UC Santa Barbara

This dissertation concerns the properties and relationships of discernible geographical features or their parts, particularly local extrema. As distinctive cases of rapid change, their local variation imbues a high degree of uncertainty. Scale is involved with this uncertainty, partially by generalization with samples, but also by cross-scale variation of spatial dependence. This research investigates whether geographical features and their parts are classifiable by attribute values and also by patterns of spatial dependence with respect to scale. The first chapter, on surface network features, evaluated classifications of terrain data as either peak, pit, pass, ridge, or course features, all local extrema. Results were reviewed with regard to spatial resolution, terrain variability, and algorithms. Large differences in algorithm results were nearby potential features. Quantitative measures found smaller differences for the crisp features of high variability terrain compared to ambiguous low variability terrain. Due to multi-scale characteristics, every location had a degree of membership in every feature class. Membership values enabled the extraction of dominant features and a quantification of uncertainty. The second chapter focuses on spatial outliers, similar to peaks as locally extreme values. A controlled study was performed to extract, with three algorithms, spatial outliers simulated as Gaussian forms of various heights and widths. Raster grids of outliers were created with various resolutions and assignment operators. Results varied most by outlier width and spatial resolution. The algorithms missed the top regions of wide outliers, spanning multiple raster cells, likely due to the presence of high local spatial autocorrelation with a mismatch between the scale of analysis and the scale of an outlier. The third chapter investigated whether non-random patterns of high local spatial autocorrelation exist in wide outliers. Simulations of sets of outliers in variable fields enabled a quantitative comparison with regard to various outlier shapes, fields, and methods. Results of three common random sampling strategies were compared to another method that employed higher probabilities for locations with high local spatial autocorrelation. The latter method resulted in higher rates of both samples on outliers and unique outliers found. Intermediary data revealed patches of high spatial autocorrelation around the outlier tops. The fourth chapter evaluated characteristics of parts of wide spatial outliers. With similar synthetic outliers and fields, patterns of local spatial autocorrelation were compared across classes of the top, side, and base parts of outliers. Small samples and correlated proxy variables were also considered. For samples in each class, a novel multi-scale Local Moran's I "distogram" was computed: a series of values representing the local spatial autocorrelation within each of several non-overlapping spatial bands outward from the point of analysis. The results indicated that the top and side classes have distinctive signatures, while the base co-mingles with the background. In challenging scenarios of small outliers in highly variable fields, differentiation was maintained in bands at about the scale of an outlier. Small samples and proxy variables maintained various degrees of distinction between the signatures. In conclusion, this dissertation investigated the properties and relationships of discernible geographical features and spatial outliers with special regard to representation across scales. Multi-scale information indicates a potential for multiple feature classes at every location. Controlled experiments indicate limitations of spatial outlier detection techniques if the scale of analysis does not match the scale of the feature. Finally, distinctive patterns of local spatial autocorrelation were found for parts of spatial outliers. This research provides empirical evidence that broad-scale local variation involves spatial dependence at a finer scale. As such, this research informs the identification of geographical features or their parts by their variation and spatial dependence across scales.