Confidence Intervals for Ground‐Water Models Using Linearization, Likelihood, and Bootstrap Methods

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Abstract
Four methods were applied to obtain confidence intervals for selected dependent variables from a nonlinear regression formulation of a ground‐water flow model. The methods examined are (1) the linearization method, (2) the likelihood method, (3) a simple, percentile bootstrap method, and (4) a regression‐based percentile bootstrap method that is a generalization of the first one.The methods were applied to a highly nonlinear regression model of a hypothetical ground‐water system involving an infiltrating stream crossing a heterogeneous aquifer. Confidence intervals were computed for a hydraulic head and a parameter that is proportional to the streamflow infiltration rate. All confidence intervals computed by the likelihood method are virtually exact. Confidence intervals for hydraulic head by the linearization, likelihood, and the second bootstrap methods are similar, but confidence intervals by the first bootstrap method are very different and highly inaccurate. Confidence intervals for the streamflow infiltration parameter by the linearization and first bootstrap methods are identical and inaccurate; confidence intervals by the second bootstrap method lie between those by the likelihood and the other two methods. These characteristics of the confidence intervals result because (1) the regression model is highly nonlinear, and (2) much of the nonlinearity can be removed by some transformation of model parameters.