Transcript Comparing Intercell Distance and Cell Wall Midpoint Criteria for

Evaluating desirable geometric characteristics of
Discrete Global Grid Systems:
Revisiting the Goodchild criteria
Matthew Gregory1, A Jon Kimerling1, Denis White2 and Kevin Sahr3
2US
1Oregon
State University
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
Equal Angle 5° grid
(45° longitude x 90° latitude)
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

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
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)
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
Cell wall midpoint ratio =
B
length of d
length of BD
C

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
Midpoint of arc between cell
centers
Cell center
d
Midpoint of cell wall
Cell center
A
D
Maximum centrality criterion
Points are maximally central within areal cells
Maximum Centrality Metric
1. Calculate latitude/longitude of points
on equally-spaced densified edges
Centroid of densified edges
d
2. Convert to R3 space
3. Find mx, my, mz as R3 centroid
4. Normalize the centroid to the unit
sphere
5. Convert back to latitude/longitude
Center as defined by method
6. Find great circle distance (d) between
this point and method-specific center
DGGS design choices
Base modeling solid
Tetrahedron
Octahedron
Hexahedron
Cell shape
Triangle
Hexagon
Quadrilateral Diamond
Icosahedron
Dodecahedron
Frequency of subdivision
2-frequency
3-frequency
Quadrilateral DGGSs
Equal Angle
Kimerling et al., 1994
Tobler-Chen
Tobler and Chen, 1986
Spherical subdivision DGGSs
Direct Spherical Subdivision
Kimerling et al., 1994
Projective DGGSs
QTM
Snyder
Dutton, 1999
Kimerling et al., 1994
Fuller-Gray
Kimerling et al., 1994
Methods- Questions
 How is a cell neighbor defined?
Cell of interest
Edge neighbor
Vertex neighbor
Methods - Questions
 How is a cell center defined?
Projective methods
Spherical subdivision
Quadrilateral methods
Snyder, Fuller-Gray, QTM, ToblerChen
DSS, Small Circle subdivision
Equal Angle
Plane center
Apply projection
Sphere cell center
Sphere vertices
Find center of planar
triangle, project to
sphere
Sphere cell center
Find midpoints of spans of
longitude and latitude
Sphere cell
center
Methods - Normalizing Statistics

Intercell distance criterion
 standard

Cell wall midpoint criterion
 mean

deviation of all cells / mean of all cells
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
0.1
0.09
0.08
SD / mean
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
2
3
4
5
6
7
8
recursion level
DSS
Fuller-Gray
Snyder
QTM
Intercell distance normalized ratios (SD / mean)
Sphere 2-frequency quadrilaterals, recursion levels 1-8
0.70
0.60
SD / mean
0.50
0.40
0.30
0.20
0.10
0.00
1
2
3
4
5
6
recursion level
Equal Angle
Tobler-Chen
7
8
All methods standardized intercell distances (SD / mean)
Mean intercell distance : 89.02 km
0.60
0.50
SD / mean
0.40
0.30
0.20
0.10
0.00
DSS
Fuller-Gray
Snyder
QTM
Equal Angle
Tobler-Chen
Method
Icosahedron 2-frequency triangles and sphere 2-frequency quadrilaterals
Icosahedron standardized intercell distances (SD / mean)
Mean intercell distance : 89.02 km
0.09
0.08
0.07
SD / mean
0.06
0.05
0.04
./
0.03
0.02
0.01
0.00
DSS
Fuller-Gray
Snyder
QTM
Method
Triangle 2-frequency
Triangle 3-frequency
Hexagon 2-frequency
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
Spatial pattern of intercell distance measurements
Quadrilateral 2-frequency DGGSs, recursion level 4
1183.818 km
30.678 km
Equal Angle
Tobler-Chen
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 mean
Icosahedron 2-frequency triangles, recursion levels 1-8
0.045
0.040
0.035
ratio mean
0.030
0.025
0.020
0.015
0.010
0.005
0.000
1
2
3
4
5
6
7
8
recursion level
DSS
Fuller-Gray
Snyder
QTM
Cell wall midpoint normalized ratio mean
Sphere 2-frequency quadrilaterals, recursion levels 1-8
0.070
0.060
ratio mean
0.050
0.040
0.030
0.020
0.010
0.000
1
2
3
4
5
6
7
recursion level
Equal Angle
Tobler-Chen
8
All methods standardized cell wall midpoint ratio means
Mean intercell distance : 89.02 km
0.025
ratio mean
0.020
0.015
0.010
0.005
0.000
DSS
Fuller-Gray
Snyder
Method
QTM
Equal Angle
Tobler-Chen
Icosahedron 2-frequency triangles and sphere 2-frequency quadrilaterals
Icosahedron standardized cell wall ratio means
Mean intercell distance : 89.02 km
0.004
0.004
ratio mean
0.003
0.003
0.002
0.002
0.001
0.001
0.000
DSS
Fuller-Gray
Snyder
QTM
Method
Triangle 2-frequency
Triangle 3-frequency
Hexagon 2-frequency
Spatial pattern of cell wall midpoint measurements
Icosahedron triangular 2-frequency DGGSs, recursion level 4
0.0683
DSS
Fuller-Gray
0.0000
QTM
Snyder
Spatial pattern of cell wall midpoint measurements
Quadrilateral 2-frequency DGGSs, recursion level 4
0.3471
0.0000
Equal Angle
Tobler-Chen
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 mean
Icosahedron 2-frequency triangles, recursion levels 1-8
0.050
maximum centrality
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
1
2
DSS
3
4
Fuller-Gray
5
6
Snyder
7
8
QTM
All methods standardized maximum centrality metric
Mean intercell distance : 89.02 km
maximum centrality
0.006
0.005
0.004
0.003
0.002
0.001
0.000
DSS
Fuller-Gray
Snyder
Icosahedron 2-frequency triangles
QTM
Spatial pattern of maximum centrality measurements
Icosahedron triangular 2-frequency DGGSs, recursion level 6
8.0 m
DSS
Fuller-Gray
Distance
spacing
0.0 m
QTM
Snyder
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)
