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.