Solving the Wave Equation Using Curvelets

Seismic imaging is a key step in seismic exploration to retrieve the Earth properties from seismic measurements at the surface. One needs to properly model the response of the Earth by solving the wave equation. We present how curvelets can be used in that respect. Curvelets can be seen from the geophysical point of view as the representation of local plane waves. The unknown pressure, solution of the wave equation, is decomposed in the curvelet domain. We derive the new associated equation for the curvelet coefficients and show how to solve it. In this paper, we focus on a simple homogeneous model to illustrate the feasibility of the curvelet-based method. This is a first step towards the modeling in more complex models. In particular, we express the derivative of the wave field in the curvelet domain. The simulation results show that our algorithm can give a multi-scale and multi-directional view of the wave propagation. A potential application is to model the wave motion in some specific directions.