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computedmap.1
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computedmap.1
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.\" Automatically generated by Pandoc 1.17.2
.\"
.TH "computedmap" "1" "2016\-06\-17" "Fortran 95" "SHTOOLS 3.3"
.hy
.SH ComputeDMap
.PP
Compute the space\-concentration kernel of an arbitrary mask on the
sphere.
.SH Usage
.PP
call ComputeDMap (\f[C]dij\f[], \f[C]dh_mask\f[], \f[C]n\f[],
\f[C]lmax\f[], \f[C]sampling\f[])
.SH Parameters
.TP
.B \f[C]dij\f[] : output, real*8, dimension ( (\f[C]lmax\f[]+1)**2, (\f[C]lmax\f[]+1)**2 )
The space\-concentration kernel corresponding to the mask dh_mask.
.RS
.RE
.TP
.B \f[C]dh_mask\f[] : input, integer, dimension (\f[C]n\f[], \f[C]sampling\f[]*\f[C]n\f[])
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).
.RS
.RE
.TP
.B \f[C]n\f[] : input, integer
The number of latitudinal samples in \f[C]dh_mask\f[].
The effective spherical harmonic bandwidth of this grid is
\f[C]L=n/2\-1\f[].
.RS
.RE
.TP
.B \f[C]lmax\f[] : input, integer
The maximum spherical harmonic degree of the matrix \f[C]dij\f[].
.RS
.RE
.TP
.B \f[C]sampling\f[] : input, optional, integer, default = 1
For 1 (default), \f[C]dh_mask\f[] has \f[C]n\f[] x \f[C]n\f[] samples.
For 2, \f[C]dh_mask\f[] has \f[C]n\f[] x \f[C]2n\f[] samples.
.RS
.RE
.SH Description
.PP
\f[C]ComputeDMap\f[] will calculate the space\-concentration kernel for
a generic mask defined on the sphere.
The input mask \f[C]dh_mask\f[] must be sampled according to the
Driscoll and Healy (1994) sampling theorem with \f[C]n\f[] samples in
latitude, and possess a value of 1 inside the concentration region, and
0 elsewhere.
\f[C]dh_mask\f[] can either possess \f[C]n\f[] samples in longitude
(\f[C]sampling=1\f[]) or \f[C]2n\f[] samples in longitude
(\f[C]sampling=2\f[]).
Given the approximate way in which the elements of \f[C]dij\f[] are
calculated (see below), \f[C]sampling=2\f[] should be preferred.
\f[C]dij\f[] is symmetric, and the elements are ordered according to the
scheme described in \f[C]YilmIndexVector\f[].
See Simons et al.
(2006) for further details.
.PP
The elements of DIJ are explicitly given by
.PP
\f[C]Dlm,l\[aq]m\[aq]\ =\ 1/(4pi)\ Integral_R\ Ylm\ Yl\[aq]m\[aq]\ dOmega\f[],
.PP
where \f[C]R\f[] is the concentration region.
In this routine, all values of \f[C]l\[aq]m\[aq]\f[] are calculated in a
single spherical harmonic transform for a given value of \f[C]lm\f[]
according to
.PP
\f[C]Dl\[aq]m\[aq]\ =\ 1/(4pi)\ Integral_Omega\ F\ Yl\[aq]m\[aq]\ dOmega\f[].
.PP
where
.PP
\f[C]F\ =\ Ylm\ dh_mask\f[].
.PP
The function \f[C]F\f[] is in general not a polynomial, and thus the
coefficients \f[C]Dl\[aq]m\[aq]\f[] should not be expected to be exact.
For this reason, the effective spherical harmonic degree of the input
mask (\f[C]L=n/2\-1\f[]) should be greater than \f[C]lmax\f[].
The exact value of \f[C]n\f[] should be chosen such that further
increases in \f[C]n\f[] do not alter the returned eigenvalues.
The routine prints out the fractional area of the mask computed in the
pixel domain divided by \f[C]D(1,1)\f[] (the fractional area computed by
the spherical harmonic transforms), and the ratio of the two should be
close to 1.
Experience suggests that \f[C]l\f[] should be about 5 times
\f[C]lmax\f[].
.SH References
.PP
Driscoll, J.R.
and D.M.
Healy, Computing Fourier transforms and convolutions on the 2\-sphere,
Adv.
Appl.
Math., 15, 202\-250, 1994.
.PP
Simons, F.J., F.A.
Dahlen, and M.A.
Wieczorek, Spatiospectral concentration on a sphere, SIAM Review, 48,
504\-536, 2006.
.SH See also
.PP
shreturntapersmap, yilmindexvector