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Suppose you took say 20 images fairly close together timewise for a variable star which had a longish period. To get a value for the apparent magnitude you could either stack the 20 images and then do some photometry on the single stacked image. Or you could do some photometry on each of the 20 images so that you had 20 estimates of the magnitude and then average the 20 values to get a final value.  Which method would be better?

 

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I would say - do 20 measurements rather than stack and do one measurement.

Result will be the same, but when doing 20 measurements - you can also do standard deviation to see level of error for your average value.

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I just tried this with 26 images. The magnitude from the stacked image was 11.499 the average magnitude from 26 images was 12.305 with a standard deviation of 0.15. The attached chart shows how the magnitude varied. The scale is magnified so it looks worse than it really is.  It looks like the last few samples went a bit wrong.  My feeling is that it's better to take the average rather than stack.  The star has a variation of 7.9 to 17.3 VV in magnitude and the period is 608 days so I wouldn't expect to see any change over the 30 minutes it took to take the pictures.

This was my first attempt and I can already see ways of improving my technique. So that's something to work on. I think I overdid the defocussing. Next time I'll just try to get good focus and see what happens.

2020-12-15-16-21-24.jpg

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Hi

I have to disagree with the consensus here. 

Unless you are planning to detect/remove outliers from a sequence of estimates, I don't think there is any statistical reason to prefer 20 measurements to one measurement, assuming the measurements are taken in one long sequence. That is, the statistics are the same. Essentially you are collecting the same photons whether they are collected in one long burst or N shorter bursts. Any benefit from the fact that you have 20 measurements and can compute a standard deviation is, I think, illusory. After all, why not take 20000 x 1ms measurements if having more measurements is beneficial?

Concerning the difference in magnitude estimates based on stacking and averaging, bear in mind that if you take the average of many magnitude estimates, depending on your software, you are taking the average of logarithmic-like quantities so they should be converted to linear units prior to averaging, then converted back to magnitudes. If you are computing the magnitude based on stacking, then (most likely) the stacking software is operating on linear quantities already before doing the final conversion to magnitude step. 

cheers

Martin

 

 

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11 minutes ago, Martin Meredith said:

That is, the statistics are the same

Average value is the same but statistics of it is not the same.

11 minutes ago, Martin Meredith said:

Any benefit from the fact that you have 20 measurements and can compute a standard deviation is, I think, illusory. After all, why not take 20000 x 1ms measurements if having more measurements is beneficial?

Let's say you have one readout based on stack. Stack is by the way just average value (or at least it should be if one wants controlled environment - no alignment / no pixel correlation and such - shift and add at best).

This value is some number 120, for example. How good is your measurement? Can you tell from that single number? What is probability that true value is measured value +/- 5 for example - or in rage of 115-125, can you tell?

Can you trace the source of system error if you just stack and have single value?

In this particular case, each measurement introduces some noise into result and there is a difference between 20000 x 1ms and 20 x 1s and 1 x 20s in achieved SNR, but depending on level of this noise, you might opt to actually do 200 x 0.1s measurements and then decide how much you want to stack and how much you want to save for statistics.

You can either have 20 x 1s samples or 40 x 0.5s samples.

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When you chop a sequence into N parts and measure standard deviation, you are measuring the standard deviation of a short unit (ie you are measuring statistics of short units) You are not measuring the standard deviation of the longer unit. Knowing about the standard deviation of a bunch of 1s subs does not provide any additional information concerning the equivalent 10s sub. It tells you how reliable the mean of 1s subs is.

So to answer your question of how do you know how good is your measurement: yes, you need to take multiple exposures. But multiple exposures of the same length. Simply chopping it up only tells you how good your measurement of the shorter length is.

Now in practice, if there are external factors that are likely to affect the value of the final measurement, such as clouds, satellite trails etc, of course it is worth breaking the measurement up and then rejecting outliers, hence my caveat above. But if there are no such external factors operating you are not gaining any information by splitting the longer sequence up.

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1 minute ago, Martin Meredith said:

When you chop a sequence into N parts and measure standard deviation, you are measuring the standard deviation of a short unit (ie you are measuring statistics of short units) You are not measuring the standard deviation of the longer unit. Knowing about the standard deviation of a bunch of 1s subs does not provide any additional information concerning the equivalent 10s sub. It tells you how reliable the mean of 1s subs is.

Mean of 10 x 1s has the same SNR as single 10s unit (with exception of read noise contribution).

You can figure out how big your measurement error is from 10 x 1s, but you can't do that on a single 10s measurement.

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What we have here is already several separate measurements, they have be taken and can't be 'adjusted', so we are working with fixed data. All I am saying is that just stacking ('averaging') all the results together may loose some useful information that may become apparent by analysis the individual measurements.

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Yes, you can figure out how big the measurement error is, but the measurement error is for 1s subs. In the absence of a model for what is going on, this doesn't tell us anything about measurement error for 10s subs (which is what we want, I assume)? That is, your 'additional statistical data' is for 1s subs, not for 10s subs. 

In this specific case we do have such a model (photon statistics) so we can generalise  to 10s subs. But since we have such a model anyway, we don't gain any information from taking 1s subs. (repeating again the caveat that of course having 1s subs allows us to do outlier removal).

Consider counting raindrops falling into the famous bucket. You measure every 1s and get a series of values. You average them and look at the standard deviation and work out the measurement error for 1s measurements of rain falling into a bucket. If you take a single 10s measurement, sure, you don't get a measurement error for 1s intervals, but why would you want to if you are taking a 10s measurement? If you want to know its measurement error, you take multiple 10s measurements. Its 1s apples versus 10s oranges. 

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5 minutes ago, Martin Meredith said:

Yes, you can figure out how big the measurement error is, but the measurement error is for 1s subs. In the absence of a model for what is going on, this doesn't tell us anything about measurement error for 10s subs (which is what we want, I assume)? That is, your 'additional statistical data' is for 1s subs, not for 10s subs. 

We started here with a series of measurements and question was:

Should we stack images representing measurements (which is same as averaging / summing them) and then reading off value, or should we read off individual value from each of measurement and then average those?

From perspective of actual resulting value - there is no difference, both methods will give the same result.

Reading off individual values and then calculating average but also other statistical tests will tell us something about that data set - even if we don't know exact model (we can perform different tests to see if data fits one distribution better than others). Having single number will tell us no such thing.

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