Elastic waves in isotropic media exhibit two modes of propagation. Finite speed of propagation holds in P-wave light cones (that is, if the solution vanishes on a space-like hypersurface, then the solution also vanishes in a P-wave light cone), but the unique continuation principle applies to S-wave light cones (that is, when the solution vanishes in a neighborhood of a time-like hypersurface, the solution also vanishes in an associated S-wave light cone). Both of these principles are utilized in analyzing reverse time continuation from a boundary.
The inverse source problem has a proper solution in the case of an instantaneous (point) source and can include the estimation of the centroid moment tensor. A possible algorithm is based on time reversal. If the source is time depen- dent, however, there is no reconstruction possible in general, and particular source representations need to be sought forto obtain uniqueness and that are still realistic to describe, for example, rupture properties of (large) earthquakes. We analyzed the discrete-time-dependent inverse source problem and developed an algorithm for reconstruction [preprint 5].
We assume that the sources are delta-like in time, and (roughly speaking) sufficiently positive, of limited oscil- lation, and with support of large enough volume in space. We begin by using time-reversal from the boundary to recover a full waveform related (but not precisely equal) to the solution of the original equation. In particular, this new wave will be an (a priori unknown) superposition of solutions to initial velocity problems related to our source problem via Duhamel’s principle and will include reflections across source times. Then, from this wave, we identify locations in space-time where displacement vanishes but velocity is sufficiently bounded from below, and show how these correspond to the initial velocities of the above problems, thus determining the sources.
Of course, there may be locations, not corresponding to sources, where displacement vanishes and velocity does not; we analyze how to differentiate these ‘artifacts’ from true sources. This is where we need the limited oscillation of the sources: if the source is not too badly behaved in this regard, displacement cannot vanish in a short time interval. For longer time intervals, if there is sufficient local energy decay, the kinetic energy of the wave must disperse, and so the velocity cannot be too high on large sets. Combining these two observations about single sources, and assuming the sources are reasonably separated from one another in space-time allows us to fully identify all sources and distinguish them from artifacts.