## 1. Introduction

[2] Wave diffraction by a statistically rough surface plays an important role not only in remote sensing of natural interfaces (such as bare terrain, sea surface, snow cover, and sea ice) by different probing signals (echo sounders in hydroacoustics, radio waves in radars, lasers in optics), but in many other physical problems and engineering applications. The theoretical approaches to this problem can be divided into two main classes: the geometrical (and quasi-geometrical) optics (GO) applied to the surfaces of which roughness scales are much larger than the wavelength of the incident wave, and the small perturbation method (SPM), which deals with small-scale roughness compared to the wavelength [e.g., *Beckmann and Spizzichino*, 1963; *Bass and Fuks*, 1972; *Ishimaru*, 1978; *Voronovich*, 1998]. To interpret the experimental results obtained in the intermediate scales of roughness parameters (or for multiscale roughness, such as the sea surface and other natural interfaces), some combinations of these two basic methods are used: various two-scale models (TSM), small slope approximation (SSA), full wave approach, etc. (see the latest review on these topics in *Elfouhaily and Guérin* [2004]).

[3] At small and moderate grazing angles, where specular reflections can be neglected, the resonant Bragg scattering mechanism plays the main role in backscattering from a multiscale, but not extremely rough surface: only the roughness into a narrow bandwidth of a spatial power spectrum (the wavelength of which is of the order of the wavelength of the probing signal) is responsible for backscattering. The remaining part of the power spectrum describes a large-scale component of roughness and plays the role of a modulation factor, which changes the local grazing angle of the incident wave. For this case, experimental observations of the frequency and polarization dependence of the backscattering cross section can be explained in the framework of the SPM and its various modifications (such as TSM or different orders of SSA).

[4] In contrast to the small and moderate grazing angles, where the perturbation theories or their different modifications are successfully employed, the case of steep incident angles and high-frequency probing signals can be described by the Kirchhoff approximation (KA). The equations for backscattering cross sections obtained in the GO limit of the KA do not depend on either the frequency of a probing signal or on its polarization; they are determined only by the probability density function (PDF) of surface slopes, which are responsible for the backscattering specular reflections.

[5] There are only a few papers in which some diffraction corrections to these limiting GO results, following from the KA, were obtained. In *McDaniel* [1985], the diffraction corrections to the GO were obtained by expansion of the Huygens-Kirchhoff integral in the series of the inverse power of the Rayleigh parameter *kσ* sin *ψ* ≫ 1 (where *k* = 2*π*/*λ* is the wave number of the incident field, *λ* is its wavelength, *σ* is the r.m.s. height of the surface roughness, and *ψ* is the grazing angle). Because the field and its derivatives at the rough surface were taken in the tangent plane approximation (TPA) (i.e., in the GO limit), the meaning of the corrections obtained to the GO is not clear. In *Lynch* [1970], the first-order high-frequency asymptotic corrections to the GO phase of a sound field at a pressure-release 2-D surface (with a Dirichlet boundary condition) were obtained using the variational principle. These results were extended in *Rodriguez* [1989, 1991] for a 3-D surface and electromagnetic wave scattering by a perfectly conducting surface. These derivations were based on the extinction theorem, and a solution of the boundary value problem was obtained as a perturbation expansion in series of a small momentum transfer, i.e., in a small angle scattering limit, but not as high-frequency asymptotic expansion in series of 1/*k*. In the papers cited above, only the first-order corrections ∼1/*k* to the GO fields at a rough surface were obtained, which were orthogonal to the zero-order GO fields; i.e., these corrections were related to the surface field phase, but not to the field amplitude.

[6] The problem of obtaining the solution beyond the TPA, i.e., taking into account the surface curvature effects, was considered by *Voronovich and Zavorotny* [1998] and *Fuks and Voronovich* [1999, 2002] in connection with the TSM improving: the zero-order field, which illuminates the small resonant scattering roughness, was corrected by diffraction (or multiple scattering at a concave surface in *Fuks and Voronovich* [1999]) by large-scale roughness. In *Elfouhaily et al.* [1999, 2001, 2003a], the field at a large-scale rough surface was considered as a solution of the surface integral equation, the zero-order iteration of which is a field obtained in the TPA. The next two iterations of the surface current integral equation, which give the higher orders of a small momentum transfer, were used to obtain the unified forms of SSA in *Elfouhaily et al.* [2003b, 2003c, 2004] and *Elfouhaily and Johnson* [2006].

[7] In this paper, we employ the equations for the surface values of the electric and magnetic field derived in *Fuks* [2004] by two consecutive iterations of the integral equations at a perfectly conducting 2-D (i.e., cylindrical) surface. These corrections to the GO values (the TPA solutions) were used [*Fuks*, 2005a, 2005b, 2006] to obtain the diffraction corrections to the scattering cross sections of the solitary specular points at a smooth surface. Here, we apply the results of *Fuks* [2004] to derive the diffraction corrections ∼1/*k*^{2} to the backscattering cross sections from a statistically rough surface at two linear polarizations as functions of the incident angle in a quasi-specular region, where the specular points provide the main contribution to backscattering.