Skip to main content
SpringerLink
Log in

Hydrological post-processing based on approximate Bayesian computation (ABC)

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

This study introduces a method to quantify the conditional predictive uncertainty in hydrological post-processing contexts when it is cumbersome to calculate the likelihood (intractable likelihood). Sometimes, it can be difficult to calculate the likelihood itself in hydrological modelling, specially working with complex models or with ungauged catchments. Therefore, we propose the ABC post-processor that exchanges the requirement of calculating the likelihood function by the use of some sufficient summary statistics and synthetic datasets. The aim is to show that the conditional predictive distribution is qualitatively similar produced by the exact predictive (MCMC post-processor) or the approximate predictive (ABC post-processor). We also use MCMC post-processor as a benchmark to make results more comparable with the proposed method. We test the ABC post-processor in two scenarios: (1) the Aipe catchment with tropical climate and a spatially-lumped hydrological model (Colombia) and (2) the Oria catchment with oceanic climate and a spatially-distributed hydrological model (Spain). The main finding of the study is that the approximate (ABC post-processor) conditional predictive uncertainty is almost equivalent to the exact predictive (MCMC post-processor) in both scenarios.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Beaumont MA, Zhang W, Balding DJ (2002) Approximate Bayesian computation in population genetics. Genetics 162(4):2025–2035

    Google Scholar 

  • Blackwell D, Dubins L (1962) Merging of opinions with increasing information. Ann Math Stat 33(3):882–886

    Article  Google Scholar 

  • Bogner K, Liechti K, Zappa M (2016) Post-processing of stream flows in Switzerland with an emphasis on low flows and floods. Water 8(4):115

    Article  Google Scholar 

  • Brown JD, Seo D-J (2010) A nonparametric postprocessor for bias correction of hydrometeorological and hydrologic ensemble forecasts. J Hydrometeorol 11(3):642–665

    Article  Google Scholar 

  • Butts MB, Payne JT, Kristensen M, Madsen H (2004) An evaluation of the impact of model structure on hydrological modelling uncertainty for streamflow simulation. J Hydrol 298(1):242–266

    Article  Google Scholar 

  • Coccia G, Todini E (2011) Recent developments in predictive uncertainty assessment based on the model conditional processor approach. Hydrol Earth Syst Sci 15:3253–3274

    Article  Google Scholar 

  • Csillery K, Francois O, Blum MGB (2012) abc: an R package for approximate Bayesian computation (abc). Methods Ecol Evol 3:475–479

    Article  Google Scholar 

  • Diaconis P, Freedman D (1986) On the consistency of bayes estimates. Ann Stat 14(1):1–26

    Article  Google Scholar 

  • Diks CGH, Vrugt JA (2010) Comparison of point forecast accuracy of model averaging methods in hydrologic applications. Stoch Environ Res Risk Assess 24(6):809–820

    Article  Google Scholar 

  • Drovandi CC, Pettitt AN (2011) Likelihood-free Bayesian estimation of multivariate quantile distributions. Comput Stat Data Anal 55(9):2541–2556

    Article  Google Scholar 

  • Evin G, Thyer M, Kavetski D, McInerney D, Kuczera G (2014) Comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity. Water Resour Res 50(3):2350–2375

    Article  Google Scholar 

  • Fearnhead P, Prangle D (2012) Constructing summary statistics for approximate bayesian computation: semi-automatic approximate Bayesian computation. J R Stat Soc Ser B Stat Methodol 74(3):419–474

    Article  Google Scholar 

  • Fenicia F, Kavetski D, Reichert P, Albert C (2018) Signature-domain calibration of hydrological models using approximate Bayesian computation: empirical analysis of fundamental properties. Water Resour Res 54:3958–3987

    Article  Google Scholar 

  • Francés F, Vélez JI, Vélez JJ (2007) Split-parameter structure for the automatic calibration of distributed hydrological models. J Hydrol 332(1):226–240

    Article  Google Scholar 

  • Frazier DT, Maneesoonthorn W, Martin GM, McCabe BP (2019) Approximate Bayesian forecasting. Int J Forecast 35(2):521–539

    Article  Google Scholar 

  • Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472

    Article  Google Scholar 

  • Gelman A, Stern HS, Carlin JB, Dunson DB, Vehtari A, Rubin DB (2013) Bayesian data analysis. Chapman and Hall/CRC, Boca Raton

    Google Scholar 

  • Glahn HR, Lowry DA (1972) The use of model output statistics (mos) in objective weather forecasting. J Appl Meteorol 11(8):1203–1211

    Article  Google Scholar 

  • Gupta HV, Kling H, Yilmaz KK, Martinez GF (2009) Decomposition of the mean squared error and nse performance criteria: implications for improving hydrological modelling. J Hydrol 377(1):80–91

    Article  Google Scholar 

  • Haario H, Saksman E, Tamminen J (2001) An adaptive metropolis algorithm. Bernoulli 7(2):223–242

    Article  Google Scholar 

  • Kavetski D, Fenicia F, Reichert P, Albert C (2018) Signature-domain calibration of hydrological models using approximate Bayesian computation: theory and comparison to existing applications. Water Resour Res 54:4059–4083

    Article  Google Scholar 

  • Khajehei S, Moradkhani H (2017) Towards an improved ensemble precipitation forecast: a probabilistic post-processing approach. J Hydrol 546:476–489

    Article  Google Scholar 

  • Klein B, Meissner D, Kobialka H-U, Reggiani P (2016) Predictive uncertainty estimation of hydrological multi-model ensembles using pair-copula construction. Water 8(4):125

    Article  Google Scholar 

  • Krzysztofowicz R, Kelly KS (2000) Hydrologic uncertainty processor for probabilistic river stage forecasting. Water Resour Res 36(11):3265–3277

    Article  Google Scholar 

  • Laio F, Tamea S (2007) Verification tools for probabilistic forecasts of continuous hydrological variables. Hydrol Earth Syst Sci 11(4):1267–1277

    Article  Google Scholar 

  • Li B, Liang Z, He Y, Hu L, Zhao W, Acharya K (2017) Comparison of parameter uncertainty analysis techniques for a topmodel application. Stoch Environ Res Risk Assess 31(5):1045–1059

    Article  Google Scholar 

  • Liang Z, Chang W, Li B (2012) Bayesian flood frequency analysis in the light of model and parameter uncertainties. Stoch Environ Res Risk Assess 26(5):721–730

    Article  Google Scholar 

  • Lindley DV, Smith AFM (1972) Bayes estimates for the linear model. J R Stat Soc Ser B Methodol 34(1):1–41

    Google Scholar 

  • Liu Y, Gupta HV (2007) Uncertainty in hydrologic modeling: toward an integrated data assimilation framework. Water Resour Res 43(7):W07401

    Google Scholar 

  • Madadgar S, Moradkhani H (2014) Improved Bayesian multimodeling: integration of copulas and Bayesian model averaging. Water Resour Res 50(12):9586–9603

    Article  Google Scholar 

  • Marin J-M, Pudlo P, Robert CP, Ryder RJ (2012) Approximate Bayesian computational methods. Stat Comput 22(6):1167–1180

    Article  Google Scholar 

  • Marjoram P, Molitor J, Plagnol V, Tavaré S (2003) Markov chain monte carlo without likelihoods. Proc Natl Acad Sci 100(26):15324–15328

    Article  CAS  Google Scholar 

  • Marshall L, Nott D, Sharma A (2004) A comparative study of Markov chain Monte Carlo methods for conceptual rainfall-runoff modeling. Water Resour Res 40(2):W02501

    Article  Google Scholar 

  • Mengersen KL, Pudlo P, Robert CP (2013) Bayesian computation via empirical likelihood. Proc Natl Acad Sci 110(4):1321–1326

    Article  CAS  Google Scholar 

  • Montanari A, Brath A (2004) A stochastic approach for assessing the uncertainty of rainfall-runoff simulations. Water Resour Res 40:W01106. https://doi.org/10.1029/2003WR002540

    Article  Google Scholar 

  • Montanari A, Grossi G (2008) Estimating the uncertainty of hydrological forecasts: a statistical approach. Water Resour Res 44:W00B08. https://doi.org/10.1029/2008WR006897

    Article  Google Scholar 

  • Montanari A, Koutsoyiannis D (2012) A blueprint for process-based modeling of uncertain hydrological systems. Water Resour Res 48(9):W09555

    Article  Google Scholar 

  • Moriasi DN, Arnold JG, Van Liew MW, Bingner RL, Harmel RD, Veith TL (2007) Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans ASABE 50(3):885–900

    Article  Google Scholar 

  • Nott DJ, Marshall L, Brown J (2011) Generalized likelihood uncertainty estimation (glue) and approximate Bayesian computation: what’s the connection? Water Resour Res 48(12):W12602

    Google Scholar 

  • Price LF, Drovandi CC, Lee A, Nott DJ (2018) Bayesian synthetic likelihood. J Comput Graph Stat 27(1):1–11

    Article  Google Scholar 

  • Pritchard JK, Seielstad MT, Perez-Lezaun A, Feldman MW (1999) Population growth of human y chromosomes: a study of y chromosome microsatellites. Mol Biol Evol 16(12):1791–1798

    Article  CAS  Google Scholar 

  • R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria

  • Raftery AE, Gneiting T, Balabdaoui F, Polakowski M (2005) Using Bayesian model averaging to calibrate forecast ensembles. Mon Weather Rev 133(5):1155–1174

    Article  Google Scholar 

  • Reichert P, Langhans SD, Lienert J, Schuwirth N (2015) The conceptual foundation of environmental decision support. J Environ Manag 154:316–332

    Article  Google Scholar 

  • Robert CP (2016) Approximate bayesian computation: A survey on recent results. In: Cools R, Nuyens D (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer, Cham, pp 185–205

    Chapter  Google Scholar 

  • Romero-Cuéllar J, Buitrago-Vargas A, Quintero-Ruiz T, Francés F (2018) Modelling the potential impacts of climate change on the hydrology of the Aipe river basin in Huila, Colombia. Ribagua 5(1):63–78

    Article  Google Scholar 

  • Schefzik R, Thorarinsdottir TL, Gneiting T (2013) Uncertainty quantification in complex simulation models using ensemble copula coupling. Stat Sci 28(4):616–640

    Article  Google Scholar 

  • Schoups G, Vrugt JA (2010) A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non-Gaussian errors. Water Resour Res 46(10):W10531

    Google Scholar 

  • Schoups G, van de Giesen NC, Savenije HHG (2008) Model complexity control for hydrologic prediction. Water Resour Res 44(12):W00B03

    Article  Google Scholar 

  • Shafii M, Tolson B, Matott LS (2014) Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study. Stoch Environ Res Risk Assess 28(6):1493–1510

    Article  Google Scholar 

  • Sikorska AE, Montanari A, Koutsoyiannis D (2015) Estimating the uncertainty of hydrological predictions through data-driven resampling techniques. J Hydrol Eng 20(1):A4014009

    Article  Google Scholar 

  • Sisson SA, Fan Y, Tanaka MM (2007) Sequential Monte Carlo without likelihoods. Proc Natl Acad Sci 104(6):1760–1765

    Article  CAS  Google Scholar 

  • Solomatine DP, Shrestha DL (2009) A novel method to estimate model uncertainty using machine learning techniques. Water Resour Res 45:W00B11. https://doi.org/10.1029/2008WR006839

    Article  Google Scholar 

  • Tavaré S, Balding DJ, Griffiths RC, Donnelly P (1997) Inferring coalescence times from DNA sequence data. Genetics 145(2):505–518

    Google Scholar 

  • Thomas H (1981) Improved methods for national water assessment, water resources contract: WR15249270. Technical report, Harvard University, Cambridge

  • Thyer M, Renard B, Kavetski D, Kuczera G, Franks SW, Srikanthan S (2009) Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: a case study using Bayesian total error analysis. Water Resour Res 45:W00B14. https://doi.org/10.1029/2008WR006825

    Article  Google Scholar 

  • Tian Y, Nearing GS, Peters-Lidard CD, Harrison KW, Tang L (2016) Performance metrics, error modeling, and uncertainty quantification. Mon Weather Rev 144(2):607–613

    Article  Google Scholar 

  • Todini E (2008) A model conditional processor to assess predictive uncertainty in flood forecasting. Int J River Basin Manag 6(2):123–137

    Article  Google Scholar 

  • Tran M-N, Nott DJ, Kohn R (2017) Variational bayes with intractable likelihood. J Comput Graph Stat 26(4):873–882

    Article  Google Scholar 

  • Turner BM, Van Zandt T (2012) A tutorial on approximate Bayesian computation. J Math Psychol 56(2):69–85

    Article  Google Scholar 

  • van Oijen M (2017) Bayesian methods for quantifying and reducing uncertainty and error in forest models. Curr For Rep 3(4):269–280

    Google Scholar 

  • Vélez JJ, Puricelli M, López Unzu F, Francés F (2009) Parameter extrapolation to ungauged basins with a hydrological distributed model in a regional framework. Hydrol Earth Syst Sci 13(2):229–246

    Article  Google Scholar 

  • Vrugt JA, Robinson BA (2007) Treatment of uncertainty using ensemble methods: comparison of sequential data assimilation and Bayesian model averaging. Water Resour Res 43(1):W01411

    Article  Google Scholar 

  • Vrugt JA, Sadegh M (2013) Toward diagnostic model calibration and evaluation: approximate Bayesian computation. Water Resour Res 49:4335–4345

    Article  Google Scholar 

  • Waerden BVD (1953) Order tests for the two-sample problem and their power. Indag Math Proc 56:80

    Article  Google Scholar 

  • Wagener T, Gupta HV (2005) Model identification for hydrological forecasting under uncertainty. Stoch Environ Res Risk Assess 19(6):378–387

    Article  Google Scholar 

  • Wang Q, Robertson D, Chiew FS (2009) A bayesian joint probability modeling approach for seasonal forecasting of streamflows at multiple sites. Water Resour Res 45(5):W05407

    Article  Google Scholar 

  • Weerts AH, Winsemius HC, Verkade JS (2011) Estimation of predictive hydrological uncertainty using quantile regression: examples from the national flood forecasting system (england and wales). Hydrol Earth Syst Sci 15(1):255–265

    Article  Google Scholar 

  • Wentao L, Qingyun D, Chiyuan M, Aizhong Y, Wei G, Zhenhua D (2017) A review on statistical postprocessing methods for hydrometeorological ensemble forecasting. Wiley Interdiscip Rev Water 4(6):e1246

    Article  Google Scholar 

  • Wilby RL, Harris I (2006) A framework for assessing uncertainties in climate change impacts: low-flow scenarios for the river thames, UK. Water Resour Res 42(2):W02419

    Article  Google Scholar 

  • Woldemeskel F, McInerney D, Lerat J, Thyer M, Kavetski D, Shin D, Tuteja N, Kuczera G (2018) Evaluating post-processing approaches for monthly and seasonal streamflow forecasts. Hydrol Earth Syst Sci 22:6257–6278. https://doi.org/10.5194/hess-22-6257-2018

    Article  Google Scholar 

  • Ye A, Duan Q, Yuan X, Wood EF, Schaake J (2014) Hydrologic post-processing of MOPEX streamflow simulations. J Hydrol 508:147–156

    Article  Google Scholar 

  • Yoon S, Cho W, Heo J-H, Kim CE (2010) A full bayesian approach to generalized maximum likelihood estimation of generalized extreme value distribution. Stoch Environ Res Risk Assess 24(5):761–770

    Article  Google Scholar 

  • Zhang X, Zhao K (2012) Bayesian neural networks for uncertainty analysis of hydrologic modeling: a comparison of two schemes. Water Resour Manag 26(8):2365–2382

    Article  Google Scholar 

  • Zhao L, Duan Q, Schaake J, Ye A, Xia J (2011) A hydrologic post-processor for ensemble streamflow predictions. Adv Geosci 29:51–59

    Article  Google Scholar 

  • Zhu W, Marin JM, Leisen F (2016) A bootstrap likelihood approach to Bayesian computation. Aust N Z J Stat 58(2):227–244

    Article  Google Scholar 

Download references

Acknowledgements

This study was partially supported by the Departamento del Huila Scholarship Program No. 677 (Colombia) and Colciencias, by the Spanish Research Project TETIS-MED (ref. CGL2014-58127-C3-3-R) and TETIS-CHANGE (ref. RTI2018-093717-B-I00). Also, G. Adelfio’s research has been supported by the national grant of the Italian Ministry of Education University and Research (MIUR) for the PRIN-2015 program, ‘Complex space-time modelling and functional analysis for probabilistic forecast of seismic events’. The authors also wish to thank the editor and the two anonymous reviewers for their thoughtful comments for the revision of the manuscript.

Author information

Authors and Affiliations

  1. Research Institute on Water and Environmental Engineering, Universitat Politècnica de València, Valencia, Spain

    Jonathan Romero-Cuellar & Félix Francés

  2. Department of Economics, Business and Statistics, Università degli Studi di Palermo, Palermo, Italy

    Antonino Abbruzzo & Giada Adelfio

Authors
  1. Jonathan Romero-Cuellar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonino Abbruzzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giada Adelfio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Félix Francés
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan Romero-Cuellar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Romero-Cuellar, J., Abbruzzo, A., Adelfio, G. et al. Hydrological post-processing based on approximate Bayesian computation (ABC). Stoch Environ Res Risk Assess 33, 1361–1373 (2019). https://doi.org/10.1007/s00477-019-01694-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-019-01694-y

Keywords

Search

Navigation