/
makegridglq.doc
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makegridglq.doc
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Create a 2D map from a set of spherical harmonic coefficients sampled on the
Gauss-Legendre quadrature nodes.
Usage
-----
gridglq = MakeGridGLQ (cilm, zero, [lmax, norm, csphase, lmax_calc, extend])
Returns
-------
gridglq : float, dimension (nlat, nlong)
A 2D map of the function sampled on the Gauss-Legendre quadrature nodes,
dimensioned as (lmax+1, 2*lmax+1) if extend is 0 or (lmax+1, 2*lmax+2) if
extend is 1.
Parameters
----------
cilm : float, dimension (2, lmaxin+1, lmaxin+1)
The real spherical harmonic coefficients of the function. When evaluating
the function, the maximum spherical harmonic degree considered is the
minimum of lmax, lmaxin, or lmax_calc (if specified). 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].
zero : float, dimension (lmax+1)
The nodes used in the Gauss-Legendre quadrature over latitude, calculated by
a call to SHGLQ.
lmax : optional, integer, default = lamxin
The maximum spherical harmonic bandwidth of the function. This determines
the sampling nodes and dimensions of the output grid.
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 used in evaluating the function. This
must be less than or equal to lmax.
extend : input, optional, bool, default = False
If True, compute the longitudinal band for 360 E.
Description
-----------
MakeGridGLQ will create a 2-dimensional map from a set of input spherical
harmonic coefficients sampled on the Gauss-Legendre quadrature nodes. This is
the inverse of the routine SHExpandGLQ. 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. 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 redundant longitudinal band for 360 E is excluded from the grid by default,
but this 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.
The reconstruction of the spherical harmonic function 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.
Reference
---------
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.