National Oceanic and
Atmospheric Administration
United States Department of Commerce

averages/integrals on the sphere


The U.S. government is closed. This site will not be updated; however, NOAA websites and social media channels necessary to protect lives and property will be maintained. To learn more, visit commerce.gov.

For the latest forecasts and critical weather information, visit weather.gov.


averages/integrals on the sphere

Question:

How does Ferret compute averages and intervals on the surface of the earth?

Discussion:

Computing averages and integrals on the surface of a sphere requires the application of a correction for the size of grid cells as one moves from the equator to the poles. Prior to Ferret v5.90 Ferret employed a cosine(latitude) correction but this computation has been refined in more recent versions.

We wish to compute the surface area of a grid cell on a sphere, centered at longitude x and latitude y, with angular widths dx & dy (measuring all angles in radians). The surface area of the band between two latitudes y+dy/2, y-dy/2 (see http://mathworld.wolfram.com/Zone.html) is

       2*pi*R*h

where

     h = R*sin(y+dy/2) - R*sin(y-dy/2)

is the distance in meters along the earth's axis between the north/south edges of the latitude cell. Interestingly, this area depends only on h and not explicitly on y. The fraction of the band between longitudes x1 and x2 is

     dx/(2*pi)

So the grid cell area is

     A = 2*pi*R*h * dx/(2*pi)

Since sin(a+b)-sin(a-b) = 2cos(a)sin(b), we have

     h = 2*R * cos(y) * sin(dy/2)

Thus the grid cell area is

     A = R^2 * cos(y) * dx * 2*sin(dy/2)

Prior to v5.90, Ferret computed cell areas on the sphere using a "small dy" approximation:

       limit as dy->0 of   2*sin(dy/2)   -> dy

I.e.,

   A_approx = R^2 * cos(y) * dx * dy

The ratio of this approximate area over the true area is

       F = A_approx / A
           = dy / 2*sin(dy/2)
           = 1 / sinc(dy/2)

with dy <= pi. This ratio always exceeds 1, which implies that areas were always slightly overestimated prior to v5.90. The fractional error, F-1, exceeds 1e-7 (machine precision) at dy = 0.1deg, 1e-4 at dy = 3 deg, 11% at dy=pi/2, and 57% at dy=pi (largest possible dy). As a result, all integrals prior to v5.90 were slightly too extreme. In addition, averages over grids with non-uniform dy tended to slightly overweight coarsely-gridded latitudes compared to finely-gridded ones. Averages over regions with uniform dy were unaffected by the approximation, so these did not change in v5.90.
 

Contributed by Andrew Wittenberg