HR: 0800h
AN: S31B-1041    [Abstracts]
TI: Tomography, Adjoint Methods, Time-Reversal, and Banana-Doughnut Kernels
AU: * Tape, C
EM: carltape@gps.caltech.edu
AF: California Institute of Technology, 1200 E California Blvd. MC 252-21, Pasadena, CA 91125 United States
AU: Tromp, J
EM: jtromp@gps.caltech.edu
AF: California Institute of Technology, 1200 E California Blvd. MC 252-21, Pasadena, CA 91125 United States
AU: Liu, Q
EM: lgy@gps.caltech.edu
AF: California Institute of Technology, 1200 E California Blvd. MC 252-21, Pasadena, CA 91125 United States
AB: We demonstrate that Fr\'{e}chet derivatives for tomographic inversions may be obtained based upon just two calculations for each earthquake: one calculation for the current model and a second, `adjoint', calculation that uses time-reversed signals at the receivers as simultaneous, fictitious sources. For a given model~$m$, we consider objective functions $\chi(m)$ that minimize differences between waveforms, traveltimes, or amplitudes. We show that the Fr\'{e}chet derivatives of such objective functions may be written in the generic form $\delta\chi=\int_VK_m(\mathbf{x})\,\delta\ln m(\mathbf{x})\,d^3\mathbf{x}$, where $\delta\ln m=\delta m/m$ denotes the relative model perturbation. The volumetric kernel $K_m$ is defined throughout the model volume $V$ and is determined by time-integrated products between spatial and temporal derivatives of the regular displacement field $\mathbf{s}$ and the adjoint displacement field $\mathbf{s}^\dagger$ obtained by using time-reversed signals at the receivers as simultaneous sources. In waveform tomography the time-reversed signal consists of differences between the data and the synthetics, in traveltime tomography it is determined by synthetic velocities, and in amplitude tomography it is controlled by synthetic displacements. For each event, the construction of the kernel $K_m$ requires one forward calculation for the regular field $\mathbf{s}$ and one adjoint calculation involving the fields $\mathbf{s}$ and $\mathbf{s}^\dagger$. For multiple events the kernels are simply summed. The final summed kernel is controlled by the distribution of events and stations and thus determines image resolution. In the case of traveltime tomography, the kernels $K_m$ are weighted combinations of banana-doughnut kernels. We demonstrate also how amplitude anomalies may be inverted for lateral variations in elastic and anelastic structure. The theory is illustrated based upon 2D spectral-element simulations.
DE: 7260 Theory and modeling
DE: 7203 Body wave propagation
SC: Seismology [S]
MN: 2004 AGU Fall Meeting