The use of regularized inversion in groundwater model calibration and prediction uncertainty analysis

Abstract

Groundwater models have a characteristic, common to other environmental models, whereby the desirable number of parameters required to accurately describe the physical phenomenon being modelled is often greater than the number which can be estimated with the available data. This conflict between the need to represent physical detail to accurately model system behaviour, and the sparse datasets that are typically available, leads to the underdetermined nature of model calibration problems. In underdetermined groundwater problems model parameterisation is conventionally undertaken using some kind of simplifying device, to allow a unique model solution to be obtained. Such simplifying devices are either: parameter lumping using zones of constant parameter value; or the use of mathematical regularisation, which allows a multiplicity of parameters to be defined prior to calibration, but which places constraints on parameter relationships or values, such that the calibrated field is a still a simplified version of reality. Both these simplifying devices convert the underdetermined problem to an overdetermined problem that can be solved with standard parameter estimation methods. Model predictions made by these simplified models always have error, which is caused by two sources: (i) observation errors, and (ii) model structural errors, from model inadequacy including, specifically, inadequate levels of parameter detail that occur in underdetermined models. Classical predictive uncertainty analysis methods ignore the second of these sources, however this is often the most significant error source for underdetermined problems. An alternative to the overdetermined, simplifying, approaches described above, is to retain “true” scale parameterisation detail, to allow accurate representation of the physical process being modelled. The drawback of this method is that nonunique parameter solutions result, and so Monte Carlo methods are required; whereby the model is run with multiple parameter fields that have been generated on the basis of a prior hydraulic property variability description, and which all honour calibration constraints. The output of such methods is a full prediction cumulative density function, which encompasses the true prediction and hence defines the prediction error. Monte Carlo methods are computationally very expensive for detailed groundwater parameterisations. Consequently a method which describes the parameter and prediction uncertainty of a single calibrated groundwater model is required if reporting on model prediction uncertainty is to become standard practice, as it should, to avoid the misleading reporting of model results. Regularisation enables a mathematical definition of the relationship between “true”, detailed, hydraulic properties and a simplified calibrated field to be made. This relationship is embodied in the so-called Resolution matrix. This relationship forms the basis for predictive uncertainty analysis for underdetermined models, as is discussed below. This thesis employs resolution analysis to examine the effect of hydraulic property variability on the prediction uncertainty of contaminant transport in groundwater. In particular, this prior knowledge of hydraulic property variability is quantified by means of the variogram as a geostatistical descriptor. This approach provides: • a mathematical description of the likely error of a prediction made by a calibrated model. This description takes account of the two sources of model prediction error, viz. observation errors, and parameter simplification; • the ability to assess the predictive uncertainty of a model prior to the calibration process being undertaken; • the ability to tailor the calibration process such that it can be undertaken to reduce the variance of the prediction it is required to make; • the ability to infer either the nature of true hydraulic property variability, or alternatively to infer measurement error, on the basis of the structure of the calibrated parameter field; • the ability to obtain solutions to the predictive error variance equation in a nonlinear context, via four alternative algorithms presented herein. These results demonstrate that there are far reaching benefits to discretising parameters at the level of detail on which a prediction depends, prior to calibration; despite being unable to uniquely parameterise this detail. Not least of these benefits is the enabling of decision makers to consider the root causes of a model’s predictive uncertainty; to judge what is to be gained from a calibration exercise in terms of prediction uncertainty reduction; and to determine whether such an exercise is cost-effective; and to make better judgements on where best to concentrate future data acquisition efforts.