Variance of a standard vector Monte Carlo estimate in the theory of polarized radiative transfer

Abstract

The spectral radius ρ of the matrix integral operator defining the covariance matrix of a standard vector Monte Carlo estimate in the polarized radiative transfer problem is examined. The theory of positive operators is used to analytically calculate ρ = ρ0 for transfer through an infinite homogeneous medium. For a bounded medium, it is shown that ρ is approximately equal to ρ0 times the spectral radius of the operator corresponding to radiative transfer without polarization. This is shown numerically by estimating the iterations of the corresponding resolvent and approximately analytically by using a perturbation of a special functional.