/
makegriddhc.doc
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/
makegriddhc.doc
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Create a 2D complex map from a set of complex spherical harmonic coefficients
that conforms with Driscoll and Healy's (1994) sampling theorem.
Usage
-----
griddh = MakeGridDHC (cilm, [lmax, norm, sampling, csphase, lmax_calc, extend])
Returns
-------
griddh : complex, dimension (nlat, nlong)
A 2D complex map of the input spherical harmonic coefficients cilm that
conforms to the sampling theorem of Driscoll and Healy (1994). If sampling
is 1, the grid is equally sampled and is dimensioned as (n by n), where n is
2lmax+2. If sampling is 2, the grid is equally spaced and is dimensioned as
(n by 2n). The first latitudinal band of the grid corresponds to 90 N, the
latitudinal sampling interval is 180/n degrees, and the default behavior is
to exclude the latitudinal band for 90 S. The first longitudinal band of the
grid is 0 E, by default the longitudinal band for 360 E is not included, and
the longitudinal sampling interval is 360/n for an equally sampled and 180/n
for an equally spaced grid, respectively. If extend is 1, the longitudinal
band for 360 E and the latitudinal band for 90 S will be included, which
increases each of the dimensions of the grid by 1.
Parameters
----------
cilm : complex, dimension (2, lmaxin+1, lmaxin+1)
The complex spherical harmonic coefficients of the function. The first
index specifies the coefficient corresponding to the positive and negative
order of m, respectively, with Clm=cilm[0,l,m] and Cl,-m=cilm[1,l,m)].
lmax : optional, integer, default = lmaxin
The maximum spherical harmonic degree of the function, which determines the
sampling n of the output grid.
norm : optional, integer, default = 1
1 = 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 = lmax
The maximum spherical harmonic degree used in evaluating the function. This
must be less than or equal to lmax, and does not affect the number of
samples of the output grid.
extend : input, optional, bool, default = False
If True, compute the longitudinal band for 360 E and the latitudinal band
for 90 S. This increases each of the dimensions of griddh by 1.
Description
-----------
MakeGridDHC will create a 2-dimensional complex map equally sampled (n by n) or
equally spaced (n by 2n) in latitude and longitude from a set of input complex
spherical harmonic coefficients, where N is 2lmax+2. This grid conforms with the
sampling theorem of Driscoll and Healy (1994) and this routine is the inverse of
SHExpandDHC. The function is evaluated at each longitudinal band by inverse
Fourier transforming the exponential terms for each degree l, and then summing
over all degrees. When evaluating the function, the maximum spherical harmonic
degree that is considered is the minimum of lmax, the size of cilm-1, or
lmax_calc (if specified).
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. The redundant longitudinal band for 360 E and the latitudinal band for
90 S are excluded by default, but these can be computed by specifying the
optional argument extend. 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 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.