Multi-frequency deconvolution - MFCLEAN

The *Miriad* deconvolution task mfclean
can be used to
avoid this distortion. Rather than just determining a flux density estimate,
mfclean
simultaneously solves for both a flux density and spectral
variation for each source. To understand how this is possible,
consider a noiseless observation of a single point source at the phase centre.
Assume that it has a linear spectral variation. While this is obviously
an approximation, it is an adequate one for dynamic ranges up to several
thousand to one (assuming typical spectral variation and fractional spread in
frequencies of no more than 30%). The visibility function
for this source can be described by two parameters: the source brightness at
some reference frequency, and the brightness derivative.

If we temporarily assume that the derivative is zero, then the image resulting from a multi-frequency synthesis observation of this source will just be the normal point-spread function (beam pattern) scaled by . If we now assume some non-zero derivative, we will get a different image. Because of the linearity of the system, the output image will be a sum of the two terms. The first term is just the normal point spread function scaled by as before, while the second term is caused by the derivative. The importance of the second term will be related to the size of the derivative and the range of frequencies sampled. This second term has been called the ``spectral dirty beam'' - it is a point-spread function resulting from a linear frequency variation. While it may seem that the most logical definition of the spectral dirty beam is the additional response resulting from a source with a flux density derivative of 1, there is a more convenient scaling. If we assume that the flux density variation is due to a spectral index, , where

then

Conventionally the spectral dirty beam is scaled so that

For a general source, a sum of the two beams is replaced by a convolution
relation:

(in the above equation,

But how important are the effects that spectral variation causes?
Generally you will want to ignore them if you can, because then
the deconvolution procedure has to solve only the
standard convolution equation,

and the other deconvolution tasks - clean and maxen - can be used.

Surprisingly spectral effects are not very important in a wide variety of problems. For fractional bandwidths of 10%, spectral effects need to be considered for dynamic ranges of better than 1000. For fractional bandwidths of 30%, the effects become important for dynamic ranges of a few hundred. This assumes typical spectral variations equivalent to spectral indices of order 1. When imaging beyond the half-power points of the primary beam, the frequency dependence of the primary beam will be quite substantial, and so spectral variation will need to be considered for lower dynamic range images.

Generally you will need to consider at the imaging step (task invert)
whether you *may* need to account for spectral variation in the
deconvolution step. If you *may*, then you need to form the spectral
dirty beam, , as well as the normal dirty beam, .
If so, you must have set the `sdb` option in
invert. This produces the spectral dirty beam as a second
plane of the beam dataset. If
you did produce this spectral dirty beam plane, and then use
clean
and maxen, they will issue warnings (which can be
safely ignored) about the existence of the second plane.

The inputs to mfclean
are fairly similar to clean. Task
mfclean
differs in that it does not have a Prussian helmet
or SDI CLEAN capability. Task mfclean
does not support a number
of the `options` given by CLEAN. More fundamental, though, is that,
whereas clean
can only CLEAN an area a quarter of the beam area, mfclean
can strictly
speaking CLEAN an area only one ninth of the beam area. To explain the
reason for this, we must consider some of the details of mfclean. To
aid it in finding peaks in the flux density and spectral index domains, it
correlates the residuals with the beam and the spectral dirty beam. More
strictly it correlates the residuals with a portion of the beam and spectral
dirty beam - the sub-beam. Strictly the sub-beam size should be the same
as the region being CLEANed. However in practise, provided it contains most
of the main sidelobes, a smaller portion is adequate. Task mfclean
will tell you the sub-beam size, and indeed may complain about it
being small. However provided it is either at least as big as the region
being CLEANed, or is 50-100 pixels, you are probably OK.
All the same, however, you should attempt to produce
images with larger guard bands if mfclean
is to be used.

As with clean, you will almost certainly want to restore your resultant image - see Section 14.6. The output from mfclean is a CLEAN component image with a difference - it contains two planes. The first plane is the normal CLEAN component image (like the one determined by clean), while the second plane contains the estimate of . Both restor (Section 14.6) and a number of other tasks use this second plane.

2016-06-21