Methods and guidelines for effective model calibration

Normal Probability Graphs and Correlation Coefficient R
2
N
For a valid regression, the errors in the observations and the prior information used in the 
regression need to be random and the weighted errors need to be uncorrelated (Draper and Smith, 
1981). In addition, inferential statistics such as confidence intervals generally require that the ob-
servation errors be normally distributed (Helsel and Hirsch, 1992). The actual errors are unknown, 
so the weighted residuals are analyzed. If the model accurately represents the actual system and 
the observation errors are random and the weighted errors are independent, the weighted residuals 
are expected to either be random, independent, and normally distributed, or have predictable cor-
relations. The first step is to determine whether the weighted residuals are independent and nor-
mally distributed. If they are not, further analysis is needed to determine if the violations are 
consistent with the expected correlations.
The test for independent, normal weighted residuals is conducted using normal probability 
graphs of weighted residuals. Such graphs can be constructed as discussed by Hill (1994), using 
files created by UCODE or MODFLOWP. The files are designed so that the graphs can be con-
structed using commonly available x-y plotting software using arithmetic axes. If the weighted re-
siduals are independent and normally distributed, they will fall on an approximately straight line 
in the normal probability graph. The associated summary statistic is R
2
N
the correlation coefficient 
between the weighted residuals ordered from smallest to largest and the order statistics from a 
N(0,1) probability distribution function (Brockwell and Davis, 1987, p. 304). This statistic tests for 
independent, normally distributed weighted residuals and was chosen instead of other statistics, 
such as chi-squared and Kolomogorov-Smirnov, because it is more powerful for commonly used 
sample sizes (Shapiro and Francia, 1972). The correlation coefficient is calculated as:
R
2
N
=
,
(25)
where all vectors are of length ND for R
2
N
evaluated only for the observation weighted residuals, 
and length ND+NPR for R
2
N
evaluated for the observation and prior information weighted resid-
uals; m is a vector with all components equal to the average of the weighted residuals, e
o 
is a vector 
of weighted residuals ordered from smallest to largest, and 
τ
is a vector with the ith element equal 
to the ordinate value of a N(0,1) probability distribution function for a cumulative probability equal 
to u
i 
= (i-0.5)/ND. A normal probability table (as in Cooley and Naff, 1990, p. 44, or any standard 
statistics text) can be used to determine that, for example, if u
= 0.853l, then 
τ
i 
= 1.05. UCODE 
and MODFLOWP print the ordered weighted residuals of e
o
and R
2
N
.
If R
2
N
is too much less than its ideal value of 1.0, the weighted residuals are not likely to 
e
o
m
–
(
)
T
τ
[
]
2
e
o
m
–
(
)
T
e
o
m
–
(
)
[
] τ
T
τ
(
)
---------------------------------------------------------------

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24
be independent and normally distributed. The critical values for R
2
N
for significance levels 0.05 
and 0.10 are shown in Appendix D and the relevant critical values are printed by MODFLOWP 
and UCODE with R
2
N
.