Pure and Applied Geophysics

, Volume 170, Issue 3, pp 353–371 | Cite as

Sensitivity Kernel for the Weighted Norm of the Frequency-Dependent Phase Correlation

  • Zhiguo Xu
  • Po ChenEmail author
  • Youlin Chen


Full-3D waveform tomography (F3DT) is often formulated as an optimization problem, in which an objective function defined in terms of the misfit between observed and model-predicted (i.e., synthetic) waveforms is minimized by varying the earth structure model from which the synthetic waveforms are calculated. Because of the large dimension of the model space and the computational cost for solving the 3D seismic wave equation, it is often mandatory to use Newton-type local optimization algorithms; in which case, spurious local optima in the objective function can prevent the global convergence of the descent algorithm if the initial estimate of the structure model is not close enough to the global optimum. By appropriate design of the objective function, it is possible to enlarge the attraction domain of the global optimum so that Newton-type local optimization algorithms can achieve global convergence. In this article, an objective function based on a weighted L 2 norm of the frequency-dependent phase correlation between observed and synthetic waveforms is proposed and studied, and its full-3D Fréchet kernel is constructed using the adjoint state method. The relation between the proposed objective function and the conventional frequency-dependent group-delay is analyzed and illustrated using numerical examples. The methodology has been successfully applied on a set of ambient-noise Green’s function observations collected in northern California to derive a full-3D crustal structure model.


Seismic tomography phase correlation misfit measure sensitivity kernel objective function nonlinearity group-delay 



This work is partly supported by the United States Geological Survey (USGS) National Earthquake Hazard Reduction Program (NEHRP) under the award number G10AP00032. The first author was partly supported by the China Scholarship Council (CSC). The second author was supported by the School of Energy Resources at University of Wyoming. The seismic waveform data used in this study was provided by the USArray of the EarthScope project, Northern California Earthquake Center and the IRIS Data Management Center. Some figures used in this article were generated using the Generic Mapping Tools (GMT) developed by Wessel and Smith (1991). Comments from two anonymous reviewers improved the manuscript and we greatly appreciate the time and effort they spent reviewing our manuscript.


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Chinese Earthquake AdministrationBeijingChina
  2. 2.Department of Geology and GeophysicsUniversity of WyomingWyomingUSA
  3. 3.ARRAY Information TechnologyGreenbeltUSA

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