Computation of synthetic seismograms and their partial derivatives for heterogeneous media with arbitrary natural boundary conditions using the Direct Solution Method

A new method is presented for calculating synthetic seismograms and their partial derivatives for laterally and vertically heterogeneous media with arbitrary natural boundary conditions. The formulation is derived by adding appropriate surface integrals to the weak form (Galerkin formulation) of the elastic equation of motion to enforce the natural boundary and continuity conditions, and inhomogeneous boundary conditions. Results applicable to media consisting of a combination of fluid and solid regions are presented. The method is called the Direct Solution Method (DSM) (Geller et al. 1990c) because the synthetic seismograms and partial derivatives are computed directly by solving a system of linear equations. In contrast, almost all previous applications of Galerkin methods in seismology have first computed the modes of free oscillation, and only then computed the synthetic seismograms and partial derivatives by summing the modes. As an example of the application of our method, we calculate synthetic seismograms for heterogeneous media which are terminated at the bottom by a thin homogeneous layer with a radiation (energy-absorbing) boundary condition. This method is well suited to computing the quantities necessary to perform linearized inversion for earth structure with respect to a laterally heterogeneous earth model (Geller & Hara 1993). It thus becomes possible to formulate iterative linearized waveform inversion for laterally heterogeneous earth structure on a local and regional scale following the same basic approach used by Hara, Tsuboi & Geller (1993) to invert waveform data for global laterally heterogeneous structure.