Application of Formal and Informal Bayesian Methods for Water Distribution Hydraulic Model Calibration
Publication: Journal of Water Resources Planning and Management
Volume 140, Issue 11
Abstract
Water distribution system model parameter calibration is an important step to obtain a representative system model, such that it may be applied to understand system operational performance, often in real time. However, few approaches have attempted to quantify uncertainty in calibrated parameters, model predictions, and consider the sensitivity of model predictions to uncertain parameters. A probabilistic Bayesian approach is applied to calibrate and quantify uncertainty in the pipe roughness groups of an Epanet2 hydraulic model of a real-life water distribution network. Within the applied Bayesian framework, the relative performance of formal and informal Bayesian likelihoods in implicitly quantifying parameter and predictive uncertainty is considered. Both approaches quantify posterior parameter uncertainty with similar posterior distributions for parameter values (mean and standard deviation). However, the uncertainty intervals identified with the informal likelihood are too narrow, regardless of the behavioral threshold applied to derive these bounds. In contrast, the formal Bayesian approach produces more realistic 95% prediction intervals based on their statistical coverage of the observations. This results as the error model standard deviation is jointly inferred during calibration, which also helps to avoid potential overconditioning of the posterior parameter distribution. However, posterior diagnostic checks reveal that the prediction intervals are not valid at percentiles other than the 95% interval as the assumptions of normality, residual homoscedasticity, and noncorrelation, often assumed in hydraulic model calibration, do no hold. More robust calibration requires the development of error models better suited to the nature of residual errors found in water distribution system models.
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Acknowledgments
The work presented in this paper was partially supported by ‘PREPARED, Enabling Change’, an ongoing European Commission Seventh Framework funded large scale integrating interdisciplinary project (Grant agreement no.: 244232, 2010-2014).
References
Beven, K., and Binley, A. (1992). “The future of distributed models—Model calibration and uncertainty prediction.” Hydrol. Processes, 6(3), 279–298.
Beven, K., and Freer, J. (2001). “Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology.” J. Hydrol., 249(1–4), 11–29.
Beven, K. J., and Brazier, R. E. (2011). “Dealing with uncertainty in erosion model predictions.” Handbook of erosion modelling, R. P. C. Morgan and M. A. Nearing, eds., Wiley, Chichester, U.K., 52–79.
Beven, K. J., Smith, P. J., and Freer, J. E. (2008). “So just why would a modeller choose to be incoherent?” J. Hydrol., 354(1–4), 15–32.
Blasone, R. S., Vrugt, J. A., Madsen, H., Rosbjerg, D., Robinson, B. A., and Zyvoloski, G. A. (2008). “Generalized likelihood uncertainty estimation (GLUE) using adaptive Markov chain Monte Carlo sampling.” Adv. Water Resour., 31(4), 630–648.
Brazier, R. E., Beven, K. J., Freer, J., and Rowan, J. S. (2000). “Equifinality and uncertainty in physically based soil erosion models: Application of the glue methodology to WEPP-the water erosion prediction project-for sites in the U.K. and U.S.” Earth Surf. Processes Landforms, 25(8), 825–845.
Bush, C. A., and Uber, J. G. (1998). “Sampling design methods for water distribution model calibration.” J. Water Resour. Plann. Manage., 334–344.
Dotto, C. B. S., et al. (2012). “Comparison of different uncertainty techniques in urban stormwater quantity and quality modelling.” Water Res., 46(8), 2545–2558.
Dotto, C. B. S., Deletic, A., and Fletcher, T. D. (2009). “Analysis of parameter uncertainty of a flow and quality stormwater model.” Water Sci. Technol., 60(3), 717–725.
Draper, D. (1995). “Assessment and propagation of model uncertainty.” J. R. Stat. Soc. Series B-Method., 57(1), 45–97.
Fisher, I., Kastl, G., and Sathasivan, A. (2011). “Evaluation of suitable chlorine bulk-decay models for water distribution systems.” Water Res., 45(16), 4896–4908.
Freni, G., and Mannina, G. (2010). “Bayesian approach for uncertainty quantification in water quality modelling: The influence of prior distribution.” J. Hydrol., 392(1–2), 31–39.
Freni, G., Mannina, G., and Viviani, G. (2008). “Uncertainty in urban stormwater quality modelling: The effect of acceptability threshold in the GLUE methodology.” Water Res., 42(8–9), 2061–2072.
Freni, G., Mannina, G., and Viviani, G. (2009a). “Uncertainty in urban stormwater quality modelling: The influence of likelihood measure formulation in the GLUE methodology.” Sci. Total Environ., 408(1), 138–145.
Freni, G., Mannina, G., and Viviani, G. (2009b). “Urban runoff modelling uncertainty: Comparison among Bayesian and pseudo-Bayesian methods.” Environ. Modell. Software, 24(9), 1100–1111.
Gelman, A., and Shalizi, C. R. (2012). “Philosophy and the practice of Bayesian statistics (with discussion).” Br. J. Math. Stat. Psychol., 66(1), 8–38.
Hall, J. W., Manning, L. J., and Hankin, R. K. S. (2011). “Bayesian calibration of a flood inundation model using spatial data.” Water Resour. Res., 47(5), W05529.
Huang, J. J., and McBean, E. A. (2007). “Using Bayesian statistics to estimate the coefficients of a two-component second-order chlorine bulk decay model for a water distribution system.” Water Res., 41(2), 287–294.
Hutton, C. J., Brazier, R. E., Nicholas, A. P., and Nearing, M. (2012a). “On the effects of improved cross-section representation in one-dimensional flow routing models applied to ephemeral rivers.” Water Resour. Res., 48(4), W04509.
Hutton, C. J., Kapelan, Z., Vamakeridou-Lyroudia, L. S., and Savic, D. (2012b). “Dealing with uncertainty in water distribution systems’ models: A framework for real-time modelling and data assimilation.” J. Water Resour. Plann. Manage., 140(2), 169–183.
Jamieson, D. G., Shamir, U., Martinez, F., and Franchini, M. (2007). “Conceptual design of a generic, real-time, near-optimal control system for water-distribution networks.” J. Hydroinf., 9(1), 3–14.
Kang, D. S., and Lansey, K. (2011). “Demand and roughness estimation in water distribution systems.” J. Water Resour. Plann. Manage., 20–30.
Kapelan, Z. S., Savic, D. A., and Walters, G. A. (2005). “Multiobjective design of water distribution systems under uncertainty.” Water Resour. Res., 41(11), W11407.
Kapelan, Z. S., Savic, D. A., and Walters, G. A. (2007). “Calibration of water distribution hydraulic models using a Bayesian-type procedure.” J. Hydraul. Eng., 927–936.
Lansey, K. E., El-Shorbagy, W., Ahmed, I., Araujo, J., and Haan, C. T. (2001). “Calibration assessment and data collection for water distribution networks.” J. Hydraul. Eng., 270–279.
Liu, Y. L., Freer, J., Beven, K., and Matgen, P. (2009). “Towards a limits of acceptability approach to the calibration of hydrological models: Extending observation error.” J. Hydrol., 367(1–2), 93–103.
McMillan, H., and Clark, M. (2009). “Rainfall-runoff model calibration using informal likelihood measures within a Markov chain Monte Carlo sampling scheme.” Water Resour. Res., 45(4), W04418.
McMillan, H., Jackson, B., Clark, M., Kavetski, D., and Woods, R. (2011). “Rainfall uncertainty in hydrological modelling: An evaluation of multiplicative error models.” J. Hydrol., 400(1–2), 83–94.
Preis, A., Whittle, A. J., Ostfeld, A., and Perelman, L. (2010). “On-line hydraulic state estimation in urban water networks using reduced models.” Integr. Water Syst., 319–324.
Romano, M., Kapelan, Z., and Savic, A. D. (2012). “Automated detection of pipe bursts and other events in water distribution systems.” J. Water Resour. Plann. Manage., 457–467.
Romanowicz, R., Beven, K. J., and Tawn, J. (1994). “Evaluation of predictive uncertainty in nonlinear hydrological models using a Bayesian approach.” Statistics for the environment II: Water related issues, V. Barnett and K. F. Turkman, eds., Wiley, Chichester, England, 297–317.
Rossman, L. A. (2000). EPANET2 users manual national risk management research laboratory, U.S. Environmental Protection Agency, Cincinnati, OH, 〈http://www.image.unipd.it/salandin/IngAmbientale/Progetto_2/EPANET/EN2manual.pdf〉, 45268 (Mar. 12, 2014).
Savic, D. A., Kapelan, Z. S., and Jonkergouw, P. M. R. (2009). “Quo vadis water distribution model calibration?” Urban Water J., 6(1), 3–22.
Schoups, G., and Vrugt, J. A. (2010). “A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and nonGaussian errors.” Water Resour. Res., 46(10), W10531.
Smith, P., Beven, K. J., and Tawn, J. A. (2008). “Informal likelihood measures in model assessment: Theoretic development and investigation.” Adv. Water Resour., 31(8), 1087–1100.
Stedinger, J. R., Vogel, R. M., Lee, S. U., and Batchelder, R. (2008). “Appraisal of the generalized likelihood uncertainty estimation (GLUE) method.” Water Resour. Res., 44(12), W00B06.
Sumer, D., and Lansey, K. (2009). “Effect of uncertainty on water distribution system model design decisions.” J. Water Resour. Plann. Manage., 38–47.
Thyer, M., Renard, B., Kavetski, D., Kuczera, G., Franks, S. W., and 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(12)W00B14.
Vrugt, J. A., Gupta, H. V., Bouten, W., and Sorooshian, S. (2003). “A shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters.” Water Resour. Res., 39(8), 1201.
Vrugt, J. A., ter Braak, C. J. F., Gupta, H. V., and Robinson, B. A. (2009). “Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?” Stochastic Environ. Res. Risk Assess., 23(7), 1011–1026.
Wilkinson, R. D. (2013). “Approximate Bayesian Computation (ABC) gives exact results under the assumption of model error.” Stat. Appl. Genet. Mol. Biol., 12(2), 129–141.
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© 2014 American Society of Civil Engineers.
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Received: Mar 27, 2013
Accepted: Oct 1, 2013
Published online: Oct 3, 2013
Discussion open until: Oct 19, 2014
Published in print: Nov 1, 2014
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