A comparison of intercell metrics on discrete global grid systems

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Computers, Environment and Urban Systems 32 (2008) 188–203 www.elsevier.com/locate/compenvurbsys

A comparison of intercell metrics on discrete global grid systems Matthew J. Gregory a,*, A. Jon Kimerling b, Denis White c, Kevin Sahr d a

Department of Forest Science, Oregon State University, Corvallis, OR 97331, USA b Department of Geography, Oregon State University, Corvallis, OR 97331, USA c US Environmental Protection Agency, Corvallis, OR 97331, USA d Department of Computer Science, Southern Oregon University, Ashland, OR 97520, USA

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. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Discrete global grid system; Intercell distance; Cell wall midpoint; Goodchild criteria; Distortion analysis

1. Introduction In recent years, the development of global analytical methods, data collection instruments and increased attention on global science issues have combined to spur growth in global dynamic modeling applications. Atmospheric motion, meta-population simulations, and coupled climate-vegetation models are just a few examples of dynamic systems studied using a global context (Heikes & Randall, 1995a; Murray, 1967; Neilson & Running, 1996). Many of these applications rely on data structures that partition the globe into areal cells and associated cell centers, which we *

Corresponding author. Tel.: +1 541 758 7778; fax: +1 541 758 7760. E-mail addresses: [email protected] (M.J. Gregory), [email protected] (A.J. Kimerling), white.denis@epamail. epa.gov (D. White), [email protected] (K. Sahr). 0198-9715/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compenvurbsys.2007.11.003

refer to as discrete global grid systems (DGGSs) (Sahr, White, & Kimerling, 2003). Movement between cells is an important function in these dynamic systems and simulated motion may be confounded by underlying distortion in the DGGS being used. A DGGS ideally would have cell centers equidistant from one another and maximally central within the areal cell, which would ensure that movement from a cell to any of its neighbors is equally probable. To this end, there is growing interest in creating DGGSs that exhibit high geometric regularity among the cells and associated point lattices across the entire globe (Heikes & Randall, 1995a; White, Kimerling, & Overton, 1992). Total geometric regularity (simultaneous equivalence of shape, surface area and intercell distance) can only be achieved on the spherical models of the five Platonic solids (Sahr et al., 2003) (Fig. 1). Any subsequent partitioning of these (or any other) spherical models necessarily introduces

M.J. Gregory et al. / Computers, Environment and Urban Systems 32 (2008) 188–203

Tetrahedron

Hexahedron

Octahedron

Dodecahedron

189

Icosahedron

Fig. 1. The five Platonic solids and their spherical equivalents.

variations in shape and/or surface area in the component cells. As has been noted in similar studies, the most commonly used DGGS is the traditional latitude–longitude graticule, upon which global data sets such as GTOPO30 and MODIS Blue Marble are based (Kimerling, Sahr, White, & Song, 1999). As a data model, the graticule benefits from established algorithms and a well-defined relationship with the geographic coordinate system. In terms of structuring spatial data, it is a relatively simple method of storage and processing. At any spatial resolution, however, the graticule suffers from extreme shape and surface area distortion among its cells. The graticule has proven inadequate for many modeling applications and survey sampling designs and its problems have been the catalyst for many alternative grid systems. In order to establish a framework for comparing DGGSs, researchers have proposed sets of ideal properties on which to compare these systems (Clarke, 2002; Goodchild, 1994; Kimerling et al., 1999; Sahr et al., 2003). Goodchild (1994) proposed the first set of these properties and these have since become known as the ‘‘Goodchild criteria”. Kimerling et al. (1999) revised and added to these criteria and proposed analytical metrics which could be used for evaluation (Table 1). Of these criteria, two relate directly to ‘‘neighborhood” properties of DGGSs – intercell distance (criterion #11) and the ‘‘cell wall midpoint” property (criterion #7) – and are especially important in dynamic modeling applications (Kimerling et al., 1999). Our objective was to develop metrics for evaluating the performance of various DGGSs based on these two criteria. This study builds on previous research which compares DGGSs based on a subset of the Goodchild criteria, most notably surface area and shape distortion measures (Kimerling et al., 1999; White, Kimerling, Sahr, & Song, 1998).

Table 1 Discrete global grid system evaluation criteria as proposed by Goodchild (1994) and modified by Kimerling et al. (1999) 1. Areal cells constitute a complete tiling of the globe, exhaustively covering the globe without overlapping 2. Areal cells have equal areas 3. Areal cells have the same topology 4. Areal cells have the same shape 5. Areal cells are compact 6. Edges of cells are straight in a projection 7. The midpoint of an arc connecting two adjacent cells coincides with the midpoint of the edge between the two cells 8. The points and areal cells of the various resolution grids which constitute the grid system form a hierarchy which displays a high degree of regularity 9. A single areal cell contains only one grid reference point 10. Grid reference points are maximally central within areal cells 11. Grid reference points are equidistant from their neighbors 12. Grid reference points and areal cells display regularities and other properties which allow them to be addressed in an efficient manner 13. The grid system has a simple relationship to the latitude and longitude graticule 14. The grid system contains grids of any arbitrary defined spatial resolution

challenge in mathematics, where researchers have studied optimal methods of distributing equidistant points across the sphere (Saff & Kuijlaars, 1997). Equidistance between cell centers is most important when considering processes which operate as a function of distance. For example, movement from a cell to any of its neighbors should be equally probable, a condition which is assured by equidistance of lattice points. Clearly, this property is useful in dynamic modeling applications (Murray, 1967; van Baalen, 2000).

1.1. Intercell distance

1.2. ‘‘Cell wall midpoint” property

This property stipulates that cell centers should be equidistant from all neighboring cell centers. This criterion forces geometric regularity of areal cells on the plane, where equidistant points (a triangular lattice) produce a Voronoi tessellation of regular hexagons. No such relationship exists, however, on the sphere. This has been a classic

This criterion was adapted from Heikes and Randall (1995a) who modeled atmospheric processes through use of a discrete global grid. To solve the shallow water equations, three finite difference operators must be calculated. In order to obtain these operators through the use of a global grid, all cells must conform to this cell wall midpoint

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property. This measure can be determined by setting a ratio defined as ‘‘the distance between the midpoint of a cell wall and the midpoint of the segment connecting two grid points divided by the length of the cell wall” (Heikes & Randall, 1995a, 1883) (Fig. 2). If the maximum value for this ratio among all cells does not converge to zero as the resolution of the grid increases, these finite difference operators are inaccurate. An important distinction must be made here. Heikes and Randall use an optimized point lattice and construct cells from a Voronoi tessellation of these points. By definition, the midpoint of the great circle arc between lattice points in a Voronoi tessellation must lie on the common cell wall. None of the DGGSs considered in this study use a Voronoi tessellation to model grid cells and it is probable that great circle arc midpoints between cell centers will not lie on cell walls. This measure, therefore, is included as a desirable shape characteristic and adherence to the property does not ensure that the finite difference operators can be obtained. 1.3. DGGS partitioning methods Discrete global grids are differentiated from one another on five different properties. These are:  geometric model (polyhedron or sphere);  change in density of the partition between recursion levels;  predominant tessellating shape;  method of transformation from the initial model to the sphere;  level of recursion. Sahr et al. (2003) note that the construction of an individual DGG is a series of design choices which utilize a unique combination of the above properties.

C

B Cell center point F

Midpoint of geodesic arc connecting cell centers d E Cell center point

Cell wall midpoint

A D Cell Wall Midpoint Ratio =

length of d length of BD

Fig. 2. The cell wall midpoint ratio as detailed in Heikes and Randall (1995b). The arc between cell centers (EF) and the connecting segment (d) are defined as geodesic arcs.

The geometric model refers to the three dimensional object on which a grid is based, which is almost always either the sphere itself or one of the aforementioned Platonic solids. Kimerling et al. (1999) describe those DGGSs based on the sphere as direct spherical tessellations and further divide these into quadrilateral based DGGSs (which are developed using the latitude–longitude graticule), subdivisions of the spherical Platonic solids and Voronoi tessellations. An alternative possibility is to base a DGGS upon one of the five Platonic solids and specify a transformation from the polyhedral face to the sphere. For the DGGSs in the study which are based on the Platonic solids, it is possible to model the systems on either the octahedron or icosahedron using the same method of transformation. The octahedron has the advantageous orientation of its six vertices at the two poles and four cardinal points on the equator, which yields a direct relationship with the graticule. The icosahedron, on the other hand, has the smallest interior angles of any of the Platonic solids when projected to the sphere and its faces more closely approximate planar equilateral triangles. Change in density denotes the n-fold increase of cells between recursion levels. We follow the research design of White et al. (1998) and implement both 4-fold and 9-fold partitions for each of the DGGSs studied. Partition density can best be illustrated when taken in conjunction with another design choice – predominant tessellating shape. On the plane, regular triangles and quadrilaterals are tessellating shapes which can be subdivided to create similar ‘‘children” that completely tile the ‘‘parent”, creating congruent, aligned hierarchies (Sahr et al., 2003) (Fig. 3). Extending this idea to the sphere, spherical triangles are able to tile the surface completely when based upon the octahedron and icosahedron. Spherical quadrilaterals (i.e., the graticule induced tessellation) are able to tile the sphere except for those cells which share a common vertex at the two poles, where the cells become spherical isosceles triangles. We also studied hexagons created from the aggregation of spherical triangles. On the plane, the hexagon is the most-sided regular polygon able to completely tile the surface. Hexagons do not form a congruent hierarchy, although they do exhibit regularity in decomposition (Sahr et al., 2003). Fig. 4a shows three recursion levels of a 4-fold triangle subdivision being aggregated into hexagons by connecting the center points of the triangles, known as a dual hexagon network (White et al., 1998). Creating a dual hexagon network of each recursion level of the triangle subdivision yields a 4-fold hexagonal grid system, whereby each parent is comprised of one whole and six one-half hexagon children. Another possible hexagon hierarchy is illustrated from 9-fold triangle subdivision, where hexagons are formed by aggregation of triangles (Fig. 4b). This produces a 9-fold hexagon grid system, where each parent is decomposed into seven whole and six one-third hexagon children. For either hexagonal grid system, there will be discontinuities of shape at the vertices of the spherical

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4-fold partition density

191

9-fold partition density

Fig. 3. Three levels of hierarchical recursion for 4- and 9-fold partition densities on planar triangles and quadrilaterals.

(a) 4-fold dual hexagons

(b) 9-fold aggregated hexagons

Fig. 4. Three levels of hierarchical recursion for 4-fold (dual) and 9-fold (aggregate) hexagons.

Platonic solids, resulting in pentagons at the 12 icosahedron vertices and quadrilaterals at the six octahedron vertices. For transformation method, we chose six DGGSs that encompass a broad range of designs, including current and proposed global data structures. The most familiar of these is the Equal Angle DGGS where grid cells are formed by taking equal intervals of latitude and longitude over the domain of the sphere. The initial resolution grid is eight cells of 90° latitude by 90° longitude from which 4and 9-fold grid systems can be derived. Cell edges are modeled as great circles (lines of longitude) and as small circles (lines of latitude). Each cell center is taken to be the midpoint (in degrees) of the spans of longitude and latitude. In direct spherical subdivision (DSS), the initial triangles of the spherical icosahedron or spherical octahedron are recursively subdivided by connecting cell wall midpoints with great circle arcs for the 4-fold case (Fekete & Treinish, 1990). For the 9-fold case, the center of the initial spherical triangle is found by finding the center point of the planar triangle defined by the spherical triangle’s three vertices and projecting this point to the surface of the sphere through use of the Gnomonic projection (Fig. 5). This center point is connected with geodesic arcs to the trisection points of a 9-fold subdivision of each cell wall. The subdivision is completed by linking adjacent breakpoints with great circle arcs. For both partition densities, the hierarchical grid is established by recursive subdivision of each new spherical triangle. The four other methods studied use an octahedron or icosahedron face as a map projection surface (Fig. 6). In these DGGSs, subdivision is first carried out on the planar faces of the base polyhedron then projected to the sphere.

Fig. 5. Geometric construction of the 9-fold direct spherical subdivision (DSS) method.

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(a) Gnomonic

(b) Fuller-Gray

(c) Snyder

(d) QTM

Fig. 6. The four polyhedral projective DGGSs. Each figure shows a 4-fold, level 4 triangle subdivision on an icosahedron face projected to geographic coordinates. Surfaces are shown in an orthographic projection to accentuate the distortion patterns present in each projection.

Each of these methods provides an exact mapping of the planar equilateral triangle to the spherical triangle so that the sphere is completely tiled with no gaps or overlaps. The simplest of these methods is the Gnomonic projection (Fig. 6a), which is a geometric projection from the center of the sphere and has the well-known property of mapping all straight lines on the plane to geodesic lines on the sphere. The projection is neither conformal nor equal area and previous studies have shown great variation among the surface area of its cells, but relatively little shape distortion (White et al., 1998). We included a DGGS based on R. Buckminster Fuller’s Dymaxion design, which Gray (1995) later adapted into an analytical projection (Fuller, 1982) (Fig. 6b). Like the Gnomonic, the Fuller–Gray projection is neither conformal nor equal area, but its surface area distortion is considerably less than the Gnomonic while exhibiting relatively minimal shape distortion (White et al., 1998). In response to a need for an equal-area projection for polyhedral globes, Snyder (1992) developed a map projection that could be applied to any of the five Platonic solids or the truncated icosahedron (Fig. 6c). While maintaining equivalent surface areas for the cells, this DGGS exhibits

moderate to severe shape distortion of its cells along radial lines extending from a polyhedron face center to its vertices. The final method considered is a DGGS developed by Dutton (1984) called the Quaternary Triangular Mesh (Fig. 6d). Besides the Equal Angle grid, Quaternary Triangular Mesh (QTM) is possibly the most established global data structure in terms of algorithms developed for its use (e.g. see Zhao, Chen, & Li (2002)). QTM is based on the 4fold recursive subdivision of the octahedron. Two projections have been published to transform QTM from the plane to the sphere: Dutton’s Zenithial OrthoTriangular (ZOT) projection (1991) and Goodchild and Yang’s projection (1992) for use in their hierarchical spatial data structure. Although these two projections use different planar surfaces (Dutton uses an isosceles right triangle, whereas Goodchild and Yang use an equilateral triangle), both give identical geographic coordinates when taken to the sphere. It is also possible to model QTM on the icosahedron and at 9-fold partition density by adapting selected projection parameters, which we have considered in this study. Tissot’s maximum angular deformation and scale variation diagrams for these four projections are presented in Fig. 7.

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Maximum Angular Deformation (ω)

193

Scale Factor (s)

11 9

1.30

7 1.10

5

1.00 3

Gnomonic

1

0.90 0.85 0.82

1.03

11

1.01 9

0.99

7

0.97

Fuller-Gray

5

0.95

3 1

15 13 11 9

Snyder

7

1.00 (Equal Area)

5

0.90 0.95

QTM

1.00 1

15

3

1.05

13

5

11 7

1.10

9 1.14

Fig. 7. Tissot maximum angle deformation and scale factor contours for the four projective DGGSs. The scale figures have been normalized to their mean scale factor for each method so that scale factors are comparable between methods.

2. Methods 2.1. Analysis design considerations Each DGGS was constructed to measure the minimum number of cells needed to obtain statistics for the entire

grid. For methods based on the icosahedron and octahedron, we constructed one triangular face’s cells, as all faces share a similarly defined method. For the Equal Angle method, a spherical triangle with vertices at (0°, 0°), (0°, 90°) and (90°, 0°) was used as the initial area. We chose this initial area to roughly correspond to cell sizes at similar

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recursion levels for those based on the spherical octahedron. For the methods based on the spherical Platonic solids, we performed recursive subdivision on the initial polyhedral face to create a triangular tessellation. We then aggregated these cells to create a hexagonal tessellation. Following the research design of White et al. (1998), 4-fold subdivision was computed to recursion level 8 and 9-fold subdivision was computed to recursive level 5, which represents an approximate difference of four orders of magnitude in cell size. There were 52 possible combinations of base modeling solid (two types), predominant tessellating shape (two types), change in partition density (two types), and level of recursion (8 or 5 levels) for each of the five methods based on the Platonic solids. The Equal Angle DGGS presented fewer combinations of grids as only the partition density and level of recursion could vary, creating 13 possible grids (8 for 4-fold partition density, 5 for 9-fold partition density). Table 2 shows the number of cells and standardized cell sizes for each DGGS considered. A number of methodological issues needed to be resolved for this analysis. First, cell neighbors are poorly defined in many DGGSs. Triangle and quadrilateral grids have cell neighbors that share only a vertex and not an edge. For the purposes of this study, we considered only edge neighbors for each DGGS. Obviously, including distances from vertex neighbors would increase the range and standard deviation of the statistics. Additionally, a cell center is somewhat of a misnomer, in that determination of the center point on the sphere is method-specific and is not necessarily the geometric centroid of the cell region. This is a limitation forced by the modeling of the cell edges either as great circle arcs, small circle arcs or projected curves. Consistency of methods then becomes important. For direct spherical subdivision, we determined triangle cell centers by finding the center of the planar triangle defined by the vertices of the spherical triangle cell. This point was then projected to the surface of the sphere using the inverse

equations for the Gnomonic projection. For the projective methods (Gnomonic, Snyder, Fuller–Gray, and QTM), we determined the center point of a cell on the plane and projected to the sphere using the inverse equations of the defined method. We defined cell centers in the Equal Angle DGGS to be the midpoints of the spans of latitude and longitude. We defined a hexagon center as the one triangle vertex within a hexagonal cell after aggregation from triangles (Fig. 8). Another issue was how to find intercell distances or cell wall midpoint ratios in those methods where cells fell on different faces of the spherical polyhedra. Since only one face of a polyhedron was constructed, we needed to define the geographic coordinate of the cell center on the adjacent face. Because of symmetry across polyhedral faces, we obtained this cell center by rotating the polyhedron edge to lie on the equator. The cell center on the adjacent face was calculated as the southern hemisphere coordinate equivalent of the initial cell center (i.e., latitude, longitude). This point was then back-rotated to its initial position and the great circle distance was calculated. 2.2. Performance measures 2.2.1. Intercell distance metrics In order to compare between all combinations of design choices, we needed to establish a normalized metric. Obviously, if the range of cell distances in a particular DGGS is zero, it must be a spherically equidistant grid. The measure of divergence from this ideal was best evaluated by the ratio of the standard deviation of the distances to the mean distance, or ‘coefficient of variation’. This provided a metric that should ideally be equal to zero and gave a standardized ratio that could be applied to all DGGSs. The coefficient of variation (CV) gave additional information about the distribution of intercell distances, which would ideally be grouped closely around the mean. We also examined the ratio of the range of distances divided by

Table 2 Cell counts and standardized surface areas for DGGSs based on the spherical octahedron and icosahedron Partition density

Recursion level

Number of cells (octahedron)

Standardized cell area (octahedron) km2

Number of cells (icosahedron)

Standardized cell area (icosahedron) km2

4-fold

1 2 3 4 5 6 7 8

32 128 512 2,048 8,192 32,768 131,072 522,688

15,939,514.747 3,984,878.687 996,219.672 249,054.918 62,263.729 15,565.932 3,891.483 975.849

80 320 1,280 5,120 20,480 81,920 327,680 1,306,720

6,375,805.899 1,593,951.475 398,487.869 99,621.967 24,905.492 6,226.373 1,556.593 390.340

9-fold

1 2 3 4 5

72 648 5,832 52,488 472,392

7,084,228.777 787,136.531 87,459.615 9,717.735 1,079.748

180 1,620 14,580 131,220 1,180,980

2,833,691.511 314,854.612 34,983.846 3,887.094 431.899

Standardized surface area is derived by dividing the surface area of a sphere with radius = 6371 km by number of cells.

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9-fold aggregated hexagons

4-fold dual hexagons

Triangle cells

195

Hexagon cells

Hexagon (and pentagon) centers created from triangle vertices

Fig. 8. Hexagon (and pentagon) cell center determination from triangle aggregation.

the mean distance as a gauge for identifying extreme values. Although the CV revealed the general performance of a DGGS, we needed to eliminate the effect of recursion level through a further standardizing procedure. Following the approach of White et al. (1998), we standardized the DGGSs at a given mean intercell distance. Because mean distance was different for each method at the same level of resolution, we selected a mean intercell distance which corresponded to the finest resolution grid for which there were data available for all DGGSs, either by calculation of intercell distances at that level or through linear interpolation between two resolutions which bounded that level, i.e.    MIDSTD  MIDn CVSTD ¼ CVn  1:0  MIDm  MIDn   MIDSTD  MIDn þ CVm  MIDm  MIDn where: MIDm mean intercell distance at coarser bounding recursion level MIDn mean intercell distance at finer bounding recursion level MIDSTD normalized mean intercell distance CVm coefficient of variation at coarser bounding recursion level CVn coefficient of variation at finer bounding recursion level CVSTD coefficient of variation at normalized distance For this study, the standardizing measure corresponded to the mean intercell distance for the Snyder 9-fold octahedron hexagons at recursion level 5 (89.02 km). 2.2.2. Cell wall midpoint metrics On the plane, the cell wall midpoint criterion measures maximum centrality of a lattice point within a cell, where the center point of each cell is defined to be the point of

intersection of all lines passing through the cell wall midpoints and perpendicular to the cell walls. Likewise, all planar cell wall midpoints are equidistant from their cell center. Therefore, the cell wall midpoint and the midpoint of the line connecting two cell centers coincide for all cells. Extending this measure to the sphere measures the deviation from this ideal. To obtain this measurement, we calculated the distance between the midpoint of a cell wall and the midpoint of the geodesic line between the corresponding cell centers. Following Heikes and Randall (1995b), this was made into a normalized ratio by dividing this distance by the length of the cell wall. Where cell walls were modeled as great circle or small circle arcs, midpoints and lengths were calculated through spherical trigonometric principles. For the projective methods, planar edges were first split into 218-recursion level equal segments and each of these breakpoints was projected to the sphere using the defined transformation. We then calculated the cell wall distance as the sum of the geodesic arc distances connecting these breakpoints. We defined the cell wall midpoint as the spherical coordinate at which half the total length had accumulated. Cell centers were defined as in the intercell distance measures. We used the ratio mean as the standardized metric to evaluate DGGS performance for the cell wall midpoint criterion. As with the intercell distance metrics, we further standardized this metric to a common resolution (Snyder 9-fold octahedron hexagons at recursion level 5) to yield more straightforward comparisons of performance. 2.3. Graphical analyses of distortion Computing statistical measures of a grid’s performance for a specific criterion gives only a general sense of how suitable that grid is for a certain application. By examining these patterns spatially, one can understand how this distortion is distributed. We constructed distortion graphics to symbolize intercell distances and cell wall midpoint

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Triangles

Hexagons

Cell walls

Quadrilaterals

Quadrilaterals used in spatial variation diagrams

Cell centers

Fig. 9. Intercell quadrilateral construction for spatial variation diagrams.

measurements, forming quadrilaterals from adjacent cells’ centers and the endpoints of their common cell wall (Fig. 9). 3. Results 3.1. Intercell distances The coefficient of variation (CV) was the basic metric for characterizing each DGGS’s performance. As in previous studies on standardized surface area and shape metrics, the CV converged asymptotically for most grids as a function of increasing recursion level, where differences in the CV between recursion levels 5–8 for 4-fold subdivision and recursion levels 3–5 for 9-fold subdivision was less than 5% (Kimerling et al., 1999; White et al., 1998) (Table 3). Each method’s performance for all subsequent recursion levels beyond those measured would likely give similar results and we suggest that conclusions of performance can be confidently extrapolated for these methods.

When each possible combination of method, partition density, tessellating shape and base modeling solid was further standardized at the mean distance of the Snyder 9fold, level 5 octahedron grid, patterns of performance emerged (Fig. 10). Among the polyhedral mapping methods, Fuller–Gray averaged the lowest normalized CVs on the icosahedron followed by DSS, Snyder, Gnomonic and QTM. When based on the octahedron, normalized CVs were considerably higher, averaging a 124% increase across all combinations of method, partition density and tessellating shape. The difference between polyhedra, however, was relatively insignificant when compared against the Equal Angle DGGS. The average icosahedron-based DGGS had a CV which was only 18% of the average CV for Equal Angle DGGSs (for the average octahedronbased DGGS, this value is 41%). The difference between 4- and 9-fold subdivision methods was minimal for the projection methods, as they had nearly equal statistics for both partition densities, especially on the icosahedron. The DSS method, however,

Table 3 Intercell distance coefficient of variation (CV) for 4- and 9-fold triangles on the icosahedron (4- and 9-fold quadrilaterals for the Equal Angle method) Partition density

Recursion level

Gnomonic

Fuller–Gray

Snyder

DSS

QTM

Equal Angle

4-fold

1 2 3 4 5 6 7 8

0.0410 0.0698 0.0744 0.0744 0.0737 0.0731 0.0728 0.0726

0.0275 0.0413 0.0457 0.0473 0.0479 0.0482 0.0483 0.0484

0.0806 0.0629 0.0643 0.0617 0.0613 0.0609 0.0607 0.0606

0.0103 0.0418 0.0543 0.0605 0.0636 0.0652 0.0660 0.0664

0.0684 0.0846 0.0865 0.0844 0.0819 0.0799 0.0787 0.0781

0.3464 0.3579 0.3549 0.3514 0.3491 0.3478 0.3471 0.3468

9-fold

1 2 3 4 5

0.0640 0.0746 0.0739 0.0730 0.0726

0.0373 0.0461 0.0478 0.0482 0.0484

0.0683 0.0631 0.0614 0.0608 0.0606

0.0312 0.0488 0.0545 0.0565 0.0571

0.0807 0.0863 0.0825 0.0795 0.0781

0.3568 0.3543 0.3495 0.3475 0.3468

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197

0.35 4

9

0.34 0.20

Coefficient of variation

0.18

4

9

4

9

0.16

4

0.14

4

4

9

4

9

4

9

4

9

4 4

0.12

4

9

9

9

9

9

9 4

0.10 0.08 4

9

4

9

0.06

4 4

9

4

9

4

9

4

4 9

9

9

0.04 0.02 0.00 Gnomonic

Fuller-Gray

Snyder

DSS

QTM

Equal Angle

Method Fig. 10. Normalized intercell distance coefficient of variation (CV) for all design choice combinations. Metrics for triangles, hexagons and quadrilaterals are represented by shape, 4- and 9-fold partition density are represented by a ‘‘4” or ‘‘9” within the shape and icosahedrons (gray) and octahedrons (white) are represented by shading. Metrics are standardized at 89.02 km (the mean intercell distance for Snyder 9-fold hexagon octahedron DGGS at recursion level 5) and correspond to a linear interpolated value between bounding recursion levels for each design choice combination. For example, Gnomonic 4fold triangles on the icosahedron correspond to the interpolated CV between recursion levels 5 (mean intercell distance: 137.89 km; CV: 0.0731) and 6 (mean intercell distance: 69.04 km; CV: 0.0728).

averaged lower CVs with 9-fold subdivision for both triangles and hexagons. Conversely, the differences between tessellating shapes seemed to have little impact on the CV metric for the DSS method, whereas 4-fold hexagons on the octahedron varied slightly from triangles and 9-fold hexagons in the projection methods. Frequency histograms and spatial variation diagrams clarified how intercell distances were distributed statistically and spatially for the DGGSs studied. Across all recursion levels, basic tessellating shape and mapping method seemed to differentiate these distributions from one another while partition density and base modeling solid had less impact on both the shape of the histogram and the pattern in the spatial variation diagram. As an example, we present results from 4-fold partition density at recursion level 5 for both triangles and dual hexagons (Fig. 11). For triangle subdivision, each method had a distinct frequency distribution. The Gnomonic and Fuller–Gray methods had relatively gradual negatively skewed distributions, whereas the Snyder method showed an interesting bimodal distribution with intermediate distances completely absent. The DSS method had the most disjoint distribution with sharp modes and large variation among distance classes. The QTM method displayed a relatively continuous distribution but for one sharp mode. The Equal Angle method had the broadest range of any method and an overwhelming majority of the intercell distances fell into the last class, due to constant latitudinal distances in this DGGS. The characteristic patterns in the spatial variation diagrams generally followed those for surface area and cell shape distortion in the studies of White et al. (1998) and Kimerling et al. (1999). The most notable difference from

these previous studies was the high local variability in these diagrams resulting from basing metrics on comparisons between adjacent cells rather than on the cell alone. This high local variability is represented in each method as a banding pattern visible to a different extent in each graphic. The Gnomonic method was the most continuous graphic, with distances smoothly ranging along a radial gradient from high in the center of the octahedron triangle to the absolute minimum distances at the three vertices. A similar pattern was present in the Fuller–Gray method, although higher local variability seemed to be more evident. The Snyder method closely followed its Tissot angular deformation diagram, concentrating its maximum and minimum intercell distances along radial lines extending from the center of the triangle to its vertices. The DSS method was marked by a strong delineation of the initial 4-fold subdivision in all subsequent recursion levels, with larger intercell distances in the central triangle. The QTM method departed from the other projective methods in that distances were not symmetric about each of the three primary axes of the triangle, but rather about its ‘‘central meridian”, due to its horizontal edges being modeled as lines of latitude. Generally, its extreme intercell distances were concentrated near the longitudinal edges of the triangle. The Equal Angle method had the highest local variability where the minimum and maximum distances on each grid came from neighboring cells near the poles. The Equal Angle grid maintained equivalent latitudinal spacing throughout the grid whereas its longitudinal spacing decreased rapidly in the polar regions. When considering dual hexagon grids created from 4fold triangle tessellation, the Fuller–Gray, Snyder and

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Fig. 11. Spatial variation diagrams and frequency histograms of intercell distances for 4-fold, recursion level 5 octahedron triangles and hexagons for polyhedral methods and sphere quadrilaterals for the Equal Angle method. Variation diagrams use a 30-class equal interval classification over the full range of distances in each mapping method. Each quadrilateral in the spatial variation diagrams represents the intercell distance between adjacent cells where vertices of the quadrilateral are comprised of the two cell centers and their common cell wall endpoints. Grayscale levels in the variation diagrams correspond to similar levels in the frequency histograms.

QTM methods showed quite different distributions from their triangular grid counterparts. The Fuller–Gray histo-

gram approached a uniform distribution, whereas the Snyder dual hexagon distribution appeared to be the mirror

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image of its triangle distribution. While the QTM method still had one sharp mode, its position shifted to the minimum distance class. The Gnomonic method maintained its histogram shape from triangle tessellation, although its distance range was greatly expanded when compared against the other methods. For all methods, the dual hexagon spatial variation patterns generally behaved similarly to the triangle diagrams. 3.2. Cell wall midpoints In the studies of Heikes and Randall (1995a, 1995b), from which this criterion was derived, a test was made for the convergence of the maximum normalized ratio for all grid cells to zero as the resolution of the grid increased. This convergence was necessary if a grid was to be used to solve the finite difference operators. This study, however, made no similar test for convergence as no claim was made for any grid’s suitability as an atmospheric modeling aid. Instead, this study focused on the mean of each grid’s ratios as an indicator of general performance. As with the other geometric Goodchild criteria, the mean ratios among DGGSs for the cell wall midpoint criterion generally followed an asymptotic path which, in this case, approached zero as recursion level increased (Table 4). For triangle subdivision, the Snyder method had consistently low ratios throughout all recursion levels for both 4- and 9-fold partition densities on both the icosahedron and octahedron. This was an unanticipated result, as previous studies had shown the Snyder method to have the highest normalized measures of shape distortion (Kimerling et al., 1999; White et al., 1998). The other projective methods (Fuller–Gray, Gnomonic and QTM) also converged to similarly low mean ratios at the highest recursion levels. The DSS method behaved similarly to the projective methods, although mean ratios converged to a slightly higher mean ratio. The Equal Angle DGGS converged quickly to a very low mean ratio. The 4- and 9-fold subdivision methods for the triangle tessellation did not have appreciably different patterns through increased recursion. Hexagon

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DGGSs followed a similar asymptotic behavior, converging near zero for all methods. Where the base modeling solid had made a significant impact on standardized metrics for the intercell distances, that factor became less of a defining difference for the cell wall midpoint criterion. While methods based on the octahedron still had higher mean ratios than those based on the icosahedron when method, tessellating shape, and partition density were held constant, there were a number of cases where octahedron mean ratios were lower than icosahedron mean ratios within specific mapping methods when other design choices were allowed to vary (Fig. 12). Most notably, 4-fold dual hexagon aggregation for the Gnomonic, Fuller–Gray, DSS and QTM methods produced much lower mean ratios than 4- and 9-fold triangles or 9-fold hexagons. For the Snyder method, the reverse was true, as aggregation into 4-fold dual hexagons from triangles sharply increased the mean cell wall midpoint ratio. Among all mapping methods, the Equal Angle DGGS averaged substantially lower mean ratios than any of the polyhedral mapping methods. Among the polyhedral methods, the Snyder averaged the lowest mean ratios, followed by Fuller–Gray, DSS, Gnomonic and QTM. As with the intercell distance metric, we constructed frequency distributions and spatial variation diagrams for cell wall midpoint ratios. We again focus our results from 4fold partition density on the octahedron at recursion level 5 for both triangles and dual hexagons (Fig. 13). We chose to log-transform the cell wall midpoint ratio in order to accentuate the patterns in the spatial variation diagrams. In the case of triangle tessellation, the projective DGGSs (Gnomonic, Fuller–Gray, Snyder and QTM) display the same basic pattern – log normal distributions with a few values on the upper tail. In contrast, the DSS method again shows a dispersed frequency distribution which is evident in its spatial variation diagram. In addition to the lowest mean ratios, the Equal Angle DGGS had the smallest range of ratios of all methods examined. For polyhedral methods, the Snyder DGGS had noticeably smaller ranges than any other method for triangle and 9-fold hexagon

Table 4 Cell wall midpoint ratio means for 4- and 9-fold triangles on the icosahedron (4- and 9-fold quadrilaterals for the Equal Angle method) Partition density

Recursion level

Gnomonic

Fuller–Gray

Snyder

DSS

QTM

Equal Angle

4-fold

1 2 3 4 5 6 7 8

0.053891 0.030092 0.015840 0.008127 0.004119 0.002073 0.001053 0.000749

0.028914 0.017457 0.009661 0.005092 0.002620 0.001350 0.000813 0.000768

0.007504 0.004784 0.004326 0.002576 0.001445 0.000766 0.000446 0.000517

0.040909 0.027205 0.015937 0.008662 0.004521 0.002310 0.001168 0.000587

0.023684 0.020849 0.014071 0.008332 0.004588 0.002422 0.001260 0.000866

0.021384 0.008965 0.004169 0.002016 0.000992 0.000492 0.000245 0.000122

9-fold

1 2 3 4 5

0.036028 0.014062 0.004866 0.001641 0.000551

0.019625 0.008598 0.003081 0.001053 0.000371

0.006720 0.004173 0.001703 0.000616 0.000219

0.027372 0.012478 0.004614 0.001593 0.000537

0.022600 0.012939 0.005334 0.001936 0.000669

0.012611 0.003678 0.001179 0.000388 0.000129

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Cell wall midpoint ratio

0.006

9

0.005

4 4

9

9 4

0.004

9

9

4

9

9

4

9

9

4

0.003 4

4

4

9

9

9

4

0.002 4 4

9

9 4

4

9

4

9

4 4

9

9

9

9

4 4

0.001 4

4

9

0.000 Gnomonic

Fuller-Gray

Snyder

DSS

QTM

Equal Angle

Me thod Fig. 12. Normalized mean cell wall midpoint ratio for all design choice combinations. Metrics for triangles, hexagons and quadrilaterals are represented by shape, 4- and 9-fold partition density are represented by a ‘‘4” or ‘‘9” within the shape and icosahedrons (gray) and octahedrons (white) are represented by shading. See Fig. 11 for a description and example of the normalization method.

tessellations. One interesting result was the drastically expanded range of the Snyder method for dual hexagon aggregation, where the maximum ratio value was over 50 times larger than either the Gnomonic or Fuller–Gray methods. Distortion in the triangle spatial variation graphics appeared in areas familiar to each method. The DSS method again showed fractal patterning based on recursive subdivision. The Snyder method was again characterized by high variability concentrated on radial axes from the center to the octahedron vertices. The highest cell wall midpoint ratios for all polyhedral triangle DGGSs occurred at face edges, where neighboring cells across polyhedron faces were mirror images of one another. When these triangles have substantive shape distortion, the geodesic arc from cell center to cell center crosses the cell wall relatively far from the cell wall bisector, leading to large ratios. This pattern is especially prevalent in the Fuller–Gray, DSS and QTM methods, where the lighter tones of the higher ratios stand out against the darker tones of lower ratios and a sawtooth pattern emerges at these edges. This can be contrasted against the Equal Angle spatial variation diagram which shows very little deviation from minimal ratios. The spatial patterns for the 4-fold dual hexagons of the Gnomonic, Fuller–Gray, DSS and QTM methods appeared very similar to those based on triangles. The only noticeable difference was the absence of high cell wall midpoint ratios at the octahedron edges for the Fuller–Gray, DSS and QTM approaches. In the dual hexagon case, cell centers are always contained within one polyhedral face, so the high ratios encountered in the 4-fold triangles from using cell centers on two facets on a polyhedron are absent in this case. The Snyder method, however, had extreme distortion in ratio values where a hexagon cell edge fell on an octahedron triangle axis. As the distance from the center of the octahedron face increased, so too did these values.

4. Discussion The statistical measures of performance for intercell distances and the cell wall midpoint criterion yielded generally decisive (and sometimes non-intuitive) results. As with previous studies, patterns of distortion were established at early recursion levels which would most likely be repeated at finer scales of analysis (Kimerling et al., 1999; White et al., 1998). This implies that the standardized measures of each criterion were good predictors of conditions at finer levels than those studied. It is also important to note, as White et al. (1998) point out, that most global science applications will likely use DGGSs toward the higher resolutions of those studied. Another important consideration when interpreting results of the study is to realize the effect of center point placement within a cell. This factor is a crucial element to both criteria measured and, while the placement of a point was not arbitrary, it was method-specific. This study chose center points which were defined in the plane for projective grids and as functions of cell vertices for methods defined on the sphere. This represents a simple approach to the issue and one, we believe, which would likely be used in the implementation of any of these methods. 4.1. Interpretation of intercell distance results While no formal statistical assessment of variation due to design choice was done, clear differences do emerge for the normalized intercell distance coefficient of variation (CV). As we anticipated, the Equal Angle method performs poorly for this criterion, as the convergence of meridians at the poles greatly expanded the range and standard deviation of this DGGS at all recursion levels. Among the methods based on the spherical Platonic solids, the base modeling solid appeared to be the design choice which most clearly differentiated DGGSs from one another for this

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Fig. 13. Spatial variation diagrams and frequency histograms of cell wall midpoint ratios for 4-fold, recursion level 5 octahedron triangles and hexagons for polyhedral methods and sphere quadrilaterals for the Equal Angle method. Variation diagrams use a 30-class equal interval classification over a log10 transformation of the cell wall midpoint ratio. Each method is based on the same scale from 105 to 100. Grayscale levels in the variation diagrams correspond to similar levels in the frequency histograms.

criterion. For all combinations of method, tessellating shape and partition density, every normalized CV on the icosahedron was lower than any normalized CV on the

octahedron. Even with this wide discrepancy, methods based on the octahedron still outperformed the Equal Angle DGGS for this criterion. Clearly, methods based on the Pla-

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tonic solids, and especially the icosahedron, would be preferential choices to optimize intercell distances. As in the study of compactness and surface area (White et al., 1998), partition density appeared to have little impact on the overall performance of each method for intercell distances. Even in the case of different methods of aggregation (4-fold dual versus 9-fold aggregated hexagons), partition density explained very little of the variation between grids. What was unexpected, however, was the lack of variation between normalized ratios of different tessellating shapes. We anticipated that hexagons would have lower standardized CVs than triangles due to smoothing effects of aggregation. This pattern was not generally seen. This may be a positive result for methods based on the Platonic solids, in that aggregation and disaggregation between tessellating shapes has little impact on the general performance of intercell distances. Among all methods studied, the Fuller–Gray DGGS produced the lowest standardized intercell distance CVs for all combinations of design choices. For polyhedral methods, the QTM and Gnomonic methods had the highest CVs, although these were still considerably lower than the CVs of the Equal Angle method. When the intercell distance metrics are compared against surface area and compactness measures done in previous studies, the ranking of methods for the intercell distance criterion appears to correspond more closely with rankings of compactness (Kimerling et al., 1999; White et al., 1998). The exception to this was the Gnomonic method which had the best performance for compactness, but had relatively poor performance among polyhedral methods for the intercell distance criterion. 4.2. Interpretation of cell wall midpoint results For the cell wall midpoint criterion, the results showed less of a distinction between design choices. All DGGSs had mean ratios which decreased as a function of recursion level. Although DGGSs based on the icosahedron generally had lower standardized mean ratios than those based on the octahedron, the difference between these base modeling solids was noticeably less than in the case of the intercell distance measure. We had expected that differences between the icosahedron and octahedron would explain much of the variability between design choices. However, for some DGGSs, such as the Gnomonic, Fuller–Gray and QTM, aggregation from 4-fold triangles into dual hexagons actually explained more of the variation than did the base modeling solid. Partition density appeared to have a greater impact as a design choice, although this was predominantly due to differences in hexagon aggregation. Even though the Equal Angle DGGS had the lowest mean ratios across methods, we were surprised by the relatively good performance of projective DGGSs, especially the Snyder method. We initially thought that the shape distortion modeled into a projected cell might skew the cell wall midpoint away from the midpoint of the great circle

route between cell centers. A possible explanation of the relatively good performance of projective DGGSs is the methodology used to define cell centers. In these constructions, we have defined cell centers on the plane and then projected to the sphere, such that they are skewed in the same direction and with the same magnitude as the cell walls. At a scale of adjacent cells (i.e. the scale at which the cell wall midpoint metric is based), the planar spatial relationship between cell centers and cell walls is maintained relatively well through projection, leading to low cell wall midpoint ratios. This result can be contrasted against the compactness metric used in White et al. (1998) and Kimerling et al. (1999), where any increase in cell wall perimeter without a corresponding increase in cell area make cells less compact. It is important to note that it is possible to construct Voronoi tessellations from the lattice of cell centers for all of the DGGSs we have considered in this study. Our goal in this study, however, was to be consistent with similar studies and model cell walls in the same manner to facilitate comparison among DGGSs. In general, the projective DGGS constructions we considered are intended to meet specific criteria that would not met by the Voronoi diagrams (e.g. area equivalence). Obviously, if cell walls were modeled as Voronoi diagrams, the results presented here may be appreciably different.

5. Conclusions The results from this study can be used in many different ways. First, if trying to optimize for a specific criterion, such as minimum variation in intercell distances or low mean cell wall midpoint ratios, the normalized ratios presented give a good indication of which grids perform well. If interested in a general all-purpose DGGS which has relatively good geometric characteristics, these results can be combined with other studies, such as White et al. (1998) and Kimerling et al. (1999), to determine a suitable grid system. If given the constraint to satisfy one criterion, such as equivalence of areal cells, this research can help answer which of the available grids would best meet the other desirable properties. This study is meant to be integrated with similar global grid research which, when taken as a whole, can yield valuable information on choosing a specific DGGS for global applications. With this research, most of the metrics of the Goodchild criteria have now been developed and studied on a number of different DGGSs. It follows that the next step in this research field is to identify which grid systems best meet the criteria for specific applications. The greatest obstacle for many of these DGGSs, however, is the lack of familiar data structures and algorithms which can support their implementation. At this stage, it seems important to explore the technical issues that limit the use of DGGSs which incorporate these desirable geometric properties.

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