Methods and guidelines for effective model calibration

The AIC and BIC Statistics
The calculated error variance and standard error are sometimes criticized for not sufficient-
ly representing the drawbacks associated with increasing the number of estimated parameters. The 
AIC and BIC statistics were developed in the time-series literature to address this criticism (Brock-
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20
well & Davis, 1989). To reflect the fact that adding too many parameters produces unreliable pa-
rameter estimates, the following two statistics equal the sum of the maximum-likelihood objective 
function (eq. 3) evaluated at the optimal parameter values, S’(b’), and terms that become large as 
more parameters are added. Although these statistics were developed for time-series problems, 
Carrera and Neuman (1986) successfully used them to discriminate between different parameter-
izations of a test case of ground-water flow. The statistics are stated below; see the cited references 
for their derivations and additional discussion. 
The statistic AIC was developed by Akaike (1974) and equals:
AIC(b’) = S’(b’) + 2 
×
NP.
(16)
The statistic BIC also was developed by Akaike (1978) as a response to concern that AIC some-
times promoted use of more parameters than was required. The version of this statistic used by 
Carrera and Neuman (1986) is:
BIC(b’) = S’(b’) + NP 
× 
ln(ND+NPR).
(17)
For both statistics, smaller values indicate a more accurate model. If the statistics for a mod-
el with fewer parameters are only slightly larger than the statistics of another model, however, it 
may be better to select the model with fewer parameters, unless the investigator has other informa-
tion indicating the validity of the more complicated model.