On non-uniqueness of the inverse problem for a seismic source-II. Treatment in terms of polynomial moments

Abstract

SUMMARY In a companion paper (Pavlov 1994) we demonstrated and examined non-uniqueness of the inverse problem for a 3-D seismic source of indigenous type in terms of equivalent force fi(x, t). In this paper we study the non-uniqueness of the inverse problem in terms of polynomial moments (PM) with respect to spatial and temporal variables of both the rate moment tensor mij(x, t) and the rate equivalent force fi(x, t). The study includes two steps. In the first step we obtain the representation for general solution of the inverse problem for the PM of fi(x, t) of total degree n. To do this we use the linear equations for the PM of fi(x, t) derived here from the representation theorem for displacements in an infinite homogeneous isotropic elastic medium. Because of linearity of the problem its general solution is represented as the sum of two members: (1) a particular solution of the inhomogeneous problem and (2) the general solution of the homogeneous one. We obtain the explicit expression for the second member of the sum. It is equal to zero for n= 1, 2 and contains n(n2 - 1)/2 - 6 free parameters for n×3. In the second step we obtain the general representation for the solution of the inverse problem in terms of the PM of mij(x, t) of total degree n. The solution is unique for n= 0, 1 and contains (n+ 2)(n2+n - 3) free parameters for n× 2. We also consider the inverse problem for a static source. Non-uniqueness of the problem is characterized by the set of null sources (i.e. a static source that does not produce displacements outside the source region). Representations are obtained for the PM of a null source. The PM of the final equivalent force fsti(x) of a null source of degree n are equal to zero for n= 0, 1 and are expressed through 3n(n+ 1)/2 arbitrary parameters for n× 2. The PM of degree n of the final-moment density tensor mstij(x) of a null source is equal to zero for n= 0 and is expressed through n2+n - 1 arbitrary parameters for n× 1.