Moment Tensors and other Phenomenological Descriptions of Seismic Sources—I. Continuous Displacements

Summary. In most seismic calculations the real physical stress tensor S is replaced by a linearized isentropic elastic model stress tensor 9. Backus & Mulcahy showed how the stress glut r = 9- S gives rise to all the usual phenomenological descriptions of earthquake sources, the equivalent forces, the moment tensor densities, the stress-free strain and the seismic moment tensors of various types and degrees. In our earlier paper the displacement field s was supposed to be continuously differentiable, so discontinuous faulting could not be discussed. The present work extends our earlier paper to cover the possibility that 9 is a singular generalized function, so that s can be discontinuous or singular. Even when s is smooth, generalized functions simplify the notation. Surface and volume forces, real or equivalent, can be combined in a single distribution; boundary conditions are automatically included in the field equations; and representing a localized source by certain moment tensors of low degree amounts to approximating the source by a point source with the same moments of low degree. It is shown that all the results in our earlier paper can be extended to discontinuous s and singular 9. New results, not obtainable in our earlier paper, include these: a simple description of the most general seismic point source; a proof that every tensor with the obvious appropriate index symmetries is a forcemoment or glut-moment tensor of a seismic point source; a catalogue of point sources with no seismic effects; a new, direct derivation of the classical phenomenological descriptions of ideal fault sources which Burridge & Knopoff, Dahlen and Walton obtained from the reciprocity theorem and the impulse response; a discussion of how, unlike ideal faults, a real fault might have a seismic moment tensor with non-zero trace; and a description of a source type, the simple surface source, which may be a more accurate phenomenological description of real faults than are the classical ideal fault descriptions. It can be shown that a simple surface source with fault surface Z is uniquely determined by the motion it produces if it is developable or if the Gaussian curvature of Z never vanishes, but not if Z contains a piece of a plane.