CHAPTER EIGHT Wavelets in Rough Surface Scattering In this chapter we will study scattering of electromagnetic waves from rough surfaces numerically, using the Coifman wavelets. Owing to the orthogonality, vanishing mo- ments, and multiresolution analysis, a very sparse moment matrix is obtained. In ad- dition the wavelet bases are continuous. Hence the sampling rate for wavelet bases is reduced to one-half the rate of the pulse cases, allowing the same computer resource to deal with quadruple the truncated surface area. More important, the Coiflets allow the development of one-point quadrature formula, which reduces the computational effort in filling matrix entries to O(n). As a result the wavelet-Galerkin method with twofold integrals is faster than the traditional pulse-collocation approach with one- fold integrals. 8.1 SCATTERING OF EM WAVES FROM RANDOMLY ROUGH SURFACES Rough surface scattering has potential applications in remote sensing, semiconduc- tor processing, radar, and sonar, among others. Figure 8.1 demonstrates a computer generated random surface, which will be discussed in Section 8.2. Scattering of electromagnetic waves from rough surfaces has been studied by an- alytical [1, 2], numerical [3–6], and experimental means [7–9]. Analytic methods provide fast solutions and allow users to foresee the effects and trends of the solution due to individual parameters in the formulas. However, there are many geometric and physical limitations restricting the utility of analytical models in general applications. For instance, the tangential plane approximation, known as the Kirchhoff model, works only for undulating surfaces without shadowing, while the small perturbation method, known as the Rice model, is valid only for small roughness. Attempts were made to extend these analytical models, including the iterated Kirchhoff [10, 11] and Wiener–Hermite expansion [12], among others. Nevertheless, the modified ana- lytical models still operate under certain assumptions and conditions. Experimental 366 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X SCATTERING OF EM WAVES FROM RANDOMLY ROUGH SURFACES 367 FIGURE 8.1 Computer generated random surface with Gaussian distribution σ = 0.2λ and Gaussian correlation  x =  y = 0.6λ. method requires fabrication of rough surfaces with specified statistical parameters, and it requires high-tech equipment that is costly and is not versatile. With advances in today’s computers, it seems ideas to develop numerical methods that are accu- rate, versatile and relatively inexpensive. In the numerical approaches, the 1D Monte Carlo was developed several decades ago using the MoM [3]. In the Monte Carlo simulation, many sample surfaces with desired roughness statistics are generated and then the scattering solution for each sample surface, or realization is obtained using the MoM. These solutions are then averaged numerically to approximate the required statistical quantities. Clearly, from the nature of physics and statistics, rough surface scattering problems are electrically large problems. Traditional MoM in conjunction with the Galerkin procedure requires that the computation time be on the order of n 2 for matrix filling and n 3 for matrix inversion if Gaussian elimination is employed. Tsang et al. reported the band matrix iterative method (BMIA) [6] and applied the method to 3D scattering problems. Nevertheless, in the BMIA computation, humans must have interact with computers to set up the strong or weak terms in the system matrix. Recently wavelets have appeared in applied mathematics [13] and have been suc- cessfully used to solve integral equations [14]. In electromagnetics, wavelets have been applied to guidedwave, radiation, object scattering, nonlinear device model- ing, and target identification [15–17]. Wavelets have also been employed in rough surface scattering [18, 19]. In [18] the Daubechies wavelets were employed as a 368 WAVELETS IN ROUGH SURFACE SCATTERING transformation matrix that converts the dense matrix generated from the MoM into a sparse matrix. This approach follows the idea in [16, 17, 20]. In [19] wavelets are directly used as the basis and testing functions to create a sparse impedance matrix, bypassing the MoM computation to fill the matrix. Despite the differences in the two approaches, both of them require massive computation to fill the entire entries of the impedance matrix on the order of O(n 2 ). Here we apply wavelets to the 2D and 3D scattering of electromagnetic waves from perfectly conducting random surfaces. The integral equations for both the HH and VV polarizations are solved using the Galerkin procedure. More specifically, we choose the Coifman wavelets, which are orthogonal and compactly supported with zero moments of both the wavelets and scalets. As a consequence, a property similar in nature to the Dirac δ is evolved that allows fast computation of the most off-diagonal elements in the impedance matrix using the single-point quadrature formula. Hence only the “strong” elements around the diagonal of the matrix need to be evaluated via numerical quadrature; they are on the order of O(n). The resultant impedance matrix is sparse and can be solved with iterative methods (e.g., conjugate gradient) or newly developed nonstandard LU fac- torization [21] on the order of O(n). As a result, the wavelet-Galerkin method with twofold integrals is faster than the traditional pulse-collocation approach with one- fold integrals. Numerical examples of the wavelet-Galerkin method are compared with those obtained from the standard MoM that employs pulse basis and a point match in scheme. Excellent agreement was observed between new approach and previously published results. 8.2 GENERATION OF RANDOM SURFACES In order to perform numerical simulations, a realization has to be generated in a ran- domly rough surface with prescribed surface distribution and autocorrelation func- tions. The spectral method [22] for the generation of a random surface profile has been found more convenient than the autoregressive (AR) method used in [23], es- pecially for surface derivatives. The description of the method for the case of the 1D random surface can be found in [24] and for the 2D case in [9]. A surface is called simple if its correlation function has only one correlation length parameter; it is called composite if more than one parameter is required to describe its correlation function. In most research articles, the rough surface profile is described in terms of its deviation from a flat “reference plane.” In general, the reference plane is assumed to be located at z = 0. The random fluctuations from this reference plane are described by the probability density function (p.d.f.). For analytical convenience, one usually uses the Gaussian type p.d.f. p(z) = 1 σ √ 2π exp  − z 2 2σ 2  , (8.2.1) GENERATION OF RANDOM SURFACES 369 0.0 5.0 10.0 15.0 20.0 25.0 30.0 distance (in wavelength) −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 height (in wavelength) σ = 0.3183 λ, l = 0.8881 λ σ = 0.3183 λ, l = 1.2732 λ σ = 0.1592 λ, l = 0.8881 λ FIGURE 8.2 Random surfaces with different standard deviations and correlation lengths. where we have assumed a zero mean  z=0 and variance z 2 =σ 2 . In the previous case the rough surface is generated by a 1D stationary (in the wide sense), normal, random process with zero mean and standard deviation σ . The height coordinate z of the surface is a realization of the random process z(x), which is a function of the x coordinate. The relations between surface points z 1 = z(x 1 ) and z 2 = z(x 2 ) are specified by the correlation function, which we consider also to be a Gaussian-type R(τ ) =z(x 1 ), z(x 2 )=σ 2 exp  − τ 2 l 2  , (8.2.2) where · denotes the ensemble average, τ = x 1 −x 2 ,andl is a correlation length in the x direction. We will describe two methods of generating a random surface profile, the autocor- relation approach and spectral domain approach. In Fig. 8.2 we plotted the random surface profiles generated with a Gaussian probability density function p.d.f. and Gaussian correlation function. We used different parameters of standard deviation σ and correlation length l in the figure. Plotted in Fig. 8.3 is the p.d.f. of the height estimated from the actual profile. In order to compare the obtained numerical results we also plotted in Fig. 8.3 the p.d.f. calculated by using (8.2.1). In Fig. 8.4 two Gaus- sian correlation functions with different parameters l are shown. As for the case of the p.d.f., we estimated the correlation functions from a numerically generated ran- dom surface profile and plotted the corresponding correlation functions using (8.2.2). To create Fig. 8.5a, we used the Gaussian p.d.f. and two different correlation func- tions, namely the Gaussian and exponential functions. The small-scale roughness in Fig. 8.5a of the random surface profile with the exponential correlation function gives rise to the high-frequency tail of the exponential spectrum. Figure 8.5b depicts these correlation functions that are calculated from the actual random surface pro- files by using theoretical expressions. All curves in Fig. 8.5b have been normalized 370 WAVELETS IN ROUGH SURFACE SCATTERING −1.5 −1 −0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 height in λ probability density σ = 0.3183 λ , l = 0.8881 λ σ = 0.1592 λ , l = 0.8881 λ theoretical theoretical FIGURE 8.3 Probability density function of height for simple surface. to the maximum value of unity. In Fig. 8.6 we also illustrate simple and composite random surface profiles. A composite surface is a superposition of two surfaces with clearly distinct vertical and horizontal scales. 8.2.1 Autocorrelation Method This method was suggested in [23]. We begin with a numerically generated sequence of independent Gaussian variables {X k } with zero mean and a standard deviation 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 distance in λ σ = 0.3183 λ , l = 0.8881 λ σ = 0.3183 λ , l = 1.2732 λ theoretical theoretical correlation function FIGURE 8.4 Normalized correlation function of simple surface. GENERATION OF RANDOM SURFACES 371 0 0.5 1 1.5 2 2.5 0 0.2 0. 4 0.6 0.8 1 correlation function Gaussian Exponential theoretical theoretical (a) (b) distance in λ 0 5 10 15 20 25 30 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 height in λ Gaussian Exponential x direction in λ FIGURE 8.5 Random surfaces of Gaussian distribution with Gaussian and exponential cor- relation functions: (a) 1D rough surfaces, (b) corresponding correlation functions. of unity. This sequence can be obtained utilizing a commercial software package such as the IMSL, Matlab, or NAG. From this uncorrelated sequence of normally distributed samples, a sequence of correlated normal samples {C k } can be obtained by digitally filtering in the manner C k = N  j=−N W j X j+k , (8.2.3) where W j are the correlation weights yet to be determined. The expectation E{C k C k+i }=  j  n W j W n E{X j+k X n+k+i }, (8.2.4) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 distance (in wavelength) −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 height (in wavelength) simple surface composite surface FIGURE 8.6 Simple and composite random surfaces. 372 WAVELETS IN ROUGH SURFACE SCATTERING and {X k } is an independent sequence, satisfying E{X j+k X n+k+i }=  0, j = n + i 1, j = n + i. (8.2.5) Hence E{C k C k+i }=  j W j W j−i . (8.2.6) The previous equation states that the autocorrelation function of the correlated nor- mal sample {C k } is identical to the convolution of the digital weights. It follows also that the Fourier transform of the correlation is equal to the product of the Fourier transforms of the digital filter weights. Thus the inverse transform of the square root of the prescribed spectrum is the filter weight. For instance, let the correlation func- tion be Gaussian ρ = exp  − j 2 l 2  . (8.2.7) Its spectrum is ρ s =  l √ π  exp  − l 2 f 2 4  , (8.2.8) and the square root of ρ s is (ρ s ) 1/2 = (l √ π) 1/2 exp  − l 2 f 2 8  . (8.2.9) The inverse Fourier transform of (8.2.9) is the filter weight and can be written as W j =  2 √ πl  1/2 exp  −2 j 2 l 2  . (8.2.10) Notice that expression (8.2.3) with W j as defined in (8.2.10) produces correlated samples of z with standard deviation of unity and with a sampling interval of unity in the x direction. For a general case where the correlated samples of z create a random surface with a standard deviation σ , correlation length l, and a sampling interval x units, we will have the following modified expression for the weight W j : W j =  2σ 2 x √ πl  1/2 exp  −2 ( j x) 2 l 2  . (8.2.11) A realization of a random surface {C k } with the properties above will be generated at points x k = k x (k = 0, ,N) with standard deviation σ , correlation length l, and root mean square (rms) slope ρ x = √ 2σ/l.Thefirst derivative of the surface at GENERATION OF RANDOM SURFACES 373 each sampling point can be approximated using the finite difference scheme  dz dx  x=x k ≈ C k+1 − C k x . (8.2.12) The derivative will be stored for future numerical computations. 8.2.2 Spectral Domain Method The second method, described in [24], imposes a roughness spectral density since the inverse Fourier transform can be done very quickly by the implementation of the standard fast Fourier transform (FFT) algorithm. For this method we use a cor- responding roughness spectral density of the correlation function to generate a real- ization of a random surface profile. If we assume a Gaussian correlation function of (8.2.2), then the corresponding roughness spectral density is W (k) = σ 2 l √ 4π exp  − k 2 l 2 4  = 1 2π  +∞ −∞ R(τ )e ikτ dτ. (8.2.13) An alternative correlation function, such as the exponential function, more precisely describes surfaces with very sharp peaks. This correlation has the form R(τ ) = σ 2 exp  − |τ | l  (8.2.14) and the corresponding roughness spectral density W (k) = σ 2 l √ 4π  1 1 + k 2 l 2  . (8.2.15) In turbulence modeling, the power law spectrum is used to model the random fluctu- ation of the propagation characteristics for the medium. Its corresponding spectrum is given by W (k) = σ 2 l √ 4π  1 + π  (2n − 3)!! (2n − 2)!!  2 k 2 l 2 4  −n , (8.2.16) where (2n − 2)!! = 2 × 4 × ···(2n − 2), (2n − 3)!! = 1 × 3 × ···(2n − 3), (−1)!! = 1andn is the order of the power law spectrum. The power law spectrum converges to the Gaussian spectrum for large order n, and is almost equivalent to the Laurentzian spectrum for order n = 1. Moreover, for any given order, the power law spectrum reduces to k −2n for large k. No closed-form expression is available for the autocorrelation of a surface with the power law spectrum. Suppose that we have a roughness spectrum W (k). For the scattering computation, surface realization (heights and first derivatives) are needed as a set of N points with spacing x over length L = N x. Realizations with the desired properties can be 374 WAVELETS IN ROUGH SURFACE SCATTERING generated at points x k = (k +0.5)x (k = 0, ,N −1) using the discrete Fourier transform (DFT) method. The rough surface profile z = f (x k ) is related to the 1D DFT of the surface spectrum by f (x) = 1 L N/2−1  n=−N/2 F(K n ) exp(iK n x), (8.2.17) where F(K n ) =  2π LW(K n )      N (0, 1) + iN(0, 1) √ 2 , n = 0, N/2 N (0, 1), n = 0, N/2 K n = 2πn L , i = √ −1, and N(0, 1) denotes an independent sample taken from a zero mean with unit stan- dard variance Gaussian distribution. For the Fourier coefficients of the first derivative of a random surface profile we have F ∂ x (K n ) := F(K n ) × iK n . (8.2.18) The first derivative of a rough surface profile at each sampling point can be obtained by using the DFT in the same manner as in (8.2.17). Equation (8.2.17) can be computed by means of a fast Fourier transform (FFT), as can the first derivative of f (x). For a p.d.f. of height with another distribution, such as a gamma distribution, it suffices to replace N (0, 1) by such an appropriate distribution. The two-point statistics are governed by the magnitude of the Fourier spectrum, which follows the surface spectrum W (k). Since the surface must be rep- resented by a sequence of real numbers, the phase of the Fourier coefficients must satisfy certain requirements. In order to generate a real sequence, the Fourier coeffi- cients must be Hermitian, namely F(K n ) = F ∗ (−K n ). (8.2.19) The requirement above is also important in the synthesis of 2D surfaces. The use of the DFT in rough surface generation requires that the surface lengths be at least five correlation lengths so that no spectral aliasing is present in the resulting surface. Furthermore the resulting rough surface is a periodic function in which the surface height and the slope are periodic in space. It is important to note that due to a finite surface length in the discrete synthesis process, the surface autocorrelation does not completely decay to zero and some oscillations are presented. In practice, the surface spectrum can be estimated from the actual surface profile by the expression W (k) = 1 2π L        L/2 −L/2 g(x) f (x)e −ikx dx      2  . (8.2.20) GENERATION OF RANDOM SURFACES 375 The purpose of the window function g(x) with an appropriate tapering is to minimize spectral sidelobes, also known as the “Gibbs phenomenon” in the Fourier analysis, due to the finite length involved. Most of the statistics used to describe 1D rough surfaces can be extended in the 2D case. The 2D rough surface is described by z = f (x, y), which is a random function of position (x, y). Various two-dimensional spectra and autocorrelations, which are basically extensions of the one-dimensional case, can be used to gener- ate the 2D rough surface. For reasons of practicality in surface manufacturing, only surfaces with Gaussian roughness and Gaussian spectrum are considered. The cor- relation function R(τ x ,τ y ) that describes the coherence between different points on the surface separated by the distance d =  τ 2 x + τ 2 y and is given by R(τ x ,τ y ) = σ 2 exp  − τ 2 x 2l 2 x − τ 2 y 2l 2 y  , (8.2.21) where τ x and τ y describe the separation between any two points along the x and y directions. The coherence length of the surface profiles is given by l x and l y .The power spectral density function of the surface W (k x , k y ) is related to the correlation function via a 2D Fourier transform. For a Gaussian correlation function given by (8.2.21), we have W (k x , k y ) = l x l y σ 2 4π exp  − k 2 x l 2 x 4 − k 2 y l 2 y 4  . (8.2.22) It is important to note that in (8.2.22), there are two distinct correlation lengths, l x and l y . The surface is isotropic when l x = l y , and anisotropic if l x = l y . In the other extreme, if one of the correlation lengths is much greater than the other, the 2D surface becomes essentially a 1D surface for the purpose of the experiments and numerical calculations. The corresponding rms slopes are defined respectively by ρ x = √ 2σ/l x and ρ y = √ 2σ/l y . Similarly to the 1D case, the rough surface profile z = f (x, y) is related to the 2D DFT of the power spectrum as f (x, y) = 1 L x L y (N x /2)−1  m=−(N x /2) (N y /2)−1  n=−(N y /2) F(K xm , K yn ) exp(iK xm x +iK yn y), (8.2.23) where F(K xm , K yn ) = 2π  L x L y W (K xm , K yn )      N (0, 1) + iN(0, 1) √ 2 , m = 0, N x /2, n = 0, N y /2 N (0, 1), m = 0, N x /2orn = 0, N y /2 (8.2.24) [...]... resolution level j0 to create a system matrix and then apply the fast wavelet transform to go down a few resolution levels [27] By doing that, we introduce wavelets into the expansion for the unknown current Ji (x) The combination of scalets and wavelets makes the system matrix extremely sparse These sparse matrices can be solved with iterative methods, or newly developed nonstandard LU factorization... calculate backscattering coefficient Note that fair comparison between the Coiflet and pulse in Table 8.1 to 8.4 should be in terms of numerical accuracy, that is, 512 pulses versus 256 wavelets, 2048 pulses versus 1024 wavelets, and so on The threshold level of 10−3 and 4 resolution levels are employed to get the sparse standard matrix form We settle on a relative error of 10−2 as the stopping criterion... distribution, the FFT leads to a systematic phase distortion 8.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets The method of moments (MoM) is employed for this numerical study The basis and testing functions are the Coifman scalets, as in the 2D cases The formulation here 392 WAVELETS IN ROUGH SURFACE SCATTERING 6 1.750 4 1.531 1.063 y 2 0.5938 0.1250 0 − 0.3438 − 0.8125 − 1.281 − 1.750... Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992 [14] G Beylkin, R Coifman, and V Rokhlin, “Fast wavelet transforms and numerical algorithm I,” Comm Pure Appl Math., 44, 141–183, 1991 [15] B Steinberg and Y Leviatan, “On the use of wavelet expansions in the method of moments,” IEEE Trans Ant Propg., 41(5), 610–119, 1993 [16] R Wagner and W Chew, “A study of wavelets for the solution of... New York, 1993 [26] A Fung and M Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J Opt Soc Am., 2(12), 2274–2284, Dec 1985 [27] G Pan, “Orthogonal wavelets with applications in electromagnetics, ” IEEE Trans Magn., 32, 975–983, 1996 [28] E Thorsos, “Backscattering enhancement with the Dirichlet boundary condition,” Workshop on Enhanced Backscatter, Boston University,... incidence (degrees) 60 20 pulse basis wavelet basis 10 0 −10 −20 −30 −40 0 10 20 30 40 50 angle of incidence (degrees) 60 FIGURE 8.11 Backscattering coefficient of the simple surface in HH and VV polarization WAVELETS IN ROUGH SURFACE SCATTERING HH polarization 20 backscattering coefficient (dB) backscattering coefficient (dB) 384 pulse basis wavelet basis 10 0 −10 −20 −30 −40 0 10 20 30 40 50 60 VV polarization... results in terms of number of unknowns and corresponding computational time for the simple surface The impedance matrix obtained from the Coifman scalets can be further sparsified by the introduction of wavelets This fact is due to the vanishing moment property, localization, and multiresolution analysis of the wavelet basis There are two kinds of matrix representation in the wavelet basis, namely the... segment into P subsegments with widths x = D/P, integral equation (8.3.2) is solved by the method of moments [25], which converts (8.3.2) into a matrix equation of the form [Q][I ] = [V ], (8.3.3) 378 WAVELETS IN ROUGH SURFACE SCATTERING where the mnth element of the impedance matrix [Q] is given by Q m,n = k0 η 4 n x+xc −D/2 (2) (n−1) x+xc −D/2 · 1+ dz dx H0 k0 (xm − x )2 + (z m − z )2 2 dx (8.3.4)... 14.4 15.7 TABLE 8.4 Computational Time: Simple Surface, Wavelet Basis Wavelet basis, VV Time (s) Number of Unknowns LU Decomposition Bi-CGSTAB Sparsity (%) 2048 1024 80574 9190 8259 2250 10.4 13.2 386 WAVELETS IN ROUGH SURFACE SCATTERING FIGURE 8.13 Standard form of the impedance matrix in HH polarization We should note here that the wavelet solution in Fig 8.14 is obtained using five resolution levels... vectortapered waves For ease of reference, the main vector tapering formulation is briefly z Hi ki Ei ki ρ θ i = 44 plane of incidence footprint ϕ i = 90 y x FIGURE 8.15 Configuration of 3D scattering 388 WAVELETS IN ROUGH SURFACE SCATTERING summarized in the next subsection For detailed derivations, discussions, and error analysis, the reader is referred to [32] 8.4.1 Tapered Wave of Incidence An ideal . set up the strong or weak terms in the system matrix. Recently wavelets have appeared in applied mathematics [13] and have been suc- cessfully used to solve integral equations [14]. In electromagnetics, . ana- lytical models still operate under certain assumptions and conditions. Experimental 366 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons,. CHAPTER EIGHT Wavelets in Rough Surface Scattering In this chapter we will study scattering of electromagnetic waves from rough surfaces numerically, using the Coifman wavelets. Owing to