Perform a localized multitaper spectral analysis using spherical cap windows.

## Usage

call SHMultiTaperSE (`mtse`

, `sd`

, `sh`

, `lmax`

, `tapers`

, `taper_order`

, `lmaxt`

, `k`

, `alpha`

, `lat`

, `lon`

, `taper_wt`

, `norm`

, `csphase`

, `exitstatus`

)

## Parameters

`mtse`

: output, real(dp), dimension (`lmax`

-`lmaxt`

+1)- The localized multitaper power spectrum estimate.
`sd`

: output, real(dp), dimension (`lmax`

-`lmaxt`

+1)- The standard error of the localized multitaper power spectral estimates.
`sh`

: input, real(dp), dimension (2,`lmax`

+1,`lmax`

+1)- The spherical harmonic coefficients of the function to be localized.
`lmax`

: input, integer- The spherical harmonic bandwidth of
`sh`

. `tapers`

: input, real(dp), 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(dp), 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 window functions will not be rotated, and the spectral analysis will be centered at the north pole. `lat`

: input, optional, real(dp)- 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 window functions will not be rotated, and the spectral analysis will be centered at the north pole. `lon`

: input, optional, real(dp)- 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 window functions will not be rotated, and the spectral analysis will be centered at the north pole. `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.

## Description

`SHMultiTaperSE`

will perform a localized multitaper spectral analysis of an input function expressed in spherical harmonics. The maximum degree of the localized multitaper cross-power spectrum estimate is `lmax-lmaxt`

. 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, 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, 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.

## See also

shmultitapercse, shreturntapers, shreturntapersm, shmtvaropt

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