A Local Level Set Method for Paraxial Geometrical Optics

Abstract

We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two-dimensional (2-D) acoustic wave equations. The geometrical optics term consists of two multivalued functions: a travel-time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally. The multivalued travel-times are obtained by solving a level set equation and a travel-time equation with a forcing term. The multivalued amplitudes are computed by a newEulerian formula based on the gradients of travel-times and takeoff angles. As a byproduct the method is also able to capture the caustic locations. The proposed Eulerian method has complexity of $O(N^2{\rm Log }N)$, rather than $O(N^4)$ as typically seen in the Lagrangian ray tracing method. Several examples including the well-known Marmousi synthetic model illustrate the accuracy and efficiency of the Eulerian method.