"Prior information" is a fancy term for "observations that are linear functions of parameters". The fact that they are linear (rather than nonlinear) functions of parameters has certain consequences.

Firstly, these functions can be expressed as equations. Each equation is a summation of parameter values multiplied by constant coefficients.

Secondly, the sensitivity of the outcome of an equation to parameters that feature in the equation does not change from iteration to iteration. Hence elements of the Jacobian matrix that record these sensitivities can be filled at the beginning of an inversion process, and left untouched from then on.

Thirdly, if these equations are short and do not cite too many parameters, they can be written directly in the PEST control file. So we do not have to go to the trouble of building a model component that evaluates them.

Like any other observation, a prior information equation must be provided with a weight. Furthermore, the equation must be assigned to an observation group (often referred to as a "prior information group"). If desired, a covariance matrix can be assigned to all prior information equations that comprise a group instead of individual weights. This is handy when prior information equations are used to implement Tikhonov regularisation.

Prior information equations must be recorded in the "prior information" section of a PEST control file. The protocol should be obvious from the following excerpt from that section. See the PEST manual for full details.

Part of the "prior information" section of a PEST control file.

Each line of the prior information section of a PEST control file begins with the name of an equation. (This name has the same role as the name of an observation). The equation follows; it must end with the "observed value" of the equation. This is followed by the equation's weight, and then by the name of the prior information group to which the equation belongs. If the equation is long, it can be continued onto multiple lines.

Where a parameter is declared as log-transformed in the "parameter data" section of a PEST control file, then PEST estimates its log. To satisfy linearity requirements, prior information equations that cite this parameter must therefore cite its log (to base 10).

Because each prior information equation is basically an observation, the PEST-calculated objective function increases when parameter values force the outcomes of equations to differ from their observed values. Of course, other components of the objective function are calculated on the basis of model-to-measurement misfit. So the situation often arises that model-to-measurement misfit is traded-off against respect for preferred parameter conditions that are encapsulated in prior information equations. This trade-off can be expressed using an "L curve".

You can add prior information to a PEST control file using a text editor. If you make a mistake, PESTCHEK will find it for you.

Constraints on parameter values form the backbone of Tikhonov-regularised inversion. These constraints are easily formulated using prior information equations. PEST and associated utility programs provide plenty of help in adding prior information equations to a PEST control file.

ADDREG1, ADDREG2 and ADDPRED1 (these utility programs are supplied with PEST) are the easiest of these utilities to use. Each of them writes one equation for each adjustable parameter that is featured in the "parameter data" section of a PEST control file. This equation states that the parameter (or its log) is equal to the initial value (or its log) of the parameter. An arbitrary weight of 1.0 is assigned to each equation. The names of groups to which the prior information equations are assigned are formulated from the names of parameter groups to which the parameters that feature in them belong. A prefix of "regul_" is added to the parameter group name. So it is a good idea to keep parameter group names short.

Programs GENREG, PPKREG and PPKREG1 from the groundwater utility suite also add prior information equations to a PEST control file. These are particularly useful where parameterisation is based on pilot points. The equations that they introduce can embody more than one parameter. So they can, for example, express preferred parameter value differences. If these differences are ubiquitously zero, then the preferred parameter condition is one of homogeneity. Weights that are assigned to these equations can be uniform, or can be calculated from the distances between pilot point parameters which are featured in the equation. (Note that PPKREG and PPKREG1 provide only a subset of the functionality of GENREG.)

The SUBREG1 utility that is supplied with PEST can remove prior information equations from a PEST control file.