Ir. Joughin et al., PROBABILITY DENSITY-FUNCTIONS FOR MULTILOOK POLARIMETRIC SIGNATURES, IEEE transactions on geoscience and remote sensing, 32(3), 1994, pp. 562-574
We derive closed-form expressions for the probability density function
s (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-distrib
uted. The resulting copolar-phase PDF is formally identical to the pha
se PDF arising in the study of SAR interferometry, so our results also
apply in that setting. By direct simulation, we verify the closed-for
m PDF's. We show that estimation of signatures from averaged covarianc
e matrices results in smaller biases and variances than averaging sing
le-look signature estimates. We then generalize our derivation to cert
ain cases in which back-scattered intensities and amplitudes are K-dis
tributed. We find in a range of circumstances that the PDF's of polari
metric signatures are unchanged from hose derived in the Gaussian case
. We verify this by direct simulation, and also examine a case that fa
ils to satisfy an important assumption in our derivation. The forms of
the signature distributions continue to describe data well in the lat
ter case, but parameters in distributions fitted to (simulated) data d
iffer from those used to generate the data. Finally, we examine sample
s of K-distributed polarimetric SAR data from Arctic sea ice and find
that our theoretical distributions describe the data well with a plaus
ible choice of parameters. This allows us to estimate the precision of
polarimetric-signature estimates as a function of the number of SAR l
ooks and other system parameters.