# Mosaic sensitivity

As of 2011-February (it has since been fixed using the formulae on this page), the ALMA sensitivity calculator reports rms values for the central position of a single pointing. For well-sampled mosaics, it is thus an underestimate. Here we estimate the true sensitivity at a point inside the region of regular sampling, assuming a Gaussian primary beam of FWHM=1.13*lambda/D, and equal time sampling at all points.

## Hexagonal patterns

Logic: Consider a point source at the center of the mosaic. Taking the example of case (3) below, there are six surrounding points each 0.5109*FWHM away, where the S/N is 0.4850*(S/N)_center. Thus, if the central pointing is worth Time = 1 second, then each of the surrounding 6 pointings is worth Time = 0.4850**2 = 0.2352 second. Six of the 12 points in the surrounding ring will be 1.022 beam diameter away which contributes at the 5.53% level, worth 0.003058 seconds each. The other six are at 0.885 beams away, which is the 11.4% level and each worth 0.01300 seconds. The total effective time on source is then = 1 + 6*0.2352 + 6*0.003058 + 6*.01300 = 2.5075. The effective sensitivity is then sqrt(2.5075) = 1.583 better than the ALMA calculator predicts.

Anything coarser than NVSS is not recommended. It is too wide for the automatic selection of the imsize parameter in the Cycle 6 pipeline:

CAS-12249.
My python script (au.mosaicSensitivity) yields:

- 7-pt: 1.012
- 19-pt: 1.012
- 37-pt: 1.012

#### (1) Spacing equal to the 25% level of the beam = FWHM / sqrt(2) = 0.7071 * FWHM = 0.7989 * lambda/D

This spacing is derived in the NVSS paper by Condon et al. as the coarsest pattern that provides uniform sensitivity across
the mosaic. (But note that they actually used a somewhat coarser pattern in the end for their survey).
My python script yields:

- 7-pt: 1.173
- 19-pt: 1.173
- 37-pt: 1.173

#### (2) Spacing equal to 0.60 * lambda/D = 0.5746 * FWHM = Coarsest recommended value

- 7-pt: 1.502
- 19-pt: 1.524
- 37-pt: 1.524

#### (3) Spacing equal to sqrt(1/3) * lambda/D = 0.5774 * lambda/D = 0.5109 * FWHM = ALMA OT definition of "Nyquist" for hexagonal grid

- 7-pt: 1.553
- 19-pt: 1.583
- 37-pt: 1.583

#### (4) Spacing equal to 0.50 * lambda/D = 0.4425 * FWHM = Full Nyquist for rectangular grid

- 7-pt: 1.740
- 19-pt: 1.826
- 37-pt: 1.828

#### (5) Spacing equal to 0.447 * lambda/D = 0.3752 * FWHM = oversampled (for better sensitivity over less area)

- 7-pt: 1.876
- 19-pt: 2.037
- 37-pt: 2.045

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ToddHunter - 2011-08-08