Finite-difference methods (FDM) are the most common ones used to simulate seismic ground motion in sedimentary basins. Using a 2D finite-difference method, Vidale and Helmberger [46] approximated the 1971 San Fernando earthquake as a point source and propagated wave (0.1-0.5Hz) along a cross section of the San Fernando and Los Angeles Basins. The same approach has been applied to other earthquakes; Schrivner and Helmberger [43] simulated ground motion in the Las Angeles basin from the 4 October 1987 Whittier Narrows aftershock (M 5.3) and the 28 June 1991 Sierra Madre main-shock (M 5.8); Pitarka et al [36] simulated ground motion in the Ashigara Valley from the 5 August 1990 Odawara, Japan, earthquake. The 2-D finite difference method was extended to 3-D by Olsen and Schuster in 1992 [34], who simulated the wave propagation in the Salt Lake Basin mine blasts and plane waves. Frankel and Vidale [17] simulated 3-D wave propagation in the Santa Clara Valley by using a far-field point source. Yomogida and Etgen [50] applied this method to the Los Angeles Basin in simulating the 1987 Whittier-Narrows earthquake. Frankel [16] took the next step when he simulated 3-D elastic waves in the San Bernardino basin from a hypothetic M6.5 earthquake on the San Andreas fault. Although the source model was nearly a line source (30-km long and 3-km wide) and the model depth was limited to 7 km, Frankel included both a 3-D velocity structure and a propagating rupture over a finite fault area. The first integration of the full 3-D velocity structure of the Los Angeles area with an earthquake rupture on a finite fault was carried out by Olsen et al [33]. Using a parallel supercomputer, they simulated ground motion for an M 7.75 earthquake along a 170-km-long stretch of San Andreas fault from Tejon Pass to San Bernardino for frequencies up to 0.4 Hz. In 1996, Olsen and Archuleta [32] simulated ground motion for three hypothetical M 6.75 earthquakes on the Los Angeles fault system by using a complex 3-D geometrical model of the Los Angeles area developed by Magistrale et al [31] and modeling the kinematic earthquake rupture.
Although FDMs have contributed to the simulation of ground motion in
the full-size LA basin, uniform grid domains prevent them from getting
results with good spatial and temporal resolution. To see why uniform grids
are impractical, consider the Los Angeles Basin.
For a shear wave velocity of 0.4 km/s and a frequency of 2 Hz, a
regular discretization of the elasticity operator would place grid
points 0.02 km apart to achieve second order accuracy. The region of
interest has dimensions 140 km 
100 km 
20 km; thus, a
regular discretization, governed by the softest layer, requires 35
billion grid points with three displacement components per grid
point. At least a terabyte of primary memory would be needed, and on
the order of 
operations would be required at each time
step. The stability condition associated with explicit time
integration of the semidiscrete equations of motion imposes a time
increment at least as small as 0.004s. Thus, a computer would have to
perform at a sustained teraflop per second for two days to simulate a
minute of shaking.