PEST employs a least squares objective function. The lower is the objective function, the better is the fit between model outputs and field measurements (and between the preferred and observed values of prior information equations if these are featured in the inverse problem).

Mathematically, there are plenty of other ways to define an objective function. However least squares is convenient when working in highly-parameterised contexts. It is a natural consequence of seeking uniqueness and numerical stability through singular value decomposition. Furthermore, where parameters and measurement noise have normal distributions, the outcomes of the history-matching process have neat Bayesian interpretations.

We define a residual as the difference between a field measurement and a corresponding model output. It is also the difference between the desired value of a prior information equation and the value of that equation when calculated using current parameter values. Generally a residual is taken to be positive if a measurement exceeds the partnered model output.

Next we associate a weight with each observation (or prior information equation). The objective function is then calculated by first multiplying each residual by its weight, squaring the product of the two, and then summing. That is, the objective function is the sum of squared weighted residuals.

In the above equation wi is the weight assigned to the i'th observation while ri is the i'th residual.

Observations (and prior information equations) can be subdivided into groups. An objective function can be calculated for each group. Group objective functions are summed to calculate the total objective function.

Monitoring group objective functions can be a very handy way of ensuring that each group is "heard" when history-matching a model. If, at the start of the history-matching process, one group's contribution to the total objective function dominates that of others, then PEST may scarcely see the other groups. This will prevent it from harvesting the information that they hold. Alternatively, if the contribution to the objective function made by one particular observation group is tiny at the start of the history-matching process, then PEST will take little notice of that group as it attempts to reduce the total objective function. This is because far greater gains in objective function reduction can be realised by reducing residuals that are associated with other groups.

It is the job of a modeller to subdivide observations and prior information equations into different groups him/herself.

However when running PEST in "regularisation" mode, a natural subdivision of observation groups occurs. This is between observations that have field measured counterparts on the one hand, and observations and prior information equations that are used for regularisation purposes on the other hand. Residuals from the first of these groups contribute to the "measurement objective function" whereas residuals from the second of these groups contribute to the "regularisation objective function". PEST is allowed to alter weights associated with members of the latter group.

A modeller may decide to calculate a group objective function using a covariance matrix instead of weights. Let C be this matrix. Then the objective function for that group is calculated as:

Φ=rTC-1r

In this equation bold letters indicate vectors, the "T" superscript indicates transpose and the "-1" superscript indicates matrix inversion.

The subject of measurement weights is a complex one. We only touch on it here and save a more detailed discussion for other pages.

Theoretically, the weight that is assigned to each observation should be inversely proportional to the standard deviation of measurement noise that is associated with that observation. The same constant of proportionality should hold for all observations.

Similar considerations apply to use of a covariance matrix instead of weights. The covariance matrix that is assigned to a group of observations should be the covariance matrix of measurement noise that is associated with that group (or be proportional to it).

Use of a covariance matrix instead of weights implies that noise is correlated between measurements. Where a covariance matrix is used with prior information equations that ascribe preferred values to parameters, the covariance matrix is a measure of prior parameter correlation. This is important for spatial parameters such as pilot points; spatial correlation expresses suspected spatial continuity of subsurface hydraulic properties.

It can be shown that pursuit of this philosophy of weights and covariance matrix assignment leads to minimisation of parameter and predictive bias. This makes sense. The less credible is an observation, the less prominently should it feature in the inversion process. The lower, therefore, should be its ascribed weight.

However while this philosophy of weights assignment is logical, it also ignores some important issues. One of these is the "noise" that is generated by simulator structural imperfections. This is often referred to as "structural noise". This noise definitely IS correlated; however we do not know its correlation matrix. Furthermore, structural noise:

•can affect some model output types more than others;

•can sometimes be filtered out by "playing to a model's strengths", in particular the fact that many models are better at simulating temporal and spatial differences than absolutes.

If you wish to remove an observation from the history-matching process, you do not need to re-build a PEST control file. Simply set its weight to zero.