# Uncertainty Quantification of Complex System Models: Bayesian Analysis

## Abstract

This chapter summarizes the main elements of Bayesian probability theory to help reconcile dynamic environmental system models with observations, including prediction in space (interpolation), prediction in time (forecasting), assimilation of data, and inference of the model parameters. Special attention is given to the treatment of parameter uncertainty (first-order approximations and Bayesian intervals), the prior distribution, the formulation of the likelihood function (using first-principles), the marginal likelihood, and sampling techniques used to estimate the posterior target distribution. This includes rejection sampling, importance sampling, and recent developments in Markov chain Monte Carlo simulation to sample efficiently complex and/or high-dimensional target distributions, including limits of acceptability. We illustrate the application of Bayes’ theorem and inference using three illustrative examples involving the flow and storage of water in the surface and subsurface. At least some level of calibration of these models is required to match their output with observations of system behavior and response. Algorithmic recipes of the different methods are provided to simplify implementation and use of Bayesian analysis.

## Keywords

Hypothesis testing Bayesian analysis Prior distribution Likelihood function Posterior distribution Monte Carlo sampling Markov chain Monte Carlo simulation Data assimilation Hydrologic modeling## Notes

### Acknowledgments

The first author is supported by funding from the UC-Lab Fees Research Program Award 237285. The material presented in this chapter is part of the first author’s graduate course on “Merging Models and Data” (CEE-290) taught at the University of California, Irvine. An animated presentation of this material can be found online at https://www.youtube.com/watch?v=bhA9vtiHxZ0. The DREAM family of algorithms discussed in this chapter are implemented in DREAM Suite, an easy to use, plug-and-play, Windows program. This program can be found online at www.dreamsuite.eu and simplifies considerably Bayesian analysis and its application to uncertainty quantification of mathematical models.

## References

- B.C. Bates, E.P. Campbell, A Markov chain Monte Carlo scheme for parameter estimation and inference in conceptual rainfall-runoff modeling. Water Resour. Res.
**37**(4), 937–947 (2001)CrossRefGoogle Scholar - T. Bayes, R. Price, An essay towards solving a problem in the doctrine of chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, A.M.F.R.S. Philos. Trans. R. Soc. Lond.
**53**(0), 370–418 (1763). https://doi.org/10.1098/rstl.1763.0053CrossRefGoogle Scholar - J.O. Berger,
*Statistical Decision Theory and Bayesian Analysis*(Springer, New York, 1985)CrossRefGoogle Scholar - J.O. Berger, J.M. Bernardo, D. Sun, The formal definition of reference priors. Ann. Stat.
**37**(2), 905–938 (2009). https://doi.org/10.1214/07-AOS587CrossRefGoogle Scholar - J.M. Bernardo, Reference posterior distributions for Bayesian inference (with discussion). J. R. Stat. Soc. Ser. B
**41**, 113–147 (1979)Google Scholar - K. Beven, A manifesto for the equifinality thesis. J. Hydrol.
**320**(1), 18–36 (2006)CrossRefGoogle Scholar - K.J. Beven, A.M. Binley, The future of distributed models: Model calibration and uncertainty prediction. Hydrol. Process.
**6**, 279–298 (1992)CrossRefGoogle Scholar - K.J. Beven, A.M. Binley, GLUE: 20 years on. Hydrol. Process.
**28**, 5879–5918 (2014). https://doi.org/10.1002/hyp.10082CrossRefGoogle Scholar - S. Blazkova, K.J. Beven, A limits of acceptability approach to model evaluation and uncertainty estimation in flood frequency estimation by continuous simulation: Skalka catchment, Czech Republic. Water Resour. Res.
**45**, W00B16 (2009). https://doi.org/10.1029/2007WR006726CrossRefGoogle Scholar - G.E.P. Box, G.C. Tiao,
*Bayesian Inference in Statistical Analysis*(Wiley, New York, 1992), 588 ppCrossRefGoogle Scholar - S.P. Brooks, A. Gelman, General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat.
**7**, 434–455 (1998)Google Scholar - M. Clark, D. Kavetski, F. Fenicia, Pursuing the method of multiple working hypotheses for hydrological modeling. Water Resour. Res.
**47**(9), 1–16 (2011). https://doi.org/10.1029/2010WR009827CrossRefGoogle Scholar - S. Dean, J.E. Freer, K.J. Beven, A.J. Wade, D. Butterfield, Uncertainty assessment of a process-based integrated catchment model of phosphorus (INCA-P). Stoch. Env. Res. Risk A.
**23**, 991–1010 (2009). https://doi.org/10.1007/s00477-008-0273-zCrossRefGoogle Scholar - Q. Duan, S. Sorooshian, V. Gupta, Effective and efficient global optimization for conceptual rainfall-runoff models. Water Resour. Res.
**28**(4), 1015–1031 (1992)CrossRefGoogle Scholar - G. Evin, D. Kavetski, M. Thyer, G. Kuczera, Pitfalls and improvements in the joint inference of heteroscedasticity and autocorrelation in hydrological model calibration. Water Resour. Res.
**49**, 4518–4524 (2013). https://doi.org/10.1002/wrcr.20284CrossRefGoogle Scholar - C. Fernandez, M.J.F. Steel, On Bayesian modeling of fat tails and skewness. J. Am. Stat. Assoc.
**93**, 359–371 (1998)Google Scholar - J. Freer, H. McMillan, J.J. McDonnell, K.J. Beven, Constraining dynamic TOPMODEL responses for imprecise water table information using fuzzy rule based performance measures. J. Hydrol.
**291**, 254–277 (2004)CrossRefGoogle Scholar - A.G. Gelman, D.B. Rubin, Inference from iterative simulation using multiple sequences. Stat. Sci.
**7**, 457–472 (1992)CrossRefGoogle Scholar - A.G. Gelman, G.O. Roberts, W.R. Gilks,
*Bayesian Statistics*(Oxford University Press, Oxford, 1996), pp. 599–608Google Scholar - J. Geweke, Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, in
*Bayesian Statistics 4*, ed. by J.M. Bernardo, J.O. Berger, A.P. Dawid, A.F.M. Smith (Oxford Oxford University Press, 1992), pp. 169–193Google Scholar - W.R. Gilks, G.O. Roberts, Strategies for improving MCMC, in
*Markov Chain Monte Carlo in Practice*, ed. by W.R. Gilks, S. Richardson, D.J. Spiegelhalter (Chapman & Hall, London, 1996), pp. 89–114Google Scholar - W.R. Gilks, G.O. Roberts, E.I. George, Adaptive direction sampling. Underst. Stat.
**43**, 179–189 (1994)Google Scholar - H.V. Gupta, T. Wagener, Y. Liu, Reconciling theory with observations: Elements of a diagnostic approach to model evaluation. Hydrol. Process.
**22**(18), 3802–3813 (2008)CrossRefGoogle Scholar - H. Haario, E. Saksman, J. Tamminen, Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat.
**14**, 375–395 (1999)CrossRefGoogle Scholar - H. Haario, E. Saksman, J. Tamminen, An adaptive Metropolis algorithm. Bernoulli
**7**, 223–242 (2001)CrossRefGoogle Scholar - H. Haario, E. Saksman, J. Tamminen, Componentwise adaptation for high dimensional MCMC. Stat. Comput.
**20**, 265–274 (2005)CrossRefGoogle Scholar - H. Haario, M. Laine, A. Mira, E. Saksman, DRAM: Efficient adaptive MCMC. Stat. Comput.
**16**, 339–354 (2006)CrossRefGoogle Scholar - H. Hastings, Monte Carlo sampling methods using Markov chains and their applications. Biometrika
**57**, 97–109 (1970)CrossRefGoogle Scholar - T.J. Heimovaara, W. Bouten, A computer-controlled 36-channel time domain reflectometry system for monitoring soil water contents. Water Resour. Res.
**26**, 2311–2316 (1990). https://doi.org/10.1029/WR026i010p02311CrossRefGoogle Scholar - J. Hoeting, D. Madigan, A. Raftery, C. Volinsky, Bayesian model averaging: A tutorial. Stat. Sci.
**14**(4), 382–417 (1999)CrossRefGoogle Scholar - H. Jeffreys, An invariant form for the prior probability in estimation problems. Proc. R. Soc. Lond. A Math. Phys. Sci.
**186**(1007), 453–461 (1946). https://doi.org/10.1098/rspa.1946.0056CrossRefGoogle Scholar - D. Kavetski, G. Kuczera, S.W. Franks, Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory. Water Resour. Res.
**42**(3), W03407 (2006a). https://doi.org/10.1029/2005WR004368CrossRefGoogle Scholar - D. Kavetski, G. Kuczera, S.W. Franks, Bayesian analysis of input uncertainty in hydrological modeling: 2. Application. Water Resour. Res.
**42**(3), W03408 (2006b). https://doi.org/10.1029/2005WR004376CrossRefGoogle Scholar - K. Keesman, Membership-set estimation using random scanning and principal component analysis. Math. Comput. Simul.
**32**, 535–543 (1990)CrossRefGoogle Scholar - T. Krueger, J.N. Quinton, J. Freer, C.J. Macleod, G.S. Bilotta, R.E. Brazier, P.M. Haygarth, Uncertainties in data and models to describe event dynamics of agricultural sediment and phosphorus transfer. J. Environ. Qual.
**38**(3), 1137–1148 (2009)CrossRefGoogle Scholar - G. Kuczera, Improved parameter inference in catchment models, 1. Evaluating parameter uncertainty. Water Resour. Res.
**19**(5), 1151–1162 (1983). https://doi.org/10.1029/WR019i005p01151CrossRefGoogle Scholar - G. Kuczera, D. Kavetski, S. Franks, M. Thyer, Towards a Bayesian total error analysis of conceptual rainfall-runoff models: Characterising model error using storm-dependent parameters. J. Hydrol.
**331**(1), 161–177 (2006)CrossRefGoogle Scholar - E. Laloy, J.A. Vrugt, High-dimensional posterior exploration of hydrologic models using multiple-try DREAM
_{(ZS)}and high-performance computing. Water Resour. Res.**48**, W01526 (2012). https://doi.org/10.1029/2011WR010608CrossRefGoogle Scholar - J.S. Liu, F. Liang, W.H. Wong, The multiple-try method and local optimization in metropolis sampling. J. Am. Stat. Assoc.
**95**(449), 121–134 (2000). https://doi.org/10.2307/2669532CrossRefGoogle Scholar - Y. Liu, J.E. Freer, K.J. Beven, P. Matgen, Towards a limits of acceptability approach to the calibration of hydrological models: Extending observation error. J. Hydrol.
**367**, 93–103 (2009). https://doi.org/10.1016/j.jhydrol.2009.01.016CrossRefGoogle Scholar - H. McMillan, J. Freer, F. Pappenberger, T. Krueger, M. Clark, Impacts of uncertain river flow data on rainfall-runoff model calibration and discharge predictions. Hydrol. Process.
**24**(10), 1270–1284 (2010)Google Scholar - N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys.
**21**, 1087–1092 (1953)CrossRefGoogle Scholar - J.E. Nash, A unit hydrograph study with particular reference to British catchments. Proc. Inst. Civ. Eng.
**17**, 249–282 (1960)Google Scholar - J.E. Nash, J.V. Sutcliffe, River flow forecasting through conceptual models part I – A discussion of principles. J. Hydrol.
**10**(3), 282–290 (1970)CrossRefGoogle Scholar - T. Page, K.J. Beven, J. Freer, A. Jenkins, Investigating the uncertaintyin predicting responses to atmospheric deposition using the model of acidification of groundwater in catchments (MAGIC) within a generalised likelihood uncertainty estimation (GLUE) framework. Water Soil Air Pollut.
**142**, 71–94 (2003)CrossRefGoogle Scholar - T. Page, K.J. Beven, D. Whyatt, Predictive capability in estimating changes in water quality: Long-term responses to atmospheric deposition. Water Soil Air Pollut.
**151**, 215–244 (2004)CrossRefGoogle Scholar - T. Page, K.J. Beven, J. Freer, Modelling the chloride signal at the Plynlimon catchments, Wales using a modified dynamic TOPMODEL. Hydrol. Process.
**21**, 292–307 (2007)CrossRefGoogle Scholar - F. Pappenberger, K. Beven, M. Horritt, S. Blazkova, Uncertainty in the calibration of effective roughness parameters in HEC-RAS using inundation and downstream level observations. J. Hydrol.
**302**, 46–69 (2005)CrossRefGoogle Scholar - F. Pappenberger, K. Frodsham, K.J. Beven, R. Romanovicz, P. Matgen, Fuzzy set approach to calibrating distributed flood inundation models using remote sensing observations. Hydrol. Earth Syst. Sci.
**11**(2), 739–752 (2007)CrossRefGoogle Scholar - K.V. Price, R.M. Storn, J.A. Lampinen,
*Differential Evolution, A Practical Approach to Global Optimization*(Springer, Berlin, 2005)Google Scholar - V.C. Radu, J. Rosenthal, C. Yang, Learn from the thy neighbor: Parallel-chain and regional adaptive MCMC. J. Am. Stat. Assoc.
**104**(488), 1454–1466 (2009)CrossRefGoogle Scholar - A.E. Raftery, S.M. Lewis, One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo. Stat. Sci.
**7**, 493–497 (1992)CrossRefGoogle Scholar - A.E. Raftery, T. Gneiting, F. Balabdaoui, M. Polakowski, Using Bayesian model averaging to calibrate forecast ensembles. Mon. Weather Rev.
**133**, 1155–1174 (2005)CrossRefGoogle Scholar - P. Reichert, J. Mieleitner, Analyzing input and structural uncertainty of nonlinear dynamic models with stochastic, timeâĂŘdependent parameters. Water Resour. Res.
**45**, W10402 (2009). https://doi.org/10.1029/2009WR007814CrossRefGoogle Scholar - B. Renard, D. Kavetski, E. Leblois, M. Thyer, G. Kuczera, S.W. Franks, Toward a reliable decomposition of predictive uncertainty in hydrological modeling: Characterizing rainfall errors using conditional simulation. Water Resour. Res.
**47**(11), W11516 (2011). https://doi.org/10.1029/2011WR010643CrossRefGoogle Scholar - C.P. Roberts, G. Casella,
*Monte Carlo Statistical Methods*, 2nd edn. (Springer, New York, 2004)CrossRefGoogle Scholar - G.O. Roberts, W.R. Gilks, Convergence of adaptive direction sampling. J. Multivar. Anal.
**49**, 287–298 (1994)CrossRefGoogle Scholar - G.O. Roberts, J.S. Rosenthal, Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab.
**44**, 458–475 (2007)CrossRefGoogle Scholar - G.O. Roberts, A. Gelman, W.R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab.
**7**, 110–120 (1997)CrossRefGoogle Scholar - M. Sadegh, J.A. Vrugt, Approximate Bayesian computation using Markov chain Monte Carlo simulation: DREAM
_{(ABC)}. Water Resour. Res.**50**(2014). https://doi.org/10.1002/2014WR015386CrossRefGoogle Scholar - M. Sadegh, J.A. Vrugt, C. Xu, E. Volpi, The stationarity paradigm revisited: Hypothesis testing using diagnostics, summary metrics, and DREAM
_{(ABC)}. Water Resour. Res.**51**, 9207–9231 (2015). https://doi.org/10.1002/2014WR016805CrossRefGoogle Scholar - M.G. Schaap, F.J. Leij, M.T. van Genuchten, Neural network analysis for hierarchical prediction of soil water retention and saturated hydraulic conductivity. Soil Sci. Soc. Am. J.
**62**, 847–855 (1998)CrossRefGoogle Scholar - M.G. Schaap, F.J. Leij, M.T. van Genuchten, Rosetta: A computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. J. Hydrol.
**251**, 163–176 (2001)CrossRefGoogle Scholar - B. Scharnagl, J.A. Vrugt, H. Vereecken, M. Herbst, Bayesian inverse modeling of soil water dynamics at the field scale: Using prior information about the soil hydraulic properties. Hydrol. Earth Syst. Sci.
**15**, 3043–3059 (2011). https://doi.org/10.5194/hess-15-3043-2011CrossRefGoogle Scholar - B. Scharnagl, S.C. Iden, W. Durner, H. Vereecken, M. Herbst, Inverse modelling of in situ soil water dynamics: Accounting for heteroscedastic, autocorrelated, and non-Gaussian distributed residuals. Hydrol. Earth Syst. Sci. Discuss.
**12**, 2155–2199 (2015)CrossRefGoogle Scholar - A. Schöniger, T. Wöhling, L. Samaniego, W. Nowak, Model selection on solid ground: Rigorous comparison of nine ways to evaluate Bayesian model evidence. Water Resour. Res.
**50**(12), W10530, 9484–9513 (2014). https://doi.org/10.1002/2014WR016062CrossRefGoogle Scholar - G. Schoups, J.A. Vrugt, A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic and non-Gaussian errors. Water Resour. Res.
**46**, W10531 (2010). https://doi.org/10.1029/2009WR008933CrossRefGoogle Scholar - J. Šimůnek, M. Šejna, H. Saito, M. Sakai, M.T. van Genuchten,
*The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat and Multiple Solutes in Variably-Saturated Media (Version 4.0)*(Department of Environmental Sciences, University of California Riverside, Riverside, 2008)Google Scholar - T. Smith, A. Sharma, L. Marshall, R. Mehrotra, S. Sisson, Development of a formal likelihood function for improved Bayesian inference of ephemeral catchments. Water Resour. Res.
**46**, W12551 (2010). https://doi.org/10.1029/2010WR009514CrossRefGoogle Scholar - S. Sorooshian, J.A. Dracup, Stochastic parameter estimation procedures for hydrologic rainfall-runoff models: Correlated and heteroscedastic error cases. Water Resour. Res.
**16**(2), 430–442 (1980)CrossRefGoogle Scholar - S.M. Stigler, Who discovered Bayes’s theorem? Am. Stat.
**37**(4 Part 1), 290–296 (1983)CrossRefGoogle Scholar - R. Storn, K. Price, A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**11**, 341–359 (1997)CrossRefGoogle Scholar - C.J.F. ter Braak, A Markov chain Monte Carlo version of the genetic algorithm differential evolution: Easy Bayesian computing for real parameter spaces. Stat. Comput.
**16**, 239–249 (2006)CrossRefGoogle Scholar - C.J.F. ter Braak, J.A. Vrugt, Differential evolution Markov chain with snooker updater and fewer chains. Stat. Comput.
**18**(4), 435–446 (2008). https://doi.org/10.1007/s11222-008-9104-9CrossRefGoogle Scholar - M. Thiemann, M. Trosset, H. Gupta, S. Sorooshian, Bayesian recursive parameter estimation for hydrologic models. Water Resour. Res.
**37**(10), 2521–2535 (2001)CrossRefGoogle Scholar - G.C. Topp, J.L. Davis, A.P. Annan, Electromagnetic determination of soil water content: Measurements in coaxial transmission lines. Water Resour. Res.
**16**, 574–582 (1980). https://doi.org/10.1029/WR016i003p00574CrossRefGoogle Scholar - M.T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**44**(5), 892–898 (1980). https://doi.org/10.2136/sssaj1980.03615995004400050002xCrossRefGoogle Scholar - E. Volpi, G. Schoups, G. Firmani, J.A. Vrugt, Sworn testimony of the model evidence: Gaussian Mixture Importance (GAME) sampling. Water Resour. Res.
**53**, 6133–6158 (2017). https://doi.org/10.1002/2016WR020167CrossRefGoogle Scholar - J.A. Vrugt, Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation. Environ. Model. Softw.
**75**, 273–316 (2016). https://doi.org/10.1016/j.envsoft.2015.08.013CrossRefGoogle Scholar - J.A. Vrugt, K.J. Beven, Embracing equifinality with efficiency: Limits of acceptability sampling using the DREAM
_{(LOA)}algorithm. J. Hydrol.**559**, 954–971 (2018). https://doi.org/10.1016/j.jhydrol.2018.02.026, In PressCrossRefGoogle Scholar - J.A. Vrugt, E. Laloy, Reply to comment by Chu et al. on High-dimensional posterior exploration of hydrologic models using multiple-try DREAM
_{t}*ext*(*ZS*) and high-performance computing. Water Resour. Res.**50**, 2781–2786 (2014). https://doi.org/10.1002/2013WR014425CrossRefGoogle Scholar - J.A. Vrugt, B.A. Robinson, Treatment of uncertainty using ensemble methods: Comparison of sequential data assimilation and Bayesian model averaging. Water Resour. Res.
**43**, W01411 (2007). https://doi.org/10.1029/2005WR004838CrossRefGoogle Scholar - J.A. Vrugt, M. Sadegh, Toward diagnostic model calibration and evaluation: Approximate Bayesian computation. Water Resour. Res.
**49**(2013). https://doi.org/10.1002/wrcr.20354CrossRefGoogle Scholar - J.A. Vrugt, C.J.F. ter Braak, DREAM
_{(D)}: An adaptive Markov chain Monte Carlo simulation algorithm to solve discrete, noncontinuous, and combinatorial posterior parameter estimation problems. Hydrol. Earth Syst. Sci.**15**, 3701–3713 (2011). https://doi.org/10.5194/hess-15-3701-2011CrossRefGoogle Scholar - J.A. Vrugt, H.V. Gupta, W. Bouten, S. Sorooshian, A Shuffled Complex Evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters. Water Resour. Res.
**39**(8), 1201 (2003). https://doi.org/10.1029/2002WR001642CrossRefGoogle Scholar - J.A. Vrugt, C.G.H. Diks, W. Bouten, H.V. Gupta, J.M. Verstraten, Improved treatment of uncertainty in hydrologic modeling: Combining the strengths of global optimization and data assimilation. Water Resour. Res.
**41**(1), W01017 (2005). https://doi.org/10.1029/2004WR003059CrossRefGoogle Scholar - J.A. Vrugt, C.J.F. ter Braak, M.P. Clark, J.M. Hyman, B.A. Robinson, Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation. Water Resour. Res.
**44**, W00B09 (2008). https://doi.org/10.1029/2007WR006720CrossRefGoogle Scholar - J.A. Vrugt, C.J.F. ter Braak, C.G.H. Diks, D. Higdon, B.A. Robinson, J.M. Hyman, Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int. J. Nonlinear Sci. Numer. Simul
**10**(3), 273–290 (2009)CrossRefGoogle Scholar - L. Wasserman, Bayesian model selection and model averaging. J. Math. Psychol.
**44**(1), 92–107 (2000). https://doi.org/10.1006/jmps.1999.1278CrossRefGoogle Scholar - I.K. Westerberg, J.-L. Guerrero, P.M. Younger, K.J. Beven, J. Seibert, S. Halldin, J.E. Freer, C.-Y. Xu, Calibration of hydrological models using flow-duration curves. Hydrol. Earth Syst. Sci.
**15**, 2205–2227 (2011). https://doi.org/10.5194/hess-15-2205-2011CrossRefGoogle Scholar - J. Yang, P. Reichert, K.C. Abbaspour, Bayesian uncertainty analysis in distributed hydrologic modeling: A case study in the Thur River basin (Switzerland). Water Resour. Res.
**43**, W10401 (2007). https://doi.org/10.1029/2006WR005497CrossRefGoogle Scholar - M. Ye, P. Meyer, S.P. Neuman, On model selection criteria in multimodel analysis. Water Resour. Res.
**44**, 1–12 (2008). https://doi.org/10.1029/2008WR006803CrossRefGoogle Scholar - S.L. Zabell, The rule of succession. Erkenntnis
**31**(2–3), 283–321 (1989)CrossRefGoogle Scholar