A Complete Theoretical Derivation of the Level of Random Height Noise in Interferometric Unfocused SAR Swath Altimetry

We derive the level of random height error that affects ocean topography measurements from interferometric unfocused SAR swath altimeters, such as the ka-band radar interferometer (KaRIn) instrument onboard the surface water ocean topography (SWOT) mission. The sources of noise considered here are thermal noise, the stochastic backscatter from the rough sea surface, and spatially white surface waves. The classical approach to predict the level of height noise resorts to approximations (e.g., independence between Doppler beams) whose impact does not seem to be discussed in any reference, and most importantly, it invokes an effective number of looks to address the variance reduction obtained through multilook averaging of the interferogram for which a motivated derivation appears to be lacking. We derive explicit expressions for the standard deviation of the height measurement in terms of the instrumental and processing parameters and show that they are accurate to the percent level across parameter space by comparing our results to numerical simulations. By contrast, the common prescription leads to errors of about 10% for SWOT and becomes increasingly wrong as the SNR increases, a failure that our formalism allows us to explain. In addition to filling a gap in the literature by providing a complete theoretical derivation from the first principles of the level of random height error, our work provides tools to validate numerical simulations and easily explore the impact of hardware and processing choices for future wide-swath altimeters. We leave for future work the treatment of the additional random error generated by the presence of spatial correlations in the waves.