Asymptotic ray theory is widely used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases. Unfortunately, these singularities are often the points of greatest interest as they are caused by inhomogeneities in the model. Transform methods, e.g., the reflectivity method and WKBJ seismograms, are used to investigate waves at these singular points but are restricted to laterally homogeneous media. Maslov asymptotic theory uses the ideas of asymptotic ray theory and transform methods, combining the advantages—simplicity and generality—of both techniques. In this paper, Maslov asymptotic theory is developed for the computation of body-wave seismograms.
The eikonal equation of asymptotic ray theory is equivalent to Hamilton's canonical equations, and the ray trajectories can be considered in the phase space of position and slowness. Normal asymptotic ray theory gives the wave solution in the spatial domain. However, the asymptotic solution for other generalized coordinates in phase space can also be found. For instance, normal transform methods find the solution in the mixed domain where the horizontal slowness replaces the coordinate. Maslov asymptotic theory extends this idea to inhomogeneous media, and the asymptotic solution in a mixed domain (position and slowness) is obtained by a canonical transformation from the spatial domain. The method is useful as the singularities in the mixed and spatial domains are at different locations, and Maslov theory provides a uniform result, combining the solutions in the different domains. These transforms between the mixed-frequency and spatial-time domains are evaluated exactly using the WKBJ seismogram algorithm. This avoids the oscillatory integrals of asymptotic theory and stabilizes the numerical solution by providing the smoothed, discrete seismograms directly. The result is a rigorous but simple method for computing body-wave seismograms in inhomogeneous media. The theory is developed in outline, and numerical examples are included.