Methods and guidelines for effective model calibration

Graphical Analysis of Model Fit and Related Statistics
The assumed model can be analyzed to determine if the simulated dependent-variable val-
ues indicate a valid regression using the methods described below. The methods were suggested 
for ground-water inverse modeling by Cooley and Naff (1990), using the work of Draper and Smith 
(1981), and are discussed in Hill (1992, 1994). Data files to support these analyses are produced 
by UCODE and MODFLOWP. Examples of many of the graphs described are presented later in 
this report in the context of the guideline it is likely to support, as listed in the section 
“
Example 
Figures.
”
Weighted Residuals Versus Weighted Simulated Values and Minimum, Maximum, 
and Average Weighted Residuals
It can be shown (Draper and Smith, 1981) that, in most situations, weighted residuals and 
weighted simulated values should be independent, so it is informative to consider graphs that as-
sess the independence of these two variables. Ideally, weighted residuals are scattered evenly about 
0.0, and their size is not related to the simulated values. In ground-water problems, for example, 
ideally the weighted residuals are not consistently larger in areas of high hydraulic head than in 
areas of low hydraulic head. Examples of such graphs, a discussion of the theory behind them, and 
some situations in which adjustments are needed because the two data sets are not independent are 

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presented in Hill (1994).
Statistics printed by UCODE and MODFLOWP that summarize the distribution of the 
weighted residuals are the minimum, maximum, and average weighted residuals, and the observa-
tions for which the minimum and maximum weighted residuals occur. The minimum and maxi-
mum weighted residuals display the range of weighted residuals at a glance. In practice, especially 
in the initial stages of calibration, the minimum and maximum weighted residuals often identify 
observations that are misrepresented in the simulation, suffer from incorrect data interpretation, or
simply have not been entered incorrectly. The average weighted residual is a simple arithmetic av-
erage of the weighted residuals and ideally equals zero. In linear regression the average always 
equals zero for the optimized parameter values; in nonlinear regression the value of the average 
weighted residual generally approaches zero as calibration proceeds.