Models for microwave thermal emission from a rough surface are currently of interest due to the goal of improved sea surface wind vector retrievals from polarimetric brightness temperature measurements. Models based on either a small slope approximation or on a physical optics approach have been proposed and have shown some success in matching observations. Both of these models involve series solutions, but computation of higher order terms typically is difficult, particularly for multi-scale sea surface models. Knowledge of higher order term contributions, however, would assist in understanding the limitations of the low-order methods applied in practice. In this paper, higher order results from both the small slope and physical optics methods are studied and compared for a simple bi-sinusoidal surface model (i.e. height profile = Asin(2πx/Px) sin(2πy/Py), where Px and Py are the surface periods in the x and y directions, respectively). Results show both methods to provide good predictions for moderate slope "large scale" surfaces (i.e. periods large compared to the observing electromagnetic wavelength) when shadowing and multiple scattering effects are negligible, while only the small slope theory correctly predicts emission from "small scale" profiles. The influence of both shadowing and multiple scattering effects is examined, and the "binary" shadowing behavior used in the physical optics method is suggested as a source of larger errors observed as shadowing effects increase.
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