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.
call SHSCouplingMatrixCap (kij
, galpha
, galpha_order
, lmax
, nmax
, exitstatus
)
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, dimension (kmax
)
: The angular orders of the spherical-cap Slepian functions in galpha
.
lmax
: input, integer
: The spherical harmonic bandwidth of the Slepian functions.
nmax
: input, integer
: The number of Slepian functions used in reconstructing the function.
exitstatus
: output, optional, integer
: 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.
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').
shreturntapers, shscouplingmatrix, shslepianvar, shmtcouplingmatrix