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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.

Table 1. Polar Azimuthal Equal-Area Map Parameters
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:

Table 2. Description of Variables in Projection Equations
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.


Table 3. Projection 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.


Figure 1. TOVS and AVHRR Polar Pathfinder Northern Hemisphere Grid Subsets
 
Figure 1. Pathfinder Northern Hemisphere Grid thumbnail
Figure 2. TOVS and AVHRR Polar Pathfinder Southern Hemisphere Grid Subsets

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.


Table 4. 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.


Table 5. Grid Dimensions and Latitude Extent
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 various Polar Pathfinder grid resolutions, in the center TOVS 100 km pixel.
Figure 3. Relationships of Polar Pathfinder Grid Resolutions Within the TOVS 100 km Pixel

References [top]