PROBABILITY DENSITY-FUNCTIONS FOR MULTILOOK POLARIMETRIC SIGNATURES

We derive closed-form expressions for the probability density functions (PDF's) for copolar and cross-polar ratios and for the copolar phase difference for multilook polarimetric SAR data, in terms of elements of the covariance matrix for the backscattering process. We begin with the case in which scattering-matrix data are jointly Gaussian-distributed. The resulting copolar-phase PDF is formally identical to the phase PDF arising in the study of SAR interferometry, so our results also apply in that setting. By direct simulation, we verify the closed-form PDF's. We show that estimation of signatures from averaged covariance matrices results in smaller biases and variances than averaging single-look signature estimates. We then generalize our derivation to certain cases in which back-scattered intensities and amplitudes are K-distributed. We find in a range of circumstances that the PDF's of polarimetric signatures are unchanged from hose derived in the Gaussian case. We verify this by direct simulation, and also examine a case that fails to satisfy an important assumption in our derivation. The forms of the signature distributions continue to describe data well in the latter case, but parameters in distributions fitted to (simulated) data differ from those used to generate the data. Finally, we examine samples of K-distributed polarimetric SAR data from Arctic sea ice and find that our theoretical distributions describe the data well with a plausible choice of parameters. This allows us to estimate the precision of polarimetric-signature estimates as a function of the number of SAR looks and other system parameters.