A Simplified Formulation for Rough Surface Cross-Polarized Backscattering Under the Second-Order Small-Slope Approximation

We present simplified expressions for the cross-polarized backscatter of a randomly rough surface predicted by the second-order small-slope approximation (SSA2). The simplification is based on appropriate polynomial approximations of the SSA2 kernel function. We obtain numerically efficient expressions for the cross-polarized backscattering amplitude of a deterministic surface in the form of a single space integral involving only the surface elevation and the second (mixed) derivative of the surface elevation. The ensemble average normalized radar cross section is then derived under a Gaussian random process assumption for the surface. The resulting expression has the form of a Kirchhoff integral involving the roughness correlation function and its second- and fourth-order cross-derivatives. Further simplification is achieved for off-nadir observations using a high-frequency approximation; the result is an analytical formula involving only the resonant curvature and the radar-filtered mean square slope in the out-of-plane direction. A numerical validation of the simplified expressions is provided by comparison with exact SSA2 predictions in representative test cases. The dependence of cross-polarized backscattering on the incidence angle as well as wind speed and direction is then investigated for the case of a directional sea surface model. At near nadir incidence, a clear maximum in azimuth of the cross-polarized backscatter is observed for radar look directions 45° from the wind direction.