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I would like to calculate the Signal-to-Noise ratio (SNR) of long-slit or integral field unit (IFU) observations of H$\alpha$ emission. I can calculate the SNR or each individual spectrum, just fine. However, how can I assign a single SNR value to an observation? Do I just quote the higher SNR or do I calculate the median/mean SNR of all spectra? Is there a convention for this? My data is low-redshift galaxy IFU observations, and thus the SNR is higher in the center of the galaxy and lower at larger radii.

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I'd say quote the central value (and indicate whether this is S/N per Angstrom or whatever spectral binning you're using). If you bin the rest of your IFU data spatially, then you can quote whatever limiting S/N you used to determine the binning (e.g., "we binned the data spatially using Voronoi binning so that each bin had a minimum S/N of $x$").

Quoting a mean or median S/N is sensible if you spatially bin your spectra, so that the (small) central bin and the (large) outer bins have roughly similar S/N values. Or you could describe how the S/N varies as you go out from the galaxy center (some people even produce 2D maps of the S/N for their IFU data).

If you have lots of galaxies to talk about, I'd probably just quote the central spectral S/N.

[Edited to clarify the "mean" S/N idea.]

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Unless you in fact find a "standard" definition for the mean or collective SNR, there is no way to answer this. As you noted, the signal strength varies with view angle (as well as wavelength), so there's really no single value. If you have a defined algorithm which processes all your spectral data to produce a final image or analysis value, then you can use standard statistical techniques to calculate the SNR of the output as a function of the SNR of all the contributing factors. (Here "factor" means each independent input variable, i.e. spectral line). I recommend getting a copy of Bevington to learn how to do this. Note- that appears to be a not-necessarily-legal-copy; you can purchase the real thing anywhere, e.g. amazon

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There are different types of signal to noise ratios ($S/N$) that you might need to consider.

  1. Read Noise dominated $S/N$

    • Extremely faint sources and background
    • Or a really poor detector

      When you are RN dominated

      • $N\approx\sigma_{\rm RN}$

which means that your $S/N$ is proportionate to the exposure time $t$. $$\frac{S}{N}\Bigg|_{\rm RN}=\frac{f_{\lambda}A_{\rm eff}\Delta\lambda\eta_{\lambda}/h\nu)t}{\sigma_{\rm RN}} $$

where $f$ is the specific flux, $A$ is the area of the primary mirror area, $\Delta\lambda$ is the wavelength range of your filter/optics, $\eta$ is the end-to-end efficiency of the system, $h\nu$ is your energy, $t$ is the exposure time, and $\sigma_{\rm RN}$ is the read noise error.

  1. Source dominated $S/N$
    • Source count rate significantly exceeds that of all others
    • Typically for bright sources
      • $N\approx\sigma_{\rm obj}$

Here your $S/N$ is proportional to $(f_{\lambda}t)^{1/2}$ $$\frac{S}{N}\Bigg|_{\rm obj}=\sqrt{\frac{f_{\lambda}A_{\rm eff}\Delta\lambda\eta_{\lambda}t}{h\nu}} $$

  1. Background dominated $S/N$
    • Common for faint sources or very bright backgrounds
    • $N\propto\sigma_{\rm sky}$

like the previous, $S/N$ is proportional to $t^{1/2}$ $$\frac{S}{N}\Bigg|_{\rm obj}=\frac{f_{\lambda}t^{1/2}\sqrt{A_{\rm eff}\Delta\lambda\eta_{\lambda}}}{\sqrt{\mu_{\lambda}A_B}} $$

Where $\mu$ is surface brightness and $A_B$ is the background area.


Another option, which might be a TON easier, depending upon the reason you're trying to calculate the $S/N$...

For HST and Keck, there is software on their respective sites, that estimates the $S/N$ for an observation, is intended to capture all of the factors for a given instrument+telescope (ie: $\eta_{\lambda}$, $A_{\rm eff}$).

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