026511-1-T Derivation of Phase Statistics of Distributed Targets from the Averaged Mueller Matrix Author: Kamal Sarabandi April 1991 26511-1-T = RL-2409

026511-1-T DERIVATION OF PHASE STATISTICS OF DISTRIBUTED TARGETS FROM THE AVERAGED MUELLER MATRIX by Kamal Sarabandi Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan April 1, 1991 PI: Fawwaz T. Ulaby

AbstractTo understand the importance of radar polarimetry for remote sensing of random media, statistics of the phase difference of the scattering matrix elements must be studied. In this paper the probability density function of the phase differences is derived from the averaged Mueller matrix assuming that the elements of the scattering matrix are jointly Gaussian. It is shown that the probability density functions of the co-polarized and cross-polarized phase differences are similar in form and can be obtained independently. The expressions for the probability density functions are verified by comparing the histograms, the mean, and the standard deviations of phase differences derived directly from polarimetric measurements of variety of a rough surfaces to the probability density function and its mean and standard deviation derived from the averaged Mueller matrices of the same data. The expressions for the probability density functions are of special interest for non-coherent polarimetric radars and non-coherent polarimetric models for random media such as vector radiative transfer. i

Contents 1 Introduction 1 2 Theoretical Derivation of Phase Difference Statistics 3 3 Comparison with Measurements 12 4 Conclusions 14 ii

List of Figures 1 The probability density function of the co-polarized phase difference for a fixed value of a (degree of correlation) and five values of C (coherentphase-difference)................................. 18 2 The probability density function of the co-polarized phase difference for a fixed value of C (coherent-phase-difference) and four values of a (degree of correlation).................................. 19 3 The mean value of the co-polarized phase difference as a function of a (degree of correlation) and C (coherent-phase-difference)...............20 4 The standard deviation of the co-polarized phase difference as a function of a (degree of correlation) and 4 (coherent-phase-difference)............ 20 5 The histogram of the real and imaginary parts of Svv and Shh for a rough surface with rms height 0.32 cm and correlation length 9.9 cm at C-band and 30~ incidence angle............................. 21 6 The histogram and p.d.f. of the co-polarized phase difference for a rough surface with rms height 0.32 cm and correlation length 9.9 cm at C-band and 30~ incidence angle............................. 22 7 The histogram and p.d.f. of the cross-polarized phase difference for a rough surface with rms height 0.32 cm and correlation length 9.9 cm at C-band and 30~ incidence angle........................... 23 iii

8 Angular dependency of the mean of the co-polarized phase difference for a dry rough surface with rms height 0.4 cm and correlation length 8.4 cm at L- and X-band............................... 24 9 Angular dependency of the standard deviation of the co-polarized phase difference for a dry rough surface with rms height 0.4 cm and correlation length 8.4 cm at L- and X-band......................... 25 iv

List of Tables 1 Normalized covariance matrix of co-polarized terms of scattering matrix for a surface with rms height 0.3 cm and correlation length 9 cm at C-band and at 30 degrees incidence angle........................ 17 2 Normalized Mueller matrix for a surface with rms height 0.3 cm and correlation length 9 cm at C-band and at 30 degrees incidence angle........ 17 v

1 Introduction In the past decade substantial effort within the microwave remote sensing community has been devoted to the development and improvement of polarimetry science. Polarimetric radars are capable of synthesizing the radar response of a target to any combination of the receive and transmit polarizations from coherent measurements of the target with two orthogonal channels. Polarimetric radars have demonstrated their abilities in improving point-target detection and classification [Ioannidis and Hammers, 1979]. That is, for a point target in a clutter background the transmit and receive polarizations can be chosen such that the target to clutter response is maximum. Also, different point targets in the radar scene can be classified according to their optimum polarization. Although radar polarimeters have shown a great potential in point-target detection and classification, their capabilities in remote sensing of distributed targets is not completely understood yet. Considering the complexity involved in designing, manufacturing, and processing the data of an imaging polarimeter as opposed to a conventional imaging radar, it is necessary to examine the advantages that the imaging polarimeter provides about the targets of interest. For example, in retrieving the biophysical parameters from the polarimetric radar data one should ask whether there exists a dependency between the parameters and the measured phase of the scattering matrix components. If the answer is negative, obviously gathering polarimetric data for inversion of that parameter is a waste of effort. One way of confirming this question is by collecting data over a range of the desired parameter while keeping other influential parameters constant. This procedure, 1

if not impossible, is very difficult to conduct because of problems in repeatability of the experiment and difficulties in controlling the environmental conditions. Moreover at high frequencies (millimeter-wave frequencies and higher) coherent measurement of the scattering matrix is impossible because of instabilities of local oscillators and relative movements of the target and the radar platform [Meads and McIntosh, 1991]. At these frequencies non-coherent radars are employed which provide the Mueller matrix of the target. Another approach to examine the dependency of the radar response to the desired parameters of the targets is the application of theoretical models. One of the most successful polarimetric models for random media is the vector radiative transfer theory [Tsang et al., 1985]. This model is based on conservation of energy and the single scattering properties of the constituent particles. The solution of the radiative transfer equation relates the scattered-wave Stokes vector to the incident-wave Stokes vector via the Mueller matrix. The Mueller matrix, as computed by this method, is an ensembleaveraged quantity because of the inherent nature of the radiative transfer theory. Since the Mueller matrix is related to the scattering matrix through a nonlinear process and the components of the scattering matrix are statistically dependent, the information about the phase difference of the scattering matrix components cannot be obtained from the Mueller matrix directly. To achieve information about the phase statistics, one may resort to the Monte Carlo-type models [Chuah and Tan, 1989] which are computationally inefficient and in general inaccurate. Experimental observations of phase difference statistics from a polarimetric SAR at L-band [Ulaby et al., 1987; Zebker et al., 1987] over agricultural terrain and bare soil 2

surfaces indicate that the statistics of the co-polarized phase difference depends on the target type and its conditions. Recent measurements of bare soil surfaces by polarimetric scatterometers show that the variance of the co-polarized phase difference is a function of the roughness parameters and incidence angle but is less sensitive to moisture content [Sarabandi et al., 1991]. In view of difficulties in measuring the scattering matrix at high frequencies and performing controlled experiments, it is necessary to establish a relationship between the averaged Mueller matrix and the statistics of the phase differences of the scattering matrix elements. In the next section we derive the probability density function of the coand cross-polarized polarized phase difference in terms of the averaged Mueller matrix elements assuming that the scattering matrix elements are jointly Gaussian. Then the assumptions and final results are compared with the experimental data acquired by polarimetric scatterometers in Section 3. 2 Theoretical Derivation of Phase Difference Statistics The polarimetric response of a point or distributed target can be obtained by simultaneously measuring both the amplitude and phase of the scattered field using two orthogonal channels. If the incident and scattered field vectors are decomposed into their horizontal and vertical components, the polarimetric response can be represented by the scattering matrix S, which, for plane wave illumination we can write e ikr SVV Svh E = IS (l) [ShV Shhj 3

where r is the distance from the radar to the center of the distributed target. It should be noted that in the backscattering case reciprocity implies Syh = Sh. Each element of the scattering matrix, in general is a complex quantity characterized by an amplitude and a phase. When the radar illuminates a volume of a random medium or an area of a random surface, many point scatterers contribute to the total scattered energy received by the radar and therefore each element of the scattering matrix may be represented by N Spq = \Spq ei5Pq = - pqe p,q = v,h. (2) n=l Here N is the total number of scatterers each having scattering amplitude iSq and phase qpnq. It should be mentioned that the phase of each scatterer, as given in (2), includes a phase delay according to the location of the scatterer with respect to the center of the distributed target. Without loss of generality all multiple scattering over the surface or in the medium can be included in (2). Since the location of the scatterers within the illuminated area (volume) is random, the process describing the phasor Spq is a Wiener process (random walk) [Davenport, 1970]. If N is large enough, application of the central limit theorem shows that the real and imaginary parts of the scattering matrix element Spq are independent identically distributed zero-mean Gaussian random variables. Equivalently it can also be shown that ISpq I and qpq are, respectively, Rayleigh and uniform independent random variables. The three elements of the scattering matrix, in general, can be viewed as a six-element random vector and it is again reasonable to assume that the six components are jointly Gaussian. Observation of polarimetric data for a variety of distributed targets such as bare soil surfaces and different kinds of vegetation-covered terrain all indicate that the cross 4

polarized component of the scattering matrix (Shv) is statistically independent of the copolarized terms (S,, and Shh). Therefore the statistical behavior of Sh, can be obtained from a single parameter, namely the variance (o2) of the real or imaginary part of Shy = X5 + iX6, that is 2 2 \ 1x + x6 fx5,X (x5,X6) = 2 exp[- + ] or equivalently the joint density function IShl and dbvh is 1 ISh 12 fIsvl,,vh(lvhl, vh)- = 2 aexp[- 2-2 1 (3) which indicates that vh is uniformly distributed between (-T, +Xr). Since measurement of the absolute phase of the scattering matrix elements is very difficult, it is customary to factor out the phase of one of the co-polarized terms, for example Sv,, and therefore the phase difference statistics are of concern as opposed to the absolute phases. Since Shv is assumed to be independent of Svv and both hv and,,vv are uniformly distributed, it can be easily shown that the cross-polarized phase difference Oc = bvh - Ovv is also uniformly distributed between (-7r, +r). The co-polarized elements of the scattering matrix, however, are dependent random variables which can be denoted by a four-component jointly Gaussian random vector X. Let us define Svv = Xl + iX2 Shh = X3 + iX4 and since X1,.. X4 are Gaussian their joint probability density function can be fully determined by a 4 x 4 symmetric positive definite matrix known as covariance matrix 5

(A) whose entries are given by [2] Aij = Aji =< XiXj > i,j E {1,.,4} The joint probability density function in terms of the covariance matrix takes the following form: 1expl —XA -'I fX(X'***,X4)= 4r2A[ exp - [ XA-X] (4) where X is transpose of the column vector X. To characterize the covariance matrix the following observations are in order. First, it was shown that the real and imaginary parts of the scattering matrix elements are mutually independent and identically distributed zero-mean random variables, therefore Al = A22 =< X2 >=< X2 > (5) A12 =< X1X2 >= (6) A33 = A44 =< X2 >=< X4 > (7) x4 =< X3X4 >= 0 (8) Second, it was shown that the absolute phase Opp is uniformly distributed and is independent of ISppI. Thus the random variable q, + >hh is also uniformly distributed and is independent of ISvvIlShhl from which it can be concluded that < ISvv IShhI coS(+vv + hh) >= 0 (9) < ISvvIIShh sin(bvv + Ithh) >= 0. In fact, the complex random variable SsvShh is obtained from a similar Wiener process 6

which led to the random variables S,, and Shh. On the other hand X1X3 - X2X4 = ISvvllShh CoS( + hh) (10) X1X4 + X2X3 = ISvIllShhl sin(vv- + bhh). In view of (9) and (10) it can easily be seen that A13 = A24 (11) A14 = -A23 (12) The properties derived for the entries of the covariance matrix, as given by (5)-(8) and (11)-(12), indicates that there are only four unknowns left in the covariance matrix. The unknowns, namely All, A13, A14, and A33, can be obtained directly from the averaged Mueller matrix of the target as will be shown next. The Mueller matrix relates the scattered-wave Stokes vector to the incident-wave Stokes vector by [van Zyl and Ulaby, 1990] FV = 2 MF r2 where F',s the modified incident- and scattered-wave Stokes vector defined by IEvl2 F= Eh12 23[EvE?] 2Q[EvE*] The Mueller matrix can be expressed in terms of the elements of the scattering matrix 7

as follows [6] I Svv2 ISvh12 [Sh SVV] -2[S[h Svv] IShv12 IShhI2 R[Sh Shv] -![S[hh Shv] M = 2[SvvS 2R[ShvSh] [SVVSh + ShvSh] -a[SvvS-h - ShvS*h] a2[SvvShv] 29[ShvShh] ~[SvvShh + ShVSh] J[SVShh - ShvSvh] In the case of a random medium we are dealing with a partially polarized scattered wave and the quantity of interest is the ensemble averaged Mueller matrix. Using the fact that the co- and cross-polarized terms of the scattering matrix are independent and employing the properties given by (5)-(8) and (11)-(12), the averaged Mueller matrix in terms of the entries of the covariance matrix is given by 2A11 22 0 0 2oa2 2A33 0 0 M =< M >= (13) 0 0 2A13 + 2a2 2A14 0 0 -2A14 2A13- 22 Equation (13) provides enough equations to determine the unknown elements of the covariance matrix and variance of the cross-polarized component, i.e. 0i- M1L c 2 2 All,' A33 = "22 2 ~ -33 -- 2 A13 M- +M4 14 = 3 4 ' 2 With the covariance matrix, the joint density function of X1,, X4 can be obtained as given by (4). Using a rectangular to polar transformation, i.e. X1 = pl cos vv, X2 = Pi sin qw X3 = p2 cos qhh, X4 = 2 sin qhh 8

the joint probability density function of the amplitudes and phases takes the following form 11 +22 2 l (14) fP1,P2,',vv hh (Pl i P2, Ivv, 'hh) = 4 2x/p1p2exp -- [apl1 + a2p - 2a3plp2]j (14) where = Al = (A1133- A3 - 4)2 al = A33/%/f, a2 = A11/VA a3 = [A13 coS(hh - vv) + A14 sin(hh - 4Om)] /V/ To obtain the co-polarized phase difference statistics the joint density function of Ovv and qhh is needed which can be obtained from f',VV hha (vv, ' hh) = J fp1,P2,,vv,hh (P1, P2, vv ihh)dpldp2 (15) Noting that a1 is a positive real number, the integration with respect to pi can be carried out which results in d1 l1 fo~ -pePdp2 f'v 'hh (Ofvv, hh) =- 7r 2- a P2e 2 e2dP2 +0-a3] p2 l1~erf( p2-aa2-a2)p2 +1 3^a3 \ 2 P ^ 2) e-a 1 2P2( )a 2Jo d p (16) where erf(.) is the error function and the plus or minus sign is used according to the sign of a3. To evaluate the integrals in (16), we need to show that both a2 and a1a2 - a3 are positive numbers. By definition, a2 is positive and to show ala2 - a2 is positive we note that A is a symmetric positive definite matrix, therefore its eigenvalues must be positive. It can be shown that A has two distinct eigenvalues 71 and 72 each with multiplicity 2 and their product is given by 7172 = All33 - A3 - A24 > 0. 9

Thus ala2 - a= 7172 + [A13 cos()hh - vv) - A14 sin(qhh - vv)]2 is positive. After integrating the first integral and the first term of the second integral in (16) directly and using integration by parts on the second term of the second integral, (16) becomes fivv,hh (vv, 7hh) = 2- + - 3 42vrA 1/ a2 ala2(ala2 - a - 32) +.2la)- 2 erf( p2)e-2La - 32dp2 -8/li(ala2 - ao)J o e /r 8 / By expanding the error function in terms of its Taylor series, interchanging the order of summation and integration, then using the definition of the Gamma function it can be shown that ( I [oerf( la31 p2)e- 2al(lP2-)3 dp2 = 2 2 tan-1 ( a O ir( ala2 - a) \2/ala2 -a The joint density function of q,, and 'hh is a periodic function of 4 = qhh - vv and therefore the random variable X, after some algebraic manipulation, can be shown to have the following probability density function over the interval (-7r, +r) A 11A33- A12 -- A4 2 1 D [ 2 D ]} f ) = 21 (1A 313- D {1 + D- [+ tan-11 D (17) 2r%(A11 A33- D2) 1133D2 where we recall that D = A13 cos X + A14 sin X and the elements of the covariance matrix in terms of the Mueller matrix elements are 10

given by Mll M22 A11 = 2A33 - ' 2 2 M33 + M44 A34 - -43 A13 = A14 = Some limiting cases can be considered in order to check the validity of (17). For example, when Sv,, and Shh are uncorrelated, then both A13 and A14 are zero for which fs(e) = 1/(27r), as expected. Also, for the case of completely polarized scattered wave where Svv and Shh are completely correlated, the determinant of A is zero and so f<(O) is a delta function. It is interesting to note that the p.d.f. of the phase difference is only a function of two parameters defined by..1 ( = tan-14 V 11i 33 A13 where a and C can vary from 0 to 1 and 0 to 27r respectively. In fact if the wave were completely polarized, C would have been the phase difference between the co-polarized terms. The parameter 4 will, henceforth, be referred to as the polarized-phase-difference. In terms of these parameters (17) can be written as 1- a2 =(0) 2 [1 - 2 cos2( - )] a cos(+-) ai acos(2) - -] -11 { 1 l-c2COS(-O [-2 +tan - COS(-o It can be shown that the maximum of the probability density function occurs at q = C independent of a. However, the width of the p.d.f. (e.g. the 3 dB angular width) is only a function of a which will be referred to as the degree of correlation. Figure 1 shows the p.d.f. for different values of 4 while keeping a constant, and Fig. 2 shows the p.d.f. for a fixed value of a while changing C as a parameter. The calculated mean and standard deviation of the phase difference as a function of both the polarized-phase-difference and 11

the degree of correlation are depicted in Figs. 3 and 4 respectively. Lastly, it is necessary to point out that the formulation of the co-polarized phase difference p.d.f., as given in (17), is not restricted to the backscattering case or to the co- and cross-polarized components being uncorrelated. In fact we can derive the crosspolarized phase difference statistics in a similar manner and the p.d.f. in this case for the backscattering case can be obtained from (17) upon the following substitution for the elements of the cross-polarized covariance matrix A11 M1 A-33 M12 2 2 213= 14= _ 2 2 3 Comparison with Measurements Using the polarimetric data gathered by scatterometers from a variety of natural targets, the assumptions leading to the probability density function of phase differences as derived in the previous section are examined. Also by generating the histograms, means, and standard deviations of the phase differences from the data and comparing them with the results based on the p.d.f. derived from the measured averaged Mueller matrices validity of the model is also examined. The polarimetric radar measurements of bare soil surfaces were performed at L-, C-, and X-band frequencies for a total of eight different soil surface conditions (four roughness and two moisture conditions). For this experiment we tried to preserve the absolute phase of the measured scattering matrix by calibrating the surface data with a metallic sphere located at the same distance from the radar as the center of the surface target. For each frequency, surface condition, and incidence angle a minimum of 700 independent samples were collected. The detailed procedure of 12

the data collection and calibration is given in reference [5]. By generating the histograms of the real and imaginary parts of the elements of the scattering matrix for all surfaces, it was found that they have a zero-mean Gaussian distribution as we assumed. Figure 5 represents a typical case where the histogram of the real and imaginary parts of S,, and Shh of a dry surface with rms height 0.32 cm and correlation length 9.9 cm at C-band have a bell-shaped distribution. The properties of the covariance matrix as given by (5)-(8) and (11)-(12) are verified by calculating the covariance matrices of the data for all cases. Table 1 represents a typical situation where the covariance matrix for the same rough surface at C-band possesses the mentioned properties approximately, that is A 11 A22, A12 A34; 0, A33 A44, A13 w A24, and A14 -A23. The small discrepancies are due to the fact that the measurement of the scattering matrix with absolute phase has an uncertainty of ~30 degrees. Table 2 gives the averaged Mueller matrix of the typical surface (Table 1) at C-band from which the co- and cross-polarized phase difference probability density functions are calculated using (17) and are compared with the measured phase histograms in Figs. 6 and 7 respectively. Similar comparisons were also made for the rest of surfaces, frequencies, and incidence angles and it was found that the expression (17) predicts the density functions very accurately. Some example of these comparisons are shown in Figs. 8 and 9. Figures 8 and 9 compare the mean and standard deviation of the co-polarized phase difference versus incidence angle at L- and X-band for a surface with rms height 0.4 cm and correlation length 8.4 cm in dry condition using the results based on the direct calculation and the results derived from (17). 13

4 Conclusions The statistical behavior of the phase difference of the scattering matrix elements for distributed targets is studied. The probability density functions of the phase differences are derived from the averaged Mueller matrix of the target. The derivation of the density functions assumes that the real and imaginary parts of the co- and cross-polarized terms of the scattering matrix are jointly Gaussian and their covariance matrices are found in terms of the averaged Mueller matrix. The functional form of the co- and cross-polarized density functions are similar and are obtained independently. The assumptions and final expressions are verified by using a set of polarimetric data acquired by scatterometers from rough surfaces. 14

References [1] Chuah, H. T., and H. S. Tan, "A monte Carlo method for radar backscatter from a half-space random medium," IEEE Trans. Geosci. Remote Sensing, vol. 27, no. 1, 1989. [2] Davenport, W. B., Probability and random processes, New York: McGraw-Hill, 1970. [3] loannidis, G.A. and D.E. Hammers, "Optimum antenna polarization for target discrimination in clutter," IEEE Trans. Antennas Propagat., vol. 27, no. 5, 1979. [4] Meads, J. B., and R. E. McIntosh, "Polarimetric backscatter measurements of deciduous and coniferous trees at 225 GHz," IEEE Trans. Geosci. Remote Sensing, vol. 29, no. 1, 1991. [5] Sarabandi, K., Y. Oh, F.T. Ulaby, "Polarimetric radar measurement of bare soil surfaces at microwave frequencies," Proc. of IEEE Geosci. Remote Sens. Symp., Espoo, June 1991. [6] Tsang, L., J.A. Kong, and R.T. Shin, Theory of Microwave Remote Sensing, New York: John Wiley and Sons, 1985. [7] Ulaby, F. T., D. Held, M.C. Dobson, K.C. McDonald, and T.B.A. Senior, "Relating polarization phase difference of SAR signals to scene properties," IEEE Trans. Geosci. Remote Sensing, vol.25, no. 1, 1987. 15

[8] van Zyl, J.J., and F.T. Ulaby, Radar Polarimetry for Geoscience Applications, Chapter 2, ed. F.T. Ulaby and C. Elachi, Norwood, MA: Artech house, 1989. [9] Zebker, H.A.,.J. van Zyl, and D.N. Held, "Imaging radar polarimetry from wave synthesis," J. Geophys. Res., vol. 92, no. B1, 1987. 16

1.00 0.03 0.75 -0.12 0.03 0.90 0.08 0.68 0.75 0.08 0.77 0.05 -0.12 0.68 0.05 0.69 Table 1: Normalized covariance matrix of co-polarized terms of scattering matrix for a surface with rms height 0.3 cm and correlation length 9 cm at C-band and at 30 degrees incidence angle. 1.000 0.028 0.000 0.000 0.030 0.767 0.000 0.000 0.000 0.000 0.770 0.110 0.000 0.000 -0.11 0.711 Table 2: Normalized Mueller matrix for a surface with rms height 0.3 cm and correlation length 9 cm at C-band and at 30 degrees incidence angle. 17

9MISSIN

> 0 -a) 4 -a) *"2 C2 >.,.0 0 0.020.... I.... i a=0.8 0.015 C=180 0.010 0.005 0.000.. -180. -135. -90. I -45. 0. 45. ohh-0w (Degrees) 90. 135. 180. Figur: The probability density function of the co-polarized phase difference for a fixed value of a (degree of correlation) and five values of ( (coherent-phase-difference). 19

0.020., =45 1 /\ I I /I 0.015 - Ia I. ) I I - I 1 = 0.010 - I >I 'a=O.61. I l 2 - '/ \ E!I/' -\\ L=0 /- '\'-,. -1 / _ \\,.0.000 \ -180. -135. -90. -45. 0. 45. 90. 135. 180. Q(hh-Dw (Degrees) Figure 2: The probability density function of the co-polarized phase difference for a fixed value of C (coherent-phase-difference) and four values of a (degree of correlation). 20

I 300 0.75 Figure 3: The mean value of the co-polarized phase difference as a function of a (degree of correlation) and ( (coherent-phase-difference). 4 --o (degree of correlation) and (coherent-phase-1difference).00 00 a 0.7 (degree of correlation) and ( (coherent-phase-difference). 21

0.500;.0N En d) aI 0 m z0 0 1 — 91 0.400 0.300 0.200 0.100 0.000 IC a-. I I. -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 Normalized Magnitude Distribution 3.0 4.0 Figure 5: The histogram of the real and imaginary parts of S,, and Shh for a rough surface with rms height 0.32 cm and correlation length 9.9 cm at C-band and 30~ incidence angle. 22

-, bX. VcD C:) 4-A.o,, -0 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 _-r -180. ~ II I '.T I " ' - ll Measured (Mean=-5.0~, Std. Dev=47.7~) - Calculated (Mean=-7.8~, Std. Dev=47.2~) - r r, i!!! r r r r r -I --- — I 8 j i 0 0 - 120. -60. -120. -60.!. I - 'r' l I I I I I I I ~ I ~~~ ~ -.- - - -.. 0. (Dhh - DVV 60. 120. 180. Figure 6: The histogram and p.d.f. of the co-polarized phase difference for a rough surface with rms height 0.32 cm and correlation length 9.9 cm at C-band and 30~ incidence angle. 23

0.020 0.018 - Measured (Mean=-1.0~, Std. Dev=102.5~) --------- Calculated (Mean=0.8~, Std. Dev=103.7~) 0.016, 0.014 e 0.012 = 0.010.4^ 0.008 0 0.006 0.004 0.002 0.000 -180. -120. -60. 0. 60. 120. 180. )hv - vv Figure 7: The histogram and p.d.f. of the cross-polarized phase difference for a rough surface with rms height 0.32 cm and correlation length 9.9 cm at C-band and 300 incidence angle. 24

60. 50. 40. I I I I I I X-band (Calculated) 0 X-band (Measured) --------- L-band (Calculated) E L-band (Measured) I 0 co 4 - 30. - 20. - 8 ---- --- - " El; E - -- 10. 1 o.[ D El -20. - -30. I a I I 0. 10. 20. 30. 40. 50. 60. 70. 80. Incidence Angle (Degrees) Figure 8: Angular dependency of the mean of the co-polarized phase difference for a dry rough surface with rms height 0.4 cm and correlation length 8.4 cm at L- and X-band. 25

90. I 80. CA 0 ti To 0.o 70. 60. /,," Q0 /e ", ---[D n E)~I~ I"-..~ I] 50. - 40. 30. 20. X-band (Calculated) 0 X-band (Measured) --------- L-band (Calculated) o L-band (Measured) I I I I I I, 10. 1 0. I 0. 10. 20. 30. Incidence 40. 50. 60. 70. 80. Angle (Degrees) Figure 9: Angular dependency of the standard deviation of the co-polarized phase difference for a dry rough surface with rms height 0.4 cm and correlation length 8.4 cm at L- and X-band. 26