## Appendix DSCUBA-2 Matched Filter

In order to optimally find sources that are the size of the telescope beam, and suppress this residual large-scale noise, the Picard recipe SCUBA2_MATCHED_FILTER may be used. If there were no large-scale noise in the map, the filtered signal map would be calculated as follows:

 $\mathsc{ℳ}=\frac{\left[M\left(x,y\right)/{\sigma }^{2}\left(x,y\right)\right]\otimes P\left(x,y\right)}{\left[1/{\sigma }^{2}\left(x,y\right)\right]\otimes \left[{P}^{2}\left(x,y\right)\right]},$ (D.1)

where $M\left(x,y\right)$ and $\sigma \left(x,y\right)$ are the signal and RMS noise maps respectively produced by Smurf, and $P\left(x,y\right)$ is a map of the PSF. Here $\otimes$ denotes the 2-dimensional cross-correlation operator. Similarly, the variance map would be calculated as

 ${\mathsc{𝒩}}^{2}=\frac{1}{\left[1/{\sigma }^{2}\left(x,y\right)\right]\otimes \left[{P}^{2}\left(x,y\right)\right]}.$ (D.2)

This operation is equivalent to calculating the maximum-likelihood fit of the PSF centered over every pixel in the map, taking into account the noise. Presently $P\left(x,y\right)$ is simply modelled as an ideal Gaussian with a FWHM set to the diffraction limit of the telescope.

However, since there is large-scale (and therefore correlated from pixel to pixel) noise, the recipe also has an additional step. It first smooths the map by cross-correlating with a larger Gaussian kernel to estimate the background, and then subtracts it from the image. The same operation is also applied to the PSF to estimate the effective shape of a point-source in this background-subtracted map.

Before running Picard, a simple parameters file called smooth.ini may be created.

[SCUBA2_MATCHED_FILTER]
SMOOTH_FWHM = 15

where SMOOTH_FWHM = 15 indicates that the background should be estimated by first smoothing the map and PSF with a 15 arcsec FWHM Gaussian. The recipe is then executed as follows:

% picard -recpars smooth.ini SCUBA2_MATCHED_FILTER map.sdf

The output of this operation is a smoothed image called map_mf.sdf. By default, the recipe automatically normalizes the output such that the peak flux densities of point sources are conserved. Note that the accuracy of this normalization depends on how closely the real PSF matches the 7.5 arcsec and 14 arcsec full-width at half-maximum (FWHM) Gaussian shapes assumed at 450$\mu$m and 850$\mu$m, respectively (an explicit PSF can also be supplied using the PSF_MATCHFILTER recipe parameter).