Beam Convolution

Dealing with data from different telescopes is a common activity for astronomers. Here is a rough method for convolving a beam (i.e. PACS, Herschel) to another beam (i.e. SCUBA-2 at 850um).

A very rough and ready method is to assume that both beams are Gaussian. Let’s say PACS has an FWHM of “A” arc-seconds and SCUBA-2 has an FWHM  of “B” arc-seconds. If “A” is larger than “B”, then you need to smooth the SCUBA-2 map using a Gaussian kernel of FWHM equal to sqrt(A*A – B*B) arc-seconds. If the SCUBA-2 map has a pixel size of Pa arc-seconds, then first convert the above size into pixels by calculating:

width = sqrt( A*A – B*B )/Pa

and then smooth the SCUBA-2 map using the “gausmooth” command in the starlink kappa package:

gausmooth in=<your scuba-2 map> out=<smoothed map> fwhm=<your "width" value>

The smoothed map is put into the file specified by the “out” parameter. Alternatively, if “B” is larger than “A” smooth the PACS map inthe same way, using a width of:

width = sqrt( B*B – A*A )/Pb

where Pb is the pixel size in the PACS map.

You will need to know what the A and B values are (at 850 um “B” is about 13.5 arc-seconds). If your maps have point sources in them, then you could determine A and B by measuring the widths of the point sources in your maps. For instance, the “psf” command (“Point Spread Function”) in the starlink kappa package allows you to determine a mean beam shape from one or more point sources in an image. It does this by fitting a generalized Gaussian function to the mean radial profile of the indicated point sources.

The above assumes that both beam shapes are Gaussian. The SCUBA-2 beam shape is not quite Gaussian and so the above method can be improved, but it involves a lot more time and effort. You need first to get accurate models for the two beams (either as analytical functions or as 2D images), then you smooth the SCUBA-2 map using the PACS beam, and then smooth the PACS map using the SCUBA-2 beam. This approach requires no deconvolution, but results in maps that have lower resolution than either the PACS or SCUBA-2 maps. The kappa package includes the “convolve” command that will smooth a map using a beam shape specified as a 2D image (the kappa “maths” command can be used to generate a 2D image if your beam shape is expressed as an analytical expression). The details of this method depend on the form in which you obtain the beam shape information.

Print Friendly, PDF & Email

Category: JCMT Data, Starlink software


Comments are closed.