Topics in interpolation and smoothing of spatial data

Abstract

This thesis addresses a number of special topics in spatial interpolation and smoothing. The motivation for the thesis comes from two projects, one being to extend the availability of a daily rainfall model for southern Africa to sites at which little or no rainfall data is available, using data from nearby sites, and the other arising from a need to improve the species abundance estimates used to produce maps for the Southern African Bird Atlas Project in areas where the original presence/absence data is sparse. Although problems of spatial interpolation and smoothing have been the subject of much research in recent years, leading to the development of the specialised discipline of geostatistics, these two problems have features which render the available methodology inappropriate in certain respects. The semi-variogram plays a central role in geostatistical work. In both of the applications considered here, the raw semi-variogram is 'contaminated' by error, but the error variance varies widely between data points, so that the spatial autocorrelation structure of the underlying error-free variable is blurred. An adjusted semi-variogram, which removes the effect of the measurement error, is defined and incorporated into the kriging equations. A number of measures have been proposed for kriging in the presence of trend, ranging from explicit modelling of a deterministic trend function to 'moving window' kriging, which assumes local stationarity as an approximation. The former approach is often inappropriate over large non-homogenous regions, while the latter approach tends to underestimate the kriging variance. As an alternative strategy it is proposed here that the trend function be considered as another random variable, with a long-range spatial autocorrelation. This approach is simple to implement, and can also be used as a basis for filtering the data to separate trend from local or high-frequency variation. The daily rainfall model is based on a Fourier series representation giving rise to amplitude and phase parameters; the latter are circular in nature, and not amenable to analysis by standard techniques. This thesis describes a method of interpolation and smoothing, analogous to kriging, which is appropriate for unit vector data available at a number of spatial locations. The cumulated values of species counts in the SABAP are essentially binomially distributed and thus again specialised techniques are required for interpolation. New geostatistical methods which cater for both binomial and Poisson data are presented. Another problem arises from the need to improve interpolated values of the rainfall model parameters by incorporating information on altitude. Although a number of approaches are possible, for example, using co-kriging or kriging with external drift, difficulties are caused by the complexity of the relationship between the rainfall at a point and the surrounding topography. This problem is overcome by the use of orthogonal functions of altitude to model the patterns of topography.