Stochastic Cross-Sections Based on the Small Slope Approximation: Theory

摘要

The small slope approximation is widely used to model the incoherent scattering cross-section per unit area from stochastic rough interfaces. It consists of an integral multiplied by a prefactor. The prefactor involves a closed-form algebraic expression, but evaluation of the integral is nontrivial. This paper develops tractable methods for evaluating the integral. The rough- surface scenarios considered generally assume spectra that have tails that decrease according to a single specified power law. This assumption is typically valid for the air-sea interface and ocean bottom at frequencies below 10 and 40 kHz respectively. The scenarios involve various tradeoffs, but the most significant tradeoff involves tractability vs broad applicability. In the most tractable scenario, it is assumed that only the isotropic tail of the spectrum is relevant. Alternately, information about the spectral peak and the power-law tail can be used to generate an isotropic 'difference spectrum' formed by taking the difference between two power laws. The difference spectrum peak can be sharpened either by grafting it to a decaying exponential at low- wavenumber, or by replacing it entirely with a Pierson-Moskowitz (isotroptic) spectrum modified by substituting its exponential factor exp(-a/k(caret)2) with exp(- a/k). The latter spectrum is tractable, and while incorporating very sharp spectral peaks, it is rather inflexible. A typical azimuthal dependence is also considered.