Perform a localized multitaper cross-spectral analysis using arbitrary windows derived from a mask.

## Usage

`mtse`

, `sd`

= SHMultiTaperMaskCSE (`sh1`

, `sh2`

, `tapers`

, [`lmax1`

, `lmax2`

, `lmaxt`

, `k`

, `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)**2,`kin`

)- 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`

. `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.
`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

`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 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.

## See also

shmultitapermaskse, shreturntapersmap, shcilmtovector

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