Spatial prediction and spatio-temporal modelling on river networks

The application of existing geostatistical theory to the context of stream networks provides a number of interesting and challenging problems. The most important of these is how to adapt existing theory to allow for stream, as opposed to Euclidean, distance to be used. Valid stream distance based models for the covariance structure have been denied in the literature, and this thesis explores the use of such models using data from the River Tweed. The data span a period of twenty-one years, beginning in 1986. During this time period, up to eighty-three stations are monitored for a variety of chemical and biological determinands. This thesis will focus on nitrogen, a key nutrient in determining water quality, especially given the Nitrates Directive (adopted in 1991) and the Water Framework Directive(adopted in 2002). These are European Union legislations that have set legally enforcable guidelines for controlling pollution which national bodies must comply with. The focus of analysis is on several choices that must be made in order to carry out spatial prediction on a river network. The role of spatial trend, whether it be based on stream or Euclidean distance, is discussed and the impact of the bandwidth of the estimate of nonparametric trend is explored. The stream distance based "tail-up" covariance model structure of Ver Hoef and Peterson (2010) is assessed and combined with a standard Euclidean distance based structure to form a mixture model. This is then evaluated using crossvalidation studies in order to determine the optimum mixture of the two covariance models for the data. Finally, the covariance models used for each of the elements of the mixture model are explored to determine the impact they have on the lowest root mean squared error, and the mixing proportion at which it is found. Using the predicted values at unobserved locations on the River Tweed, the distribution of yearly averaged nitrate levels around the river network is predicted and evaluated. Changes through the 21 years of data are noted and areas exceeding the limits set by the Nitrates Directive are highlighted. The differences in fitted values caused by using stream or Euclidean distance are evident in these predictions. The data is then modelled through space and time using additive models. A novel smoothing function for the spatial trend is defined. It is adapted from the tail-up model in order to retain its core features of flow connectivity and flow volume based weightings, in addition to being based on stream distance. This is then used to model all of the River Tweed data through space and time and identify temporal trends and seasonal patterns at different locations on the river