-
Notifications
You must be signed in to change notification settings - Fork 106
/
shmtvaropt.html
628 lines (302 loc) · 21.2 KB
/
shmtvaropt.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="">
<meta name="keywords" content="fortran, spherical harmonics software package, spherical harmonic transform, legendre functions, multitaper spectral analysis, fortran, Python, gravity, magnetic field">
<title>SHMTVarOpt (Fortran) | SHTOOLS - Tools for working with spherical harmonics</title>
<link rel="stylesheet" href="css/syntax.css">
<link rel="stylesheet" type="text/css" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.7.0/css/font-awesome.min.css">
<!--<link rel="stylesheet" type="text/css" href="css/bootstrap.min.css">-->
<link rel="stylesheet" href="css/modern-business.css">
<!-- Latest compiled and minified CSS -->
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" integrity="sha384-BVYiiSIFeK1dGmJRAkycuHAHRg32OmUcww7on3RYdg4Va+PmSTsz/K68vbdEjh4u" crossorigin="anonymous">
<link rel="stylesheet" href="css/customstyles.css">
<link rel="stylesheet" href="css/boxshadowproperties.css">
<!-- most color styles are extracted out to here -->
<link rel="stylesheet" href="css/theme-blue.css">
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.1.4/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery-cookie/1.4.1/jquery.cookie.min.js"></script>
<script src="js/jquery.navgoco.min.js"></script>
<!-- Latest compiled and minified JavaScript -->
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha384-Tc5IQib027qvyjSMfHjOMaLkfuWVxZxUPnCJA7l2mCWNIpG9mGCD8wGNIcPD7Txa" crossorigin="anonymous"></script>
<!-- Anchor.js -->
<script src="https://cdnjs.cloudflare.com/ajax/libs/anchor-js/2.0.0/anchor.min.js"></script>
<script src="js/toc.js"></script>
<script src="js/customscripts.js"></script>
<link rel="shortcut icon" href="images/favicon.ico">
<!-- HTML5 Shim and Respond.js IE8 support of HTML5 elements and media queries -->
<!-- WARNING: Respond.js doesn't work if you view the page via file:// -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
<script src="https://oss.maxcdn.com/libs/respond.js/1.4.2/respond.min.js"></script>
<![endif]-->
<link rel="alternate" type="application/rss+xml" title="" href="https://shtools.oca.eu/feed.xml">
<script>
$(document).ready(function() {
// Initialize navgoco with default options
$("#mysidebar").navgoco({
caretHtml: '',
accordion: true,
openClass: 'active', // open
save: false, // leave false or nav highlighting doesn't work right
cookie: {
name: 'navgoco',
expires: false,
path: '/'
},
slide: {
duration: 400,
easing: 'swing'
}
});
$("#collapseAll").click(function(e) {
e.preventDefault();
$("#mysidebar").navgoco('toggle', false);
});
$("#expandAll").click(function(e) {
e.preventDefault();
$("#mysidebar").navgoco('toggle', true);
});
});
</script>
<script>
$(function () {
$('[data-toggle="tooltip"]').tooltip()
})
</script>
<script>
$(document).ready(function() {
$("#tg-sb-link").click(function() {
$("#tg-sb-sidebar").toggle();
$("#tg-sb-content").toggleClass('col-md-9');
$("#tg-sb-content").toggleClass('col-md-12');
$("#tg-sb-icon").toggleClass('fa-toggle-on');
$("#tg-sb-icon").toggleClass('fa-toggle-off');
});
});
</script>
</head>
<body>
<!-- Navigation -->
<nav class="navbar navbar-inverse navbar-static-top">
<div class="container topnavlinks">
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target="#bs-example-navbar-collapse-1">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="fa fa-home fa-lg navbar-brand" href="index.html"> <span class="projectTitle"> SHTOOLS</span></a>
</div>
<div class="collapse navbar-collapse" id="bs-example-navbar-collapse-1">
<ul class="nav navbar-nav navbar-right">
<!-- toggle sidebar button -->
<li><a id="tg-sb-link" href="#"><i id="tg-sb-icon" class="fa fa-toggle-on"></i> Nav</a></li>
<!-- entries without drop-downs appear here -->
<li><a href="https://github.com/SHTOOLS/SHTOOLS/" target="_blank">GitHub</a></li>
<li><a href="https://github.com/SHTOOLS/SHTOOLS/releases" target="_blank">Releases</a></li>
<!-- entries with drop-downs appear here -->
<!-- conditional logic to control which topnav appears for the audience defined in the configuration file.-->
<li>
<a class="email" title="Submit feedback" href="#" onclick="javascript:window.location='mailto:mark.wieczorek@oca.eu?subject=SHTOOLS feedback&body=I have some feedback about SHMTVarOpt (Fortran) page: ' + window.location.href;"><i class="fa fa-envelope-o"></i> Feedback</a>
</li>
<!--comment out this block if you want to hide search-->
<li>
<!--start search-->
<div id="search-demo-container">
<input type="text" id="search-input" placeholder="search...">
<ul id="results-container"></ul>
</div>
<script src="js/jekyll-search.js" type="text/javascript"></script>
<script type="text/javascript">
SimpleJekyllSearch.init({
searchInput: document.getElementById('search-input'),
resultsContainer: document.getElementById('results-container'),
dataSource: 'search.json',
searchResultTemplate: '<li><a href="{url}" title="SHMTVarOpt (Fortran)">{title}</a></li>',
noResultsText: 'No results found.',
limit: 10,
fuzzy: true,
})
</script>
<!--end search-->
</li>
</ul>
</div>
</div>
<!-- /.container -->
</nav>
<!-- Page Content -->
<div class="container">
<div id="main">
<!-- Content Row -->
<div class="row">
<!-- Sidebar Column -->
<div class="col-md-3" id="tg-sb-sidebar">
<ul id="mysidebar" class="nav">
<li class="sidebarTitle">SHTOOLS 4.5</li>
<li>
<a href="#">Getting started</a>
<ul>
<li><a href="index.html">Overview</a></li>
<li><a href="installing.html">Installing</a></li>
<li><a href="using-with-f95.html">Using with Fortran 95</a></li>
<li><a href="using-with-python.html">Using with Python</a></li>
</ul>
</li>
<li>
<a href="#">Spherical harmonics</a>
<ul>
<li><a href="real-spherical-harmonics.html">Real spherical harmonics</a></li>
<li><a href="complex-spherical-harmonics.html">Complex spherical harmonics</a></li>
<li><a href="implementation-details.html">Implementation details</a></li>
</ul>
</li>
<li>
<a href="#">Fortran 95 Routines</a>
<ul>
<li><a href="f95-modules.html">Modules</a></li>
<li><a href="legendre-functions.html">Legendre functions</a></li>
<li><a href="spherical-harmonic-transforms.html">Spherical harmonic transformations</a></li>
<li><a href="io.html">I/O, storage, conversions</a></li>
<li><a href="global-spectral-analysis.html">Global spectral analysis</a></li>
<li><a href="localized-spectral-analysis.html">Localized spectral analysis</a></li>
<li><a href="spherical-harmonic-rotations.html">Spherical harmonic rotations</a></li>
<li><a href="gravity-magnetics.html">Gravity and magnetics</a></li>
<li><a href="miscellaneous-routines.html">Miscellaneous</a></li>
</ul>
</li>
<li>
<a href="#">Python components</a>
<ul>
<li><a href="python-classes.html">Python classes</a></li>
<li><a href="python-legendre-functions.html">Legendre functions</a></li>
<li><a href="python-spherical-harmonic-transforms.html">Spherical harmonic transformations</a></li>
<li><a href="python-io.html">I/O, storage, conversions</a></li>
<li><a href="python-spectral-analysis.html">Global and localized spectral analysis</a></li>
<li><a href="python-spherical-harmonic-rotations.html">Spherical harmonic rotations</a></li>
<li><a href="python-gravity-magnetics.html">Gravity and magnetics</a></li>
<li><a href="python-utilities.html">Utilities</a></li>
<li><a href="python-constants.html">Constants</a></li>
</ul>
</li>
<li>
<a href="#">Examples programs</a>
<ul>
<li><a href="python-examples.html">Python examples</a></li>
<li><a href="fortran-examples.html">Fortran 95 examples</a></li>
</ul>
</li>
<li>
<a href="#">Support</a>
<ul>
<li><a href="faq.html">FAQ</a></li>
<li><a href="fortran-95-problems.html">Fortran 95 problems</a></li>
<li><a href="how-to-contribute.html">How to contribute</a></li>
</ul>
</li>
<li>
<a href="#">Release Notes</a>
<ul>
<li><a href="release-notes-v4.html">Version 4</a></li>
<li><a href="release-notes-v3.html">Version 3</a></li>
<li><a href="license.html">License</a></li>
<li><a href="contributors.html">Contributors</a></li>
</ul>
</li>
<!-- if you aren't using the accordion, uncomment this block:
<p class="external">
<a href="#" id="collapseAll">Collapse All</a> | <a href="#" id="expandAll">Expand All</a>
</p>
-->
</ul>
<!-- this highlights the active parent class in the navgoco sidebar. this is critical so that the parent expands when you're viewing a page. This must appear below the sidebar code above. Otherwise, if placed inside customscripts.js, the script runs before the sidebar code runs and the class never gets inserted.-->
<script>$("li.active").parents('li').toggleClass("active");</script>
</div>
<!-- Content Column -->
<div class="col-md-9" id="tg-sb-content">
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
TeX: { equationNumbers: { autoNumber: "AMS" } }
});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
displayAlign: "left",
displayIndent: "2em"
});
</script>
<script type="text/javascript" async
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-MML-AM_CHTML">
</script>
<div class="post-header">
<h1 class="post-title-main">SHMTVarOpt (Fortran)</h1>
</div>
<div class="post-content">
<p>Calculate the theoretical minimum variance of a localized multitaper spectral estimate and the corresponding optimal weights to apply to each localized spectrum.</p>
<h2 id="usage">Usage</h2>
<p>call SHMTVarOpt (<code class="highlighter-rouge">l</code>, <code class="highlighter-rouge">tapers</code>, <code class="highlighter-rouge">taper_order</code>, <code class="highlighter-rouge">lwin</code>, <code class="highlighter-rouge">kmax</code>, <code class="highlighter-rouge">sff</code>, <code class="highlighter-rouge">var_opt</code>, <code class="highlighter-rouge">var_unit</code>, <code class="highlighter-rouge">weight_opt</code>, <code class="highlighter-rouge">unweighted_covar</code>, <code class="highlighter-rouge">nocross</code>, <code class="highlighter-rouge">exitstatus</code>)</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><code class="highlighter-rouge">l</code> : input, integer</dt>
<dd>The spherical harmonic used degree to determine the minimum variance and optimal weights.</dd>
<dt><code class="highlighter-rouge">tapers</code> : input, real(dp), dimension (<code class="highlighter-rouge">lwin</code>+1, <code class="highlighter-rouge">kmax</code>)</dt>
<dd>A matrix of localization functions obtained from <code class="highlighter-rouge">SHReturnTapers</code> or <code class="highlighter-rouge">SHReturnTapersM</code>.</dd>
<dt><code class="highlighter-rouge">taper_order</code> : input, integer, dimension (<code class="highlighter-rouge">kmax</code>)</dt>
<dd>The angular order of the windowing coefficients in <code class="highlighter-rouge">tapers</code>.</dd>
<dt><code class="highlighter-rouge">lwin</code> : input, integer</dt>
<dd>The spherical harmonic bandwidth of the localizing windows.</dd>
<dt><code class="highlighter-rouge">kmax</code> : input, integer</dt>
<dd>The maximum number of tapers to be used when calculating the minimum variance and optimal weights.</dd>
<dt><code class="highlighter-rouge">sff</code> : input, real(dp), dimension (<code class="highlighter-rouge">l</code>+<code class="highlighter-rouge">lwin</code>+1)</dt>
<dd>The global unwindowed power spectrum of the function to be localized.</dd>
<dt><code class="highlighter-rouge">var_opt</code> : output, real(dp), dimension (<code class="highlighter-rouge">kmax</code>)</dt>
<dd>The minimum variance of the multitaper spectral estimate for degree <code class="highlighter-rouge">l</code> using 1 through <code class="highlighter-rouge">kmax</code> tapers.</dd>
<dt><code class="highlighter-rouge">var_unit</code> : output, real(dp), dimension (<code class="highlighter-rouge">kmax</code>)</dt>
<dd>The variance of the multitaper spectral estimate using equal weights for degree <code class="highlighter-rouge">l</code> using 1 through <code class="highlighter-rouge">kmax</code> tapers.</dd>
<dt><code class="highlighter-rouge">weight_opt</code> : optional, output, real(dp), dimension (<code class="highlighter-rouge">kmax</code>, <code class="highlighter-rouge">kmax</code>)</dt>
<dd>The optimal weights (in columns) that minimize the multitaper spectral estimate’s variance using 1 through <code class="highlighter-rouge">kmax</code> tapers.</dd>
<dt><code class="highlighter-rouge">unweighted_covar</code> : optional, output, real(dp), dimension (<code class="highlighter-rouge">kmax</code>, <code class="highlighter-rouge">kmax</code>)</dt>
<dd>The unweighted covariance matrix of the <code class="highlighter-rouge">kmax</code> tapers (i.e., Fij in Wieczorek and Simons 2007).</dd>
<dt><code class="highlighter-rouge">nocross</code> : optional, input, integer, default = 0</dt>
<dd>If 1, only the diagonal terms of the covariance matrix Fij will be computed. If 0, all terms will be computed.</dd>
<dt><code class="highlighter-rouge">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>
<h2 id="description">Description</h2>
<p><code class="highlighter-rouge">SHMTVarOpt</code> will determine the minimum variance that can be achieved by a weighted multitaper spectral analysis, as is described by Wieczorek and Simons (2007). The minimum variance is output as a function of the number of tapers utilized, from 1 to a maximum of <code class="highlighter-rouge">kmax</code>, and the corresponding variance using equal weights is output for comparison. The windowing functions are assumed to be solutions to the spherical-cap concentration problem, as determined by a call to <code class="highlighter-rouge">SHReturnTapers</code> or <code class="highlighter-rouge">SHReturnTapersM</code>. The minimum variance and weights are dependent upon the form of the global unwindowed power spectrum, <code class="highlighter-rouge">Sff</code>.</p>
<p>If the optional argument <code class="highlighter-rouge">weight_opt</code> is specified, then the optimal weights will be returned as a function of the number of tapers employed, from 1 to <code class="highlighter-rouge">kmax</code>. If <code class="highlighter-rouge">unweighted_covar</code> is specified, then the unweighted covariance matrix of the <code class="highlighter-rouge">kmax</code> tapers (i.e., Fij) will be output. If the optional argument <code class="highlighter-rouge">nocross</code> is set to 1, then only the diagnonal terms of <code class="highlighter-rouge">Fij</code> will be computed.</p>
<h2 id="references">References</h2>
<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>
<h2 id="see-also">See also</h2>
<p><a href="shmtvar.html">shmtvar</a>, <a href="shreturntapers.html">shreturntapers</a>, <a href="shreturntapersm.html">shreturntapersm</a>, <a href="shmultitaperse.html">shmultitaperse</a>, <a href="shmultitapercse.html">shmultitapercse</a>, <a href="shlocalizedadmitcorr.html">shlocalizedadmitcorr</a>, <a href="shbiasadmitcorr.html">shbiasadmitcorr</a>, <a href="shbiask.html">shbiask</a>, <a href="shmtdebias.html">shmtdebias</a></p>
<div class="tags">
<b>Tags: </b>
<a href="tag_fortran.html" class="btn btn-default navbar-btn cursorNorm" role="button">fortran</a>
</div>
<a target="_blank" href="https://github.com/SHTOOLS/SHTOOLS/tree/develop/src/fdoc/shmtvaropt.md" class="btn btn-default githubEditButton" role="button"><i class="fa fa-github fa-lg"></i> Edit me</a>
</div>
<hr class="shaded"/>
<footer>
<div class="row">
<div class="col-lg-12 footer">
©2019 SHTOOLS. All rights reserved. <br />
<a href="https://www.oca.eu/?lang=en" target="_blank" rel="noopener">Observatoire de la Côte d'Azur</a><br />
<a href="https://lagrange.oca.eu/fr/accueil-lagrange" target="_blank" rel="noopener">Laboratoire Lagrange</a><br />
<p><img src="images/company_logo.png" alt="SHTOOLS"/></p>
</div>
</div>
</footer>
</div>
<!-- /.row -->
</div>
<!-- /.container -->
</div>
<!-- /#main -->
</div>
</body>
</html>