# Grid Relationships

## Summary of NOAA/NASA Polar Pathfinder Grid Relationships

by M. J. Brodzik, NSIDC

All NOAA/NASA Polar Pathfinder grids are based on either a north or south polar azimuthal equal-area map, and differ only in grid resolution. This document describes the map coordinate transformations and the relationships of the various resolution grids. The following sections are covered in this document:

Map Parameters

Grid Parameters

References

## Map Parameters

The polar azimuthal equal-area maps are defined by
the equations listed in Table 1. Table 2 describes the variables in
the projection equations.

North Map Parameters | South Map Parameters |
---|---|

r = 2*R/C * sin(lambda) * sin(PI/4 - phi/2) + r0 | r = 2*R/C * sin(lambda) * cos(PI/4 - phi/2) + r0 |

s = -2*R/C * cos(lambda) * sin(PI/4 - phi/2) + s0 | s = 2*R/C * cos(lambda) * cos(PI/4 - phi/2) + s0 |

h = cos(PI/4 - phi/2) | h = sin(PI/4 - phi/2) |

k = sec(PI/4 - phi/2) | k = csc(PI/4 - phi/2) |

Where:

Variable | Description |
---|---|

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 |

C | nominal cell size |

r0 | map origin column |

s0 | map origin row |

Both North and South projections are based on a spherical model of the Earth with radius R = 6371.228 km. This radius gives a sphere with the same surface area as an ellipsoid using the International Datum. Table 3 lists the aspect ratios.

Azimuthal Equal-Area | |
---|---|

latitude |
k/h |

90 | 1.00 |

75 | 1.02 |

60 | 1.07 |

45 | 1.17 |

30 | 1.33 |

15 | 1.59 |

0 | 2.00 |

## Grid Parameters [top]

The SSM/I North and South azimuthal grids (NL and SL, respectively) are defined with a nominal cell size of 25 km x 25 km, and all other Polar Pathfinder grid resolutions are calculated as exact multiples or factors of this cell size. The actual cell size C=25.067525 km was chosen to make the SSM/I cylindrical 25 km grid exactly span the equator, and was then used for the azimuthal projections for the sake of SSM/I EASE-Grid consistency. Of course, few "25 km" cells actually have these dimensions, but they all have the same area.

The SSM/I grids represent full hemispheric coverage, and the TOVS and AVHRR grids are proper subsets of the full hemispheres.

The TOVS and AVHRR subsets of the SSM/I full Northern Hemisphere grid are shown in Figure 1. Figure 2 displays the TOVS and AVHRR subsets of the SSM/I full Southern Hemisphere grid.

Grid coordinates (r,s) start in the upper left corner with r increasing to the right and s increasing downward. Rounding the grid coordinates up at .5 yields the grid cell number. Grid cell i,j is centered at grid coordinates (i,j) and bounded by:

i - 0.5 < r ≤ i + 0.5, j - 0.5 < s ≤ j + 0.5

With the exception of the AVHRR 1.25 km grids (Na1 and Sa1), each Polar Pathfinder grid is defined such that the grid coordinates are coincident, or bore-centered, at the respective pole. All of these grids are square, symmetric about the pole, with an odd number of rows and columns, and the location of the center pixel of each grid is the (North or South) pole. The AVHRR 1.25 km grids are the only exceptions, being exact multiples of the AVHRR 5 km grids, and so location of the pole in each of these is at the intersection of the four center pixels.

There are four SSM/I grids, six AVHRR polar grids, and two TOVS grids. Table 4 lists grid details by sensor.

Grid Name | Sensor | Resolution (km) | Hemisphere |
---|---|---|---|

NpathP | TOVS | 100 | Northern |

SpathP | TOVS | 100 | Southern |

TOVS | TOVS | 100 | Northern |

NL | SSM/I | 25 | Northern |

SL | SSM/I | 25 | Southern |

NA25 | AVHRR | 25 | Northern |

SA25 | AVHRR | 25 | Southern |

NH | SSM/I | 12.5 | Northern |

SH | SSM/I | 12.5 | Southern |

NA5 | AVHRR | 5 | Northern |

SA5 | AVHRR | 5 | Southern |

NA1 | AVHRR | 1.25 | Northern |

SA1 | AVHRR | 1.25 | Southern |

The dimensions and latitude extent of each grid are summarized in Table 5. The map projections used for all grids are defined with the origin at the respective pole. For the AVHRR and TOVS grids, the latitude extent listed is the latitude of the center of the corner pixels on the grid. For the SSM/I grids, the latitude extent listed is the latitude of the center of the center pixel on a side.

Grid Name | Dimensions | Latitude Extent | ||
---|---|---|---|---|

Width | Height | Minimum | Maximum | |

NpathP | 67 | 67 | 46.90928 N | 90.00000 N |

SpathP | 89 | 89 | 90.00000 S | 30.63221 S |

NL | 721 | 721 | 0.33836 S | 90.00000 N |

SL | 721 | 721 | 90.00000 S | 0.33836 N |

NA25 | 361 | 361 | 29.89694 N | 90.00000 N |

SA25 | 321 | 321 | 90.00000 S | 37.13584 S |

NH | 1441 | 1441 | 0.25845 S | 90.00000 N |

SH | 1441 | 1441 | 90.00000 S | 0.25845 N |

NA5 | 1805 | 1805 | 29.74956 N | 90.00000 N |

SA5 | 1605 | 1605 | 90.00000 S | 36.99339 S |

NA1 | 7220 | 7220 | 29.72191 N | 90.00000 N |

SA1 | 6420 | 6420 | 90.00000 S | 36.96667 S |

The relationships of the various grids are shown in Figure 3.

**Figure 3.**Relationships of Polar Pathfinder Grid Resolutions Within the TOVS 100 km Pixel

## References [top]

**Brodzik, M.J. 1998.**The EASE-Grid, A Versatile Set of Equal-Area Projections and Grids. Special report to the National Snow and Ice Data Center, Boulder, CO.**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, CO.- DMSP SSM/I Pathfinder Daily EASE-Grid Brightness Temperatures online guide document
- TOVS Pathfinder Path-P Daily Arctic Gridded Atmospheric Parameters online guide document
- AVHRR Polar 1 km Level 1b Data Set online guide document
- .mpp files
*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