Methods and guidelines for effective model calibration

Lack of Limits on Estimated Parameter Values
Upper and lower limits on parameters that constrain possible estimated values are com-
monly available in inverse models (for example, PEST, Doherty, 1994) and are suggested by, for 
example, Sun (1994, p. 35). While such limiting constraints on parameter values may, at first, ap-
pear to be necessary given the unrealistic parameter values that can be estimated through inverse 
modeling, Hill and others (1998), using a complex synthetic test case, demonstrate that this prac-
tice can disguise more fundamental modeling errors. Poeter and Hill (1996) use a simple synthetic 
test case to further demonstrate the concept, and in Anderman and others (1996), unrealistic opti-
mized values of recharge in a field problem revealed important model construction inaccuracies. 
As discussed in the section "Guideline 5: Use Prior Information Carefully," unrealistic estimated 
parameter values are likely to indicate either (1) that the data do not contain enough information to 
estimate the parameters, or (2) there is a more fundamental model error. In the first circumstance, 
the best response is to use prior information on the parameter value, which will tend to produce an 
estimate that is close to the most likely value, instead of at the less likely value that generally con-
stitutes the imposed upper or lower limit. In the second circumstance, the best response is to find 
and resolve the error. UCODE, like MODFLOWP, does not support constraining limits on param-
eter values because a circumstance in which the use of such limits is the best way to proceed has 
not been identified.
Weights for Observations and Prior Information
Observations and prior information typically need to be weighted in the regression. In most 
circumstances, diagonal weight matrices are used, and it is useful to introduce weighting in this 
simpler context. Hill (1992) presents a detailed discussion of the assumptions implied by using a 
diagonal weight matrix. 
Weighting performs two related functions. The most fundamental function is that the 
weighting needs to produce weighted residuals that have the same units so that they can be squared 

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and summed using equation 1 or 2. Obviously, summing numbers with different units produces a 
nonsensical result. The second function of the weighting is to reduce the influence of observations 
that are less accurate and increase the influence of observations that are more accurate. 
These two functions are directly related to the theoretical requirement of the weighting, as 
derived in Appendix C. This requirement is that the weight matrix be proportional to the inverse 
of the variance-covariance matrix of the observation errors. For a diagonal weight matrix, this re-
quirement means that the weights of equation 1 need to be proportional to 1 divided by the variance 
of the measurement error. More detail on how to determine values for weights and interpret regres-
sion results relative to the weighting is presented in the section "Guideline 6: Assign weights which 
reflect measurement errors."