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Finite-Frequency Kernels Based on Adjoint Methods

Liu, Qinya and Tromp, Jeroen (2006) Finite-Frequency Kernels Based on Adjoint Methods. Bulletin of the Seismological Society of America, 96 (6). pp. 2383-2397. ISSN 0037-1106. doi:10.1785/0120060041.

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We derive the adjoint equations associated with the calculation of Fréchet derivatives for tomographic inversions based upon a Lagrange multiplier method. The Fréchet derivative of an objective function χ(m), where m denotes the Earth model, may be written in the generic form δχ = ∫ K_m(x) δ ln m(x) d^3x, where δ ln m = δm/m denotes the relative model perturbation and K_m the associated 3D sensitivity or Fréchet kernel. Complications due to artificial absorbing boundaries for regional simulations as well as finite sources are accommodated. We construct the 3D finite-frequency “banana-doughnut” kernel K_m by simultaneously computing the so-called “adjoint” wave field forward in time and reconstructing the regular wave field backward in time. The adjoint wave field is produced by using time- reversed signals at the receivers as fictitious, simultaneous sources, while the regular wave field is reconstructed on the fly by propagating the last frame of the wave field, saved by a previous forward simulation, backward in time. The approach is based on the spectral-element method, and only two simulations are needed to produce the 3D finite-frequency sensitivity kernels. The method is applied to 1D and 3D regional models. Various 3D shear- and compressional-wave sensitivity kernels are presented for different regional body- and surface-wave arrivals in the seismograms. These kernels illustrate the sensitivity of the observations to the structural parameters and form the basis of fully 3D tomographic inversions.

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Tromp, Jeroen0000-0002-2742-8299
Additional Information:© 2006 The Seismological Society of America. Manuscript received 22 February 2006. We would like to thank Carl Tape, Jesper Spetzler, and an anonymous reviewer for helpful comments and suggestions. This research was funded by the National Science Foundation under grant EAR-0309576 and by the Southern California Earthquake Center (SCEC). SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. The SCEC contribution number for this article is 1006. This is contribution no. 9150 of the Division of Geological & Planetary Sciences (GPS) of the California Institute of Technology. The numerical simulations for this research were performed on Caltech’s Division of Geological and Planetary Sciences Dell cluster.
Funding AgencyGrant Number
Southern California Earthquake Center (SCEC)UNSPECIFIED
NSF Cooperative AgreementEAR-0106924
USGS Cooperative Agreement02HQAG0008
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Other Numbering System NameOther Numbering System ID
Division of Geological & Planetary Sciences9150
Issue or Number:6
Record Number:CaltechAUTHORS:20140804-101116394
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Official Citation:Liu, Q., & Tromp, J. (2006). Finite-Frequency Kernels Based on Adjoint Methods. Bulletin of the Seismological Society of America, 96(6), 2383-2397. doi: 10.1785/0120060041
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:47879
Deposited On:04 Aug 2014 18:29
Last Modified:10 Nov 2021 17:51

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