EASE-Grid Projection

Table of Contents

Overview

EASE-Grid projections
Figure 1. EASE-Grid map projections: The Northern Hemisphere Azimuthal (top left), the Southern Hemisphere Azimuthal (top right), and the Global Cylindrical (bottom).

EASE-Grid is a versatile format for global-scale 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 equal-area, azimuthal) and full global (cylindrical, equal-area). Figure 1 shows the regions of the Northern Hemisphere Azimuthal EASE-Grid (top left), the Southern Hemisphere Azimuthal EASE-Grid (top right), and the Global Cylindrical EASE-Grid (bottom). EASE-Grid users may specify any subset area of these three projections and any grid resolution. With EASE-Grid, visualization and intercomparison operations are greatly simplified, making analysis and intercomparison more convenient.

There are two versions of EASE-Grid; the original EASE-Grid was defined in 1992 and has been used for many data sets, while EASE-Grid 2.0 was defined in 2011 and is recommended for new data sets.

Differences between the original EASE-Grid and EASE-Grid 2.0

EASE-Grid 2.0 is defined with the WGS 84 ellipsoid rather than the spherical Earth model as in the original EASE-Grid, and offers the following enhancements and improvements:

  • GeoTIFF compatibility
  • Exact azimuthal grid scale
  • Simplified nested grids
  • No undefined "off-the-Earth" grid cells at corners of azimuthal grids
  • With reprojection, comparisons can still be made to data in original EASE-Grid format

Table 1 summarizes the differences between the original EASE-Grid and EASE-Grid 2.0.

Table 1. Comparison of EASE-Grid Versions
Feature Original EASE-Grid EASE-Grid 2.0
Datum International 1924 Authalic Sphere WGS 84
Pole location Center of cell Intersection of the four center cells
Scale (data set-specific) Azimuthal/Cylindrical coupled (such as Nl/Sl/Ml 25.067525 km) Azimuthal: Exact (such as 25.0 km or 36.0 km)

Cylindrical: Integer-multiples across latitude of true scale
Dimensions Odd-numbered (721 x 721) Even-numbered (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 reprojection1
Software Issues Usually requires user to understand custom projection settings Most software supported1
1 Achieved by setting projection ellipsoid to reference datum
 

Importing EASE-Grid 2.0 Data into a GIS

To import these data into a GIS, refer to the Importing NSIDC Passive Microwave EASE-Grid Data into a GIS page.

Land-Ocean-Coast-Ice Masks

Corresponding Land-Ocean-Coast-Ice (LOCI) masks derived from the BU-MODIS land cover data are also available. These Land-Ocean-Coastline-Ice (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 EASE-Grid and EASE-Grid 2.0 azimuthal and global projections, at various spatial resolutions ranging from 3 km to 100 km.

Map Projection Parameters and Grid Definitions

The Equal-Area Scalable Earth Grid (EASE-Grid) consists of a set of 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, 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 EASE-Grid projections comprise two azimuthal equal-area projections for the Northern and Southern hemispheres, and a global cylindrical equal-area 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 EASE-Grid 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.

Table 1. Projection Equations for Northern Hemisphere (NH) and Southern Hemisphere (SH) Azimuthal Equal-Area Projections and Global Cylindrical Equal-Area Projection
Variable Definition NH Azimuthal Equal-Area Projection SH Azimuthal Equal-Area Projection Global Cylindrical Equal-Area 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:

Table 2. Definitions of Variables in Projection Equations
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 EASE-Grid Family of Grid Definitions section of this document for details.

Why use Equal-Area Maps?

Each projection has different properties and thus different best uses. Sometimes the question is raised as to why we chose equal-area projections over the other possibilities for the EASE-Grids, 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 equal-area. No map projection is both, and some are neither." On equal-area 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 EASE-Grids, 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 EASE-Grid, 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 low-latitudes.

Table 2. Aspect Ratios (a measure of shape distortion) of the EASE-Grid Projections
Azimuthal Equal-Area Cylindrical Equal-Area
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.

Table 3. Areal Distortion of the Polar Stereographic Map True at 70° N
Polar Stereographic, (true at 70° N)
latitude kh - 1
90 -6%
45 29%
0 276%

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. Refer to Table 4.

Table 4. Areal and Shape Distortion
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 equal-area projections for the EASE-Grids 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. 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 look-up 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 Equal-Area 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 EASE-Grid 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 EASE-Grid 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 (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 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.
 

Table 5. Dimensions, Center, and Extent of the original SSM/I grids
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 EASE-Grid 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 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. Figure 5 shows the grid extent for SSM/I, TOVS Polar, and AVHRR Polar grids.

For more information on the relationships between the Polar Pathfinder EASE-Grids, please see Summary of NOAA/NASA Polar Pathfinder Grid Relationships.
Polar Pathfinder Northern Hemisphere Grids Extent

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 EASE-grid (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 bore-centered, 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 EASE-Grids.

Upper Left Corner of AVHRR and AARI EASE-Grids

Figure 6. Grid Extent Boundaries at the Upper Left Corner of the AARI and NA25 EASE-Grids

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 EASE-Grid definitions.

    Software Support for EASE-Grid Versions

    Table 7 summarizes the current state of support for EASE-Grid 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 EASE-Grid 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.

    Table 7. Software Support for EASE-Grid Versions
    Software
    & Codes
    Original EASE-Grid
    (Spherical Projection)
    EASE-Grid 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 EASE-Grid 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 EASE-Grid 2.0 data. Both original EASE-Grid and EASE-Grid 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 EASE-Grid 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 EASE-Grid 2.0 cylindrical.
    ENVI Azimuthal supported.
    T. Haran (NSIDC) has written procedure to add cylindrical equal-area to ENVI. Requires simple modifications to support header information for EASE-Grid data sets. ENVI cannot read/write user-defined projection information from/to GeoTIFF, but it can do so with its own ENVI header format.
    Azimuthal supported.
    Cylindrical equal-area requires user-defined projection information; simple modifications to support header information for EASE-Grid 2.0 data sets are forthcoming.
    Simple modifications are required for cylindrical equal-area in both EASE-Grid 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

      Table 8. .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 10-Day Arctic Ocean EASE-Grid 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 EASE-Grid Brightness Temperatures

      TOVS Pathfinder Path-P Daily and Monthly Polar Gridded Atmospheric Parameters

      Citing this Application

      Please use the following citations when referencing data using the EASE-Grid projection:

      EASE-Grid 1.0

      Brodzik, M. J. and K. W. Knowles. 2002. “Chapter 5: EASE-Grid: A Versatile Set of Equal-Area 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.

      EASE-Grid 2.0

      Brodzik, M. J., B. Billingsley, T. Haran, B. Raup, M. H. Savoie. 2012. EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets. ISPRS International Journal of Geo-Information, 1(1):32-45, doi:10.3390/ijgi1010032. http://www.mdpi.com/2220-9964/1/1/32.

      Brodzik, M. J., B. Billingsley, T. Haran, B. Raup, M. H. Savoie. 2014. Correction: Brodzik, M. J. et al. EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets. ISPRS International Journal of Geo-Information 2012, 1, 32-45. ISPRS International Journal of Geo-Information, 3(3):1154-1156, doi:10.3390/ijgi3031154. http://www.mdpi.com/2220-9964/3/3/1154