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# Geostatistical approach to bayesian inversion of geophysical data: Markov chain Monte Carlo method

## Abstract

This paper presents a practical and objective procedure for a Bayesian inversion of geophysical data. We have applied geostatistical techniques such as kriging and simulation algorithms to acquire a prior model information. Then the Markov chain Monte Carlo (MCMC) method is adopted to infer the characteristics of the marginal distributions of model parameters. Geostatistics which is based upon a variogram model provides a means to analyze and interpret the spatially distributed data. For Bayesian inversion of dipole-dipole resistivity data, we have used the indicator kriging and simulation techniques to generate cumulative density functions from Schlumberger and well logging data for obtaining a prior information by cokriging and simulations from covariogram models. Indicator approaches make it possible to incorporate non-parametric information into the probabilistic density function. We have also adopted the Markov chain Monte Carlo approach, based on Gibbs sampling, to examine the characteristics of a posterior probability density function and marginal distributions of each parameter. The MCMC technique provides a robust result from which information given by the indicator method, that is fundamentally non-parametric, is fully extracted. We have used the a prior information proposed by the geostatistical method as the full conditional distribution for Gibbs sampling. And to implement Gibbs sampler, we have applied the modified Simulated Annealing (SA) algorithm which effectively searched for global model space. This scheme provides a more effective and robust global sampling algorithm as compared to the previous study.

## References

• Bayes, T., An essay towards solving a problem in the doctrine of chances, reprinted in Biometrika, 45(1958), 293â€“315, 1763.

• Besag, J., Spatial interaction and the statistical analysis of lattice systems (with discussion), J. R. Statist. Soc. B, 36, 192â€“236, 1974.

• Carlin, B. P. and T. A. Louis, Bayes and Empirical Bayes Methods for Data Analysis, Champman & Hall, 1996.

• Chunduru, R. K., M. K. Sen, and P. L. Stoffa, 2-D resistivity inversion using spline parameterization and simulated annealing, Geophysics, 61, 151â€“161, 1996.

• Deutsch, C. V. and A. G. Journel, GSLIB: Geostatistical Software Library and Userâ€™s Guide, Oxford University Press, New York, 1992.

• Duijndam, A. J. W., Bayesian estimation in seismic inversion, part i: Principles, Geophys. Prosp., 36, 878â€“898, 1988a.

• Duijndam, A. J. W., Bayesian estimation in seismic inversion, part ii: Uncertainty analysis, Geophys. Prosp., 36, 899â€“918, 1988b.

• Gelfand, A. E. and A. F. M. Smith, Sampling-based approaches to calculating marginal densities, J. Am. Statist. Ass., 85, 398, 409, 1990.

• Gelfand, A. E., S. E. Hills, A. Racine-Poon, and A. F. M. Smith, Illustration of Bayesian inference in normal data models using Gibbs sampling, J. Am. Statist. Ass., 85, 972â€“985, 1990.

• Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, Champman & Hall, 1995.

• Geman, S. and D. Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine Intelligence, v. PAMI-6, 721â€“741, 1984.

• Gilks, W. R., S. Richardson, and D.J. Spiegelhalter, Markov Chain Monte Carlo in Practice, Champman & Hall, 1996.

• Glacken, I., Change of Support by Direct Conditional Block Simulation, Masterâ€™s thesis, Stanford Univ., Stanford, CA., 1996.

• Gomez-Hernandez, J., A stochastic approach to the simulation of block conductivity fields conditional upon data measured at a smaller scale, Doctoral Dissertation, Stanford University, Stanford, CA, 1991.

• Goovaerts, P., Geostatistics for Natural Resources Evaluation, Oxford University Press, New York, 1997.

• Gouveia, W. P., Bayesian seismic waveform data inversion: Parameter estimation and uncertainty analysis, Ph.D. thesis, Colo. Sch. of Mines, Golden, Co, 1996.

• Grandis, H., M. Menvielle, and M. Roussignol, Bayesian inversion with Markov chainsâ€”I. The magnetotelluric one-dimensional case, Geophys. J. Int., 138, 757â€“768, 1999.

• Hastings, W. K., Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97â€“109, 1970.

• Isaaks, E. H., The application of Monte Carlo Methods to the Analysis of Spatially Correlated Data, Doctoral Dissertation, Stanford University, Stanford, CA, 1990.

• Isaaks, E. H. and R. M. Srivastava, An Introduction to Applied Geostatistics, Oxford University Press, New York, 1989.

• Jaynes, E. T., Probability Theory: The Logic of Science, http://bayes.wustl.edu/etj/prob.html, 1994.

• Lines, L. R. and S. Treitel, Tutorial: A review of least-squares inversion and its application to geophysical problems, Geophys. Pros., 32, 159â€“186, 1984.

• Loredo, T. J., From laplace to supernova SN 1987A: Bayesian inference in astrophysics, in Maximum Entropy and Bayesian Methods, edited by P. F. Fougere, pp. 81â€“142, Kluwer Academic Publishers, The Netherlands, 1990.

• Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equations of state calculations by fast computing machine, J. Chem. Phys., 21, 1087â€“1091, 1953.

• Moraes, F. S., The application of marginalization and local distributions to multidimensional Bayesian inverse problems, Ph.D. Thesis, Colorado School of Mines, Golden, Co, 1996.

• Mosegaard, K. and A. Tarantola, Monte Carlo sampling of solutions to inverse problem, J. Geophys. Res., 100, 12,431â€“12,447, 1995.

• MPI Forum, MPI: A Message-Passing Interface Standard, University of Tennesse, Knoxville, Tennesee, 1995.

• Olea, R., Optimum mapping technique: Series on spatial analysis No. 2, Kansas Geol. Survey, Lawrence, KA, 1975.

• Pacheco, P. S., Parallel Programming with MPI, Morgan Kaufmann Publisher, Inc., San Francisco, CA, 1997.

• Roberts, G. O., Markov chain concepts related to sampling algorithms, in Markov Chain Monte Carlo in Practice edited by W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, pp. 45â€“57, Champman & Hall, 1995.

• Rothman, D. H., Nonlinear inversion, statistical mechanics, and residual statics estimation, Geophysics, 50, 2784â€“2796, 1985.

• Rothman, D. H., Automatic estimation of large statics corrections, Geophysics, 51, 332â€“346, 1986.

• Sambridge, M., Geophysical inversion with a neighbourhood algorithm-II. Appraising the ensemble, Geophys. J. Int., 138, 727â€“746, 1999.

• Scales, J. A. and A. Tarantola, An example of geologic prior information in a Bayesian seismic inverse calculation, Center for Wave Phenomena 159, Golden, Co, 1994.

• Scales, J. A. and L. Tenorio, Prior information and uncertainty in inverse problems, Mathematical Geophysics Summer School, Stanford University, 1998.

• Sen, M. and P. L. Stoffa, Global optimization methods in geophysical inversion, Advances in Exploration Geophysics, v.4, Elsevier, Amsterdam, 1995.

• Srivastava, R. M., Minimum variance or maximum profitability?, CIM Bulletin, 80, 63â€“68, 1987.

• Szu, H. and R. Heartley, Fast simulated annealing, Phys. Lett., 122, 157â€“162, 1987.

• Tarantola, A., Inverse problem theory: Methods for data fitting and model parameter estimation, Elsevier, 1987.

• Tierney, L., Introduction to general state-space Markov chain theory, in Markov Chain Monte Carlo in Practice, edited by W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, pp. 59â€“74, Champman & Hall, 1995.

• Wijk, K., J. A. Scales, W. Navidi, and K. Roy-Chowdhury, Estimating data uncertainty for least square optimization, Center for Wave Phenomena 264, Golden, Co, 1997.

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Oh, SH., Kwon, BD. Geostatistical approach to bayesian inversion of geophysical data: Markov chain Monte Carlo method. Earth Planet Sp 53, 777â€“791 (2001). https://doi.org/10.1186/BF03351676