Calculate the eigenfunctions and eigenvalues of the space-concentration problem for an arbitrary region.
eigenvalues = SHReturnTapersMap (
tapers: float, dimension ((
- The spherical harmonic coefficients of the tapers, arranged in columns, from best to worst concentrated. The spherical harmonic coefficients in each column are indexed according to the scheme described in
eigenvalues: float, dimension (
- The concentration factor for each localization window specified in the columns of
dh_mask: integer, dimension (
- 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).
- The spherical harmonic bandwidth of the localization windows.
n: optional, integer
- The number of latitudinal samples in
dh_mask. The effective spherical harmonic bandwidth of this grid is
ntapers: optional, integer, default = (
- The number of best concentrated eigenvalues and corresponding eigenfunctions to return in
eigenvalues. The default value is to return all tapers.
sampling: optional, integer, default determined from
- For 1,
n x nsamples. For 2,
n x 2nsamples. The default it to determine
samplingfrom the dimensions of
degrees: integer, optional, dimension (
lmax+1), default = 1
- List of degrees to use when computing the eigenfunctions. Only those degrees where
degrees[l]is non-zero will be employed.
SHReturnTapersMap will calculate the eigenfunctions (i.e., localization windows) of the space-concentration problem for an arbitrary concentration region specified in
dh_mask (see Simons et al. (2006) for further details). 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 (
2n samples in longitude (
sampling=2). Given the approximate way in which the elements of the space-concentration kernel are calculated (see
ComputeDMap for details),
sampling=2 should be preferred. The effective spherical harmonic bandwidth (L=N/2-1) of the grid
dh_mask determines the accuracy of the results, and experience shows that this should be about 3 times larger than
The spherical harmonic coefficients of each window are given in the columns of
tapers, and the corresponding concentration factors are given in
eigenvaules. The spherical harmonic coefficients are ordered according to the scheme described in
YilmIndexVector, which can be converted to matrix form using
SHVectorToCilm, and the columns of
tapers are ordered from best to worst concentrated. The localization windows are normalized such that they have unit power. If the optional parameter
ntapers is specified, then only the
ntapers largest eigenvalues and corresponding eigenfunctions will be calculated and returned. If the optional vector
degrees is specified, then the eigenfunctions will be computed using only those degrees where
degrees(l) is not zero.
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.