Compute the space-concentration kernel of an arbitrary mask on the sphere.
Usage
dij = ComputeDMap (dh_mask, lmax, [n, sampling, degrees])
Returns
- dij : float, dimension ( (lmax+1)**2, (lmax+1)**2 )
- The space-concentration kernel corresponding to the mask dh_mask.
Parameters
- dh_mask : integer, dimension (nin, sampling*nin)
- A Driscoll and Healy (1994) sampled grid describing the concentration region R. All elements should either be 1 (for inside the concentration region) or 0 (for outside R).
- lmax : integer
- The maximum spherical harmonic degree of the matrix dij.
- n : optional, integer, default = nin
- The number of latitudinal samples in dh_mask. The effective spherical harmonic bandwidth of this grid is L=n/2-1.
- degrees : integer, optional, dimension (lmax+1), default = 1
- List of degrees to use when computing the space-concentration kernel. Only those degrees where degrees[l] is non-zero will be employed.
- sampling : optional, integer, default determined by dimensions of dh_mask
- For 1, dh_mask has n x n samples. For 2, dh_mask has n x 2n samples.
Description
ComputeDMap will calculate the space-concentration kernel for a generic mask defined on the sphere. The input mask dh_mask must be sampled according to the Driscoll and Healy (1994) sampling theorem with n samples in latitude, and possess a value of 1 inside the concentration region, and 0 elsewhere. dh_mask can either possess n samples in longitude (sampling=1) or 2n samples in longitude (sampling=2). Given the approximate way in which the elements of dij are calculated (see below), sampling=2 should be preferred. dij is symmetric, and the elements are ordered according to the scheme described in YilmIndexVector. See Simons et al. (2006) for further details. If the optional vector degrees is specified, then the matrix will be computed only for elements where degrees(l) is not zero.
The elements of DIJ are explicitly given by
Dlm,l'm' = 1/(4pi) Integral_R Ylm Yl'm' dOmega,
where R is the concentration region. In this routine, all values of l’m’ are calculated in a single spherical harmonic transform for a given value of lm according to
Dl'm' = 1/(4pi) Integral_Omega F Yl'm' dOmega.
where
F = Ylm dh_mask.
The function F is in general not a polynomial, and thus the coefficients Dl’m’ should not be expected to be exact. For this reason, the effective spherical harmonic degree of the input mask (L=n/2-1) should be greater than lmax. The exact value of n should be chosen such that further increases in n do not alter the returned eigenvalues. The routine prints out the fractional area of the mask computed in the pixel domain divided by D(0,0) (the fractional area computed by the spherical harmonic transforms), and the ratio of the two should be close to 1. Experience suggests that l should be about 5 times lmax.
References
Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.
Simons, F.J., F.A. Dahlen, and M.A. Wieczorek, Spatiospectral concentration on a sphere, SIAM Review, 48, 504-536, 2006.
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