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Methods are presented for computing three types of simultaneous confidence and prediction intervals (exact, likelihood ratio, and linearized) on output from nonlinear regression models with normally distributed residuals. The confidence intervals can be placed on individual regression parameters or on the true regression function at any number of points in the domain of the independent variables, and the prediction intervals can be placed on any number of future observations. The confidence intervals are analogous to simultaneous Scheffè intervals for linear models and the prediction intervals are analogous to the prediction intervals of Hahn (1972). All three types of intervals can be computed efficiently by using the same straightforward Lagrangian optimization scheme. The prediction intervals can be treated in the same computational framework as the confidence intervals by including the random errors as pseudoparameters in the Lagrangian scheme. The methods are applied to a hypothetical groundwater model for flow to a well penetrating a leaky aquifer. Three different data sets are used to demonstrate the effect of sampling strategies on the intervals. For all three data sets, the linearized confidence intervals are inferior to the exact and likelihood ratio intervals, with the latter two being very similar; however, all three types of prediction intervals yielded similar results. The third data set (time drawdown data at only a single observation well) points out many of the problems that can arise from extreme nonlinear behavior of the regression model.

Persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable, yet efficient, method for determining the positions of seepage surfaces. New techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations. A series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems.

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A new nonlinear least squares solution for the Hydrogeologic parameters, sources and sinks, and boundary fluxes contained in the equations approximately governing two‐dimensional or radial steady state groundwater motion was developed through use of a linearization and iteration procedure applied to the finite element discretization of the problem. Techniques involving (1) use of an iteration parameter to interpolate or extrapolate the changes in computed parameters and head distribution at each iteration and (2) conditioning of the least squares coefficient matrix through use of ridge regression techniques were proven to induce convergence of the procedure for virtually all problems. Because of the regression nature of the solution for the parameter estimation problem, classical methods of regression analysis are promising as an aid to establishing approximate reliability of computed parameters and predicted values of hydraulic head. Care must be taken not to compute so many parameters that the stability of the estimates is destroyed. Reduction of the error variance by adding parameters is desirable provided that the number of degrees of freedom for error remains large.