Weakly nonlinear theory and sea state bias estimations

The theory of "weakly" nonlinear (WNL) waves is commonly used in generating higher statistical moments of a random surface wave field. These moments can be used, for example, to estimate the sea state bias (SSB) in radar altimetry under a geometric optical assumption. The present investigation suggests that several previous SSB studies appear to have misapplied this WNL theory by violating its condition of validity. As a result, a pronounced inconsistency appears even for lower-order moments. This inconsistency appears as a difference between the statistical moments of the nonlinear surface generated by the application of the WNL theory and those of the linear input (or bare) spectrum. If measured spectra, which yield measured moments (such as rms elevation and slope), are chosen as an input to WNL theory, then the corresponding moments of resulting output spectra may be severely overestimated. To strictly avoid the inconsistency, WNL theory must only be applied to long gravity waves where the wave-wave interactions are weak, hence the WNL epithet. To further illustrate this problem, we present an inversion scheme that determines the proper input spectrum by forcing the low-order moments of the output spectrum to equal the measured moments. Analytical solutions are given for this inversion based on an explicit formulation of the low-order nonlinear moments and a simplified one-dimensional power law spectrum. The solutions show that the high-frequency portion (wave components shorter than about 10 m) of the input (or bare) spectrum must be significantly less energetic than that of the output spectrum. Our results emphasize the importance of the shorter-scale waves in the SSB mechanism.