The Computation of Seismograms in 3D Heterogeneous Media Using Maslov Theory

The most efficient methods for modeling seismic waves in heterogeneous media are asymptotic. However, the simplest of these methods (asymptotic ray theory or ART) breaks down near caustics. Caustics are abundant in heterogeneous media. It is therefore necessary to replace ART by a method that is more generally valid. Such methods typically consist of one or two-dimensional integrals over initial slowness or take-off angles. One such method is Maslov asymptotic ray theory (MART). We present a two-dimensional integration method based on Maslov theory. We show how a Maslov amplitude for the integrand can be derived using the method of stationary phase. Then, away from caustics where ART is valid, MART agrees asymptotically with ART. The amplitude function has the desirable property that it is zero at pseudo-caustics so artifacts are of lower order. The terms needed for the Maslov integrand can all be obtained by standard kinematic and dynamic ray tracing. In order to minimize the number of rays shot, triangular ray tubes are formed using the Delaunay algorithm, and an error criterion imposed in each tube. Rays are added until the paraxial approximation is accurate for interpolation in all tubes. Thus the integration domain is irregularly sampled and the integration reduces to an integral over irregular triangles. The resultant algorithm is both efficient and accurate. The method is applied to a number of heterogeneous models (both 1D and 3D). It is shown that care must be taken in choosing the integration domain large enough so that all the relevant rays are taken into account.