Bias and uncertainty in regression-calibrated models of groundwater flow in heterogeneous media

Abstract

Groundwater models need to account for detailed but generally unknown spatial variability (heterogeneity) of the hydrogeologic model inputs. To address this problem we replace the large, m-dimensional stochastic vector β that reflects both small and large scales of heterogeneity in the inputs by a lumped or smoothed m-dimensional approximation γθ*, where γ is an interpolation matrix and θ* is a stochastic vector of parameters. Vector θ* has small enough dimension to allow its estimation with the available data. The consequence of the replacement is that model function f(γθ*) written in terms of the approximate inputs is in error with respect to the same model function written in terms of β, γ,f(γ), which is assumed to be nearly exact. The difference f(β) - f(γθ*), termed model error, is spatially correlated, generates prediction biases, and causes standard confidence and prediction intervals to be too small. Model error is accounted for in the weighted nonlinear regression methodology developed to estimate θ* and assess model uncertainties by incorporating the second-moment matrix of the model errors into the weight matrix. Techniques developed by statisticians to analyze classical nonlinear regression methods are extended to analyze the revised method. The analysis develops analytical expressions for bias terms reflecting the interaction of model nonlinearity and model error, for correction factors needed to adjust the sizes of confidence and prediction intervals for this interaction, and for correction factors needed to adjust the sizes of confidence and prediction intervals for possible use of a diagonal weight matrix in place of the correct one. If terms expressing the degree of intrinsic nonlinearity for f(β) and f(γθ*) are small, then most of the biases are small and the correction factors are reduced in magnitude. Biases, correction factors, and confidence and prediction intervals were obtained for a test problem for which model error is large to test robustness of the methodology. Numerical results conform with the theoretical analysis. © 2005 Elsevier Ltd. All rights reserved.