Regular Hierarchical Surface Models: A conceptual model of scale variation in a GIS and its application to hydrological geomorphometry

Environmental and geographical process models inevitably involve parameters that vary spatially. One example is hydrological modelling, where parameters derived from the shape of the ground such as flow direction and flow accumulation are used to describe the spatial complexity of drainage networks. One way of handling such parameters is by using a Digital Elevation Model (DEM), such modelling is the basis of the science of geomorphometry. A frequently ignored but inescapable challenge when modellers work with DEMs is the effect of scale and geometry on the model outputs. Many parameters vary with scale as much as they vary with position. Modelling variability with scale is necessary to simplify and generalise surfaces, and desirable to accurately reconcile model components that are measured at different scales. This thesis develops a surface model that is optimised to represent scale in environmental models. A Regular Hierarchical Surface Model (RHSM) is developed that employs a regular tessellation of space and scale that forms a self-similar regular hierarchy, and incorporates Level Of Detail (LOD) ideas from computer graphics. Following convention from systems science, the proposed model is described in its conceptual, mathematical, and computational forms. The RHSM development was informed by a categorisation of Geographical Information Science (GISc) surfaces within a cohesive framework of geometry, structure, interpolation, and data model. The positioning of the RHSM within this broader framework made it easier to adapt algorithms designed for other surface models to conform to the new model. The RHSM has an implicit data model that utilises a variation of Middleton and Sivaswamy (2001)’s intrinsically hierarchical Hexagonal Image Processing referencing system, which is here generalised for rectangular and triangular geometries. The RHSM provides a simple framework to form a pyramid of coarser values in a process characterised as a scaling function. In addition, variable density realisations of the hierarchical representation can be generated by defining an error value and decision rule to select the coarsest appropriate scale for a given region to satisfy the modeller’s intentions. The RHSM is assessed using adaptions of the geomorphometric algorithms flow direction and flow accumulation. The effects of scale and geometry on the anistropy and accuracy of model results are analysed on dispersive and concentrative cones, and Light Detection And Ranging (LiDAR) derived surfaces of the urban area of Dunedin, New Zealand. The RHSM modelling process revealed aspects of the algorithms not obvious within a single geometry, such as, the influence of node geometry on flow direction results, and a conceptual weakness of flow accumulation algorithms on dispersive surfaces that causes asymmetrical results. In addition, comparison of algorithm behaviour between geometries undermined the hypothesis that variance of cell cross section with direction is important for conversion of cell accumulations to point values. The ability to analyse algorithms for scale and geometry and adapt algorithms within a cohesive conceptual framework offers deeper insight into algorithm behaviour than previously achieved. The deconstruction of algorithms into geometry neutral forms and the application of scaling functions are important contributions to the understanding of spatial parameters within GISc

Flooding countries and destroying dams

In many applications of terrain analysis, pits or local minima are considered artifacts that must be removed before the terrain can be used. Most of the existing methods for local minima removal work only for raster terrains. In this paper we consider algorithms to remove local minima from polyhedral terrains, by modifying the heights of the vertices. To limit the changes introduced to the terrain, we try to minimize the total displacement of the vertices. Two approaches to remove local minima are analyzed: lifting vertices and lowering vertices. For the former we show that all local minima in a terrain with n vertices can be removedintheoptimalwayinO(n log n) time. For the latter we prove that the problem is NP-hard, and present an approximation algorithm with factor 2 ln k, wherek is the number of local minima in the terrain