Spatio-temporal models for the analysis and optimisation of groundwater quality monitoring networks

Commonly groundwater quality data are modelled using temporally independent spatial models. However, primarily due to cost constraints, data of this type can be sparse resulting in some sampling events only recording a few observations. With data of this nature, spatial models struggle to capture the true underlying state of the groundwater and building models with such small spatial datasets can result in unreliable predictions. This highlights the need for spatio-temporal models which `borrow strength' from earlier sampling events and which allow interpolations of groundwater concentrations between sampling points. To compare the relative merits of analysing groundwater quality data using spatial compared to spatio-temporal statistical models, a comparison study is presented using data from a hypothetical contaminant plume along with a real life dataset. In this study, the estimation accuracy of spatial p-spline and Kriging models are compared with spatio-temporal p-spline models. The results show that spatio-temporal methods can increase prediction efficiency markedly so that, in comparison with repeated spatial analysis, spatio-temporal methods can achieve the same level of performance but with smaller sample sizes. For the comparison study, in the spatio-temporal p-splines model, differing levels of variability over space and time were controlled using different numbers of basis functions rather than separate smoothing parameters due to the computational expense of their optimisation. However, deciding on the number of basis functions for each dimension is subjective due to space and time being measured on different scales, and thus methodology is developed to efficiently tune two smoothing parameters. The proposed methodology exploits lower resolution models to determine starting points for the optimisation procedure allowing for each parameter to be tuned separately. Working with spatio-temporal models can, however, pose their own problems. Due to the sporadic layout of many monitoring well networks, due to built-up urban areas and transport infrastructure, ballooning can be experienced in the predictions of these models. `Ballooning' is a term used to describe the event where high or low predictions are made in regions with little data support. To determine when this has occurred a measure is developed to highlight when ballooning may be present in the models predictions. In addition to the measure, to try to eliminate ballooning from happening in the first place, a penalty based on the idea that the total contaminant mass should not change significantly over time is proposed. However, the preliminary results presented here indicate that further work is needed to make this effective. It is shown that by adopting a spatio-temporal modelling framework a smoother, clearer and more accurate prediction through time can be achieved, compared to spatial modelling of individual time steps, whilst using fewer samples. This was shown using existing sampling schemes where the choice of sampling locations was made by someone with little knowledge or experience in sampling design. Sampling designs on fixed monitoring well networks are then explored and optimised through the minimisation two objective functions; the variance of the predicted plume mass (VM) and the integrated prediction variance (IV). Sampling design optimisations, using spatial and spatio-temporal p-spline models, are carried out, using a variety of numbers of wells and at various future sampling time points. The effects of well-specific sampling frequency are also investigated and it is found that both objective functions tend to propose wells for the next sampling design which have not been sampled recently. Often, an existing monitoring well network will need to be changed, either by adding new wells or by down-scaling and removing wells. The decision to add wells to the network comes at a financial expense, so it is of paramount importance that wells are added into areas where the gain in knowledge of the region is maximised. The decision to remove a well from the network is equally important and involves a trade-off between costs saved and information lost. The design objective functions suggest a well should be added in an area where the distance to the nearest neighbouring wells is greatest. Finally, consideration is given to optimal sampling designs when it is assumed the recorded data has multiplicative error - a common assumption in groundwater quality data. When modelling with this type of data, the response is normally log transformed prior to modelling and the predictions are then transformed back onto the original scale for interpretation. Assuming a log transformed response, the objective functions, initially presented, can be used if computation of the objective function is also on the log scale. However, if the desired scale of interpretation of the objective functions is the original scale but modelling was performed on the log scale, the resulting objective function values cannot simply be exponentiated to give an interpretation on the original scale. Modelling on the log scale while interpreting the objective function on the original scale can be achieved by adopting a lognormal distribution for the predicted response and subsequently numerically integrating its variance to compute the IV objective function. The results indicate that the designs do differ depending on which scale interpretation of the objective function is to be made. When interpreting on the original scale the objective function favours sampling from wells where higher values were previously estimated. Unfortunately, computation of the VM objective function when assuming a lognormal distribution has not been achieved so far