This routine provides a means to fit and remove the dipole and monopole from a HEALPix map. The fit is obtained by solving the linear system

$\begin{displaymath} \sum_{j=0}^{d^2-1}\ A_{ij}\ f_j = b_i \end{displaymath}$ (15)

with, d=1 or 2, and

$\begin{displaymath} b_i = \sum_{p \in \cal{P}} s_i(p) w(p) m(p), \end{displaymath}$ (16)

$\begin{displaymath} A_{ij} = \sum_{p \in \cal{P}} s_i(p) w(p) s_j(p), \end{displaymath}$ (17)

where $\cal{P}$ is the set of valid, unmasked pixels, m is the input map, w is pixel weighting, while s₀(p) = 1 and $s_1(p)=x,\ s_2(p)=y,\ s_3(p)=z$ are respectively the monopole and dipole templates. The output map is then

$\begin{displaymath} m'(p) = m(p) - \sum_{i=0}^{d^2-1} f_i s_i(p). \end{displaymath}$ (18)

Location in HEALPix directory tree: src/f90/mod/pix_tools.f90

FORMAT

call remove_dipole*( nside, map, ordering, degree, multipoles, zbounds [, fmissval, mask, weights] )

ARGUMENTS

name & dimensionality	kind	in/out	description

nside	I4B	IN	value of ${N_{\rm side}}$ resolution parameter for input map
map(0:12nsidenside-1)	SP/ DP	INOUT	HEALPix map from which the monopole and dipole will be removed. Those are removed from all unflagged pixels, even those excluded by the cut `zounds` or the `mask`.
ordering	I4B	IN	HEALPix scheme 1:RING, 2: NESTED
degree	I4B	IN	multipoles to fit and remove. It is either 0 (nothing done), 1 (monopole only) or 2 (monopole and dipole).
multipoles(0:degree*degree-1)	DP	OUT	values of best fit monopole and dipole. The monopole is described as a scalar in the same units as the input map, the dipole as a 3D cartesian vector, in the same units.
zbounds(1:2)	DP	IN	section of the map on which to perform the fit, expressed in terms of $z=\sin({\rm latitude}) = \cos(\theta)$ . If zbounds(1)<zbounds(2), the fit is performed on the strip zbounds(1)<z<zbounds(2); if not, the fit is performed outside of the strip zbounds(2)<z<zbounds(1).
fmissval (OPTIONAL)	SP/ DP	IN	value used to flag bad pixel on input default:-1.6375e30. Pixels with that value are ignored during the fit, and left unchanged on output.
mask(0:12nsidenside-1) (OPTIONAL)	SP/ DP	IN	mask of valid pixels. Pixels with \|mask\|<10^-10 are not used for fit. Note: the map is not multiplied by the mask.
weights(0:12nsidenside-1) (OPTIONAL)	SP/ DP	IN	weight to be given to each map pixel before doing the fit. By default pixels are given a uniform weight of 1. Note: the output map is not multiplied by the weights.

EXAMPLE:

s = sin(15.0_dp * PI / 180.0_dp) call remove_dipole*(128, map, 1, 2, multipoles, ( s, -s ) )

Will compute and remove the best fit monopole and dipole from a map with ${N_{\rm side}}=128$ in RING ordering scheme. The fit is performed on pixels with |b|>15^o.

MODULES & ROUTINES

This section lists the modules and routines used by remove_dipole*.

pix_tools: module, containing:

RELATED ROUTINES

This section lists the routines related to remove_dipole*

add_dipole: routine to add a dipole and monopole to a map.

Eric Hivon 2010-06-18

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