# shaeC

Computes spherical harmonic analysis of a scalar field on a fixed grid via spherical harmonics.

## Prototype

function shaeC ( g : numeric ) return_val : float or double

## Arguments

*g*

discrete function to be analyzed (input, array with two or more
dimensions). The last two dimensions must be *nlat* x *nlon*.
Note:

- input values must be in ascending latitude order
- input arrays must be on a global grid

## Return value

If the last two dimensions of *g* are *nlat* x *nlon*,
then the last two dimensions of the return array *ab* will be
*nlat* x *N*, where *N* is determined as follows:

N = minimum(nlat, (nlon+2)/2) if nlon is evenThe first dimension of

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

*ab*will be 2, which addresses the real and imaginary parts.

For example, if *g* is a two-dimensional array
(*nlat*,*nlon*), then the dimensions of *ab* will be 2
x *nlat* x *N*, where *ab*(0,*nlat*,*N*)
contains the "real" coefficients and *ab*(1,*nlat*,*N*)
contains the "imaginary" coefficients.

If *g* is a three-dimensional array
(*nt*,*nlat*,*nlon*), then
*ab*(0,*nt*,*nlat*,*N*) will contain the "real"
coefficients, and *ab*(1,*nt*,*nlat*,*N*) will
contain the "imaginary" coefficients.

If *g* is a four-dimensional array
(*nt*,*nlvl*,*nlat*,*nlon*), then
*ab*(0,*nt*,*nlvl*,*nlat*,*N*) will contain
the "real" coefficients, and
*ab*(1,*nt*,*nlvl*,*nlat*,*N*) will contain
the "imaginary" coefficients.

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

## Description

**shaeC** performs the spherical harmonic analysis on
the array *g*. In general, **shaeC** (performs
spherical harmonic analysis) is used in conjunction with
**shseC** (performs spherical harmonic synthesis).
Note that both **shaeC** and **shseC**
operate on a fixed grid.

NOTE: This function 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 Legendre index, n, is sometimes referred to as the total wave number.

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 input array

*g*is on a gaussian grid,

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

**shaeC**is the function version of

**shaec**.

## See Also

**shaec**, **shseC**, **shsec**, **shagC**, **shagc**,
**shsgc**, **shsgC**, **rhomb_trunC**, **tri_trunC**

## Examples

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

**Example 1**

g(nlat,nlon):

ab =shaeC(g) [do something with the coefficients] g =shseC(ab,nlon)

**Example 2**

g(nt,nlat,nlon):

ab =shaeC(g) ab =rhomb_trunC(ab,15) g =shseC(ab,nlon)

**Example 3**

g(nt,nlvl,nlat,nlon):

ab =Note: ifshaeC(g) [do something with the coefficients] g =shseC(ab,nlon)

*g*has dimensions, say, nlat = 73 and nlon = 145, where the "145" 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 on a fixed grid that contains cyclic points. (Remember NCL subscripts start at zero.)

**Example 4**

g(nlat,nlon)

nlon1 = nlon -1 ab =shaeC(g(:,0:nlon1-1)) ; only use the non-cyclic data [do something with the coefficients] g(:,0:nlon1-1) =shseC(ab,nlon1) ; reconstruct g(:,nlon1) = g(:,0) ; add new cyclic pt

**Example 5**

g(nt,nlat,nlon)

nlon1 = nlon -1 ab =shaeC(g(:,:,0:nlon1-1)) [do something with the coefficients] g(:,:,0:nlon1-1) =shseC(ab,nlon1) ; reconstruct g(:,:,nlon1) = g(:,:,0) ; add new cyclic pt

**Example 6**

g(nt,nlvl,nlat,nlon)

nlon1 = nlon-1 ab =shaeC(g(:,:,:,0:nlon1-1)) [do something with the coefficients] g(:,:,:,0:nlon1-1) =shseC(ab,nlon1) g(:,:,:,nlon1) = g(:,:,:,0) ; add new cyclic pt

**Example 7**

Given the spherical harmonic coefficients, to get the (amplitude^2)/2 to plot a spatial spectrum normalized by 0.25 for m = 0 and 0.5 for m ne 0.

Let g(nlat,nlon) and "ntr" be the maximum truncation (or as big as nlat) since spec is 0 beyond "ntr":

ab =Or, using array syntax and the built-in functionshaeC(g) ; for clarity place in separate arrays cr = ab(0,:,:) ; real coef (nlat,nlat) ci = ab(1,:,:) ; imaginary (nlat,nlat) pwr = (cr^2 + ci^2)/2. ; (nlat,nlat) array spc =new( nlat,typeof(cr) )delete(spc@_FillValue) ; for clarity use do loops do n=1,ntr spc(n) = pwr(n,0) do m=1,n spc(n) = spc(n) + 2.*pwr(n,m) end do spc(n) = 0.25*spc(n) end do

**sum**:

do n=1,ntr spc(n) = 0.25*(pwr(n,0) + 2.*sum( pwr(n,1:n) ) ) end do

## 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)