An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis

Abstract

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.

Acknowledgments

This work has been supported by a Research Grant in Commemoration of the 60th Anniversary of the Radiation Science Society of Japan. The author would like to thank the reviewer and the editor for their valuable comments and suggestions on improvements to this paper. Finally, the author would like to express his gratitude to Junichi Nakayama and Kazuhiro Hattori, Kyoto Institute of Technology, for their helpful discussions.

Notes

^† Similarly, the stochastic functional approach is available for analysis of scattering from a random periodic surface that is deformed from a periodic surface with Gaussian height randomness or binary randomness. The ‘stochastic Floquet form’ becomes a product of an unknown periodic stationary random process and an exponential phase factor. In this case, by applying an adequate stochastic orthogonal functional expansion, e.g. a Wiener–Hermite expansion on a Gaussian random process or a binary expansion on a binary random process[41], to such an unknown periodic stationary random process, the scattering problem reduces to the determination of the unknown expansion coefficients, e.g. Wiener kernels or binary kernels. We call the determination of the unknown binary kernels ‘binary analysis’[41].

^† By means of the D ^a -Fourier transformation[13], an inhomogeneous random field derived from a homogeneous random field can be described as a homogeneous random field in the spectral domain.

^‡ Only three orthogonal functional expansions have been found: the Wiener–Hermite expansion for Gaussian random fields[39, 40], the Poisson–Charlier expansion for Poisson process[52] and the binary expansion for binary random sequences[41].

^† In classical stochastic analysis, this solution procedure has been used with paper and pencil under several approximations [9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 38].

^‡ However, it should be noted that this adjective ‘simplest’ is not for numerical but analytical solution procedures. The case of a perfectly conductive surface is a limiting case where the resistivity vanishes or the dielectric constant is −∞. In general, such a case often gives rise to critical problems for numerical analysis when resonance phenomena are considered.

^‡ Formalisms without Rayleigh's hypothesis have been formally presented in [46, 47]. However, concrete discussions were not performed.

^† Formalisms with the exact boundary condition have been formally discussed in [46, 47] but concretely in [28, 29].

This is not strange. According to [45], first- and second-iterative solutions of the Bethe–Salpeter equation combined with the iterative mass operator (29), where they are given as scattering cross-sections per unit surface, are constructed to satisfy exactly the optical theorem. Therefore, the optical theorem of such iterative solutions exactly holds. It should be noted that it is impossible to obtain other statistical properties corresponding to such iterative solutions, e.g. the variance of the random wavefield or the error of the boundary value. Of course, a corresponding realization of the random wavefield cannot be obtained. These are remarkable disadvantages to multiple scattering theory.

^† A ‘matrix with n variables’ means a matrix like an ‘n-dimensional multiple array’ appearing in Fortran, Basic and C. Therefore, the nth-order linear equation becomes an ‘M_L+M_U+1)-dimensional matrix equation with n variables’. In a practical application, however, such a matrix with n variables is replaced by the usual matrix with sorting in lexicographic order.

^† It is strictly noted that such a diagonal element involves a factor |F _m |²γ^[2] _mm γ^[1] _m ▵ _m from the third term of the left-hand side of (43). However, when the number of partitions M_L+M_U becomes large and ▵ _m becomes small, the contribution of such an element obviously vanishes since the corresponding diagonal element 1+γ^[1] _m m ^(1)[1] _Dm is invariant. Therefore, it is found that the non-diagonal term in the original Equation (23) becomes −F(λ₁)∫^∞ _−∞ F*(λ)γ(λ₀+λ₁+ λ)γ(λ₀+λ)A ₁(λ|λ₀)dλ.

^† For example, we have introduced two minor techniques for the numerical calculation of M ⁽²⁾ _D in figure 8 in [17]. One is a concrete determination of λ _m with high accuracy from a minimum value finding of |Δ⁽¹⁾ _D (λ)| by means of the golden section search method[55]. The other is necessity of ingenuity on numerical integration because the half bandwidth of the spike of the resonance factor 1/|Δ⁽¹⁾ _D (λ)| becomes too narrow to calculate M _D ⁽²⁾ usually. For example, 1/|Δ⁽¹⁾ _D (λ)| has a half bandwidth as 1.79×10⁻¹⁴ k where it takes its maximum at λ _m /k = 3.3374482892180346k for kκ = 2. A large number of calculation points for the numerical integration for M ⁽²⁾ _D (λ) are needed to obtain a highly accurate value. Thus, we have divided the original integral intervals into subintervals centred at two singular points λ _m −λ,−λ _m −λ and applied the double exponential formula[58] with guaranteed accuracy processing[59] to each subinterval. These two techniques have been producing results with good accuracy and much smaller CPU times compared with results without those techniques.

^‡ On the other hand, it be should noted that this form for the coherent wavefield is naturally derived from (7) and (10) in the stochastic functional approach.

^† The error of the optical theorem was discussed in [9, 10, 14, 15, 16], and the mean square of the boundary value error only in [15]. On the other hand, the error of the reciprocal theorem has not been concretely performed yet. It should be carefully noted that these error evaluations have not been treated at all by other methods.

^‡ Another analytical proof is given by multiple scattering theory, where our NDA⁽¹⁾ solution perfectly agrees with the first-order solution[45].

^† A prototype of this work was presented at the Technical Meeting on Electromagnetic Theory Symposium 1997 [1].