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Comparing Intercell Distance and Cell Wall Midpoint Criteria for Discrete Global Grid Systems Matt Gregory, A Jon Kimerling, Denis White and Kevin Sahr Department of Geosciences Oregon State University

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Objectives Develop criteria to get a general understanding of neighborhood metrics 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

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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 Equal Angle 5° grid (45° longitude x 90° latitude)

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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

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Intercell distance criterion 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) Points are equidistant from their neighbors

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A B D C Cell center Cell wall midpoint criterion 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 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 Cell wall midpoint ratio = length of d length of BD d Midpoint of arc between cell centers Midpoint of cell wall

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Cell shape DGGS design choices Base modeling solid TetrahedronHexahedronOctahedronDodecahedronIcosahedron TriangleHexagonQuadrilateral Diamond Frequency of subdivision 2-frequency3-frequency

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Hexagons induced from triangle tessellations 2-frequency dual hexagons3-frequency aggregate hexagons

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

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Spherical subdivision DGGSs Direct Spherical SubdivisionSmall Circle Subdivision Kimerling et al., 1994Song, 1998

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Projective DGGSs SnyderFuller-GrayQTM Dutton, 1999Kimerling et al., 1994

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

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

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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 Further standardization to common resolution linear interpolation based on mean intercell distance

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Spatial pattern of intercell distance measurements Icosahedron triangular 2-frequency DGGSs, recursion level 4 DSS Small CircleSnyder QTMFuller-Gray 354.939 km 205.638 km

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

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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 high SD/mean ratios Triangles, hexagons and diamonds show little variation from one another

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Spatial pattern of cell wall midpoint measurements Icosahedron triangular 2-frequency DGGSs, recursion level 4 DSS Small CircleSnyder QTMFuller-Gray 0.0683 0.0000

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

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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 Small Circle subdivision and 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 and diamonds

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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

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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 Examine the plausibility of using these DGGSs in global modeling applications

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