On non-uniqueness of the inverse problem for a seismic source-I. Treatment in terms of equivalent force

Abstract

SUMMARY The inverse problem of the earthquake-source theory is considered. It is shown that for a seismic source of general type distributed over a 3-D spatial region in infinite homogeneous isotropic elastic medium the problem has no unique solution not only in terms of moment density tensor (Backus & Mulcahy 1976a,b), but also in terms of equivalent force. This fact is related to the existence of non-radiating seismic sources. (A non-radiating seismic source is the one that has non-zero equivalent force but produces no motion outside some ball containing the source region: the acoustic case was considered by Friedlander 1973). We show that in the case when the input data are specified to be the far-field displacements, the only solutions of the homogeneous inverse problem are non-radiating sources. This means that far-field displacements uniquely determine the near-field ones. Therefore, the adding of near-field data cannot improve the statement of the problem to avoid non-uniqueness. To constrain the class of admissible sources we consider 2-D indigenous seismic sources that have a moment density tensor distributed over a known smooth surface. We show that for an unclosed surface the inverse problem has a unique solution in terms of equivalent force. A counter example for the unit sphere shows that if a surface is closed the inverse problem may have no uniqueness.