A Comparison of Scattering Model Results for Two-Dimensional Randomly Rough Surfaces

Bistatic radar cross sections are calculated using two modern scattering models: the small slope approximation (both firstand second-order) and the phase perturbation technique. The problem is limited to scalar-wave scattering from two-dimensional, randomly rough Diricblet surfaces with a Gaussian roughness spectrum. Numerical results for the cross sections are compared to those found using the classical Kirchhoff, or physical optics, approximation and perturbation theory. Over a wide range of scattering angles, the new results agree well with the classical results when the latter are considered to be accurate. A comparison between the new results shows that the phase perturbation method gives better results in the backscattering region for correlation lengths greater than approximately one wavelength, while both the firstand secondorder small slope approximations yield greater accuracy in the forward scattering direction at low grazing angles. 1. INTRODUCT~ON N the theory of rough surface scattering, the two I most widely used approximate models are the perturbation and Kirchhoff approximations [11-[41. Recent numerical studies of scattering from one-dimensional (1-D) Dirichlet surfaces with both a Gaussian roughness spectrum and a Pierson-Moskowitz spectrum have been useful in illustrating the limitations of these two classical methods [5]-[91. In an effort to expand the regions of validity for the two solutions, new approximate models have been proposed. Among them are the phase perturbation technique [lo], [ 111 and the small slope approximation [ 121. A significant number of numerical studies for the phase perturbation method for 1-D surfaces have been reported 1131-[16], but few are available for the small slope approximation [17]. Numerical results have not been reported for two-dimensional (2-D), randomly rough surfaces using either method. In this paper, we present a comparison study of the bistatic radar cross section for 2-D randomly rough Dirichlet surfaces satisfying a 2-D, isotropic Gaussian roughness spectrum using both the phase perturbation and firstand second-order small slope approximations. Numerical results are compared with those of the two classical methods. It is found that both Manuscript received December 27, 1991; revised June 19, 1992. This The authors are with the School of Electrical Engineering and ComIEEE Log Number 92049 11. work was supported by the Office of Naval Research, Code 11250A. puter Science, Washington State University, Pullman, WA 99164. new techniques bridge the classical solutions, and both give lower scattering levels than for the equivalent 1-D surface cases. The phase perturbation technique performs better in the backscattering region for correlation lengths greater than approximately one radiation wavelength, while both the firstand second-order small slope approximations yield higher accuracy in the forward scattering direction at low grazing angles. Both the phase perturbation and second-order small slope results agree well with that of fourth-order (in the intensity) perturbation theory for a case when second-order perturbation theory (firstorder in the field) and the Kirchhoff approximation fail. 11. THE PHASE PERTURBATION TECHNIQUE In phase perturbation theory, the unknown surface source density excited on the surface by the incident field is written in exponential form, and the complex phase is expanded in a perturbation series in the small roughness parameter kh, where k is the radiation wavenumber and h is the rms surface height. This is similar to the Rytov method for wave propagation in random media [3]. The bistatic radar cross section, as defined in [3, ch. 211, is found by taking averages of the incoherent and coherent T-matrices, and the results are given by