Mathematical Geology

, Volume 36, Issue 2, pp 187–216 | Cite as

Using Bayesian Statistics to Capture the Effects of Modelling Errors in Inverse Problems

  • J. N. Carter


When the parameters of a numerical model are adjusted, so that the predictions of the model match measurements from the real system, we need to take account of two sources of errors. These being measurement errors and modelling errors. Measurement errors are commonly considered, and a number of different approaches are in general usage, the most common being the weighted sum of squares method. In this paper the standard Bayesian equation, used for inverse problems, is reformulated so as to make it more intuitive to use. This allows the inclusion of both a modelling error and correlations between measurements to be carried out easily. The results are tested on a simple one-parameter numerical model and a cross-sectional model of a petroleum reservoir. In the first case the proposed error model appears to work well. In the second case it appears that the objective function is multimodal, leading to multiple acceptable solutions. The results of this paper are important to those whose numerical models are thought to contain significant modelling error. This encompasses many areas of modelling related to earth science and engineering.

likelihood uncertainty sum of squares Bayesian analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aanonsen, S. I., Cominelli, A., Gosselin, O., Aavatsmark, I., and Barkve, T., 2002, Integration of 4D data in the history match loop by investigating scale dependent correlations in the acoustic impedance cube, in Proceedings of the 8th European Conference on Maths of Oil Recovery, p. E59. European Association of Geoscientists and Engineers, Houten, The Netherlands.Google Scholar
  2. Bush, M. D., and Carter, J. N., 1996, Applications of a modified genetic algorithm to parameter estimation in the petroleum industry: Intelligent engineering systems through artificial neural networks, v. 6, p. 397. ASME Press, New York.Google Scholar
  3. DeVolder, B., Glimm, J., Grove, J. W., Kamg, Y., Lee, Y., Poa, K., Sharp, D. H., and Ye, K., 2002, Uncertainty quantification for multiscale simulations: J. Fluids Eng., v. 124, p. 1-13.Google Scholar
  4. Floris, F. J. T., Bush, M. D., Cuypers, M., Roggero, F., and Syversveen, A. R., 2001, Methods for quantifying the uncertainty of production forecasts—a comparative study: Pet. Geosci., v. 7, p. S87-S96.Google Scholar
  5. Glimm, J., Hou, S., Kim, H., Lee, Y., Sharp, D. H., and Zou, Q., 2001, Risk management for petroleum reservoir production: A simulation-based study of prediction: Comp. Geosci., v. 5, p. 173-197.Google Scholar
  6. Hauge, R., Arntzen, O. J., and Soleng, H., 2002, Choosing objective function for conditioning on production data, in Proceedings of the 8th European Conference on Maths of Oil Recovery, p. E47.Google Scholar
  7. Sivia, D. S., 1996, Data analysis: A Bayesian tutorial: Oxford University Press, Oxford.Google Scholar
  8. Tarantola, A., 1987, Inverse problem theory: Methods for data fitting and model parameter estimation: Elsevier Science, Amsterdam.Google Scholar
  9. Wu, Z., Reynolds, A. C., and Oliver, D. S., 1999, Conditioning geostatistical models to two-phase production data: SPE J., v. 4, p. 142-155.Google Scholar

Copyright information

© International Association for Mathematical Geology 2004

Authors and Affiliations

  • J. N. Carter
    • 1
  1. 1.Department of Earth Sciences and EngineeringImperial CollegeLondonUnited Kingdom

Personalised recommendations