|Northern Hemisphere EASE-Grid projection||Southern Hemisphere EASE-Grid projection|
The North azimuthal equal-area map is defined as
The South azimuthal equal-area map is defined as
|Global EASE-Grid projection|
The modified cylindrical equal-area map is defined as
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.
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|
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|
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:
|Shape Distortion||Areal Distortion||Angular Deformation|
|latitude||k/h||kh - 1||w|
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."
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.
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 EASE-Grid family of grid definitions includes, but is not limited to, the following specific 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|
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).
| Relative Northern hemisphere grid extents of Polar Pathfinders
(SSM/I (full hemisphere), AVHRR and TOVS).
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:
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.
| Mean air temperature (degrees Centigrade), February, 1981 - 1990.
(Dots indicate the EWG EASE-Grid lattice points.)
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.
| Arctic Satellite-derived Sea Ice, June 13, 1994. Top left image
(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).
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.