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shexpandglq.doc
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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.