Generalized least-squares solutions to quasi-linear inverse problems with a priori information

Abstract

A quasi-linear inverse problem with a priori information about model parameters is formulated in a stochastic framework by using a singular value decomposition technique for arbitrary rectangular matrices. In many geophysical inverse problems, we have a priori information from which a most plausible solution and the statistics of its probable error can be guessed. Starting from the most plausible solution, the optimization of model parameters is made by the successive iteration of solving a set of standardized linear equations for corrections at each step. In under-determined cases, the solution depends inherently on the initial guess of model parameters, and then uncertainties in the solution are evaluated by the covariances of estimation error which results not only from the random noise in data but also from the probable error in the initial guess. A best linear inverse operator which minimizes the variances of estimation errors within the framework of a generalized least-squares approach is directly obtained from the “natural inverse” of Lanczos for a coefficient matrix by regarding an eigenvalue in the inverse as zero if it is smaller than unity. This provides a theoretical basis on the “sharp cutoff approach” of Wiggins and also Jackson in their general inverse formalisms. © 1982, The Seismological Society of Japan, The Volcanological Society of Japan, The Geodetic Society of Japan. All rights reserved.