[CF-metadata] Gaussian Lat/Lon grids

[John Caron]

thanks Brian!

would you also recommend storing the gaussian weights in the file? somehow otherwise documenting this?

Brian Eaton wrote:
> 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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