Abstract
We propose a new method to separate the incident and reflected arrivals and correct for the post-critical phase shift in wide angle reflection imaging situations. Such a situation arises, for example, in quasi-vertical geological fault imaging using data from small earthquakes. Major faults are often associated with a high degree of seismic activity. There is a contrast in impedance across the fault surface due to the shift of the bounding structures, so that the fault surface can generate reflections of seismic waves visible on seismograms. These two factors make it possible to use reflections of waves from small earthquakes to perform seismic imaging of the fault. Two major challenges arise due to the earthquake sources being very close to the fault: (1) the incident and reflected waves are not well separated on seismograms so that muting of the incident wave is not possible, and (2) most of the waves are reflected post-critically, which causes a distortion in the reflected waveforms. In this paper we present a new technique to simultaneously separate the incident and reflected arrivals, and compute the phase correction for the post-critically reflected waves by formulating these two steps as a single optimization problem. Our implementation of the method is acoustic and 2-D. The method is based on asymptotic representation of the incident and reflected acoustic waves from point sources in two dimensions, and assumes that the source time function of the source is known. The minimization problem is highly non-linear and the objective function is very oscillatory. We propose to solve it by a particle swarm optimization method. We present synthetic numerical examples of fault reconstructions from separated and phase-corrected reflections obtained by our method.