PUTTING THE PARAMETERS IN PARAMETER ESTIMATION

A key decision made as part of any model parameter estimation effort - in groundwater or otherwise - is the selection of a parameterization strategy. This decision has profound ramifications on many aspects of the parameter estimation including numerical stability, computational expense, and solution quality. This work focuses on benefits gained from parameter flexibility and illustrates important implications for the use of prediction uncertainty for the acquisition of new observations.

Hadamard (1902) discussed three aspects which, when met, result in well-posed problems. These are existence, uniqueness, and stability of a solution. Existence is generally met in groundwater applications, but both uniqueness and stability pose challenges.

To overcome a lack of uniqueness and stability, practitioners have traditionally reduced the number of unknown parameters to fewer than the number of field observations. Rather than regarding these as discrete, it is useful to consider the amount of information contained in observations that is imparted on parameters through the estimation process. In this way, even with a large and distributed parameter field, a combination of field observations and soft-knowledge (expressed as pseudo-observations) can be specified to meet the conditional uniqueness requirement in that an estimated set of parameters is unique conditional upon the data used for the estimation process. This conditionality is explicitly considered in a Bayesian context, but extends beyond mathematical formalism.

Retaining distributed parameter fields can result in great computational expense and flexibility is accompanied by the possibility of overfitting, resulting in unrealistic parameter estimates. Luckily, the computational expense can be overcome through parallel computing owing to the embarrassingly parallel nature of the computations, and recent advances in cloud computing, accompanied by appropriate tools, have made this power accessible to all practitioners. The problem of overfitting is addressed by balancing soft knowledge with the field observations in such a way that the hydrogeologic understanding of the practitioner plays a direct role in guiding the parameter estimation.

Recent work has explored the role played by parameterization in the use of prediction uncertainty in designing efficient strategies for acquiring new information to calibrate the model. Following a Bayesian approach, a prediction made by a model is also conditional on the data used to calibrate the model. This can be formulated such that data are evaluated using their effect on the certainty of model predictions. This serves two important purposes: first, it recognizes that a model should be designed with a specific objective (prediction) in mind; and second, field campaigns can be designed such that observations that best meet the objective of the model are given priority. While several approaches to this are possible, and some are more expensive than others, linear propagation of uncertainty has been shown to be a useful compromise between expense and rigor. However, when a model is restricted in its flexibility through limiting the number of parameters, spurious results are possible.

References

Hadamard, J. (1902). "Sur les problèmes aux dérivées partielles et leur signification physique." Princeton University Bulletin 13: 49-52.