We use single-station recordings of widely distributed earthquake sources to identify heterogeneities in the Earth's crust that scatter energy into the early part of the observed S-wave codas. We assume that ray theory accurately describes the propagation of body waves from the earthquake sources to the scatterers, but we assume no propagation mechanism between the scatterers and the seismic station. A grid of hypothetical scatterers is assumed to cover the Earth's surface, and the strength of each scatterer is determined by an iterative inversion method. Our data are recordings of aftershocks of the 1986 North Palm Springs earthquake recorded at a seismic station in the Coachella Valley, SMP, and at a station in the Morongo Valley, SMC. The technique is applied separately to each horizontal component of motion at each station. We identify a particular spot at the northwest edge of the Quaternary alluvium in the Coachella Valley that scatters waves into the S coda observed at SMP 8 km distant in the valley. The geologic structure that causes the strong scattering seems to be the basin-edge structure rather than the mountainous terrain 3 km farther west. The early coda at SMC is dominated by scattering within a few kilometers of SMC, either in the basin itself or in the Morongo Valley fault zone. This study shows that laterally propagating waves are sometimes observable in the early coda, casting doubt on the use of 1-D techniques for estimating site responses. The northwest edge of the Coachella Valley seems to be an especially strong scatterer of incident waves. If other basins have similar loci of strong scattering, perhaps these places can be identified using microearthquake seismograms and used to predict strong ground motions in the basins.