A frequency-domain method is presented in which the Fourier spectral amplitudes of observed earthquake ground motions are used as data to constrain the space-time dependence of slip on the fault. Performing the temporal deconvolution in the frequency domain allows the spatial dependence of slip at each frequency to be computed independently. This greatly reduces the computational effort and allows the grid spacing to be chosen sufficiently fine enough to form an accurate numerical approximation to the continuous problem, thereby eliminating spatial and temporal discretization effects. Time-domain methods require the specification of rupture time as a function of position on the fault in order to reduce the number of parameters in inversion. Some non-linear inversion methods iterate on the rupture time in order to find a set of rupture times which provides the best fit to the data. This non-linear restriction, and the potential bias it may introduce, is eliminated in the frequency-domain formulation.The method is applied to synthetic test data calculated using Haskell's model of a uniform rupture in a homogeneous full-space. Three different recording geometries with characteristics comparable to current strong motion arrays are considered. Inversion for the slip function is demonstrated to be non-unique and a particular solution is found which minimizes the square of the slip velocity averaged over the fault surface. The minimum norm solutions have systematically lower peak slip velocities and spectra which fall off much faster than the input model ( f P 3 vs. f -' ) . This discrepancy is shown to result from a trade-off between the spectral amplitude of slip at a point on the fault and the local phase velocity of slip propagation. The trade-off is quite large for the arrays considered, a factor of 10 to 100 at frequencies between 1.0 Hz and 2.5 Hz.

A new method is presented for computing the complete elastic response of a vertically heterogeneous half-space. The method utilizes a discrete wavenumber decomposition for the horizontal dependence of the wave motion in terms of a Fourier-Bessel series. The series representation is exact if summed to infinity and consequently eliminates the need to integrate a continuous Bessel transform numerically. In practice, a band-limited solution is obtained by truncating the series at large wavenumbers. The vertical and time dependence of the wave motion is obtained as the solution to a system of partial differential equations. These equations are solved numerically by a combination of finite element and finite difference methods which accommodate arbitrary vertical heterogeneities. By using a reciprocity relation, the wave motion is computed simultaneously for all source-observer combinations of interest so that the differential equations need only be solved once. A comparison is made, for layered media, between the solutions obtained by discrete wavenumber/finite element, wavenumber integration, axisymmetric finite element, and generalized rays.

Using a representation theorem from elastodynamics, subsurface slip on a known fault is formulated as the solution to an inverse problem in which recorded surface ground motion is the data. Two methods of solution are presented: the least-squares method, which minimizes the squared differences between theory and data, and the constrained least-squares method which simultaneously maintains a set of linear inequalities. Instabilities in the solution are effectively eliminated in both methods, and the sensitivity of the solution to small changes in the data is quantitatively stated. The inversion methodology is applied to 77 components of near-field ground acceleration recorded during the 15 October 1979 Imperial Valley earthquake. The faulting is constrained to propagate bilaterally away from the epicenter at an average velocity of 90 per cent of the shear wave speed on a vertical fault plane extending from the surface to 10 km depth. Inequality constraints are used to keep the faulting sequence physically reasonable by maintaining right-lateral motion and positive slip velocity. The preferred solution is stable and provides a good fit to the data; it is also realistic and consistent with observed surface offsets and independent estimates of seismic moment