# Original EASE-Grid Format Description

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

Authors: Mary J. Brodzik and Ken Knowles, NSIDC

A version of this paper was presented at the NCGIA International Conference on Discrete Global Grids, March 28, 2000, Santa Barbara, California USA. The proceedings of that meeting were subsequently published by NCGIA. The version of this paper published by NCGIA is available at:

Brodzik, M. J. and K. W. Knowles. 2002.
EASE-Grid: A Versatile Set of Equal-Area Projections and Grids
in M. Goodchild (Ed.) *Discrete Global Grids*. Santa Barbara,
California USA: National Center for Geographic Information & Analysis.

- Introduction: The NSIDC EASE-Grid Map Projection Parameters and Grid Definitions
- EASE-Grid Map Parameters
- EASE-Grid Family of Grid Definitions
- Other Grid Definitions in the EASE-Grid Family
- References

## Introduction: The NSIDC EASE-Grid 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, 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 by following the Mapx link on the Data Analysis and Imaging Tools Web page, and contains software to convert among many projections, but this document is intended as 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 tool for users of global-scale gridded data, specifically remotely sensed data, although it is gaining popularity as a common gridding scheme for data from other sources as well. Data from various sources can be expressed as digital arrays of varying grid resolutions, which are defined in relation to one of the three possible projections. The user will find that visualization and intercomparison operations are then greatly simplified, and that the tasks of analysis and intercomparison can be more readily accomplished.

## EASE-Grid Map Parameters

The three EASE-Grid projections comprise two azimuthal equal-area projections, for the Northern or Southern hemisphere, respectively, and a global cylindrical equal-area projection. All projections are based on a spherical model of the Earth with 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).

### Northern Hemisphere Azimuthal Equal-Area Projection

The North azimuthal equal-area map is defined by the following equations:

- r = 2*R/C * sin(lambda) * sin(PI/4 - phi/2) + r0
- s = 2*R/C * cos(lambda) * sin(PI/4 - phi/2) + s0
- h = cos(PI/4 - phi/2)
- k = sec(PI/4 - phi/2)

where:

Variable | Definition |
---|---|

r | Column coordinate |

s | Row coordinate |

h | Particular scale along meridians |

k | Particular scale along parallels |

lambda | Longitude in radians |

phi | Latitude in radians |

R | Radius of the Earth = 6371.228 km |

C | Nominal cell size |

r0 | Map origin column |

s0 | Map origin row |

**Note**: 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.

Figure 1 shows the region of the Northern Hemisphere Azimuthal EASE-Grid.

Figure 1. Northern Hemisphere Azimuthal Equal-Area Projection |

### Southern Hemisphere Azimuthal Equal-Area Projection

The South azimuthal equal-area map is defined by the following equations:

- r = 2*R/C * sin(lambda) * cos(PI/4 - phi/2) + r0
- s = -2*R/C * cos(lambda) * cos(PI/4 - phi/2) + s0
- h = sin(PI/4 - phi/2)
- k = csc(PI/4 - phi/2)

See Table 1 for definitions of the variables in these equations.

Figure 2 shows the region of the Southern Hemisphere Azimuthal EASE-Grid.

Figure 2. Southern Hemisphere Azimuthal Equal-Area Projection |

### Global Cylindrical Equal-Area Projection

The cylindrical equal-area map is defined as true at 30° N/S and by the following equations:

- r = r0 + R/C * lambda * cos(30)
- s = s0 - R/C * sin(phi) / cos(30)
- h = cos(phi) / cos(30)
- k = cos(30) / cos(phi)

See Table 1 for definitions of the variables in these equations.

Figure 3 shows the region of the global cylindrical EASE-Grid.

Figure 3. Global Cylindrical Equal-Area Projection |

### Why Equal-Area Maps?

Discussion of map projections is often unnecessarily lengthy and sidetracked by ignorance or disregard for the fact that there is no one best map projection. 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.

"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" (Knowles, 1993). 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. 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 (see details in the following table). 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.

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.

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.

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, as follows:

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 Points, Pixels, Grids, and Cells -- A Mapping and Gridding Primer document.

### 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. 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.

## 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 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, anchoring the map origin to the center of the grid, etc.).

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:

### 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, 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.

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 25 and 12.5 km 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 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. The figure below 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.

#### The AARI Sea Ice Data in EASE-Grid

The AARI EASE-Grid sea ice 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

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 the image above. Here is an example of the 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 Points, Pixels, Grids, and Cells: A Mapping and Gridding Primer document for details on defining custom EASE-Grid definitions.

### References

- Knowles, Kenneth W. 1993.
*Points, Pixels, Grids, and Cells -- a Mapping and Gridding Primer*. Unpublished report to the National Snow and Ice Data Center, Boulder, Colorado USA. - NOAA/NASA Pathfinder EASE-Grid Brightness Temperatures
- AARI 10-Day Arctic Ocean EASE-Grid Sea Ice Observations
- TOVS Pathfinder Path-P Daily and Monthly Polar Gridded Atmospheric Parameters
- .mpp files
- M200correct.mpp (Used for SSM/I grids)
- N200correct.mpp (Used for AVHRR and SSM/I grids)
- S200correct.mpp (Used for AVHRR and SSM/I grids)
- NpathP.mpp (Used for TOVS grid)

- .gpd files