Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation

Abstract

[1] There is increasing consensus in the hydrologic literature that an appropriate framework for streamflow forecasting and simulation should include explicit recognition of forcing and parameter and model structural error. This paper presents a novel Markov chain Monte Carlo (MCMC) sampler, entitled differential evolution adaptive Metropolis (DREAM), that is especially designed to efficiently estimate the posterior probability density function of hydrologic model parameters in complex, high-dimensional sampling problems. This MCMC scheme adaptively updates the scale and orientation of the proposal distribution during sampling and maintains detailed balance and ergodicity. It is then demonstrated how DREAM can be used to analyze forcing data error during watershed model calibration using a five-parameter rainfall-runoff model with streamflow data from two different catchments. Explicit treatment of precipitation error during hydrologic model calibration not only results in prediction uncertainty bounds that are more appropriate but also significantly alters the posterior distribution of the watershed model parameters. This has significant implications for regionalization studies. The approach also provides important new ways to estimate areal average watershed precipitation, information that is of utmost importance for testing hydrologic theory, diagnosing structural errors in models, and appropriately benchmarking rainfall measurement devices.

Bayesian model selection in hydrogeophysics: Application to conceptual subsurface models of the South Oyster Bacterial Transport Site, Virginia, USA

“…The posterior distribution and the MAP solution that is used by the LM ( Section 2.2.2 ) and GMIS ( Section 2.2.3 ) methods are derived from MCMC simulation using the DREAM (ZS) algorithm ( Laloy and Vrugt, 2012;Vrugt, 2016;Vrugt et al, 2008 ). This multi-chain method creates proposals on the fly from an historical archive of past states using a mix of parallel direction and snooker updates.…”

“…For this purpose, we compare evidence estimates computed by (1) the brute force Monte Carlo method ( Hammersley and Handscomb, 1964 ), (2) the Laplace-Metropolis method ( Lewis and Raftery, 1997 ) and (3) the Gaussian mixture importance sampling (GMIS) estimator of Volpi et al (2016) . This latter method approximates the evidence by importance sampling from a Gaussian mixture model fitted to a large sample of posterior solutions generated with the DREAM (ZS) algorithm ( Laloy and Vrugt, 2012;Vrugt, 2016;Vrugt et al, 2008 ). Then, we present an application of Bayesian model selection to subsurface modeling using geophysical data from the South Oyster Bacterial Transport Site in Virginia (USA) ( Chen et al, 20 01;20 04;Hubbard et al, 2001;Linde et al, 2008;Linde and Vrugt, 2013 ).…”

The role of uncertainty in bedrock depth and hydraulic properties on the stability of a variably-saturated slope

“…Bayesian analysis was used to reconcile the DTB model with distributed bedrock depth measurements. We used Markov chain Monte Carlo simulation (MCMC) with the DREAM algorithm [36,37] for posterior inference in the Bayesian framework. This approach searched the DTB model parameter space for posterior solutions that "best" honor the observed bedrock depth data.…”

“…We use the calibrated geomorhologic model of [33] to generate plausible maps of the bedrock depth. The model was calibrated against a rich data set of distributed bedrock depth measurements using Bayesian inference with the DREAM algorithm [36,37]. Posterior maps of the simulated bedrock depth topography serve as input to a threedimensional finite-element (FE) water flow model of the bedrock-soil domain, and are used to evaluate hillslope stability for a 22-day rainfall record via numerical limit analysis [38] using the ensemble of simulated transient pore water pressures.…”

Hypothetico‐inductive data‐based mechanistic modeling of hydrological systems

“…In other words, like most rainfall-flow modeling studies in hydrology, the example has not considered the input ''errors-in-variables'' problem (see the supporting information for further discussion on this, as well as the related problems of inverse estimation and the separation of aleatory and epistemic uncertainty). And even where this problem has been considered [see e.g., Kuczera et al, 2006;Vrugt et al, 2009b;Kirchner, 2009], the basic ambiguity that exists when there are both input and output noise effects present on the data [see e.g., Söderström, 2007, and the references therein] has not been addressed fully. So there is a clear need for more theoretical and practical research on this problem before a satisfactory solution is obtained.…”