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pyshmultitapercse.html
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pyshmultitapercse.html
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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="shmultitapercse">SHMultiTaperCSE</h1>
<p>Perform a localized multitaper cross-spectral analysis using spherical cap windows.</p>
<h1 id="usage">Usage</h1>
<p><code>mtse</code>, <code>sd</code> = pyshtools.SHMultiTaperCSE (<code>sh1</code>, <code>sh2</code>, <code>tapers</code>, <code>taper_order</code>, [<code>lmax1</code>, <code>lmax2</code>, <code>lmaxt</code>, <code>k</code>, <code>lat</code>, <code>lon</code>, <code>taper_wt</code>, <code>norm</code>, <code>csphase</code>])</p>
<h1 id="returns">Returns</h1>
<dl>
<dt><code>mtse</code> : float, dimension (<code>lmax</code>-<code>lmaxt</code>+1)</dt>
<dd>The localized multitaper cross-power spectrum estimate. <code>lmax</code> is the smaller of <code>lmax1</code> and <code>lmax2</code>.
</dd>
<dt><code>sd</code> : float, dimension (<code>lmax</code>-<code>lmaxt</code>+1)</dt>
<dd>The standard error of the localized multitaper cross-power spectral estimates. <code>lmax</code> is the smaller of <code>lmax1</code> and <code>lmax2</code>.
</dd>
</dl>
<h1 id="parameters">Parameters</h1>
<dl>
<dt><code>sh1</code> : float, dimension (2, <code>lmax1in</code>+1, <code>lmax1in</code>+1)</dt>
<dd>The spherical harmonic coefficients of the first function.
</dd>
<dt><code>sh2</code> : float, dimension (2, <code>lmax2in</code>+1, <code>lmax2in</code>+1)</dt>
<dd>The spherical harmonic coefficients of the second function.
</dd>
<dt><code>tapers</code> : float, dimension (<code>lmaxtin</code>+1, <code>kin</code>)</dt>
<dd>An array of the <code>k</code> windowing functions, arranged in columns, obtained from a call to <code>SHReturnTapers</code>. Each window has non-zero coefficients for a single angular order that is specified in the array <code>taper_order</code>.
</dd>
<dt><code>taper_order</code> : integer, dimension (<code>kin</code>)</dt>
<dd>An array containing the angular orders of the spherical harmonic coefficients in each column of the array <code>tapers</code>.
</dd>
<dt><code>lmax1</code> : optional, integer, default = <code>lmax1in</code></dt>
<dd>The spherical harmonic bandwidth of <code>sh1</code>. This must be less than or equal to <code>lmax1in</code>.
</dd>
<dt><code>lmax2</code> : optional, integer, default = <code>lmax2in</code></dt>
<dd>The spherical harmonic bandwidth of <code>sh2</code>. This must be less than or equal to <code>lmax2in</code>.
</dd>
<dt><code>lmaxt</code> : optional, integer, default = <code>lmaxtin</code></dt>
<dd>The spherical harmonic bandwidth of the windowing functions in the array <code>tapers</code>.
</dd>
<dt><code>k</code> : optional, integer, default = <code>kin</code></dt>
<dd>The number of tapers to be utilized in performing the multitaper spectral analysis.
</dd>
<dt><code>lat</code> : optional, float, default = 90</dt>
<dd>The latitude in degrees of the localized analysis. The default is to perform the spectral analysis at the north pole.
</dd>
<dt><code>lon</code> : optional, float, default = 0</dt>
<dd>The longitude in degrees of the localized analysis.
</dd>
<dt><code>taper_wt</code> : optional, float, dimension (<code>kin</code>), default = -1</dt>
<dd>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 <code>SHMTVarOpt</code>. The default value specifies not to use <code>taper_wt</code>.
</dd>
<dt><code>norm</code> : optional, intger, default = 1</dt>
<dd>1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
</dd>
<dt><code>csphase</code> : optional, integer, default = 1</dt>
<dd>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.
</dd>
</dl>
<h1 id="description">Description</h1>
<p><code>SHMultiTaperCSE</code> will perform a localized multitaper cross-spectral analysis of two input functions expressed in spherical harmonics, <code>SH1</code> and <code>SH2</code>. The maximum degree of the localized multitaper power spectrum estimate is <code>lmax-lmaxt</code>, where <code>lmax</code> is the smaller of <code>lmax1</code> and <code>lmax2</code>. The coefficients and angular orders of the windowing coefficients (<code>tapers</code> and <code>taper_order</code>) are obtained by a call to <code>SHReturnTapers</code>. If <code>lat</code> and <code>lon</code> are 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.</p>
<p>If the optional array <code>taper_wt</code> is specified, then these weights will be used in calculating a weighted average of the individual <code>k</code> tapered estimates (<code>mtse</code>) and the corresponding standard error of the estimates (<code>sd</code>). If not present, the weights will all be assumed to be equal. When <code>taper_wt</code> is not specified, the mutltitaper spectral estimate for a given degree is calculated as the average obtained from the <code>k</code> individual tapered estimates. The standard error of the multitaper estimate at degree l is simply the population standard deviation, <code>S = sqrt(sum (Si - mtse)^2 / (k-1))</code>, divided by sqrt(<code>k</code>). See Wieczorek and Simons (2007) for the relevant expressions when weighted estimates are used.</p>
<p>The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments <code>norm</code> and <code>csphase</code>; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m.</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>
<h1 id="see-also">See also</h1>
<p><a href="pyshmultitaperse.html">shmultitaperse</a>, <a href="pyshreturntapers.html">shreturntapers</a>, <a href="pyshreturntapersm.html">shreturntapersm</a>, <a href="pyshmtvaropt.html">shmtvaropt</a></p>
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