pycomputedmap.1

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.\" ========================================================================
.\"
.IX Title "COMPUTEDMAP 1"
.TH COMPUTEDMAP 1 "2014-11-10" "SHTOOLS 3.0" "SHTOOLS 3.0"
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.SH "ComputeDMap"
.IX Header "ComputeDMap"
.IP "ComputeDMap \-" 15
.IX Item "ComputeDMap -"
Compute the space-concentration kernel of a mask defined on the sphere.
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.IP "\s-1SUBROUTINE\s0 ComputeDMap (" 25
.IX Item "SUBROUTINE ComputeDMap ("
\&\s-1DIJ\s0, \s-1DH_MASK\s0, N, \s-1LMAX\s0, \s-1SAMPLING\s0 )
.RS 4
.IP "REAL*8" 10
.IX Item "REAL*8"
\&\s-1DIJ\s0( (\s-1LMAX+1\s0)**2, (\s-1LMAX+1\s0)**2 )
.IP "\s-1INTEGER\s0" 10
.IX Item "INTEGER"
\&\s-1DH_MASK\s0( N, N ) or \s-1DH_MASK\s0( N, 2*N ), N, \s-1LMAX\s0
.IP "\s-1INTEGER\s0, \s-1OPTIONAL\s0" 10
.IX Item "INTEGER, OPTIONAL"
\&\s-1SAMPLING\s0
.RE
.RS 4
.RE
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
\&\fIComputeDMap\fR will calculate the space-concentration kernel for a generic mask defined on the sphere. The input mask \s-1DH_MASK\s0 must be sampled according to the Driscoll and Healy (1994) sampling theorem with N samples in latitude, and possess a value of 1 inside the concentration region, and 0 elsewhere. \s-1DH_MASK\s0 can either possess N samples in longitude (SAMPLING=1) or 2*N samples in longitude (SAMPLING=2). Given the approximate way in which the elements of \s-1DIJ\s0 are calculated (see below), SAMPLING=2 should be preferred. \s-1DIJ\s0 is symmetric, and the elements are ordered according to the scheme described in \fIYilmIndex\fR. See Simons et al. (2006) for further details.
.PP
The elements of \s-1DIJ\s0 are explicitly given by
.PP
Dlm,l'm' = 1/(4pi) Integral_R Ylm Yl'm' dOmega,
.PP
where R is the concentration region. In this routine, all values of l'm' are calculated in a single spherical harmonic transform (\fISHExpandDH\fR) for a given value of lm according to
.PP
Dl'm' = 1/(4pi) Integral_Omega F Yl'm' dOmega.
.PP
where
.PP
F = Ylm \s-1DH_MASK\s0.
.PP
The function F is in general not a polynomial, and thus the coefficients Dl'm' should not be expected to be exact. For this reason, the effective spherical harmonic degree of the input mask (L = N/2 \- 1) should be greater than \s-1LMAX\s0. The exact value of N should be chosen such that further increases in N do not alter the returned eigenvalues. The routine prints out the fractional area of the mask computed in the pixel domain divided by D(1,1) (the fractional area computed by the spherical harmonic transforms), and the ratio of the two should be close to 1. Experience suggests that L should be about 5 times \s-1LMAX\s0.
.SH "ARGUMENTS"
.IX Header "ARGUMENTS"
.IP "\s-1DIJ\s0" 10
.IX Item "DIJ"
(output) REAL*8, \s-1DIMENSION\s0 ( (\s-1LMAX+1\s0)**2, (\s-1LMAX+1\s0)**2 )
.Sp
The space-concentration kernel corresponding to the mask \s-1DH_MASK\s0.
.IP "\s-1DH_MASK\s0" 10
.IX Item "DH_MASK"
(input) \s-1INTEGER\s0, \s-1DIMENSION\s0 (N, N) or \s-1DIMENSION\s0 (N, 2*N)
.Sp
A Driscoll and Healy (1994) sampled grid describing the concentration region R. All elements should either be 1 (for inside the concentration region) or 0 (for outside R).
.IP "N" 10
.IX Item "N"
(input) \s-1INTEGER\s0
.Sp
The number of latitudinal samples in \s-1DH_MASK\s0. The effective spherical harmonic bandwidth of this grid is L = N/2 \- 1.
.IP "\s-1LMAX\s0" 10
.IX Item "LMAX"
(input) \s-1INTEGER\s0
.Sp
The maximum spherical harmonic degree of the matrix \s-1DIJ\s0.
.IP "\s-1SAMPLING\s0" 10
.IX Item "SAMPLING"
(input) \s-1INTEGER\s0, \s-1OPTIONAL\s0
.Sp
For 1 (default), \s-1DH_MASK\s0 has N x N samples. For 2, \s-1DH_MASK\s0 has N x 2N samples.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
\&\fIshreturntapersmap\fR\|(1), \fIyilmindex\fR\|(1)
.PP
<http://shtools.ipgp.fr/>
.SH "REFERENCES"
.IX Header "REFERENCES"
Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2\-sphere, \fIAdv. Appl. Math.\fR, 15, 202\-250, 1994.
.PP
Simons, F.J., F.A. Dahlen, and M.A. Wieczorek, Spatiospectral concentration on a sphere, \fI\s-1SIAM\s0 Review\fR, 48, 504\-536, 2006.
.SH "COPYRIGHT AND LICENSE"
.IX Header "COPYRIGHT AND LICENSE"
Copyright 2012 by Mark Wieczorek <wieczor@ipgp.fr>.
.PP
This is free software; you can distribute and modify it under the terms of the revised \s-1BSD\s0 license.