Bistatic Electromagnetic Scattering from Randomly Rough Surfaces

In this paper we propose a bistatic model for electromagnetic scattering from a Gaussian rough surface with small to moderate heights. It is based on the integral equation formulation where the spectral representations of the Green’s function and its gradient are in complete forms, a general approach similar to those used in the advanced integral equation model (AIEM) and the integral equation model for second-order multiple scattering (IEM2M). Yet this new model can be regarded as an extension to these two models on two accounts: first it has made fewer and less restrictive assumptions in evaluating the complementary scattering coefficient for single scattering, and second it contains a more rigorous analysis by the inclusion of the error function related terms for the cross and complementary scattering coefficients, which stems from the absolute phase term in the spectral representation of the Green’s function. It is expected that our result for complementary scattering coefficient is more accurate and more general, even when the effect of the error function related terms is neglected. As a result, the proposed model is expected to have wider applicability with a better accuracy. The validity of this extended bistatic scattering model is demonstrated through the excellent agreements between model predictions and measurement data. DOI: 10.2529/PIERS061113040233 The original IEM model [1] which used simplified surface current estimate has shown to provide good predictions for forward and backward scattering coefficients. Concerns over the assumptions have prompted several modifications and variations of IEM in the literature. Regarding the spectral representation of the Green’s function, the simplification was discarded and full form was restored, resulting in a modification to the complementary components. The resulting model is the so called improved IEM model (I-IEM) [2]. Additional restoration of the spectral representation of the gradient of the Green’s function in its full form leads to the advanced IEM model (AIEM) [3] and the IEM2M model [4]. However, there are some technical subtleties in connection with the restoration of the full Green’s function that have not been adequately reflected in these models. For example, in evaluating the average scattered complementary field over height deviation z, a split of the domain of integration into two semi-infinite ones is required due to the absolute phase term present in the spectral representation of the Green’s function. This operation will lead to an expression containing the error function. Inclusion of the error function related terms is also encountered when one evaluates the incoherent powers that involve the scattered complementary field. Thus, a complete expression for the cross scattering coefficient or for the complementary scattering coefficient should have two parts: one does not contain the error function and the other includes its effect. The latter can be regarded as a correction term and an analysis of its effect is desirable. Roughly speaking, for the case where both the media above and below the rough surface are lossless, it can be shown that the correction term vanishes for the cross scattering coefficient, but not for the complementary scattering coefficient; for the case where either medium is of lossy nature, the correction term due to this lossy medium will contribute to both the cross and complementary scattering coefficients. Inclusion of the correction terms associated with the error function and an according discussion of their effect, to the best knowledge of the authors, are still missing in the literature. Supplement of this information forms one of the two contributions of the present work. The other contribution is a new treatment of the complementary incoherent power, where fewer and less restrictive assumptions are made than previous work. In the analysis to follow immediately, we shall highlight the key development of our study, in particular on how the error function is introduced and how it is used in evaluating the bistatic scattering coefficient. PIERS ONLINE, VOL. 3, NO. 5, 2007 638 In calculating the Kirchhoff-complementary incoherent power, we need to evaluate the following quantity IF = (1) where ksz and kz are the z-components of the scattered and incident wave vectors, respectively. q is the z-related term in the spectral representation of the Green’s function. Now we introduce the transformation of variables, z = y1+y2 √ 2 and z′ = y1−y2 √ 2 to obtain the decoupling IF = IF1IF2 (2) where IF1 involves integration over the variable y1 while IF2 involves integration over the variable y2 which is expressed as IF2 = 1 √ 2πσ √ 1− ρ ∫ +∞ −∞ dy2 { exp { −i [ kszy2 √ 2 − kzy2 √ 2 − √ 2q |y2| ]} exp [ − y 2 2 2σ2(1− ρ) ]} (3) where σ is the rms height, ρ is the surface correlation function. The quantity IF2 can be decomposed as IF2 = I F2 + I − F2 (4) where I F2 = 1 √ 2πσ √ 1− ρ ∫ +∞ 0 dy2 exp [ − y 2 2 2σ2(1− ρ) − i ( ksz − kz √ 2 − √ 2q ) y2 ] (5) and I− F2 = 1 √ 2πσ √ 1− ρ ∫ 0 −∞ dy2 exp [ − y 2 2 2σ2(1− ρ) − i ( ksz − kz √ 2 + √ 2q ) y2 ] (6) Carrying out the operation yields I F2 = 1 2 exp [ − 4 (ksz − kz − 2q) (1− ρ)σ ]( 1 + erf [ σ √ 1− ρ 2 (ksz − kz − 2q) ]) (7)