Body Force Equivalents for Stress - Drop Seismic Sources

The equivalent body forces for a stress-drop seismic source are found. When the isotropic stress drop and one of the three principal stress drops are zero, then the equivalent body forces are the same double couple without moment which would result from a shear dislocation. In general however, all six stress-drop components must be specified as independent functions of time. THE STRESS-DROP SOURCE Gilbert (1970) introduced the concept of a seismic moment tensor, which he defined as the volume integral of the stress drop. Gilbert's "stress drop" does not consider the change in stress resulting from the elastic waves radiated by the source. Thus his stress drop, z(x, t) serves only to define the kinematic body forces Z = ~ j j (1) which in turn generate the dynamic displacement field through the equation of motion a i j j p ~ = f i . (2) On the other hand, several investigators, e.g., Richards (1973), have considered the problem of finding the dynamic dislocation on a fault plane which is compatible with a prescribed stress drop (total change in the stress field at each point on the fault, rather than Gilbert's definition). The dislocation field then is used with the representation theorem to calculate displacements everywhere in the medium. In this paper, we consider Gilbert's source and hence "stress drop" refers to his definition. The displacements f roman arbitrary body-force distribution are given by the representation theorem of de Hoop (1958) and Burridge and Knopoff (1964), ui(x, t) = Sv~ Go[fj]dV(q) (3) where Vs is a volume completely enclosing the region of nonzero body forces, and Gij is the Green's tensor operator. For the stress-drop source (3) becomes ui(x, t) = f. vs --'Gij['Cjk,k(~' t)]dV(~). (4) But by Green's theorem (4) becomes us(x, t) = J'as-Gij[zjknk]dA(~) +I~ ~ a,j[~jkldV(~). (5) Because zij = 0 everywhere outside the source region and A~ is a surface completely enclosing the source, the surface integral vanishes, thus ui(x, t) = Sv, Gij.k[Tjk(¢, t)]dV(¢) (6) 1801 1802 ROBERT J. GELLER POINT STRESS-DRoP SOURCES Gilbert (1970) introduced the concept of using a moment tensor (volume integral of stress drop) to represent a point source. Gilbert (1973) gives the moment tensor elements for an isotropic source, a shear dislocation and a compensated linear vector dipole. McGarr (1976) uses the moment tensor representation to study earthquakes resulting from volume changes. Randall (1971) showed that seismic moment of a "generalized dislocation" is a tensor. In general a moment tensor representation has degrees of freedom (the six stress-drop elements or alternatively the three principal axes and the three principal stress drops) and thus six independent time functions. One cannot find "the" source function for a seismic event without first having determined from observations that all six time functions are identical. In this section we will consider the simplified case of a point stress-drop source with fixed principal axes and only two independent principal stress time functions--one for the isotropic dilatational part and one for the deviatoric part. When represented in the principal axes coordinate system the stress-drop tensor is (i0 0) = ~ 2 2 ~ ( x x o ) 0 %3 = AIg(t)+ DEE 0 h(t) f i (x -xo) (7a) 0 D 3 3