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Strategies in Adjoint Tomography

  • Yang Luo
  • Ryan Modrak
  • Jeroen Tromp
Reference work entry

Abstract

We investigate issues of convergence, resolution, and nonlinearity related to the feasibility of adjoint tomography in regional and global tomography and exploration geophysics. Most current methods of adjoint tomography, whether based on adjoint methods or other formulations, suffer from slow convergence in that only the gradient (not the Hessian) is readily available for computing model updates. As an alternative to working with the unpreconditioned gradients, we examine the speed-up offered by various preconditioners that can be computed in the framework of adjoint methods. We show that each preconditioner bears some similarity to the Hessian, thus motivating and justifying its use for accelerating convergence. Next, we examine the role of the Hessian in resolution analysis. Recalling that the action of the Hessian on an arbitrary model perturbation relates to the classical point spread function concept, we introduce a scalar quantity termed the average eigenvalue that provides a good overall representation of resolution. Whereas a point-spread function reveals the orientation of misfit contours, the average eigenvalue describes the sharpness of the misfit function along the direction of the chosen model perturbation. Finally, we provide an example in which we directly compare the results of travel time and waveform tomography, illustrating the resolution limits of the former and the nonlinearity pitfalls of the latter.

Notes

Acknowledgements

Numerical simulations for this article were performed on a Dell cluster built and maintained by the Princeton Institute for Computational Science & Engineering (PICSciE). This research was partly sponsored by TOTAL, and by the U.S. National Science Foundation under grants EAR-1112906 and DMS-1025418.

References

  1. Akçelik V, Biros G, Ghattas O (2002) Parallel multiscale Gauss–Newton–Krylov methods for inverse wave propagation. In: Proceedings of the ACM/IEEE supercomputing SC’02 conference. Published on CD-ROM and at www.sc-conference.org/sc2002
  2. Aki K, Christoffersson A, Husebye ES (1977) Determination of the three-dimensional seismic structure of the lithosphere. J Geophys Res 82:277–296CrossRefGoogle Scholar
  3. Brenders AJ, Pratt RG (2007) Full waveform tomography for lithospheric imaging: results from a blind test in a realistic crustal model. Geophys J Int 168:133–151CrossRefGoogle Scholar
  4. Brossier R, Operto S, Virieux J (2009) Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion. Geophysics 74:WCC105–WCC118CrossRefGoogle Scholar
  5. Bunks C, Saleck FM, Zaleski S, Chavent G (1995) Multiscale seismic waveform inversion. Geophysics 60:1457–1473CrossRefGoogle Scholar
  6. Byrd RH, Nocedal J, Schnabel R (1994) Representations of quasi-Newton matrices and their use in limited memory methods. Mathematical Programming 63:129–156MathSciNetCrossRefMATHGoogle Scholar
  7. Červenỳ V (2005) Seismic ray theory. Cambridge University Press. ISBN:9780521018227, http://www.cambridge.org/9780521018227
  8. Chavent G (1974) Identification of function parameters in partial differential equations. In: Goodson RE, Polis M (eds) Identification of parameter distributed systems. American Society Of Mechanical Engineers, New York (1974)Google Scholar
  9. Dahlen F, Nolet G, Hung S (2000) Fréchet kernels for finite-frequency travel time – I. Theory. Geophys J Int 141:157–174CrossRefGoogle Scholar
  10. Dahlen FA (2005) Finite-frequency sensitivity kernels for boundary topography perturbations. Geophys J Int 162:525–540CrossRefGoogle Scholar
  11. Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics. ISBN:9780898712742, http://books.google.com/books?id=Nxnh48rS9jQC
  12. Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math 57:1413–1457CrossRefMATHGoogle Scholar
  13. Dziewonski AM, Hager BH, O’Connell RJ (1977) Large-scale heterogeneities in the lower mantle. J Geophys Res 82:239–255CrossRefGoogle Scholar
  14. Fichtner A, Kennett BLN, Igel H, Bunge H-P (2009) Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods. Geophys J Int 179:1703–1725CrossRefGoogle Scholar
  15. Fichtner A, Trampert J (2011a) Hessian kernels of seismic data functionals based upon adjoint techniques. Geophys J Int 185:775–798CrossRefGoogle Scholar
  16. Fichtner A, Trampert J (2011b) Resolution analysis in full waveform inversion. Geophys J Int 187:1604–1624CrossRefGoogle Scholar
  17. Fornberg B (1999) A practical guide to pseudospectral methods. Cambridge University Press. ISBN:9780521645645, http://www.cambridge.org/9780521645645
  18. Guitton A, Symes WW (2003) Robust inversion of seismic data using the Huber norm. Geophysics 68(4):1310–1319 (2003)CrossRefGoogle Scholar
  19. Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall. ISBN:9780133170252, http://books.google.com/books?id=pF-IQgAACAAJ
  20. Komatitsch D, Tromp J (1999) Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J Int 139:806–822CrossRefGoogle Scholar
  21. Komatitsch D, Tromp J (2002a) Spectral-element simulations of global seismic wave propagation – I. Validation. Geophys J Int 149:390–412CrossRefGoogle Scholar
  22. Komatitsch D, Tromp J (2002b) Spectral-element simulations of global seismic wave propagation – II. Three-dimensional models, oceans, rotation and self-gravitation. Geophys J Int 150:308–318CrossRefGoogle Scholar
  23. Komatitsch D, Vilotte J-P (1998) The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull Seismol Soc Am 88:368–392Google Scholar
  24. Krebs J, Anderson J, Hinkley D, Neelamani R, Baumstein A, Lacasse MD, Lee S (2009) Fast full-wavefield seismic inversion using encoded sources. Geophysics 74:WCC177–WCC188CrossRefGoogle Scholar
  25. Lailly P (1983) The seismic inverse problem as a sequence of before stack migration. In: Bednar J (ed) Conference on inverse scattering: theory and application. Society for Industrial and Applied Mathematics, Philadelphia, pp 206–220Google Scholar
  26. Liu Q, Tromp J (2006) Finite-frequency kernels based on adjoint methods. Bull Seismol Soc Am 96:2383–2397CrossRefGoogle Scholar
  27. Loris I, Nolet G, Daubechies I, Dahlen FA (2007) Tomographic inversion using l1-norm regularization of wavelet coefficients. Geophys J Int 170:359–370CrossRefGoogle Scholar
  28. Luo Y, Schuster GT (1991) Wave-equation travel time inversion. Geophysics 56:645–653CrossRefGoogle Scholar
  29. Madariaga R (1976) Dynamics of an expanding circular fault. Bull Seismol Soc Am 65:163–182Google Scholar
  30. Maggi A, Tape C, Chen M, Chao D, Tromp J (2009) An automated time window selection algorithm for seismic tomography. Geophys J Int 178:257–281CrossRefGoogle Scholar
  31. Marquering H, Dahlen FA, Nolet G (1999) Three-dimensional sensitivity kernels for finite-frequency travel times: the banana-doughnut paradox. Geophys J Int 137:805–815CrossRefGoogle Scholar
  32. Martin GS, Marfurt KJ, Larsen S (2002) Marmousi-2: an updated model for the investigation of AVO in structurally complex areas. In: Proceedings of 72nd annual international meeting, Tulsa, pp 1979–1982. Society of Exploration GeophysicistsGoogle Scholar
  33. Moghaddam PP, Herrmann FJ (2010) Randomized full-waveform inversion: a dimensionality-reduction approach, vol 29, pp 977–982. SEG Technical Program Expanded AbstractsGoogle Scholar
  34. Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Math Comput 35(151):773–782MathSciNetCrossRefMATHGoogle Scholar
  35. Plessix R (2006) A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys J Int 167:495–503CrossRefGoogle Scholar
  36. Plessix R-E, Baeten G, de Maag JW, ten Kroode F, Zhang R (2012) Full waveform inversion and distance separated simultaneous sweeping: a study with a land seismic data set. Geophys Prospect 60:733–747CrossRefGoogle Scholar
  37. Pratt RG, Shin CS, Hicks GJ (1998) Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion. Geophys J Int 133:341–362CrossRefGoogle Scholar
  38. Ravaut C, Operto S, Improta L, Virieux J, Herrero A, Dell’Aversana P (2004) Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt. Geophys J Int 159:1032–1056CrossRefGoogle Scholar
  39. Romero LA, Ghiglia DC, Ober CC, Morton SA (2000) Phase encoding of shot records in prestack migration. Geophysics 65:426–436CrossRefGoogle Scholar
  40. Schuster GT (2009) Seismic interferometry, vol 1. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  41. Talagrand O, Courtier P (1987) Variational assimilation of meteorological observations with the adjoint vorticity equation. I: theory. Q J R Meteorol Soc 113:1311–1328CrossRefGoogle Scholar
  42. Tape C, Liu Q, Tromp J (2007) Finite-frequency tomography using adjoint methods – methodology and examples using membrane surface waves. Geophys J Int 168:1105–1129CrossRefGoogle Scholar
  43. Tape C, Liu Q, Maggi A, Tromp J (2009) Adjoint tomography of the Southern California crust. Science 325:988–992CrossRefGoogle Scholar
  44. Tape C, Liu Q, Maggi A, Tromp J (2010) Seismic tomography of the southern California crust based on spectral-element and adjoint methods. Geophys J Int 180:433–462CrossRefGoogle Scholar
  45. Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49(8):1259–1266CrossRefGoogle Scholar
  46. Tromp J, Tape C, Liu QY (2005) Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys J Int 160:195–216CrossRefGoogle Scholar
  47. Tromp J, Luo Y, Hanasoge S, Peter D (2010) Noise cross-correlation sensitivity kernels. Geophys J Int 183:791–819CrossRefGoogle Scholar
  48. van der Hilst RD, Engdahl ER, Spakman W, Nolet G (1991) Tomographic imaging of subducted lithosphere below northwest pacific island arcs. Nature 353:37–43CrossRefGoogle Scholar
  49. Virieux J (1986) P-sv wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51:889–901CrossRefGoogle Scholar
  50. Zhao L, Jordan TH, Chapman CH (2000) Three-dimensional Fréchet differential kernels for seismic delay times. Geophys J Int 141:558–576CrossRefGoogle Scholar
  51. Zhao L, Jordan TH, Olsen KB, Chen P (2005) Fréchet kernels for imaging regional earth structure based on three-dimensional reference models. Bull Seismol Soc Am 95:2066–2080MathSciNetCrossRefGoogle Scholar
  52. Zhu H, Bozdag E, Peter D, Tromp J (2012) Structure of the European upper mantle revealed by adjoint tomography. Nat Geosci. doi: 10.1038/NGEO1501Google Scholar
  53. Zienkiewicz OC (1977) The finite element method. McGraw-Hill, London. ISBN 9780070840720, http://books.google.com/books?id=S8lRAAAAMAAJ

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of GeosciencesPrinceton UniversityPrincetonUSA
  2. 2.Program in Applied & Computational MathematicsPrinceton UniversityPrincetonUSA

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