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A. Precision of the parameter estimate
Poor: Large parameter
composite scaled sensitivity,
coefficient of variation, or
confidence interval
Good: Small parameter
composite scaled sensitivity,
coefficient of variation, or
confidence interval
Importance
of the
parameter
to
predictions
of interest
Not important:
Small prediction
scaled sensitivity
I. Acceptable
1
II. Acceptable
1
Important:
Large prediction
scaled sensitivity
IV. Improve estimation of
this parameter and associated
system features.
2
III. Acceptable
1
B. Uniqueness of the parameter estimate
Poor: The absolute value of
some of this parameter’s
correlation coefficients are
close to 1.0.
3
Good: All of this parameter’s
correlation coefficients have
absolute values less than
about 0.95.
3
Importance
of the
parameter
to
predictions
of interest
Not important:
The same parameter
pairs are extremely
correlated.
4
I. Acceptable
1
II. Acceptable
1
Important:
Previously correlated
parameter pairs are
uncorrelated.
4
IV. Improve estimation of
this parameter and associated
system features .
2
III. Acceptable
1
64
Parameter correlation coefficients are cited in figure 16 as measures both of the uniqueness
of the parameter estimate and the importance of parameters to the predictions of interest. In both
cases, the correlation coefficients are variations of the parameter correlation coefficients printed at
the end of most regression runs, as discussed above in the sections “Variances and Covariances”
and “Correlation Coefficients.” An example of the utility of such correlation coefficients can be
found in the following ground-water modeling example. Consider a ground-water flow model cal-
ibrated with hydraulic-head and streamflow gain or loss observation data. The calibrated model is
being developed to predict (a) hydraulic head at a location where no measurement can be obtained,
and (b) advective transport from the site of a contaminant spill. Correlation coefficients for all pa-
rameters are obtained using the calibrated model using all defined parameters (see section “Vari-
ances and Covariances”); the prediction correlation coefficients are obtained by adding the
prediction hydraulic-head location and advective transport as ‘observations’ in the input file and
again calculating the correlation matrix for the same set of parameters. A similar calculation is re-
ported by Anderman and others (1996), showing that advective-travel was affected by individual
parameter values, while hydraulic heads were not. In this circumstance, prediction of hydraulic
heads did not require uncorrelated parameter estimates while prediction of advective travel did.
An example analysis of predictions is presented in figure 17. Prediction scaled sensitivities
calculated using equation 12 are compared to parameter composite scaled sensitivities of equation
10. In the example, the predictions of interest are the cartesian components of advective travel sim-
ulated by particle tracking using the ADV Package of Anderman and Hill (1997). The figure shows
the range and mean of the prediction scaled sensitivities for eight transported particles. The predic-
tion scaled sensitivity is defined to equal the percent change in the advective transport caused by a
one-percent change in parameter value. The figure clearly shows that parameters T3 and T4 are
most important to the determination of advective-transport distance in all three coordicate direc-
tions, and that the observations used in the regression provide more information for parameter T3
than for parameter T4. This type of information can be invaluable for understanding model
strengths and weaknesses and for planning additional modeling and data collection efforts.
65
Figure 17: Composite scaled sensitivities for estimated parameters and prediction scaled sensitiv-
ities for the spatial components of predicted advective transport. The composite scaled
sensitivites for parameters estimated in the regression are shown using black bars; those
not estimated in the regression are shown using gray bars. The prediction scaled sensi-
tivities are defined as the percent change in the prediction given a one-percent change in
the parameter value, so ‘Percent change’ is used to label the vetical axes.
0
2
4
6
8
10
12
14
T1
T2
T3
T4
AN
IV
RC
H1
RC
H2
GHB
1
GHB
2
AN
I
Parameter label
C
o
mp
osi
te sc
al
ed
se
nsi
tiv
it
y
East-west advective-transport distance
-60
-30
0
30
Perc
en
t c
han
ge
North-south advective-transport distance
-60
-30
0
30
Pe
rc
e
n
t c
h
a
nge
Vertical advective-transport distance
-300
-200
-100
0
100
200
300
T1
T2
T3
T4
ANIV
R
CH1
R
CH2
GH
B1
GH
B2
ANI
Parameter label
Pe
rc
e
n
t c
h
a
nge
66
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