Methods and guidelines for effective model calibration

ISSUES OF COMPUTER EXECUTION TIME
Computer execution time is often a problem when using regression methods. Thus, a set of 
hints for effective use of regression also needs to include a few ideas about model construction as 
it affects execution time. The suggestion about starting with a relatively simple model of the 
ground-water system and building complexity as warranted by the system and by the available da-
ta, as discussed in guideline 1, also is relevant to the issue of minimizing execution time. Starting 
with a simple model often results in shorter execution times.
Execution times for regression, or inverse, simulations can be estimated using execution 
times for forward simulations (a simulation for hydraulic heads in a ground-water flow problem) 
as:
T
i
= 2(NP) T
f
(1+NP)
(38) 
where 
T
i
is the execution time for the regression (inverse) solution;
T
f
is the execution time for the forward solution; and 
NP is the number of parameters being estimated by regression. 
This assumes that the number of parameter-estimation iterations approximately equals twice the 
number of parameters, that is, 2(NP), which is, on average, typical. The (1 + NP) term is for one 
forward simulation and one simulation to calculate sensitivities for each of the NP parameters. The 
NP sensitivity simulations solve a slight variation of the forward problem for the forward- or back-
ward-difference sensitivities of UCODE, or sensitivity equations that result from taking the deriv-
ative of the forward equation with respect to the parameter in MODFLOWP. In both cases, each 
of the sensitivity simulations take, on average, the same amount of execution time as a forward 
simulation.
Experience indicates that inverse model execution times that exceed about 15 hours (an 
overnight simulation) commonly occur when the forward execution time exceeds 30 minutes. The 
number of grid rows, columns, and layers, and the number of time steps this execution time allows 
depends on the speed of the computer and the characteristics of the simulated system, including the 
contrasts present in the hydraulic-conductivity field. 
Sometimes simple changes in the simulation can dramatically improve execution times. 
For example, the initial hydraulic conductivity structure of the model described by D’Agnese and 
others (1998, in press) was characterized by values in bordering finite-difference cells that differed 
by more than five orders of magnitude in many parts of the model. Introducing single cells of mod-
erate hydraulic conductivity between the high and low valued cells in most of the model resulted 
in about a 6-fold decrease in execution time with little effect on simulated results. 

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Another simplification that can dramatically reduce execution time is to replace nonlinear 
forward problems with linear approximations as much as possible without substantial diminish-
ment of accuracy. In ground-water flow simulations, for example, water-table and convertible lay-
ers (as they are called in MODFLOW and MODFLOWP) often can be replaced by confined layers 
with approximate thicknesses. This is nearly always good practice for steady-state simulations, but 
can be too inaccurate for transient simulations in which layers are substantially de-watered during 
the calibration period. The inaccuracy produced by this simplification can be evaluated by compar-
ing forward simulations that include the water-table and convertible layers with those that include 
the approximate thicknesses.