Aportaciones al estudio de la anisotropía y modelado espacial de información

Resum

When the visualization and observation of the analyzed process does not adequately describe his behaviour in space, it is necessary to design a mathematical model that explains and summarize it. Usually, the space phenomena are the result of the sum of two types of effects: global and local. The overall effects appear when there are significant variations in the space of the average value of the process. The local effects are shown when the average value at one point is influenced by the average value at neighbouring points. The design of spatial models is only possible under assumption of homogeneity in the global and/or local effects. If no kind of stationarity is assumed, it would be extremely complex to model the phenomenon due to the high number of necessary parameters for the mathematical formalization of its behaviour. Moreover, it is necessary to determine if the phenomenon presents anisotropic tendencies, since this fact would invalidate the employment of models defined for isotropic functions, and would make necessary to use space transformations to get anisotropic functions from isotropic models. For this reason, the spatial processes modelling requires not only an analysis of stationarity in global and local variations, but also a directional study to determine the existence of an anisotropic component. Many and varied tools are used for analysing stationarity: Delaunay triangulation, spatial movable average, k-functions and covariance functions, variograms, statistics of spatial autocorrelation (Moran´s I, Geary´s C, Geti´s G), autocorrelograms, spatial trend.... Techniques used for the analysis of anisotropy are not many, and they just detect directional effects by inspecting experimental variograms in different directions. The examination of these variograms �by rough estimate� only provides an informal verification of the existence of anisotropy and the approximated determination of the direction in which this one is pronounced. As a result of this, in this report appears: � The analysis of the problems associated with determining the existence and quantifying the component in directional patterns. � A study on the state-of-the-art of anisotropy, especially in the aspect related to the data spatial modelling. � The analysis of the problems associated with determining the existence and quantifying of the component in directional patterns. � The adaptation of the existing circulars statistics to analyze the directional behaviour of natural phenomena within the overall process of analysing the first order properties (homogeneity) and second order ones (autocorrelation and anisotropy) in patterns. � A test for the existence of addressing in patterns that present spatial dependence assuming normality of the population. � A method for the design of generating ovals in which the surface is distributed as Normal form around its dominant direction. These methods have been distinguished and adapted to digital images, and more specifically to satellite images.