A Bayesian approach to improved calibration and prediction of groundwater models with structural error

Abstract

Numerical groundwater flow and solute transport models are usually subject to model structural error due to simplification and/or misrepresentation of the real system, which raises questions regarding the suitability of conventional least squares regression‐based (LSR) calibration. We present a new framework that explicitly describes the model structural error statistically in an inductive, data‐driven way. We adopt a fully Bayesian approach that integrates Gaussian process error models into the calibration, prediction, and uncertainty analysis of groundwater flow models. We test the usefulness of the fully Bayesian approach with a synthetic case study of the impact of pumping on surface‐ground water interaction. We illustrate through this example that the Bayesian parameter posterior distributions differ significantly from parameters estimated by conventional LSR, which does not account for model structural error. For the latter method, parameter compensation for model structural error leads to biased, overconfident prediction under changing pumping condition. In contrast, integrating Gaussian process error models significantly reduces predictive bias and leads to prediction intervals that are more consistent with validation data. Finally, we carry out a generalized LSR recalibration step to assimilate the Bayesian prediction while preserving mass conservation and other physical constraints, using a full error covariance matrix obtained from Bayesian results. It is found that the recalibrated model achieved lower predictive bias compared to the model calibrated using conventional LSR. The results highlight the importance of explicit treatment of model structural error especially in circumstances where subsequent decision‐making and risk analysis require accurate prediction and uncertainty quantification.

Groundwater Flow Determination Using an Interval Parameter Perturbation Method

“…Calculating |ψ|∆α, ∆H (1) could be obtained; Calculating above-described steps 1-4, the current time head can be obtained ∆H (1) In the whole calculation, when the hydrogeological elements change within a certain interval, determining the head interval response is essentially a matter of interval optimization. In the following sections, numerical examples will be used to analyze the effectiveness of the proposed interval method.…”

“…This theory involves three aspects including sampling methods, likelihood functions and convergence criteria [1]. A crucial step of the Bayesian theory is the sampling algorithm.…”

“…where ∆H (1) is the first-order approximation of ∆H. That is, replacing ∆H with ∆H (1) in Equation (18), where H is…”

“…Over the past 40 years, numerical models of groundwater flow and solute transport have been widely used in studies of water resources management, migration of pollutants, sea water intrusion and many other applications [1][2][3][4]. However, because of the complexity of hydrogeological conditions and the limited budget and time, scientists often find it difficult to collect sufficient data to completely describe the hydrogeological systems in great details.…”

Gaussian process modelling for uncertainty quantification in convectively-enhanced dissolution processes in porous media

“…The 95% CIs are computed as in [45], i.e., by using the intervals (m j 2s j , m j + 2s j ) where m j and s j are respectively the predictive mean (32) and square root (standard deviation) of the variance (33) for a given design point⇠ j . The first covariance function (the most commonly used, see e.g., [59]) is the squared exponential (SE). This function is infinitely di↵erentiable, which means that the associated GP has mean-square derivatives to all orders.…”