Hi John,
In order to construct latitude cell bounds so that the cell areas give
identical integrals over the surface of a unit sphere to those that would
be obtained by Gaussian integration, we do the following:
Let phi(j), j = 1, nlat be the Gauss points ordered from S.P. to N.P.
Let phi_bnd(j), j = 1, nlat+1 be the latitude cell edges which satisfy
phi_bnd(j) < phi(j) < phi_bnd(j+1) for all j = 1, nlat
and phi_bnd(1) = -pi/2, phi_bnd(nlat+1) = pi/2.
Let gw(j), j = 1, nlat be the Gauss weights corresponding to Gauss points
phi(j).
In psuedo code:
sin_phi_bnd(1) = -1.0
do j = 1, nlat
sin_phi_bnd(j+1) = sin_phi_bnd(j) + gw(j)
end do
Then compute corresponding latitudes by taking arc sine.
Note that since sin_phi_bnd(1) = -1, and since the Gauss weights sum to
2.0, then sin_phi_bnd(nlat+1) will equal 1.0 to within the accuracy that
the Gauss weights sum to 2.0. We generally set sin_phi_bnd(nlat+1) = 1.0
and let the accumulated error in the sum of the Gauss weights (hopefully at
the double precision roundoff level) be included in the last latitude cell.
Brian
On Tue, Jun 26, 2007 at 10:08:57AM -0600, John Caron wrote:
> Hi Karl:
>
> Is there a standard way that the latitude bounds should be calculated that reflects the gaussian?
>
> If I do include the weights, sounds like I should include a description of how they were calculated. I used "the roots of the ordinary Legendre polynomial of degree NLAT using Newton's iteration method". Does that seem sufficient and useful?
>
> BTW, it appears you didnt send to list, so i forwarded.
>
> Karl Taylor wrote:
> > Dear John,
> >
> > I think that CF doesn't bother identifying gaussian grids or weights as
> > such, but simply stores the latitude values and the latitude bounds.
> > This avoids the problem that there is no unique way of defining weights
> > for a given spherical harmonic truncation (although conventionally
> > nearly everyone does this in the same way).
> >
> > I hope someone will correct me if I've said anything incorrect.
> >
> > cheers,
> > Karl
> >
> > John Caron wrote:
> >> I dont see a standard for encoding gaussian weights for a gaussian
> >> Latitude grid (eg GRIB GDS type 4 "Gaussian Lat/Lon"). Does this seem
> >> needed? If so, any suggestions?
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Received on Tue Jun 26 2007 - 19:26:28 BST
