Calculate the theoretical variance of a multitaper spectral estimate for a given input power spectrum.

## Usage

call SHMTVar (l, tapers, taper_order, lwin, kmax, sff, variance, taper_wt, unweighted_covar, nocross, exitstatus)

## Parameters

l : input, integer
The spherical harmonic degree used to calculate the theoretical variance.
tapers : input, real(dp), dimension (lwin+1, kmax)
A matrix of localization functions obtained from SHReturnTapers or SHReturnTapersM.
taper_order : input, integer, dimension (kmax)
The angular order of the windowing coefficients in tapers.
lwin : input, integer
The spherical harmonic bandwidth of the localizing windows.
kmax : input, integer
The maximum number of tapers to be used when calculating the variance.
sff : input, real(dp), dimension (l+lwin+1)
The global unwindowed power spectrum of the function to be localized.
variance : output, real(dp)
The theoretical variance of the multitaper spectral estimate for degree l.
taper_wt : optional, input, real(dp), dimension (kmax)
The weights to be applied to the multitaper spectral estimates.
unweighted_covar : optional, output, real(dp), dimension (kmax, kmax)
The unweighted covariance matrix of the kmax tapers (i.e., Fij in Wieczorek and Simons 2007).
nocross : optional, input, integer, default = 0
If 1, only the diagonal terms of the covariance matrix Fij will be computed. If 0, all terms will be computed.
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

SHMTVar will determine the theoretical variance of a multitaper spectral estimate for a given input power spectrum, degree l, and optionally, a set of taper weights (see eq. C21 of Wieczorek and Simons 2007). The windowing functions are assumed to be solutions to the spherical-cap concentration problem, as determined by a call to SHReturnTapers or SHReturnTapersM. If unweighted_covar is specified, then the unweighted covariance matrix of the kmax tapers (i.e., Fij) will be output. If the optional argument nocross is set to 1, then only the diagnonal terms of Fij will be computed.

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.