Expand a 2D grid sampled on the Gauss-Legendre quadrature nodes into spherical harmonics.

## Usage

cilm = SHExpandGLQ (gridglq, w, zero, [norm, csphase, lmax_calc])

## Returns

- cilm : float, dimension (2, lmax+1, lmax+1) or (2, lmax_calc+1, lmax_calc+1)
- The real spherical harmonic coefficients of the function. The coefficients C0lm and Cilm refer to the “cosine” (Clm) and “sine” (Slm) coefficients, respectively, with Clm=cilm[0,l,m] and Slm=cilm[1,l,m].

## Parameters

- gridglq : float, dimension (lmax+1, 2*lmax+1)
- A 2D grid of data sampled on the Gauss-Legendre quadrature nodes. The latitudinal nodes correspond to the zeros of the Legendre polynomial of degree lmax+1, and the longitudinal nodes are equally spaced with an interval of 360/(2*lmax+1) degrees. See also GLQGridCoord.
- w : float, dimension (lmax+1)
- The Gauss-Legendre quadrature weights used in the integration over latitude. These are obtained from a call to SHGLQ.
- zero : float, dimension (lmax+1)
- The nodes used in the Gauss-Legendre quadrature over latitude, calculated by a call to SHGLQ.
- norm : optional, integer, default = 1
- 1 (default) = Geodesy 4-pi normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
- csphase : optional, integer, default = 1
- 1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.
- lmax_calc : optional, integer, default = lmax
- The maximum spherical harmonic degree calculated in the spherical harmonic expansion.

## Description

SHExpandGLQ will expand a 2-dimensional grid of data sampled on the Gauss-Legendre quadrature nodes into spherical harmonics. This is the inverse of the routine MakeGridGLQ. The latitudinal nodes of the input grid correspond to the zeros of the Legendre polynomial of degree lmax+1, and the longitudinal nodes are equally spaced with an interval of 360/(2*lmax+1) degrees. It is implicitly assumed that the function is bandlimited to degree lmax. If the optional parameter lmax_calc is specified, the spherical harmonic coefficients will be calculated up to this degree, instead of lmax.

The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments norm and csphase; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m. The normalized legendre functions are calculated in this routine using the scaling algorithm of Holmes and Featherstone (2002), which are accurate to about degree 2800. The unnormalized functions are only accurate to about degree 15.

The spherical harmonic transformation may be speeded up by precomputing the Legendre functions on the Gauss-Legendre quadrature nodes in the routine SHGLQ. However, given that this array contains on the order of lmax**3 entries, this is only feasible for moderate values of lmax.

## References

Holmes, S. A., and W. E. Featherstone, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, J. Geodesy, 76, 279- 299, 2002.

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