Impact of vertical water particle motions on focused SAR altimetry

New in this work is a new analytical derivation of a Delay-Doppler map model for Fully-Focused SAR, which includes the vertical wave particle motion and a more general surface representation. It was indeed already noticed in Egido and Smith (2017) that the open ocean surfaces decorrelate within the integration time interval, which is used in the FF-SAR processing to form the synthetic antenna. However, this was not further discussed. The impact of surface motion is expected to be larger on FF-SAR than for unfocused SAR, since a longer integration time is used in the first processing. We introduce vertical wave particle velocities and accelerations as random variables in the analytical formulation, and relate this variables to ocean-wave spectral parameters, such as the mean zero up-crossing and crest period. Further on, we show and discuss for the first time that the flat sea surface response and the probability density function of the height of scattering surface elements are not the same as in unfocused SAR. Moreover, the response of a point target differs significantly and depends on the location of a scattering surface element, the satellite orbit parameters and the tracking range. This work provides the analytical expressions and a thorough discussion of these issues, including numerical results using CryoSat-2 mission parameters.