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shmultitapercse.doc
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shmultitapercse.doc
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Perform a localized multitaper cross-spectral analysis using spherical cap
windows.
Usage
-----
mtse, sd = SHMultiTaperCSE (sh1, sh2, tapers, taper_order, [lmax1, lmax2, lmaxt,
k, lat, lon, taper_wt, norm, csphase])
Returns
-------
mtse : float, dimension (lmax-lmaxt+1)
The localized multitaper cross-power spectrum estimate. lmax is the smaller
of lmax1 and lmax2.
sd : float, dimension (lmax-lmaxt+1)
The standard error of the localized multitaper cross-power spectral
estimates. lmax is the smaller of lmax1 and lmax2.
Parameters
----------
sh1 : float, dimension (2, lmax1in+1, lmax1in+1)
The spherical harmonic coefficients of the first function.
sh2 : float, dimension (2, lmax2in+1, lmax2in+1)
The spherical harmonic coefficients of the second function.
tapers : float, dimension (lmaxtin+1, kin)
An array of the k windowing functions, arranged in columns, obtained from a
call to SHReturnTapers. Each window has non-zero coefficients for a single
angular order that is specified in the array taper_order.
taper_order : integer, dimension (kin)
An array containing the angular orders of the spherical harmonic
coefficients in each column of the array tapers.
lmax1 : optional, integer, default = lmax1in
The spherical harmonic bandwidth of sh1. This must be less than or equal to
lmax1in.
lmax2 : optional, integer, default = lmax2in
The spherical harmonic bandwidth of sh2. This must be less than or equal to
lmax2in.
lmaxt : optional, integer, default = lmaxtin
The spherical harmonic bandwidth of the windowing functions in the array
tapers.
k : optional, integer, default = kin
The number of tapers to be utilized in performing the multitaper spectral
analysis.
lat : optional, float, default = 90
The latitude in degrees of the localized analysis. The default is to perform
the spectral analysis at the north pole.
lon : optional, float, default = 0
The longitude in degrees of the localized analysis.
taper_wt : optional, float, dimension (kin), default = -1
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 SHMTVarOpt. The default value
specifies not to use taper_wt.
norm : optional, intger, default = 1
1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-
normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
csphase : optional, integer, 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)^m to the associated Legendre functions.
Description
-----------
SHMultiTaperCSE will perform a localized multitaper cross-spectral analysis of
two input functions expressed in spherical harmonics, SH1 and SH2. The maximum
degree of the localized multitaper power spectrum estimate is lmax-lmaxt, where
lmax is the smaller of lmax1 and lmax2. The coefficients and angular orders of
the windowing coefficients (tapers and taper_order) are obtained by a call to
SHReturnTapers. If lat and lon are specified, then 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.
If the optional array taper_wt is specified, then these weights will be used in
calculating a weighted average of the individual k tapered estimates (mtse) and
the corresponding standard error of the estimates (sd). If not present, the
weights will all be assumed to be equal. When taper_wt is not specified, the
mutltitaper spectral estimate for a given degree is calculated as the average
obtained from the k individual tapered estimates. The standard error of the
multitaper estimate at degree l is simply the population standard deviation, S =
sqrt(sum (Si - mtse)^2 / (k-1)), divided by sqrt(k). See Wieczorek and Simons
(2007) for the relevant expressions when weighted estimates are used.
The employed spherical harmonic normalization and Condon-Shortley phase
convention can be set by the optional arguments norm and csphase; if not set,
the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-
Shortley phase of (-1)^m.
References
----------
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.