Matthew Gregory 1 , A Jon Kimerling 1 , Denis White 2 and Kevin Sahr 3
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Evaluating desirable geometric characteristics of Discrete Global Grid Systems: Revisiting the Goodchild criteria Matthew Gregory 1 , A Jon Kimerling 1 , Denis White 2 and Kevin Sahr 3 1 Oregon State University 2 US Environmental Protection Agency 3 Southern Oregon University
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Evaluating desirable geometric characteristics of Discrete Global Grid Systems: Revisiting the Goodchild criteria. Matthew Gregory 1 , A Jon Kimerling 1 , Denis White 2 and Kevin Sahr 3 1 Oregon State University 2 US Environmental Protection Agency 3 Southern Oregon University. Objectives. - PowerPoint PPT Presentation
Transcript of Matthew Gregory 1 , A Jon Kimerling 1 , Denis White 2 and Kevin Sahr 3
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)
