shscouplingmatrixcap(1)

SHSCouplingMatrixCap
This routine returns the spherical harmonic coupling matrix for a given set of spherical-cap Slepian basis functions. This matrix relates the power spectrum expectation of the function expressed in a subset of the best-localized Slepian functions to the expectation of the global power spectrum.

Usage
call SHSCouplingMatrixCap (kij, galpha, galpha_order, lmax, nmax, exitstatus)

Parameters

kij : output, real(dp), dimension (lmax+1, lmax+1)
    The coupling matrix that relates the power spectrum expectation of the function expressed in a subset of the best-localized spherical-cap Slepian functions to the expectation of the global power spectrum.

galpha : input, real(dp), dimension (lmax+1, nmax)
    An array of spherical-cap Slepian functions arranged in columns from best to worst localized and obtained from a call to SHReturnTapers.

galpha_order : input, integer(int32), dimension (kmax)
    The angular orders of the spherical-cap Slepian functions in galpha.

lmax : input, integer(int32)
    The spherical harmonic bandwidth of the Slepian functions.

nmax : input, integer(int32)
    The number of Slepian functions used in reconstructing the function.

exitstatus : output, optional, integer(int32)
    If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.

Description
SHSCouplingMatrixCap returns the spherical harmonic coupling matrix that relates the power spectrum expectation of the function expressed in a subset of the best-localized spherical-cap Slepian functions to the expectation of the global power spectrum (assumed to be stationary). The spherical-cap Slepian functions are determined by a call to SHReturnTapers and each row of galpha contains the (lmax+1) spherical harmonic coefficients for the single angular order as given in galpha_order.

The relationship between the global and localized power spectra is given by:

< S_{\tilde{f}}(l) > = \sum_{l'=0}^lmax K_{ll'} S_{f}(l')

where S_{\tilde{f}} is the expectation of the power spectrum at degree l of the function expressed in Slepian functions, S_{f}(l') is the expectation of the global power spectrum, and < ... > is the expectation operator. The coupling matrix is given explicitly by

K_{ll'} = \frac{1}{2l'+1} Sum_{m=-mmax}^mmax ( Sum_{alpha=1}^nmax g_{l'm}(alpha) g_{lm}(alpha) )**2

where mmax is min(l, l’).

See also
shreturntapers, shscouplingmatrix, shslepianvar, shmtcouplingmatrix