Inverse Problems in Geophysics

  • R. Snieder
  • J. Trampert
Part of the International Centre for Mechanical Sciences book series (CISM, volume 398)


An important aspect of the physical sciences is to make inferences about physical parameters from data. In general, the laws of physics provide the means for computing the data values given a model. This is called the “forward problem”, see figure 1. In the inverse problem, the aim is to reconstruct the model from a set of measurements. In the ideal case, an exact theory exists that prescribes how the data should be transformed in order to reproduce the model. For some selected examples such a theory exists assuming that the required infinite and noise-free data sets would be available. A quantum mechanical potential in one spatial dimension can be reconstructed when the reflection coefficient is known for all energies [Marchenko, 1955; Burridge, 1980]. This technique can be generalized for the reconstruction of a quantum mechanical potential in three dimensions [Newton, 1989], but in that case a redundant data set is required for reasons that are not well understood. The mass-density in a one-dimensional string can be constructed from the measurements of all eigenfrequencies of that string [Borg,1946], but due to the symmetry of this problem only the even part of the mass-density can be determined. If the seismic velocity in the earth depends only on depth, the velocity can be constructed exactly from the measurement of the arrival time as a function of distance of seismic waves using an Abel transform [Herglotz, 1907; Wiechert, 1907]. Mathematically this problem is identical to the construction of a spherically symmetric quantum mechanical potential in three dimensions [Keller et al., 1956]. However, the construction method of Herglotz-Wiechert only gives an unique result when the velocity increases monotonically with depth [Gerver and Markushevitch, 1966]. This situation is similar in quantum mechanics where a radially symmetric potential can only be constructed uniquely when the potential does not have local minima [Sabatier, 1973].


Inverse Problem Rayleigh Wave Love Wave Model Vector Seismic Tomography 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldridge, D.F., Linearization of the eikonal equation, Geophysics, 59, 1631–1632, 1994.CrossRefGoogle Scholar
  2. 2.
    Alsina, D., R.L. Woodward, and R.K. Snieder, Shear-Wave Velocity Structure in North America from Large-Scale Waveform Inversions of Surface Waves, J. Geophys. Res., 101, 15969–15986, 1996.CrossRefGoogle Scholar
  3. 3.
    Aki, K., and P.G. Richards, Quantitative Seismology (2 volumes), Freeman and Co. New York, 1980.Google Scholar
  4. 4.
    Backus, G., and J.F. Gilbert, Numerical applications of a formalism for geophysical inverse problems, Geophys. J.R. Astron. Soc., 13, 247–276, 1967.CrossRefGoogle Scholar
  5. 5.
    Backus, G., and J.F. Gilbert, The resolving power of gross earth data, Geophys. J.R. Astron. Soc., 16, 169–205, 1968.CrossRefGoogle Scholar
  6. 6.
    Backus, G. E. and F. Gilbert, Uniqueness in the inversion of inaccurate gross earth data, Philos. Trans. R. Soc. London, Ser. A, 266, 123–192, 1970.Google Scholar
  7. 7.
    Ben-Menahem, A. and S.J. Singh, Seismic waves and sources, Springer Verlag, New York, 1981.CrossRefGoogle Scholar
  8. 8.
    Borg, G., Eine Umkehrung der Sturm-Liouvillischen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math., 78, 1–96, 1946.CrossRefGoogle Scholar
  9. 9.
    Burridge, R., The Gel’fand-Levitan, the Marchenko and the Gopinath-Sondi integral equations of inverse scattering theory, regarded in the context of the inverse impulse response problems, Wave Motion, 2, 305–323, 1980.CrossRefGoogle Scholar
  10. 10.
    Cara, M., Regional variations of higher-mode phase velocities: A spatial filtering method, Geophys. J.R. Astron. Soc., 54, 439–460, 1978.CrossRefGoogle Scholar
  11. 11.
    Claerbout, J.F., Fundamentals of Geophysical data processing, McGraw-Hill, New York, 1976.Google Scholar
  12. 12.
    Claerbout, J.F., Imaging the Earth’s interior, Blackwell, Oxford, 1985.Google Scholar
  13. 13.
    Clayton, R. W. and R. P. Comer, A tomographie analysis of mantle heterogeneities from body wave travel time data, EOS, Trans. Am. Geophys. Un., 64, 776, 1983.Google Scholar
  14. 14.
    Constable, S.C., R.L. Parker, and C.G. Constable, Occam’s inversion: a practical algorithm for generating smooth models from electromagnetic sounding data, Geophysics, 52, 289–300, 1987.CrossRefGoogle Scholar
  15. 15.
    Dahlen, F.A., and J. Tromp, Theoretical global seismology, Princeton University Press, Princeton, 1998.Google Scholar
  16. 16.
    Dorren, H.J.S., E.J. Muyzert, and R.K. Snieder, The stability of one-dimensional inverse scattering, Inverse Problems, 10, 865–880, 1994.CrossRefGoogle Scholar
  17. 17.
    Douma, H., R. Snieder, and A. Lomax, Ensemble inference in terms of Empirical Orthogonal Functions, Geophys. J. Int., 127, 363–378, 1996.CrossRefGoogle Scholar
  18. 18.
    Dziewonski, A.M., and D.L. Anderson, Preliminary Reference Earth Model, Phys. Earth. Plan. Int., 25, 297–356, 1981.CrossRefGoogle Scholar
  19. 19.
    Gerver, M.L. and V. Markushevitch, Determination of a seismic wave velocity from the travel time curve, Geophys. J. Royal astro. Soc., 11 165–173, 1966.CrossRefGoogle Scholar
  20. 20.
    Gilbert, F., Ranking and winnowing gross Earth data for inversion and resolution, Geophys. J. Royal astro. Soc., 23 125–128, 1971.Google Scholar
  21. 21.
    Gouveia, W.P., and J.A. Scales, Bayesian seismic waveform inversion: parameter estimation and uncertainty analysis, J. Geophys. Res., 103, 2759–2779, 1998.CrossRefGoogle Scholar
  22. 22.
    Gutenberg, B., Dispersion und Extinktion von seismischen Oberflächenwellen und der Aufbau der obersten Erdschichten, Physikalische Zeitschrift, 25, 377–382, 1924.Google Scholar
  23. 23.
    Herglotz, G. Über das Benndorfsche Problem des Fortpflanzungsgeschwindigkeit der Erdbebenstrahlen, Zeitschrift fur Geophys., 8 145–147, 1907.Google Scholar
  24. 24.
    Keller, J.B., I. Kay, and J. Shmoys, Determination of a potential from scattering data, Phys. Rev., 102, 557–559, 1956.CrossRefGoogle Scholar
  25. 25.
    Kircpatrick, S., C. Gelatt, and M.P. Vechhis, Optimization by simulated annealing, Science, 220, 671–680, 1983.CrossRefGoogle Scholar
  26. 26.
    Lanczos, C., Linear Differential Operators, Van Nostrand, London, 1961.Google Scholar
  27. 27.
    Levenberg, K., A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2, 164–168, 1944.Google Scholar
  28. 28.
    Lomax, A., and R. Snieder, The contrast in upper-mantle shear-wave velocity between the East European Platform and tectonic Europe obtained with genetic algorithm inversion of Rayleigh-wave group dispersion, Geophys. J. Int., 123, 169–182, 1995.CrossRefGoogle Scholar
  29. 29.
    Marchenko, V.A., The construction of the potential energy from the phases of scattered waves, Dokl. Akad. Nauk, 104, 695–698, 1955.Google Scholar
  30. 30.
    Matsu’ura M. and N. Hirata, Generalized least-squares solutions to quasi-linear inverse problems with a priori information, J. Phys. Earth, 30, 451–468, 1982.CrossRefGoogle Scholar
  31. 31.
    Mayer, K., R. Marklein, K.J. Langenberg and T.Kreutter, Three-dimensional imaging system based on Fourier transform synthetic aperture focussing technique, Ultrasonics, 28, 241–255, 1990.CrossRefGoogle Scholar
  32. 32.
    Menke, W., Geophysical data analysis: discrete inverse theory, Academic Press, San Diego, 1984.Google Scholar
  33. 33.
    Merzbacher, E., Quantum mechanics ( 2nd ed. ), Wiley, New York, 1970.Google Scholar
  34. 34.
    Montagner, J.P., and H.C. Nataf, On the inversion of the azimuthal anisotropy of surface waves, J. Geophys. Res., 91, 511–520, 1986.CrossRefGoogle Scholar
  35. 35.
    Mosegaard, K., Resolution analysis of general inverse problems through inverse Monte Carlo sampling, Inverse Problems, 14, 405–426, 1998.CrossRefGoogle Scholar
  36. 36.
    Mosegaard, K., and A. Tarantola, Monte Carlo sampling of solutions to inverse problems, J. Geophys. Res., 100, 12431–12447, 1995.CrossRefGoogle Scholar
  37. 37.
    Muyzert, E., and R. Snieder, An alternative parameterization for surface waves in a transverse isotropic medium, Phys. Earth Planet Int. (submitted), 1999.Google Scholar
  38. 38.
    Natterer, F., H. Sielschott, and W. Derichs, Schallpyrometrie, in Mathematik–Sclzisseltechnologie fzbr die Zukunft, edited by K.H. Hoffmann, W. Jäger, T. Lochmann and H. Schunk, 435–446, Springer Verlag, Berlin, 1997.CrossRefGoogle Scholar
  39. 39.
    Newton, R.G., Inversion of reflection data for layered media: A review of exact methods, Geophys. J.R. Astron. Soc., 65, 191–215, 1981.CrossRefGoogle Scholar
  40. 40.
    Newton, R.G., Inverse Schrödinger scattering in three dimensions, Springer Verlag, Berlin, 1989.CrossRefGoogle Scholar
  41. 41.
    Nolet, G., The upper mantle under Western-Europe inferred from the dispersion of Rayleigh wave modes, J. Geophys., 43, 265–285, 1977.Google Scholar
  42. 42.
    Nolet, G., Linearized inversion of (teleseismic) data, in The Solution of the Inverse Problem in Geophysical Interpretation, edited by R.Cassinis, Plenum Press, New York, 1981.Google Scholar
  43. 43.
    Nolet, G., Solving or resolving inadequate and noisy tomographic systems, J. Comp. Phys., 61, 463–482, 1985.CrossRefGoogle Scholar
  44. 44.
    Nolet, G., Seismic wave propagation and seismic tomography, in Seismic Tomography, edited by G.Nolet, pp. 1–23, Reidel, Dordrecht, 1987.Google Scholar
  45. 45.
    Nolet, G., Partitioned waveform inversion and two-dimensional structure under the Network o f Autonomously Recording Seismographs, J. Geophys. Res., 95, 84998512, 1990.Google Scholar
  46. 46.
    Nolet, G., S.P. Grand, and B.L.N. Kennett, Seismic heterogeneity in the upper mantle, J. Geophys. Res., 99, 23753–23766, 1994.CrossRefGoogle Scholar
  47. 47.
    Nolet, G, and R. Snieder, Solving large linear inverse problems by projection, Geophys. J. Int., 103, 565–568, 1990.CrossRefGoogle Scholar
  48. 48.
    Paige, C.G., and M.A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least-squares, ACM Trans. Math. Software, 8, 43–71, 1982.CrossRefGoogle Scholar
  49. 49.
    Paige, C.G., and M.A. Saunders, LSQR: Sparse linear equations and least-squares problems, ACM Trans. Math. Software, 8, 195–209, 1982.CrossRefGoogle Scholar
  50. 50.
    Parker, R.L., Geophysical Inverse Theory, Princeton University Press, Princeton, New Jersey, 1994.Google Scholar
  51. 51.
    Passier, M.L., and R.K. Snieder, Using differential waveform data to retrieve local S-velocity structure or path-averaged S-velocity gradients, J. Geophys. Res., 100, 24061–24078, 1995.CrossRefGoogle Scholar
  52. 52.
    Passier, M.L., and R.K. Snieder, Correlation between shear wave upper mantle structure and tectonic surface expressions: Application to central and southern Germany, J. Geophys. Res., 101, 25293–25304, 1996.CrossRefGoogle Scholar
  53. 53.
    Passier, T.M., R.D. van der Hilst, and R.K. Snieder, Surface wave waveform inversions for local shear-wave velocities under eastern Australia, Geophys. Res. Lett, 24, 1291–1294, 1997.CrossRefGoogle Scholar
  54. 54.
    Press, W.H., Flannery, B.P., Teukolsky, S.A. and W.T. Vetterling, Numerical Recipies, Cambridge University Press, Cambridge, 1989.Google Scholar
  55. 55.
    Rothman, D.H., Nonlinear inversion, statistical mechanics and residual statics estimation, Geophysics, 50, 2784–2796, 1985.CrossRefGoogle Scholar
  56. 56.
    Sabatier, P.C., Discrete ambiguities and equivalent potentials, Phys. Rev. A, 8, 589–601, 1973.CrossRefGoogle Scholar
  57. 57.
    Sambridge, M., Non-linear arrival time inversion: constraining velocity anomalies by seeking smooth models in 3-D, Geophys. J.R. Astron. Soc., 102, 653–677, 1990.CrossRefGoogle Scholar
  58. 58.
    Sambridge, M., and G. Drijkoningen, Genetic algorithms in seismic waveform inversion, Geophys. J. Int., ’109, 323–342, 1992.Google Scholar
  59. 59.
    Scales, J., and R. Snieder, To Bayes or not to Bayes?, Geophysics, 62, 1045–1046, 1997.Google Scholar
  60. 60.
    Scales, J., and R. Snieder, What is noise?, Geophysics, 63, 1122–1124, 1998.CrossRefGoogle Scholar
  61. 61.
    Sen, M.K., and P.L. Stoffa, Rapid sampling of model space using genetic algorithms: examples of seismic wave from inversion, Geophys. J. Int., 198, 281–292, 1992.CrossRefGoogle Scholar
  62. 62.
    Sluis, A. van der, and H.A. van der Vorst, Numerical solution of large, sparse linear algebraic systems arising from toniographic problems, in Seismic tomography, with applications in global seismology and exploration geophysics, edited by G. Nolet, Reidel, Dordrecht, 1987.Google Scholar
  63. 63.
    Snieder, R., 3D Linearized scattering of surface waves and a formalism for surface wave holography, Geophys. J. R. astron. Soc., 84, 581–605, 1986a.CrossRefGoogle Scholar
  64. 64.
    Snieder, R., The influence of topography on the propagation and scattering of surface waves, Phys. Earth Planet. Inter., 44, 226–241, 1986b.CrossRefGoogle Scholar
  65. 65.
    van Heijst, H.J. and J.H. Woodhouse, Measuring surface wave overtone phase velocities using a mode-branch stripping technique, Geophys. J. Int., 131, 209–230, 1997.CrossRefGoogle Scholar
  66. 66.
    Snieder, R., Surface wave holography, in Seismic tomography, with applications in global seismology and exploration geophysics, edited by G. Nolet, pp. 323–337, Reidel, Dordrecht, 1987.Google Scholar
  67. 67.
    Snieder, R., Large-Scale Waveform Inversions of Surface Waves for Lateral Heterogeneity, 1, Theory and Numerical Examples, J. Geophys. Res., 93, 12055–12065, 1988.CrossRefGoogle Scholar
  68. 68.
    Snieder, R., Large-Scale Waveform Inversions of Surface Waves for Lateral Heterogeneity, 2, Application to Surface Waves in Europe and the Mediterranean, J. Geophys. Res., 93, 12067–12080, 1988.CrossRefGoogle Scholar
  69. 69.
    Snieder, R., A perturbative analysis of nonlinear inversion, Geophys. J. Int., 101, 545–556, 1990.CrossRefGoogle Scholar
  70. 70.
    Snieder, R., The role of the Born-approximation in nonlinear inversion, Inverse Problems, 6, 247–266, 1990.CrossRefGoogle Scholar
  71. 71.
    Snieder, R., An extension of Backus-Gilbert theory to nonlinear inverse problems, Inverse Problems, 7, 409–433, 1991.CrossRefGoogle Scholar
  72. 72.
    Snieder, R., Global inversions using normal modes and long-period surface waves. in Seismic tomography, edited by H.M. Iyer and K. Hirahara, pp. 23–63, Prentice-Hall, London, 1993.Google Scholar
  73. 73.
    Snieder, R., and D.F. Aldridge, Perturbation theory for travel times, J. Acoust. Soc. Am., 98, 1565–1569, 1995.CrossRefGoogle Scholar
  74. 74.
    Snieder, R.K., J. Beckers, and F. Neele, The effect of small-scale structure on normal mode frequencies and global inversions, J. Geophys. Res., 96, 501–515, 1991.CrossRefGoogle Scholar
  75. 75.
    Snieder, R., and A. Lomax, Wavefield smoothing and the effect of rough velocity perturbations on arrival times and amplitudes, Geophys. J. Int., 125, 796–812, 1996.CrossRefGoogle Scholar
  76. 76.
    Snieder, R., and G. Nolet, Linearized scattering of surface waves on a spherical Earth, J. Geophys., 61, 55–63, 1987.Google Scholar
  77. 77.
    Snieder, R., and M. Sambridge, The ambiguity in ray perturbation theory, J. Geophys. Res., 98, 22021–22034, 1993.CrossRefGoogle Scholar
  78. 78.
    Spakman, W., S. Van der Lee, and R.D. van der Hilst, Travel-time tomography of the European-Mediterranean mantle down to 1400 km, Phys. Earth Planet. Int., 79, 3–74, 1993.CrossRefGoogle Scholar
  79. 79.
    Strang, Linear algebra and its applications, Harbourt Brace Jovanovisch Publishers, Fort Worth, 1988.Google Scholar
  80. 80.
    Takeuchi, H. and M. Saito, Seismic surface waves, in Seismology: Surface waves and earth oscillations, (Methods in computational physics, 11), Ed. B.A. Bolt, Academic Press, New York, 1972.Google Scholar
  81. 81.
    Tanis, E., 1921. Uber Fortplanzungsgeschwindigkeit der seismis- chen Oberffii,chenwellen kings kontinentaler und ozeanischer Wege. Centralblatt für Mineralogie, Geologic und Palriontologle, ` 9 -9, 44–52, 1921.Google Scholar
  82. 82.
    Tanimoto, T., Free oscillations in a slighly anisotropie earth, Geophys. J.R. Astrun. Soc., 87, 493–517, 1986.CrossRefGoogle Scholar
  83. 83.
    Tarantola, A., Linearized inversion of seismic reflection data, Geophys. Prosp. 32, 998–1015, 1984.CrossRefGoogle Scholar
  84. 84.
    Tarantola, A., Inverse problem. theory, Elsevier, Amsterdam, 1987.Google Scholar
  85. 85.
    Tarantola, A. and B. Valette, Inverse problems = quest for information, J. Geophys., 50, 159–170, 1982a.Google Scholar
  86. 86.
    Tarantola, A., and B. Valette, Generalized nonlinear inverse problems solved using the least squares criterion, Rev. Geophys. Space Phys. 20, 219–232, 1982b.CrossRefGoogle Scholar
  87. 87.
    Trampert, J., Global seismic tomography: the inverse problem and beyond, Inverse Problems, 14, 371–385, 1998.CrossRefGoogle Scholar
  88. 88.
    Trampert, J., and J.J. Lévcque, Simultaneous Iterative Reconstruction Technique: Physical interpretation based on the generalized least squares solution, J. Geophys. Res., 95, 12553–12559, 1990.Google Scholar
  89. 89.
    Trampert, J.,.T.J. LOvînue., and M. Cara, Inverse problems in seismology, in Inverse problems in scattering and imaging, edited by M. Bertero and E.R. Pike, pp. 131145, Adam Hilger, Bristol, 1992.Google Scholar
  90. 90.
    Tramspert,.T., and R. Snieder, Model estimations based on truncated expansions: Possible artifacts in seismic tomograph y, Science, 271, 1257–1260, 1996.Google Scholar
  91. 91.
    Trampert, J., ami J.H. Woodhouse, Global phase velocity maps of Love and Rayleigh waves between 40 and 150 seconds, Geophys. J. Int., 122, 675–690, 1995.CrossRefGoogle Scholar
  92. 92.
    Trasnpert,.1., and J.H. Woodhouse, High resolution global phase velocity distributions, Geophys. Res. Lett., 23, 21–24, 1996.Google Scholar
  93. 93.
    VanDecar, J.C., and R. Snieder, Obtaining smooth solutions to large linear inverse problems, Geophysics, 59, 818–829, 1994.CrossRefGoogle Scholar
  94. 94.
    van der Hilst, R.D., S. Widiyantoro, and E.R. Engdahl, Evidence for deep mantle circulation from global tomography, Nature, 386, 578–584, 1997.CrossRefGoogle Scholar
  95. 95.
    van der Hilst, R.D., and B.L.N. Kennett, Upper mantle structure beneath Australia from portable array deployments, American Geophysical Union ‘Geodynamics Series’, 38, 39–57, 1998.CrossRefGoogle Scholar
  96. 96.
    van Heijst, H.J. and J.H. Woodhouse, Measuring surface wave overtone phase velocities using a mode-branch stripping technique, Geophys. J. Int., 131, 209–230, 1997.CrossRefGoogle Scholar
  97. 97.
    Weidelt, P., The inverse problem of geomagnetic induction, J. Geophys., 38, 257289, 1972.Google Scholar
  98. 98.
    Wiechert, E., Über Erdbebenwellen. I. Theoretisches über die Ausbreitung der Erdbebenwellen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse, 415–529 1907.Google Scholar
  99. 99.
    Woodhouse, J. H. and A. M. Dziewonski, Mapping the upper mantle: Three dimensional modelling of Earth structure by inversion of seismic waveforms, J. Geophys. Res, 89, 5953–5986, 1984.CrossRefGoogle Scholar
  100. 100.
    Yilmaz, O., Seismic data processing, Investigations in geophysics, 2, Society of Exploration Geophysicists, Tulsa, 1987.Google Scholar

Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • R. Snieder
    • 1
  • J. Trampert
    • 1
  1. 1.Utrecht UniversityUtrechtThe Netherlands

Personalised recommendations