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shmultitaperse.1
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shmultitaperse.1
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.TH "shmultitaperse" "1" "2021-02-15" "Fortran 95" "SHTOOLS 4.10"
.hy
.SH SHMultiTaperSE
.PP
Perform a localized multitaper spectral analysis using spherical cap
windows.
.SH Usage
.PP
call SHMultiTaperSE (\f[V]mtse\f[R], \f[V]sd\f[R], \f[V]cilm\f[R],
\f[V]lmax\f[R], \f[V]tapers\f[R], \f[V]taper_order\f[R],
\f[V]lmaxt\f[R], \f[V]k\f[R], \f[V]alpha\f[R], \f[V]lat\f[R],
\f[V]lon\f[R], \f[V]taper_wt\f[R], \f[V]norm\f[R], \f[V]csphase\f[R],
\f[V]exitstatus\f[R])
.SH Parameters
.TP
\f[V]mtse\f[R] : output, real(dp), dimension (\f[V]lmax\f[R]-\f[V]lmaxt\f[R]+1)
The localized multitaper power spectrum estimate.
.TP
\f[V]sd\f[R] : output, real(dp), dimension (\f[V]lmax\f[R]-\f[V]lmaxt\f[R]+1)
The standard error of the localized multitaper power spectral estimates.
.TP
\f[V]cilm\f[R] : input, real(dp), dimension (2, \f[V]lmax\f[R]+1, \f[V]lmax\f[R]+1)
The spherical harmonic coefficients of the function to be localized.
.TP
\f[V]lmax\f[R] : input, integer(int32)
The spherical harmonic bandwidth of \f[V]cilm\f[R].
.TP
\f[V]tapers\f[R] : input, real(dp), dimension (\f[V]lmaxt\f[R]+1, \f[V]k\f[R])
An array of the \f[V]k\f[R] windowing functions, arranged in columns,
obtained from a call to \f[V]SHReturnTapers\f[R].
Each window has non-zero coefficients for a single angular order that is
specified in the array \f[V]taper_order\f[R].
.TP
\f[V]taper_order\f[R] : input, integer(int32), dimension (\f[V]k\f[R])
An array containing the angular orders of the spherical harmonic
coefficients in each column of the array \f[V]tapers\f[R].
.TP
\f[V]lmaxt\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 to be utilized in performing the multitaper
spectral analysis.
.TP
\f[V]alpha\f[R] : input, optional, real(dp), dimension(3)
The Euler rotation angles used in rotating the windowing functions.
\f[V]alpha(1) = 0\f[R], \f[V]alpha(2) = -(90-lat)*pi/180\f[R],
\f[V]alpha(3) = -lon*pi/180\f[R].
Either \f[V]alpha\f[R] or \f[V]lat\f[R] and \f[V]lon\f[R] can be
specified, but not both.
If none of these are specified, the window functions will not be
rotated, and the spectral analysis will be centered at the north pole.
.TP
\f[V]lat\f[R] : input, optional, real(dp)
The latitude in degrees of the localized analysis.
Either \f[V]alpha\f[R] or \f[V]lat\f[R] and \f[V]lon\f[R] can be
specified but not both.
If none of these are specified, the window functions will not be
rotated, and the spectral analysis will be centered at the north pole.
.TP
\f[V]lon\f[R] : input, optional, real(dp)
The longitude in degrees of the localized analysis.
Either \f[V]alpha\f[R] or \f[V]lat\f[R] and \f[V]lon\f[R] can be
specified, but not both.
If none of these are specified, the window functions will not be
rotated, and the spectral analysis will be centered at the north pole.
.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 and
standard error.
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]norm\f[R] : input, optional, integer(int32), default = 1
1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt
semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal
harmonics.
.TP
\f[V]csphase\f[R] : input, optional, integer(int32), default = 1
1 (default) = do not apply the Condon-Shortley phase factor to the
associated Legendre functions; -1 = append the Condon-Shortley phase
factor of (-1)\[ha]m to the associated Legendre functions.
.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]SHMultiTaperSE\f[R] will perform a localized multitaper spectral
analysis of an input function expressed in spherical harmonics.
The maximum degree of the localized multitaper cross-power spectrum
estimate is \f[V]lmax-lmaxt\f[R].
The coefficients and angular orders of the windowing coefficients
(\f[V]tapers\f[R] and \f[V]taper_order\f[R]) are obtained by a call to
\f[V]SHReturnTapers\f[R].
If \f[V]lat\f[R] and \f[V]lon\f[R] or \f[V]alpha\f[R] is specified, the
symmetry axis of the localizing windows will be rotated to these
coordinates.
Otherwise, the localized spectral analysis will be centered over the
north pole.
.PP
If the optional array \f[V]taper_wt\f[R] is specified, these weights
will be used in calculating a weighted average of the individual
\f[V]k\f[R] tapered estimates \f[V]mtse\f[R] and the corresponding
standard error of the estimates \f[V]sd\f[R].
If not present, the weights will all be assumed to be equal.
When \f[V]taper_wt\f[R] is not specified, the mutltitaper spectral
estimate for a given degree will be calculated as the average obtained
from the \f[V]k\f[R] individual tapered estimates.
The standard error of the multitaper estimate at degree \f[V]l\f[R] is
simply the population standard deviation,
\f[V]S = sqrt(sum (Si - mtse)\[ha]2 / (k-1))\f[R], divided by
\f[V]sqrt(k)\f[R].
See Wieczorek and Simons (2007) for the relevant expressions when
weighted estimates are used.
.PP
The employed spherical harmonic normalization and Condon-Shortley phase
convention can be set by the optional arguments \f[V]norm\f[R] and
\f[V]csphase\f[R]; if not set, the default is to use geodesy 4-pi
normalized harmonics that exclude the Condon-Shortley phase of
(-1)\[ha]m.
.SH References
.PP
Wieczorek, M.
A.
and F.
J.
Simons, Minimum-variance multitaper spectral estimation on the sphere,
J.
Fourier Anal.
Appl., 13, doi:10.1007/s00041-006-6904-1, 665-692, 2007.
.SH See also
.PP
shmultitapercse, shreturntapers, shreturntapersm, shmtvaropt