## Signal-to-noise as a function of spectral index

rlw@stsci.edu
Posts: 13
Joined: Mon Mar 03, 2014 6:42 pm

### Signal-to-noise as a function of spectral index

The signal-to-noise (SNR) for detecting a source in a VLA sky survey depends on the spectral index α, where F(ν) ~ ν**α. The dependence on spectral index is a bit more complicated than you might think because the effective exposure time is a function of frequency in wide-band sky surveys. In either a pointed survey or an OTF survey, the exposure time at any point in the sky is directly proportional to the primary beam area at that frequency. At the low-frequency end of the band, where the primary beam is large, there is a lot of overlap between adjacent pointings and the exposure time is large. At the high-frequency end, where the primary beam is much smaller, the beams from adjacent pointings overlap minimally and the effective exposure time is shorter. Since the radius of the primary beam is inversely proportional to frequency, the PB area and the effective exposure time both go as 1/ν².

Note that the VLA exposure calculator calculates the noise for a flat-spectrum source (α=0) at the center of a single pointing. So it assumes that the exposure time is the same across the bandpass. That means the exposure calculator does not give appropriate rms values for a mosaicked survey.

However, the rms from the exposure calculator can be used as the basis of a SNR calculation that includes the 1/ν² variation in exposure time for a survey. Assume that the intrinsic receiver noise is constant across the bandpass. (An rms that varies with frequency can also be included but is more complicated, so I'm not doing that at the moment.) In the continuum limit, the flux density averaged across the bandpass is

dν F(ν) / Δν

where Δν=ν2-ν1 is the bandpass range. Then using propagation of errors, the noise σ(ν) = σ sqrt(Δν), where σ is the rms value from the VLA exposure calculator.

The spectrum of a source, F(ν) = F0 (ν/ν0)**α, determines the SNR at each frequency. The optimal frequency-dependent weight to compute the flux F0at the reference frequency is W(ν) = SNR(ν)². Propagation of errors can be used to calculate the noise in the estimate of F0. That leads to this expression for the SNR of the detection of a source of flux F0 with spectral index α:

SNR = (F0/σ) xc g(3α-1,x1,x2) / sqrt(Δx g(4α-1,x1,x2))

where
F0 = flux at reference frequency ν0
α = spectral index
σ = rms noise from VLA exposure calculator
x = ν/ν0 is a normalized frequency
x1 = ν1/ν0 lower edge of bandpass
x2 = ν2/ν0 upper edge of bandpass
xc = (x1+x2)/2
Δx = x2 - x1

and the function g is defined as

g(p,x1,x2) = dx x**(p-1) = (x2**p - x1**p)/p

The plots below show the SNR as a function of spectral index for the S-band (2-4 GHz) and the Ku-band (12-18 GHz). Both plots use a reference frequency ν0 in the middle of the band (3 GHz for S, 15 GHz for Ku). The Ku-band dependence with spectral index is flatter than S-band because the Ku band covers a narrower fractional bandwidth (less than the factor of 2 for S-band). Since these plots show SNR, a larger value is better.

Note that the above plots, while useful for assessing how sensitivity to sources varies within the survey, do not allow a direct comparison of the sensitivity of the two bands. That's because they use different reference frequencies for the spectra. Clearly for a given inverted spectrum source (say α=+2), the Ku-band sensitivity is greatly enhanced compared to S-band because the flux density is 5**α=25 times greater at 15 GHz than at 3 GHz.

But the above equations can be used to determine the sensitivity of surveys in different frequency bands to the properties of the source populations. The keys to doing this comparison are (1) use the same reference frequency ν0 for all the bands being compared, and (2) adjust the SNR estimates by multiplying by the square root of the survey speed from Table 1 of Steve Myers' VLA Sky Survey Prospectus white paper. Scaling by the survey speed accounts for the fact that the lower frequencies cover sky much more quickly, enabling longer exposure times.

The figure below shows the scaled sensitivity as a function of spectral index for all the L, S, C, and Ku bands. The reference frequency is ν0=4 GHz. Changing the reference frequency would skew all the lines but would leave the relative ratios for different bands unchanged at each spectral index.

Note that the y-axis showing the SNR is a log scale. For a spectral index α=0 the sensitivity of the different bands is essentially set by the survey speed (which measures the rate at which sky is covered to a rms depth of 100 μJy). But for higher and lower spectral indices, the variations in sensitivity are enormous, exceeding a factor of 30 between L and Ku band at α=-1 and α=+2.

The table below gives the values for the curves plotted in the above figure.

Code: Select all

`  α     L     S     C     Ku-1.00  9.89  5.39  1.49  0.28-0.75  7.47  4.84  1.66  0.39-0.50  5.66  4.37  1.84  0.54-0.25  4.31  3.95  2.05  0.74+0.00  3.28  3.58  2.29  1.02+0.25  2.51  3.26  2.56  1.41+0.50  1.92  2.97  2.86  1.95+0.75  1.48  2.71  3.20  2.69+1.00  1.14  2.49  3.59  3.73+1.25  0.88  2.29  4.03  5.16+1.50  0.69  2.12  4.54  7.16+1.75  0.54  1.97  5.11  9.95+2.00  0.42  1.84  5.77 13.84`

claw
Posts: 26
Joined: Mon Mar 03, 2014 7:18 pm

### Re: Signal-to-noise as a function of spectral index

Great stuff again, Rick.
For context, it would be nice to overplot a histogram of the spectral index of all FIRST/NVSS sources. Do you know the best reference for such a data product?

casey

rlw@stsci.edu
Posts: 13
Joined: Mon Mar 03, 2014 6:42 pm

### Re: Signal-to-noise as a function of spectral index

claw wrote:Great stuff again, Rick.
For context, it would be nice to overplot a histogram of the spectral index of all FIRST/NVSS sources. Do you know the best reference for such a data product?

I don't! Of course FIRST and NVSS could not measure spectral indices (that's one of the nice things that will come out of a VLASS). I know there is information available, I just don't know off-hand what the best source is.

I agree, it would be an interesting comparison. Given such a distribution it would be straightforward to integrate it over the sensitivity to estimate the source counts and observed spectral index distribution for the surveys. (Although there will be additional complications from correlations between spectral indices and the presence of extended emission.)

It would also be helpful to have the SI distribution for compact sources in the Galactic plane, which is certainly different than the extragalactic population.

claw
Posts: 26
Joined: Mon Mar 03, 2014 7:18 pm

### Re: Signal-to-noise as a function of spectral index

After a quick look, one of the best is from Kimball & Ivezic (2008): http://iopscience.iop.org/1538-3881/136 ... _2_684.pdf. They construct a master catalog of FIRST, NVSS, GB6, WENSS, and SDSS. They include a number of tests and discriminations of morphology, etc.
The most relevant plot for our discussion is this:
spectral index histogram of FIRST/NVSS sources from Kimball & Ivezic (2008)
Screen Shot 2014-05-05 at 10.20.50 AM.png (18.77 KiB) Viewed 3205 times

This is the histogram of mean spectral index from WENSS through GB6, so from 92 to 6 cm (from the top left panel of Figure 14 from that paper). The most obvious selection effect here is that it is in the overlap region of all three surveys, which is a patch of extragalactic sky. It covers 2955 sq deg from Dec from 30--60 deg and RA from 150--250 deg.

casey