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 Agency

3Southern Oregon University

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

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)

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

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

A

B

D

C

Cell center

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 on the plane

The midpoint of an edge between any two adjacent cells is the midpoint of the great circle arc connecting the

centersof those two cells

Cell wall midpoint ratio =length of d

length of BD

d

Midpoint of arc between cell centers

Midpoint of cell wall

Center as defined by method

Maximum centrality criterionPoints are maximally central within areal cells

Maximum Centrality Metric1. Calculate latitude/longitude of points

on equally-spaced densified edges

2. Convert to R3 space

3. Find x, y, z as R3 centroid

4. Normalize the centroid to the unit sphere

5. Convert back to latitude/longitude

6. Find great circle distance (d) between this point and method-specific center

Centroid of densified edges

d

Cell shape

DGGS design choicesBase modeling solid

Tetrahedron

HexahedronOctahedron DodecahedronIcosahedron

Triangle Hexagon

Quadrilateral

Diamond

Frequency of subdivision

2-frequency 3-frequency

Quadrilateral DGGSs

Kimerling et al., 1994

Equal Angle

Tobler and Chen, 1986

Tobler-Chen

Spherical subdivision DGGSsDirect Spherical Subdivision

Kimerling et al., 1994

Projective DGGSs

Snyder Fuller-GrayQTM

Dutton, 1999 Kimerling et al., 1994 Kimerling et al., 1994

How is a cell neighbor defined?

Methods- Questions

Cell of interest Edge neighbor Vertex neighbor

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 subdivisionSpherical subdivision

Sphere cell center

Find center of planar triangle, project to sphere

Sphere vertices

Equal AngleQuadrilateral methods

Find midpoints of spans of longitude and latitude

Sphere cell center

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

Intercell distance normalized ratios (SD / mean)Icosahedron 2-frequency triangles, recursion levels 1-8

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 2 3 4 5 6 7 8

recursion level

SD /

mea

n

DSS Fuller-Gray Snyder QTM

Intercell distance normalized ratios (SD / mean)Sphere 2-frequency quadrilaterals, recursion levels 1-8

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

1 2 3 4 5 6 7 8

recursion level

SD

/ m

ean

Equal Angle Tobler-Chen

All methods standardized intercell distances (SD / mean)Mean intercell distance : 89.02 km

0.00

0.10

0.20

0.30

0.40

0.50

0.60

DSS Fuller-Gray Snyder QTM Equal Angle Tobler-Chen

Method

SD /

mea

n

Icosahedron 2-frequency triangles and sphere 2-frequency quadrilaterals

Icosahedron standardized intercell distances (SD / mean)Mean intercell distance : 89.02 km

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

DSS Fuller-Gray Snyder QTM

Method

SD

/ m

ean

Triangle 2-frequency Triangle 3-frequency Hexagon 2-frequency

./

Spatial pattern of intercell distance measurementsIcosahedron triangular 2-frequency DGGSs, recursion level 4

DSS

SnyderQTM

Fuller-Gray

354.939 km

205.638 km

Spatial pattern of intercell distance measurementsQuadrilateral 2-frequency DGGSs, recursion level 4

Equal Angle Tobler-Chen

1183.818 km

30.678 km

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

Cell wall midpoint normalized ratio meanIcosahedron 2-frequency triangles, recursion levels 1-8

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

1 2 3 4 5 6 7 8

recursion level

ratio

mea

n

DSS Fuller-Gray Snyder QTM

Cell wall midpoint normalized ratio meanSphere 2-frequency quadrilaterals, recursion levels 1-8

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

1 2 3 4 5 6 7 8

recursion level

ratio

mea

n

Equal Angle Tobler-Chen

All methods standardized cell wall midpoint ratio meansMean intercell distance : 89.02 km

0.000

0.005

0.010

0.015

0.020

0.025

DSS Fuller-Gray Snyder QTM Equal Angle Tobler-ChenMethod

ratio

mea

n

Icosahedron 2-frequency triangles and sphere 2-frequency quadrilaterals

Icosahedron standardized cell wall ratio meansMean intercell distance : 89.02 km

0.000

0.001

0.001

0.002

0.002

0.003

0.003

0.004

0.004

DSS Fuller-Gray Snyder QTM

Method

ratio

mea

n

Triangle 2-frequency Triangle 3-frequency Hexagon 2-frequency

Spatial pattern of cell wall midpoint measurementsIcosahedron triangular 2-frequency DGGSs, recursion level 4

DSS

SnyderQTM

Fuller-Gray

0.0683

0.0000

Spatial pattern of cell wall midpoint measurementsQuadrilateral 2-frequency DGGSs, recursion level 4

Equal Angle Tobler-Chen

0.3471

0.0000

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

Maximum centrality metric normalized ratio meanIcosahedron 2-frequency triangles, recursion levels 1-8

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

1 2 3 4 5 6 7 8

max

imum

cen

tral

ity

DSS Fuller-Gray Snyder QTM

All methods standardized maximum centrality metricMean intercell distance : 89.02 km

0.000

0.001

0.002

0.003

0.004

0.005

0.006

DSS Fuller-Gray Snyder QTM

max

imum

cen

tral

ity

Icosahedron 2-frequency triangles

Spatial pattern of maximum centrality measurements

Icosahedron triangular 2-frequency DGGSs, recursion level 6

DSS

SnyderQTM

Fuller-Gray

8.0 m

0.0 m

Distancespacing

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

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

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)