Skip to main content
SpringerLink
Log in

Strategies in Adjoint Tomography

  • Living reference work entry
  • First Online:
Handbook of Geomathematics

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Similar content being viewed by others

References

  • 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

  • Aki K, Christoffersson A, Husebye ES (1977) Determination of the three-dimensional seismic structure of the lithosphere. J Geophys Res 82:277–296

    Article  Google Scholar 

  • Brenders AJ, Pratt RG (2007) Full waveform tomography for lithospheric imaging: results from a blind test in a realistic crustal model. Geophys J Int 168:133–151

    Article  Google Scholar 

  • Brossier R, Operto S, Virieux J (2009) Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion. Geophysics 74:WCC105–WCC118

    Article  Google Scholar 

  • Bunks C, Saleck FM, Zaleski S, Chavent G (1995) Multiscale seismic waveform inversion. Geophysics 60:1457–1473

    Article  Google Scholar 

  • Byrd RH, Nocedal J, Schnabel R (1994) Representations of quasi-Newton matrices and their use in limited memory methods. Mathematical Programming 63:129–156

    Article  MATH  MathSciNet  Google Scholar 

  • Červenỳ V (2005) Seismic ray theory. Cambridge University Press. ISBN:9780521018227, http://www.cambridge.org/9780521018227

  • 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 

  • Dahlen F, Nolet G, Hung S (2000) Fréchet kernels for finite-frequency travel time – I. Theory. Geophys J Int 141:157–174

    Article  Google Scholar 

  • Dahlen FA (2005) Finite-frequency sensitivity kernels for boundary topography perturbations. Geophys J Int 162:525–540

    Article  Google Scholar 

  • Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics. ISBN:9780898712742, http://books.google.com/books?id=Nxnh48rS9jQC

  • Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math 57:1413–1457

    Article  MATH  Google Scholar 

  • Dziewonski AM, Hager BH, O’Connell RJ (1977) Large-scale heterogeneities in the lower mantle. J Geophys Res 82:239–255

    Article  Google Scholar 

  • 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 179:1703–1725

    Article  Google Scholar 

  • Fichtner A, Trampert J (2011a) Hessian kernels of seismic data functionals based upon adjoint techniques. Geophys J Int 185:775–798

    Article  Google Scholar 

  • Fichtner A, Trampert J (2011b) Resolution analysis in full waveform inversion. Geophys J Int 187:1604–1624

    Article  Google Scholar 

  • Fornberg B (1999) A practical guide to pseudospectral methods. Cambridge University Press. ISBN:9780521645645, http://www.cambridge.org/9780521645645

  • Guitton A, Symes WW (2003) Robust inversion of seismic data using the Huber norm. Geophysics 68(4):1310–1319 (2003)

    Article  Google Scholar 

  • 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

  • Komatitsch D, Tromp J (1999) Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J Int 139:806–822

    Article  Google Scholar 

  • Komatitsch D, Tromp J (2002a) Spectral-element simulations of global seismic wave propagation – I. Validation. Geophys J Int 149:390–412

    Article  Google Scholar 

  • 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 150:308–318

    Article  Google Scholar 

  • 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 88:368–392

    Google Scholar 

  • 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–WCC188

    Article  Google Scholar 

  • 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 206–220

    Google Scholar 

  • Liu Q, Tromp J (2006) Finite-frequency kernels based on adjoint methods. Bull Seismol Soc Am 96:2383–2397

    Article  Google Scholar 

  • Loris I, Nolet G, Daubechies I, Dahlen FA (2007) Tomographic inversion using l1-norm regularization of wavelet coefficients. Geophys J Int 170:359–370

    Article  Google Scholar 

  • Luo Y, Schuster GT (1991) Wave-equation travel time inversion. Geophysics 56:645–653

    Article  Google Scholar 

  • Madariaga R (1976) Dynamics of an expanding circular fault. Bull Seismol Soc Am 65:163–182

    Google Scholar 

  • Maggi A, Tape C, Chen M, Chao D, Tromp J (2009) An automated time window selection algorithm for seismic tomography. Geophys J Int 178:257–281

    Article  Google Scholar 

  • Marquering H, Dahlen FA, Nolet G (1999) Three-dimensional sensitivity kernels for finite-frequency travel times: the banana-doughnut paradox. Geophys J Int 137:805–815

    Article  Google Scholar 

  • 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 Geophysicists

    Google Scholar 

  • Moghaddam PP, Herrmann FJ (2010) Randomized full-waveform inversion: a dimensionality-reduction approach, vol 29, pp 977–982. SEG Technical Program Expanded Abstracts

    Google Scholar 

  • Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Math Comput 35(151):773–782

    Article  MATH  MathSciNet  Google Scholar 

  • Plessix R (2006) A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys J Int 167:495–503

    Article  Google Scholar 

  • 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 60:733–747

    Article  Google Scholar 

  • Pratt RG, Shin CS, Hicks GJ (1998) Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion. Geophys J Int 133:341–362

    Article  Google Scholar 

  • 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 159:1032–1056

    Article  Google Scholar 

  • Romero LA, Ghiglia DC, Ober CC, Morton SA (2000) Phase encoding of shot records in prestack migration. Geophysics 65:426–436

    Article  Google Scholar 

  • Schuster GT (2009) Seismic interferometry, vol 1. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Talagrand O, Courtier P (1987) Variational assimilation of meteorological observations with the adjoint vorticity equation. I: theory. Q J R Meteorol Soc 113:1311–1328

    Article  Google Scholar 

  • Tape C, Liu Q, Tromp J (2007) Finite-frequency tomography using adjoint methods – methodology and examples using membrane surface waves. Geophys J Int 168:1105–1129

    Article  Google Scholar 

  • Tape C, Liu Q, Maggi A, Tromp J (2009) Adjoint tomography of the Southern California crust. Science 325:988–992

    Article  Google Scholar 

  • 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 180:433–462

    Article  Google Scholar 

  • Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49(8):1259–1266

    Article  Google Scholar 

  • Tromp J, Tape C, Liu QY (2005) Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys J Int 160:195–216

    Article  Google Scholar 

  • Tromp J, Luo Y, Hanasoge S, Peter D (2010) Noise cross-correlation sensitivity kernels. Geophys J Int 183:791–819

    Article  Google Scholar 

  • van der Hilst RD, Engdahl ER, Spakman W, Nolet G (1991) Tomographic imaging of subducted lithosphere below northwest pacific island arcs. Nature 353:37–43

    Article  Google Scholar 

  • Virieux J (1986) P-sv wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51:889–901

    Article  Google Scholar 

  • Zhao L, Jordan TH, Chapman CH (2000) Three-dimensional Fréchet differential kernels for seismic delay times. Geophys J Int 141:558–576

    Article  Google Scholar 

  • 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 95:2066–2080

    Article  MathSciNet  Google Scholar 

  • Zhu H, Bozdag E, Peter D, Tromp J (2012) Structure of the European upper mantle revealed by adjoint tomography. Nat Geosci. doi: 10.1038/NGEO1501

    Google Scholar 

  • Zienkiewicz OC (1977) The finite element method. McGraw-Hill, London. ISBN 9780070840720, http://books.google.com/books?id=S8lRAAAAMAAJ

Download references

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.

Author information

Authors and Affiliations

  1. Department of Geosciences, Princeton University, Princeton, NJ, USA

    Yang Luo, Ryan Modrak & Jeroen Tromp

  2. Program in Applied & Computational Mathematics, Princeton University, Princeton, NJ, USA

    Jeroen Tromp

Authors
  1. Yang Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryan Modrak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jeroen Tromp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Luo .

Editor information

Editors and Affiliations

  1. Geomathematische Gruppe, TU Kaiserslautern FB Mathematik, Kaiserslautern, Germany

    Willi Freeden

  2. University of Central Florida Dept. Mathematics, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Inst. Analysis, TU Braunschweig FB 1 Mathematik und Informatik, Braunschweig, Germany

    Thomas Sonar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Luo, Y., Modrak, R., Tromp, J. (2013). Strategies in Adjoint Tomography. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27793-1_96-2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27793-1_96-2

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Online ISBN: 978-3-642-27793-1

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation