Effects of model sensitivity and nonlinearity on nonlinear regression of ground water flow

Nonlinear regression is increasingly applied to the calibration of hydrologic models through the use of perturbation methods to compute the Jacobian or sensitivity matrix required by the Gauss-Newton optimization method. Sensitivities obtained by perturbation methods can be less accurate than those obtained by direct differentiation, however, and concern has arisen that the optimal parameter values and the associated parameter covariance matrix computed by perturbation could also be less accurate. Sensitivities computed by both perturbation and direct differentiation were applied in nonlinear regression calibration of seven ground water flow models. The two methods gave virtually identical optimum parameter values and covariances for the three models that were relatively linear and two of the models that were relatively nonlinear, but gave widely differing results for two other nonlinear models. The perturbation method performed better than direct differentiation in some regressions with the nonlinear models, apparently because approximate sensitivities computed for an interval yielded better search directions than did more accurately computed sensitivities for a point. The method selected to avoid overshooting minima on the error surface when updating parameter values with the Gauss-Newton procedure appears for nonlinear models to be more important than the method of sensitivity calculation in controlling regression convergence.