. The returned signal, and the prirciiles of signal processing based on free space operation are outlined. Various quantities involved in this "free space" nodel that nay be effected by the atmosphere are pointed out. In order to bring out primarily the effect of atmosphere on its operation, this simple system is idealized, and the syster noise is ignored. In Section 3 we review briefly the problem of wave propagation in inhomogeneous media. Several "popular models" of atnosphere are described, and oir present knowledge on waves in such atnosphere nodels is reviewed. Results useful to the analysis of the operation of SAR systen are outlined. In Section 4, the results of Section 3 are introduced to the signal processing scheme of the sirmple SAR system given in Section 2. The effect of the atmosphere on the operation of this systen are then deduced. 1

015239-1-F Dje to the uncertainties and variations of the parameters of atmosphere involved, this report emphasizes the basic approach to the probler and the use of mutual cohence fjnction and two frequency coherence function in the analysis of SAR resolution. For the case of weak turbulence, approximate sample calculations appears to indicate that the effect o" turbulence on rarge resolution are negligible while the azimuth resolution is deteriorated slightly due to turbulence. This conclusion, of course, is based on the particular set of parameters used. For the rainy and storny weather, the theory of weak turbulence does not apply. In Section 4.4 we formulate the approach that could be used to analyze the SAR resolution. Due to excessive nunerical work that is involved and lack of time, we did not proceed further. 2

015239-1-F 2. A SIMPLE SAR SYSTEM 2.1 Geonetry of a Simple SAR System A simple model of a synthetic aperture side-looking radar system is used in this chapter to illustrate the basic operating principles of SAR system. The idealized geometry involved in the operation of this system is given in Figure 2-1. A moving transmitter A emits pulsed signals, illuminating a patch of ground. The transmitter is moving with velocity v << c, in the x-direction at a fixed height h. Thus, the vector position of the transmitter is given by {x. = vt, 0, h}. The antenna of the transmitting beam is pointed toward the broadside with a depression angle Y, so that the direction of the center of the beam is s = a cosY - a sinY. (2-1) The antenna radiation pattern nay be expressed in terms elevation angle a and azimuth angle 3. As illustrated in Figure 2-1, the direction of any ray from the transmitter B is given by s = a sin 3 + a cos 3 cos (Y - a) - a cos 3 sin (Y - a) (2-2) x x y z For simplicity, the ground is taken as the plane z = 0, and the ground property is characterized by the ground reflectivity o(x, y). For free space propagation, a ray in direction s would be reflected by a point on the ground with coordinates x = x(a, 3) = h tan 3/sin (y - a) + x. i (2-3) y = y(a, 3) = h cot (Y - a) 3

015239-1-F z / (a) i / Z J-2^ / / / y. / x z 13r Iy (b) 3= 0 Y A a = constant (c) x=O Xi Fi ure 2-1. (a) (b) (c) Geometry for a side looking radar system. Beans in y-z plane through A. Beams in a slant plane. 4

315239-1-F In particular, for the center of the bean, x= x(D, O) = x (2-4) YO = y(O, 3) = h cotY The slant range of any reflecting point on the direction sa3 is h R = R(a, 3) = sin (a- sec 3. (2-5) In particular, for the center of the beam, R =R(0 ~) h (2-6) ' sin ' and in any slant plane with fixed a (or y), the shortest slant range is Ry = R(a, 0) = h/sin (Y - a) (2-7) In most analysis, we assure a,6 to be small and employ the following approximate relation in signal processing a. The azinuth range is x - x. = Ra3 R (2-8a) R R b. The broadside range y - yo iy a (2-8b) (x - x.i2 c. For fixed a, R R (2-8c) y 2R d. In the direction 3 = 0, (y - y )2 0 R(a, 3) = RQ + 2Ro cos Y (Y - g) + 2R (2-8) These ate commonly used in relating the transmitter radiation pattern (function of a, 3), the ground reflection point, and estirating the time delay in the 5

015239-1-F analysis of the returned signal. However, in an inhorogeneous medium, the rays may be curved, and the above relations must be modified according to the variation of the rean index of re-ractior of the atmosphere. 2.2 Transmitted and Returned Signal The moving transmitter A in Figure 2-1 emits a sequence of pulses, and the returned signals for each pulse, after the carrier is removed, (or converted to IF) is stored for processing. Let us consider the mth 3ulse, the transmitted signal way be represented by -iw t' S (t') = R A (t') e, (2-9) where wo is the carrier frequency (Ra/sec), and A (t') is the complex pulse modulation. The tire t' is a shift in t such that t' = 3, the transritter is at x. = x. Thus, in terms of t', 1 T xi = xm + vt', (2-10) and the expression S (t') is same for each pulse. The returned pilse depends on the reflectivity of the ground and the medium between the transmitter and the ground. Let us assume that for a two way transmission from tie transmitter A to a ground point (x', y') introduces phase of 9(x., x', y') to the RF signal and modifies the comolex amplitude (due to delay and oossible dispersion) to ft', xi, x', y'). Then the returned signal from the illurinated patch of ground may be formally written as fi (x i, x ') Sm(t') = dx' dy' G(a, 3) A(t', X, x, y')e D(x, y') (2-11) where G(a, 3) is the power pattern of the transmitting antenna, aid a, 3 are functions of x' and y'. For simplicity, we shall assume that G (a, 3) = 1. 6

015239-1-F The returned signal for all the pulses are stored in two dimensional format and may be considered as a function of two variables, xn and t'. S(x, t) fdx' fdy' (t,, ',x, x y') Exp[i[(xm, ', y')]D(x', y') (2-12) In (2-12), t' takes continuous values from - 2 to +, where T is the pulse length while xm takes discrete values nvT, where T is the interval between pulses and n are integers. In signal processing, we want to obtain the best estimation of o(x', y') from the information S(x,t). This is usually accomplished optically or electronically using the principle of matched filters. 2.3 The Ideal Resolution The two dinensional signal S(xm, t') nay be considered as a two dimensional napping of o(x', y') with Kernel K(xm, t', x', y') = (t', xm, x', y')Exp[i6(t', xm, x', y')] (2-13) Basec on the principles of matched filters, the resolution of the estimation of o(x, y) may be obtained from the generalized ambiguity function X (x, y, x', y') = I dt' dx Kt', x', y') (*, x x, y) (2-14) where N is the number of pulses summed to gether during processinc (for convenience of analysis, the summation is approximately replaced by integration). The integrand in (2-14) may be factored into KK* = gv * 9h (2-15) 7

015239-1-F where 9v (t'9, xm, x, y) *(t, xm, x, y), (2-16a) and gh = exp[i6(x m x, y') - i6(x, x, y')]. (2-16b) It is seen that 9h depends on the RF phase of the signal and gv depends on the modulation envelope distortion. In free space, the (t, xm, x', y') = A(t' - 2R' (2-17) where R' is the distance from the transmitter at {x m, 9, h} to the ground point (x', y', 0). From (2-8c) we may approximate (x - x )2 R' - R + 2 m (2-18) y In the near forward direction, the second term can be neglected, so that gv may be considered as a function of R y, only. The phase delay, for free soace orooagation, is given by 2 = 2w 2w (x' - x )2 = R -= R,+ c I. +(2-19) c y c y c 2R,21 From (2-18) and (2-19), it is seem that for simplicity, the analysis of the resolution near the forward direction ray be decoupled into two one-dinensional problems. a. In the forward direction, x' = x = x m, the range resolution is obtained from the range ambiguity function 8

315239-1-F x, y, Y' ) dt' gSft', Y, y')dt',yy d (2-20) b. For fixed y (range Ry), the azimuth resolution may be determined from Xh(X, x') - vT dxm 9h(X' X', X ) - - - 2vT (2-21) It is expected that in real atmosphere, such decoupling is also valid, hence the azimuth resolution depends on the structure function of RF phase shift and the range resolution depends on the envelope distortion. In free space, from (2-19) and (2-16b) 2O 2 2 h = Exp[-i c-y (x' - x)x ]exp[icR0 (x - x )] y y (2-22) Thus D = exp[i (x2 Xh(x' x') = xh(x - x') = exp[i cRi(x: y' - x2)] -vT 2 exp[-i c (x' - x)x]dx -_vT Y.4 (2-23) This equation indicates that the synthetic aperture in the azimuth direction is equivalent to an aperture of effective length Leff = 2NvT (2-24) 9

015239-1-F To estimate the resolution, we see that the pattern function of this array except and imaginary multiplicative constant, is given by p-vT 2 Fh(x') exp[-i cR x x']dx v y n n j2 If we define the resolution in terms of the equivalent rectangular pattern, then the azinith resolution is given by | F(x') 2dx' XR = - __L (2-25) h F() 2 2\vT F(0) The range resolution is usually achieved by chirping. If we let 2 A(t) =e (2-26) then 2R 2R 2 v= exp[ia(t' - -2 ia(t' - )2] 4ia R Itl (R2 2 = exp[- - (R - R )]exp[ R 2 - R 2 (2-27) c y y' 2 y Y so that v(Ry, Ry) = exp[4 ( 2 (R )] exp[ (Ry - Ry)t]dt'. (2-28) c T T Thus, the range resolution is, = 2-c (2-29) v 2a' The deterioration of these ideal resolutions due to atmospheric effect s all be discussed in Section 4. 10

01 5239-1-F 3. WAVE PR9PAGATION IN IMHOMOGENEDUS MEDIA 3.1 Introduction The Droblew of wave propagation in an inhonogeneous medium (such as atmosphere) has been investigated extensively in the past few decades. Substantial useful results have been obtained from the early investigation of the propagation problers in stratified media, in weak turbulent media and ir the more recent works involving nedia with stronc turbulence and discrete scattering particles. The results of recent works are sumrarized in a paper by A. Ishinaru (1977). In this section, sone of the results that are directly applicable to SAR operations are given. Fron the analysis of Section 2, the resolution of SAR systen is effected nost by the phase shift of the carrier and distortion of the pulse modulation, the results we quote in this section therefore concern these two types of problems. It is to be noted that due to complicated spatial and temporal index of refraction of the atmosphere, most results are aooroximate, and the validity of each approximate approach depends on the frequency and distance of propagation. In this work, we are arbitrarily limiting our application to a oropagatinc distance of 10 kn and in the frequency range of 8 GHz to 20.5 GHz. In this range the parameters that may decide the particular choice of iodel are tabulated below. Frequency 8 x 109 Hz 26.5 x 109 Hz Aavelength X 0.0375 m 0.011 m k 167.55 1/w 550.0 (1/n) x 1 * 19.36 ff 10.63 i dithin this range of parameters, it aopears that the ray and weak turbulence approach nay be appropriate. 11

015239-1-F 3.2 Stratified Vedia It is known since early days of radar operation that the index of refraction (r) of atmosphere, on the average is nearly unity but exhibits variations with neight due to pressure, temperature and moisture variations. The variation of r with height (h) is usually very small, and may be functionally represented in the forr r = 1 + 6f(z), (3-1) where - is a small quantity, and f(z) is the profile of the variation of r. In the exponential atrosohere nodel, we have -6 6 = 313 x 10-6 and f(z) = exp[-(z)], (3-2) with c = 0.1439/kn. In general, due to excess moisture near sea and clouds, and large temperature gradient near deserts, f(z) takes different forrs, and ray be approximated by sections of straight lines. Exact solutions for wave propagation in a stratified nedium is difficult and ray theory (geometric optics) is commonly used. For a transmitter located at (xi, i', h), the solution of the ray equation indicates that a ray starts fron the transmitter in the direction s = s a + s a - s a (3-3) x x y y z z would be reflected from a point on the ground plane (z=O) with coordinates: 12

C15239-1-F l s xn(h)dz x = x. + x-x2 =J [r2(z) - r (h) (s 2 0 I 2 11/2 -s )] y (3-4) y =- y. + fh Syr(h) dz J [r (Z) - r2(h)(s2 + s Y)]1/2 The phase delay of the ray from the transmitter to the grojnd along this ray is w 0 -C fh I r (z)dz (3-5), [r2(z) - 2 (h)(s2 + s 2)]1/2, o x y For r decreesing with height, the ray bends toward the transritter as illustrated in Figure 3 1. This refraction phenorena has an important effect on astronomical observations, and computer programs have been developed [Garifinkel (1957)] for the integration of the ray path and angle of arrival. For approximate analysis of the ray path (3-3) through (3-5) may be approximately evaluated to the first order or 6, the results can be easily shown 5 sh 1fh x - x. = x h + 6 [flh) - I y sz Jo y -y = sy h + 6s [f(h) - ~ ss h + h 0 =c sz o= h + 6 h [1 - s 2]f(h) z 3 z f(z)dz], f(z)dz], - (1 - 2s 2)o f(z)dz]. (3-6) (3-7) and (3-8) These ecuations shall be utilized in the next section in the investigation of the distortion of the ground image. 13

01 5239-1 -F ri (Xi Yi, ) a - - in stratified atmosphere Ray in free space h \1 \ U I point of reflection (x, y, o) point of reflection for ray in free space Figure 3-1. Bending of a ray due to refraction. 14

015239-1-F 3.3 Models of Turbulent Atmosphere The index of refraction of a turbulent atmosphere is represented by r(r, t) =r+ r(^, t) when r is the nean value (used in (3-2) and rl(r, t) is a random function of zero mean. In wave oroDagation problems, for simplicity, rl(r, t) is assumed to be isotropic, stationary and may statistically be described by correlation fjnction, structure function or spectral density function. To avoid confusion we briefly state the notations used: a) Given any isotropic random function, f(r, t), we represent f(r, t) = fc() - f (, t), (3-9) where f(V) = <f(r, t)> (3-10) is the ensemble average and f1(r, t) is a random function of zero mean. b) The correlation function is denoted by the statistical average B f(l - r2) = <f(rlt) f(2 t)>.) For stationary and spatially homogeneous random field, Bf( - 1 ) = B f( - ir2) = Bf (r), (3-12) where r= r1 -r2 The mean square variation of fl(r, t) is denoted by <fl > = Bf(O) (3-13) 15

01 5239-1-F c) If a random function is not homogeneous, it is sometimes reaningful [Tarkarsi (1961)] to introduce a structure function defined by + =- 2 Df), t) - r2 t)>. (3-14) Dr' r2) ' f1 (rl t) f1 (r2 For stationary, locally horoceneous isotropic rardom field, Df( 1 ) 2l = D(r) (3-15) d) For a homogeneous randor field we have Df(r) = 2B (C) - 2Bf(r), (3-16) Bf(r) 1 Dfr - - Df() (3-17) e) The three dimensional Fourier transforr of Bf(r) is the spectral density o(K) - 2k rBf(r) sin(Kr)dr 3-18) The inverse is Bf(r) = - K o6K) sin Kr dK (3-19) Jo In zerms of spectral density, Df(r) = 4 1 siK r) o(<) K dK. (3-20) Various models of correlation functions are postulated for atmosphere in the study of tropospheric scattering [see for exarDle Staras and Jheelon (1959)]. Solution of the proorooation problems however has carried out in detail only for the Gaussian model [Chernov (1963)]. For this model, 16

C15239-1-F - r 2A2 B (r = B (D) e. (3-21) Chernov assumed the value of Bn(C) = <nl2> = 5 x109, n and o = 0.6 n. The model of locally horogeneous turbulence is a more realistic description of the atmosphere. Correlating with the theory of homogeneous turbulence and meteorological measurements, the model postulates the stricture functior of the index of refraction to be [Tartarski (1961)] Dn(-) = C2 2/3 '3-22) D (r C2 2/3 (r2 Dn(r Cn o r << 0 Lo is the outer scale of the turbulence, o is smallest size of eddies, and Vn is known as the structure constant. These parameters are usua'ly inferred from meteorological measurements and varies over a wide range. L0 is generally on the order of 100 m and k is on the order of 10 to 10 m [Strohbehn (1968)]. 0 The measured values of structure "constant" (n varies with altitude and depends on the frequency (microwave or optical). Sore data of Cn2 was given by Hafnagel (1966). In sample calculation for this work, we shall arbitrarily take the value Cn2 - 2x 1013/m23. 3.4 have Propagation in Weak Turbulence Feaningful approximate solutions of wave propagation in atmosphere with weak turbulence were obtained by the method of smooth perturbations. [Tartaski (1961']. Using this approximation, the structure functions of the phase ard amplitude fluctuations of a plane wave (or spherical waves) at points transverse to the direction of propagation were expressed formally in terrrs 17

015239-1-F of the spectral density of the index of refraction. A summary of available results are given by Lawrence and Strohbehn (1970). Because these results aopear to be directly applicable (approximately) in investigations of azimuth resolution, a brief description of the method of snooth perturbation is outlined in this section. We also derived the two frequency choerence function which is to be used in the analysis of range resolution. -he essential steps in studying the fluctuation of EM wave in turbulent media are given below. a) Neglecting polarization effects, and consider the solut'on of the scalar Helmholtz equation 2 2 2 2 u + k (1 + n1) u = C. (3-23) b) For plane wave propagation in the x-direction, assuming u = exp[-iwt + ikx] exp V (3-24) If we write a = x + is, (3-25) then x is the log-amplitude fluctuation and s is the phase fluctuation. 2 c) Assuming n << 1, Ivyl << k, and neglecting the term a3 (parabolic approximation), it is shown that V satisfies ax,2o 2 2+ 2 + 2ik x + 2i. (3-26) ay az 18

u15239-1-F Thus, using Born's approximation, u(ri can be expressed by (2 (x',y',z') 2 + 2 xjx,y,,J X 1( exp[ik 2;x-x' ] dx'dy'dz' (3-27) d) From (3-27) we may express the spectral density of the random functions x and s in terms of the spectral density of 1. If we define the two dimensional partial Fourier transform of any random function f(r) by 1 fcc of(x, j) 2 - j o(k-)Bf(, ) d (3-28) o0 then, it can be shown that for two points {L,O} and {L + AL, 3 = S}, L L o(AL, t)= k2dx dx" o (x'-x", <.0 0 K 2(L + AL - x') K 2(L x") sin[ 2k ]sin[ - 2 ] (3-29) For phase fluctuations, the expression for o (AL, i ) is obtained by S replacing the sine function in (3-29) with cosine functions. Equation (3-29) is a slight generalization of the result given by Tartarski, who considered the special case of AL = 3. e) If <D << Lo, (3-29) can be simplified to K 2AL K (L + AL) K L (AL, ): -k2L[cos- +- 2 cos- sin k o-0, 2k K 2 2k 2k n (3-30) we denote r = {x, D}, and t = {K1, i } to separate the effects in transverse and longitudinal direction. 19

015239-1-F f) Fron (3-30) we nay obtain the correlation and structure functions for phase and amplitude by the integrals given below B (AL, D_ = 2r{Jok D) - (AL, j_ Kd< (3-31) 0 o and Dx(AL, D) = 4r [oX(0, J D)_X(AL, K dk- (3-32) s s For the case L = 0, (i.e., for points lie in a plane transverse to the direction of propagation), various expressions have been derived 'or the correlation and structure functions for log-amplitude and phase fluctuations. The results given by Tartarski, using the structure function given in (3-22), corresponding to the structure function 0.333C 2 K 11/3 K < K = 5.48/Z o (K) = n n 0 (3-33) 0 K > K appears to correlate fairly well with some experiments. de shall therefore use then in the present study. The results that shall be used in this report are <X2> = BX(C, 0) = 0.31 C 2 k7/6Lll/6 > (3-34) 3.44 k2 1/2 2 n3o o D (0, ) = 2.91 k 2L o5/3 o > AT (3-35) 1.46 k2C 2L o5/3 < < o < /n o For the range of frequency and distance of interest in this work, X = 0.0375 - 0.011 m, and for a distance of L - 104 n, vlA = 19.36 to 10.63 m, 20

015239-1-F so that the last expression seems to be valid. For this range of parameters, 2 it is also to be noted that the highest value of <x > is about 2.2L x 1C3, hence the theory of weak turbulence is valid, and the attenuation may be, to the first order, neglected. In investigating pulse propagation in random media, it is usually convenient to use the two frequency mutual coherence function [see for example Ishimaru and Hong (1975)]. Although recent investigations on the mutual coherence function are rostly for the case of strong turbulence and discrete scatters, for weak turbulence, the two frequency coherence functions that we shall use in Section 4 can be derived by a method of sriooth perturbations. For a description of coherence function we need, let us consider a plane wave propagating into a random medium. At any distance L, the phase fluctuation at two different frequencies, S(y1, L) and s(w2, L) are different, and we need for pulse prooagation, the two frequency coherence-structure function -s(l 2' L, = < s(l, L) - s(12 L) > From the spectral density function of s(w1, L) obtained by the method of smooth perturbations, it is easily seen that oC s 1' 2' L = 3) = 2IK dK (s 2' L, K ), (3-36) 0o and s(T 1 2' L' K-) k 2 3 L k3 K 2L n (0, j{k1 L + e sin(k ) + k L + sin( kk 2 'k K 1 K 2 2 2k k12k2 K 2L(kl+k2) 4 k 2k K -2k -k2) -.-. 2 sin[ 2 - 1 2 sin[ ]> 1 2 ] (3-37) (kl+k2) K 2 [ 2klk2 (l- 2 in 2k1k2 1(klk2)K 12 To simplify the integration we choose the Gaussian rrodel for B (r) [(3-21)], the spectral density for which is 21

015239-1-F n 2>, 3 K2, 2 K) 1 ~ exp [- 4] 8 Tr T-r (3-38) Using (3-38) and carry out the integration, we have 2 3 1n L 2 4 3 -1 4L kW1 L,o=) = - k L + 2 + tan 2 s 2L /8 { 2k1 L 2k2 +k1 tan o o 0 k 2 k + k = k+ Ak, (3-4C) k2 tko akn (3-39) k2=k -Ak and expand r as series of Ak. Approximately, then r is represented by the first term of the series which is given by <1 2> k + k 2 0 ( L ) ' g ~ Ak24p + 16ko tan,1 ^ - 70 o (p 1k; and expand ~ as s e r i es ~~~ ~of k prxmtlte srpeetdb h is 4 k 0p + 4 22 p +k ) 0 2 p3 3k 2 0 '3-41) (3-42) where 4L P= 2 0 is the distance parameter, while the other factors in (3-41) depend on frequency only. Equation (3-41) revealsthat within this approximate formulation 3 - varies as L for large L, and varies linearly with L for small L. Moreover, for the case that 4L P =- - << k p 2 0 so (3-43) 22

015239-1-F is alnost independent of frequency. It is interesting to note that the condition given in (3-13) is precisely the condition often quoted for the validity of geometric optics. For p >> k, (3-41l nay be silolified to 0 (+2 2 1 L lO7- 2 8L 2=) <n L12 - (0sD,1 =0) 2/ <2 L >1 0 + 3k I A(3-44) 3k 2z 0 0 On the otier iand for p << k, (3-41) nay be aDproxinated by s'l' 1 2' L, D = 0) 7/7 <n1> LAk2 (3-45) We shall use this equation in connection with the discussion of range resolution in Section r It is to be noted that (3-41) is obtained by using the Gaussian rodel because the Tartarski rodel has a singularity at K = 0 to cause the integral in (3-36) to diverge. I- we assume a value of the outer scale L and use a modified spectrun suggested by Strohbehn (1968), the integral of (3-36) converges, but no sirple analytical for ris possible. In our numerical calculation, we 2 shall choose nl and Lo for the Gaussian -odel such that the Gaussian spectrui is approxinately equivalent to the modified spectrum for small va'ues of K. The condition to be satisfied for such a choice can be shown to be 2 -2/3 = 26.13 <n >O (3-46 n 1 o For the sample computation carried out in Section 1 we choose Q = 8 - and <n2> = 1.1 x 10-13 For this choice of parameters, for L = 10 "i, p < k, for the frequency range of our interest, hence (3-15) is used in Section 1. 23

C15239-1-F 4. ATMOSPHERIC EFFECTS ON SAR OPERATION 4.1 Introduction In this section, the results of wave propagation given in Section 3 are applied to the analysis of the operation of the simple SAR system postulated in Section 2. The effect of bending of rays due to straitification on the error in imaging is discussed in section 4.2. The effect of turbulence on the return signal in general is formulated, and the deterioration of resolution in azimuth and range are analyzed in section 4.3. Considerations were given for the effects of rain and storm on the SAR operation. General formulation of the problem are outlined in section 4.4. But due to complicated numerical procedures involved, no attempt has been made to carry out a numerical solution. This would be an appropriate area for further research. 4.2 Atmospheric Stratification The effect of atmospheric straitification on the wave propagation has been studied extensively in problems of the angle of arrival in astronomical observations [see for example Weil (1973)]. If scattering effect is neglected, then fron the ray theory, the bending of the rays causes deviations of the position of the groind point to be rapped [Eq. (3-4)] and derivation of tine delay and phase shift [Eq. (3-5). If n(z) is known exactly, then for a given a, 3 s = sin8 s = cos3cos(y - a) (4-1) y = COS3sin( - a) s5 = cos3sin(y -c) I 24

015239-1-F one nay integrate (3-4) to obtain the relation between the image Do0nts x, y and the apparent image points [Eq. (2-3)] htan3 X -. r- - -+ X. sin(y - o, 1 and (4-2) y = hcot(y- a) Similarly given any ground point x, y and direction of main beam y, we may determine numerically the corresponding values of a and 3 fror (2-4). These values of a, 3 are then used in (3-4) to determine the correct tine delay and RF phase shift for proper signal Processing. This probably could be done, in principle, by digital processing. If n(z) is not known exactly, but we know that the deviation of n(z) from unity is small, then the approximate equations [Eq. (3-6), (3-7) and (3-8)] can be used to estimate the errors. Let us consider a specific example for which the transmitter is at a height h = 5 km and the direction of the main bean is at y = 20~. For this case, assuming a, 3 small, S = 3 X Sy cosy (4-3) s = siny 9 z the reflection point is approximately given by x3 3h - x- x. - h + [f(h) - f(z)dz] 1 siny sin y (4-4) y tany [f(h) - f(z)dz] sny 25

015239-1-F and h R = h+ 6- [cos2 h) - (1 - 2sin2y) 1 f(z)dz. (4-5) ino 3 s yt[C 2sino) ssiny Jo In the above equations, the second term (involving 6) is due to atmospheric effects. For standard exponential variation f(z) - exp[-0.1439 x 10"3 z] and 6 = 313 x 1305, we find that AX = -8.842 3 n, Ay = -8.309 m, and AR = 4.545 r. It should be noted that (4-4) and (4-5' can also be used to estimate the azinuth and range errors due to cloud layers between the transmitter and the ground. From the water content of the cloud, if the increase in the index of refraction due to each layer is represented by 6.f.(z), the increase of the errors in Ax, Ay and AR are inc 3 i tT inc sin Y i 1JT. Aycote T f (z)dz inc sin3 (1 - 2sin Y) P (z)dz where T. is the thickness of each layer. 26

015239-1-F L.3 Effect Due to weak Turbulence Ir Section 3.2 we postulated that for a transritted signal A(t)Exo(-iw t), the part of the returned signal reflected by the ground at point {x',y'} takes the form A(t,R')ExD[-io t + i6(xrR')] where R' is the distance from the transmitter to the reflecting point when the nth pulse is radiated. For azimuth processing, y' is constant, so that approximately (x' - xn)2 " y I+ 2R (4-6) while for range Drocessing, x' = x R' R + (y' - y)cosY. (4-7) 0 The funclioral foris of A ard 9 are now examined based on tnie results of Section 3-4. Let js represent the transmitted signal by its Fourier transform. - iot A(t) e = jd a(w) exp [-i (w + wo)t (4-8) The returned signal can then be expressed in terms of a log-amplitude fluctuation x and a phase fluctuation s. The result is A(t) exp [-io t + o(xr, )] o r - d2R' = d) a(w) exp [-i'n + o )t + 2 (W + o)] x exp [X(xm, a + o R') + i s(x, w + W0, R')]. (4-9) 27

015239-1-F For azimuth processing R ' is constant, and we may write (4-9) in the fori 2R'w A(t)exp[-iw t + io(x, R')] exp[-iw t + c + x(x, R') 0 c m 0 J. 2R' + is(x,, R')] Jdu a(u)exp[-iot + i c -- (4-10) Approximately, therefore, for azimuth processing, if the variation of x and s with the frequency is ignored, the distortion of the pulse shame is negligible. Since it has been noted that for fixed y, the term involved in the integration is not sensitive to the variation of x', the azimuth resolution is determined by the function [see Equation (2-22)] 2w(x' - x')xn h =Exp-i' + x(xn, W0, R') + x(x, o' R) y + i s(xw, Ro, R') - i s(xr, w, R)] [4-11) [Here we have neglected the quadratic terms in x and x' since they can be eliminated by focussing.] Therefore, the study of the azimuth resolution is equivalent to the study of the pattern of a linear aperture with random error. The pattern of the linear array is 2w Fhx') - 2 exp [-icR0 x'xr + x(x, o, R') + i s(xm, w R')]dxm (4-12) I NvT y 2 For range resolution, x' = x we -ay write (L-12) in the forr 2Rt -(t), ') = exp [-i-wt + i(x R exp [ f 2R' 1 dw a(w) exp [iwt + i c ] exD [x(xm, w+wO, R') - i s(xn, +ao, R')]} 28 (4-13) 28

015239-1-F Consider the terr in the function xv becomes curled bracket of (4-12) as A(t, y'), the arbiguity f 2iw x(R' - Ro R - RQ) = f dw a(w)a*(w) exo [2 (R' - R)] exp [x(xr, u + w0, R') + x(xn, + o0, R)] exp [i s(x, o + O, R')- is(xr, m + o, R)]. For chirpped signal, A(t) =e i t2], (4-14) (4-15) so that, except for a constant be approximated by Rect (2- ). fror the pattern function CaT v(R' - Ro) = Exp[i 2w -C '~- T (assune large time-bandwidth product), a(o) nay Thus, the resolution in range may be estimated (R' - Ro) + x(x, t + o, R') 0m 0 + i s(xm, w + w, R')]do. (4-16) For no pertdroation, (4-16) yields the same resolution as given by (2-27). Since both v and ah are now randoi functions, we iay only estimate the epxected values of the resolution by defining [see Equation (2-20)] rh(xl dx' h> h 2 (4-17) hhO Si-ilarly rb (4-18) 29

015239-1-F In general, in order to find the expected values of A, we need rore statistical descriptions of the random variables x and s. If we assume that they are Gaussian, then in principle we -ay find their correlation coefficient, fror the joint probability density function, and compute the expected values of 3v and 3h. However, since <x2> is very srall within the range of parameter we are interested (see Section 3.4), they ray be neglected. The resolution probler is then reduced to the problei of pattern deterioration due to randor phase variation across a linear aperture. This problem is discussed thoroughly by Brown and Riordan (1970), Their approach can be readily adopted in the present investigation. By neglecting x, we have fron (L-12), fvT 2 i% h(x) = exp[-cR- x'x + i s(xm,, R')]dx 4-19) -J' vT y 2 From Parseval's relation, NvT 2 2 '(xT) dx' = = NvT (4-20) - NvT 2 which is independent of the random phase shift. Thus it is seen from '2-5) that the expected resolution due to randor phase is 2 h IFh(~) (vT * (4-21) <3h>: 2 - (4-21) h <h(O) > < h( O) 2 > Now, rOvT NvT o) 2= Jdx| dx1' exp[i s(xr, R ')] (L-22) - NvT NvT 2 2 30

015239-1-F hence, the expected value of ilh(o) depends on the phase structure ijnction between two points xn and x'm. If we approximate the antenna bear in the forward direction as a part of tqe plane wave, then these two poirts are transverse to the direction of propagation, with L = Ry' and a separation distance of y and a separation distance of o = X - x' 2 The expected value of Fh(o)I can then be expressed in terms of the phase structure function D (AL, o) [see Equation (3-35)]. The result is 5k (4-24),NvT rNvT <lh(o) > =, dx N dx ' exp[- 2 Ds(o, Ix,' NvT NvT 2 2 - x I) m. (4-25) Similarly, neglecting x in (4-16) > v(O) 2 = d, - I IT -aT dw2 exp[i s(xn, 1 + o)o YO) - i s(xm, 2 + Wo' yo) 2T ~ Yl-iSXVw 5Y~ (4-26) Ferce, the expected value of <I!v(o) > nay be expressed as <? v() > = I J -OC dw JJ- OtT dw2 exp[- sIl' w2' yo)] (4-27) where -s(l' -2' Y ) is the two frequency coherence structdre junction derived in (3-45). 31

015239-1-F <nowing the functional forns of Ds and r, (4-22) and (4-25) grated at least numerically. From (3-35), we may represent can be inte D (o, x ' - x ) - b x' - xl S n 5 m f where. n = 5/3 and for two-way propagation b - 4 x 1.46 k2C2L. n (4-28) (4-29) Fo the parameters of interest for this work, b gration may be carried ojt by series expansion. deterioration of azimuth resolution is given by -3 % 10, hence approximate Approximately therefore, intethe <3h > n bln" -1 h + = nb(vT -1 3h (n+l-)(n+2] (4-30) Similarly, for the azi-uth resolution (3-44) may be represented by n s- W1' W2' -) = b' w1 - m2 (4-31) where r' = 2. and Cu 28A 2 <nI A2 b' - 28/ <n 2 L a2 c2 1 o C (4-32) Again, b' is a very small quantity for the ranges of parameter of the present investigation. Therefore the deterioration of the range resolution is given by <3v> - [1 - b'(2T"n 3 (n'+l)(n'+2) V -1 (4-33) 32

015239-1-F Based on (4-30) and (4-33), one may also investigate the problem of optimal choice of the syster parameters (NvT) and (Tr) to achieve the best expected resolution. Since AR 3 2 (4-34) r 2NvT and 3 =, (4-35) V 2a ' the mathematical problen of ootimization is the same. The result is 1 (N T ) n (4-36) Similarly, 1 1 n'+ 2 r' ( op 2 '- (4-37) ("T)opt -b The oDtirur resolutions are given by AR, 1 <3 pt =2Y b (n) (4-38) h opt 2 (n) and 1n <3v>opt = b'n Cln') (4-39) where C(r) = 1(n+ (4-40) n(n+2)p For r = 2, C(n) = 0.75 while for r = 5/3, C(n) = 0.0733. For a sa-ple calcjlation using the above results, let us consider an idealized syster with the following oarameters: f = 8 x 109 Hz and L = 104 n, and the ideal resolution ignoring the turbulence effect are 33

015239-1-F 3 n 3v 3h 3 corresponding to values of NvT = 62 n and AT - 0.524 c (sec-1) Under the influence of weak turbulence specified by the following parameters 2 -13 2/3 C2 = 2 x 101 (n-213) n 2 -13 <n1 > = 1.4 x 1013 and z= 6 -. 0 0 The following numerical results may be easily conputed a' b = 3.28 x 103 [Eq. (4-29)] b' 556x 10-6 [Eq. (4-32)] 2 C b) Deterioration of the expected azinuth resolution (Eq. (4-30)] 3h - = 1.03 c) Deterioration of the expected range resolution [Eq. (4-33)] =1 d) Dptimu" choice of MvT for best expected resolution [Eq. (4-35)] (NvT) 268.,5 n oot 34

315239-1-F e) Optirui range resolution [Eq. (4-38)] h opt = 2.2 m f) Optinum choice of aT for best expected resolution [Eq. (4-37)] (a)opt = 424 c g) OQtiwum range resolution [Eq. (4-39)] <3 > opt = 7.4 x 103 n Although the nurerical values concerning the range resolution appears to be unrealistic due to the set of turbulence parameters we assumed, it appears that in general, weak turbulence has negligible effect in range resolution and slight effect on the azimuth resolution. 4.4 Effects of Strong Turbulence and Discrete Scatters Based on the analysis of Section 4.3, it is seen that the two functions relevant to the analysis of SAR resolution are: a) The spatial correlation between the fields u(r) at two points transverse to the direction of propagation. This is a spatial rutual coherence function commnonly denoted by <u(L, p) j*(L, o*) = -(a, L, o -p ). i(4-41 ) b) The frequency correlation of the field u(r) at two different frequencies, w1 and T2' -his is commonly known as two frequency nutual coherence function [Hong, Screenivasiah and Ishinaru (1977)], denoted by r(wl' w2, rl, r2 tl' t2) = <u(wl,1' tl)u*(W2, r2, t2)>. (4-42) 35

015239-1-F (In our application, we need only the finction for z = = L, = and 12 1 2 t = t2.) In our analysis, due to the assumption of weak turbulence, we find approximate solution for u(r, t) first by Born approximation, and then obtain the I's by assuming Gaussian distribution of the fluctuation. For strong turbulence, and for the analysis in a medium with discrete scatters (rain, snow), this analysis is inadequate. Improving the approximate solution for u by using iteration would be too complicated to get meaningful results- Recent research work in the field, therefore is concerned in obtaining approximate differential equations 4or F. The resulting equations are more complicated than the Helrholtz equation and numerical techniques are required to obtain the solutions. In this section, the equations for r of our interest are briefly reviewed. Let us start with tie scalar Felmholtz equation V 2+ k2(1 + e1) = O. (4-43) For plane wave propagation,'let = u eik, (4-44) and obtain approximately ik au U + v u + k. (4-45) The average field <u> = u cannot be obtained directly without further approximations. The jsual assumption is that u is aMlarkov process in z, or equivalently E1 is delta correlated in x, i.e., 3 (z- z', D) = 6(z- z') A(I), (4-46) A(D) is related to the spectral density of ~ by A(D) = 2 J (K) e d. (4-47) 36

015239-1-F Lsing this assumption we can deduce that 2ik < u+> + A(o2<u> = O, (4-48) ~(z, od) 2 ik3 2ik z d + vd (z, d ) + [A(o) - A(Od)] r(z, O) = (C-49) where =d = P 2 - Eouation (4-49) introduces an attenuation to the average field which is consistent to the physical picture o- that coherert field is attenuated dje to scattering. Equation (0-49) -ay be used for the problem involving discrete scatterers by proper interpretation of A(o) and A(0) for such nedium. Fron '4-48), we see that <u> is atterjated according to exp[-A(okz] and froT transport theory, we know that <u> is attenuated according to not 2 e where at is extinction cross section of a particle, and not is the average extinction cross section per unit volume. Given a distribution of scatterers, we can therefore corrpute nat, and identify 4nat A(o) =, (4-50) k the furction A(p) is not directly identified for discrete scatterers without further assjrptions. Fron cross section theory of scattering, we know at k f(i i) (4-51) k' 37

315239-1-F where f(s, i) is the complex arplitude of single scattering of waves into direction s due to a wave of unit amplitude incident from direction i. Also, from transport theory, we know the field satisfies approximately V2<> + 2<> = O, (4-52) where K = k + 2 n f(i, i)/k (4-53) A = K +i K. r 1 which is related to the scattered amplitude in the forward direction. By considering f(s, i) as the angular spectrum of scattered field, and A(D) as the spatial tlransfor- of the angular distribution, one can then argue that approximately, A ^ ^ ^ i #, d 1. A(d) = nj f(s, i) f*(s, i) e d. (4-54) Thus, knowing the size and density distribution of scatterers, one iay compute A(D) and solve r fron (4-49). It is to be noted, however, due to the assumption that a is a Markov process in z, we are physically limiting our problem to forward scattering, so that (4-49) is to be used for large (compared to wave length) particles. Differential equations for the two frequency mutual coherence functions are deduced from (4-L9). The resulting equation is [- 2 )\v( L - iK - (iK - tA(d)](, 2, z, dd ) = 0. (4-55) 1 2 - 1 2 I 2 ) A ( d - ] ( d In literature, available solutions of (4-49) and (4-55) are all by numerical methods, and not enough data is available for innediate application to our problem. Some consideration has been given to choosing a model for the size and particle distributions of rain, and then carrying out a numerical solution of Equations (4-49) and (4-55) for the analysis of SAR resolution, but the task is too elaborate to accomplish in this period of research. 38

015239-1-F 5. REFERENCES Brown, W.M. and J.F. Riordan, "Resolution limits with propagation phase errors", IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-6, N'o. 5, Dp. 657-662, September 1970. Brown, W.M., "Synthetic aperture radar", IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-3, pp. 217-229, March 1967. Cutrona, L.J., W.E. Vivian, E.1. Leith and G.O. Hall, "A high resolution radar corbat system", IRE -rars. on Military Electronics, Vol. MIL-5, op. 127 -131, April 1961. Chernov, L.A., Wave Propagation in a Random Medium, McGraw-Hill Book Co., New York, N"., 1960. Garfinkel, B., "Astronomical refraction in a polytropic atmosphere", Astron. J. Vol. 72, pp. 235-254, 1967. Hong, T.H., I. Screenivasiah and A. Ishiiari, "Plane wave pulse propagation throUgh random media", IEEE Trans. on Antenna and Propagation, Vol. AP-25, No., pp. 822-828, November 1977. Ishinaru, A., "Theory and aDplication of wave propagation and scattering in randor media", Proc. IEEE, Vol. 65, po. 1030-1061, July 1977. Ishimaru, A. and S.T. Hong, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles", Rad. Sci. 13, Vol. 6, pp. 637-644, June 1975. Lawrence, R.T. and J.A. Strohbehn, "A survey of clear air propagation effects relevant to optical communications", Proc. IEEE, Vol. 58, No. 10, pp. 1523 -1545, 1970. 39

01 5239-1 -F Staras, F. ard A.D.,heelon, "Theoretical research on tropospheric scatter propagation in the United States", IRE Trans. AP, Vol. AP-7, pp. 80-87, January 1959. Tartarski, V.I., "Light Propagation in a rediur with randon reflective index inhonogeneouneities in the Markov random process approximation", Soviet Physics, JETP, Vol. 29, No. 6, pp. 1133-1138, Deceibe' 1969. Tartarski, V.I., Wave Propagatior in a Turbulent Mediun, McGraw-Fill Book Co., fiew York, 'Y, 1951. Weil, T.A., "Atmospheric lens effect, another loss for the radar range equation", IEEE Trans. Aerospace and Electronic Systems, Vol. AES-9, pr. 51-54, January 1973. 40