Perform a localized multitaper cross-spectral analysis using spherical cap windows.

## Usage

call SHMultiTaperCSE (mtse, sd, sh1, lmax1, sh2, lmax2, tapers, taper_order, lmaxt, k, alpha, lat, lon, taper_wt, norm, csphase, exitstatus)

## Parameters

mtse : output, real*8, dimension (lmax-lmaxt+1)
The localized multitaper cross-power spectrum estimate. lmax is the smaller of lmax1 and lmax2.
sd : output, real*8, 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*8, 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*8, 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*8, dimension (lmaxt+1, k)
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 : input, integer, dimension (k)
An array containing the angular orders of the spherical harmonic coefficients in each column of the array tapers.
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.
alpha : input, optional, real*8, dimension (3)
The Euler rotation angles used in rotating the windowing functions. alpha(1)=0, alpha(2)=-(90-lat)*pi/180, alpha(3)=-lon*pi/180. Either alpha or lat and lon can be specified, but not both. If none of these are specified, the spectral analysis will be centered at the north pole.
lat : input, optional, real*8
The latitude in degrees of the localized analysis. Either alpha or lat and lon can be specified but not both. If none of these are specified, the spectral analysis will be centered at the north pole.
lon : input, optional, real*8
The longitude in degrees of the localized analysis. Either alpha or lat and lon can be specified, but not both. If none of these are specified, the spectral analysis will be centered at the north pole.
taper_wt : input, optional, real*8, 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.

## 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 or alpha is 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 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.

## 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.