ISEA discrete global grids

Abstract

This article describes a recently proposed standard, ISEA discrete global grids, for gridding information on the surface of the earth. The acronym ISEA stands for icosahedral Snyder equal area. The grid cells not only have equal areas, they are hexagons when projected onto an icosahedron! Being an advocate of hexagon binning, and corresponding graphics, my (Dan) enthu-siasm is such that I want to call attention to this new approach. Jon Kimerling, a geosciences professor, won the Ore-gon State University (OSU) Milton Harris Award for his research on the topic. Since I was peripherally involved in the development of the grids, I asked Kevin, Ralph and Tony to help me with portions of this article. They all interacted with Jon in the re-search and we all shared the desire to promote ISEA grids. Kevin, an admirer of Buckminster Fuller, de-veloped many of the ISEA algorithms and graph-ics. Along with OSU collaborators Mathew Gre-gory and Larry Hughes, he developed a web site, http://bufo. geo. orst. edu/tc/firma/gg/, that contains foldable figures, descriptions and three reference lists. At the Milton Harris Seminar, Ralph provided an excellent overview on the relevance to summarization and presentation of data from the Earth Observing System (EOS). EOS is a series of NASA satellites designed to detect and monitor global cli-mate change, starting in summer 1998.Material from Ralph's talk is available at the above web site and we include portions here. Tony was the instigator of the whole development with his push to develop a globally consistent environmental sampling methodology. Tony and his EMAP co-workers helped EPA Regions, states and nations develop environmental sampling plans us-ing an EMAP grid that was developed in the early stages of the research. The structure of this article is as follows. Section 2 traces some of the history behind the ISEA grids. Sec-tions 3 describes the ISEA grid. Section 4 introduces a potential application, storage of summaries derived from Earth Observing System sensors. Section 5 dis-

Full text

TOPICS IN INFORMATION VISUALIZATION
ISEADiscreteGlobal
Grids
By Dan Carr, Ralph Kahn, Kevin Sahr,
and Tony Olsen
1. Introduction
This article describes a recently proposed standard,
ISEA discrete global grids, for gridding information on
the surface of the earth. The acronym ISEA stands for
icosahedral Snyder equal area. The grid cells not only
have equal areas, they are hexagons when projected
onto an icosahedron! Being an advocate of hexagon
binning, and corresponding graphics, my (Dan) enthu-
siasm is such that I want to call attention to this new
approach.
Jon Kimerling, a geosciences professor, won the Ore-
gon State University (OSU) Milton Harris Award for
his research on the topic. Since I was peripherally
involved in the development of the grids, I asked
Kevin, Ralph and Tony to help me with portions of
this article. They all interacted with Jon in the re-
search and we all shared the desire to promote ISEA
grids. Kevin, an admirer of Buckminster Fuller, de-
veloped many of the ISEA algorithms and graph-
ics. Along with OSU collaborators Mathew Gre-
gory and Larry Hughes, he developed a web site,
http://bufo.geo.orst.edu/tc/firma/gg/,
that contains foldable figures, descriptions and three
reference lists. At the Milton Harris Seminar, Ralph
provided an excellent overview on the relevance to
summarization and presentation of data from the Earth
Observing System (EOS). EOS is a series of NASA
satellites designed to detect and monitor global cli-
mate change, starting in summer 1998. Material from
Ralph’s talk is available at the above web site and we
include portions here. Tony was the instigator of the
whole development with his push to develop a globally
consistent environmental sampling methodology. Tony
and his EMAP co-workers helped EPA Regions, states
and nations develop environmental sampling plans us-
ing an EMAPgrid that was developed in the early stages
of the research.
The structure of this article is as follows. Section 2
traces some of the history behind the ISEA grids. Sec-
tions 3 describes the ISEA grid. Section 4 introduces
a potential application, storage of summaries derived
from Earth Observing System sensors. Section 5 dis-
cusses graphics for hexagon grids, Splus algorithms for
low resolution ISEA grid smoothing on the globe, and
new, resolution 9 (about 200,000 cells) binned map of
global elevation data. Section 6 closes with challenges
for future research.
2. The Recent Historical Development of
ISEA Grids
The impetus for the global grid system came from what
many would call an unusual perspective - survey sam-
pling. In 1989, Denis White and Scott Overton, a ge-
ographer and a statistician from Oregon State Univer-
sity, held a workshop in Corvallis, Oregon, to discuss
the geographic requirements for a general survey de-
sign. The survey design would be the foundation for all
surveys conducted as part of the Environmental Moni-
toring and Assessment Program (EMAP) (Messer et al.,
1991; Stevens, 1994). Scott Overton, leader of the sur-
vey design effort, recommended basing the design on
a systematic grid with a random start (Overton et al.,
1990). We all know how to accomplish that for pla-
nar surfaces but when the design must cover the United
States, we are faced with a non-planar surface - the
earth. Jon Kimerling, along with the other geographers
at the workshop, devised a discrete grid system that sat-
isfied the needs of EMAP at that time (White et al.,
1992). The system used a truncated icosahedron model
of the earth with a triangular point grid applied to the
large hexagon plates. It worked for the contiguous 48
states. However, the initial discrete grid system did not
solve all the underlying issues and the embedded tri-
angular grid structure had elements that were arbitrary.
As an example, the EMAP team also applied the grid
to China, Russia, and Indonesia. The team knew prob-
lems would exist for China and Russia, as a single large
hexagon plate would not cover either country. Although
small in area, Indonesia is stretched out and also is not
covered! The initial discrete grid system had problems
at the boundaries of the plates.
In 1993, Tony Olsen, faced with these inadequacies,
initiated a research effort with Jon Kimerling, Kevin
Sahr, and Denis White in the OSU Geosciences depart-
ment to investigate an alternative discrete global grid
system. Tony required the system to be truly global
and result in an equal area tessellation. He also had a
preference for compact areas, minimal shape distortion,
a triangular point grid, and a hierarchical grid structure
allowing multiple grid densities. These characteristics
would enable global implementation of survey designs
for continuous spatial populations (Stevens, 1997).
Vol.8 No.2/3 Statistical Computing & Statistical Graphics Newsletter 31

My (Dan) formal involvement did not start until Oregon
State researchers held a workshop on discrete global
grids at Santa Barbara in 1994. Others in attendance
were Denis White, Jon Kimerling, Michael Goodchild,
Waldo Tobler, Tony Olsen, Geoff Dutton, Frank Davis,
and David Mark. Many in the group had already de-
veloped their own approaches for global grids. Waldo
Tobler was already using his methodology to show pop-
ulations on the global. Geoff Dutton had developed a
gridding system that modeled the earth as an octahe-
dron with an appropriate map projection. Kimerling
and White presented their icosahedral alternative to the
EMAP (truncated) icosahedron model. There are of
course additional approaches that work more directly
on the globe. (For a recent discussion of distributing
points on a sphere, see Saff and Kuijlaars 1997). All
methods must deal with the fact that there is no perfect
regular partition for the surface of a sphere. One mem-
ber noted that there is always at least one singularity,
as he humorously pointed to the bald spot on his head.
Michael Goodchild suggested that the meeting produce
a list of desirable properties for gridding systems. The
list appears below. Tony knew my objective when I pro-
posed cells being “compact” (having a small dimension-
less second central moment - see Conway and Sloane
1988 ). It was my attempt to promote hexagon cells.
At the Santa Barbara workshop Michael Goodchild pro-
posed a prioritized attribute list for a discrete global grid
system. The elements of the list are: the domain is
the globe (sphere, spheroid), areas exhaustively cover
the domain, areas are equal in size, areas are compact,
areas are equal in shape, areas have same number of
edges, edges of areas are of equal length, edges of ar-
eas are straight on some projection, areas form a hierar-
chy preserving some properties for
areas, each
area is associated with only one point, points are maxi-
mally central within areas, points are equidistant, points
form a hierarchy preserving some properties for
points, addresses of points and areas are regular and re-
flect other properties.
With methods and evaluation criteria at hand, the group
planned two sessions at the GIS/LIS94 meeting. To
have something to contribute, I, on the spur of the mo-
ment, concocted a method based on projecting 3-D lat-
tice points “near” a sphere surface onto the surface. My
subsequent attempts with different lattices, packings,
and notions of near, did not lead to hexagon patterns
over the whole sphere. The redeeming features of my
talk at GIS/LIS94 turned out to be the color anaglyph
stereo viewgraphs and brevity. The other presentations
carried the two sessions.
After GIS/LIS94, work proceeded on the icosahedron
model (see Kimerling, Sahr, Song, White, and Iltis,
1995). I called the research to the attention of Ralph
Kahn (NASA-JPL) who was looking for better ways to
summarize the global data expected from EOS. Tony
sought to involve Noel Cressie for dealing with spatial
estimation issues.
Jon Kimerling subsequently won the Milton Harris
Award. In May 1997, he held the Milton Harris Award
Symposium on Global Grids: New Approaches to
Global Data Analysis. In addition to presentations by
team members Kevin Sahr and Denis White, he invited
Ralph Kahn (NASA -JPL), Noel Cressie (Iowa State
University), Ross Kiester (USDA-Forest Science Lab-
oratory), Tony Olsen (USEPA-Corvallis) and myself to
make presentations. The following sections cover se-
lected topics from the Symposium and Kevin’s web site.
3. Icosahedral Snyder Equal Area
(ISEA) Grids
The S in ISEA refers to John P. Snyder. He came out of
retirement specifically to address projection problems
with the original EMAP grid (see Snyder, 1992). He
developed the equal area projection that underlies the
gridding system. His work at the U.S. Geological Sur-
vey on map projections is known by all who spend any
time with map projections. John Snyder died this year.
By all reports, he was a modest man who would not
seek to have procedures named after him. Nonetheless,
in honor of his contributions to the field of map pro-
jections, those developing the gridding system have de-
sired to use his name.
ISEA grids are simple in concept. Begin with a Snyder
Equal Area projection to a regular icosahedron (see the
stereo pairs in Figure 1) inscribed in a sphere. In each
of the 20 equilateral triangle faces of the icosahedron
inscribe a hexagon by dividing each triangle edge into
thirds (see the large gray hexagon in Figure 2). Then
project the hexagon back onto the sphere using the In-
verse Snyder Icosahedral equal area projection. This
yields a coarse-resolution equal area grid called the res-
olution 1 grid. It consists of 20 hexagons on the surface
of the sphere and 12 pentagons centered on the 12 ver-
tices of the icosahedron.
To form higher resolution grids, tessellate each equi-
lateral triangle in the planar view with more hexagons
and use the inverse projection back to the sphere. The
details of the regular tessellation are as follows: Al-
ways center a hexagon about the center point of the
32 Statistical Computing & Statistical Graphics Newsletter Vol.8 No.2/3

Figure 1. Stereo pairs of a regular icosahedron.
equilateral triangle. For odd resolution grids, orient the
hexagon so its base is parallel to the base of the trian-
gle. For even resolution grids orient the hexagon so a
vertex points at the base of the triangle. (Figure 2 shows
the central hexagons for resolutions 1 and 2 in gray and
black, respectively.) Select the edge length of a resolu-
tion
hexagon so it is times the edge length of
a resolution
hexagon. Thus, the area of a hexagon re-
duces by a factor of 3 with each increase in resolution.
As the resolution increases by 1, the tessellation pro-
cedure produces a hexagon centered on each hexagon
vertex and center point of the lower resolution tessella-
tion.
As illustrated in Figure 2, the procedure partitions a
lower resolution hexagon cell into one central cell and
six fractional (1/3) cells. This is not as simple as parti-
tioning a large square into exactly four smaller squares.
While the merits of strictly nesting cells within cells de-
pend on the context, one clear merit is aggregation sim-
plicity. The ISEAfractional cells create aggregation and
disaggregation problems that are currently under inves-
tigation.
The orientation of the icosahedron relative to the globe
is an important consideration. The selected orientation
for the ISEA grid creates symmetry about the equa-
tor. This is desirable for numerical modeling purposes.
There are always 12 pentagon cells about the vertices of
the icosahedron. The selected orientation places 11 of
the pentagon cells over water areas, so that most land
mass views will be completely composed of hexagons.
Table 1 on the next page (taken from Kevin’s web site)
gives the number of cells and characteristic hexagon
edge lengths for ISEA grids of increasing resolution.
The advantages of the ISEA grids are (1) they have ir-
regularities (12 pentagon cells) that are minor nuisances
rather than being pathological singularities, (2) they are
suitable for modeling on all parts of the globe includ-
ing the poles, (3) they preserve symmetry about the
equator, (4) they provide an infinite nesting of equal-
area sub-grids, and (5) they provide a basis for uni-
form global density of sampling for data at all spatial
resolutions. The grid facilitates comparisons between
high and low latitude data and high and low spatial-
resolution data. The grid also improves the isotropy
of finite-difference quantities compared to those cal-
culated for rectangular grid schemes. For example
Fisch, Hasslacher and Pomeau (1986) note that two-
dimensional Navier-Stokes implementations are opti-
mal with hexagons. Finally, no ambiguity exists about
nearest neighbors as all nearest neighbor cells share an
edge with a reference cell and their distances to the cen-
ter of a reference cell are nearly equal.
4. EOS and the Potential Application of
ISEA Grids
There are many potential applications of ISEA grids.
We are particularly mindful of NASA’s Earth Observ-
ing System and the wealth of global earth science data
that it will collect. The EOS AM-1 Platform is sched-
uled for launch in June 1998. The summarization of this
data provides a rapidly approaching opportunity to use
ISEA grids.
Vol.8 No.2/3 Statistical Computing & Statistical Graphics Newsletter 33

Resolution Number of Cells Length Scale (km)
1 32 4,684.2571
2 92 2,694.2932
3 272 1,553.6212
4 812 896.6139
5 2,432 517.5892
6 7,292 298.8166
7 21,872 172.5192
8 65,612 99.6035
9 196,832 57.5060
10 590,492 33.2011
11 1,771,472 19.1687
12 5,314,412 11.0670
13 15,943,232 6.3896
14 47,829,692 3.6890
15 143,489,072 2.1299
16 430,467,212 1.2297
17 1,291,401,632 0.7100
18 3,874,204,892 0.4099
Table 1. The number of cells and the characteristic
hexagon edge lengths for ISEA grids of increasing res-
olution.
More specifically, ISEA grids are relevant to Level 3
Products in the EOS Data Product Classification. Level
1 Products involve raw radiances with geometric and
radiometric calibration. Level 2 Products are geophysi-
cal parameters at the highest resolution available. These
data sets preserve the non-uniform spatial and temporal
sampling of the satellite instruments. Level 3 products
are globally and temporally uniform data sets. Level
3 products are needed where large-scale, uniform cov-
erage is required (e.g., global-scale budgets, and prob-
lems that depend on data sets from multiple sources).
Various tradeoffs will drive the selection of spatial and
temporal scales chosen for Level 3 standard products so
a multiple-resolution equal-area global grid system is
immediately relevant.
The massive amount of data and the resolution issues
drive the need for professional algorithms. Forexample,
one instrument on the platform (MISR) will help char-
acterize, on a global basis, atmospheric aerosol type and
optical depth, surface bi-directional reflectance proper-
ties, and cloud properties. The amount of data to be
collected from this one sensor is enormous. With a
spatial resolution of 16 values per km
and 36 chan-
nels, a global description will involve
basic
measurements. The MISR collection rates will be 40
Gbytes/day of raw data, 300 Gbytes/day total data, and
Figure 2. Subdividing the faces of a regular icosa-
hedron: Gray and black regions represent the central
hexagons for resolutions 1 and 2, respectively.
15-100 Tbytes/yr for at least 5 years. The computing
tools developed for the graphics in this article will not
handle such data.
Of course there is the old alternative to handle Level 3
gridding, equal angle grids. The equal angle grid re-
lies on the global latitude-longitude system and uses
a cylindrical map projection. It typically has a spa-
tial resolution of
( 112 km) and sub-grids based
on equal-angle divisions at
( 56 km) and
( 28 km). The advantages of the equal angle grid are
that the latitude-longitude system is convenient, famil-
iar, and entrenched. Also very important is the fact that
the results are easy to represent in 2-D arrays. How-
ever, the equal angle approach leads to several issues
such as rapidly changing spatial resolution at high lati-
tudes, non-uniform resolution for fine scales, ambiguity
of nearest neighbor operations and problems in repre-
senting data at multiple scales. The current solutions to
the multiple scale problem are discipline-specific vari-
ations, for example, specialized grids for polar and for
local high-resolution applications. The ISEA approach,
among other things, would provide compatible grids
across disciplines.
Those seeking additional information on alternative
grids and EOS sensors can access Ralph’s descrip-
tions at
http://bufo.goe.orst.edu/tc/firma/
gg/kahntoc.html. Of particular interest is an ex-
ample that shows the huge discrepancies that can re-
sult from changing from one grid to another and back.
Those seeking more information on Level 2 products
or discussion of problems in validating satellite derived
parameters can start with Kahn et al. (1991).
34 Statistical Computing & Statistical Graphics Newsletter Vol.8 No.2/3

Vol.8 No.2/3 Statistical Computing & Statistical Graphics Newsletter 35

36 Statistical Computing & Statistical Graphics Newsletter Vol.8 No.2/3

5. Graphics for Hexagon Cells,
Global Binning and Foldable Figures
Many graphics are available for hexagon cells. Some
of these graphics involve spatial smoothing. The figure
on page 35 (adapted from Yang and Carr, 1995) shows
a breeding bird diversity map based on smoothing to
the previous EMAP grid. The brute force smoothing
of ten year prevalence data for 615 bird species to the
13000 cell grid involved close to 8 million local logis-
tic regressions! Soon after an article on mortality map
smoothing (Carr and Pickle, 1993), Andrew Carr and I
created a point and click Splus function (UNIX only)
for selecting U.S. cancer mortality rates and smoothing
the rates to hexagon grids. The smoother in that con-
text was loess. This collection of functions is avail-
able as an Splus data.dump file, nchs.dmp, by
anonymous ftp to galaxy.gmu.edu. It is located in
pub/dcarr/newsletter/nchs. While hexagon
cell maps are relatively uncommon, the general notion
of choropleth maps is, of course, not new.
There are several sources for innovative hexagon graph-
ics. Carr et al. (1987), and Carr (1991) present various
density representations and a practical bivariate gener-
alization of box plots. Kevin and Ron Keister (per-
sonal communication) have developed an approach for
showing the change from cell to cell by coloring trian-
gles within the hexagons. Papers of Carr (1989), Carr
(1991), and Carr, Olsen and White (1992) address sym-
bol congestion control with the first showing a stereo
regression diagnostic and the last two papers focusing
attention on maps. The idea is to partition the map (or
plot) using hexagon cells and provide symbols to repre-
sent the summary for each cell with data. For example,
the angle of a ray glyph can represent a continuous vari-
able, such as a trend estimated from a time series. The
rays can point down (below horizontal) for small values
or negative trends and up for large values or positive
trends. The rays can plot on top of confidence arcs that
represent associated confidence bounds. Two rays with
common origin, one pointing to the left and one to the
right can easily represent two continuous variables on a
map.
Splus derivatives of my 2-D lattice functions now facil-
itate hexagon binning, gray level erosion, smoothing,
hexagon plotting and ray plotting. Familiarity and con-
venience suggested following the conventions in this
software when developing binning, smoothing and dis-
play procedures for global grids. The result is a set of
closely related Splus functions for low resolution grids.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Figure 3. A flattened icosahedron foldable figure for
resolution three.
The task addressed here is binning of global ETOPO
minute elevation data into an ISEA resolution 9 grid.
Conceptually the computation of a cell id for a latitude
longitude pair involves four steps. First, the Snyder
Equal Area projection produces coordinates in one of
the 20 triangles of the icosahedron. Second, an affine
transformation (one for each of the twenty triangles)
maps the coordinates into a flattened icosahedron fold-
able figure as shown for resolution three in Figure 3.
Next, a hexagon index routine (like xy2cellin Splus)
produces a planar cell id. The last step uses a look-up
table (a vector) to convert planar cell id into a globe cell
id. (Globe cell ids are integers ranging from 1 to the
number of globe cells.) Given globe cell ids, compu-
tation of summary statistics for data falling in the cells
is straightforward. The work was in generating the pla-
Vol.8 No.2/3 Statistical Computing & Statistical Graphics Newsletter 37

nar cell id to globe cell id conversion vector. The re-
indexing omits unused hexagons that cover the Figure 3
rectangle. The re-indexing also accounts for split fold-
able figure planar cells (for example those containing
the five left triangles tips) that are really parts of the
same cell on the globe.
Rather than reading the large ETOPO file into Splus, I
modified one of Kevin’s programs. The program im-
ports the resolution 9 re-indexing vector generated in
Splus and the bins on the fly. After reading the binned
results into Splus, I used three additional bookkeeping
vectors to compute hexagon boundaries and colors for
all cells (196832 globe cells and the 832 cloned cells)
in the foldable figure. The figure on page 36 shows
the average elevation for each cell. Jon Kimerling sug-
gested the basic elevation and depth coloring scheme.
A further refinement requiring additional data would be
to distinguish land hexagons that are slightly below sea
level from ocean floor hexagons. I had problems pro-
ducing the whole postscript file for the figure on page
36 so I wrote out pieces and connected them using Unix
tools. A procedure that reads a value for a location and
writes a hexagon directly to a file would be better for
graphics output.
My Splus routines for odd resolution ISEA grids are
available via anonymous ftp to
galaxy.gmu.edu.
Change directory to
pub/dcarr/newsletter/isea.
There is a README document describing the vari-
ous functions. For example, one function produces a
globe cell near neighbor pointer matrix (for low reso-
lutions). Another function uses this matrix for smooth-
ing values on the globe. (More sophisticated smoothers
could restrict domains to land masses, oceans land-
ocean boundaries, or address flow constraints.) A script
file shows the process of producing a foldable icosahe-
dron. The script starts by randomly generating a vec-
tor of 2432 values that implicitly correspond to globe
cells in a resolution 5 grid. After smoothing the values
and converting them into colors for hexagons, the script
plots the hexagons along with tabs for gluing. Creasing
along the lines shown in Figure 3 helps in the construc-
tion. I have made several figures for the holiday season.
Postscript files for different examples and sizes are in
the above directory. Kevin’s web site contains more ex-
amples including one of my favorites. The favorite is an
amazing gift from the antiquity of basketry, a six great
circle weave.
For stereo presentations on a globe, a simple approach
partitions each hexagon into six triangles. The plotting
step then renders triangles whose vertices result from
the inverse Snyder equal area projection.
6. Additional Challenges and Closing
Remarks
This article describes a 1-D indexing system that is vi-
able for modest odd resolution grids. The basic index-
ing is for hexagon cells that cover a rectangle bound-
ing the planar icosahedron view. A re-indexing vector,
whose length is the number of covering cells, removes
the unused and redundant indices. After binning with
the new indices, pre-computed
and vectors provide
plotting positions for the planar icosahedron view. The
binned results correspond to cells on the globe so a short
subscript vector extracts values corresponding to split
cells in the planar view. The result of concatenating the
two vectors corresponds to the planar
and coordi-
nates. No doubt a similar approach will work for even
resolution grids but the bookkeeping to handling unused
and split cells will require some work.
When the grid involves billions of cells, the indexing
based on the rectangle bounding the foldable icosahe-
dron planar view may be too wasteful. A first chal-
lenge is to develop a more efficient indexing system.
Quite possibly this will just cover the twenty triangles
with hexagons and handle the cells that cross the edges
of touching triangles. A second challenge is to move
from a demonstration system to professional quality al-
gorithms for high resolution grids.
There are many tasks to be addressed for a collection of
algorithms to be complete. Tony is interested in index-
ing optimized for subsets of the globe such as the con-
tinental U.S. Perhaps the most crucial task is to provide
fast, conceptually acceptable algorithms for changing
resolutions. As indicated earlier, lack of strictly nested
cells at different resolutions poses a problem. The equal
area projection approach easily adapts to strictly nested
triangles, but that would give up some of the merits of
hexagon cells.
A second challenge area is to consider the use of spa-
tial models in producing cell summaries. For exam-
ple, Ralph has noted that the current procedure for pro-
ducing pixel values for satellite images involves a sim-
ple near neighbor averaging process. Noel Cressie ad-
dressed some of the spatial modeling possibilities in his
talk at the Harris Seminar.
Assuming the computation issues are solved, we will
then face the biggest challenge of all, institutional iner-
tia. Proposing a standard is one thing. Getting scientists
in different nations and different disciplines to use it is
another.
38 Statistical Computing & Statistical Graphics Newsletter Vol.8 No.2/3

Acknowledgments
Research related to this article was supported by EPA
under cooperative agreement No. CR820820-01-0. The
article has not be subjected to the review of the EPA and
thus does not necessarily reflect the view of the agency
and no official endorsement should be inferred.
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Dan Carr
Institute for Computational Sciences
and Informatics
George Mason University
dcarr@voxel.galaxy.gmu.edu
Ralph Kahn
Jet Propulsion Laboratory
ralph.kahn@jpl.nasa.go
Kevin Sahr
University of Oregon
sahrk@cs.uoregon.edu
Tony Olsen
EPA National Health and Environmental
Effects Research Laboratory
tolsen@mail.cor.epa.gov
Vol.8 No.2/3 Statistical Computing & Statistical Graphics Newsletter 39