DSM SYNTHETIC SEISMOGRAMS USING ANALYTIC TRIAL FUNCTIONS - PLANE-LAYERED, ISOTROPIC, CASE

The Direct Solution Method (DSM) (Geller et al. 1990; Geller & Ohminat o 1994) is a Galerkin weak-form method for solving the elastic equatio n of motion. In previous applications of the DSM to both laterally hom ogeneous and laterally heterogeneous media the vertically dependent pa rt of the trial functions has been either linear splines (e.g. Cummins et al. 1994a, b) or the vertically dependent part of modal eigenfunct ions (e.g. Hara, Tsuboi & Geller 1993). In this paper we formulate the DSM using analytic trial functions which are solutions of the homogen eous (source-free) equation of motion in locally homogeneous portions of the medium. We present explicit formulations for SH and P-SV wave p ropagation in an isotropic, laterally homogeneous, plane-layered model . The trial functions so that it is easy to satisfy continuity of disp lacement at internal interfaces. For the laterally homogeneous SH prob lem we first find a set of R + 1 analytic trial functions that satisfy continuity of displacement at the R - 1 internal interfaces between t he R layers. Each of the trial functions is non-zero in at most two la yers. We then solve the weak form of the equation of motion, which in effect enforces the upper and lower boundary conditions, and continuit y of traction at the internal interfaces. The trial functions are chos en so that the equation of motion becomes a tridiagonal (R + 1) x (R 1) system of linear equations. For the P-SV problem we define a set o f 2R + 2 analytic trial functions that satisfy continuity of displacem ent, but not continuity of traction, at internal interfaces; the trial functions are chosen so that the equation of motion then becomes a (2 R + 2) x (2R + 2) system with a bandwidth of 7. In contrast, previous global solution methods (e.g. Chin, Hedstrom & Thigpen 1984; Schmidt & Tango 1986), which solve simultaneously for both internal continuity of displacement and traction as well as the external boundary conditio ns, solve a 2R x 2R system of linear equations for the SH problem or a 4R x 4R system for the P-SV problem, each having approximately twice the bandwidth of our systems of equations. We show that through an app ropriate choice of the form of the homogeneous solutions in each layer our approach can also be readily incorporated into strong-form global solution methods, thereby leading to exactly the same system of equat ions obtained by our weak form derivation. We also present the DSM equ ation of motion for a plane-layered medium composed of a combination o f fluid and solid layers. The dependent variable in the fluid layers i s a scalar quantity proportional to the pressure change, while the dep endent variable in the solid is the displacement. Continuity of displa cement and traction at fluid-solid boundaries is enforced by augmentin g the weak-form operator by appropriate surface integrals.