Sensitivity analysis of wave-equation tomography: a multi-scale approach. (English) Zbl 1200.35180

Summary: Earthquakes, viewed as passive sources, or controlled sources, like explosions, excite seismic body waves in the earth. One detects these waves at seismic stations distributed over the earth’s surface. Wave-equation tomography is derived from cross correlating, at each station, data simulated in a reference model with the observed data, for a (large) set of seismic events. The times corresponding with the maxima of these cross correlations replace the notion of residual travel times used as data in traditional tomography. Using first-order perturbation, we develop an analysis of the mapping from a wavespeed contrast (between the “true” and reference models) to these maxima. We develop a construction using curvelets, while establishing a connection with the geodesic X-ray transform. We then introduce the adjoint mapping, which defines the imaging of wavespeed variations from “finite-frequency travel time” residuals. The key underlying component is the construction of the Fréchet derivative of the solution to the seismic Cauchy initial value problem in wavespeed models of limited smoothness. The construction developed in this paper essentially clarifies how a wavespeed model is probed by the method of wave-equation tomography.


35L15 Initial value problems for second-order hyperbolic equations
35L05 Wave equation
35R05 PDEs with low regular coefficients and/or low regular data
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
35R30 Inverse problems for PDEs
86A15 Seismology (including tsunami modeling), earthquakes


Full Text: DOI


[1] Bijwaard, H., Spakman, W., Engdahl, E.R.: Closing the gap between regional and global travel time tomography. J. Geophys. Res. 103, 30055–30078 (1998)
[2] Brytik, V., de Hoop, M.V., Smith, H.F., Uhlmann, G.: Parametrix construction following polarization for the elastic wave equation with stiffness of limited smoothness: A multi-scale approach. Preprint (2009)
[3] Candès, E.J., Donoho, D.: New tight frames of curvelets and optimal representations of objects with piecewise-C 2 singularities. Commun. Pure Appl. Math. 57, 219–266 (2004) · Zbl 1038.94502
[4] Candès, E.J., Donoho, D.: Continuous curvelet transform: I. Resolution of the wavefront set. Appl. Comput. Harmon. Anal. 19, 162–197 (2005) · Zbl 1086.42022
[5] Candès, E.J., Donoho, D.: Continuous curvelet transform: II. Discretization and frames. Appl. Comput. Harmon. Anal. 19, 198–222 (2005) · Zbl 1086.42023
[6] Dahlen, F.A., Hung, S.-H., Nolet, G.: Fréchet kernels for finite-frequency traveltimes–I. Theory. Geophys. J. Int. 141, 157–174 (2000)
[7] Dahlen, F.A., Nolet, G.: Comment on ’On sensitivity kernels for ’wave-equation’ transmission tomography’. Geophys. J. Int. 163, 949–951 (2005)
[8] De Hoop, M.V., Van der Hilst, R.D.: On sensitivity kernels for ’wave-equation’ transmission tomography. Geophys. J. Int. 160, 621–633 (2005)
[9] De Hoop, M.V., Van der Hilst, R.D.: Reply to comment by F.A. Dahlen and G. Nolet on ’On sensitivity kernels for ’wave-equation’ transmission tomography’. Geophys. J. Int. 163, 952–955 (2005)
[10] De Hoop, M.V., Van der Hilst, R.D., Shen, P.: Wave-equation reflection tomography: Annihilators and sensitivity kernels. Geophys. J. Int. 167, 1332–1352 (2006)
[11] Evans, L.C.: Partial Differential Equations, vol. 19. Am. Math. Soc., Providence (2002).
[12] Greenleaf, A., Uhlmann, G.: Estimates for singular Radon transforms and pseudodifferential operators with singular symbols. J. Funct. Anal. 89, 202–232 (1990) · Zbl 0717.44001
[13] Kennett, B.L.N., Widiyantoro, S., Van der Hilst, R.D.: Joint seismic tomography for bulk-sound and shear wavespeed. J. Geophys. Res. 103, 12469–12493 (1998)
[14] Kingsbury, N.: Image processing with complex wavelets. Philos. Trans. R. Soc. Lond. A357, 2543–2560 (1999) · Zbl 0976.68527
[15] Kingsbury, N.: Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmon. Anal. 10, 234–253 (2002) · Zbl 0990.94005
[16] Loris, I., Nolet, G., Daubechies, I., Dahlen, F.A.: Tomographic inversion using l 1-norm regularization of wavelet coefficients. Geophys. J. Int. 170, 359–370 (2007)
[17] Luo, Y., Schuster, G.T.: Wave-equation travel time inversion. Geophysics 56, 645–653 (1991)
[18] Marquering, H., Nolet, G., Dahlen, F.A.: Three-dimensional waveform sensitivity kernels. Geophys. J. Int. 132, 521–534 (1998)
[19] Montelli, R., Nolet, G., Dahlen, F.A.: Comment on ’Banana-doughnut kernels and mantle tomography’. Geophys. J. Int. 167, 1204–1210 (2006)
[20] Pestov, L., Uhlmann, G.: Two-dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 161, 1089–1106 (2005) · Zbl 1076.53044
[21] Salo, M.: Stability for solutions of wave equations with C 1,1 coefficients. Inverse Probl. Imaging 1, 537–556 (2007) · Zbl 1134.35071
[22] Sieminski, A., Liu, Q., Trampert, J., Tromp, J.: Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods. Geophys. J. Int. 171, 368–389 (2007)
[23] Sirgue, L., Pratt, R.: Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies. Geophysics 69, 231–248 (2004)
[24] Smith, H.F.: A parametrix construction for wave equations with C 1,1 coefficients. Ann. Inst. Fourier 48, 797–835 (1998) · Zbl 0974.35068
[25] Smith, H.F.: Spectral cluster estimates for C 1,1 metrics. Am. J. Math. 128, 1069–1103 (2006) · Zbl 1284.35149
[26] Stolk, C.C.: On the modeling and inversion of seismic data. Ph.D. thesis, Utrecht University (2000) · Zbl 1074.86516
[27] Stolk, C.C., De Hoop, M.V.: Microlocal analysis of seismic inverse scattering in anisotropic, elastic media. Commun. Pure Appl. Math. 55, 261–301 (2002) · Zbl 1018.86002
[28] Stolk, C.C., De Hoop, M.V.: Seismic inverse scattering in the downward continuation approach. Wave Motion 43, 579–598 (2006) · Zbl 1231.35315
[29] Su, W.-J., Dziewonski, A.M.: Simultaneous inversion for 3-D variations in shear and bulk velocity in the mantle. Phys. Earth Planet. Inter. 100, 135–156 (1997)
[30] Tataru, D.: Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. Am. J. Math. 122, 349–376 (2000) · Zbl 1057.11506
[31] Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II. Am. J. Math. 123, 385–423 (2001) · Zbl 0988.35037
[32] Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. J. Am. Math. Soc. 15, 419–442 (2002) · Zbl 0990.35027
[33] Van der Hilst, R.D., De Hoop, M.V.: Banana-doughnut kernels and mantle tomography. Geophys. J. Int. 163, 956–961 (2005)
[34] Van der Hilst, R.D., De Hoop, M.V.: Reply to comment by R. Montelli, G. Nolet and F.A. Dahlen on ’Banana-doughnut kernels and mantle tomography’. Geophys. J. Int. 167, 1211–1214 (2006)
[35] Van der Hilst, R.D., Widyantoro, S., Engdahl, E.R.: Evidence for deep mantle circulation from global tomography. Nature 386, 578–584 (1997)
[36] Woodward, M.J.: Wave-equation tomography. Geophysics 57, 12–26 (1992)
[37] Zhao, L., Jordan, T., Chapman, C.: Three-dimensional Fréchet differential kernels for seismic delay times. Geophys. J. Int. 141, 558–576 (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.