This chapter extends the derivation of the shadowing function to a two-dimensional (2D) anisotropic random rough surface [7,25]. In polar coordinates (x = R cos φ,y = R sin φ) and the term `anisotropic' means that the statistical properties of the surface depends on the azimuthal direction φ. Typically, for a sea surface, φ stands for the wind direction. We will show that the 2D case is obtained from the one-dimensional (1D) case by making a rotation of an angle φ and next by calculating the marginal probability along this direction. First, the monostatic shadowing function is derived, and next it is generalized to the bistatic case. Unlike the 1D case [8], the latter configuration requires a special attention and needs to account for the correlation with respect to the direction φ to avoid a discontinuity when the transmitter and the receiver are not in the same plane (the azimuthal directions differ).

When dealing the general problem of the wave scattering (emitted from an optics sensor or a Radar) by an object (in this chapter, it is a random rough surface), the calculation of the scattered field is not straightforward. For canonical geometries, like spheres, circular cylinders, etc., a closed-form expression of the scattered field can be found by introducing special functions [37]. Full-wave techniques, such as the method of moments, have been developed that rigorously incorporate all electromagnetic scattering mechanisms (edge diffraction, multiple reflection, shadowing effect, etc.) and can provide highly accurate results for rough surface scattering [15,16,18,38] (for instance). However, full-wave techniques are computationally intensive and, as a result, it is usually not feasible to employ a full-wave approach when we want to get results quickly. Consequently, approximate techniques are often the only practical alternative. One class of such methods is the geometrical optics (GO) approximation, which can be obtained from the well-known Kirchhoff approximation [17] (for instance). This approach assumes that the frequency tends to infinity and the incident field can be modelled as a collection of rays. Here, the rays are assumed to be not complex, that is the contribution of the creeping waves is neglected. In other words, the field in the shadow zone vanishes [39]. For incidence angles near the horizon, GO can be corrected by accounting for the fact that a part of the rough surface is not illuminated from the transmitter or/and not observed from the receiver. Then, from the shadowing function derived in Chapters 1 and 2, three applications are addressed in this chapter: · The microwave radar scattering from a rough sea surface at low grazing angles. · The sea infrared radiation (emissivity and reflectivity). · The connection between the sea infrared reflectivity and the BRDF (bidirectional reflectance distribution function) employed in the computer graphics domain.