Linearization of a scalar matrix operator method radiative transfer model with respect to aerosol and surface properties

Abstract

In this paper, we review the radiative transfer formalism of the matrix operator method, and present the analytic form for its differentiation with respect to aerosol optical thickness, microphysical parameters and surface parameters. This “linearization” is an exact method that allows for an accurate and speedy computation of the Jacobian matrix, which is key to most optimization-based retrieval methods. We define an aerosol in terms of its optical thickness, complex refractive index and lognormal size distribution. We consider a bimodal aerosol distribution, consisting of a fine and coarse mode, such that the two modes also differ in their respective complex refractive indices. Three types of surfaces have been considered, viz. a purely Lambertian surface, a modified Rahman–Pinty–Verstraete bidirectional reflecting surface, and a Fresnel-reflecting ocean surface. We verify our results by comparing our linearized Jacobians of normalized intensities calculated at four different wavelengths in the visible (VIS) and near-infrared (NIR) and viewing angles ranging from −75° through 0° to 75° with those computed by the method of finite differences. We demonstrate the guaranteed accuracy of the linearized approach by contrasting it with the finite difference method which can only be used as a rough estimate due to its sensitivity to step size, especially for derivatives with respect to aerosol microphysical parameters. We also report that the computational speed-up due to linearization improves with the number of parameters involved, parity being achieved with the finite difference method for just one parameter. Finally, we discuss the features of the illustrated Jacobians as a function of viewing angle and wavelengths.