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The phase change of repeat-pass Interferometric Synthetic Aperture Radar (InSAR) observations has been applied to monitoring of surface deformation, forest biomass, soil moisture and snow water equivalent in the past two decades. The InSAR phase measurement accuracy depends on the interferometric coherence of radar signals from repeat passes. In this article we presented theoretical modeling of the interferometric coherence of SAR observations, applicable to bistatic forward looking Signals of Opportunity (SoOp) as well as the backscatter geometries with primary focus on low microwave frequency (<5 GHz) for terrestrial snow remote sensing. The theoretical expression can account for the temporal and spatial variability of surface roughness and snow depth within a footprint and the domain for multi-look averaging. We applied our model to the analysis of interferometric coherence caused by snow depth changes for a range of surface roughness. The temporal and spatial characteristics of snow depth were based on a set of airborne LIDAR snow depth acquired at the Grand Mesa Colorado, which showed that snow depth can vary from a few cm to 50 cm in standard deviation and a spatial correlation length of a few to 40 meters. In addition, the characteristics of soil surface roughness were also estimated based on airborne LIDAR surveys. According to the observed characteristics of snow depth and rough surface variability, we find that the theoretical interferometric coherence varies with the correlation length and root mean square (rms) of the snow depth and also soil surface roughness. An increase in soil surface roughness will reduce the area of contribution to the interferometric coherence integral, thus reducing the effect of subfootprint variability and leading to an increase in coherence. Our theoretical formulation goes beyond the independent scattering assumption used in the past theoretical modeling for temporal decorrelation of repeat pass radar observations and can be used to assess the confounding effects of soil surface roughness, snow depth and spatial resolution. Numerical examples have been presented for two SAR configurations, including a L-band backscatter SAR and a P-band forward scattering SAR.
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