At this step, choose which regularization options you want to use in the PEST Run.

One of the great advantages of using pilot points is that we can distribute a superfluity of those points throughout the model domain and then ask PEST to find for itself those regions within the study area where transmissivity must be greater or less than average in order to ensure that there is good agreement between model outputs and field measurements. If we had based our parameterization of the model domain solely on zones, we might not have placed those zones in the correct position for the calibration process to properly infer the existence or extent of such heterogeneity.

The introduction of regularization into the calibration process serves two purposes. Firstly it brings a high degree of numerical stability to a parameter estimation problem which would otherwise be highly susceptible to the deleterious effects of a singular normal matrix (you might have noticed when inspecting hcal.rec that PEST was not able to calculate any parameter statistics due to singularity of the normal matrix.) Secondly, if regularization constraints are appropriately defined, model calibration can proceed with a “homogeneous unless proven otherwise” philosophy; that is, in spite of the number of parameters at its disposal, PEST will make each zone within the model domain as uniform as it can in terms of the distribution of the estimated hydraulic property, introducing heterogeneity into a zone only where this is necessary in order to allow a good of fit between model outputs and field data to be achieved. Hence any heterogeneity which is introduced as an outcome of the calibration process is “there because it has to be there”. In many modeling contexts this philosophy of model calibration has a large intuitive appeal, allowing a modeler to use zones to characterize the distribution of some hydraulic property within a model domain while, at the same time, removing the inflexibility that accompanies the characterization of a model domain by areas of piecewise parameter constancy.

Relationships between pairs of parameter values can be introduced into the calibration process as prior information equations. The weight assigned to each of these prior information equations can be the same. Alternatively, if the weight is proportional to the inverse of the square root of the variogram calculated for the distance between the respective pilot points, then it can be shown that this is in harmony with the geostatistical characterization of the area as encapsulated in the variogram. What this characterization says, in short, is that “the closer are two points together, the more likely are the hydraulic properties at those points to be the same”. By calculating weights on the basis of the inverse of the variogram, we are enforcing the “zero difference” condition more strongly for points which are closer together than for those which are farther apart.

When run in this mode, a number of control variables are required in the PEST control file, in addition to those required when PEST is run in “parameter estimation” mode. One of these variables is PHIMLIM. This specifies the degree of model-to-measurement misfit that is allowed to occur in the present optimization process. Because the attainment of a good model-to-measurement fit, and the simultaneous enforcement of homogeneity constraints, may place conflicting requirements on parameter values, a compromise between the two must be reached. The user determines the “compromise level” by setting a maximum model-to-measurement misfit that he/she will tolerate, this misfit being expressed in terms of the “measurement objective function”. The maximum permissible value of the measurement objective function (ie. PHIMLIM) should be set a little higher than the objective function than it is possible to achieve without any regularization constraints being enforced.

Each prior information equation included in the parameter estimation process must be assigned a weight. As was discussed above, weights are calculated on the basis of geostatistical information available (or assumed) for the model area. If an observation or prior information equation is used for regularization purposes, then it is assigned to the observation group “regul”. As part of its regularization functionality, PEST adjusts the weights assigned to all members of this group during each iteration of the optimization process; however the relative weight values within this group remain the same. The “regularization weight factor” by which the initial weights of all members of the group “regul” are multiplied during each optimization iteration is calculated in such a way as to respect the PHIMLIM value provided by the user as the maximum tolerable model-to-measurement misfit for the current case. An initial regularization weight factor needs to be supplied by the user.

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