HR: 0800h
AN: S11A-0998    [Abstracts]
TI: Modeling Finite Faults Using the Adjoint Wave Field
AU: * Hj\"orleifsd\'ottir, V
EM: vala@gps.caltech.edu
AF: Caltech, MC 252-21 1200 E California Blvd, Pasadena, CA 91125
AU: Liu, Q
EM: lqy@gps.caltech.edu
AF: Caltech, MC 252-21 1200 E California Blvd, Pasadena, CA 91125
AU: Tromp, J
EM: jtromp@gps.caltech.edu
AF: Caltech, MC 252-21 1200 E California Blvd, Pasadena, CA 91125
AB: Time-reversal acoustics, a technique in which an acoustic signal is recorded by an array of transducers, time-reversed, and retransmitted, is used, e.g., in medical therapy to locate and destroy gallstones (for a review see \textit{Fink},~1997). As discussed by \textit{Tromp et al.}~(2004), time-reversal techniques for locating sources are closely linked to so-called `adjoint methods' (\textit{Talagrand and Courtier}, 1987), which may be used to evaluate the gradient of a misfit function. \textit{Tromp et al.}~(2004) illustrate how a (finite) source inversion may be implemented based upon the adjoint wave field by writing the change in the misfit function, $\delta\chi$, due to a change in the moment-density tensor, $\delta \mathbf{m}$, as an integral of the adjoint strain field $\epsilon^\dagger({\mathbf x},t)$ over the fault plane $\Sigma$: $\delta \chi = \int_0^T\!\!\!\int_\Sigma \epsilon^\dagger({\mathbf x},T-t) \!:\!\delta {\mathbf m}({\mathbf x},t)\:d^2{\mathbf x}dt$. We find that if the real fault plane is located at a distance $\delta h$ in the direction of the fault normal $\hat{{\mathbf n}}$, then to first order an additional factor of $\int_0^T\!\!\!\int_\Sigma \delta h ({\mathbf x}) \partial_n \epsilon^\dagger({\mathbf x},T-t)\!:\!{\mathbf m}({\mathbf x},t) \:d^2{\mathbf x}dt$ is added to the change in the misfit function. The adjoint strain is computed by using the time-reversed difference between data and synthetics recorded at all receivers as simultaneous sources and recording the resulting strain on the fault plane. In accordance with time-reversal acoustics, all the resulting waves will constructively interfere at the position of the original source in space and time. The level of convergence will be deterimined by factors such as the source-receiver geometry, the frequency of the recorded data and synthetics, and the accuracy of the velocity structure used when back propagating the wave field. The terms $\epsilon^\dagger({\mathbf x},T-t)$ and $\partial_n \epsilon^\dagger({\mathbf x},T-t)\!\!:\!\!{\mathbf m}({\mathbf x},t)$ can be viewed as sensitivity kernels for the moment density and the faultplane location respectively. By looking at these quantities we can make an educated choice of fault parametrization given the data in hand. The process can then be repeated to invert for the best source model, as demonstrated by \textit{Tromp et al.}~(2004) for the magnitude of a point force. In this presentation we explore the applicability of adjoint methods to estimating finite source parameters. \begin{thebibliography}{3} \bibitem{Fink1997} Fink, M. (1997), Time reversed acoustics, \textit{Physics {T}oday}, \textit{50}(3), 34--40. \bibitem{TalagrandCourtier1987} Talagrand, O., and P.~Courtier (1987), Variational assimilation of meteorological observations with the adjoint vorticity equatuation. {I}: {T}heory, \textit{Q. J. R. Meteorol. Soc.}, \textit{113}, 1311--1328. \bibitem{TrompTapeLiu2004} Tromp, J., C.~Tape, and Q.~Liu (2004), Waveform tomography, adjoint methods, time reversal, and banana-doughnut kernels, \textit{Geophys. Jour. Int., in press} \end{thebibliography}
DE: 7260 Theory and modeling
DE: 7215 Earthquake parameters
SC: Seismology [S]
MN: 2004 AGU Fall Meeting