INTERACTIVE EFFECTS OF HETEROGENEITY AND PARAMETER NONLINEARITY ON BIAS IN PREDICTIONS OF REGRESSION-CALIBRATED GROUND-WATER MODELS

Standard ground-water models have been criticized because they do not explicitly include small-scale heterogeneity in system properties such as hydraulic conductivity, specific storage, and recharge/discharge; model parameters used are lumped or smoothed approximations of the system properties. It has been shown by others that failure to explicitly include this heterogeneity can produce bias in model predictions because the dependent variables of ground-water models (such as hydraulic heads and fluxes) are nonlinear in system properties and model parameters such as hydraulic conductivity and specific storage. This bias can be reduced or even eliminated by formulating the model as a generalized nonlinear regression model in which the weight matrix of the objective function is the inverse of the second-moment matrix (or an approximation of this matrix) that is a function of both observation (measurement) errors and model errors resulting from replacing system properties with lumped or smoothed model parameters. Bias terms in predictions made using the regression model are dependent mainly on a property known in the statistics literature as intrinsic nonlinearity, which describes the degree to which transformations of system properties and model parameters can reduce model nonlinearity. If intrinsic nonlinearity is small (as may often be the case), the bias terms are small. The parameters used do not have to be the transformations that reduce model nonlinearity, and the transformations do not have to be known. The magnitude and importance of intrinsic nonlinearity can be measured so that the potential bias in predictions can be estimated.