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SHMultiTaperMaskCSE
Perform a localized multitaper cross-spectral analysis using arbitrary windows derived from a mask.
Usage
mtse
, sd
= SHMultiTaperMaskCSE (sh1
, sh2
, tapers
, [lmax1
, lmax2
, lmaxt
, k
, taper_wt
, norm
, csphase
])
Returns
mtse
: float, dimension (lmax
-lmaxt
+1)- The localized multitaper cross-power spectrum estimate.
lmax
is the smaller oflmax1
andlmax2
. sd
: float, dimension (lmax
-lmaxt
+1)- The standard error of the localized multitaper cross-power spectral estimates.
lmax
is the smaller oflmax1
andlmax2
.
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)**2,kin
)- An array of the
k
windowing functions, arranged in columns, obtained from a call toSHReturnTapersMap
. The spherical harmonic coefficients are packed according to the conventions inSHCilmToVector
. lmax1
: optional, integer, default =lmax1in
- The spherical harmonic bandwidth of
sh1
. This must be less than or equal tolmax1in
. lmax2
: optional, integer, default =lmax2in
- The spherical harmonic bandwidth of
sh2
. This must be less than or equal tolmax2in
. 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 usetaper_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
SHMultiTaperMaskCSE
will perform a localized multitaper cross-spectral analysis of two input functions expressed in spherical harmonics, SH1
and SH2
, using an arbitrary set of windows derived from a mask. The maximum degree of the localized multitaper power spectrum estimate is lmax-lmaxt
, where lmax
is the smaller of lmax1
and lmax2
. 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, 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.
See also
shmultitapermaskse, shreturntapersmap, shcilmtovector
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