# EASE-Grid 2.0: Incremental but Significant Improvements for Earth-Gridded Data Sets

^{*}

## Abstract

**:**

## 1. EASE-Grid History and Attributes

**Scalable**Earth-Grid”, to emphasize the versatility of applications it enjoyed. Today, the term EASE-Grid refers to the three original projections and associated gridding scheme, but does not include prescriptions for binning or interpolation methods.

**Figure 1.**Northern and Southern EASE-Grid projections. See second figure in Section 2 for zoomed corner subset.

## 2. EASE-Grid Limitations

**ellipsoid**is a closed surface formed by the rotation of an ellipse about its shorter (minor) axis.

**coordinate system**(CS) is a set of rules to define how coordinates are assigned to points, usually by means of associated axes.

**coordinate reference system**(CRS) is a CS that defines position, scale and orientation of its axes defined with respect to an object, which for our purposes is the Earth.

**Geodetic latitude**is the angle from the equatorial plane to the perpendicular to the ellipsoid through a given point.

**Geodetic longitude**is the angle from the prime meridian plane to the meridian plane of a given point.

**ellipsoidal coordinate system**(ECS) is a CS specified by geodetic latitude and longitude.

**geodetic datum**defines the relationship of a 2- or 3-dimensional CS to the Earth. A geodetic datum defines the position of the origin, the scale, and the orientation of the axes of an ECS with respect to a particular Earth ellipsoid. The characteristics of several Earth ellipsoids are included in Table 1.

**geodetic coordinate reference system**(GCRS) is an ECS defined for a particular geodetic datum.

**coordinate conversion**is a change of coordinates from one CRS to another, in which the CRSs are either based on the same datum or, if they are based on different datums, no algorithm has been applied to transform the coordinates from one datum to the other.

**coordinate transformation**is a change of coordinates from one CRS to another in which the CRSs are based on different datums. In this case, a coordinate transformation algorithm is applied to convert the coordinates of one CRS to conform to the datum of the other CRS.

**map projection**is a coordinate conversion from an ECS to a plane. A map projection is defined by various projection parameters, including the projection ellipsoid, the ECS coordinates of the projection origin in the plane, the orientation of the projected axes in the plane with respect to the ECS, and other projection-specific parameters.

**projected coordinate reference system**(PCRS) is a CRS derived from a GCRS by applying a specified map projection. Note that the projection ellipsoid specified in the map projection may or may not match the datum specified in the GCRS.

**Table 1.**Characteristics of Earth ellipsoids (Table 2.1, p. 10 in [21] and Appendix B) discussed in the text.

Earth Ellipsoids | ||||
---|---|---|---|---|

Name | Equatorial Radius (m) | Flattening | Calculated Polar Radius | Calculated Eccentricity |

a | f | b = a(1 − f ) | e = $\sqrt{2f-{f}^{2}}$ | |

International 1924 Ellipsoid | 6 378 388 | 1/297.0 | 6 356 911.946 | 0.081 991 889 979 0 |

International 1924 Authalic Sphere | 6 371 228 | 0 | 6 371 228 | 0.0 |

WGS 84 | 6 378 137 | 1/298.257 223 563 | 6 356 752.314 | 0.081 819 190 842 6 |

**Figure 3.**Example of NASA Operation IceBridge flight tracks overlaid on NASA Blue Marble basemap of central eastern Greenland. Blue Marble data are produced in a Plate-Carrée projection referenced to the WGS 84 datum and stored as WMS layers in the original EASE-Grid. The left-hand base map was derived by coordinate transformation into the original EASE-Grid, with an inappropriate datum shift. The right-hand base map was derived correctly by coordinate conversion only (reprojection without the datum shift). The error in the basemap was not identified until the flight manager notified us that the flight track actually traversed down the middle of Daugaard-Jensen Gletscher in the upper Scoresby Sund fjord (top center circle in each image) and that the flightlines parallel to Blosseville Kyst (lower two circles in each image) were performed over sea ice along the coast, not inland.

**Figure 4.**Zoomed area of lower right corner of Northern EASE-Grid projection in Figure 1, indicating 25-kilometer corner pixels with undefined (latitude, longitude) coordinates. All corners are symmetric.

## 3. EASE-Grid 2.0 Definition

**Figure 5.**Relative gridding schemes for representative azimuthal 25 km and 12.5 km original EASE-Grid ((

**Left**), bore-centered) vs. EASE-Grid 2.0 ((

**Right**), nested) cells near the pole.

**Figure 6.**Relationship of equator to right edge of grid coverage in Northern Hemisphere azimuthal 25-kilometer original EASE-Grid ((

**Left**), equator enclosed within spatial coverage) vs. EASE-Grid 2.0 ((

**Right**), equator slightly outside spatial coverage). Southern Hemisphere grids are defined likewise. Curvature of equator is greatly exaggerated relative to edge of grid.

## 4. Conclusions

## Acknowledgements

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## Appendix A: EASE-Grid 2.0 Projection Formulae

**Definition.**

_{0}= map reference latitude

_{0}= map reference longitude

_{1}= latitude of true scale

_{0}= 0.0°

_{0}= 0.0°

_{1}= 30.0°

_{0}= 90.0°

_{0}= 0.0°

_{0}= -90.0°

_{0}= 0.0°

**Definition**

**Definition**

_{0}from Equation (1), use q(ϕ) from Equation (2), and let

_{p}= q(ϕ = 90°)

^{2}/3 + 31e

^{4}/180 + 517e

^{6}/5040) sin 2β

+ (23e

^{4}/360 + 251e

^{6}/3780) sin 4β

+ (761e

^{6}/45360) sin 6β

**Definition**

_{p}= q(ϕ = 90°)

_{0})

**Definition**

_{p}= q(ϕ = 90°)

^{2}/3 + 31e

^{4}/180 + 517e

^{6}/5040)sin 2β

+ (23e

^{4}/360 + 251e

^{6}/3780)sin 4β

+ (761e

^{6}/45360)sin 6β

## Appendix B: Authalic Sphere Definition

## Appendix C: EASE-Grid 2.0 Projection Parameters

EASE-Grid 2.0 Projection Parameters | |||||||
---|---|---|---|---|---|---|---|

Name | Map Origin | Latitude of True Scale | Reference Datum | Projection Ellipsoid | PROJ.4 Arguments | ||

ϕ_{0} | λ_{0} | ϕ_{1} | |||||

North | 90°N | 0°E | 90°N | WGS 84 | WGS 84 | +proj=laea | +lat_0=90 |

+lon_0=0 | +x_0=0 | ||||||

+y_0=0 | +ellps=WGS84 | ||||||

+datum=WGS84 +units=m | |||||||

South | 90°S | 0°E | 90°S | WGS 84 | WGS 84 | +proj=laea | +lat_0=90 |

+lon_0=0 | +x_0=0 | ||||||

+y_0=0 | +ellps=WGS84 | ||||||

+datum=WGS84 +units=m | |||||||

Cylindrical | 0°N | 0°E | 30°N/S | WGS 84 | WGS 84 | +proj=laea | +lat_0=90 |

+lon_0=0 +lat 1=30 | +x_0=0 | ||||||

+y_0=0 | +ellps=WGS84 | ||||||

+datum=WGS84 +units=m |

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Brodzik, M.J.; Billingsley, B.; Haran, T.; Raup, B.; Savoie, M.H.
*EASE-Grid 2.0*: Incremental but Significant Improvements for Earth-Gridded Data Sets. *ISPRS Int. J. Geo-Inf.* **2012**, *1*, 32-45.
https://doi.org/10.3390/ijgi1010032

**AMA Style**

Brodzik MJ, Billingsley B, Haran T, Raup B, Savoie MH.
*EASE-Grid 2.0*: Incremental but Significant Improvements for Earth-Gridded Data Sets. *ISPRS International Journal of Geo-Information*. 2012; 1(1):32-45.
https://doi.org/10.3390/ijgi1010032

**Chicago/Turabian Style**

Brodzik, Mary J., Brendan Billingsley, Terry Haran, Bruce Raup, and Matthew H. Savoie.
2012. "*EASE-Grid 2.0*: Incremental but Significant Improvements for Earth-Gridded Data Sets" *ISPRS International Journal of Geo-Information* 1, no. 1: 32-45.
https://doi.org/10.3390/ijgi1010032