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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
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<title>SHTOOLS - Localized spectral analysis</title>
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<h1 id="shlocalizedadmitcorr">SHLocalizedAdmitCorr</h1>
<p>Calculate the localized admittance and correlation spectra of two functions at a given location using spherical cap localization windows.</p>
<h1 id="usage">Usage</h1>
<p>call SHLocalizedAdmitCorr (<code>tapers</code>, <code>taper_order</code>, <code>lwin</code>, <code>lat</code>, <code>lon</code>, <code>gilm</code>, <code>tilm</code>, <code>lmax</code>, <code>admit</code>, <code>corr</code>, <code>k</code>, <code>admit_error</code>, <code>corr_error</code>, <code>taper_wt</code>, <code>mtdef</code>, <code>k1linsig</code>, <code>exitstatus</code>)</p>
<h1 id="parameters">Parameters</h1>
<dl>
<dt><code>tapers</code> : input, real*8, dimension (<code>lwin</code>+1, <code>k</code>)</dt>
<dd>A matrix of spherical cap localization functions obtained from <code>SHReturnTapers</code> or <code>SHReturnTapersM</code>.
</dd>
<dt><code>taper_order</code> : input, integer, dimension (<code>k</code>)</dt>
<dd>The angular order of the windowing coefficients in <code>tapers</code>.
</dd>
<dt><code>lwin</code> : input, integer</dt>
<dd>The spherical harmonic bandwidth of the localizing windows.
</dd>
<dt><code>lat</code> : input, real*8</dt>
<dd>The latitude of the localized analysis in degrees.
</dd>
<dt><code>lon</code> : input, real*8</dt>
<dd>The longitude of the localized analysis in degrees.
</dd>
<dt><code>gilm</code> : input, real*8, dimension (2, <code>lmax</code>+1, <code>lmax</code>+1)</dt>
<dd>The spherical harmonic coefficients of the function G.
</dd>
<dt><code>tilm</code> : input, real*8, dimension (2, <code>lmax</code>+1, <code>lmax</code>+1)</dt>
<dd>The spherical harmonic coefficients of the function T.
</dd>
<dt><code>lmax</code> : input, integer</dt>
<dd>The maximum spherical harmonic degree of the input functions corresponding to <code>gilm</code> and <code>tilm</code>.
</dd>
<dt><code>admit</code> : output, real*8, dimension (<code>lmax</code>-<code>lwin</code>+1)</dt>
<dd>The admittance function, which is equal to <code>Sgt/Stt</code>.
</dd>
<dt><code>corr</code> : output, real*8, dimension (<code>lmax</code>-<code>lwin</code>+1)</dt>
<dd>The degree correlation function, which is equal to <code>Sgt/sqrt(Sgg Stt)</code>.
</dd>
<dt><code>k</code> : input, integer</dt>
<dd>The number of tapers to be used in the multitaper spectral analysis.
</dd>
<dt><code>admit_error</code> : output, optional, real*8, dimension (<code>lmax</code>-<code>lwin</code>+1)</dt>
<dd>The standard error of the admittance function.
</dd>
<dt><code>corr_error</code> : output, optional, real*8, dimension (<code>lmax</code>-<code>lwin</code>+1)</dt>
<dd>The standard error of the degree correlation function.
</dd>
<dt><code>taper_wt</code> : input, optional, real*8, dimension (<code>k</code>)</dt>
<dd>The weights to be applied to the spectral estimates when calculating the admittance, correlation, and their associated errors. This must sum to unity.
</dd>
<dt><code>mtdef</code> : input, optional, integer, default = 1</dt>
<dd>1 (default): Calculate the multitaper spectral estimates Sgt, Sgg and Stt first, and then use these to calculate the admittance and correlation functions. 2: Calculate admittance and correlation spectra using each individual taper, and then average these to obtain the multitaper admittance and correlation functions.
</dd>
<dt><code>k1linsig</code> : input, optional, integer</dt>
<dd>If equal to one, and only a single taper is being used, the errors in the admittance function will be calculated by assuming that the coefficients of <code>gilm</code> and <code>tilm</code> are related by a linear degree-dependent transfer function and that the lack of correlation is a result of uncorrelated noise. This is the square root of eq. 33 of Simons et al. 1997.
</dd>
<dt><code>exitstatus</code> : output, optional, integer</dt>
<dd>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.
</dd>
</dl>
<h1 id="description">Description</h1>
<p><code>SHLocalizedAdmitCorr</code> will calculate the localized admittance and degree correlation spectra of two functions at a given location. The windowing functions are solutions to the spherical-cap concentration problem (as calculated by <code>SHReturnTapers</code> or <code>SHReturnTapersM</code>), of which the best <code>k</code> concentrated tapers are utilized. If <code>k</code> is greater than 1, then estimates of the standard error for the admittance and correlation will be returned in the optional arrays <code>admit_error</code> and <code>corr_error</code>. The symmetry axis of the localizing windows are rotated to the coordinates (<code>lat</code>, <code>lon</code>) before performing the windowing operation.</p>
<p>The admittance is defined as <code>Sgt/Stt</code>, where <code>Sgt</code> is the localized cross-power spectrum of two functions <code>G</code> and <code>T</code> expressed in spherical harmonics. The localized degree-correlation spectrum is defined as <code>Sgt/sqrt(Sgg Stt)</code>, which can possess values between -1 and 1. Two methods are available for calculating the multitaper admittance and correlation functions. When <code>mtdef</code> is 1 (default), the multitaper estimates and errors of Sgt, Stt, and Sgg are calculated by calls to <code>SHMultiTaperSE</code> and <code>SHMultiTaperCSE</code>, and these results are then used to calculate the final admittance and correlation functions. When <code>mtdef</code> is 2, the admitance and correlation are calculated invidivually for each individual taper, and these results are then averaged.</p>
<p>If the optional parameter <code>k1linsig</code> is specified, and only a single taper is being used, the uncertainty in the admittance function will be calculated by assuming the two sets of coefficients are related by a linear degree-dependent transfer function and that the lack of correlation is a result of uncorrelated noise.</p>
<p>When <code>mtdef</code> is 1, by default, the multitaper spectral estimates are calculated as an unweighted average of the individual tapered estimates. However, if the optional argument <code>taper_wt</code> is specified, a weighted average will be employed using the weights in this array. Minimum variance optimal weights can be obtained from the routines <code>SHMTVarOpt</code> if the form of the underlying global power spectrum is known. Taper weights can not be used when <code>mtdef</code> is 2</p>
<p>This routine assumes that the input functions and tapers are expressed using geodesy 4-pi normalized spherical harmonic functions that exclude the Condon-Shortley phase factor of (-1)^m.</p>
<h1 id="see-also">See also</h1>
<p><a href="shreturntapers.html">shreturntapers</a>, <a href="shreturntapersm.html">shreturntapersm</a>, <a href="shmultitaperse.html">shmultitaperse</a>, <a href="shmultitapercse.html">shmultitapercse</a></p>
<h1 id="references">References</h1>
<p>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.</p>
<p>Simons, F. J., F. A. Dahlen and M. A. Wieczorek, Spatiospectral concentration on the sphere, SIAM Review, 48, 504-536, doi:10.1137/S0036144504445765, 2006.</p>
<p>Wieczorek, M. A. and F. J. Simons, Localized spectral analysis on the sphere, Geophys. J. Int., 162, 655-675, 2005.</p>
<p>Simons, M., S. C. Solomon and B. H. Hager, Localization of gravity and topography: constrains on the tectonics and mantle dynamics of Venus, Geophys. J. Int., 131, 24-44, 1997.</p>
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