On the effective inversion by imposing a priori information for retrieval of land surface parameters

Abstract

The anisotropy of the land surface can be best described by the bidirectional reflectance distribution function (BRDF). As the field of multiangular remote sensing advances, it is increasingly probable that BRDF models can be inverted to estimate the important biological or climatological parameters of the earth surface such as leaf area index and albedo. The state-of-the-art of BRDF is the use of the linear kernel-driven models, mathematically described as the linear combination of the isotropic kernel, volume scattering kernel and geometric optics kernel. The computational stability is characterized by the algebraic operator spectrum of the kernel-matrix and the observation errors. Therefore, the retrieval of the model coefficients is of great importance for computation of the land surface albedos. We first consider the smoothing solution method of the kernel-driven BRDF models for retrieval of land surface albedos. This is known as an ill-posed inverse problem. The ill-posedness arises from that the linear kernel driven BRDF model is usually underdetermined if there are too few looks or poor directional ranges, or the observations are highly dependent. For example, a single angular observation may lead to an under-determined system whose solution is infinite (the null space of the kernel operator contains nonzero vectors) or no solution (the rank of the coefficient matrix is not equal to the augmented matrix). Therefore, some smoothing or regularization technique should be applied to suppress the ill-posedness. So far, least squares error methods with a priori knowledge, QR decomposition method for inversion of the BRDF model and regularization theories for ill-posed inversion were developed. In this paper, we emphasize on imposing a priori information in different spaces. We first propose a general a priori imposed regularization model problem, and then address two forms of regularization scheme. The first one is a regularized singular value decomposition method, and then we propose a retrieval method in I 1 space. We show that the proposed method is suitable for solving land surface parameter retrieval problem if the sampling data are poor. Numerical experiments are also given to show the efficiency of the proposed methods.