An inverse problem in seismology: derivation of the seismic source parameters from P and S seismic waves

摘要

This paper presents the solution of an inverse problem in Seismology, which aims at deriving the seismic source parameters from P and S seismic waves. In particular, the paper gives the deduction of the seismic-moment tensor. The problem is tackled in this paper under three particular circumstances. First, we use the amplitude of the far-field (P and S) seismic waves as input data. We use the analytical expression of the seismic waves in a homogeneous isotropic body with a seismic-moment source of tensorial forces, the source being localized both in space and time. We assume that the position of the seismic source is known. The far-field waves provide three equations for the six unknown parameters of the general tensor of the seismic moment, such that the system of equations is under-determined. Second, the Kostrov vectorial (dyadic) representation of the seismic moment for a shear faulting is used. This representation relates the seismic moment to the focal displacement in the fault and the orientation of the fault (moment-displacement relation); it reduces the seismic moment to four unknown parameters. Third, the fourth missing equation is derived from the energy conservation and the covariance condition. The four equations derived here are solved and the seismic moment is determined, as well as other parameters of the seismic source, like focal volume, focal slip, fault orientation, and duration of the seismic activity in the source. It turns out that the seismic moment is traceless, its magnitude is of the order of the elastic energy stored in the focal region (as expected), and the solution is governed by the unit quadratic form associated with the seismic-moment tensor (related to the magnitude of the longitudinal displacement in the P wave). A useful picture of the seismic moment is the conic represented by the associated quadratic form, which is a hyperbola (seismic hyperbola). This hyperbola provides an image for the focal region: its asymptotes are oriente