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A review of Markov chain Monte Carlo and information theory tools for inverse problems in subsurface flow. (English) Zbl 1254.49021

Summary: Parameter identification is one of the key elements in the construction of models in geosciences. However, inherent difficulties such as the instability of ill-posed problems or the presence of multiple local optima may impede the execution of this task. Regularization methods and Bayesian formulations, such as the maximum a posteriori estimation approach, have been used to overcome those complications. Nevertheless, in some instances, a more in-depth analysis of the inverse problem is advisable before obtaining estimates of the optimal parameters. The Markov Chain Monte Carlo (MCMC) methods used in Bayesian inference have been applied in the last 10 years in several fields of geosciences such as hydrology, geophysics or reservoir engineering. In the present paper, a compilation of basic tools for inference and a case study illustrating the practical application of them are given. Firstly, an introduction to the Bayesian approach to the inverse problem is provided together with the most common sampling algorithms with MCMC chains. Secondly, a series of estimators for quantities of interest, such as the marginal densities or the normalization constant of the posterior distribution of the parameters, are reviewed. Those reduce the computational cost significantly, using only the time needed to obtain a sample of the posterior probability density function. The use of the information theory principles for the experimental design and for the ill-posedness diagnosis is also introduced. Finally, a case study based on a highly instrumented well test found in the literature is presented. The results obtained are compared with the ones computed by the maximum likelihood estimation approach.

MSC:

49N45 Inverse problems in optimal control
93E12 Identification in stochastic control theory
65C05 Monte Carlo methods
62F15 Bayesian inference
62B10 Statistical aspects of information-theoretic topics
62G07 Density estimation
62P12 Applications of statistics to environmental and related topics
62P30 Applications of statistics in engineering and industry; control charts

Software:

PEST; RFortran; QSIMVN
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carrera, J., Alcolea, A., Medina, A., Hidalgo, J., Slooten, L.J.: Inverse problem in hydrogeology. Hydrogeol. J. 13(1), 206–222 (2005) · doi:10.1007/s10040-004-0404-7
[2] Burnham, K.P., Anderson, D.R.: Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach, 2nd edn. Springer, New York (2002) · Zbl 1005.62007
[3] Fisher, R.A.: On the mathematical foundations of theoretical statistics. Philos. trans. R. Soc. Lond., A 222, 309–368 (1922) · JFM 48.1280.02 · doi:10.1098/rsta.1922.0009
[4] Akaike, H.: A new look at the statistical model identification. IEEE Trans. Automat. Contr. AC-19(6), 716–723 (1974) · Zbl 0314.62039 · doi:10.1109/TAC.1974.1100705
[5] de Leeuw, J.: Information theory and an extension of the maximum likelihood principle by Hirotogu Akaike. In: Kotz, S., Johnson, N.L. (eds.) Breakthroughs in Statistics, pp. 599–609. Springer, Berlin (1990)
[6] Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978) · Zbl 0379.62005 · doi:10.1214/aos/1176344136
[7] Rissanen, J.: Stochastic complexity. J. R. Stat. Soc., B 49(3), 223–239 (1987) · Zbl 0654.62008
[8] Kashyap, R.L.: Optimal choice of Ar and Ma parts in augoregressive moving average models. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4(2), 99–104 (1982) · Zbl 0514.62009 · doi:10.1109/TPAMI.1982.4767213
[9] Young, P., Wallis, S.: Recursive estimation: a unified approach to the identification estimation, and forecasting of hydrological systems. Appl. Math. Comput. 17(4), 299–334 (1985) · Zbl 0583.93058 · doi:10.1016/0096-3003(85)90039-6
[10] Young, P.C., Beven, K.J.: Data-based mechanistic modelling and the rainfall-flow non-linearity. Environmetrics 5(3), 335–363 (1994) · doi:10.1002/env.3170050311
[11] Neuman, S.P., Carrera, J.: Maximum-likelihood adjoint-state finite-element estimation of groundwater parameters under steady- and nonsteady-state conditions. Appl. Math. Comput. 17(4), 405–432 (1985) · doi:10.1016/0096-3003(85)90043-8
[12] Noakes, D.J., McLeod, A.I., Hipel, K.W.: Forecasting monthly riverflow time series. Int. J. Forecast. 1(2), 179–190 (1985) · doi:10.1016/0169-2070(85)90022-6
[13] Ledesma, A., Gens, A., Alonso, E.E.: Parameter and variance estimation in geotechnical backanalysis using prior information. Int. J. Numer. Anal. Methods Geomech. 20(2), 119–141 (1996) · Zbl 0860.73050 · doi:10.1002/(SICI)1096-9853(199602)20:2<119::AID-NAG810>3.0.CO;2-L
[14] Di Baldassarre, G., Laio, F., Montanari, A.: Design flood estimation using model selection criteria. Phys. Chem. Earth, A/B/C 34(10–12), 606–611 (2009) · doi:10.1016/j.pce.2008.10.066
[15] Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer, New York (1985) · Zbl 0572.62008
[16] Diciccio, T.J., Kass, R.E., Raftery, A., Wasserman, L.: Computing Bayes factors by combining simulation and asymptotic approximations. J. Am. Stat. Assoc. 92(439), 903–915 (1997) · Zbl 1050.62520 · doi:10.1080/01621459.1997.10474045
[17] Kass, R.E., Raftery, A.E.: Bayes factors. J. Am. Stat. Assoc. 90(430), 773–895 (1995) · Zbl 0846.62028 · doi:10.1080/01621459.1995.10476572
[18] Perreault, L., Bernier, J., Bobée, B., Parent, E.: Bayesian change-point analysis in hydrometeorological time series. Part 2. Comparison of change-point models and forecasting. J. Hydrol. 235(3–4), 242–263 (2000) · Zbl 0961.62109 · doi:10.1016/S0022-1694(00)00271-7
[19] Poeter, E., Anderson, D.: Multimodel ranking and inference in ground water modeling. Ground Water 43(4), 597–605 (2005) · doi:10.1111/j.1745-6584.2005.0061.x
[20] Marshall, L., Nott, D., Sharma, A.: Hydrological model selection: a Bayesian alternative. Water Resour. Res. 41(10), W10422 (2005)
[21] Frost, A.J., Thyer, M.A., Srikanthan, R., Kuczera, G.: A general Bayesian framework for calibrating and evaluating stochastic models of annual multi-site hydrological data. J. Hydrol. 340(3–4), 129–148 (2007) · doi:10.1016/j.jhydrol.2007.03.023
[22] Neuman, S.P.: Maximum likelihood Bayesian averaging of uncertain model predictions. Stoch. Environ. Res. Risk Assess. 17(5), 291–305 (2003) · Zbl 1039.00514 · doi:10.1007/s00477-003-0151-7
[23] Ajami, N.K., Duan, Q., Sorooshian, S.: An integrated hydrologic Bayesian multimodel combination framework: confronting input, parameter, and model structural uncertainty in hydrologic prediction. Water Resour. Res. 43(1), W01403 (2007)
[24] Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: I. Maximum likelihood method incorporating prior information. Water Resour. Res. 22(2), 199–210 (1986) · doi:10.1029/WR022i002p00199
[25] Carrera, J., Neuman, S.P.: Estimation of aquifer parameters under transient and steady state conditions: 2. Uniqueness, stability, and solution algorithms. Water Resour. Res. 22(2), 211–227 (1986) · doi:10.1029/WR022i002p00211
[26] McLaughlin, D., Townley, L.R.: A reassessment of the groundwater inverse problem. Water Resour. Res. 32(5), 1131–1161 (1996) · doi:10.1029/96WR00160
[27] Young, P.: Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Comput. Phys. Commun. 117(1–2), 113–129 (1999) · doi:10.1016/S0010-4655(98)00168-4
[28] Young, P.C., McKenna, P., Bruun, J.: Identification of non-linear stochastic systems by state dependent parameter estimation. Int. J. Control 74(18), 1837–1857 (2001) · Zbl 1023.93064 · doi:10.1080/00207170110089824
[29] Young, P.C.: The data-based mechanistic approach to the modelling, forecasting and control of environmental systems. Annu. Rev. Control 30(2), 169–182 (2006) · doi:10.1016/j.arcontrol.2006.05.002
[30] Kuczera, G., Kavetski, D., Franks, S., Thyer, M.: Towards a Bayesian total error analysis of conceptual rainfall-runoff models: characterising model error using storm-dependent parameters. J. Hydrol. 331(1–2), 161–177 (2006) · doi:10.1016/j.jhydrol.2006.05.010
[31] Romanowicz, R.J., Young, P.C., Beven, K.J.: Data assimilation and adaptive forecasting of water levels in the river Severn catchment, United Kingdom. Water Resour. Res. 42(6), W06407 (2006)
[32] Reichert, P., Mieleitner, J.: Analyzing input and structural uncertainty of nonlinear dynamic models with stochastic, time-dependent parameters. Water Resour. Res. 45(10), W10402 (2009)
[33] Aldrich, J.: R. A. Fisher and the making of maximum likelihood 1912–1922. Stat. Sci. 12(3), 162–176 (1997) · Zbl 0955.62525 · doi:10.1214/ss/1030037906
[34] Edwards, A.W.F.: Likelihood, Expanded edn. Johns Hopkins University Press, Baltimore (1992)
[35] Kay, S.M.: Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Prentice Hall, Upper Saddle River (1993) · Zbl 0803.62002
[36] Yeh, T.-C.J., Simunek, J.: Stochastic fusion of information for characterizing and monitoring the vadose zone. Vadose Zone J. 1(2), 207–221 (2002)
[37] Yeh, T.C.J., Lee, C.H., Hsu, K.C., Tan, Y.C.: Fusion of active and passive hydrologic and geophysical tomographic surveys: the future of subsurface characterization. Data integration in subsurface hydrology. Geophys. Monogr. 171, 109–120 (2007)
[38] Renard, B., Kavetski, D., Kuczera, G., Thyer, M., Franks, S.W.: Understanding predictive uncertainty in hydrologic modeling: the challenge of identifying input and structural errors. Water Resour. Res. 46(5), W05521 (2010)
[39] Gelfand, A.E., Sahu, S.K.: Identifiability, improper priors and Gibbs sampling for generalized linear models. J. Am. Stat. Assoc. 94(445), 247–253 (1999) · Zbl 1072.62611 · doi:10.1080/01621459.1999.10473840
[40] Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, New York (1977) · Zbl 0354.65028
[41] Aster, R.C., Borchers, B., Thurber, C.H.: Parameter Estimation and Inverse Problems. Academic Press, San Diego (2005) · Zbl 1088.35081
[42] Howie, D.: Interpreting Probability: Controversies and Developments in the Early Twentieth Century. Cambridge Studies in Probability, Induction and Decision Theory. Cambridge University Press, Cambridge (2002) · Zbl 1031.01012
[43] Jaynes, E.T., Bretthorst, G.L.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003)
[44] Jeffreys, H.: Theory of Probability, 3rd edn. Oxford Classic Texts in the Physical Sciences. Oxford University Press, Oxford (1998)
[45] Jaynes, E.T.: How does the brain do plausible reasoning? In: Erickson, G.J., Smith, C.R. (eds.) Maximum-Entropy and Bayesian Methods in Science and Engineering, vol. 1, pp. 1–24. Kluwer, Dordrecht (1988)
[46] Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics, Philadelphia, USA (2005) · Zbl 1074.65013
[47] Tarantola, A., Valette, B.: Generalized nonlinear inverse problems solved using the least squares criterion. Rev. Geophys. Space Phys. 20(2), 219–232 (1982) · doi:10.1029/RG020i002p00219
[48] Hanna, S., Yeh, T.C.J.: Estimation of co-conditional moments of transmissivity, hydraulic head, and velocity fields. Adv. Water Resour. 22(1), 87–95 (1998) · doi:10.1016/S0309-1708(97)00033-X
[49] Lehman, E.L., Casella, G.: Theory of Point Estimation, 2nd edn. Springer, New York (1998) · Zbl 0916.62017
[50] Woodbury, A., Sudicky, E., Ulrych, T.J., Ludwig, R.: Three-dimensional plume source reconstruction using minimum relative entropy inversion. J. Contam. Hydrol. 32(1–2), 131–158 (1998) · doi:10.1016/S0169-7722(97)00088-0
[51] Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, Cambridge (2008)
[52] Vogel, C.R.: Computational methods for inverse problems. Frontiers in Applied Mathematics. SIAM, University City Science, Center, PATraduce (2002) · Zbl 1008.65103
[53] Zhdanov, M.: Geophysical inverse theory and regularization problems, vol. 36. Methods in Geochemistry and Geophysics. Elsevier, Amsterdam (2002)
[54] Mosegaard, K., Tarantola, A.: Probabilistic approach to inverse problems. In: Lee, W.H.K., Kanamori, H., Jennings, P., Kisslinger, C. (eds.) International Handbook of Earthquake and Engineering Seismology, vol. A, pp. 237–265. Academic Press, London (2002)
[55] Skahill, B.E., Doherty, J.: Efficient accommodation of local minima in watershed model calibration. J. Hydrol. 329(1–2), 122–139 (2006) · doi:10.1016/j.jhydrol.2006.02.005
[56] Kavetski, D., Clark, M.P.: Ancient numerical Daemons of conceptual hydrological modeling: 2. Impact of time stepping schemes on model analysis and prediction. Water Resour. Res. 46(10), W10511 (2010)
[57] Beven, K., Binley, A.: The future of distributed models: model calibration and uncertainty prediction. Hydrol. Process. 6(3), 279–298 (1992) · doi:10.1002/hyp.3360060305
[58] Box, G.E.P., Tiao, G.C.: Bayesian Inference in Statistical Analysis, Reprint edn. Wiley, New York (1992) · Zbl 0850.62004
[59] Ulrych, T.J., Sacchi, M.D., Woodbury, A.: A Bayes tour of inversion: a tutorial. Geophysics 66(1), 55–69 (2001) · doi:10.1190/1.1444923
[60] Hadamard, J.: Sur les problèmes aux dérivées partielles et leur signification physique, pp. 49–52. Princeton University Bulletin (1902)
[61] Bennet, J.H. (ed.) R. A. Fisher. Statistical Inference and Analysis. Selected correspondence of R.A. Fisher. Oxford University Press, Oxford (1990)
[62] Kavetski, D., Kuczera, G., Franks, S.W.: Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory - art. no. W03407. Water Resour. Res. 42(3), 3407–3407 (2006)
[63] Kavetski, D., Kuczera, G., Franks, S.W.: Bayesian analysis of input uncertainty in hydrological modeling: 2. Application - art. no. W03408. Water Resour. Res. 42(3), 3408–3408 (2006)
[64] Bretthorst, G.L.: An Introduction to Parameter-Estimation Using Bayesian Probability-Theory. In: Fougère, P.F. (ed.) Maximum Entropy and Bayesian Methods ///, vol. 39. Fundamental Theories of Physics, pp. 53–79. Kluwer, Dordrecht (1990) · Zbl 0735.62033
[65] Amar, J.G.: The Monte Carlo method in science and engineering. Comput. Sci. Eng. 8(2), 9–19 (2006) · Zbl 05091882 · doi:10.1109/MCSE.2006.34
[66] Beichl, I., Sullivan, F.: Monte Carlo methods. Comput. Sci. Eng. 8(2), 7–8 (2006) · Zbl 0779.58024 · doi:10.1109/MCSE.2006.27
[67] Sorensen, D.: Likelihood. Bayesian and MCMC Methods in Genetics. Springer, New York (2002) · Zbl 1013.62105
[68] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953) · doi:10.1063/1.1699114
[69] Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970) · Zbl 0219.65008 · doi:10.1093/biomet/57.1.97
[70] Kuczera, G., Parent, E.: Monte Carlo assessment of parameter uncertainty in conceptual catchment models: the Metropolis algorithm. J. Hydrol. 211(1–4), 69–85 (1998) · doi:10.1016/S0022-1694(98)00198-X
[71] Vrugt, J.A., Bouten, W., Gupta, H.V., Sorooshian, S.: Toward improved identifiability of hydrologic model parameters: the information content of experimental data. Water Resour. Res. 38(12), 481–4813 (2002)
[72] Fu, J., Jaime Gómez-Hernández, J.: Uncertainty assessment and data worth in groundwater flow and mass transport modeling using a blocking Markov Chain Monte Carlo method. J. Hydrol. 364(3–4), 328–341 (2009) · doi:10.1016/j.jhydrol.2008.11.014
[73] Chen, J., Hubbard, S., Rubin, Y., Murray, C., Roden, E., Majer, E.: Geochemical characterization using geophysical data and Markov Chain Monte Carlo methods: a case study at the South Oyster bacterial transport site in Virginia. Water Resour. Res. 40(12), 1–14 (2004)
[74] Chen, J., Hubbard, S., Peterson, J., Williams, K., Fienen, M., Jardine, P., Watson, D.: Development of a joint hydrogeophysical inversion approach and application to a contaminated fractured aquifer. Water Resour. Res. 42(6), W06425 (2006)
[75] Fu, J., Gómez-Hernández, J.J.: A blocking Markov Chain Monte Carlo method for inverse stochastic hydrogeological modeling. Math. Geosci. 41(2), 105–128 (2009) · Zbl 1205.86044 · doi:10.1007/s11004-008-9206-0
[76] Oliver, D.S., Cunha, L.B., Reynolds, A.C.: Markov Chain Monte Carlo methods for conditioning a permeability field to pressure data. Math. Geol. 29(1), 61–91 (1997) · doi:10.1007/BF02769620
[77] Efendiev, Y., Datta-Gupta, A., Ginting, V., Ma, X., Mallick, B.: An efficient two-stage Markov Chain Monte Carlo method for dynamic data integration. Water Resour. Res. 41(12), 1–6 (2005)
[78] Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6(6), 721–741 (1984) · Zbl 0573.62030 · doi:10.1109/TPAMI.1984.4767596
[79] Gelfand, A.E., Hills, S.E., Racine-Poon, A., Smith, A.F.M.: Illustration of Bayesian inference in normal data models using gibbs sampling. J. Am. Stat. Assoc. 85(412), 972–985 (1990) · doi:10.1080/01621459.1990.10474968
[80] Kuczera, G., Kavetski, D., Renard, B., Thyer, M.: A limited-memory acceleration strategy for MCMC sampling in hierarchical Bayesian calibration of hydrological models. Water Resour. Res. 46(7), W07602 (2010)
[81] Baragona, R., Battaglia, F., Poli, I.: Evolutionary Statistical Procedures. Statistics and Computing. Springer, Heidelberg (2011) · Zbl 1378.62005
[82] Drugan, M.M., Thierens, D.: Evolutionary Markov Chain Monte Carlo. Lect. Notes Comput. Sci. 2936, 63–76 (2004) · Zbl 1098.68650 · doi:10.1007/978-3-540-24621-3_6
[83] Ter Braak, C.J.F.: 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 (2006) · doi:10.1007/s11222-006-8769-1
[84] Vrugt, J.A., Ter Braak, C.J.F., Diks, C.G.H., Robinson, B.A., Hyman, J.M., Higdon, D.: Accelerating Markov Chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int. J. of Nonlinear Sci. &amp; Numer. Simul. 10(3), 273–290 (2009) · Zbl 06942400
[85] Vrugt, J.A., ter Braak, C.J.F., Gupta, H.V., Robinson, B.A.: Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling? Stoch. Environ. Res. Risk Assess. 23(7), 1011–1026 (2009) · doi:10.1007/s00477-008-0274-y
[86] Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2011) · Zbl 1209.86001 · doi:10.1007/s10596-010-9194-2
[87] Brooks, S.P., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7(4), 434–455 (1998)
[88] Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7(4), 457–472 (1992) · Zbl 1386.65060 · doi:10.1214/ss/1177011136
[89] Sorensen, D.A., Andersen, S., Gianola, D., Korsgaard, I.: Bayesian inference in threshold models using Gibbs sampling. Genet. Sel. Evol. 27, 229–249 (1995) · doi:10.1186/1297-9686-27-3-229
[90] Cowles, M.K., Carlin, B.P.: Markov Chain Monte Carlo convergence diagnostics: a comparative review. J. Am. Stat. Assoc. 91(434), 883–904 (1996) · Zbl 0869.62066 · doi:10.1080/01621459.1996.10476956
[91] Chen, M.H., Shao, Q.M., Ibrahim, J.G.: Monte Carlo Methods in Bayesian Computation. Springer Series in Statistics, Springer, New York (2000) · Zbl 0949.65005
[92] Gamerman, D., Freitas Lopes, H.: Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Chapman and Hall/CRC, Boca Raton (2006)
[93] Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov Chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28(2), 776–803 (2006) · Zbl 1111.65003 · doi:10.1137/050628568
[94] Ginting, V., Pereira, F., Presho, M., Wo, S.: Application of the two-stage Markov Chain Monte Carlo method for characterization of fractured reservoirs using a surrogate flow model. Comput. Geosci. 1–17 (2011). doi: 10.1007/s10596-011-9236-4 · Zbl 1237.76190
[95] Emerick, A.A., Reynolds, A.C.: Combining the Ensemble Kalman Filter with Markov Chain Monte Carlo for Improved History Matching and Uncertainty Characterization. Paper presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 21–23 February (2011)
[96] Geyer, C.J.: Practical Markov Chain Monte Carlo. Stat. Sci. 7(4), 473–511 (1992) · Zbl 0085.18501 · doi:10.1214/ss/1177011137
[97] Gelfand, A.E., Smith, A.F.M., Lee, T.M.: Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling. J. Am. Stat. Assoc. 87, 523–532 (1992) · doi:10.1080/01621459.1992.10475235
[98] Chen, M.-H.: Importance-weighted marginal Bayesian posterior density estimation. J. Am. Stat. Assoc. 89(427), 818–824 (1994) · Zbl 0804.62040 · doi:10.1080/01621459.1994.10476815
[99] Scott, D.W.: Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley Series in Probability and Statistics. Wiley, New York (1992) · Zbl 0850.62006
[100] Scott, D.W., Sain, S.R.: Multi-dimensional density estimation. In: Rao, C.R., Wegman, E.J., Solka, J. (eds.) Handbook of Statistics: 24 Data Mining and Computational Statistics, Data Mining and Computational Statistics, vol. 24, pp. 229–262. Elsevier, Amsterdam (2004)
[101] Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London (1986) · Zbl 0617.62042
[102] Botev, Z.I., Grotowski, J.F., Kroese, D.P.: Kernel density estimation via diffusion. Ann. Stat. 38(5), 2916–2957 (2010) · Zbl 1200.62029 · doi:10.1214/10-AOS799
[103] Hoeting, J.A., Madigan, D., Raftery, A.E., Volinsky, C.T.: Bayesian model averaging: a tutorial. Stat. Sci. 14(4), 382–417 (1999) · Zbl 1059.62525 · doi:10.1214/ss/1009212519
[104] Stewart, L.: Hierarchical bayesian analysis using Monte Carlo integration: computing posterior distributions when there are many possible models. Statistician 36(2/3), 211–219 (1987) · doi:10.2307/2348514
[105] Hammersley, J.M., Handscomb, D.C.: Monte Carlo Methods. Methuen’s Monographs on Applied Probability and Statistics. Methuen &amp; Co. LTD., London, UK (1964) · Zbl 0121.35503
[106] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran 77. The Art of Scientific Computing, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1992) · Zbl 0778.65002
[107] Chen, M.H.: Computing marginal likelihoods from a single MCMC output. Stat. Neerl. 59(1), 16–29 (2005) · Zbl 1069.62023 · doi:10.1111/j.1467-9574.2005.00276.x
[108] Kavetski, D.: Introduction to Bayesian inference using standard and hierarchical models. Paper presented at the Systemanalyse und Modellierung, Bayesian Uncertainty Analysis, Viena, Austria (2008)
[109] Thyer, M., Leonard, M., Kavetski, D., Need, S., Renard, B.: The open source RFortran library for accessing R from Fortran, with applications in environmental modelling. Environ. Model. Softw. 26(2), 219–234 (2011) · doi:10.1016/j.envsoft.2010.05.007
[110] Chambers, J.M., Cleveland, W.S., Kleiner, B., Tukey, P.A.: Graphical Methods for Data Analysis. Statistics/Probability Series. Wadsworth &amp; Brooks/Cole, Pacific Grove, California, USA (1983) · Zbl 0532.65094
[111] Finsterle, S., Faybishenko, B.: Inverse modeling of a radial multistep outflow experiment for determining unsaturated hydraulic properties. Adv. Water Resour. 22(5), 431–444 (1999) · doi:10.1016/S0309-1708(98)00030-X
[112] De Pauw, D.J.W., Sin, G., Insel, G., Van Hulle, S.W.H., Vandenberghe, V., Vanrolleghem, P.A.: Discussion of ”Assessing parameter identifiability of activated sludge model number 1” by Pedro Afonso and Maria da Conceição Cunha. J. Environ. Eng. 130(1), 110–112 (2004) · doi:10.1061/(ASCE)0733-9372(2004)130:1(110)
[113] Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948) · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x
[114] Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951) · Zbl 0042.38403 · doi:10.1214/aoms/1177729694
[115] Thyer, M., Frost, A.J., Kuczera, G.: Parameter estimation and model identification for stochastic models of annual hydrological data: is the observed record long enough? J. Hydrol. 330(1–2), 313–328 (2006) · doi:10.1016/j.jhydrol.2006.03.029
[116] Gelfand, A.E., Dey, D.K.: Bayesian model choice: asymptotics and exact calculations. J. R. Stat. Soc., B 56(3), 501–514 (1994) · Zbl 0800.62170
[117] Abellan, A., Noetinger, B.: Optimizing subsurface field data acquisition using information theory. Math. Geosci. 42(6), 603–630 (2010) · Zbl 1197.86049 · doi:10.1007/s11004-010-9285-6
[118] Wu, C.M., Yeh, T.C.J., Zhu, J.F., Lee, T.H., Hsu, N.S., Chen, C.H., Sancho, A.F.: Traditional analysis of aquifer tests: comparing apples to oranges? Water Resour. Res. 41(9), W09402 (2005)
[119] Moench, A.F., Garabedian, S.P., LeBlanc, D.R.: Estimation of hydraulic parameters from an unconfined aquifer test conducted in a glacial outwash deposit, Cape Cod, Massachusetts. In: US Geological Survey Professional Paper, vol. 1629, pp. 1–51. US Geological Survey, Reston, Virginia (2001)
[120] Moench, A.F.: Estimation of hectare-scale soil-moisture characteristics from aquifer-test data. J. Hydrol. 281(1–2), 82–95 (2003) · doi:10.1016/S0022-1694(03)00202-6
[121] Moench, A.F.: Importance of the vadose zone in analyses of unconfined aquifer tests. Ground Water 42(2), 223–233 (2004) · doi:10.1111/j.1745-6584.2004.tb02669.x
[122] Moench, A.F.: Combining the Neuman and Boulton models for flow to a well in an unconfined aquifer. Ground Water 33(3), 378–384 (1995) · doi:10.1111/j.1745-6584.1995.tb00293.x
[123] Moench, A.F.: Flow to a well of finite diameter in a homogeneous, anisotropic water table aquifer. Water Resour. Res. 33(6), 1397–1407 (1997) · doi:10.1029/97WR00651
[124] Leblanc, D.R.: Sewage plume in a sand and gravel aquifer, Cape Cod, Massachusetts. In: U.S. Geological Survey, Open-File Report 84-475, p. 28. U.S. Geological Survey, Washington (1984)
[125] Leblanc, D.R., Guswa, J.H., Frimpter, M.H., Londquist, C.J.: Ground-water resources of Cape Cod, Massachusetts. In: U.S. Geological Survey, Hydrologic Atlas 692, p. 4. U.S. Geological Survey, Reston, Virginia (1986)
[126] Masterson, J.P., Stone, B.D., Walters, D.A., Savoie, J.: Hydrogeologic framework of western Cape Cod, Massachusetts. In: U.S. Geological Survey, Hydrologic Atlas 741, p. 1. U.S. Geological Survey, Reston, Virginia (1997)
[127] Doherty, J., Brebber, L., Whyte, P.: PEST: Model Independent Parameter Estimation, p. 249. Watermark Numerical Computing (1994)
[128] Genz, A.: Numerical computation of multivariate normal probabilities. J. Comput. Graph. Stat. 1(1), 141–149 (1992)
[129] Tanner, M.A.: Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions. Springer Series in Statistics. Springer, New York (1996) · Zbl 0846.62001
[130] Chen, M.H., Shao, Q.M.: Monte carlo estimation of bayesian credible and HPD interval. J. Comput. Graph. Stat. 8(1), 69–92 (1999)
[131] Boonstra, J., Kselik, R.A.L.: SATEM 2002: Software for Aquifer Test Evaluation. Publication 57. p. 148. International Institute for Land Reclamation and Improvement. Wageningen, The Netherlands (2001)
[132] Massey, F.J.: The Kolmogorov–Smirnov test for goodness of fit. J. Am. Stat. Assoc. 46(253), 68–78 (1951) · Zbl 0042.14403 · doi:10.1080/01621459.1951.10500769
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