Implications of frequency-domain inversion of earthquake ground motions for resolving the space-time dependence of slip on an extended fault

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‒3 vs. f‒1). 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.