# shsgc

Computes spherical harmonic synthesis of a scalar quantity on a gaussian grid via spherical harmonics.

## Prototype

procedure shsgc ( a : numeric, b : numeric, g : float or double )

## Arguments

*a*

*b*

Spherical harmonic coefficients (input, array with two or more
dimensions). If the last two dimensions of *g* are *nlat* x
*nlon*, then the last two dimensions of *a* and *b*
must be *nlat* x *N*, where *N* is
determined as follows:

N = minimum(nlat, (nlon+2)/2) if nlon is evenIn general,

N = minimum(nlat, (nlon+1)/2) if nlon is odd

*a*and

*b*were created in the process of doing spherical harmonic analysis via

**shagc**. Note:

- input values must be in ascending latitude order
- input arrays must be global

*g*

The result of spherical harmonic synthesis of *a* and *b*
(output, last two dimensions must be *nlat* x *nlon*). This
must be the same size as the array(s) upon which the spherical
harmonic analysis was performed. The space for *g* must be created
prior to invoking the above procedures.

## Description

**shsgc** performs the spherical harmonic
synthesis on the arrays *a* and *b* and stores the
results in the array *g*.

In general, **shagc** (performs spherical harmonic
analysis) will have to be invoked prior to calling
**shsgc** (performs spherical harmonic synthesis). Note
that both **shagc** and **shsgc**
operate on a gaussian grid.

NOTE: This procedure does not allow for missing data (defined by the _FillValue attribute)
to be present. *g* should **not** include the cyclic (wraparound) points, as
this procedure uses spherical harmonics. (NCL procedures/functions that
use spherical harmonics should never be passed input arrays that include cyclic points.)

Normalization: Let m be the Fourier wave number (rightmost dimension) and let n be the Legendre index (next-to-last dimension). Then ab = 0 for n < m.

The associated Legendre functions are normalized such that:

sum_lat sum_lon { [ Pmn(lat,lon)e^im lon ]^2 w(lat)/mlon } = 0.25 (m=0) sum_lat sum_lon { { [ Pmn(lat,lon)e^im lon ]^2 + [ Pmn(lat,lon)e^i-m lon ]^2 } w(lat)/mlon } = 0.5 (m /= 0)where w represents the Gaussian weights:

sum_lat { w(lat) } = 2.If the original input array is on a fixed grid,

**shsec**should be used. Also, note that

**shsgc**is the procedural version of

**shsgC**.

## See Also

**shsgC**, **shagc**, **shagC**, **shsec**, **shseC**,
**shaeC**, **shaec**, **rhomb_trunc**, **tri_trunc**

## Examples

In the following, assume *g* is on a gaussian grid, and no cyclic points
are included.

**Example 1**

g(nlat,nlon):

N = nlat if (nlon%2 .eq.0) then ; note % is NCL's modulus operator N =min((/ nlat, (nlon+2)/2 /)) else ; nlon must be odd N =min((/ nlat, (nlon+1)/2 /)) end if T = 19 a =new( (/nlat,N/), float) b =new( (/nlat,N/), float)shagc(g,a,b)tri_trunc(a,b,T)shsgc(a,b,g)

**Example 2**

g(nt,nlat,nlon):

[same "if" test as in example 1] a =new( (/nt,nlat,N/), float) b =new( (/nt,nlat,N/), float)shagc(g,a,b) [do something with the coefficients]shsgc(a,b,g)

**Example 3**

g(nt,nlvl,nlat,nlon):

[same "if" test as in example 1] T = 19 a =Note: ifnew( (/nt,nlvl,nlat,N/), float) b =new( (/nt,nlvl,nlat,N/), float)shagc(g,a,b)rhomb_trunc(a,b,T)shsgc(a,b,g)

*g*has dimensions, say, nlat = 64 and nlon = 129, where the "129" represents the cyclic points, then the user should pass the data to the procedure such that the cyclic points are not included. In the following examples,

*g*is a gaussian grid that contains cyclic points. (Remember NCL subscripts start at zero.)

**Example 4**

g(nlat,nlon):

N = nlat M = nlon-1 ; test using the dimension without cyclic point if (M%2 .eq.0) then ; use M to determine appropriate dimension N =min((/ nlat,(M+2)/2 /)) else ; nlon must be odd N =min((/ nlat,(M+1)/2 /)) end if a =new( (/nlat,N/), float) b =new( (/nlat,N/), float)shagc(g(:,0:M-1), ,a,b) ; only use the non-cyclic data [do something with the coefficients]shsgc(a,b, g(:,0:M-1)) g(:,M) = g(:,0) ; add new cyclic point

**Example 5**

g(nt,nlat,nlon) where nlat=64 and nlon=129 and the "129" represents the cyclic points:

[same "if" test as in example 4] a =new( (/nt,nlat,N/), float) b =new( (/nt,nlat,N/), float)shagc(g(:,:,0:nlon-2), a,b) [do something with the coefficients]shsgc(a,b, g(:,:,0:nlon-2)) g(:,:,nlon-1) = g(:,:,0) ; add new cyclic point

**Example 6**

g(nt,nlvl,nlat,nlon) where nlat=64 and nlon=129 and the "129" represents the cyclic points:

[same "if" test as in example 4] a =new( (/nt,nlvl,nlat,N/), float) b =new( (/nt,nlvl,nlat,N/), float)shagc(g(:,:,:,0:nlon-2), a,b) [do something with the coefficients]shsgc(a,b, g(:,:,:,0:nlon-2)) g(:,:,:,nlon-1) = g(:,:,:,0) ; add new cyclic point

## Errors

If *jer* or *ker* is equal to:

1 : error in the specification ofnlat

2 : error in the specification ofnlon

4 : error in the specification ofN(jeronly)