A norm minimization criterion for the inversion of earthquake ground motion

Abstract

SUMMARY A method is presented for obtaining a unique temporal and spatial slip distribution on a prescribed fault plane from inversion of earthquake ground motion. The inverse problem is formulated in the frcquency domain where the spatial distribution of slip is found for a set of frequency values. The time dependence of slip is then recovered by Fourier synthesis. A new norm minimization condition is presented, based upon the scalar wave equation, which produces the model having the most nearly constant rupture velocity and the most nearly constant slip amplitude distribution. Although a preferred rupture velocity is specified, no assumption on the geometrical shape of the rupture front is made. While time-domain inversion methods prescribe the hypocentre location, source time function, and rupture front geometry, the frequency-domain method actually solves for these important source properties. The wave equation minimization condition is tested using synthetic data from Haskell-type dislocation models in a homogeneous full-space. These solutions are compared with the traditional two-norm minimization solutions. The two-norm minimization models exhibit a strong dependence on the recording array geometry. The wave equation minimization models are smoother and devoid of many of the spurious features that are not actually required by the data but that, nevertheless, appear in the two-norm minimization models. The results of numerical experiments indicate that the solution need not conform to the specified rupture velocity value used in the construction of the wave equation norm if the data are sufficient to warrant otherwise.