-
Notifications
You must be signed in to change notification settings - Fork 103
/
shmultitapermaskcse.doc
83 lines (76 loc) · 4.01 KB
/
shmultitapermaskcse.doc
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
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.