The idea has been around for decades that as you increase the SNR, you get much better positions, so you don't really need high resolution to do optical identifications. That is true for perfect point sources (and perfect data). But how well does it work for real data? I have a mini-research paper here, so apologies for the length. If you don't care about the details, the bottom line is that for NVSS at least, you need to use a matching radius of 20 arcsec (40% of the FWHM beam size) even for SNR = 100 sources. The same is true for the FIRST survey: you need a matching radius of 40% of the FWHM beam size = 2 arcsec to find optical counterparts. The same thing will be true of the SKA pre-cursor surveys: WODAN (15" resolution) will require a matching radius of 6", and ASKAP-EMU (10" resolution) will require a matching radius of 4". Higher SNR certainly does not magically give you better ability to identify optical counterparts.

Why are the positions so inaccurate compared with the SNR model predictions? Real radio sources are not symmetrical objects. They have lobes, jets, cores; star-forming galaxies have spiral arms. And there can be confusion where multiple radio sources get mixed together in the low resolution beam. With a low resolution survey, you do indeed get a very accurate measurement of the mean flux-weighted position as the SNR increases. However,

**the flux-weighted centroid is not where the optical counterpart lies**. In many cases the counterpart is associated with some sharp structure within the radio source, and that structure may be far from the flux-weighted center.

Conclusion: We need high resolution to get the accurate positions required for optical identifications. Deeper imaging is not a substitute.

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Below I provide data in support of these statements based on a comparison of the FIRST and NVSS data. The NVSS resolution is 45 arcsec FWHM. The FIRST resolution is 5.4 arcsec FWHM. I did a large-scale test that as far as I know has not been done before: I selected a sample of all the FIRST sources that have an SDSS match within 0.7 arcsec and that have an NVSS match within 100 arcsec. For all these ~135,000 sources, I computed the distance to the nearest NVSS source. The important thing about this sample is that the FIRST source matches the optical source position. That means that if NVSS is to identify the same counterpart, it needs to converge to the FIRST source position. There may be several FIRST source components associated with a single NVSS source, but only the FIRST sources that match optical counterparts are included.

The prediction of what I'll call the "SNR model" is that as the flux density increases for the NVSS source, the positional error will decrease as 1/SNR, allowing the optical counterpart to be matched. Specifically, the NVSS description gives this formula for the noise in RA or Dec for point sources:

sigma = BeamSize / (SNR * sqrt(2*ln 2))

The BeamSize is the FWHM (45") and the SNR is the signal-to-noise ratio. The median NVSS rms noise for these matched sources is 0.47 mJy. Note that this noise equation already has been increased by an emprical factor of sqrt(2) compared with the theoretical equation "to adjust the errors into agreement with the more accurate FIRST positions". This predicts sigma ~ 7.6 arcsec at the catalog detection limit (SNR=5) and sigma ~ 1 arcsec at a flux density of 18 mJy.

To test this, the first plot shows the position difference between the NVSS and FIRST positions as a function of the NVSS flux density. The positional differences do tend to decrease as the flux densities increase. There are 3 lines on the plot:

Blue line (bottom): Shows the above sigma value, simply assuming that all objects are point sources with the median NVSS rms value. Most of the points have separations far above this line. That's because (1) the above value is a 1-dimensional uncertainty, so it must be increased by another factor sqrt(2) to account for noise in both RA and Dec, and (2) in a 2-D distribution, many values will scatter outside the 1-sigma circle.

Red line (middle): Shows the 90% confidence separation limit computed from sigma. This is a constant factor sqrt(2*ln 10) times sigma. With that increase it is necessary to go all the way to 40 mJy (SNR=85) to get the predicted separation error down to 1 arcsec. This line looks better compared with the data; however, there are still a lot of points above the line.

Green histogram (top): Shows the empirical 90% confidence separation as a function of flux density, computed by determining the 90th percentile of the actual separations in each bin. Note that this is relatively flat all the way past 100 mJy, and it is much larger than the predicted 90% curve.

The second plot shows the same quantities with a linear separation scale to make it easier to see the value of the empirically measured 90% separations.

This empirical distribution does not look like the theoretical distribution. Remember that there is an optical counterpart near zero in these plots for every source. To find 90% of those counterparts using the NVSS positions, you need to use a matching radius of approximately 20 arcsec even for sources that are 100 times the rms noise level. The theoretical SNR model predicts that the positions ought to be much more accurate than that (sigma = 0.4", 90% radius = 0.8").

Why are the positions so inaccurate compared with the SNR model predictions? Real radio sources are not symmetrical objects. They have lobes, jets, cores; star-forming galaxies have spiral arms. And there can be confusion where multiple radio sources get mixed together in the low resolution beam. With a low resolution survey, you do indeed get a very accurate measurement of the mean flux-weighted position as the SNR increases. However, the flux-weighted centroid is not where the optical counterpart lies. In many cases the counterpart is associated with some sharp structure within the radio source, and that structure may be far from the flux-weighted center.

The bottom line, again: You really do need high resolution images to identify counterparts to radio sources. Getting higher SNR with a low resolution image simply is not nearly equivalent to having high resolution images and positions.

Yes, I know the source sizes are going to be different at sub-mJy levels. But the confusion from nearby sources is going to be much more important there. The point is this: The argument that "we don't need a high resolution survey because the rms noise is going to be so good in the low resolution survey" is simply wrong!

From this test, matching to the 45" resolution NVSS requires a matching radius of 20" = 40% of the NVSS FWHM resolution. Our experience with the FIRST survey was similar: to get a reasonably complete list of optical identifications we had to use a matching radius of 2" ~ 40% of the FIRST FWHM resolution. I argue that is a universal requirement for radio sources, at least for sources down to the sub-mJy regime: the matching radius that is required for realistic radio source morphologies is 40% of the FWHM resolution.

According to Matt Jarvis, the resolution for WODAN (which will survey the northern sky accessible to the JVLA) is of order 15x17 arcsec, while the resolution for ASKAP is 10 arcsec. WODAN will therefore require an optical matching radius of 6x7 arcsec and ASKAP will require 4 arcsec.

So is that a problem? Yes, it absolutely is. A cross-match between SDSS and FIRST shows that 34% of FIRST sources have a false (chance) SDSS counterpart within 6.5". For comparison, 33% of FIRST sources have a true match within 2". So half the optical counterparts at SDSS depth will be false matches when using a 6.5" matching radius. Of course you can reduce this somewhat by doing a careful analysis of the likelihood of association as a function of separation, but when the starting point is a sample that is 50% real matches and 50% junk, the final list of identifications is not going to be complete and reliable.

And the false matching problem will only get worse for deeper optical/IR data. For example, Pan-STARRS is about 1 magnitude fainter than SDSS in the red and also goes into the Galactic plane where the source density is much higher, so it demands better resolution than FIRST.

We need high resolution to get the accurate positions required for optical identifications. Deeper imaging is not a substitute.