The Signal-To-Noise Ratio (SNR)

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Forum_2014
Posts: 266
Joined: 03 Dec 2018, 22:33

The Signal-To-Noise Ratio (SNR)

Post by Forum_2014 »

Because of the quantum nature of light, the detection of a photon by a CCD is always a random process dominated by a relationship from the Poisson statistic: the Poissonian distribution (after the French mathematician Simèon Poisson, 1781-1840).

The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a specified unit of space or time: it is like a Gaussian distribution (bell shaped) with the width determined by the square root of the total number of counts. When we try to measure a Poissonian event such as a photon detection with a CCD camera, the counting noise associated is given by the square root of the signal; then the noise N associated at a signal S is √S and the Signal to Noise Ratio (or S/N or SNR) is:
eq1.gif
This fundamental relation is the key to understanding how noise affect our observation: actually this relation shows the natural upper limit of the SNR. It makes no difference whether an object is bright or faint or whether we take long or short exposures with big or small telescopes: it all depends on how many photons (S) we are able to collect: the SNR will never been greater than √S .
Actually there are other sources of noise in a CCD image that keep the SNR of our object of interest lower than the theoretical limit of √S: here is a list of the most important of them:

• readout noise: is the number of electrons introduced per pixel into your final signal upon readout of the CCD device. Typical values in modern CCD are within the 10 electrons/pixel;
• dark count: is the number of thermal electrons generated per second per pixel at a specified temperature. Typical values are very few electrons or fraction of electrons with cooled CCD;
• background noise: this is not an instrumental noise but it is of great importance. Natural skyglow, moonlight and light pollution all contribute to the signal collected by the CCD but these are not increasing the signal of the object of our interest. Because this background is generated by photons and because all photons measurements have an inherent uncertainty, when we subtract an uncertain value to remove the unwanted signal we actually add more noise.
• processing noise: every time you make some basic image processing such as subtracting dark frames and flat fielding you are combining uncertain numbers with other uncertain numbers on a pixel-by-pixel basis. Since in general these are independent source of noise they add their own noise in quadrature. For example, if we have noise from three sources with values N1, N2 and N3, the total noise will be:
eq1a.GIF
Let’s make an example with some real numbers to see, for example, how much the background noise could affect the overall noise of an image and degrade the quality of the image itself.

We suppose, for simplicity, that the seeing is good enough to make our source of interest (a star) falls completely within 1 pixel.

The total signal collected by the pixel is of 900 counts: 400 of them are from the sky background and 500 are from the star’s light. The star plus sky combination has a noise of √900 = 30 counts on that pixel and this gives a SNR of 900/30 = 30. But this is not the correct way to evaluate the SNR. The signal of our source it is indeed of 500 counts, then a better estimate of the SNR is 500/√900 = 16.7. Actually, the true SNR is even lower: since we have no way of knowing that the sky background is exactly of 400 counts (the sky background itself is affected by the photon noise!) we should add the noise contribution from the sky (√400=20) to that of the star plus sky background (√400+500 =30 ), then we have:
eq1b.GIF
This means that we could measure the star’s brightness with a precision of ±1/13.8 or about ±7% .

Forum_2014
Posts: 266
Joined: 03 Dec 2018, 22:33

Re: The Signal-To-Noise Ratio (SNR)

Post by Forum_2014 »

But what happens if we try to take images of the same star under a bright urban sky where the background is, say, 5000 counts instead of 400?
eq1c.GIF
Now the star is barely visible, a typical SNR value for a detectable star is ≥ 3, and we can measure its brightness with a precision of ±20% only. In practice, however, other noise sources like readout noise and thermal noise will lower even more this value.
The example above explain why we can see fainter stars (and/or nebula details) under a darker sky: it’s purely a matter of SNR of an object seen against a bright background.

A CCD detector is an array of light-sensitive silicon cells. Each cell produces a signal that is converted into a number representing a pixel in the digital image. Taking a picture of the sky with a CCD is like to take a brightness map of the sky itself and the SNR (Signal to Noise Ratio) is a way to describe how good this map it is.
How could we define the SNR and measure it? One of the most used equations to describe the SNR is that of Merline & Howell (1995, Expt. Astron., 6, 163):
eq2.GIF
Where:

Ns total number of photons (signal )per pixel collected from the object of interest
NB total number of photons per pixel from the sky background
ND total number of dark current electrons per pixel
NR total number of electrons per pixel resulting from the readout noise
npix number of pixels involved in the calculus of each noise term
nB number of background pixels used to estimate the mean sky background level
G gain of the CCD (electrons/ADU)
σf2 Estimate o the 1sigma error introduced within the A/D converter: it depends on the actual internal electrical workings of a given A/D converter; its value is approximately of 0.289.

Then we can use the following simplified formula without making a big error:
eq3.GIF

Forum_2014
Posts: 266
Joined: 03 Dec 2018, 22:33

The ideal case

Post by Forum_2014 »

From the last equation we can derive again the SNR for the ideal CCD (no thermal noise, no readout noise, etc.) under the ideal sky (total darkness):
eq4.gif
The result is surprising, there is still some noise as predicted by the poisson theory in the first post. This noise is caused by randomness. For example, with 100 photons the noise is 10, so measuring the star you may get 92, 104, 88, 107 and so on.

To understand that, you may imagine that this is caused by photons trajectory, some of them will hit the CCD and some others will hit the mount, the finder, the tripod, etc. and this is a random process. (Actually this is not 100% correct, because light is also a wave and its interaction with matter is a random process too).

So, even with with a perfect CCD under a black sky we still need a long exposure, to be able to collect thousand of samples. For example, with 50000 photons the noise is 223, so we may expect an error of about +- 223/50000 = 0.4 % which may be acceptable. The error itself follows a gaussian distribution, so there may cases where the error is much higher than 0.4 %.

Please notice that the unit in this examples is photons, while in CCD images the unit is ADU. If your ideal camera produces 1 ADU every 4 photons then a star measured 25 ADU was sampled by 100 photons. The poisson noise is 10 photons, so 2.5 ADU.

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