Bayesian calibration of a large-scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm

Water Resources Research

Volumn 47, Issue 10, 2011, Pages

  Cui, T ^a   Fox, C ^b   O'Sullivan, M J ^a  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

^b UNIVERSITY OF OTAGO   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

ADAPTIVITY; BAYESIAN CALIBRATION; BAYESIAN INFERENCE; EXPECTED VALUES; GEOTHERMAL RESERVOIR; HIGH-DIMENSIONAL; MARKOV CHAIN MONTE CARLO; MCMC SAMPLING; MEASURED DATA; METROPOLIS-HASTINGS ALGORITHM; NUMERICAL MODELS; POSTERIOR DISTRIBUTIONS; PROPOSAL DISTRIBUTION; REDUCED ORDER MODELS; SAMPLING EFFICIENCY; UNCERTAINTY QUANTIFICATIONS;

ADAPTIVE ALGORITHMS; BAYESIAN NETWORKS; EFFICIENCY; ESTIMATION; GEOTHERMAL FIELDS; INFERENCE ENGINES; INVERSE PROBLEMS; MARKOV PROCESSES; PARAMETER ESTIMATION; STOCHASTIC MODELS; THREE DIMENSIONAL;

COMPUTATIONAL EFFICIENCY;

ALGORITHM; BAYESIAN ANALYSIS; CALIBRATION; COMPUTER SIMULATION; DATA SET; ESTIMATION METHOD; GEOTHERMAL SYSTEM; MARKOV CHAIN; MONTE CARLO ANALYSIS; NUMERICAL MODEL; RESERVOIR; THREE-DIMENSIONAL MODELING; UNCERTAINTY ANALYSIS;

EID: 80055063784     PISSN: 00431397     EISSN: None     Source Type: Journal    
DOI: 10.1029/2010WR010352     Document Type: Article

References (112)

1. Amit, Y., and U. Grenander (1991), Comparing sweep strategies for stochastic relaxation, J. Multivariate Anal., 37, 197-222.
2. Andrieu, C., and E. Moulines (2006), On the ergodicity properties of some adaptive Markov chain Monte Carlo algorithms, Ann. Appl. Probab., 16(1), 462-505.
3. Arridge, S. R., J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen (2006), Approximation errors and model reduction with an application in optical diffusion tomography, Inverse Probl., 22, 175-195. (Pubitemid 43185096)
4. Atchade, Y. F., and J. S. Rosenthal (2005), On adaptive Markov chain Monte Carlo algorithms, Bernoulli, 11, 815-828. (Pubitemid 44966970)
5. Awotunde, A. A., and R. N. Horne (2011), A wavelet approach to adjoint state sensitivity computation for steady state differential equations, Water Resour. Res., 47, W03502, doi:10.1029/2010WR009165.
6. Bates, B. C., and E. P. Campbell (2001), A Markov chain Monte Carlo scheme for parameter estimation and inference in conceptual rainfallrunoff modeling, Water Resour. Res., 37(4), 937-947, doi:10.1029/2000WR900363. (Pubitemid 32516048)
7. Begg, S. H., R. R. Carter, and P. Dranfield (1989), Assigning effective values to simulator gridblock parameters for heterogeneous reservoirs, SPE Reservoir Eng., 4(4), 455-463.
8. Besag, J. E. (1974), Spatial interaction and the statistical analysis of lattice systems, J. R. Stat. Soc. Ser. B, 36(2), 192-236.
9. Besag, J. E. (1986), On the statistical analysis of dirty pictures, J. R. Stat. Soc. Ser. B, 48, 259-302.
10. Beven, K. J., and A. M. Binley (1992), The future of distributed hydrological models: Model calibration and uncertainty prediction, Hydrol. Processes, 6(3), 279-298, doi:10.1002/hyp. 3360060305.
11. Beven, K. J., and J. Freer (2001), Equifinality, data assimilation, and uncertainty estimation in mechanistic modeling of complex environmental systems using the glue methodology, J. Hydrol., 249, 11-29. (Pubitemid 32664993)
12. Caers, J., and T. Hoffman (2006), The probability perturbation method: A new look at Bayesian inverse modeling, Math. Geol., 38(1), 81-100, doi:10.1007/s11004-005-9005-9. (Pubitemid 43899127)
13. Carrera, J., and S. P. Neuman (1986), Estimation of aquifer parameters under transient and steady state conditions: 1. Maximum likelihood method incorporating prior information, Water Resour. Res., 22(2), 199-210, doi:10.1029/WR022i002p00199. (Pubitemid 16477839)
14. Carrera, J., A. Alcolea, A. Medina, J. Hidalgo, and L. J. Slooten (2005), Inverse problem in hydrogeology, Hydrogeol. J., 13(1), 206-222, doi:10.1007/s10040-004-0404-7.
15. Chib, S., and E. Greenberg (1995), Understanding the Metropolis-Hastings algorithm, Am. Stat., 49(4), 327-335. (Pubitemid 126008969)
16. Christen, J. A., and C. Fox (2005), MCMC using an approximation, J. Comput. Graph. Stat., 14(4), 795-810.
17. Christensen, O. F., G. O. Roberts, and M. Skold (2006), Robust Markov chain Monte Carlo methods for spatial generalized linear mixed models, J. Comput. Graph. Stat., 15(1), 1-17. (Pubitemid 43514409)
18. Christie, M. A., and M. J. Blunt (2001), Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE Reservoir Eng. Eval., 4, 308-317.
19. Chu, L., A. C. Reynolds, and D. S. Oliver (1995), Computation of sensitivity coefficients for conditioning the permeability field to well-test pressure data, In Situ, 19(2), 179-223.
20. Cooley, R. L. (2000), An analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields, Water Resour. Res., 36(4), 1159-1163, doi:10.1029/2000WR900008. (Pubitemid 30181396)
21. Cressie, N. A. C. (1993), Statistics for Spatial Data, Wiley, New York.
22. Cui, T., C. Fox, and M. J. O'Sullivan (2011), Adaptive error modelling in MCMC sampling for large scale inverse problem, Tech. Rep. 686, Fac. of Eng., Univ. of Auckland, Auckland, New Zealand.
23. Deutsch, C. V., and A. G. Journel (1998), GSLIB, Geostatistical Software Library User's Guide, Oxford Univ. Press, New York.
24. Doherty, J. (2005), PEST: Software for Model-Independent Parameter Estimation, Watermark Numer. Comput., Brisbane, Australia.
25. Duan, Q., S. Sorooshian, and V. Gupta (1992), Effective and efficient global optimization for conceptual rainfall-runoff models, Water Resour. Res., 28(4), 1015-1031, doi:10.1029/91WR02985.
26. Duan, Q., N. K. Ajami, X. Gao, and S. Sorooshian (2007), Multi-model ensemble hydrologic prediction using Bayesian model averaging, Adv. Water Resour., 30(5), 1371-1386, doi:10.1016/j.advwatres.2006.11.014. (Pubitemid 46323445)
27. Durlofsky, L. J. (1998), Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques, Comput. Geosci., 2, 73-92.
28. Efendiev, Y., A. Datta-Gupta, V. Ginting, X. Ma, and B. Mallick (2005), An efficient two-stage Markov chain Monte Carlo method for dynamic data integration, Water Resour. Res., 41, W12423, doi:10.1029/2004WR003764. (Pubitemid 43151138)
29. Finsterle, S. (1993), ITOUGH2 User's Guide, Lawrence Berkeley Natl. Lab., Berkeley, Calif.
30. Flath, H. P., V. Akcelik, L. C. Wilcox, J. Hill, B. van Bloemen Waanders, and O. Ghattas (2011), Fast algorithms for Bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial Hessian approximations, SIAM J. Sci. Comput., 33, 407-432, doi:10.1137/090780717.
31. Fox, C. (2008), A conjugate sampler for normal distributions, with a few computed examples, Tech. Rep. 2008-1, pp. 1172-4234, Electron. Group, Univ. of Otago, Dunedin, New Zealand.
32. Fox, C., S. Tan, and G. K. Nicholls (2009), Inverse Problems, Lecture Notes of ELEC 404, Univ. of Otago, Dunedin, New Zealand.
33. Fu, J. L., and J. J. Gomez-Hernandez (2009), Uncertainty assessment and data worth in groundwater flow and mass transport modeling using a blocking Markov chain Monte Carlo method, J. Hydrol., 364, 328-341, doi:10.1016/j. jhydrol.2008.11.014.
34. Galarza, G. A., J. Carrera, and A. Medina (1999), Computational techniques for optimization of problems involving non-linear transient simulations, Int. J. Numer. Methods Eng., 45(3), 319-334.
35. Gelman, A., and D. B. Rubin (1992), Inference from iterative simulation using multiple sequences (with discussion), Stat. Sci., 7, 457-511.
36. Gelman, A., G. O. Roberts, and W. R. Gilks (1996), Efficient Metropolis jumping rules, in Bayesian Statistics, vol. 5, edited by J. O. Berger, J. M. Bernardo, A. P. Dawid, D. V. Lindley, and A. F. M. Smith, pp. 599-608, Oxford Univ. Press, Oxford, U. K.
37. Geweke, J. (1992), Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, in Bayesian Statistics 4, edited by J. M. Bernardo et al., pp. 169-193, Oxford Univ. Press, New York.
38. Gilks, W. R., G. O. Roberts, and E. I. George (1994), Adaptive direction sampling, J. R. Stat. Soc. Ser. D, 43(1), 179-189.
39. Gomez-Hernanez, J. J., A. Sahuquillo, and J. E. Capilla (1997), Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data - I. theory, J. Hydrol., 203, 162-174, doi:10.1016/S0022-1694 (97) 00098-X. (Pubitemid 28118236)
40. Goodman, J., and A. Sokal (1989), Multigrid Monte Carlo method: Conceptual foundations, Phys. Rev. D, 40(6), 2035-2071.
41. Grant, M. A., I. G. Donaldson, and P. F. Bixley (1982), Geothermal Reservoir Engineering, Acad. Press, New York.
42. Green, P. J. (1995), Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika, 82, 711-732.
43. Green, P. J., and A. Mira (2001), Delayed rejection in reversible jump Metropolis-Hastings, Biometrika, 88, 1035-1053. (Pubitemid 33771550)
44. Haario, H., E. Saksman, and J. Tamminen (2001), An adaptive Metropolis algorithm, Bernoulli, 7, 223-242. (Pubitemid 33793059)
45. Haario, H., M. Laine, A. Mira, and E. Saksman (2006), DRAM: Efficient adaptive MCMC, Stat. Comput., 16, 339-354. (Pubitemid 44581755)
46. Hadamard, J. (1902), Sur les problemes aux derivees partielles et leur signification physique, (on the problems of partial derivatives and their physical significance), Princeton Univ. Bull., 13, 49-52.
47. Haslett, J., M. Whiley, S. Bhattacharya, M. Salter-Townshend, S. P. Wilson, J. R. M. Allen, B. Huntley, and F. J. G. Mitchell (2006), Bayesian palaeoclimate reconstruction (with discussion), J. R. Stat. Soc. Ser. A, 169(3), 395-438.
48. Hastings, W. (1970), Monte Carlo sampling using Markov chains and their applications, Biometrika, 57, 97-109.
49. Hendricks-Franssen, H. J., A. Alcolea, M. Riva, M. Bakr, N. van der Wiel, F. Stauffer, and A. Guadagnini (2009), A comparison of seven methods for the inverse modelling of groundwater flow. Application to the characterisation of well catchments, Adv. Water Resour., 32(6), 851-872, doi:10.1016/j.advwatres. 2009.02.011.
50. Hernandez, A. F., S. P. Neuman, A. Guadagnini, and J. Carrera (2006), Inverse stochastic moment analysis of steady state flow in randomly heterogeneous media, Water Resour. Res., 42, W05425, doi:10.1029/2005WR004449. (Pubitemid 43955924)
51. Higdon, D., H. Lee, and C. Holloman (2003), Markov chain Monte Carlobased approaches for inference in computationally intensive inverse problems, in Bayesian Statistics 7, edited by J. M. Bernardo et al., pp. 181-197, Oxford Univ. Press, New York.
52. Hurn, M. A., O. Husby, and H. Rue (2003), "Advances in Bayesian image analysis", in Highly Structured Stochastic Systems, pp. 302-322, Oxford Univ. Press, New York.
53. Huttunen, J. M. J., and J. P. Kaipio (2007), Approximation error analysis in nonlinear state estimation with an application to state-space identification, Inverse Probl., 23(5), 2141, doi:10.1088/0266-5611/23/5/019. (Pubitemid 351453862)
54. International Formulation Committee (1967), A Formulation of the Thermodynamic Properties of Ordinary Water Substance, IFC Secretariat, Düsseldorf, Germany.
55. Jaynes, E. T. (1968), Prior probability, IEEE Trans. Syst. Sci. Cybernetics, 4(3), 227-241.
56. Kaipio, J., and E. Somersalo (2004), Statistical and Computational Inverse Problems, Springer, New York.
57. Kaipio, J. P., and E. Somersalo (2007), Statistical inverse problems: Discretization, model reduction and inverse crimes, J. Comput. Appl. Math., 198(2), 493-504. (Pubitemid 44402950)
58. Karpouzos, D. K., F. Delay, K. L. Katsifarakis, and G. De Marsily (2001), A multipopulation genetic algorithm to solve the inverse problem in hydrogeology, Water Resour. Res., 37(9), 2291-2302, doi:10.1029/2000WR900411. (Pubitemid 32804531)
59. Kavetski, D., G. Kuczera, and S. W. Franks (2006), Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory, Water Resour. Res., 42, W03407, doi:10.1029/2005WR004368.
60. Keating, E., J. Doherty, J. A. Vrugt, and Q. Kang (2010), Optimization and uncertainty assessment of strongly non-linear groundwater models with high parameter dimensionality, Water Resour. Res., 46, W10517, doi:10.1029/ 2009WR008584.
61. Kolehmainen, V., T. Tarvainen, S. R. Arridge, and J. P. Kaipio (2011), Marginalization of uninteresting distributed parameters in inverse problems, Int. J. Uncertainty Quant., 1, 1-17.
62. Kuczera, G., and E. Parent (1998), Monte Carlo assessment of parameter uncertainty in conceptual catchment models: The Metropolis algorithm, J. Hydrol., 211, 69-85, doi:10.1016/S0022-1694 (98) 00198-X. (Pubitemid 29014557)
63. Kuczera, G., D. Kavetski, S. Franks, and M. Thyer (2006), Towards a Bayesian total error analysis of conceptual rainfall-runoff models: Characterising model error using storm-dependent parameters, J. Hydrol., 331, 161-177, doi:10.1016/j.jhydrol.2006.05.010. (Pubitemid 44646456)
64. Lehikoinen, A., M. J. Huttunen, S. Finsterle, M. B. Kowalsky, and J. P. Kaipio (2010), Dynamic inversion for hydrological process monitoring with electrical resistance tomography under model uncertainties, Water Resour. Res., 46, W04513, doi:10.1029/2009WR008470.
65. Lieberman, C., K. Willcox, and O. Ghattas (2010), Parameter and state model reduction for large-scale statistical inverse problems, SIAM J. Sci. Comput., 32(5), 2523-2542.
66. Lipponen, A., A. Seppanen, and J. P. Kaipio (2010), Reduced-order estimation of nonstationary flows with electrical impedance tomography, Inverse Probl., 26(7), 074010, doi:10.1088/0266-5611/26/7/074010.
67. Liu, J. S. (2001), Monte Carlo Strategies in Scientific Computing, Springer, New York.
68. Liu, J. S., F. M. Liang, and W. H. Wong (2000), The use of multiple-try method and local optimization in Metropolis sampling, J. Am. Stat. Assoc., 95(449), 121-134.
69. Liu, X., M. A. Cardiff, and P. K. Kitanidis (2010), Parameter estimation in nonlinear environmental problems, Stochastic Environ. Res. Risk Assess., 24(7), 1003-1022, doi:10.1007/s00477-010-0395-y.
70. Liu, Y., and H. V. Gupta (2007), Uncertainty in hydrologic modeling: Toward an integrated data assimilation framework, Water Resour. Res., 43, W07401, doi:10.1029/2006WR005756. (Pubitemid 350194456)
71. MacKay, D. J. C. (2003), Information Theory, Inference, and Learning Algorithms, Cambridge Univ. Press, New York.
72. Mariethoz, G., P. Renard, and J. Caers (2010), Bayesian inverse problem and optimization with iterative spatial resampling, Water Resour. Res., 46, W11530, doi:10.1029/2010WR009274.
73. Marshall, L., D. Nott, and A. Sharma (2007), Towards dynamic catchment modelling: A Bayesian hierarchical mixtures of experts framework, Hydrol. Processes, 21(7), 847-861, doi:10.1002/hyp. 6294. (Pubitemid 46510578)
74. Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller (1953), Equation of state calculations by fast computing machines, J. Chem. Phys., 21, 1087-1092.
75. Mondal, A., Y. Efendiev, B. Mallick, and A. Datta-Gupta (2010), Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain Monte Carlo methods, Adv. Water Resour., 33(3), 241-256.
76. Neuman, S. P. (2003), Maximum likelihood Bayesian averaging of uncertain model predictions, Stochastic Environ. Res. Risk Assess., 17, 291-305. (Pubitemid 38013163)
77. Oliver, D. S., L. B. Cunha, and A. C. Reynolds (1997), Markov chain Monte Carlo methods for conditioning a permeability field to pressure data, Math. Geol., 29(1), 61-91, doi:10.1007/BF02769620.
78. O'Sullivan, M. J. (1985), Geothermal reservoir simulation, Int. J. Energy Res., 9(3), 319-332.
79. O'Sullivan, M. J., and R. McKibbin (1989), Geothermal Reservoir Engineering, Univ. of Auckland, Auckland, New Zealand.
80. Pruess, K. (1991), TOUGH2-A General-Purpose Numerical Simulator for Multiphase Fluid and Heat Flow, Lawrence Berkeley Natl. Lab., Berkeley, Calif.
81. Raftery, A. E., and S. M. Lewis (1992), One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo, Stat. Sci., 7, 493-497.
82. RamaRao, B., A. LaVenue, G. De Marsily, and M. Marietta (1995), Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields: 1. Theory and computational experiments, Water Resour. Res., 31(3), 475-493, doi:10.1029/94WR 02258. (Pubitemid 26437835)
83. Renard, B., D. Kavetski, G. Kuczera, M. Thyer, and S. W. Franks (2010), Understanding predictive uncertainty in hydrologic modeling: The challenge of identifying input and structural errors, Water Resour. Res., 46, W05521, doi:10.1029/2009WR008328.
84. Renard, P., and G. De Marsily (1997), Calculating equivalent permeability: A review, Adv. Water Resour., 20, 253-278.
85. Roberts, G. O. (1996), 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. 45-57, Chapman and Hall, New York.
86. Roberts, G. O., and J. S. Rosenthal (2001), Optimal scaling for various Metropolis-Hastings algorithms, Stat. Sci., 16, 351-367. (Pubitemid 33636356)
87. Roberts, G. O., and J. S. Rosenthal (2007), Coupling and ergodicity of adaptive MCMC, J. Appl. Prob., 44, 458-475.
88. Roberts, G. O., and J. S. Rosenthal (2009), Examples of adaptive MCMC, J. Comput. Graph. Stat., 18(2), 349-367.
89. Roberts, G. O., and S. K. Sahu (1997), Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler, J. R. Stat. Soc. Ser. B, 59(2), 291-317. (Pubitemid 127686955)
90. Roberts, G. O., A. Gelman, and W. R. Gilks (1997), Weak convergence and optimal scaling of random walk Metropolis algorithms, Ann. Appl. Prob., 7, 110-120.
91. Rue, H., and L. Held (2005), Gaussian Markov Random Fields: Theory and Applications, Chapman and Hall, New York.
92. Schoups, G., and J. A. Vrugt (2010), A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic and non-Gaussian errors, Water Resour. Res., 46, W10531, doi:10.1029/2009WR008933.
93. Schoups, G., J. A. Vrugt, F. Fenicia, and N. C. van de Giesen (2010), Corruption of accuracy and efficiency of Markov chain Monte Carlo simulation by inaccurate numerical implementation of conceptual hydrologic models, Water Resour. Res., 46, W10530, doi:10.1029/2009WR008648.
94. Simpson, D. P., I. W. Turner, and A. N. Pettitt (2008), Fast sampling from a Gaussian Markov random field using Krylov subspace approaches, QUT ePrints ID 14376, http://eprints.qut.edu.au/14376/, Queensland Univ. of Technology, Brisbane, Australia.
95. Smith, T. J., and L. A. Marshall (2008), Bayesian methods in hydrologic modeling: A study of recent advancements in Markov chain Monte Carlo techniques, Water Resour. Res., 44, W00B05, doi:10.1029/2007WR006705.
96. Sun, N.-Z., and W. W.-G. Yeh (1990), Coupled inverse problems in groundwater modeling: 1. Sensitivity analysis and parameter identification, Water Resour. Res., 26(10), 2507-2525, doi:10.1029/WR026i010p 02507.
97. Ter Braak, C. J. F. (2006), A Markov chain Monte Carlo version of the genetic algorithm differential evolution: easy Bayesian computing for real parameter spaces, Stat. Comput., 16(3), 239-249, doi:10.1007/s11222-006-8769-1. (Pubitemid 43992692)
98. Ter Braak, C. J. F., and J. A. Vrugt (2008), Differential evolution Markov chain with snooker updater and fewer chains, Stat. Comput., 18(4), 435-446, doi:10.1007/s11222-008-9104-9.
99. Thyer, M., B. Renard, D. Kavetski, G. Kuczera, S. Franks, and S. Srikanthan (2009), Critical evaluation of parameter consistency and predictive uncertainty in hydrological modelling: A case study using Bayesian total error analysis, Water Resour. Res., 45, W00B14, doi:10.1029/2008WR 006825.
100. Tierney, L., and A. Mira (1999), Some adaptive Monte Carlo methods for Bayesian inference, Stat. Med., 18, 2507-2515. (Pubitemid 29423512)
101. Ulrych, T. J., and M. D. Sacchi (2005), Information-Based Inversion and Processing with Applications, Handbook of Geophysical Exploration, Seismic Exploration, vol. 36, Elsevier, New York.
102. Ulrych, T. J., M. D. Sacchi, and A. D. Woodbury (2001), A Bayes tour of inversion: A tutorial, Geophysics, 60(1), 55-69. (Pubitemid 32281929)
103. Van Genuchten, M. T. (1980), A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, 892-898.
104. Vrugt, J. A., H. V. Gupta, W. Bouten, and S. Sorooshian (2003), A shuffled complex evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters, Water Resour. Res., 39(8), 1201, doi:10.1029/2002WR001642.
105. Vrugt, J. A., C. J. F. ter Braak, M. P. Clark, J. M. Hyman, and B. A. Robinson (2008), Treatment of input uncertainty in hydrologic modeling: doing hydrology backward with Markov chain Monte Carlo simulation, Water Resour. Res., 44, W00B09, doi:10.1029/2007WR006720.
106. Vrugt, J. A., C. J. F. ter Braak, C. G. H. Diks, D. Higdon, B. A. Robinson, and J. M. Hyman (2009), Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling, Int. J. Nonlin. Sci. Numer. Simul., 10(3), 273-290.
107. Waagepetersen, R., and D. Sorensen (2001), A tutorial on reversible jump MCMC with a view toward applications in qtl-mapping, Int. Stat. Rev., 69(1), 49-61. (Pubitemid 33386505)
108. Wolff, U. (2004), Monte Carlo errors with less errors, Comput. Phys. Commun., 156(2), 143-153, doi:10.1016/S0010-4655 (03) 00467-3.
109. Woodbury, A. D., and T. J. Ulrych (2000), A full-Bayesian approach to the groundwater inverse problem for steady-state flow, Water Resour. Res., 36(8), 2081-2093, doi:10.1029/2000WR900086. (Pubitemid 30610899)
110. Yapo, P. O., H. V. Gupta, and S. Sorooshian (1998), Multi-objective global optimization for hydrologic models, J. Hydrol., 204, 83-97, doi:10.1016/S0022-1694 (97) 00107-8.
111. Yeh, W. W.-G. (1986), Review of parameter identification procedures in groundwater hydrology: The inverse problem, Water Resour. Res., 22(2), 95-108, doi:10.1029/WR022i002p00095.
112. Zimmerman, D. A., et al. (1998), A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow, Water Resour. Res., 34(6), 1373-1413, doi:10.1029/98WR00003. (Pubitemid 28253515)