shmtcouplingmatrix.3

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.TH "shmtcouplingmatrix" "1" "2021-02-15" "Fortran 95" "SHTOOLS 4.11"
.hy
.SH SHMTCouplingMatrix
.PP
This routine returns the multitaper coupling matrix for a given set of
power spectra of arbitrary localization windows.
This matrix relates the expectation of the localized multitaper spectrum
to the expectation of the power spectrum of the global function.
.SH Usage
.PP
call SHMTCouplingMatrix (\f[V]mmt\f[R],
\f[V]lmax\f[R],\f[V]tapers_power\f[R], \f[V]lwin\f[R], \f[V]k\f[R],
\f[V]taper_wt\f[R], \f[V]exitstatus\f[R])
.SH Parameters
.TP
\f[V]mmt\f[R] : output, real(dp), dimension (\f[V]lmax\f[R]+\f[V]lwin\f[R]+1, \f[V]lmax\f[R]+1)
The full multitaper coupling matrix that relates the expectation of the
localized multitaper spectrum to the global power spectrum of the
function.
.TP
\f[V]lmax\f[R] : input, integer(int32)
The spherical harmonic bandwidth of the global power spectrum.
.TP
\f[V]tapers_power\f[R] : input, real(dp), dimension (\f[V]lwin\f[R]+1, \f[V]k\f[R])
An array of power spectra of the k windowing functions, arranged in
columns.
.TP
\f[V]lwin\f[R] : input, integer(int32)
The spherical harmonic bandwidth of the windowing functions in the array
\f[V]tapers\f[R].
.TP
\f[V]k\f[R] : input, integer(int32)
The number of tapers utilized in the multitaper spectral analysis.
.TP
\f[V]taper_wt\f[R] : input, optional, real(dp), dimension (\f[V]k\f[R])
The weights used in calculating the multitaper spectral estimates.
Optimal values of the weights (for a known global power spectrum) can be
obtained from the routine \f[V]SHMTVarOpt\f[R].
.TP
\f[V]exitstatus\f[R] : 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.
.SH Description
.PP
\f[V]SHMTCouplingMatrix\f[R] returns the multitaper coupling matrix that
relates the expectation of the localized multitaper spectrum to the
expectation of the global power spectrum of the function (assumed to be
stationary).
This is given by eqs 4.5 and 4.6 in Wieczorek and Simons (2007):
.PP
\f[V]< S_{Phi Phi}\[ha](mt) > = M\[ha](mt) S_{ff}\f[R]
.PP
where \f[V]S_{Phi Phi}\f[R] is a vector containing the
\f[V]lmax+lwin+1\f[R] localized multitaper power spectral estiamtes,
\f[V]S_{ff}\f[R] is a vector of the global power spectrum up to degree
\f[V]lmax\f[R], and \f[V]< ... >\f[R] is the expectation operator.
The coupling matrix is given explicitly by
.PP
\f[V]M_{ij} = Sum_{l=0}\[ha]L Sum_{k=1}\[ha]K a_k S_{hh}\[ha]{k}(l) [ C_{l0j0}\[ha]{i0} ]\[ha]2\f[R]
.PP
where \f[V]a_k\f[R] are the taper weights, \f[V]S_{hh}\f[R] is the power
of the window, and \f[V]C\f[R] is a Clebsch-Gordon coefficient.
.PP
Note that this routine returns the \[lq]full\[rq] coupling matrix of
dimension (\f[V]lmax\f[R] + \f[V]lwin\f[R] + 1, \f[V]lmax\f[R] + 1).
When multiplied by a global input power spectrum with bandwidth
\f[V]lmax\f[R], it returns the output power spectrum with a bandwidth of
\f[V]lmax\f[R] + \f[V]lwin\f[R].
In doing so, it is implicitly assumed that input power spectrum is
exactly zero for all degrees greater than lmax.
If this is not the case, the ouput power spectrum should be considered
valid only for the degrees up to and including \f[V]lmax\f[R] -
\f[V]lwin\f[R].
.SH References
.PP
Wieczorek, M.
A.
and F.
J.
Simons, Minimum-variance multitaper spectral estimation on the sphere,
J.
Fourier Anal.
Appl., 13, 665-692, doi:10.1007/s00041-006-6904-1, 2007.
.SH See also
.PP
shmultitaperse, shmultitapercse, shreturntapers, shmtvaropt, shmtdebias