shexpanddh.doc

Expand an equally sampled or equally spaced grid into spherical harmonics using
Driscoll and Healy's (1994) sampling theorem.

Usage
-----
cilm = SHExpandDH (griddh, [norm, sampling, csphase, lmax_calc])

Returns
-------
cilm : float, dimension (2, n/2, n/2) or (2, lmax_calc+1, lmax_calc+1)
    The real spherical harmonic coefficients of the function. These will be
    exact if the function is bandlimited to degree lmax=n/2-1. The coefficients
    c1lm and c2lm refer to the cosine (clm) and sine (slm) coefficients,
    respectively, with clm=cilm[0,l,m] and slm=cilm[1,l,m].

Parameters
----------
griddh : float, dimension (n, n) or (n, 2*n)
    A 2D equally sampled (default) or equally spaced grid that conforms to the
    sampling theorem of Driscoll and Healy (1994). The first latitudinal band
    corresponds to 90 N, the latitudinal band for 90 S is not included, and the
    latitudinal sampling interval is 180/n degrees. The first longitudinal band
    is 0 E, the longitude band for 360 E is not included, and the longitudinal
    sampling interval is 360/n for an equally and 180/n for an equally spaced
    grid, respectively.
norm : optional, integer, default = 1
    1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-
    normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
sampling : optional, integer, default = 1
    If 1 (default) the input grid is equally sampled (n by n). If 2, the grid is
    equally spaced (n by 2n).
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 = n/2-1
    The maximum spherical harmonic degree calculated in the spherical harmonic
    expansion.

Description
-----------
SHExpandDH will expand an equally sampled (n by n) or equally spaced grid (n by
2n) into spherical harmonics using the sampling theorem of Driscoll and Healy
(1994). The number of latitudinal samples, n, must be even, and the transform is
exact if the function is bandlimited to spherical harmonic degree n/2-1. The
inverse transform is given by the routine MakeGridDH. If the optional parameter
lmax_calc is specified, the spherical harmonic coefficients will only be
calculated to this degree instead of n/2-1. The algorithm is based on performing
FFTs in longitude and then integrating over latitude using an exact quadrature
rule.

The default is to use an input grid that is equally sampled (n by n), but this
can be changed to use an equally spaced grid (n by 2n) by the optional argument
sampling.  When using an equally spaced grid, the Fourier components
corresponding to degrees greater than n/2-1 are simply discarded; this is done
to prevent aliasing that would occur if an equally sampled grid was constructed
from an equally spaced grid by discarding every other column of the input grid.

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 accurate
only to about degree 15.

References
----------
Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on
the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.

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.