Modeling Strong Motions Produced by Earthquakes with Two-dimensional Numerical Codes

We present a scheme for generating synthetic point-source seismograms for shear dislocation sources using line source (two-dimensional) theory. It is based on expanding the complete three-dimensional solution of the wave equation expressed in cylindrical coordinates in an asymptotic form which provides for the separation of the motions into SH and P-SV systems. We evaluate the equations of motion with the aid of the Cagniard·de Hoop technique and derive close-formed expressions appropriate for finite-difference source excitation. INTRODUCTION Recent strong motion modeling efforts have been restricted to plane-layered models as displayed in Figure la. Point-source shear dislocations, or double couples, are applied at each element where the seismic field is decomposed into SH and PSV type motions and the vertical and horizontal dependences separated following the approach pioneered by Harkrider (1964). Nonuniform fault slip may be simulated by summing weighted point sources distributed along the fault plane to construct realistic synthetic seismograms. Recent inversion studies based on matching these synthetics to observations such as Hartzell and Heaton (1983), Archuleta (1984), and Olson and Apsel (1982) have provided amazing detail on the complex faulting process for the Imperial Valley 1979 earthquake. Unfortunately, most geologic structures in the vicinity of earthquakes are at least as complicated as displayed in Figure lb. Separating propagational effects from complex faulting becomes much more difficult in these situations. In this paper, we address the construction of synthetics along the surface for twodimensional structures such as displayed in Figure lb. We assume that the model remains constant into and out of the plane of the paper along with line sources through each element. We design the line-source characteristics to mimic the vertical radiation pattern appropriate for double couples, where the SHand P-SV field remain decoupled along paths to the receivers. Our main objective is to derive these line-source excitation functions. In a companion paper [Vidale and Heimberger (1988)], we discuss numerical strong ground motion calculations for a two-dimensional structural model through the Los Angeles region and compare these results with observations from the San Fernando earthquake. THEORY The approach follows closely the usual shear dislocation theory developed for treating plane layered models, where the wave field is separated into vertical and horizontal functions. This separation is essential for expressing the field in terms of SH and P-SV systems and provides the key to our approach. A particularly convenient form of the solution is given by Heimberger and Harkrider (1977) in terms of Laplace-transformed displacements along the vertical, tangential, and radial directions, 109 110 DONALD V. HELMBERGER AND JOHN E. VIDALE