Bayesian Analysis

Computational methods for parameter estimation in climate models

Gabriel Huerta, Charles S. Jackson, Mrinal K. Sen, and Alejandro Villagran

Full-text: Open access


Intensive computational methods have been used by Earth scientists in a wide range of problems in data inversion and uncertainty quantification such as earthquake epicenter location and climate projections. To quantify the uncertainties resulting from a range of plausible model configurations it is necessary to estimate a multidimensional probability distribution. The computational cost of estimating these distributions for geoscience applications is impractical using traditional methods such as Metropolis/Gibbs algorithms as simulation costs limit the number of experiments that can be obtained reasonably. Several alternate sampling strategies have been proposed that could improve on the sampling efficiency including Multiple Very Fast Simulated Annealing (MVFSA) and Adaptive Metropolis algorithms. The performance of these proposed sampling strategies are evaluated with a surrogate climate model that is able to approximate the noise and response behavior of a realistic atmospheric general circulation model (AGCM). The surrogate model is fast enough that its evaluation can be embedded in these Monte Carlo algorithms. We show that adaptive methods can be superior to MVFSA to approximate the known posterior distribution with fewer forward evaluations. However the adaptive methods can also be limited by inadequate sample mixing. The Single Component and Delayed Rejection Adaptive Metropolis algorithms were found to resolve these limitations, although challenges remain to approximating multi-modal distributions. The results show that these advanced methods of statistical inference can provide practical solutions to the climate model calibration problem and challenges in quantifying climate projection uncertainties. The computational methods would also be useful to problems outside climate prediction, particularly those where sampling is limited by availability of computational resources.

Article information

Bayesian Anal., Volume 3, Number 4 (2008), 823-850.

First available in Project Euclid: 22 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Parametric Uncertainties Inverse Problems Simulated Annealing Adaptive Metropolis Climate Models


Villagran, Alejandro; Huerta, Gabriel; Jackson, Charles S.; Sen, Mrinal K. Computational methods for parameter estimation in climate models. Bayesian Anal. 3 (2008), no. 4, 823--850. doi:10.1214/08-BA331.

Export citation


  • Allen, M. (1999). "Do-it-yourself climate prediction." Nature, 401: 642.
  • Annan, J. and Hargreaves, J. (2007). "Efficient estimation and ensemble generation in climate modelling." Philosophical Transactions of the Royal Society A, 365: 2077–2088.
  • Barnett, D., Brown, S., Murphy, J., Sexton, D., and Webb, M. (2006). "Quantifying uncertainty in changes in extreme event frequency in response to doubled CO2 using a large ensemble of GCM simulations." Climate Dynamics, 26: 489–511.
  • Collins, M., Booth, B., Harris, G., Murphy, J., Sexton, D., and Webb, M. (2006). "Towards quantifying uncertainty in transient climate change." Climate Dynamics, 127–147.
  • Forest, C., Allen, M. R., Sokolov, A. P., and Stone, P. H. (2001). "Constraining climate model properties using optimal fingerprint detection methods." Climate Dynamics, 18: 277–295.
  • Forest, C., Allen, M. R., Stone, P. H., and Sokolov, A. P. (2000). "Constraining uncertainties in climate models using climate change detection techniques." Geophys. Res. Lett., 27(4): 569–572.
  • Forest, C., Stone, P. H., Sokolov, A. P., Allen, M. R., and Webster, M. D. (2002). "Quantifying uncertainties in climate system properties with the use of recent climate observations." Science, 295: 113–117.
  • Gates, W. L., Boyle, J. S., Covey, C., Dease, C. G., Doutriaux, C. M., Drach, R. S., Fiorino, M., Gleckler, P. J., Hnilo, J. J., Marlais, S. M., Phillips, T. J., Potter, G. L., Santer, B. D., Sperber, K. R., Taylor, K. E., and Williams, D. N. (1999). "An Overview of the Results of the Atmospheric Model Intercomparison Project (AMIP I)." Bulletin of the American Meteorological Society, 80(1): 29–56.
  • Gelfand, A. and Smith, A. (1990). "Sampling-Based Approaches to Calculating Marginal Densities." Journal of the American Statistical Association, 85: 398–409.
  • Gelman, A., Roberts, G., and Gilks, W. (1996). "Efficient Metropolis jumping rules." In Bernardo, J., Berger, J., Dawid, A., and Smith, A. (eds.), Bayesian Statistics, volume 5, 599–608. Oxford University Press.
  • Geman, S. and Geman, D. (1984). "Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images." IEEE Transactions PAMI-6, 721–741.
  • Haario, H., Laine, M., Lehtinen, M., Saksman, E., and Tamminen, J. (2004). "Markov Chain Monte Carlo methods for high dimensional inversion in remote sensing." Journal of Royal Statistical Society, 66: 591–607.
  • Haario, H., Laine, M., Mira, A., and Saksman, E. (2006). "DRAM: Efficient adaptive MCMC." Statistics and Computing, 16: 339–354.
  • Haario, H., Saksman, E., and Tamminen, J. (1999). "Adaptive proposal distribution for random walk Metropolis algorithm." Computational Statistics, 14: 375–395.
  • –- (2001). "An Adaptive Metropolis algorithm." Bernoulli, 7: 223–242.
  • –- (2005). "Componentwise adaptation for high dimensional MCMC." Computational Statistics, 20(2): 265–263.
  • Hastings, W. (1970). "Monte Carlo sampling methods using Markov chains and their applications." Biometrika, 57: 97–109.
  • Ingber, L. (1989). "Very fast simulated re-annealing." Mathematical Computational Modelling, 12: 967–973.
  • Jackson, C. S. and Broccoli, A. (2003). "Orbital forcing of Arctic climate: mechanisms of climate response and implications for continental glaciation." Climate Dynamics, 21: 539–557.
  • Jackson, C. S., Sen, M., Huerta, G., Deng, Y., and Bowman, K. (2008). "Error Reduction and Convergence in Climate Prediction." Journal of Climate (in press).
  • Jackson, C. S., Sen, M., and Stoffa, P. (2004). "An Efficient Stochastic Bayesian Approach to Optimal Parameter and Uncertainty Estimation for Climate Model Predictions." Journal of Climate, 17: 2828–2840.
  • Joussaume, S. and Taylor, K. (2000). "The Paleoclimate Modeling Intercomparison Project (PMIP)." In Braconnot, P. (ed.), Proceedings of the third PMIP workshop, Canada, 4-8 October 1999, WCRP-111, WMO/TD-1007, 271.
  • Kettleborough, J., Booth, B., Stott, P., and Allen, M. (2007). "Estimates of Uncertainty in Predictions of Global Mean Surface Temperature." Journal of Climate, 20: 843–855.
  • Kirkpatrick, S., Gelatt, J., and Vecchi, M. (1983). "Optimization by simulated annealing." Science, 220: 671–680.
  • Lopez, A., Tebaldi, C., New, M., Stainforth, D., Allen, M., and Kettleborough, J. (2006). "Two Approaches to Quantifying Uncertainty in Global Temperature Changes." Journal of Climate, 19: 4785–4796.
  • McAvaney, B. J., Covey, C., Joussaume, S., Kattsov, V., Kitoh, A., Ogana, W., Pitman, A., Weaver, A., Wood, R., and Zhao, Z. (2001). "Model Evaluation. Climate Change 2001: The Scientific Basis Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change." In Houghton, J., Ding, Y., Griggs, D., Noguer, M., van der Linden, P., Dai, X., Maskell, K., and Johnson, C. (eds.), Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), 881. Cambridge University Press,UK.
  • McCarthy, J. J., Canziani, O. F., Leary, N. A., Dokken, D., and White, K. S. (eds.) (2001). Contribution of Working Group II to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC). Cambridge University Press, UK.
  • Meehl, G., Boer, G., Covey, C., Latif, M., and Stouffer, R. (2000). "The Coupled Model Intercomparison Project (CMIP)." Bulletin of the American Meteorological Society, 81(2): 313–318.
  • Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E. (1953). "Equations of State Calculations by Fast Computing Machines." Journal of Chemical Physics, 21: 1087–1091.
  • Milankovitch, M. (1941). "Canon of insolation and the ice age problem (Israel Program for Scientific Translations, Jerusalem)."
  • Murphy, J., Sexton, D. M. H., Barnett, D. N., Jones, G. S., Webb, M. J., Collins, M., and Stainforth, D. A. (2004). "Quantification of modelling uncertainties in a large ensemble of climate change simulations." Nature, 430: 768–772.
  • Parry, M., Canziani, O., Palutikof, J., van der Linden, P., and Hanson, C. (eds.) (2007). IPCC, 2007: Climate Change 2007: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Fourth Assessment Report of Intergovermental panel on Climate Change. Cambridge University Press, UK.
  • Sambridge, M. and Mosegaard, K. (2002). "Monte Carlo Methods in Geophysical Inverse Problems." Review of Geophysics, 40: 1–29.
  • Sansó, B., Forest, C., and Zantedeschi, D. (2008). "Inferring Climate System Properties Using a Computer Model (with discussion)." Bayesian Analysis, 3(1): 1–62.
  • Sen, M. and Stoffa, P. (1996). "Bayesian Inference, Gibbs sampler and uncertainty estimation in geophysical inversion." Geophysical Prospecting, 44: 313–350.
  • Stainforth, D., Aina, T., Christensen, C., Collins, M., Faull, N., Frame, D. J., Kettleborough, J. A., Knight, S., Martin, A., Murphy, J. M., Piani, C., Sexton, D., Smith, L. A., Spicer, R. A., Thorpe, A. J., and Allen, M. R. (2005). "Uncertainty in predictions of the climate response to rising levels of greenhouse gases." Nature, 433: 403–406.
  • Tebaldi, C., Smith, R., Nychka, D., and Mearns, L. (2005). "Quantifying Uncertainty in Projections of Regional Climate Change: A Bayesian Approach to the Analysis of Multimodel Ensembles." Journal of Climate, 18: 1524–1540.
  • Tierney, L. and Mira, A. (1999). "Some Adaptive Monte Carlo Methods for Bayesian Inference." Statistics in Medicine, 18: 2507–2515.