Discrete Global Grids
A Web Book
Edited byMichael F. Goodchild and A. John Kimerling

Abstract

Introduction

Defining a Gridded Data Set

EASE-Grid Map Parameters

EASE-Grid Family of Grid Definitions

An Example Application

Conclusions

References

EASE-Grid:
A Versatile Set of Equal-Area Projections and Grids

by Mary J. Brodzik and Kenneth W. Knowles

National Snow & Ice Data Center
Cooperative Institute for Research in Environmental Science
University of Colorado, Boulder, CO, USA

Brodzik, M. J., & Knowles, K. (2002). EASE-Grid: a versatile set of equal-area projections and grids. In M. Goodchild and A. J. Kimerling (Eds.), Discrete Global Grids. Santa Barbara, CA, USA: National Center for Geographic Information & Analysis.
http://www.ncgia.ucsb.edu/globalgrids-book/ease_grid/


Abstract

The National Snow and Ice Data Center's (NSIDC) Equal-Area Scalable Earth Grid (EASE-Grid) comprises three equal-area projections, combined with an infinite number of possible grid definitions. The EASE-Grid is based on a philosophy of digital mapping and gridding definitions developed at the University of Colorado at Boulder.

The original, 25 kilometer grids were defined for data products generated by the NOAA/NASA Level 3 Passive Microwave Pathfinder Projects, which include gridded passive microwave brightness temperatures and related geophysical products derived from the brightness temperatures. However, the grids have been adopted by a number of other projects, including the TIROS Operational Vertical Sounder (TOVS) and Advanced Very High Resolution Radiometer (AVHRR) Polar Pathfinders, the Arctic and Antarctic Research Institute, St. Petersburg, Russia (AARI) Sea Ice data, and NSIDC's EASE-Grid versions of the Global Land Cover Classification (GLCC) data and the International Permafrost Association Permafrost and Ground Ice Map.

The gridding philosophy used to implement a library of software routines is based on the assumption that a gridded data set is completely defined by two abstractions, the map projection and an overlaid lattice of grid points, often referred to as "cells." The software may be used to define all standard projections, and convert gridded data between them. However, this presentation will be restricted to an overview of the family of specific projections and grids that we have called the NSIDC EASE-Grid, or simply EASE-Grid. The discussion will include how to define a map projection and grid definition in general, and our reasons for choosing equal-area projections and a spherical Earth model for our target applications in particular. It will include some examples of applications and an intercomparison between EASE-Grid data sets.

Introduction

The Equal-Area Scalable Earth Grid (EASE-Grid) comprises three equal-area projections, combined with an infinite number of possible grid definitions. It is based on a philosophy of digital mapping and gridding definitions that was developed at the National Snow and Ice Data Center (NSIDC), in Boulder, CO. This philosophy was used to implement a library of software routines, based on the assumption that a gridded data set is completely defined by two abstractions, the map projection and an overlaid lattice of grid points. The complete source code is available via ftp, and contains software to convert among many projections, but this paper will be restricted to an overview of the family of specific projections and grids that we have called the NSIDC EASE-Grid, or simply EASE-Grid.

The EASE-Grid is intended to be a versatile scheme for users of global-scale gridded data, specifically remotely sensed data, although it is gaining popularity as a common gridding format for data from other sources as well. We begin with a short introduction to the abstractions used in NSIDC's generic mapping and gridding software, and proceed to the specific projections and grids that comprise the EASE-Grid family of grids. We include several short descriptions of the EASE-Grid used for different projects, as examples of the flexibility of the format. We conclude with an example of two EASE-Grid data sets that are used to study sea ice concentrations in Baffin Bay.


Defining a Gridded Data Set

NSIDC's mapping software is based on the assumption that a gridded data set is completely defined by the map projection and an overlaid lattice of grid points, often referred to as "cells". It is useful to think of the projection and the grid lattice as separate but related abstractions. The projection is simply a mathematical coordinate transformation of points on the curved surface of the Earth to points on a plane. The lattice of grid points can be imagined as a transparent piece of graph paper, overlaid on the plane of projection and anchored to it at a specified point.

Once the projection is chosen, any number of grid definitions can be used to describe the effect of changing the grid, or "graph paper," for the application at hand. For example, smaller graph cells can be used when a higher resolution is needed, or the size of the graph paper (number of columns/rows) can be reduced to study a subset area of the full projection.

An array of gridded data, then, consists of one data element for each grid "cell" or lattice point. The user has complete flexibility to define the meaning of grid cell values, according to the most appropriate sampling technique for the data and application at hand. In many cases, particularly with remotely sensed imagery, it is important for a user to think of gridded data elements as values associated with the lattice points of the graph paper, rather than as associated with the "area of the grid cell." We have found the more general lattice point concept to be more useful when the data represent regularly sampled measurements in a continuous field, for example, passive microwave brightness temperatures, or visible wavelength radiances. In the sensor swath space, the data value is simply a sample measurement from the continuous field, and may or may not have any physical relationship to the size and shape of the eventual grid "cell" surrounding the eventual lattice point in the regularly gridded data array. For example, although the sampling interval of a 19 GHz scanning passive microwave radiometer might be 25 km, the effective field of view of the antenna might be an elliptical area, 40 km x 60 km. The eventual gridded data array might be defined on a 25 km grid, with data element values chosen from the nearest neighbor of the latest swath. In this case, the data element still represents a 40 km x 60 km brightness temperature.

It is entirely up to the data set producer to define the meaning of a gridded data element during the sampling process. The process of sampling data to the regular grid is sometimes referred to as binning. Examples of binning techniques include averaging all data that falls into a given cell, or taking a maximum, minimum, median or latest value in the cell. The producer might choose to involve the area and shape of the cell in the definition of a data element, but is certainly not required to do so. Alternative methods include nearest neighbor, bilinear interpolation, or otherwise weighted averages of surrounding data samples.


EASE-Grid Map Parameters

The EASE-Grid was originally developed at NSIDC for the data products generated by the NOAA/NASA Pathfinder Program Special Sensor Microwave Imager (SSM/I) project, which included gridded passive microwave brightness temperature and related geophysical products derived from the brightness temperatures at a relatively coarse 25 km resolution (Armstrong and Brodzik, 1995; Armstrong, Brodzik and Varani, 1997). However, given the flexibility of an infinite number of grid definitions for the EASE-Grid projections, the format has since been adopted by a number of other projects, with grid resolutions ranging from 1.25 km to 250 km. In addition to the original SSM/I Pathfinder data, these include:

Ancillary EASE-Grid data sets include NSIDC's EASE-Grid versions of the Global Land Cover Classification (GLCC) data and the International Permafrost Association Permafrost and Ground Ice Map. Instrument teams for the recently launched (December, 1999) Moderate Resolution Imaging Spectroradiometer (MODIS) and the upcoming Advanced Microwave Scanning Radiometer (AMSR) sensors plan to release gridded snow and ice products in EASE-Grid.

The EASE-Grid projections comprise polar aspect spherical Lambert azimuthal equal-area projections (Snyder, 1987, p. 186), for the Northern or Southern hemisphere, respectively, and a "modified" cylindrical equal-area projection (Snyder, 1987, p. 77; Maling, 1992, p. 431) for applications in mid- and low-latitude regions.

Northern Hemisphere Spatial Coverage Map Southern Hemisphere Spatial Coverage Map
Northern Hemisphere EASE-Grid projection Southern Hemisphere EASE-Grid projection

The North azimuthal equal-area map is defined as

  • r = (ncols - 1) - r0 + 2R/C sin(pi/4 - p/2) sin(l)
  • s = (nrows - 1) - s0 + 2R/C sin(pi/4 - p/2) cos(l)
  • k = sec(pi/4 - p/2)

The South azimuthal equal-area map is defined as

  • r = (ncols - 1) - r0 + 2R/C cos(pi/4 - p/2) sin(l)
  • s = (nrows - 1) - s0 - 2R/C cos(pi/4 - p/2) cos(l)
  • k = csc(pi/4 - p/2)

Global Cylindrical Spatial Coverage Map
Global EASE-Grid projection

The modified cylindrical equal-area map is defined as

  • r = (ncols - 1) - r0 + R/C l cos(ps)
  • s = (nrows - 1) - s0 - R/C sin(p) / cos(ps)
  • k = cos(ps) / cos(p)

where:

  • k = particular scale along parallels
  • h = 1/k (particular scale along meridians)
  • l = longitude in radians
  • p = latitude in radians
  • R = radius of the Earth = 6371.228 km
  • C = nominal cell size (km)
  • ncols = number of grid columns
  • nrows = number of grid rows
  • r = column coordinate (increasing to the right)
  • s = row coordinate (increasing from top to bottom)
  • r0 = map origin column
  • s0 = map origin row
  • ps = pi/6 radians = 30 degrees (cylindrical projection standard parallel)

Both projections are based on a spherical model of the Earth with radius R = 6371.228 km. This radius was chosen for historical reasons.

The values of C, ncols, nrows, r0 and s0 are determined by the grid that is chosen to overlay the projection.

Why "equal-area" maps?

We chose equal-area projections over other possibilities for the EASE-Grid, because our original application was a fixed geographic look-up table for storage and retrieval of satellite passive microwave brightness temperatures. We were using an optimal interpolation binning method to derive brightness temperatures at fixed grid locations, effectively yielding what the sensor would have seen if it had been pointed at the center of the fixed cell. Therefore, a rectangular grid lattice superimposed on an equal-area map most faithfully represented the nominal passive microwave footprint.

On equal-area maps, a small circle placed anywhere on the map will always cover the same amount of area on the globe, and, at any point on the map, the product of the scale h along a meridian of longitude and the scale k along a parallel of latitude is always one, that is, h = 1/k. The aspect ratio, k/h = k2 = 1/h2, and angular deformation, w = 2 arcsin(|h - k|/(h + k)), are measures of shape distortion.

For the Northern and Southern hemisphere EASE-Grid projections, the aspect ratio varies from 1 at the pole to 1.17 at 45N and increases to only 2 at the equator. For the global EASE-Grid projection, the aspect ratio varies more widely (see details in the following table). The selection of +/-30 degrees for the standard parallels of the cylindrical projection gives a map with minimum mean angular distortion over the continents. This projection is intended for the study of parameters in the mid- to low-latitudes.

Aspect ratios and angular deformation (measures of shape distortion) of the EASE-Grid projections:

Azimuthal Equal-Area   Cylindrical Equal-Area
latitude k/h w   latitude k/h w
90° 1.00   80° 24.90 134°
75° 1.02   75° 11.20 113°
60° 1.07   60° 3.00 60°
45° 1.17   45° 1.50 23°
30° 1.33 16°   30° 1.00
15° 1.59 26°   15° 0.80 12°
2.00 39°   0.75 16°

In contrast, on conformal maps, shapes within a small area are reproduced accurately, so a small circle on the globe will look like a small circle on the map. At any point on the map, the scale h along a meridian of longitude is equal to the scale k along a parallel of latitude, and kh - 1 is a measure of areal distortion. For example, NSIDC produces other polar gridded data products using a polar stereographic map true at 70N. The projection is a conformal map. By definition, the aspect ratio remains 1 everywhere, however, the areal distortion of this map varies from -6% at the pole to +29% at 45N and increases to +276% at the equator. Angular deformation varies from 0° at 70N/S to 4° at the pole and 71° at the equator.

Areal distortion and angular deformation of the Polar Stereographic map true at 70N:

Polar Stereographic, (true at 70N)
latitude kh - 1 w
90° -6%
45° 29% 15°
276% 71°

A very popular map that is neither equal-area nor conformal is the cylindrical equidistant map, also known as the "lat-lon grid." This map suffers from both areal and shape distortion, and angular deformation, as follows:

Cylindrical Equidistant
  Shape Distortion Areal Distortion Angular Deformation
latitude k/h kh - 1 w
89° 57 5630% 176°
80° 6 476% 141°
60° 2 100% 74°
45° 1.4 41% 39°
1 0%

In summary, given the choices of either shape distortion or areal distortion or both, we decided in favor of the equal-area projections for the EASE-Grid because they minimized the amount of distortion over the hemispheric and global scale we were attempting to portray. One convenient side effect of this choice is that calculations of areal statistics are reduced to simply summing pixels and multiplying by a constant area per pixel, so the acronym, "EASE-" takes on a secondary meaning, as in "easy to use."

Why a Spherical Earth Model?

Another question that is sometimes raised is why we chose to use a spherical earth model over an elliptical model, and how much "error" this introduces in the gridding geolocation. Keeping in mind that the EASE-Grid is simply a geographic data storage and retrieval mechanism, the answer is that no error is introduced by this model choice.

Representation of the gridded data as a fixed array of values is accomplished with a set of equations to map from geographic coordinates (latitude, longitude) to grid coordinates (column, row). In this sense, the location (column and row) of each grid "cell" can just be considered an entry in a look-up table, i.e. a place to store the data (brightness temperature, albedo, time stamp, etc.) for a specific, implicitly defined, geographic location. As long as the transformation back from grid coordinates (column,row) to geographic coordinates (latitude, longitude) is performed with the inverse transformation that uses the same Earth model, there is no error introduced by using a spherical Earth model. Choice of an elliptical model would only slow down the transformation calculations, (geographic to grid and back), with no gain in accuracy.


EASE-Grid Family of Grid Definitions

A grid is always defined in relation to a specific map projection. It is essentially the parameters necessary to define a rectangular coordinate system overlaid on a flat map and anchored to it at the map origin. The following four elements completely describe a grid:

  • the map projection
  • the numbers of columns and rows
  • the number of grid cells per map unit (the map unit is part of the projection parameters)
  • the grid cell coordinates of the map's origin

The EASE-Grid family of grid definitions includes, but is not limited to, the following specific grids.

The Original SSM/I Grids

The original 25 kilometer grids were defined for the data products generated by the SSM/I Level 3 Pathfinder Project at NSIDC. The sampling resolutions of the SSM/I brightness temperature data were 25 km and 12.5 km. The original grids were designated "low" (25 km) and "high" (12.5 km) resolution, although these are, of course, relative terms. The "low" resolution grids have a nominal cell size of 25 km x 25 km. A slightly larger actual cell size C=25.067525 km was chosen to make the full global, 25 km ("Ml", the uppercase "M" stands for "Modified" cylindrical equal-area, or "Mid-"latitude; the lowercase "l" stands for "low" resolution) grid exactly span the equator, and was then used for all three projections for the sake of data product consistency. Of course, few cells actually have these dimensions, but they all have the same area.

By convention, grid coordinates (r,s) start in the upper left corner, at cell (0,0), with r increasing to the right and s increasing downward (Knowles, 1993). Rounding the grid coordinates up at .5 yields the grid cell number. A grid cell is centered at grid coordinates (j,i) and bounded by: (j - .5) <= r < (j + .5) and (i - .5) <= s < (i + .5).

The 25 km hemispheric grids for the polar aspect azimuthal projections (aka "Nl" for "Northern Hemisphere, low resolution" and "Sl" for "Southern Hemisphere, low resolution") are defined with 721 columns, 721 rows, with the respective pole anchored at cell (360.0,360.0). The Ml grid for the cylindrical projection is defined with 1383 columns, 586 rows, with the point where the equator crosses the prime meridian at cell location (691.0,292.5).

For each 25 km grid, the set of corresponding "high" resolution (12.5 km) grids, "Nh", "Sh" and "Mh", was defined such that the grid coordinates are coincident (aka "bore-centered") and exactly double the lower resolution grid coordinates. The Ml grid is symmetrical about the prime meridian, but the Mh grid is not. The (25 km) Ml grid exactly spans the equator, from 180 W to 180 E, with 1383 grid cells. The (12.5 km) Mh grid, also exactly spans the equator, with 2766 grid cells. However, since the center of the Ml column 0 is coincident with the Mh column 0, the western edge of the Mh grid cell in column 0 row 293 (at the equator) is slightly east of 180 W, and the eastern edge of the Mh grid cell in column 2765 row 293 is slightly east of 180 E.

The dimensions, center, and extent of the original SSM/I grids are summarized below. It is important to remember that there is nothing specific to the SSM/I data in these definitions. If these grid definitions are considered appropriate for another data set, they can be used with no changes.

Original "Low" (25 km) and "High" (12.5 km) Resolution SSM/I Grids
Grid Dimensions Map Origin Map Origin Grid Extent
Name Width Height Column (r0) Row (s0) Latitude Longitude Minimum Latitude Maximum Latitude Minimum Longitude Maximum Longitude
Ml 1383 586 691.0 292.5 0.0 0.0 86.72S 86.72N 180.00W 180.00E
Mh 2766 1171 1382.0 585.0 0.0 0.0 85.95S 85.95N 179.93W 180.07E
Nl 721 721 360.0 360.0 90.0N 0.0 0.34S 90.00N 180.00W 180.00E
Nh 1441 1441 720.0 720.0 90.0N 0.0 0.26S 90.00N 180.00W 180.00E
Sl 721 721 360.0 360.0 90.0S 0.0 90.00S 0.34N 180.00W 180.00E
Sh 1441 1441 720.0 720.0 90.0S 0.0 90.00S 0.26N 180.00W 180.00E

Other Grid Definitions in the EASE-Grid Family

The Polar Pathfinders

Users of the NSIDC EASE-Grid are not limited to the grid orientation, size and resolution described above, and are free to define grids that are more appropriate for a given data set. For example, the TOVS Polar Pathfinder data were defined with the EASE-Grid Northern hemisphere map projection parameters, and a polar subset of the original hemisphere at a 100 kilometer resolution. The AVHRR Polar Pathfinder data were defined for both Northern and Southern hemisphere maps, as subsets of each, at 1.25 km, 5 km, and 25 km resolutions. The figure below shows the Northern hemisphere grid extent for SSM/I (the full hemisphere), TOVS Polar, and AVHRR Polar grids (respective subsets).

Polar Pathfinder Northern Hemisphere Grids Extent

Relative Northern hemisphere grid extents of Polar Pathfinders
(SSM/I (full hemisphere), AVHRR and TOVS).

The AARI Sea Ice Data in EASE-Grid

The Arctic and Antarctic Research Institute (AARI) 10-Day Arctic Ocean EASE-Grid Sea Ice Observations data provide another example. These data did not require hemispheric coverage, but the data set producers at NSIDC wanted to provide them in a grid that would facilitate intercomparison with sea ice data derived from SSM/I. Therefore the AARI EASE-Grid was defined to be the subset of the SSM/I Pathfinder Nh grid (Northern hemisphere, 12.5 km resolution) defined by columns 360 through 1080 and rows 360 through 1080. The resulting AARI EASE-Grid is 721 columns and 721 rows. This, in turn, relates the AARI EASE-Grid definition to the 25 km AVHRR EASE-grid (aka "NA25") subset via the following simple relationship:

  • AARIcolumn = 2 * NA25column
  • AARIrow = 2 * NA25row

The Arctic Climatology Project Arctic Meteorology and Climate Atlas in EASE-Grid

NSIDC has produced an atlas of Arctic meteorology and climatology under the auspices of the U.S.-Russian Joint Commission on Economic and Technological Cooperation's Environmental Working Group (EWG). The gridded fields produced for this atlas are defined for a subset of the full northern hemisphere above 65N, at a 250 km grid spacing. Data arrays are 23 columns by 23 rows. An example temperature field is included below.

EWG (250 km) Mean Air Temperatures, February, 1981 - 1990

Mean air temperature (degrees Centigrade), February, 1981 - 1990.
(Dots indicate the EWG EASE-Grid lattice points.)

An Example Application

The following set of images illustrates a comparison of SSM/I-derived (top left) sea ice in Baffin Bay (the water body to the west of Greenland) with AVHRR-derived albedo (top right and zoomed area, below) for June 13, 1994. Since all images are derived from various resolution grids of the same Nl EASE-Grid projection, the reader can easily visually compare the ice edge. Digital comparison and analysis are likewise easily performed using the user's favorite graphical analysis package.

Satellite-Derived Sea Ice, 1994-06-13

Arctic Satellite-derived Sea Ice, June 13, 1994. Top left image represents SSM/I-derived
(passive microwave) sea ice concentrations at 25 km resolution; top right image represents 25 km
AVHRR Albedo Browse Product; bottom image represents zoomed area of Baffin Bay, AVHRR Albedo
Product full resolution (1.25 km).

Conclusions

While originally intended for use with a single data product, the EASE-Grid has proven to be flexible and extensible to other global, gridded applications. The projection and gridding abstractions are simple and easy to apply to the requirements of a new data set. Data from diverse sources can be resampled and expressed as digital arrays of varying resolutions, which are defined in relation to one of three possible projections. Storage as a simple digital array facilitates portability and ability to be imported into a user's favorite analysis package. Users find that visualization and intercomparison operations are then greatly simplified, and that the tasks of analysis and intercomparison can be more readily accomplished.


References and Useful Web Sites

  • Armstrong, R. L. and M. J. Brodzik. 1995. An Earth-Gridded SSM/I Data Set for Cryospheric Studies and Global Change Monitoring. Adv. Space Res. 16(10):155-163.

  • Armstrong, Richard, M. J. Brodzik, A. Varani, 1997. "The NSIDC EASE-Grid: Addressing the need for a common, flexible, mapping and gridding scheme." Earth System Monitor, Vol. 7, No. 4, 3 pp.

  • Knowles, Kenneth W. 1993. Points, pixels, grids, and cells -- a mapping and gridding primer. <http://cires.colorado.edu/~knowlesk/ppgc.html>

  • Maling, D. H. 1992. Coordinate Systems and Map Projections. 2nd ed. Oxford: Pergamon Press.

  • National Snow and Ice Data Center. All About EASE-Grid. <http://nsidc.org/data/ease/>

  • National Snow and Ice Data Center. Complete source code for mapping and gridding software. <ftp://baikal.colorado.edu/pub/NSIDC/maps.tar.gz>

  • Snyder, John P. 1987. Map Projections: A Working Manual. USGS Professional Paper 1395. Washington, DC: US Government Printing Office.