 NCL Home > Documentation > Functions > Spherical harmonic routines

# shsgC

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

## Prototype

```	function shsgC (
ab    : numeric,
nlon  : byte, short, integer or long
)

return_val  :  float or double
```

## Arguments

ab

Array containing spherical harmonic coefficients (input, array with multiple dimensions). The last two dimensions must be nlat x N, where N is determined as follows:

N = minimum(nlat, (nlon+2)/2) if nlon is even
N = minimum(nlat, (nlon+1)/2) if nlon is odd
The first dimension of ab will be 2, which addresses the real and imaginary parts.

Note:

• input values must be in ascending latitude order
• input array must be global
In general, this array was created in the process of doing spherical harmonic analysis via shagC.

nlon

The longitude dimension of the array to be returned.

As of version 6.0.0, this can be of type byte, short, integer or long.

## Return value

Note:

• output values will be in ascending latitude order
• output array is global
If the last two dimensions of ab are nlat x N, then the last two dimensions of the return array g will be nlat x nlon.

For example, if ab is a three-dimensional array (2,nlat,N), then g will be dimensioned nlat x nlon.

If ab is a four-dimensional array (2,nt,nlat,N), then g will be dimensioned (nt,nlat,nlon).

If ab is a five-dimensional array (2,nt,nlvl,nlat,N), then g will be dimensioned (nt,nlvl,nlat,nlon).

The return array will be double if the input is double, and float otherwise.

## Description

shsgC performs the spherical harmonic synthesis on the array ab. 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 function does not allow for missing data (defined by the _FillValue attribute) to be present.

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 function version of shsgc.

## Examples

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

Example 1

g(nlat,nlon):

```   ab = shagC (g)
[do something with the coefficients]
g  = shsgC (ab,nlon)
```
Example 2

g(nt,nlat,nlon):

```   ab = shagC (g)
[do something with the coefficients]
g  = shsgC (ab,nlon)
```
Example 3

g(nt,nlvl,nlat,nlon):

```   ab = shagC (g)
ab = rhomb_trunC (ab,15)
g  = shsgC (ab,nlon)
```
Note: if 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 function 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)

```   nlon1 = nlon -1
ab = shagC (g(:,0:nlon1-1))         ; only use the non-cyclic data
[do something with the coefficients]
g(:,0:nlon1-1) = shsgC (ab,nlon1)  ; reconstruct
g(:,nlon1)     = g(:,0)           ; add new cyclic point
```
Example 5

g(nt,nlat,nlon)

```   nlon1 = nlon -1
ab = shagC (g(:,:,0:nlon1-1))
[do something with the coefficients]
g(:,:,0:nlon1-1) = shsgC (ab,nlon1) ; reconstruct
g(:,:,nlon1)   = g(:,:,0)          ; add new cyclic point
```
Example 6

g(nt,nlvl,nlat,nlon)

```   nlon1 = nlon-1
ab = shagC (g(:,:,:,0:nlon1-1))
[do something with the coefficients]
g(:,:,:,0:nlon1-1) = shsgC (ab,nlon1)
g(:,:,:,nlon1)   = g(:,:,:,0)  ; add new cyclic point
```

## Errors

If jer or ker is equal to:

1 : error in the specification of nlat
2 : error in the specification of nlon
4 : error in the specification of nt (jer only)