An analysis of perturbation based methods for the treatment of parameter uncertainty in numerical groundwater models

Monte Carlo methods are robust approaches for the estimation of prediction uncertainty in groundwater flow and transport modelling under uncertain model parameters. However Monte Carlo procedures estimate the prediction statistics by generating a population of solutions from random realisations of the model parameters which are consistent with the parameter statistics, and as a result are computationally demanding. Taylor series based procedures offer an alternative to Monte Carlo methods for calculating the prediction statistics. Two such approaches, the first-order second moment and McLaughlin and Wood's perturbation method, are based on using a Taylor series to derive approximate expressions for the model predictions first and second statistical moments. In this paper the perturbation method presented by McLaughlin and Wood is rederived using Vetter matrix notation. This is compared with the first-order second moment (FOSM) method and while the steady state expressions for these two approaches are shown to be equivalent, the transient forms are considerably different. A new form of the FOSM is derived, which is simpler and has a lower computational burden. However, the transient McLaughlin and Wood expression is found to have a significantly lower computational overhead than either of the FOSM methods presented.