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Inversion of Geophysical Data

  • Kalyan Kumar RoyEmail author
Chapter
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Part of the Springer Geophysics book series (SPRINGERGEOPHYS)

Abstract

In this chapter, a few topics of inversion generally used for interpreting geoelectrical data are discussed briefly. The topics included are: (i) the general introduction on the subject; (ii) Tikhnov’s regularization; (iii) very brief touch on singular value decomposition, least squares inversion, weighted least squares and weighted ridge regression; (iv) Backus–Gilbert inversion; (v) 2D Occam inversion; (vi) REBOCC inversion; (vii) rapid relaxation inversion; (viii) method of steepest descent; (ix) conjugate gradient minimisation; (x) joint inversion; and (xi) appraisal. Topics included in joint inversion are combinations of: (i) magnetotelluric and DC resistivity; (ii) DC resistivity and induced polarization; and (iii) magnetotellurics and seismics. One field example of use of multiple inversion tools for 1D inversion of the same set of data is given.

Keywords

Multidimensional Joint inversion Geoelectrical data 

References

  1. Backus, G., and F. Gilbert. 1967. Numerical application of a formalism for geophysical inverse problem. Geophysical Journal 33: 247–276.CrossRefGoogle Scholar
  2. Backus, G., and F. Gilbert. 1968. The resolving power of gross earth data. Geophysical Journal of the Royal Astronomical Society 16: 169–205.CrossRefGoogle Scholar
  3. Backus, G., and F. Gilbert. 1970. Uniqueness in the inversion of inaccurate gross earth data. Philosophical Transactions of the Royal Society 266A: 123–192.Google Scholar
  4. Benech, C., A. Tabbagh, and G. Desvignes. 2002. Joint inversion of EM and magnetic data for near surface studies. Geophysics, 67 (6): 1729–1739.CrossRefGoogle Scholar
  5. Bhattacharyya, P.K., and H.P. Patra. 1968. Direct current geoelectric sounding. Elsevier, Amsterdam: Principles and Interpretation.Google Scholar
  6. Chave, A.D. 1984. Fréchet derivatives of electromagnetic inversion. Journal of Geophysical Research, 3373–3379.Google Scholar
  7. Constable, S.C., R.L. Parker, and C.G. Constable. 1987. Occam’s inversion: a practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics 52: 289–300.CrossRefGoogle Scholar
  8. Davis, L.D. 1991. Hand book of genetic algorithm. Van Nostrand Reinhold.Google Scholar
  9. deGroot-Hedlin, C., and S. Constable. 1990. Occam’s inversion to generate smooth, two-dimensional models from magnetotelluric data. Geophysics 55 (12): 1613–1624.CrossRefGoogle Scholar
  10. deGroot-Hedlin, C., and S. Constable. 2004. Inversion of magnetotelluric data for 2D structure with sharp resistivity contrasts. Geophysics, 69 (1): 78–86.CrossRefGoogle Scholar
  11. Dieft, P., and X. Xhou. 1993. A steepest descent method for oscillatory Riemann Hilbert problems. The Annals of Mathematics 37 (2): 298–368.Google Scholar
  12. Dobroka, M., M. Kis, and E. Turai. 2001. Approximate joint inversion of MT and DC geoelectric data in case of 2-D structures. In EAEG 63rd conference and technical exhibition, Amsterdam, The Netherlands.Google Scholar
  13. Dosso, S.E., and D.W. Oldenburg. 1989. Linear and non-linear appraisal using extremal models of bounded variation. Geophysical Journal International 99: 483–495.CrossRefGoogle Scholar
  14. Dosso, S.E., and D.W. Oldenburg. 1991. Magnetotelluric appraisal using simulated annealing. Geophysical Journal International 106: 379–385.CrossRefGoogle Scholar
  15. Draper, N.R., and J. Smith. 1968. Applied regression analysis. Newyork: John Wiley and Sons Inc.Google Scholar
  16. Gallardo, L.A., and M.A. Meju. 2003. Characterization of heterogeneous near surface material by joint 2D inversion of DC resistivity and seismic data. Geophysical Research Letters 30: 1658–1661.CrossRefGoogle Scholar
  17. Gallardo, L.A, and M.A. Meju. 2004. Joint 2D resistivity and seismic inversion, EM INDIA 2004 MTDIW. Advances in integrated geoelectromagnetic and seismic interpretation (joint two dimensional resistivity and Seismic travel time inversion with cross gradient constraints). Journal of Geophysical Research 109 (B3): B03311.Google Scholar
  18. Gallardo-Delgado, L.A., M.A. Prez Flores, and E. Gomez-Trevino. 2003. A varsatile algorithm for joint 3D inversion of gravity and magnetic data. Geophysics 68 (3): 949–959.CrossRefGoogle Scholar
  19. Ghosh, D.P. 1970. The application of linear filter theory to the direct application of geoelectrical resistivity measurements, monograph. Gravenhage, The Netherlands: Drukker, J.H., PasmansGoogle Scholar
  20. Goldberg, D.E. 1989. Genetic algorithm in search, optimization and machine learning. USA, MA: Addison Wesley Reading.Google Scholar
  21. Hering, A., R. Misiek, A. Gyulai, T. Ormos, M. Dobroka, and L. Dresen. 1993. A joint Inversion algorithm to process geoelectric and surface wave seismic data Part I: basic ideas. Geophysical Prospecting, 137–156.Google Scholar
  22. Hestenes, M., and E. Stiefel. 1952. Methods of conjugate gradient for solving linear systems. Journal of Research of the National Bureau of Standards 49 (6): 409–435.CrossRefGoogle Scholar
  23. Holland, J.H. 1975. Adaptation in natural and artificial systems. USA: University of Michigan Press.Google Scholar
  24. Hopfield, J.J., and D.W. Tank. 1986. Computing with neural circuits. A Model Science 233: 625–633.Google Scholar
  25. Ingber, L. 1989. Very fast simulated annealing (VFSA). Mathematical Computation and Modeling 12 (8): 967–993.CrossRefGoogle Scholar
  26. Ingber, L. 1993. Simulated annealing, practice versus theory, statistics and computing.Google Scholar
  27. Inman, J.R. 1975. Resistivity inversion with ridge regression. Geophysics 40 (5): 798–817.CrossRefGoogle Scholar
  28. Inman, J.R., J. Ryu, and S.H. Ward. 1973. Resistivity inversion: Geophysics 38 (6): 1088–1108.Google Scholar
  29. Jackson, D.D. 1972. Interpretation of inaccurate, insufficient, and inconsistent data. Geophysical Journal of the Royal Astronomical Society 28: 97–109.CrossRefGoogle Scholar
  30. Jackson, D.D. 1973. Marginal solutions to quasilinear inverse problems in geophysics. Geophysical Journal of the Royal Astronomical Society 35: 121–136.CrossRefGoogle Scholar
  31. Jackson, D.D. 1979. The use of a priori data to resolve non-uniqueness in linear inversion. Geophysical Journal of the Royal Astronomical Society 57: 137–157.CrossRefGoogle Scholar
  32. Jupp, D.L.B., and K. Vozoff. 1975. Stable iterative methods for the inversion of geophysical data. Geophysical Journal of the Royal Astronomical Society 42: 957–976.CrossRefGoogle Scholar
  33. Jupp, D.L.B., and K. Vozoff. 1977a. Two-dimensional magnetotelluric inversion. Geophysical Journal of the Royal Astronomical Society 50: 333–352.CrossRefGoogle Scholar
  34. Jupp, D.L.B., and K. Vozoff. 1977b. Resolving anisotropy in layered media by joint inversion. Geophysical Prospecting 25: 460–470.CrossRefGoogle Scholar
  35. Keller, G.V., and F.C. Frischknecht. 1966. Electrical methods in geophysical prospecting. Oxford: Pergamon Press.Google Scholar
  36. Koefoed, O. 1979. Geosounding principles–1: resistivity sounding measurements. Amsterdam: Elsevier.Google Scholar
  37. Larsen, J.C. 1981. A new technique for layered earth magnetotelluric inversion. Geophysics 46 (9): 1247–1257.CrossRefGoogle Scholar
  38. Lanczos, C. 1941. Linear differential operator. Van Nostrand and Co. London.Google Scholar
  39. Li, Y., and D.W. Oldenburg. 2000. Joint Inversion of surface and three component bore hole magnetic data. Geophysics 65: 540–552.CrossRefGoogle Scholar
  40. Mackie, R.L., and T.R. Madden. 1993a. Conjugate direction relaxation solutions for 3-D magnetotelluric modeling. Geophysics, 58 (7): 1052–1057.CrossRefGoogle Scholar
  41. Mackie, R. L., and T.R. Madden. 1993b. Three-dimensional magnetotelluric inversion using conjugate gradients. Geophysical Journal International, 115: 215–229.CrossRefGoogle Scholar
  42. Marquardt, D.W. 1963. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics 11: 431–441.CrossRefGoogle Scholar
  43. Marquardt, D.W. 1970. Generalized inverse, ridge regression, based linear estimation and non-linear estimation. Technometrics 12 (3): 591–612.CrossRefGoogle Scholar
  44. Menke, W., 1984. Geophysical data analysis discrete inverse theory. Academic Press.Google Scholar
  45. Nabighian, M.N., and C.L. Elliott. 1976. Negative induced polarization effects from layered media. Geophysics 41: 1236–1255.CrossRefGoogle Scholar
  46. Oldenburg, D.W. 1979. One-dimensional inversion of natural source magnetotelluric observations. Geophysics 44: 1218–1244.CrossRefGoogle Scholar
  47. Oldenburg, D.W. 1990. Inversion of electromagnetic data: an overview of new techniques. Surveys In Geophysics 11: 231–270.CrossRefGoogle Scholar
  48. Oldenburg, D.W., and R.G. Ellis. 1991. Inversion of geophysical data using an approximate inverse mapping. Geophysical Journal International, 105: 325–353.CrossRefGoogle Scholar
  49. Oldenburg, D.W., P.R. McGillivray, and R.G. Ellis. 1993. Generalised subspace methods for large scale inverse problems. Geophysical Journal International, 114: 12–20.CrossRefGoogle Scholar
  50. Parker, R.L. 1970. The inverse problems of electrical conductivity of the mantle. Geophysical Journal of the Royal Astronomical Society 22: 121–138.CrossRefGoogle Scholar
  51. Parker, R.L. 1972. Inverse theory with grossly inadequate data. Geophysical Journal of the Royal Astronomical Society 29: 123–138.CrossRefGoogle Scholar
  52. Parker, R.L. 1977a. Understanding inverse theory. Annual Review of Earth and Planetary Sciences 5: 35–64.CrossRefGoogle Scholar
  53. Parker, R.L. 1977b. The Fréchet derivative for the one dimensional electromagnetic induction problem. Geophysical Journal of the Royal Astronomical Society 49: 543–547.CrossRefGoogle Scholar
  54. Parker, R.L. 1980. The inverse problem of electromagnetic induction: existence and construction of solutions based upon incomplete data. Journal Geophysical Research 85: 4421–4425.CrossRefGoogle Scholar
  55. Parker, R.L. 1983. The magnetotelluric inverse problems. Surveys In Geophysics 6: 5–25.CrossRefGoogle Scholar
  56. Parker, R.L. 1984. The inverse problem of resistivity sounding. Geophysics 49 (12): 2143–2158.CrossRefGoogle Scholar
  57. Parker, R.L. 1994. Geophysical inverse theory. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
  58. Pedersen, L.B. 1979. Constrained inversion of potential field data. Geophysical Prospecting 27: 726–748.CrossRefGoogle Scholar
  59. Pedersen, L.B., and M. Gharibi. 2000. Automatic 1D inversion of magnetotelluric data: finding the simplest possible model that fits the data. Geophysics 65 (3): 773–782.CrossRefGoogle Scholar
  60. Press, F. 1968. Earth models obtained by Monte Carlo inversion. Journal of Geophysical Research 73: 5223–5234.CrossRefGoogle Scholar
  61. Raiche, A.P., D.l.B. Jupp, H. Rutter, and K. Vozoff. 1985. The joint use of coincident loop transient electromagnetics and Schlumberger sounding to resolve the layered structures. Geophysics 50: 1618–1627.CrossRefGoogle Scholar
  62. Rijo, L. 1977. Modelling of electric and electromagnetic data. Ph.D. thesis Unpublished, University of Utah, Salt Lake City, USA.Google Scholar
  63. Rodi, W.L. 1976. A technique for improving the accuracy of finite element solution for magnetotelluric data. Geophysical Journal of the Royal Astronomical Society 44: 483–506.CrossRefGoogle Scholar
  64. Rodi, W., and R.L. Mackie. 2001. Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion. Geophysics, 66 (1): 174–187.CrossRefGoogle Scholar
  65. Roy, A. 1974. Resistivity signal partition in layered media. Geophysics 39 (2): 190–204.CrossRefGoogle Scholar
  66. Roy, K.K. 2007. Potential theory in applied geophysics. Heidelberg: Springer.Google Scholar
  67. Roy, K.K. 2014. Geophysical signatures for detection of fresh eater and saline water zones, in ‘Recent Trends in Modelling of Environmental Contaminants’, ed. D. Sengupta. New Delhi, India: Springer.Google Scholar
  68. Roy, K.K., and O.P. Rathi. 1988. Resistivity inversion using resistivity signal partitions. Gerlands Beitrage Zur Geophysik Leipzig 97 (6)S: 472–494.Google Scholar
  69. Roy, K.K., and O.P. Rathi. 1988. A new approach for inversion of resistivity and induced polarization sounding data in “Frontiers in Exploration Geophysics”, ed. B.B. Bhattacharyya, 269–291. Oxford: I.B.H.Google Scholar
  70. Roy, K.K., J. Bhattacharyya, K.K. Mukherjee, and Mahatsente. 1995. An interactive inversion of resistivity and induced polarization sounding for location of saline water pockets. Exploration Geophysics (Australia), 25: 207–211.Google Scholar
  71. Sasaki, Y. 1989. Two dimensional joint inversion of magnetotelluric and dipole-dipole resistivity data. Geophysics 54 (2): 254–262.CrossRefGoogle Scholar
  72. Scales, J.A., M.L. Smith, and S. Treitel. 2001. Introduction to geophysical inverse theory. Golden, Colorado: Sameezdat Press.Google Scholar
  73. Seigel, H.O. 1959. Mathematical formulation and type curves for induced polarization. Geophysics 24: 547–565.CrossRefGoogle Scholar
  74. Sen, M.K., and P.L. Stoffa. 1994. Global optimisation methods for geophysical inversion. Amsterdam: Elsevier.Google Scholar
  75. Sharma, S.P., E. Pracser, and K.K. Roy. 2005. Joint inversion of seismic refraction and magnetotelluric data for resolving deeper: subsurface structure. Acta Geodaetica et Geophysica, Hungarica 40 (2): 241–258.CrossRefGoogle Scholar
  76. Siripunvaraporn, W., and G. Egbert. 2000. An efficient data-subspace inversion method for 2-D magnetotelluric data. Geophysics, 65 (3): 791–803.CrossRefGoogle Scholar
  77. Smith, J.T., and J.R. Booker. 1988. Magnetotelluric inversion for minimum structure. Geophysics 53 (12): 1565–1576.CrossRefGoogle Scholar
  78. Smith, J.T., and J.R. Booker. 1991. Rapid inversion of two- and three-dimensional magnetotelluric data. Journal of Geophysical Research 96 (B3): 3905–3922.CrossRefGoogle Scholar
  79. Smith, N.C, and K. Vozoff. 1984. Two dimensional DC resistivity inversion for dipole–dipole data. IEEE Transactions on Geoscience and Remote Sensing, GE- 22 (1): 21–28.CrossRefGoogle Scholar
  80. Tarantola, A. 1987. Inverse problem theory: methods for data fitting and model parameter estimation. Amsterdam: Elsevier.Google Scholar
  81. Telford, W.M., L.P. Geldart, R.E. Sheriff, and D.A. Keys. 1981. Applied geophysics. Cambridge: Cambridge University Press.Google Scholar
  82. Tikhnov, A.N., and V.Y. Arsenin. 1977. Solution of ill-posed problems (Translated by Fritz John), New York University, Courant Institute of Mathematical Sciences. New York: Wiley.Google Scholar
  83. Tikhnov, A.N., A.V. Goncharsky, V.V. Stepanov, and A.G. Yagola. 1995. Numerical methods for solution of ill-posed problems. Dordrecht: Kluwer Academic Publisher.CrossRefGoogle Scholar
  84. Tripp, A.C., G.W. Hohmann, and C.M. Swift Jr. 1984. Two dimensional resistivity inversion. Geophysics 49 (10): 1708–1717.CrossRefGoogle Scholar
  85. Vozoff, K., and D.L.B. Jupp. 1975. Joint inversion of geophysical data. Geophysical Journal of the Royal Astronomical Society 42: 977–991.CrossRefGoogle Scholar
  86. Wannamaker, P.E. 1991. Advances in three dimensional magneto-telluric modeling using integral equations. Geophysics 56 (11): 1716–1728.CrossRefGoogle Scholar
  87. Wannamaker, P.E., J.A. Stodt, and L. Rijo. 1987. A stable finite element solution for two dimensional magnetotelluric modelling. Geophysical Journal of the Royal Astronomical Society 88: 277–296.CrossRefGoogle Scholar
  88. Weidelt, P. 1975. Inversion of two dimensional conductivity structures. Physics of the Earth and Planetary Interiors 10: 282–291.CrossRefGoogle Scholar
  89. Zhdanov, M.S. 2002. Geophysical inverse theory and regularisarion problems. Amsterdam: Elsevier.Google Scholar
  90. Zhdanov, M.S., and G.V. Keller. 1994. The geoelectrical methods in geophysical exploration, 871p. Amsterdam: Elsevier Scientific Publishing Company.Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Earth SciencesIndian Institute of Engineering Science and Technology (IIEST)HowrahIndia

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