Using Topography Statistics to Help Phase Unwrapping

Conventional techniques approach Phase Unwrapping (PU) as an optimization problem, where figures of merit like the total branch-cut length, the number of cuts, etc, is to be minimized. They disregard the properties of the field to be unwrapped: the topography, i.e. the DEM, projected in the SAR reference. The purpose of the paper is to fill in this gap by providing statistics of the “fringe maps”. We first exploit the Woodward theorem to link the interferogram Power Spectrum Density (PSD) with the Probability Density Function (PDF) of the Phase Gradient that would result in a likely topography. A parametric model of the expected, unwrapped PG PDF is then derived by exploiting the fractal properties of topography. Its parameters can be accurately estimated given the wrapped PG. This model provides useful statistic information for phase unwrapping. It is then possible, for example: (a) to estimate the number of residuals; (b) to find the best azimuth presumming factor and, (c), to find the optimal interferogram range demodulation. Finally, we exploit the second order statistics of the PG field (as a fractal) to derive a suitable approximation for the expected length of the branch cuts.