Perform a localized multitaper cross-spectral analysis using using arbitrary windows derived from a mask.
call SHMultiTaperMaskCSE (mtse, sd, sh1, lmax1, sh2, lmax2, tapers, lmaxt, k, taper_wt, norm, csphase, exitstatus)
mtse : output, real(dp), dimension (lmax-lmaxt+1)
: The localized multitaper cross-power spectrum estimate. lmax is the smaller of lmax1 and lmax2.
sd : output, real(dp), dimension (lmax-lmaxt+1)
: The standard error of the localized multitaper cross-power spectral estimates. lmax is the smaller of lmax1 and lmax2.
sh1 : input, real(dp), dimension (2, lmax1+1, lmax1+1)
: The spherical harmonic coefficients of the first function.
lmax1 : input, integer
: The spherical harmonic bandwidth of sh1.
sh2 : input, real(dp), dimension (2, lmax2+1, lmax2+1)
: The spherical harmonic coefficients of the second function.
lmax2 : input, integer
: The spherical harmonic bandwidth of sh2.
tapers : input, real(dp), dimension ((lmaxt+1)**2, k)
: 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.
lmaxt : input, integer
: The spherical harmonic bandwidth of the windowing functions in the array tapers.
k : input, integer
: The number of tapers to be utilized in performing the multitaper spectral analysis.
taper_wt : input, optional, real(dp), dimension (k)
: 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.
norm : input, optional, integer, default = 1
: 1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
csphase : input, 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.
exitstatus : output, optional, integer
: 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.
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 will be 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.