Methods and guidelines for effective model calibration

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computational effort equivalent to a full regression. The section 
“
Testing for Linearity
”
discusses 
a test with which model nonlinearity can be evaluated.
Linear intervals use the assumption of normality of the parameter estimates in their con-
struction. As discussed in the section 
“
Normal Probability Graphs and Correlation Coefficient 
R
N
2
,
”
the weighted residuals are the only quantities that can be readily tested for normality. A sam-
ple normal probability graph is shown in figure 15, along with graphs showing normally distributed 
random numbers generated with and without regression-induced correlations, as described in the 
section 
“
Determining Acceptable Deviations from Independent Normal Weighted Residuals.
” 
Fig-
ure 15 shows that most aspects of the nonlinear pattern evident in the weighted residuals can be 
explained by the regression-induced correlations.

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Figure 15: Normal probability graphs for the steady-state version of test case 1 of Hill (1992), 
including (A) weighted residuals, (B) normally distributed, uncorrelated random num-
bers, and (C) normally distributed random numbers correlated as expected given the fit-
ting of the regression. In B and C, four sets of generated numbers are shown, each with 
a different symbol.
Christensen and Cooley (1996; in press) show that in nonlinear problems, nonlinear confi-
dence intervals can be very different than linear intervals for some quantities, while they can be 
very close for others. It appears that linear confidence intervals are useful as a general indication 
of uncertainty in many circumstances, but, if at all possible given computer resources, some non-
linear intervals need to be calculated if the model is nonlinear.
Linear and nonlinear confidence intervals, along with any other method of uncertainty anal-
ysis, such as Monte Carlo methods and the methods presented by Sun (1994), are based on the as-
sumption that the model accurately represents the real system. In truth, all models are 
simplifications of real systems, and the accuracy of the uncertainty analysis is in question. Accu-
racy of uncertainty analyses is very difficult to evaluate definitively. Steen Christensen and R.L. 
Cooley (written commun., 1997) compared nonlinear prediction intervals to measured heads and 
flows indicating good correspondence between the expected and realized significance level of the 
intervals. If model fit to data indicates model bias, the theory suggests the calculated intervals do