Fourth-Order Small-Slope Theory of Sea-Surface Brightness Temperatures

A derivation of the fourth-order term in the small-slope approximation (SSA) of thermal emission from the sea surface is presented. It is shown that this term has the form of a fourfold integration over a product of two sea spectra for a Gaussian random process sea, thereby describing emission "interaction" effects among pairs of sea waves. An approximation for "long-long" wave interactions (i.e., the optical limit) is considered and shown to match the physical optics theory. Interaction effects between "long" and "short" waves are also considered, both through numerical and approximate evaluations of the fourth-order theory. The approximation developed has a form similar to an expanded "two-scale" model and enables comparisons of short-wave "tilting" effects between the two models in terms of spectrum independent "weighting" functions. The weighting functions obtained are found to be similar, but not identical, for the SSA and two-scale theories. In addition, azimuthal harmonics from the fourth-order SSA expansion of long-short wave interactions for a particular sea-surface model are compared against the full fourth-order theory and the two-scale model. Results again show the SSA and two-scale models to yield similar, but not identical, predictions