Methods and guidelines for effective model calibration

Graphs Using Independent Variables and the Runs Statistic
Evaluating weighted and unweighted residuals and weighted and unweighted observations 
and simulated values as related to the independent variables of a problem, such as space and time, 
is crucial. Ideally, weighted residuals plotted on maps or time graphs such as hydrographs show no 
discernible patterns and appear random. Distinct patterns can indicate significant model error that 
may make simulated predictions incorrect and misleading. Distinct patterns often are present, 
however, especially in time graphs. 
A summary statistic that checks for the randomness of weighted residuals is the runs test 
(Cooley, 1979; Draper and Smith, 1981, p. 157-162). A sequence of residuals of the same sign is 
called a run, and the number of runs is counted and the value assigned to the variable u. For exam-
ple, for the sequence of numbers -5, -2, 4, 3, 6, -4, 2, -3, -9, has the five runs (-5,-2), (4,3,6), (-4), 
(2), (-3, -9), so that u=5. Using the total number of positive residuals (n
1
), and the total number of 
negative residuals (n
2
), u can be defined as a random variable that depends on the order of the neg-
ative and positive values. It can be shown that if n1>10 and n2>10, u is normally distributed with 
mean, 
µ
, and variance, 
σ
2
, equal to:
µ
= [2n
1
n
2
/(n
1
+n
2
)]+1.0, (21)
σ
2
= [2n
1 
n
2 
(2n
1
n
2
-n
1
-n
2
)]/[(n
1
+n
2
)
2
(n
1
+n
2
-1)]. (22)
The actual number of runs in a data set is compared with the expected value using test statistics. 
The test statistic for too few runs equals 
z
f
= (u-
µ
+0.5)/
σ
; (23)
the test statistic for too many runs equals
z
m
= (u-
µ
-0.5)/
σ
. (24)
Critical values for z
f
and z
m
are printed by UCODE and MODFLOWP. Otherwise, critical values 
can be determined from a normal probability table available in most statistics texts. 
In UCODE and MODFLOWP, the weighted residuals are analyzed using the sequence in 
which the observations are listed in the input file. The runs test is included because it takes the order 
of the residuals into account, which is ignored in all the other summary statistics. If observations 
are grouped by location in transient simulations, too few runs commonly indicate positive serial 
correlation between residuals at individual locations.
m
y’
i
y y’
ω
m
y
i
m
y’
i

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