5.3.1 Joint Distribution of Single-Look Complex Data

The complex circular Gaussian model can be generalized to N measures that may be either N = L sub-looks (see section 5.3.4), N frequency or polarization channels, or a set of N neighboring pixels of the same image. In this case, z turns into a complex vector Z where N components are the z _n measures in the different channels. If the Goodman hypotheses are verified for each of these channels, and the imaginary and real parts of z _n verify the two following conditions:

the distribution of Z is [GOO 63]:

where ^t Z is the transposed vector of Z, and C _z = E( Z ^t Z*) is the covariance matrix of Z. The Hermitian matrix C _z generalizes the radar reflectivity concept, since its diagonal elements, i.e., z _n variances, correspond to N radar reflectivities R ₁, R ₂, etc., R _N. However, the non-diagonal elements C _nm = cov( z _n, z _m), which are covariances between the components, are proportional to the complex correlation coefficient (also referred to as the degree of coherence) between z _n and z _m:

The complex correlation module D is simply called coherence, while the phase ? is the effective phase...