EKH vs shift-and-add restoration

Two kinds of restoration algorithms for wobbler switched data are available: The EKH [Emerson et al. 1979] and shift-and-add algorithms. EKH is much more elaborated and it enables to recover all the source structure as long as the source stays undetected at the edges of the map along the scanning direction. However, it works well only for relatively small source size (i.e. $\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{sou}}}< 0.5^\circ$). Now, there are scientific projects which search for point sources (at least sources whose size is smaller than the half wobbler throw) on much larger sky areas. For these (i.e. search for point sources on very large field of views), the shift-and-add restoration method works well.

We thus offer the user the following choice:

EKH restoration

The PI must give the source size in both perpendicular directions: $\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{sou}}}\times \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{sou}}}$. The PI can request unrelated sources in this mode (the sources must share the same pointing calibrator...).

Shift-and-add restoration

The PI must give the sizes of the final mosaic: $\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{mos}}}\times \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{mos}}}$. We then assume that this area will be observed in square submaps, whose typical duration is set to so that $\ensuremath{d_\ensuremath{\mathrm{cal}}}/\ensuremath{d_\ensuremath{\mathrm{submap}}}$ is an integer number. If we define $\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{edge}}}= \ensurem... ...{throw}}}+\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}}$, we then find the size through the following equation:

$$\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\math... ...suremath{\delta}\,\ensuremath{v_\ensuremath{\mathrm{\Vert}}}.$$ (15)

This second order equation has a single physical solution

$$\ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\math... ...Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{edge}}}}{2}.$$ (16)

There are then two cases:

1. If $\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{mos}}}\,\ens... ...}}}\le \ensuremath{\Delta^\ensuremath{\mathrm{2}}_\ensuremath{\mathrm{submap}}}$, then we use the user inputs:
   

   $$\ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{tot}}} = \ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{mos}}},$$ (17)

   $$\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{tot}}} = \ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathr... ...hrow}}}+ \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}},$$ (18)

   $$\ensuremath{n_\ensuremath{\mathrm{source}}} = 1.$$ (19)

   
   

2. Else we use:
   

   $$\ensuremath{\Delta^\ensuremath{\mathrm{\bot}}_\ensuremath{\mathrm{tot}}} = \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{submap}}},$$ (20)

   $$\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{tot}}} = \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{sub... ...hrow}}}+ \ensuremath{\Delta^\ensuremath{\mathrm{}}_\ensuremath{\mathrm{base}}},$$ (21)

   $$\ensuremath{n_\ensuremath{\mathrm{source}}} = \frac{\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{... ...}}}}{\ensuremath{\Delta^\ensuremath{\mathrm{2}}_\ensuremath{\mathrm{submap}}}}.$$ (22)

   
   
   We note that in this case, is not a real number of sources in this case. It is a number of submap and it thus does not need to be an integer.