The use of partitioned matrices in geophysical inversion problems

Abstract

The parameters in many geophysical inverse problems may be partitioned so as to separate them into two distinct sets. This separation might be on the grounds of physical differences between the two sets, or it might be for computational reasons. In this paper, methods for making estimates of one or other set of parameters unbiased by uncertainty in the other set are summarized. It is shown that these procedures, which are characterized by asymmetric resolution matrices, are not equivalent to the generalized inverse solution. The properties of various matrix inverses used to obtain solutions are discussed in relation to the usual least-squares and minimum-norm conditions. Finally, a new algorithm for calculating the generalized inverse, in terms of the inverses of partitions, is given.