Matthew Gregory 1 , A Jon Kimerling 1 , Denis White 2 and Kevin Sahr 3

1. Evaluating desirable geometric characteristics of Discrete Global Grid Systems: Revisiting the Goodchild criteria Matthew Gregory1, A Jon Kimerling1, Denis White2 and Kevin Sahr3 1Oregon State University2US Environmental Protection Agency3Southern Oregon University

2. Objectives • Develop metrics to address desirable shape characteristics for discrete global grid systems (DGGSs) • Characterize the behavior of different design choices within a specific DGGS (e.g. cell shape, base modeling solid) • Apply these criteria to a variety of known DGGSs

3. Equal Angle 5° grid (45° longitude x 90° latitude) The graticule as a DGGS • commonly used as a basis for many global data sets (ETOPO5, AVHRR) • well-developed algorithms for storage and addressing • suffers from extreme shape and surface area distortion at polar regions • has been the catalyst for many different alternative grid systems

4. DGGS Evaluating Criteria • Topological checks of a grid system • Areal cells constitute a complete tiling of the globe • A single areal cell contains only one point • Geometric properties of a grid system • Areal cells have equal areas • Areal cells are compact • Metrics can be developed to assess how well a grid conforms to each geometric criterion

5. Intercell distance criterion Points are equidistant from their neighbors • on the plane, equidistance between cell centers (a triangular lattice) produces a Voronoi tessellation of regular hexagons (enforces geometric regularity) • classic challenge to distribute points evenly across a sphere • most important when considering processes which operate as a function of distance (i.e. movement between cells should be equally probable)

6. length of d Cell wall midpoint ratio = length of BD C B Midpoint of arc between cell centers Cell center d Midpoint of cell wall Cell center A D Cell wall midpoint criterion The midpoint of an edge between any two adjacent cells is the midpoint of the great circle arc connecting the centers of those two cells • derived from the research of Heikes and Randall (1995) using global grids to obtain mathematical operators which can describe certain atmospheric processes • criterion forces maximum centrality of lattice points within areal cells on the plane

7. Maximum centrality criterion Points are maximally central within areal cells Maximum Centrality Metric • Calculate latitude/longitude of points on equally-spaced densified edges • Convert to R3 space • Find mx, my, mz as R3 centroid • Normalize the centroid to the unit sphere • Convert back to latitude/longitude • Find great circle distance (d) between this point and method-specific center Centroid of densified edges d Center as defined by method

8. Tetrahedron Octahedron Hexahedron Icosahedron Dodecahedron 2-frequency 3-frequency Triangle Hexagon Quadrilateral Diamond DGGS design choices Base modeling solid Cell shape Frequency of subdivision

9. Quadrilateral DGGSs Equal Angle Tobler-Chen Tobler and Chen, 1986 Kimerling et al., 1994

10. Direct Spherical Subdivision Kimerling et al., 1994 Spherical subdivision DGGSs

11. Projective DGGSs QTM Snyder Fuller-Gray Dutton, 1999 Kimerling et al., 1994 Kimerling et al., 1994

12. Methods- Questions • How is a cell neighbor defined? Cell of interest Edge neighbor Vertex neighbor

13. Methods - Questions • How is a cell center defined? Projective methods Spherical subdivision Quadrilateral methods DSS, Small Circle subdivision Equal Angle Snyder, Fuller-Gray, QTM, Tobler-Chen Sphere vertices Find midpoints of spans of longitude and latitude Plane center Find center of planar triangle, project to sphere Sphere cell center Apply projection Sphere cell center Sphere cell center

14. Methods - Normalizing Statistics • Intercell distance criterion • standard deviation of all cells / mean of all cells • Cell wall midpoint criterion • mean of cell wall midpoint ratio • Maximum centrality criterion • mean of distances between centroid and cell center / mean intercell distance • Further standardization to common resolution • linear interpolation based on mean intercell distance

19. Spatial pattern of intercell distance measurements Icosahedron triangular 2-frequency DGGSs, recursion level 4 354.939 km DSS Fuller-Gray 205.638 km QTM Snyder

20. Spatial pattern of intercell distance measurements Quadrilateral 2-frequency DGGSs, recursion level 4 1183.818 km 30.678 km Equal Angle Tobler-Chen

21. Results - Intercell Distances • Asymptotic behavior of normalizing statistic, levels out at higher recursion levels • Fuller-Gray had lowest SD/mean ratio for all combinations • Equal Angle and Tobler-Chen methods had relatively high SD/mean ratios • Triangles and hexagons show little variation from one another

26. DSS Fuller-Gray QTM Snyder Spatial pattern of cell wall midpoint measurements Icosahedron triangular 2-frequency DGGSs, recursion level 4 0.0683 0.0000

27. Spatial pattern of cell wall midpoint measurements Quadrilateral 2-frequency DGGSs, recursion level 4 0.3471 0.0000 Equal Angle Tobler-Chen

28. Results - Cell Wall Midpoints • Asymptotic behavior approaching zero • Equal Angle has lowest mean ratios with Snyder and Fuller-Gray performing best for methods based on Platonic solids • Tobler-Chen only DGGSs where mean ratio did not approach zero • Projection methods did as well (or better) than methods that were modeled with great and small circle edges • Triangles performed slightly better than hexagons although results were mixed

31. Spatial pattern of maximum centrality measurements Icosahedron triangular 2-frequency DGGSs, recursion level 6 8.0 m DSS Fuller-Gray Distancespacing 0.0 m QTM Snyder

32. Results – Maximum Centrality • Asymptotic behavior of normalizing statistic • DSS has lowest maximum centrality measures as centroids are coincident with cell centers by definition • Snyder method has relatively large offsets along the radial axes • Tesselating shape seems to have little impact on the standardizing statistic

33. General Results • Asymptotic relationship between resolution and normalized measurement allows generalization • Relatively similar intercell distance measurements for triangles, hexagons and diamonds implies aggregation has little impact on performance for Platonic solid methods • Generally, projective DGGSs performed unexpectedly well for cell wall midpoint criterion

34. Implications and Future Directions • Grids can be chosen to optimize one specific criterion (application specific) • Grids can be chosen based on general performance of all DGGS criteria • Study meant to be integrated with comparisons of other metrics to be used in selecting suitable grid systems • Study the impact of different methods of defining cell centers • Extend these metrics to other DGGSs (e.g. EASE, Small Circle)