Kevin Sahr

A COMPARISON OF RESERVE SELECTION ALGORITHMS USING DATA ON TERRESTRIAL VERTEBRATES IN OREGON

We compare the number of species represented and the spatial pattern of reserve networks derived using five types of reserve selection algorithms on a set of vertebrate distribution data for the State of Oregon (USA). The algorithms compared are: richness-based heuristic algorithms (four variations), weighted rarity-based heuristic algorithms (two variations), progressive rarity-based heuristic algorithms (11 variations), simulated annealing, and a linear programming-based branch-and-bound algorithm. The linear programming algorithm provided optimal solutions to the reserve selection problem, finding either the maximum number of species for a given number of sites or the minimum number of sites needed to represent all species. Where practical, we recommend the use of linear programming algorithms for reserve network selection. However, several simple heuristic algorithms provided near-optimal solutions for these data. The near-optimality, speed and simplicity of heuristic algorithms suggests that they are acceptable alternatives for many reserve selection problems, especially when dealing with large data sets or complicated analyses.

Geodesic Discrete Global Grid Systems

In recent years, a number of data structures for global geo-referenced data sets have been proposed based on regular, multi-resolution partitions of polyhedra. We present a survey of the most promising of such systems, which we call Geodesic Discrete Global Grid Systems (Geodesic DGGSs). We show that Geodesic DGGS alternatives can be constructed by specifying five substantially independent design choices: a base regular polyhedron, a fixed orientation of the base regular polyhedron relative to the Earth, a hierarchical spatial partitioning method defined symmetrically on a face (or set of faces) of the base regular polyhedron, a method for transforming that planar partition to the corresponding spherical/ellipsoidal surface, and a method for assigning point representations to grid cells. The majority of systems surveyed are based on the icosahedron, use an aperture 4 triangle or hexagon partition, and are either created directly on the surface of the sphere or by using an equal-area transformation. An examination of the design choice options leads us to the construction of the Icosahedral Snyder Equal Area aperture 3 Hexagon (ISEA3H) Geodesic DGGS.

Comparing Geometrical Properties of Global Grids

Transforming raw observations into geometrically regular global grids is a fundamental data processing and storage problem underlying much of our global data analysis. The basic geometry of traditionally employed quadrilateral-based point or area grids, while well suited to array storage and matrix manipulation, may inherently hinder numerical and geostatistical modeling efforts. Several scientists have noted the superior performance of triangular point grids and associated triangular cells that can be aggregated into hexagonal surface tessellations, yet, no thorough evaluation of discrete global grid alternatives has been conducted. We present results from a global grid comparison study that focused on recursive tiling of polyhedral faces projected onto the globe. A set of evaluation criteria for global partitioning methods were developed. Of these, metrics for spheroidal surface area, compactness, and centerpoint spacing were found to be of particular importance. We present examples of these metrics app...

Finding all optimal solutions to the reserve site selection problem: formulation and computational analysis

The problem of selecting nature reserves has received increased attention in the literature during the past decade, and a variety of approaches have been promoted for selecting those sites to include in a reserve network. One set of techniques employs heuristic algorithms and thus provides possibly sub-optimal solutions. Another set of models and accompanying algorithms uses an integer programming formulation of the problem, resulting in an optimization problem known as the Maximal Covering Problem, or MCP. Solution of the MCP provides an optimal solution to the reserve site selection problem, and while various algorithms can be employed for solving the MCP they all suffer from the disadvantage of providing a single optimal solution dictating the selection of areas for conservation. In order to provide complete information to decision makers, the determination of all alternate optimal solutions is necessary. This paper explores two procedures for finding all such solutions. We describe the formulation and motivation of each method. A computational analysis on a data set describing native terrestrial vertebrates in the state of Oregon illustrates the effectiveness of each approach.

Comparing area and shape distortion on polyhedral-based recursive partitions of the sphere

Regular grid sampling structures in the plane are a common spatial framework for many studies. Constructing grids with desirable properties such as equality of area and shape is more difficult on a sphere. We studied the distortion characteristics of recursive partitions of the surface of the globe starting with the octahedron and icosahedron polyhedral models. We used five different methods for mapping from the polyhedral model to the surface of the sphere: the Gnomonic projection, Fullers Dymaxion projection, Snyders equal area polyhedral projection, direct spherical subdivision, and a recursive polyhedral projection. We increased partition density using both a 4-fold and a 9-fold ratio at each level of recursive subdivision by subdividing to the 8th level with the 4-fold density ratio (65 536 cells per polyhedral face) and to the fifth level with the 9-fold density ratio (59 049 cells per polyhedral face). We measured the area and perimeter of each cell at each level of recursion for each method on e...

Location coding on icosahedral aperture 3 hexagon discrete global grids

Discrete global grid systems (DGGSs) represent a relatively new, but increasingly popular, approach to the problem of representing geospatial location on computer systems. Despite growing interest amongst potential users in icosahedral aperture 3 hexagon DGGSs, the practical use of such systems has been hindered by a lack of efficient spatial indexing methods. In this paper we discuss the two primary approaches to developing multi-resolution location coding systems for DGGSs: pyramid addressing and path addressing. We then describe an efficient pyramid addressing system for icosahedral aperture 3 hexagon DGGSs, the quadrilateral 2-dimensional integer system. After reviewing the problems inherent in developing path addressing systems for hexagon-based DGGSs we describe a class of path-based location coding solutions for icosahedral aperture 3 hexagon DGGSs called modified generalized balanced ternary, and show how this system can be used to index vector data. We then discuss a subset of this system, the icosahedral aperture 3 hexagon tree, which can be used to index raster and bucket data structures. Conversion algorithms to/from geodetic coordinates are discussed.

Planar and spherical hierarchical, multi-resolution cellular automata

We present a new generalized definition of spatial hierarchy and use it to create a data structure for spatial hierarchies on the plane and the sphere. The data structure is then used as the basis for hierarchical, multi-resolution cellular automata which are topology-independent so that many topologies may be studied. In these systems the dynamics of a focal cell is dependent on its neighbors and also the cell above and below it in the next coarser or finer resolution. Results from a multi-resolution version of the Game of Life show complex and unexpected behavior which is dependent on the topology chosen and initial conditions. These results and the software which produced them provide a proof of concept for the new data structure and algorithms. These may be especially useful for data analysis and simulation at the global scale.

A comparison of intercell metrics on discrete global grid systems

Abstract A discrete global grid system (DGGS) is a spatial data model that aids in global research by serving as a framework for environmental modeling, monitoring and sampling across the earth at multiple spatial scales. Topological and geometric criteria have been proposed to evaluate and compare DGGSs; two of which, intercell distance and the “cell wall midpoint” criterion, form the basis of this study. We propose evaluation metrics for these two criteria and present numerical results from these measures for several DGGSs. We also consider the impact of different design choices on these metrics, such as predominant tessellating shape, base modeling solid and partition density between recursive subdivisions. For the intercell distance metric, the Fuller–Gray DGGS performs best, while the Equal Angle DGGS performs substantially worse. For the cell wall midpoint metric, however, the Equal Angle DGGS has the lowest overall distortion with the Snyder and Fuller–Gray DGGSs also performing relatively well. Aggregation of triangles into hexagons has little impact on intercell distance measurements, although dual hexagon aggregation results in markedly different statistics and spatial patterns for the cell wall midpoint property. In all cases, partitions on the icosahedron outperform similar partitions on the octahedron. Partition density accounts for little variation.

Introduction to discrete global grids

This part presents 3 papers presented at the second International Conference on Discrete Global Grids.