Encyclopedia of Solid Earth Geophysics

Living Edition
| Editors: Harsh K. Gupta

Inverse Theory, Monte Carlo Method

  • Malcolm SambridgeEmail author
  • Kerry Gallagher
Living reference work entry
DOI: https://doi.org/10.1007/978-3-030-10475-7_192-1


Monte Carlo method. A computational technique making use of random numbers to solve problems that are either probabilistic or deterministic in nature. Named after the famous Casino in Monaco.

Monte Carlo inversion method. A method for sampling a parameter space of variables representing unknowns, governed by probabilistic rules.

Markov chain Monte Carlo (McMC). A probabilistic method for generating vectors or parameter variables whose values follow a prescribed density function.


Because geophysical observations are made at (or very near) the Earth’s surface, all knowledge of the Earth’s interior is based on indirect inference. There always exists an inverse problem where models of physical properties are sought at depth that are only indirectly constrained by the available observations made at the surface. Geophysicists have been dealing with such problems for many years and in doing so have made substantial contributions to the understanding of inverse problems.

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We would like to thank FAST (French-Australia Science and Technology exchange program) for their support during the preparation of this entry. This project is supported by the Commonwealth of Australia under the International Science Linkages program.


  1. Aster R, Borchers R, Thurber CH (2005) Parameter estimation and inverse problems. International Geophysics Series, vol 90. Elsevier, AmsterdamGoogle Scholar
  2. Backus GE, Gilbert JF (1967) Numerical applications of a formalism for geophysical inverse problems. Geophys J R Astron Soc 13:247–276CrossRefGoogle Scholar
  3. Backus GE, Gilbert JF (1968) The resolving power of gross Earth data. Geophys J R Astron Soc 16:169–205CrossRefGoogle Scholar
  4. Backus GE, Gilbert JF (1970) Uniqueness in the inversion of inaccurate gross Earth data. Philos Trans R Soc Lond A 266:123–192CrossRefGoogle Scholar
  5. Bernardo JM, Smith AFM (1994) Bayesian theory. Wiley, ChichesterCrossRefGoogle Scholar
  6. Bodin T, Sambridge M (2009) Seismic tomography with the reversible jump algorithm. Geophys J Int 178:1411–1436CrossRefGoogle Scholar
  7. Charvin K, Gallagher K, Hampson G, Labourdette R (2009) A Bayesian approach to infer environmental parameters from stratigraphic data 1: methodology. Basin Res 21:5–25CrossRefGoogle Scholar
  8. Duijndam AJW (1988a) Bayesian estimation in seismic inversion part I: principles. Geophys Prospect 36:878–898CrossRefGoogle Scholar
  9. Duijndam AJW (1988b) Bayesian estimation in seismic inversion part II: uncertainty analysis. Geophys Prospect 36:899–918CrossRefGoogle Scholar
  10. Gallagher K, Charvin K, Nielsen S, Sambridge M, Stephenson J (2009) Markov chain Monte Carlo (McMC) sampling methods to determine optimal models, model resolution and model choice for Earth science problems. Mar Pet Geol 26:525–535CrossRefGoogle Scholar
  11. Geyer CJ, Møller J (1994) Simulation procedures and likelihood inference for spatial point processes. Scand J Stat 21:369–373Google Scholar
  12. Gilks WR, Richardson S, Spiegalhalter DJ (1996) Markov chain Monte Carlo in practice. Chapman & Hall, LondonGoogle Scholar
  13. Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–732CrossRefGoogle Scholar
  14. Green PJ (2003) Chapter 6: Trans-dimensional McMC. In: Green PJ, Hjort N, Richardson S (eds) Highly structured stochastic systems. Oxford statistical sciences series. Oxford University Press, Oxford, pp 179–196Google Scholar
  15. Hammersley JM, Handscomb DC (1964) Monte Carlo methods. Chapman & Hall, LondonCrossRefGoogle Scholar
  16. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109CrossRefGoogle Scholar
  17. Hopcroft P, Gallagher K, Pain CC (2009) A Bayesian partition modelling approach to resolve spatial variability in climate records from borehole temperature inversion. Geophys J Int 178:651–666CrossRefGoogle Scholar
  18. Jasra A, Stephens DA, Gallagher K, Holmes CC (2006) Analysis of geochronological data with measurement error using Bayesian mixtures. Math Geol 38:269–300CrossRefGoogle Scholar
  19. Kennett BLN, Brown DJ, Sambridge M, Tarlowski C (2003) Signal parameter estimation for sparse arrays. Bull Seismol Soc Am 93:1765–1772CrossRefGoogle Scholar
  20. Lee PM (1989) Bayesian statistics: an introduction. Edward Arnold, New York/TorontoGoogle Scholar
  21. Malinverno A (2002) Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem. Geophys J Int 151:675–688CrossRefGoogle Scholar
  22. Malinverno A, Parker RL (2005) Two ways to quantify uncertainty in geophysical inverse problems. Geophysics 71:15–27CrossRefGoogle Scholar
  23. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21:1087–1092CrossRefGoogle Scholar
  24. Mosegaard K, Sambridge M (2002) Monte Carlo analysis of inverse problems. Inverse Probl 18:R29–R54CrossRefGoogle Scholar
  25. Mosegaard K, Tarantola A (1995) Monte Carlo sampling of solutions to inverse problems. J Geophys Res 100:12431–12447CrossRefGoogle Scholar
  26. Sambridge M (1999) Geophysical inversion with a neighbourhood algorithm – I. Searching a parameter space. Geophys J Int 138:479–494CrossRefGoogle Scholar
  27. Sambridge M, Mosegaard K (2002) Monte Carlo methods in geophysical inverse problems. Rev Geophys 40:3.1–3.29CrossRefGoogle Scholar
  28. Sambridge M, Gallagher K, Jackson A, Rickwood P (2006) Trans-dimensional inverse problems, model comparison and the evidence. Geophys J Int 167:528–542CrossRefGoogle Scholar
  29. Smith AFM (1991) Bayesian computational methods. Philos Trans R Soc Lond A 337:369–386CrossRefGoogle Scholar
  30. Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Ser B 55:3–23Google Scholar
  31. Tarantola A, Valette B (1982) Inverse problems = quest for information. J Geophys 50:159–170Google Scholar

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Authors and Affiliations

  1. 1.Seismology and Mathematical GeophysicsResearch School of Earth Sciences, The Australian National UniversityCanberraAustralia
  2. 2.UMR 6118 – Géosciences RennesGêosciences, Université de Rennes 1Rennes CedexFrance