Accounting for the definition domain of the forward map in traveltime tomography—application to the inversion of prismatic reflections

Abstract

Traveltime tomography aims at recovering an earth model from the traveltimes associated with some seismic events. Traveltime tomography appears as a strange problem in which the forward modelling operator may be non-defined or multivalued. Although important work has been devoted to the solution of the latter difficulty, the possible non-definition of the forward map has not yet been investigated. However, this pathology is a major problem when events with complex kinematics are to be inverted. This leads us to propose a new algorithm to overcome the difficulty. In this algorithm the inverse problem is, by construction, restricted to the definition domain of the forward map and is based on a reformulation of the forward problem. Our formalism is described in the context of the inversion of a specific multiple reflection, namely the prismatic reflection. Yet the philosophy of our approach is quite general and could be transposed to the inversion of events with arbitrary kinematics. The effectiveness of our method is illustrated through a simple synthetic experiment in which we invert a prismatic reflection for the delineation of an overhanging salt flank.