Study of the SAR signature of internal waves by nonlinear parametric autoregressive models

In this paper, we propose a method for estimating the nonlinear differential equation governing the propagation of internal waves (IWs) in a stratified water column of the ocean. This estimation of the kind of nonlinear differential equation and the coefficients of this equation is performed from profiles of the synthetic aperture radar signature at the ocean surface. Theoretically, the equation is based upon the Korteveg-de Vries (KdV) equation or one of its improved versions. The estimation of the differential equation version is based on nonlinear polynomial autoregressive models up to the third order. The estimation of the parameters of these models is performed simply by solving a set of linear equations using higher order statistics. This general approach allows us to identify the best model of propagation, i.e., the kind of nonlinearity as well as the structure of each kernel (linear, quadratic, cubic), without determining an analytic solution to the equation of propagation. Moreover, the physical oceanic parameters (such as the thermocline depth) are deduced from the estimated coefficients of the KdV equation (assuming an underlying model of the water column). Results on simulated profiles generally lead to an exact model identification (i.e., the one used for simulation with the exact order of nonlinearity and the exact structure of each kernel) and lead to satisfying geophysical parameter estimates. Finally, the geophysical parameters estimated from three sets of ERS-1 profiles are generally coherent with the parameters observed in the IW propagation mechanism but have not been validated by "in situ" measurements.