Perform a localized multitaper spectral analysis using arbitrary windows derived from a mask.
mtse
, sd
= SHMultiTaperMaskSE (sh
, tapers
, [lmax
, lmaxt
, k
, taper_wt
, norm
, csphase
])
mtse
: float dimension (lmax
-lmaxt
+1)
: The localized multitaper power spectrum estimate.
sd
: float, dimension (lmax
-lmaxt
+1)
: The standard error of the localized multitaper power spectral estimates.
sh
: float, dimension (2, lmaxin
+1, lmaxin
+1)
: The spherical harmonic coefficients of the function to be localized.
tapers
: float, dimension ((lmaxtin
+1)**2, kin
)
: An array of the k
windowing functions, arranged in columns, obtained from a call to SHReturnTapersMap
. The spherical harmonic coefficients are packed according to the conventions in SHCilmToVector
.
lmax
: optional, integer, default = lmaxin
: The spherical harmonic bandwidth of sh
. This must be less than or equal to lmaxin
.
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.
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, integer, 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.
SHMultiTaperMaskSE
will perform a localized multitaper spectral analysis of an input function expressed in spherical harmonics using an arbitrary set of windows derived from a mask. The maximum degree of the localized multitaper cross-power spectrum estimate is lmax-lmaxt
. The matrix tapers
contains the spherical harmonic coefficients of the windows and can be obtained by a call to SHReturnTapersMap
. The coefficients of each window are stored in a single column, ordered according to the conventions used in SHCilmToVector
.
If the optional array taper_wt
is specified, 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.
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.