Image Sensitivity

Image sensitivity is the RMS thermal noise (ΔI_m) expected in a single-polarization image.   Image sensitivities for the 10-station VLBA, for typical observing parameters, are listed in column [7] of the Receiver Frequency Ranges & Performance table.

Alternatively, the image sensitivity for a homogeneous array with natural weighting can be calculated using the following formula (Wrobel 1995; Wrobel & Walker 1999).

$$\mathrm{\Delta }{I}_m=\mathrm{S}\mathrm{E}\mathrm{F}\mathrm{D}/[{\eta }_s\cdot (N\cdot (N-1)\cdot \mathrm{\Delta }\nu \cdot t_{\mathrm{i}\mathrm{n}\mathrm{t}}{)}^{1/2}]\mathrm{J}\mathrm{y}\mathrm{b}\mathrm{e}\mathrm{a}{\mathrm{m}}^{-1}$$

Parameters SEFD, ${\eta }_s$, and $\mathrm{\Delta }\nu$ are the same as those used in computing baseline sensitivity, $N$ is the number of observing stations, and $t_{\mathrm{i}\mathrm{n}\mathrm{t}}$ is the total integration time on source in seconds.

The expression for image noise becomes rather more complicated for a heterogeneous array such as the HSA, and may depend quite strongly on the data weighting that is chosen in imaging.   The EVN sensitivity calculator provides a convenient estimate.   For example, the RMS noise at 22 GHz for the 10-station VLBA in a 1-hr integration is reduced by a factor between 4 and 5 by adding the GBT and the phased VLA.

If simultaneous dual polarization data are available with the above value of ΔI_m per polarization, then for an image of Stokes I, Q, U, or V,

$$\mathrm{\Delta }I=\mathrm{\Delta }Q=\mathrm{\Delta }U=\mathrm{\Delta }V=\frac{\mathrm{\Delta }{I}_m}{\sqrt{2}}$$

For a polarized intensity image of $P=\sqrt{Q^2+U^2}$

$$\mathrm{\Delta }P=0.655\times \mathrm{\Delta }Q=0.655\times \mathrm{\Delta }U$$

It is sometimes useful to express $\mathrm{\Delta }{I}_m$ in terms of an RMS brightness temperature in Kelvins ($\mathrm{\Delta }{T}_B$) measured within the synthesized beam.  An approximate formula for a single-polarization image is

$$\mathrm{\Delta }{T}_b\sim 320\times \mathrm{\Delta }{I}_m\times (B_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{k}\mathrm{m}}{)}^2\mathrm{K}$$

where $B_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{k}\mathrm{m}}$ is as in Table 5.