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Shot Noise -another variable?


Merlin66

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Maybe because it is an 'unavoidable' noise?? There is nothing you can do about it; it's purely a function of the signal level (shot noise = sqrt(signal) for a poissonian distribution such as photon flux levels).

Of course, understanding that would be useful for people, so it's surprising it is not covered.

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Maybe because it is an 'unavoidable' noise?? There is nothing you can do about it; it's purely a function of the signal level (shot noise = sqrt(signal) for a poissonian distribution such as photon flux levels).

Of course, understanding that would be useful for people, so it's surprising it is not covered.

Well, it is true that you can't do anything about it in the strict sense but by ensuring that your exposures are as long as possible (without saturating) you can reduce its relative contribution - or such is my understanding.

Olly

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I understand now the random "arrival" distribution of the photons, and that a stack of images (ie 10 x 3 mins) will never achieve the same total "collection" of photons as a single long exposure (ie 1 x 30mins) but in the final image with darks/ flats how much does the SNR vary between them. Does the stacking give a lower signal but a better SNR?

I'm taking a series of images to see the results/ differences.

Ken

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Thats my understanding Ken. Multiple shots increases the SNR allowing you to pull more out from the stack. A single long exposure would have a lot more random noise so you would not be able to stretch as much BUT you would not need to stretch as much as the data signal would be so much stronger.

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Has anyone actually measured the comparison SNR between a single long exposure and a series of shorter exposures adding up to the same total exposure time Ie 10 x 3min comparated with 1 x 30min.

( After the usual dark/flat corrections)

Only interested in the final measured SNR. ( this is for a spectroscope application which needs maximimum SNR outcome)

Any help appreciated.

Ken

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The flux (and hence the shot noise) you get from 10x3min exposures is exactly the same as the flux (and shot noise) you get from 1x30min exposure. There is no difference at all.

The advantage of fewer longer exposures over more shorter exposures is that each exposure has a fixed amount of noise added by the readout (read-noise). You get this each time you read the detector, so obviously 1x RN is better than 30x RN (though noise sums in quadrature, so actually it is sqrt(30)x RN). However, that affect is usually overestimated by people I think.

Depending on your imaging setup, the contribution of readnoise is or isn't significant. You want to make sure your exposures are long enough that uncontrollable sources of noise (i.e. shot noise from the sky background) dominate over controllable sources of noise such as read-noise. Once this is the case, you get no (or at least very minimal) advantage from taking longer exposures.

Most amateur CCDs have readnoises of 5-10e-. That is equivalent to the shot noise from a sky background of 25-100e-/pixel (noise = sqrt(signal)) . So, if you sky background is much larger than this (say 1000e-/pixel), there is virtually no noise penalty for reading out your CCD more frequently (and huge benefits, because you're less sensitivity to guiding errors, clouds, tripping over the power cable, etc...).

Unless you have a really dark site, you'll probably find your into this regime with a minute or so for broad-band imaging. (I know from my town centre site, R band images are sky-noise limited within 30 seconds -- so there is no point taking individual exposures longer than this. I might as well take lots of 30s images and add them up afterwards.)

For spectroscopy, readnoise is typically very important, because you are spreading the light out so much that the noise contribution from the sky is typically very small. The same is true for narrow band imaging.

Contribution of noise from flats and darks. Hmmm, trickier this one. In theory, you should be able to make a 'perfect' flat and dark which does not add to the noise. In practice you can never do that, and you will always add some noise. However, you're applying the *same* noise to each frame, so the result is correlated and then it all gets a bit more tricky... Will need to think more about that one (been up too long now!).

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Interesting feedback...

The claim has been strongly made to me that due to the shot noise it's physically impossible to obtain the same flux intensity form a stack of shorter exposures ie 30 x 1min is not equal to 1 x 30min.

I could accept some loss of flux if the SNR was improved....

With the spectrum there's effectively no background to worry about the star on the slit and the slit gap determine the light being dispersed into the spectrum......so no sky fogging, light pollution etc etc just a noisy spectrum....

Hopefully the tests will show something....

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Only interested in the final measured SNR. ( this is for a spectroscope application which needs maximimum SNR outcome)

The equation for SNR is (not sure the equation will be too clear to read here!)

SNR = Signal / SQRT(Signal + Background + RN^2)

So, if you want to do it for given number of exposure (nexp), each of a given exposure time (exptime in seconds), you get;

SNR = nexp*exptime*Flux / SQRT(nexp*(exptime*(Flux + BGFlux) + RN^2))

where Flux and BGFlux are the values per second.

So, as long as;

exptime*(Flux + BGFlux + DC) >> RN^2

the value of nexp*exptime is the dominant factor, and it doesn't matter what the individual value of nexp or exptime are...

Very happy to discuss S:N estimation for spectrographs with you Ken -- it's a level more complicated than for imagers. May be easier to do offline. PM me if you're interested.

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The claim has been strongly made to me that due to the shot noise it's physically impossible to obtain the same flux intensity form a stack of shorter exposures ie 30 x 1min is not equal to 1 x 30min.

Well with respect to whoever made the claim, they are plain wrong. The shutter is open for the same amount of time in total. The same total number of photons fall onto the detector.

Another way to think about this is to take it to the other extreme, and ask what happens for very very short exposures. For example, if you have a photon counting detector with no readnoise, you could have an exposure time so short (1 millisecond, say) that you expect far fewer than 1 photon per exposure. In most exposures, I detect 0 photons. In some exposures I detect 1 photon. I can count all of these up afterwards, and see how many I detected in one million short exposures (1 second, in total). Why would I detect fewer photons this way, than if I just let the detector do the counting for 1 second??

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If the "Shot Noise" (which until this point I'd never heard of) is random, isn't that exactly the type of noise that the stacking process is supposed to remove / reduce?

Please have the conversation on the S:N estimation for spectrographs on the boards as I'd be interested in reading it!

Cheers

Ant

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However, you're applying the *same* noise to each frame, so the result is correlated and then it all gets a bit more tricky...
In fact it is the same as applying individual flats (or darks) to each frame.

So let us say (for simplicity) you take 30 lights and 30 darks. Then most people will stack their darks to form a master dark and subtract this from each light, then stack the lights. But this is the same operation mathematically as subtracting a different one of the darks from each light, then stacking the resulting 30 frames.

NigelM

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Well with respect to whoever made the claim, they are plain wrong. The shutter is open for the same amount of time in total. The same total number of photons fall onto the detector.
Absolutely correct! I see these mistaken claims over and over again on astro forums - usually expressed in the form that you 'need a long enough sub to get at least one photon otherwise it doesn't matter how many subs you add up'.

NigelM

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NigelM made this point elsewhere, and it's well taken. But it does only hold if you are going to be strictly averaging all the frames, does it not? If you do a kappa-sigma stack, it's not as straightforward, I believe.

If the "Shot Noise" (which until this point I'd never heard of) is random, isn't that exactly the type of noise that the stacking process is supposed to remove / reduce?

Yes, that's my understanding. We are faking a longer exposure which will have higher signal, higher noise but a better signal/noise ratio.

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I feel based on the information that's being presented, that the noise calculations of 1 dark/ 1 light combo then stacked v's the master dark from each that the master dark (as it contains the "noise" from all of the 30 images in the satck) subtracted from EACH light then subsequently stacked will/ would infact add MORE noise to the resulting image!!!

I have some data and need the time tonight to come up with some hard data based on:

1 Raw sum of light stacks ( say 5, 10, 15) v's a 1 x total iro SRN

2. Stack of lights with dark correction ( 1 frame per) v's 1 x total - 1 dark of the same exposure)

3. As per 2 above but using say a Master dark for each exposure time based on 10 darks/ exposure.

Are there any other variations suggested?

I will use Vspec function for SNR measured on a flat continuum part of the spectrum as the "definative" SNR achieved.....

(I should have said using the same ambient temp etc etc)

Ken

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If the "Shot Noise" (which until this point I'd never heard of) is random, isn't that exactly the type of noise that the stacking process is supposed to remove / reduce?

Stacking (i.e summing images together) does not reduce shot noise, but increases the signal-to-noise by increasing the signal faster than the shot noise increases. The actual shot noise in the stack will always be higher than in an individual sub.

I think some confusion arises because in practice "stacking" usually implies averaging as well. But averaging is essentially just dividing by a constant (assuming you aren't sigma clipping or somesuch) and has no physics in it.

NigelM

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If you do a kappa-sigma stack, it's not as straightforward, I believe.

Clipping is done to remove other sorts of noise (non-Poisson). You would never need to clip if your noise was entirely shot noise (you would end up making the signal-to-noise worse).

NigelM

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You need to express it algebraically for me to understand the difference. L1,L2,..,L15 are the lights (L is the average light) and D1,D2,...D15 are the darks (D is the average dark)

(L1-D1) + (L2-D2) + ...+ (L15 -D15) = L1+L2+...+L15 - (D1+D2+..+D15) = (L1-D) + (L2-D) + ... + (L15-D) = 15* (L-D)

In this simple situation all these operations "commute" (stack then calibrate, calibrate then stack)

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Guys,

I think for the benefit of the onlookers maybe we should explain a bit.

What we are talking about/ discussing/ debating is a mathematical model of "noise" in a image.

In a typical astrophotographic image, processed with darks/ flats etc, after stacking they look relatively smooth and sexy.... the way they should. That won't change!

Ken

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Samir does a good job showing the results but he never quantifies the SNR in the images....

Signal to Noise and Subexposure Calculations

The associated SNR write-up is also targeting AP - the graph of SNR only goes up to about 30; for ProAm spectrum data submission I need a Minimum of SNR=50 and SNR=100 for "good data"

A different league from "Ohh-Ahhh" AP images.

Ken

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for ProAm spectrum data submission I need a Minimum of SNR=50 and SNR=100 for "good data"

So this tells you immediately that you need a minimum (total) flux of at least 2500 to 10000 counts in your data. Whether you get those counts in a single integration, or by summing/averaging many doesn't matter. For this many counts, the shot noise alone from the photons will limit you to SNR=50 or 100 respectively. This is known as the 'photon noise limit' -- the best you can ever do with a given set of data. Any other noise sources (RN, flats, darks, etc) will reduce this SNR, and the holy grail of data reduction is to calibrate those other noise sources so well that you can reach the 'photon noise limit'.

Are you measuring this SNR per pixel or per resolution element? Another thing to consider for spectroscopy; and there isn't a generally accepted consistent 'standard' unfortunately ;)

BTW -- it's not unusual for the most highly cited papers to come from the lowest S:N data; the faintest objects are almost always the most interesting! Depends on your science goal of course :)

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Thanks for that!

The SNR is usually measured (Using Vspec) "on an appropriate section of the continuum containing no emission or absorption features" - this is usually a section of continuum about 50-100 A wide along side the "feature of interest".

ken

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