Methods and guidelines for effective model calibration

Prediction Scaled Sensitivities
Sensitivities can be calculated for simulated predictions as dz
’
l
/db
j
, where z
’
l
is the simu-
lated prediction. These sensitivities indicate the importance of the parameter values to these pre-
dictions, and can be scaled to produce statistics by which to compare the relative importance of 
different parameters. One useful scaling results in a statistic that indicates the percentage change 
in the prediction produced by a one-percent change in the parameter value. This is defined in this 
work as a prediction scaled sensitivitiy (pss
j
), and is calculated as:
pss
j
=(dz
’
l
/db
j
) (bj/100) (100/z
’
l
)
(12)
The prediction scaled sensitivity is the one-percent sensitivity of equation 11 calculated for predic-
tions, and multiplied by the last term of equation 12.
A different scaling is needed when z
’
l
= 0.0, or if the change in the predictive quantity rel-
ative to, perhaps, a regulatory limit, is of interest. In such circumstances, the predictive scaled sen-
stivity can be calculated as:
pss
j 
= (dz
’
l
/db
j
) (bj/100) (100/a
’
l
)
(13)
dss
ij
b
j
∂
∂
y’
i
b
j
100
---------
=

---

17
The resulting statistic is the change in the prediction caused by a one-percent change in the param-
eter value, expressed as a percentage of a
l
. The value used for a
l
could be the regulatory limit or 
another number that was relevant to a given situation.
Prediction scaled sensitivities are not calculated by UCODE or MODFLOWP, but can eas-
ily be calculated from one-percent sensitivities calculated for predictions. For UCODE, this is ac-
complished using PHASE=44; for MODFLOWP this is achieved by following the directions for 
the post-processing program YCINT (Hill, 1994). 
In some circumstances the prediction on interest is the difference between two simulations. 
For example, in ground-water problems, the prediction of interest is often the drawdown or change 
in flow to a stream caused by pumpage. These were termed differences by Hill (1994), and 
UCODE and MODFLOWP are designed to calculate sensitivities related to differences. For 
UCODE, this is accomplished using PHASE=45; for MODFLOWP, this is achieved as described 
for YCINT in Hill (1994).
There are two points that need to be considered when calculating prediction scaled senstiv-
ities. First, generally it is important to calculate prediction scaled sensitivities for the parameters 
which were not estimated by regression during model calibration well as the paramters that were 
estimated by regression. Parameters that could not be estimated by regression because of insensi-
tivity, as indicated by composite scaled senstivities, can be important to preditions, and prediction 
scaled senstivities are likely to display that importance.
Second, often it is important to calculate the prediction scaled sensitivities for other sets of 
parameter values besides the optimal parameter values. This tests the robustness of the conclusions 
drawn from the prediction scaled sensitivities with respect to model nonlinearity.