Highly-Parameterized Inversion

As well as traditional parameter estimation, PEST supports highly parameterized inversion. In doing this it employs similar technologies to those which are used in geophysical data analysis, image processing, and other fields where extracting all possible information from expensive datasets is of paramount importance.

Because of the limited information that is available in most environmental datasets, model calibration necessarily leads to a simplified parameter field compared with the complex, heterogeneous disposition of physical and chemical properties that characterizes real-world environmental systems. The trick is to do this simplification in such a way that the calibrated parameter field is of minimum potential wrongness. This does not mean that it is right - only that it is the simplest, sensible parameter set that is compatible with the data. Knowing that it will be wrong (as it must be as a simplified version of reality), it follows that in calibrating a model we must seek that parameter field for which its potential for wrongness is roughly symmetrically disposed with respect to it.

So how should the simplification that is necessary for model calibration be achieved? One option is to do this simplification manually before calibration commences. Alternatively, we can allow the simplification to take place as part of the calibration process itself. The latter is preferable for, if properly done, it can indeed guarantee a parameter set of minimum error variance.

It is through PEST's use of sophisticated mathematical regularization techniques that it is able to accommodate the use of hundreds, or even thousands, of parameters in the calibration process with unwavering numerical stability, and a guarantee of parameter reasonableness. The more parameters that are employed, the more flexibility that mathematical regularization techniques have in attaining simple parameter fields of maximum reasonableness.

Mathematical regularization techniques employed by PEST include the following.

• Tikhonov regularization;
• Truncated singular value decomposition;
• LSQR;
• "SVD-assist"
• and any combination of these.

The "SVD-assist" methodology is unique to PEST. It allows highly parameterized inversion to be employed in the calibration of complex environmental models with run-time efficiencies that are normally associated with the use of only a few parameters. Through the use of this device it is now commonplace to employ hundreds, or even thousands, of parameters in the calibration of complex three-dimensional models with heterogeneous parameter fields, even if these models take over an hour to run. Further gains in numerical efficiency are made if these model runs are parallelized.

Hydraulic conductivity field of a large regional groundwater model, calibrated using PEST's SVD-assist technique.

Advantages of a highly-parameterized approach to model calibration include the following.

• A modeller does not need to agonize over what parameters to hold fixed and what parameters to adjust during the calibration process. He/she simply declares all parameters as adjustable. PEST then adjusts those parameter combinations for which there is information in the calibration dataset, while leaving other parameter combinations alone.
• The model is not denied access to information that is resident in the calibration dataset just because adjustable parameters which could have received that information have been arbitrarily excluded from it.
• Calibrated parameter fields deviates minimally from a "preferred parameter state" set by the user in accordance with expert knowledge of the model domain. This guarantees maximum parameter reasonableness.
• Excellent fits can be obtained between model outputs and field data. At the same time, "over-fitting" is prevented as PEST enforces a sensible limit on the goodness of fit achieved through the highly parameterized inversion process.
• Spatial parameter fields that emerge from the regularized inversion process always have a "better feel" than those achieved through the use of manual parsimonization devices such as zones of piecewise constancy.
• Recognition of the existence of real-world system complexity, even if it cannot be uniquely estimated on the basis of a limited calibration dataset, is essential to the integrity of parameter and predictive uncertainty analysis.