Ever since Halley (1686) superimposed onto a map of land forms. the direction of trade winds and monsoons between and near the tropics and attempted to assign them a physical cause. homo-sapiens has attempted to develop procedures which quantify the level of change in a spatial process. or assess the relationship between associated spatially measured variables. Most spatial data. whether it be originally point. linear or areal in nature. can be converted by a suitable procedure into a continuous form and plotted as an isarithmic map i.e. points of equal height are joined. Once in that form it may be regarded as a statistical surface in which height varies over area in much the same way as the terrain varies on topographic maps. Particularly in environmental statistics. the underlying shape of the surface is unknown. and hence the use of non-parametric techniques is wholly appropriate. For most applications. the location of data points is beyond the control of the map-maker hence the analyst must cope with irregularly spaced data points. A variety of possible techniques for describing a surface are given in chapter two, with attention focusing on the methodology surrounding kernel density estimation. Once a surface has been produced to describe a set of data. a decision concerning the number of contours and how they should be selected has to be taken. When comparing two sets of data. it is imperative that the contours selected are chosen using the same criteria. A data based procedure is developed in chapter three which ensures comparability of the surfaces and hence spurious conclusions are not reached as a result of inconsistencies between surfaces. Contained within this chapter is a discussion of issues which relate to other aspects of how a contour should be drawn to minimise the potential for inaccuracies in the swface fitting methodology. Chapter four focuses on a whole wealth of techniques which are currently available for comparing surfaces. These range from the simplest method of overlaying two maps and visually comparing them to more involved techniques which require intensive numerical computation. It is the formalisation of the former of these techniques which forms the basis of the methodology developed in the following two chapters to discern whether change/association has materialised between variables.One means of quantifying change between two surfaces, represented as a contoured surface, is in terms of the transformation which would be required for the two surfaces to be matched. Mathematically, transformations are described in terms of rotation, translation and scalar change. Chapter five provides a geometrical interpretation of the three transformations in terms of area, perimeter, orientation and the centre of gravity of the contour of interest and their associated properties. Although grid resolution is fundamentally a secondary level of smoothing, this aspect of surface fitting has generally been ignored. However to ensure consistency across surfaces, it is necessary to decide firstly, whether data sets of different sizes should be depicted using different mesh resolutions and secondly, how fine a resolution provides optimal results, both in terms of execution time and inherent surface variability. This aspect is examined with particular reference to the geometric descriptors used to quantify change. The question of random noise contained within a measurement process has been ignored in the analysis to this point. However in practice, some form of noise will always be contained within a process. Quantifying the level of noise attributable to a process can prove difficult since the scientist may be over optimistic in his evaluation of the noise level. In developing a suitable set of test statistics, four situations were examined, firstly when no noise was present and then for three levels of noise, the upper bounds of which were 5, 15 and 25%. Based on these statistics, a series of hypothesis tests were developed to look at the question of change for individual contour levels Le. local analysis. or alternatively for a whole surface by combining the statistics and effectively performing a multivariate test. A number of problems are associated with the methodology. These difficulties are discussed and various remedial measures are proposed. The theoretical derivation of the test statistic, both in the absence and presence of random noise, has proved mathematically to be extremely complex, with a number of stringent assumptions required to enable the theoretical distribution to be derived. A major simulation study was subsequently undertaken to develop the empirical probability distribution function for the various statistics defining change for the four levels of noise. Also for each of the statistics, the resultant power of the test was examined.The remaining chapter explicitly examines two case studies and how the methodology developed in the preceding two chapters may be implemented. The first example cited raises the question, 'Has a seasonal temperature change resulted during the fifty year span, 1930 to 1980, within the contiguous United States of America?' The data base was provided by the United States Historical Climatology Network (HCN) Serial Temperature and Precipitation Data, Quinlan et al (1987). The second problem examines whether there is an association between background radiation levels, within three regions of the south-west England, and the location of various fonns of leukaemia or whether case location is a product of the population distribution. Differences between this example and the previous illustration materialise in terms of the spatial resolution of the data; the leukaemia data are defined as punctual data points and are extremely sparse; the population distribution is defined as areal regions; with the radiation data being of a more continuous format. The methodology developed required modification, but aside of this a preliminary set of conclusions were reached.