Tikhonov Regularization

Assuming that the initial parameter value represent the expected values of the expert modeller, equations of prior information are created that will penal­ize departures of parameter values from initial values.

As a result, calibration adjusted parameters will be close to the values pre­ferred from a modeller’s point of view.

This level of regularization suggest a preferred homogeneity in the parameter distributions between adjacent pilot points. PEST creates prior information equations, which basically contains each two adjacent pilot points, and indi­cate a preferred difference of zero.

Typically, such a regularization approach is intended for zones of the model that belong to the same geological unit.

Pilot-point type parameter values for each definition are regularized taking into account the expected correlation (i.e., using a covariance matrix). Differ­ences in parameter values of pilot points closer than their correlation length will be penalized. As a result, smoother parameter definitions will be preferred over heterogeneous ones.

The range parameter in the Kriging Settings indicated for each pilot-point parameter definition is used to compute the covariance matrix. The larger the range value is, the larger the number of pilot points influencing the regulariza­tion weights.

FePEST provides the ability to use the PEST Utility GENREG for defining advantage settings of regularization. GENREG can automatically fill the Prior Information section in the PEST Control file (*.pst) based on user-defined rules. Typical examples are preferred homogeneity, preferred ratio (e.g. for anisotropy settings of conductivity), etc.

This is the limit for the measurement objective function below which the model is considered calibrated.

The value can be calculated by summing up the (weighted) measurement noise associated with the observations. If this is not possible, a value some­what higher than the objective function that results from a calibration run with­out Tikhonov regularization can be chosen.

This additional threshold is usually set slightly (5% to 10%) above PHIMLIM. PHIMLIM and PHIMACCEPT define a buffer zone for the measurement objective function value for stability reasons. In this zone, an objective func­tion value is tolerated even though it does not meet the target value. This is necessary as the parameter upgrade vector will often "miss" the exact limit because it relies on a linearity assumption.