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

Abstract

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.

Resolution of Rise Time in Earthquake Slip Inversions: Effect of Station Spacing and Rupture Velocity

“…In principle, the model discretization in time is similar to the classical multi-timewindow method introduced by Hartzell and Heaton (1983), but the implementations of that approach are typically limited to about 10 windows (Hartzell and Langer, 1993). Our unrestricted multi-time-window approach allows for time windows to cover the whole duration of the rupture, at all fault locations, unlike other multi-time-window approaches used (Olson and Anderson, 1988;Das and Kostrov, 1990;Gallovič et al, 2009;Gallovič and Zahradník, 2011). Our approach does assume a known fault geometry, as typical in inversions (Hartzell and Heaton, 1983;Olson and Anderson, 1988;Graves and Wald, 2001;Ji et al, 2002).…”

“…The effect of station spacing on inversions has been considered in several studies. Miyatake et al (1986) and Olson and Anderson (1988) studied this effect by considering a line array of stations perpendicular and parallel to the fault. Saraò et al (1998) found that stations on the hanging wall facilitate source inversion on dip-slip faults.…”

Reducing nonuniqueness in finite source inversion using rotational ground motions

“…Furthermore, nonlinearities in the forward model, noisy data with insufficient coverage, and trade-offs between model parameters, result in a well-documented nonuniqueness of finite source models. Examples where different rupture scenarios explain observations equally well may be found, for instance, in Olson and Anderson [1988], Beresnev [2003], and Ide et al [2005], as well as in the Blindtest on Earthquake Source Inversion, initialized within the E.U. Project SPICE (Seismic wave Propagation and Imaging in Complex media: a European network) [Mai et al, 2007; http://equake-rc.info/].…”

Source Description of the 1999 Hector Mine, California, Earthquake, Part I: Wavelet Domain Inversion Theory and Resolution Analysis

“…These studies are time-domain inversions; i.e., the fault model is determined by fitting the seismic waveform data. In addition, Olson and Anderson (1988) investigated the use of a linear frequency-domain inversion, which also allows for the simultaneous solution of both slip amplitude and rupture time. Several larger earthquakes have been analyzed using a variety of the above methods (e.g., 1989 Loma Prieta earthquake, Beroza, 1991;Wald et al, 1991;1992 Landers earthquake, Cohee and Beroza, 1994;Wald and Heaton, 1994;1994 Northridge earthquake, Hartzell et al, 1996;Wald et al, 1996), and the complexity and general characteristics of these earthquakes have been used in scores of seismological studies and even to invoke a new theory of earthquake mechanics (e.g., Heaton, 1990).…”