**Phase Statistics of SAR Interferometric Stacks**
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Samiei Esfahany, Sami; Hanssen, Ramon
Delft University of Technology, NETHERLANDS
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Since the late 1990s, different methodologies have been developed for processing multi-baseline Interferometric Synthetic Aperture Radar (InSAR) data stacks. Most of these algorithms use interferometric phases as initial inputs. Therefore, the phase statistics of interferograms (i.e., noise characteristics of interferometric phases) can play an important role in performance of different InSAR methodologies. For example, these statistics can be used for maximum likelihood estimation of different parameters of interest such as deformation, deformation rate and topography, and can also be used for uncertainty (precision and reliability) description of these products.
The phase statistics are ideally described by a probability distribution function (pdf) of interferometric phases. This pdf is a function of absolute interferometric coherence, which is a measure of the accuracy of the interferometric phase. In case of a single interferogram, this pdf has been derived in the closed form for both single- and multi-looked cases in literature. However, in case of multi-baseline interferograms, we are dealing with a complicated multi-dimensional joint pdf of interferometric phases. Such a joint pdf is difficult to derive and exploit, not only because of its high dimensionality but also because of the highly nonlinear relation between the interferometric phases and the acquired complex SAR data. For simplicity, this pdf is usually computed based on the harsh assumption of mutual independence between interferograms. In this case, the joint pdf of multiple interferograms would be the product of pdfs of the single interferograms. However, this independence assumption is not always met, especially because of the correlation of noise components between interferograms. Also, two interferograms with a common SAR acquisition can never be independent, since they both depend on the common acquisitions. Some attempts have been found in literature for the closed-form evaluation of the second-order joint pdf of two single-looked interferograms obtained by three correlated SAR images. No such study has been found for more general cases, i.e., for any two interferograms and for different multi-looking factors.
Another convenient way to describe the phase statistics, especially for near-Gaussian data is using pdf second-order central moments (i.e. covariance matrix) of interferometric phases instead of the full joint pdf. In this way, a complicated *n*-dimensional joint pdf can be simplified to an *nxn* covariance matrix which is more convenient to exploit in practice. Diagonal elements of such a matrix are variances of interferometric phases and off-diagonal elements are the covariances which describe the dependency among interferometric phases. The closed-form expression of interferometric phase variances has been derived in literatures for the single-looked case. For multi-looked phases, variances should be derived numerically from the pdf. For evaluating the covariances, we need however again the joint pdf of interferometric phases which is not available in general form.