Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory

Abstract

[1] Parameter estimation in rainfall-runoff models is affected by uncertainties in the measured input/output data (typically, rainfall and runoff, respectively), as well as model error. Despite advances in data collection and model construction, we expect input uncertainty to be particularly significant (because of the high spatial and temporal variability of precipitation) and to remain considerable in the foreseeable future. Ignoring this uncertainty compromises hydrological modeling, potentially yielding biased and misleading results. This paper develops a Bayesian total error analysis methodology for hydrological models that allows (indeed, requires) the modeler to directly and transparently incorporate, test, and refine existing understanding of all sources of data uncertainty in a specific application, including both rainfall and runoff uncertainties. The methodology employs additional (latent) variables to filter out the input corruption given the model hypothesis and the observed data. In this study, the input uncertainty is assumed to be multiplicative Gaussian and independent for each storm, but the general framework allows alternative uncertainty models. Several ways of incorporating vague prior knowledge of input corruption are discussed, contrasting Gaussian and inverse gamma assumptions; the latter method avoids degeneracies in the objective function. Although the general methodology is computationally intensive because of the additional latent variables, a range of modern numerical methods, particularly Monte Carlo analysis combined with fast Newton-type optimization methods and Hessian-based covariance analysis, can be employed to obtain practical solutions.

Framework for Understanding Structural Errors (FUSE): A modular framework to diagnose differences between hydrological models

“…Moreover, the ensemble may not bracket the measurements because all models have similar weaknesses (e.g., no mechanisms for generating infiltration-excess runoff, no vegetation submodel, no representation of the spatial variability in precipitation). Future work is also necessary to both separate errors in model inputs from errors in model structure [e.g., Clark and Slater, 2006;Kavetski et al, 2006aKavetski et al, , 2006b; J. A. Vrugt et al, Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov Chain Monte Carlo simulation, submitted to Water Resources Research, 2007], and rigorously quantify the independence between different models [e.g., Abramowitz and Gupta, 2008].…”

Bayesian multiresponse calibration of TOPMODEL: Application to the Haute‐Mentue catchment, Switzerland

“…Kavetski et al [2006aKavetski et al [ , 2006b and Yang et al [2007Yang et al [ , 2008 used a Bayesian methodology for single-response calibration where input uncertainty was taken into account, and Marshall et al [2006] integrated the uncertainty of the model structure in a Bayesian model averaging methodology. Ajami et al [2007] integrated both input uncertainty and model structure in order to estimate the uncertainty of simulated discharges.…”

Improving Flood Prediction Assimilating Uncertain Crowdsourced Data into Hydrological and Hydraulic Models

“…Critical analyses of the GLUE approach are presented in Mantovan and Todini (2006) and Stedinger et al (2008). Different Bayesian approaches like standard Bayesian approaches (Kuczera and Parent, 1998;Krzysztofowicz, 1999;Feyen et al, 2007), Bayesian Recursive Estimation (Thiemann et al, 2001), Bayesian hierarchical models (Kuczera et al, 2006;Kavetski et al, 2006;Huard and Mailhot, 2008) and Bayesian model averaging (Duan et al, 2007;Marshall et al, 2007) are also used in uncertainty analysis. The last decade has witnessed a major shift from deterministic to probabilistic flood hazard assessment by means of hydrodynamic modelling.…”

“…For this reason, EWSs should be able to quantify uncertainty around a given deterministic value. A growing number of studies analysed the impact of rainfall uncertainty on flood event prediction (Kavetski et al, 2006;Moulin et al, 2009;McMillan et al, 2011). One possible way to quantify uncertainty in flood forecasting is to use an ensemble of weather predictions.…”