EASEGrid Projection
Table of Contents
Overview
EASEGrid is a versatile format for globalscale gridded data, especially remotely sensed data, but is also a common gridding scheme for other data as well. Data from various sources can be expressed as digital arrays of varying grid resolutions, which are defined in relation to one of three possible projections: Northern and Southern Hemisphere (Lambert's equalarea, azimuthal) and full global (cylindrical, equalarea). Figure 1 shows the regions of the Northern Hemisphere Azimuthal EASEGrid (top left), the Southern Hemisphere Azimuthal EASEGrid (top right), and the Global Cylindrical EASEGrid (bottom). EASEGrid users may specify any subset area of these three projections and any grid resolution. With EASEGrid, visualization and intercomparison operations are greatly simplified, making analysis and intercomparison more convenient.
Differences between the original EASEGrid and EASEGrid 2.0
EASEGrid 2.0 is defined with the WGS 84 ellipsoid rather than the spherical Earth model as in the original EASEGrid, and offers the following enhancements and improvements:
 GeoTIFF compatibility
 Exact azimuthal grid scale
 Simplified nested grids
 No undefined "offtheEarth" grid cells at corners of azimuthal grids
 With reprojection, comparisons can still be made to data in original EASEGrid format
Table 1 summarizes the differences between the original EASEGrid and EASEGrid 2.0.
Feature  Original EASEGrid  EASEGrid 2.0 

Datum  International 1924 Authalic Sphere  WGS 84 
Pole location  Center of cell  Intersection of the four center cells 
Scale (data setspecific)  Azimuthal/Cylindrical coupled (such as Nl/Sl/Ml 25.067525 km)  Azimuthal: Exact (such as 25.0 km or 36.0 km) Cylindrical: Integermultiples across latitude of true scale 
Dimensions  Oddnumbered (721 x 721)  Evennumbered (720 x 720) 
Nested Grids  Must choose between total coverage and nested cells  Coverage can stay the same, only number of cells changes 
Corner Points  Corner cell locations in azimuthal grids are undefined  No undefined corner cells 
GeoTIFF  Requires reprojection  Supported without reprojection^{1} 
Software Issues  Usually requires user to understand custom projection settings  Most software supported^{1} 
^{1} Achieved by setting projection ellipsoid to reference datum 
Importing EASEGrid 2.0 Data into a GIS
To import these data into a GIS, refer to the Importing NSIDC Passive Microwave EASEGrid Data into a GIS page.
LandOceanCoastIce Masks
Corresponding LandOceanCoastIce (LOCI) masks derived from the BUMODIS land cover data are also available. These LandOceanCoastlineIce (LOCI) files provide land classification masks derived from the Boston University MOD12Q1 V004 MODIS/Terra 1 km Land Cover Product. The masks are available in various EASEGrid and EASEGrid 2.0 azimuthal and global projections, at various spatial resolutions ranging from 3 km to 100 km.
 EASEGrid LandOceanCoastlineIce Masks Derived from Boston University MODIS/Terra Land Cover Data
 EASEGrid 2.0 LandOceanCoastlineIce Masks Derived from Boston University MODIS/Terra Land Cover Data
Map Projection Parameters and Grid Definitions
The EqualArea Scalable Earth Grid (EASEGrid) consists of a set of three equalarea 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, Colorado, USA. This philosophy was used to implement a library of software routines, which are 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 on the What is the Map Transformations Library (Mapx)? Web page, and contains software to convert between many projections.
Map Projection Details
The three EASEGrid projections comprise two azimuthal equalarea projections for the Northern and Southern hemispheres, and a global cylindrical equalarea projection (Fig. 1). All projections are based on a spherical model of the Earth with a radius R = 6371.228 km. This radius defines a sphere with the same surface area as the 1924 International Ellipsoid (also known as the International 1924 Authalic Sphere). In the following, the three EASEGrid map projections and their defining parameters are described in more detail.
Map Projection Parameters and Equations
The three maps are defined by a set of four equations, describing the column and row coordinates, and the particular scales along the meridians and parallels, for example, longitude and latitude lines. Table 1 gives an overview of these equations for all three projections, while Table 2 lists the variables needed to solve these equations.
Variable  Definition  NH Azimuthal EqualArea Projection  SH Azimuthal EqualArea Projection  Global Cylindrical EqualArea Projection 

r 
Column coordinate  2*R/C * sin(lambda) * sin(PI/4  phi/2) + r0  2*R/C * sin(lambda) * cos(PI/4  phi/2) + r0  r0 + R/C * lambda * cos(30) 
s 
Row coordinate  2*R/C * cos(lambda) * sin(PI/4  phi/2) + s0  2*R/C * cos(lambda) * cos(PI/4  phi/2) + s0  s0  R/C * sin(phi) / cos(30) 
h 
Particular scale along meridians  cos(PI/4  phi/2)  sin(PI/4  phi/2)  cos(phi) / cos(30) 
k 
Particular scale along parallels  sec(PI/4  phi/2)  csc(PI/4  phi/2)  cos(30) / cos(phi) 
Where:
Variable  Definition 

R 
Radius of the Earth = 6371.228 km 
C 
Nominal cell size 
lambda 
Longitude in radians 
phi 
Latitude in radians 
r0 
Map origin column 
s0 
Map origin row 
The values of C, r0, and s0 are determined by the grid that is chosen to overlay the projection. See the EASEGrid Family of Grid Definitions section of this document for details.
Why use EqualArea Maps?
Each projection has different properties and thus different best uses. Sometimes the question is raised as to why we chose equalarea projections over the other possibilities for the EASEGrids, and the answer relies on a basic understanding of projection characteristics.
Knowles (1993) states that "Two of the most important characteristics of maps are whether they are conformal or equalarea. No map projection is both, and some are neither." On equalarea maps, a small circle placed anywhere on the map will always cover the same 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 equal to one. The aspect ratio k:h is a measure of shape distortion.
For the Northern and Southern hemisphere EASEGrids, the aspect ratio varies from 1:1 at the pole to 1.17:1 at 45N and increases to only 2:1 at the equator. For the global EASEGrid, the aspect ratio varies more widely. Refer to Table 2. The selection of +/30 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 lowlatitudes.
Azimuthal EqualArea  Cylindrical EqualArea  

latitude  k/h  latitude  k/h 
90  1.00  80  24.90 
75  1.02  75  11.20 
60  1.07  60  3.00 
45  1.17  45  1.50 
30  1.33  30  1.00 
15  1.59  15  0.80 
0  2.00  0  0.75 
In contrast, on conformal maps, angles 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 hk  1 is a measure of areal distortion. Refer to Table 3.
For example, the Polar Stereographic map true at 70N that is used for the SMMR and SSM/I polar gridded data distributed by NSIDC is a conformal map. By definition, the aspect ratio remains 1:1 everywhere; however, the areal distortion of this map varies from 6 percent at the pole to +29 percent at 45° N and increases to +276 percent at the equator.
Polar Stereographic, (true at 70° N)  

latitude  kh  1 
90  6% 
45  29% 
0  276% 
A very popular map that is neither equalarea nor conformal is the cylindrical equidistant map, also known as the latlon grid. This map suffers from both areal and shape distortion. Refer to Table 4.
Shape Distortion  Areal Distortion  

latitude  k/h  kh  1 
89  57  5630% 
80  6  476% 
60  2  100% 
45  1.4  41% 
0  1  0% 
In summary, given the choices of either shape distortion or areal distortion or both, we decided in favor of the equalarea projections for the EASEGrids 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 easytouse. Users wishing a more general discussion of projection characteristics should also read the A Mapping and Gridding Primer: Points, Pixels, Grids, and Cells document.
Why use 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. 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 lookup table  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.
The fastest calculations of course would simply involve mapping to the cylindrical equidistant projection that was mentioned in the previous section. Since in that projection, the latitude and longitude values are in effect the column and row coordinates. However, that projection choice was rejected for reasons of unacceptable distortion in the output gridded data. Please see the previous section discussion, Why EqualArea Maps?, for more information.
Grid Definitions
A grid is always defined in relation to a specific map projection. It is essentially the parameters necessary to define a transparent piece of graph paper that is overlaid on a flat map and then 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 that corresponds to the map's origin
Any number of grid definitions can therefore be used to describe the effect of changing the "graph paper." For example, using fewer columns or rows, a higher resolution, or anchoring the map origin to the center of the grid. 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 each grid cell value according to the most appropriate binning technique for the data and application at hand.
The EASEGrid family of grid definitions includes, but is not limited to, the following specific grids:
 Original SSM/I Grid
 Polar Pathfinder Grids
 AARI Sea Ice Grid
In the following, these three grids are discussed in more detail.
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, which includes gridded Passive Microwave Brightness Temperatures and a set of geophysical products derived from the Brightness Temperatures. However, subsets of the grids for the azimuthal projections have been adopted by a number of other projects, including the TOVS and AVHRR Polar Pathfinders, and the AARI (Arctic and Antarctic Research Institute, St. Petersburg, Russia) Sea Ice data that have been regridded to EASEGrid by NSIDC.
These 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 grid (ML) 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. Rounding the grid coordinates up at .5 yields the grid cell number. Grid cell is centered at grid coordinates (j,i) and bounded by: j .5 <= R < J +.5, I .5 <= S < I +.5.
The 25 km hemispheric grids for the North and South azimuthal projections (NL and SL, respectively) are defined with 721 columns, 721 rows, and the respective pole anchored at cell (360.0,360.0). The ML grid for the cylindrical projection is defined with 1383 columns, 586 rows, and is defined 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 12.5 km grids was defined such that the grid coordinates are coincident (borecentered) 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 global 12.5 km grid (MH) also exactly spans the equator, with 2766 grid cells. However, since the center of the ML column 0 is coincident with the ML 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 is slightly east of 180° E.
Table 5 summerizes the dimensions, center, and extent of the original SSM/I grids. 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 25 km and 12.5 km Grids



Grid Name  Projection/ Resolution 
Dimensions  Map Origin  Map Origin  Grid Extent  
Width  Height  Column (r0)  Row (s0)  Latitude  Longitude  Minimum Latitude  Maximum Latitude  Minimum Longitude  Maximum Longitude  
ML  Global 25 km 
1383  586  691.0  292.5  0.0  0.0  86.72S  86.72N  180.00W  180.00E 
MH  Global 12.5 km 
2766  1171  1382.0  585.0  0.0  0.0  85.95S  85.95N  179.93W  180.07E 
NL  Northern Hemisphere 25 km 
721  721  360.0  360.0  90.0N  0.0  0.34S  90.00N  180.00W  180.00E 
NH  Northern Hemisphere 12.5 km 
1441  1441  720.0  720.0  90.0N  0.0  0.26S  90.00N  180.00W  180.00E 
SL  Southern Hemipshere 25 km 
721  721  360.0  360.0  90.0S  0.0  90.00S  0.34N  180.00W  180.00E 
SH  Southern Hemisphere 12.5 km 
1441  1441  720.0  720.0  90.0S  0.0  90.00S  0.26N  180.00W  180.00E 
Polar Pathfinders
Users of the NSIDC EASEGrid are not limited to the grid orientation, size, and resolution used in the Original SSM/I Grid, and are free to define grids that are more appropriate to a given data set. For example, the TOVS Polar Pathfinder data were defined with the EASEGrid 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. Figure 5 shows the grid extent for SSM/I, TOVS Polar, and AVHRR Polar grids.
Figure 5. Grid Extent for SSM/I, TOVS Polar, and AVHRR Polar Grids.
AARI Sea Ice Grid
The AARI Sea Ice Grid provides 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 Sea Ice 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 Sea Ice Grid is 721 columns and 721 rows. This, in turn, relates the AARI Sea Ice Grid definition to the 25 km AVHRR EASEgrid (aka "NA25") subset via the following simple relationship:
 AARIcolumn = 2 * NA25column
 AARIrow = 2 * NA25row
For example, the center of the (12.5 km) AARI grid cell at (column,row)=(0,0) corresponds to the center of the (25 km) NA25 grid cell (0,0). The AARI grid cell at (1,0) corresponds to the NA25 grid cell (0.5,0), etc. Since these grids were based on the original SSM/I grids, which were defined to be borecentered, the extent of the AARI grid is therefore one half of one 12.5 km cell inward from the extent of the NA25 grid displayed in Figure 5. Figure 6 is an example of the grid extent boundaries at the upper left corner of the AARI and NA25 EASEGrids.
Figure 6. Grid Extent Boundaries at the Upper Left Corner of the AARI and NA25 EASEGrids
Users are encouraged to explore the versatility of this format for their own applications. Please refer to the A Mapping and Gridding Primer: Points, Pixels, Grids, and Cells document for details on defining custom EASEGrid definitions.
Software Support for EASEGrid Versions
Table 7 summarizes the current state of support for EASEGrid format data when used with current popular software packages. Table 7 also lists geodetic parameter data set codes from the European Petroleum Survey Group (EPSG) for both the original EASEGrid and 2.0 versions. EPSG codes are helpful shortcuts for PROJ.4 software, and for software that depends on PROJ.4, such as GDAL Tools.
Software & Codes 
Original EASEGrid (Spherical Projection) 
EASEGrid 2.0 (WGS 84 Projection) 
Current Status 

HDFEOS 2.xx HDFEOS 5.xx 
Supported  Cylindrical projection is supported. Azimuthal is not supported.  HDF depends on GCTP. NSIDC may propose a change to GCTP to support Lambert Azimuthal with ellipsoid. 
PROJ.4  Supported  Azimuthal and cylindrical EASEGrid 2.0 supported in PROJ 4.8.0 (13 March 2012). Azimuthal projection was supported in PROJ 4.7.0 (23 September 2009). Bug in PROJ 4.7.0 resulted in error of ~15 km when transforming points near equator for cylindrical EASEGrid 2.0 data.  Both original EASEGrid and EASEGrid 2.0 are fully supported as of PROJ 4.8.0 (13 March 2012). 
GDAL Tools  Supported  Supported, with correct version of PROJ.4.  User must ensure correct PROJ.4 is installed; see PROJ.4 notes. 
EPSG Codes  North: 3408 South: 3409 Global: 3410 
North: 6931 South: 6932 Global: 6933 Note1: Use these ProjectedCRS Codes for EPSG Version 8.6 or later. Note2: GDAL tools built with the correct version of PROJ4 will work on the EASEGrid 2.0 projections with PROJ4 strings even if it's built with EPSG Codes earlier than 8.6. 
PROJ 4.8.0 is required to obtain correct values for EASEGrid 2.0 cylindrical. 
ENVI  Azimuthal supported. T. Haran (NSIDC) has written procedure to add cylindrical equalarea to ENVI. Requires simple modifications to support header information for EASEGrid data sets. ENVI cannot read/write userdefined projection information from/to GeoTIFF, but it can do so with its own ENVI header format. 
Azimuthal supported. Cylindrical equalarea requires userdefined projection information; simple modifications to support header information for EASEGrid 2.0 data sets are forthcoming. 
Simple modifications are required for cylindrical equalarea in both EASEGrid versions; see respective column for current status. 
Mapx  Supported  Supported  N/A 
ArcGIS  Supported  Supported  N/A 
ERDAS  Supported  Supported  N/A 
.mmp and .gpd File Information
.mpp files  gpd files 

M200correct.mpp (Used for SSM/I grids)  NpathP.gpd (TOVS grid) 
N200correct.mpp (Used for AVHRR and SSM/I grids)  Ml.gpd (ML grid) 
S200correct.mpp (Used for AVHRR and SSM/I grids)  Nl.gpd (NL grid) 
NpathP.mpp (Used for TOVS grid)  Sl.gpd (SL grid) 
Na25.gpd (NA25 grid)  
Sa25.gpd (SA25 grid)  
Mh.gpd (MH grid)  
Nh.gpd (NH grid)  
Sh.gpd (SH grid)  
Na5.gpd (NA5 grid)  
Sa5.gpd (SA5 grid)  
Na1.gpd (NA1 grid)  
Sa1.gpd (SA1 grid) 
References
AARI 10Day Arctic Ocean EASEGrid Sea Ice Observations
Knowles, Kenneth W. 1993. A Mapping and Gridding Primer: Points, Pixels, Grids, and Cells. Unpublished report to the National Snow and Ice Data Center, Boulder, Colorado USA.
NOAA/NASA Pathfinder EASEGrid Brightness Temperatures
TOVS Pathfinder PathP Daily and Monthly Polar Gridded Atmospheric Parameters
Citing this Application
Please use the following citations when referencing data using the EASEGrid projection:
EASEGrid 1.0
Brodzik, M. J. and K. W. Knowles. 2002. “Chapter 5: EASEGrid: A Versatile Set of EqualArea Projections and Grids.” in Michael F.Goodchild (Ed.) Discrete Global Grids: A Web Book. Santa Barbara, California USA: National Center for Geographic Information & Analysis. https://escholarship.org/uc/item/9492q6sm.
EASEGrid 2.0
Brodzik, M. J., B. Billingsley, T. Haran, B. Raup, M. H. Savoie. 2012. EASEGrid 2.0: Incremental but Significant Improvements for EarthGridded Data Sets. ISPRS International Journal of GeoInformation, 1(1):3245, doi:10.3390/ijgi1010032. http://www.mdpi.com/22209964/1/1/32.
Brodzik, M. J., B. Billingsley, T. Haran, B. Raup, M. H. Savoie. 2014. Correction: Brodzik, M. J. et al. EASEGrid 2.0: Incremental but Significant Improvements for EarthGridded Data Sets. ISPRS International Journal of GeoInformation 2012, 1, 3245. ISPRS International Journal of GeoInformation, 3(3):11541156, doi:10.3390/ijgi3031154. http://www.mdpi.com/22209964/3/3/1154