A Statistical Kirchhoff Model for EM Scattering from Gaussian Rough Surface

Transcription

1 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August A Statistical Kirchhoff Model for EM Scattering from Gaussian Rough Surface Yang Du 1, Tao Xu 1, Yingliang Luo 1, and J. A. Kong 2 1 Zhejiang University, China 2 Massachusetts Institute of Technology, USA Abstract In this paper we propose a statistical Kirchhoff model (SKM) for shadow-corrected EM scattering from a rough surface. It treats the local coordinates and Fresnel reflection coefficients statistically over the orientation distribution of surface unit norm as characterized by the joint probability distribution function of its two directional slopes. In calculating the incoherent scattered power, for a Gaussian rough surface, the joint probability distribution function of surface unit norms at two different surface points is shown to follow a joint Gaussian distribution with zero mean and covariance matrix of special form. Decomposition of such covariance matrix into uncorrelated term and fully correlated terms of different types not only assists a better understanding of the interaction between any pair of points on the surface, but also enables the simplification of calculation of the expectation of the product of Kirchhoff term at one point and the conjugate Kirchhoff term at another point. The validity of SKM is demonstrated through the good agreements between model predictions and method of moment (MoM) simulations for statistically known surfaces. More importantly, all the simulated cases are outside the validity regions of small perturbation model (SPM) and conventional Kirchhoff model (KM), which means that SKM can bridge the gap between SPM and KM. I. Introduction The complexity and challenge inherent in modeling of EM scattering from rough surface has led to a tenet of models, which range from traditional ones such as the small perturbation method (SPM) and the Kirchhoff model (KM) to the recently developed integral equation model (IEM) and its various variations[1-5]. Each model has its fair share of strengths and weaknesses. For the SPM model, it is required that surface height variance be much smaller than incident wavelength and surface slopes also comparably small. The KM is applicable for a rough surface whose mean surface curvature is large; its two asymptotic approximations, the Physical Optics (PO) and Geometrical Optics (GO) models, are valid for small surface slope and very high frequency, respectively. Yet there is a vast gap between the validity regions of SPM and KM. To fill the gap, IEM and its variations make use of the so called complementary fields and seem to work well for certain cases. In this paper we propose a statistical Kirchhoff model (SKM) for shadow-corrected EM scattering from a rough surface. It treats the local coordinates and Fresnel reflection coefficients statistically over the orientation distribution of surface unit norm as characterized by the joint probability distribution function of its two directional slopes. In calculating the incoherent scattered power, for a Gaussian rough surface, the joint probability distribution function of surface unit norms at two different surface points is rigorously derived. Decomposition of the covariance matrix into uncorrelated term and fully correlated terms of different types enables the simplification of calculation of the correlated power term. The validity of SKM is demonstrated through the good agreements between model predictions and method of moment (MoM) simulations for statistically known surfaces. This paper is organized as follows. In Section II, the shadow-corrected tangential Kirchhoff fields are formulated. The scattered fields at far zone are also given. The scattering coefficient is computed in Section III, where statistical treatment based on assumptions of the surface norm distribution and height distribution is carried out in detail. Comparisons between SKM theoretic predictions and MoM simulations for some known statistical surfaces are given in section IV. Section V concludes this paper. II. Shadow-corrected Surface Tangential Fields Consider a harmonic plane wave incident in free space upon a random rough surface as suggested in Fig. 1. The time-factor e iωt is suppressed.the incident electric and magnetic fields are given by E i = ˆpE 0 e ik r

2 188 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August and H i = 1 η k Ei, where ˆp is the unit polarization vector, E 0 is the amplitude of the electric field, k = ˆk i k, k is the wave number in free space and is given by k = ω η 0 ǫ 0, ˆk i is the unit propagation vector given by ˆk i = ˆxsin θ i cosφ i + ŷ sin θ i sin φ i + ẑ cosφ i, and η is the intrinsic impedance in free space. For an arbitrary point r on the rough surface, where r = ˆxx + ŷy + ẑz, its surface norm is ˆn = ˆxZx ŷzy+ẑ, 1+Z 2 x +Z 2 y where Z x and Z y are surface slopes along x-axis and y-axis respectively. When expressed in terms of the spherical coordinates (θ n, φ n ), ˆn is given by ˆn = ˆx sinθ n cosφ n + ŷ sin θ n sin φ n + ẑ cosφ n The local coordinate system (ˆk i, ˆt, ˆd) for Fresnel reflection coefficients is defined as ˆt = ˆk i ˆn and ˆd = ˆk ˆk i ˆt. i ˆn The Fresnel reflection coefficients R and R are calculated using the local incidence angle. At far zone, the scattered Kirchhoff electric field E s,k of polarization ˆq is given by where f qp (r, ˆn) is given by: E s,k qp = ike 0 4πR eikr f qp (r, ˆn)e i(ks ki) r dxdy (1) f qp (r, ˆn) ={ˆq k s ˆn [(1 + R )(ˆp ˆt)ˆt + (1 R )(ˆp ˆd) ˆd] + ˆq ˆn [(1 R )(ˆp ˆt) ˆd (1 + R )(ˆp ˆd)ˆt]}I (2) k (r )/ cosθ n and I k (r ) is the shadow function with r = ˆxx + ŷy. In the simple case where I k (r ) depends only on the intersection angle between ˆk i and ˆn, f qp (r, ˆn) can be written as f qp (ˆn). Figure 1: Scattering of plane wave incident in free space upon a random rough surface Figure 2: ρ xx and ρ xy as functions of x = x x for a given y = y y III. The Scattering Coefficients To calculate the scattering coefficient, we need to determine the incoherent power first. The Kirchhoff incoherent power is given by: Pqp k =< Ek qp Ek qp > < Ek qp >< Ek qp > = E0 2 (4πR) 2 {< f qp (ˆn)f qp(ˆn )exp[ i(k s k i ) (r r )]dx dy dxdy > < f qp (ˆn)exp[ i(k s k i ) r]dxdy > 2 } (3) To carry out the involved expectations, for a Gaussian surface, the following needed properties are readily established: (i) the surface slopes Z x and Z y are uncorrelated with height z; (ii) the joint statistics of Z x and Z y is specified by the joint probability distribution function of vector µ = [Z x Z y Z x Z y ]t, which follows a joint Gaussian distribution with zero mean and covariance matrix C given by: C = σ 2 s 1 0 ρ xx ρ xy 0 1 ρ xy ρ yy ρ xx ρ xy 1 0 ρ xy ρ yy 0 1 (4)

3 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August where σ 2 s = 2σ2 L 2 (5) ρ xx (r,r ) = {1 2(x x ) 2 L 2 } (6) ρ xy (r,r ) = { 2(x x )(y y ) L 2 } (7) ρ yy (r,r ) = {1 2(y y ) 2 L 2 } (8) Fig. 2 shows plots of ρ xx and ρ xy as functions of x = x x for a given y = y y. It is seen that as x > 3L, ρ xx and ρ xy approach zero. In view of the above, Pqp k can be calculated as follows, Pqp k = E0A 2 0 (4πR) 2 exp( dzσ 2 ){ f qp (ˆn)f qp(ˆn )g(ˆn, ˆn )exp[ i(k dx ζ + k dy ς)]dz x dz y dz xdz ydζdς < f qp (ˆn) > 2 exp[ i(k dx ζ + k dy ς)]dζdς} (9) where A 0 is the illuminated area, k dχ = k sχ k χ, χ = x, y, z, ζ = x x, ς = y y, and g(ˆn, ˆn ) is the joint pdf given by g(ˆn, ˆn 1 ) = (2π) 2 C exp( 1 2 µt C 1 µ). It is clear that calculation of the first term of Pqp k involves a 6-fold integration, a procedure rigorous yet time consuming. The complexity stems from correlation between ˆn and ˆn, which from Fig. 2 can be regarded as concentrated in a disk of radius R = 3L formally defined as D 0 = {(ζ, ς) : ζ 2 + ς 2 3L}. To simplify such calculation, decomposition of the covariance matrix is considered. First it is normalized with respect to the slope variance σs 2, and the resultant matrix C = C/σs 2 is decomposed into four matrices representing different correlation relations between the surface norms ˆn(Z x, Z y ) and ˆn(Z x, Z y ) as follows: C = a b c d (10) where b = ρxx+ρyy 2, c = ρxx ρyy 2, d = ρ xy, and a = 1 b c d. Among the decomposed four matrices, the first matrix represents uncorrelated surface norm ˆn(Z x, Z y ) and ˆn (Z x, Z y ), and the other three represent fully correlated ˆn(Z x, Z y ) and ˆn (Z x, Z y ) in different senses, as indicated by the corresponding correlation coefficient of 1 or -1. For instance, the last matrix denotes fully correlated cross-directional pairs (Z x, Z y ) and (Z x, Z y). This decomposition makes it clear that the surface norm ˆn (Z x, Z y ) is related to ˆn(Z x, Z y ) through a weighted combination of four mechanisms: uncorrelated, two co-directionally correlated and one cross-directionally correlated. The relative weight of each mechanism depends on the relative position of r to r: when r approaches r, a 0, c 0, d 0, b 1, which means the correlation coefficient between ˆn(Z x, Z y ) and ˆn (Z x, Z y) approaches 1; when r is far away from r, b 0, c 0, d 0, a 1, which means ˆn(Z x, Z y ) and ˆn (Z x, Z y) are uncorrelated, just as intuition would indicate. Such interpretation suggests a way to calculate the expectation of f qp (ˆn)f qp(ˆn ) which is essential in determining the incoherent scattered power. The new method is formulated as follows: < f qp (θ n, φ n )fqp(θ n, φ n) > { b < = (1 b c d ) < f qp (θ n, φ n ) > 2 fqp (θ + n, φ n (θ n, φ n ) > (if b 0) { b < f qp (θ n, φ n )fqp(θ n, π + φ n ) > (if b < 0) c < fqp (θ + n, φ n )fqp (θ n, π 2 φ { n) > (if c 0) d < fqp (θ c < f qp (θ n, φ n )fqp (θ n, 3π 2 φ + n, φ n )fqp(θ n, φ n ) > (if d 0) n) > (if c < 0) d < f qp (θ n, φ n )fqp (θ n, π φ n ) > (if d < 0) (11)

4 190 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August The scattering coefficient is defined as: The backscattering scattering coefficient of SKM is given by σ 0 qp = 4πR2 P qp E 2 0 A 0 cosθ i (12) σ 0 qp = σ 0 qp1 + σ 0 qp2 + σ 0 qp3 (13) where σqp1 0 = exp[ (k sz k z ) 2 σ 2 ]{ f qp (θ n, φ n )fqp 4π cosθ (θ n, φ n ) exp[(k sz k z ) 2 σ 2 ρ 2 exp( i L 2 )] D 0 exp[ i(k sx k x )ρ cosφ i(k sy k y )ρ sin φ]ρdρdφ D 0 f qp (θ n, φ n ) 2 exp[ i(k sx k x )ρ cosφ i(k sy k y )ρ sinφ]ρdρdφ (14) σ 0 qp2 = 2 cosθ i exp[ (k sz k z ) 2 σ 2 ] f qp (θ n, φ n ) 2 where J 0 ( ) is the 0-th order Bessel function, k dρ = (k sx k x ) 2 + (k sy k y ) 2 and R 0 ρj 0 (k dρ ρ){exp[(k sz k z ) 2 σ 2 exp( ρ2 )] 1}dρ (15) L2 σ 0 qp3 = 2 cosθ i exp[ (k sz k z ) 2 σ 2 ] f qp (θ n, φ n ) 2 n=1 σ (2n) (k sz k z ) n 2 W (n) (k sx k x, k sy k y ) n! (16) where W (n) (α, β) is the roughness spectrum of the n-th power of the surface correlation function given by W (n) (α, β) = 1 ρ n (x, y)e i(αx+βy) dxdy (17) 2π IV. Numerical Simulations To validate the SKM, we compare its simulation results with that of method of moment (MoM) for rough surfaces with Gaussian height and Gaussian power spectrum. Various surface roughness parameters are used with kl ranging from 3 to 5 and kσ from 0.4 to 0.8, all outside the traditional validity ranges of KM and SPM, as shown in Fig. 3. The complex dielectric constant ǫ r is fixed to be i. Figure 3: Validity regions of backscattering models Fig. 4 and 5 represent the typical angular behaviors of backscattering coefficients predicted by the SKM model.

5 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August In the figures, SKM shows superior performance to that of KM as benchmarked against MoM. For VV polarization, backscattering coefficients predicted by KM suffer from the Brewster angle effect when incident angle approaches 60 0 while SKM is completely free from such adversary effect. For HH polarization, the KM predicts backscattering coefficients uniformly overestimated over all the incidence angle range, while SKM provides predictions reasonably close to that of MoM. It should be noted that at small incident angles, especially near zero degree, SKM and KM backscattering coefficients all exceed MoM by around 3 db. Such discrepancy is currently under investigation. Figure 4: Comparisons among MoM, KM and SKM simulations at 5GHz. Figure 5: Comparisons among MoM, KM and SKM simulations at 7GHz. V. Conclusion A statistical Kirchhoff model (SKM) is developed for shadow-corrected EM scattering from rough surface. In the model we treat the local coordinates and Fresnel reflection coefficients statistically over the orientation distribution of surface unit norm ˆn. Furthermore, in calculating the incoherent scattered power, for a Gaussian rough surface, we investigate the joint probability distribution function of surface unit norms at two different surface points. Decomposition of the covariance matrix is carried out. The validity of SKM is demonstrated through the good agreements between model predictions and method of moment (MoM) simulations for statistically known surfaces. More importantly, all the simulated cases are outside the validity regions of small perturbation model (SPM) and conventional Kirchhoff model (KM), which means that SKM can bridge the gap between SPM and KM. Extensions to non-gaussian surface statistics, application of more realistic shadow functions and reduction of discrepancy at small incident angles are currently under investigation. REFERENCES 1. Fung, A. K., LI Zongqi and K. S. Chen, Backscattering from a Randomly Rough Dielectric Surface, IEEE Transactions on Geoscience and Remote Sensing, Vol. 30, No. 2, Chen, K. S. and A. K. Fung, Comparison of Backscattering Models for Rough Surfaces, IEEE Transactions on Geoscience and Remote Sensing, Vol. 33, No. 1, Chen, M. F. and A. K. Fung, A Numerical Study of the Regions of Validity of the Kirchhoff and Perturbation Rough Surface Scattering Models, Radio Science, Vol. 23, No. 2, , Sancer, M. I., Shadow-Corrected Electromagnetic Scattering from a Randomly Rough Surface, IEEE Transactions on Antenna and Propagation, Vol. 17, , Fung, A. K., Theory of Cross-Polarized Power Returned from a Random Surface, Applied Science Research, Vol. 18, 50-60, 1967a.