Richard L. Cooley

Evaluation of prediction intervals for expressing uncertainties in groundwater flow model predictions

We tested the accuracy of 95% individual prediction intervals for hydraulic heads, streamflow gains, and effective transmissivities computed by groundwater models of two Danish aquifers. To compute the intervals, we assumed that each predicted value can be written as the sum of a computed dependent variable and a random error. Testing was accomplished by using a cross-validation method and by using new field measurements of hydraulic heads and transmissivities that were not used to develop or calibrate the models. The tested null hypotheses are that the coverage probability of the prediction intervals is not significantly smaller than the assumed probability (95%) and that each tail probability is not significantly different from the assumed probability (2.5%). In all cases tested, these hypotheses were accepted at the 5% level of significance. We therefore conclude that for the groundwater models of two real aquifers the individual prediction intervals appear to be accurate.

Evaluation of confidence intervals for a steady-state leaky aquifer model

Abstract The fact that dependent variables of groundwater models are generally nonlinear functions of model parameters is shown to be a potentially significant factor in calculating accurate confidence intervals for both model parameters and functions of the parameters, such as the values of dependent variables calculated by the model. The Lagrangian method of Vecchia and Cooley [Vecchia, A.V. & Cooley, R.L., Water Resources Research, 1987, 23(7), 1237–1250] was used to calculate nonlinear Scheffe-type confidence intervals for the parameters and the simulated heads of a steady-state groundwater flow model covering 450 km2 of a leaky aquifer. The nonlinear confidence intervals are compared to corresponding linear intervals. As suggested by the significant nonlinearity of the regression model, linear confidence intervals are often not accurate. The commonly made assumption that widths of linear confidence intervals always underestimate the actual (nonlinear) widths was not correct. Results show that nonlinear effects can cause the nonlinear intervals to be asymmetric and either larger or smaller than the linear approximations. Prior information on transmissivities helps reduce the size of the confidence intervals, with the most notable effects occurring for the parameters on which there is prior information and for head values in parameter zones for which there is prior information on the parameters.

An analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields

An analysis of the pilot point method for automated calibration of an ensemble of conditionally simulated transmissivity fields was conducted on the basis of the simplifying assumption that the flow model is a linear function of log transmissivity. The analysis shows that the pilot point and conditional simulation method of model calibration and uncertainty analysis can produce accurate uncertainty measures if it can be assumed that errors of unknown origin in the differences between observed and model-computed water pressures are small. When this assumption is not met, the method could yield significant errors from overparameterization and the neglect of potential sources of model inaccuracy. The conditional simulation part of the method is also shown to be a variant of the percentile bootstrap method, so that when applied to a nonlinear model, the method is subject to bootstrap errors. These sources of error must be considered when using the method.

Practical Scheffé‐type credibility intervals for variables of a groundwater model

Simultaneous Scheffe-type credibility intervals (the Bayesian version of confidence intervals) for variables of a groundwater flow model calibrated using a Bayesian maximum a posteriori procedure were derived by Cooley [1993b]. It was assumed that variances reflecting the expected differences between observed and model-computed quantities used to calibrate the model are known, whereas they would often be unknown for an actual model. In this study the variances are regarded as unknown, and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance. The credibility intervals are calculated from the posterior distribution, which was developed by considering each group variance to be a random variable about which nothing is known a priori, then eliminating it by integration. Numerical experiments using two test problems illustrate some characteristics of the credibility intervals. Nonlinearity of the statistical model greatly affected some of the credibility intervals, indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems. The parameter characterizing the probability level for the credibility intervals was, however, accurately computed using a linear model approximation, as compared with values calculated using second-order and fully nonlinear formulations. This allows the credibility intervals to be computed very efficiently.