The SAR Image of Short Gravity Waves On a Long Gravity Wave

A SAR imaging model appropriate to oceanographic applications is derived, unifying fundamental models of hydrodynamics, rough surface scattering, and SAR imaging of time-variant scenes. The sea surface is a sinusoidal long gravity wave upon which short gravity waves propagate and are modified by the long wave in accordance with a recent theory of Phillips; the electromagnetic scattering is described by the two-scale approximation appropriate to long wave and short wave ensembles that are, respectively, smoothly varying and not too rough with respect to the radar wavelength. The resulting model, accurate to first order in the long wave slope, for the first time fundamentally characterizes the nonlinear hydrodynamic and scattering interactions of the long and short waves and their effect, along with temporal variation, on the SAR image. Of particular importance, the long wave enters (among other ways) as a phasemodulated waveform that, when filtered by the SAR system, can be, for large-amplitude long waves, the principal determinant of the image nature. The numerical analysis of the model is discussed and an approximation describing the image of a delimited scene area is derived and exemplified. (1) When the small waves are a range-directed ensemble and the long wave is azimuth directed, the latter’s temporal variation “blurs,” in azimuth, the image due, primarily, to the SAR system’s narrowband filtering of the aforementioned phase- modulated waveform and, secondarily, the nonlinear hydrodynamic interaction; it is shown that this “blurring” is, at higher long wave amplitudes, due to a quadratic phase error proportional to the phase velocity of the long wave. That part of the short wave ensemble allowed influential by SAR system (wavenumber) filtering is approximately nondispersive during the SAR azimuth integration time, its concerted effect being a rigid azimuth image translation, proportional to the short wave’s mean phase velocity. (2) More briefly treated, when the long wave is ranged directed and the short wave ensemble is simply a single range-directed sinusoid (“Bragg-matched”), the image is solely range variant and its nature is primarily determined by the aforementioned narrowband filtering effect, the secondary effects of nonlinear hydrodynamics and physical optics (i.e., surface slope) being evident. Therefore, the present model, as thus far elaborated, contradicts predictions of models based on the SAR response to a point scatterer in motion in accordance with the “orbital motion” of the long wave: e.g., no “azimuth bunching” attributed to such motion is observed.