Methods and guidelines for effective model calibration

Nonlinear Confidence and Prediction Intervals
Accurate evaluation of parameter and prediction uncertainty for nonlinear models requires 
nonlinear confidence and prediction intervals, as discussed by Veccia and Cooley (1987), Cooley 
(1997) and Christensen and Cooley (in press). Calculation of nonlinear confidence intervals re-
quires the equivalent of a full regression for each limit of each interval, so can entail substantial 
additional computer execution time. For many nonlinear problems, a practical approach is to cal-
culate linear confidence and(or) prediction intervals, and then to calculate nonlinear intervals for 
selected predictions. Unfortunately, nonlinear intervals are not calculated with the present versions 
of UCODE and MODFLOWP.
Testing for Linearity
The methods presented in this section are only applicable if the model is sufficiently linear. 
Although the modified Gauss-Newton optimization method and many of the statistical methods 
discussed are useful even for problems which are quite nonlinear, more stringent requirements on 
linearity are needed for the linear confidence and prediction intervals to adequately represent pa-
rameter and prediction uncertainty. The assumption of linearity upon which the linear confidence 
intervals are based can be tested using the modified Beale’s measure (also called Linssen’s mea-
sure) described by Cooley and Naff (1990) and Hill (1994). Although the modified Beale’s mea-
sure indicates nonlinearity of the confidence region of the parameters, and does not directly 
measure nonlinearity of confidence intervals, no better indicator of nonlinearity is available. In-
creasingly problematic situations occur as predictive quantities or situations differ more from cal-

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ibration observations and situations. 
The modified Beale’s measure can be calculated using MODFLOWP and the computer 
program BEALEP of Hill (1994); a slightly modified version of BEALEP can be executed by 
UCODE using PHASE=33. Many practical problems are nonlinear, in which case the linear inter-
vals are inaccurate.
Example Figures
Table 1 lists most of the statistics and graphical analyses discussed in this section of the re-
port, and the figures and guidelines in which they are presented and discussed in the next section. 
Note that the use of these statistics and graphs is not restricted to the suggested application.

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Table 1: Statistics and graphical analyses, related figures, and the guidelines in which the figures 
are presented. 
1
1. The statistics and graphs often are useful for other guidelines as well. See Table 2.
2. Unless otherwise indicated.
3. No example is provided in this report.
4. Repeated because of their frequent application for two purposes.
5. No example is provided in this report. See Anderman (1996) and Yager (in press).
6. No example is provided in this report. An example is shown by Christensen and Cooley (in press).