Motions Near a Shallow Rupturing Fault: Evaluation of Effects Due to the Free Surface

The displacement of the surface of a half space near a shallow rupturing fault is, generally, approximated poorly by doubling the amplitude calculated for the same source in an infinite space. To obtain this result, the motions of a half space were calculated using a Green's function which is a solution to Lamb's problem, and the motions of an infinite space were calculated using the formulae of Haskell. A good approximation to the half-space displacement caused by a P- or S-wave incident at most angles from a point source is numerical correction of the displacement resulting from the point source in an infinite space for the amplification and phase shift of a plane-wave incident at a free surface. This correction approximately doubles the amplitude of the infinite space displacement for SV-waves with angles of incidence within 30 degrees of vertical, for P-waves within 70 degrees of vertical, and for all SH-waves. The static offset of the free surface from a point source is not, in general, twice the offset calculated in the corresponding infinite space case. The displacement from an extended fault is calculated by superposition of point sources on the fault plane; when the infinite space amplitudes may be doubled for all (or most) of these point sources, it may also be doubled for the extended source.