I wanted to understand better the way the ensemble reference star system worked. I took a star and selected 5 reference stars from the chart. Then I analyzed it using the ensemble of the 5 reference stars. Then I did it again 5 times each with just a single individual reference star. I was expecting that the ensemble magnitude would be around about the average of the 5 individual magnitudes but it didn't turn out like that. The ensemble magnitude was 10.323, the individual magnitude weres 10.319, 10.320, 10.343, 10.365, 10.369 giving an average of 10.342. So the ensemble magnitude was close to the two lowest valued magnitudes. I thought it might be something to do with the SNR of the reference stars but the two (10.319 and 10.320) had the lowest SNRs by quite a long way. I wondered if anyone knows what's going on here.
I think it would be helpful if you provided the info for the five individual comp magnitudes (magnitude, snr and color), that yielded the corresponding target magnitudes given above. Comp color vs target color will impact the calculated target magnitude. Any weighting of the comps would also impact the mean target mag.
What software did you use for these calculations? Did you use equal weighting or snr weighting?
Thanks Ken, I'm using Maxim DL. I found a document in its help system which details the calculations and I'm working my way through that at the moment. I've set it up in excel so I can see what's going on and I get the same result as MaximDL which is encouraging. It does a weighting based on SNR. The final equation is like this -
V = w1xV1/W + w2xV2/W + w3xV3/W + w4xV4/W + w5xV5/W
V is the final magnitude.
w1,w2,w3,w4,w5 are the uncertainties for each of the reference stars
W= w1+w2+w3+w4+w4 (total uncertainty)
SO the weightings are w1/W, w2/W, w3/W, w4/W, w5/W (which add up to 1 as you'd expect)
V1, V2, V3, V4, V5 are the magnitudes calculated for the object star from each of the reference stars.
I can now see where I don't understand it. The uncertainties (w1.. etc) are basically 1/SNR so the smaller the SNR the bigger the uncertainty and the bigger the uncertainty the bigger the weighting. To my brain this seems counter-intuitive. I would have expected that it would give greater weighting to reference stars which had a higher SNR and therefore lower uncertainty.
Check the definition of 'weight' wi.
I believe 'weight' is generally not defined as the uncertainty (sigma) BUT 1/sigma^2.
Reference stars with the higher SNR have a smaller error/sigma (1/SNR) but this yields a larger weight by this definition.
However, beware that this MAY NOT always be correct IF the accuracy of the brighter comp is POOR for some unrelated systematic error. In such cases, an equal weight MAY give a more accurate mean? Despite what I just warned, I normally use a weighted mean calculation.
Remember that precision and accuracy are NOT the same thing!
Ken, this is a link to the MaximDL photometry calculations-
You are correct about the weight. The weight factor is equal to the uncertainty squared. But the uncertainty is equal to (2.5 log10(e)) / SNR. It means that the weight factor increases with lower SNR. Using MAXIMDL to measure the SNR of the reference stars I find that the brighter the star the higher the SNR. That always seems to be the case whatever star I'm looking at. The table below shows the chart reference, magnitude and SNR of my 5 reference stars.
101 10.095 1851
106 10.608 1285
109 10.856 1093
115 11.452 639
120 12.007 386
The reference and magnitude come from the chart and the SNR is measured by MAXIMDL.
Because the weighting depends on SNR squared the result is heavily weighted towards the two dimmest references 115 and 120. That doesn't seem right. I'm thinking that an unweighted average would be better.
I looked at the link (see below) but do not follow/agree with it? Someone might explain? I've seen different equations. Not sure what the first equation means? What is e?
<<Let the 1-sigma SNR-based uncertainty be defined as
Let be the weight factor arising from the SNR of reference star .
Let , where N is the number of reference stars.
Then the standard magnitude of the object is given by
Ken, I'm glad I'm not the only one puzzled. The link I gave is identical to the help information page available directly from MAXIMDL. As far as I can see e is just the mathematical e. It doesn't matter anyway because in the calculation for Vbar all those 2.5log10(e) terms cancel out. I've reproduced the calculation in excel and I get exactly the same result as MAXIMDL so it appears that those are the calculations that maximdl is using even if they are wrong.
Again, we don't have the details, but it almost sounds as though, given the offsets when using the comp stars separately, that the adopted magnitudes of the stars themselves have some jitter in their zero-points. Often the magnitudes shown via AAVSO charts have nominal uncertainties of ~0.05 mag, so simply picking _one_ of those stars versus another could give the offsets described. Ken notes the possible color issue, and perhaps one or more of the comps is near saturation, or some other problem(s) may be involved.
Just to convince myself of the expected target magnitude on a single image, I went through the calculation of a target magnitude based on an ensemble of four equally weighted (i.e. unweighted) comps in VPhot compared to the calculation of a target magnitude based on the unweighted (i.e. equally weighted) mean from the same four individual/single comps in VPhot.
As expected/should be true, the mean target mag was identical in both procedures in my experiment. It doesn't matter if the comp errors are different or the comp colors are different because the SAME comps are used in both procedures.
The only reason I can understand the different target mag mean results you obtained with your target is IF the estimated target magnitudes based on the ensemble of identical comps are NOT weighted equally.
Since you observed that MaximDL does weight the ensemble comps on the basis of their individual uncertainties, I think that explains the difference you found for the mean target mags.
You could confirm this if you can force MaximDL to weight the ensemble comps equally? I would think this is possible?
Were the magnitudes transformed?
Mention has been made of the colours of the stars. Differences among B-V values for the variable and the various comp stars would be more important if the results were non-transformed. Tv_bv can be very small (e.g., -0.01 or less) and in this case would result in relatively small non-transformed V mag errors. But if Tv_bv is not very small (e.g., if it were closer to -0.05) it would clearly be a source of more obvious difference between non-transformed magnitudes determined from comp stars of varying colours.
This issue of course becomes much more important for photometry on images from DSLR and other colour CMOS cameras, where transformation coefficients are much larger than the example values quoted for Johnson V filters above.
Roy, they were not transformed and I'm using a V filter on a monochrome CMOS camera. I'm hoping to start doing transforms this season. I understand the issue about the differences in B-V between the stars but even allowing for that it still seems wrong that the dimmer stars end up with a higher weighting because of their lower SNR. Suppose those stars had exactly the same spectrum. It would still be the case that stars with low SNR have a higher weighting. More so because the weighting is based on SNR squared which amplifies the difference.
Yes, I know my…
Yes, I know my question was not specifically related to your issue, but could be a possible reason for the range of mags determined from the individual comp stars. I understand that your concern re SNR and weightings is a different matter, and agree that it would still arise with stars of the same colour but different SNRs.
Thanks Roy, you could be right. I'm looking at V0919 CEP which has spectral type B8 which according to a list I find has a B-V value of -0.11. The reference stars have B-V values between 0.502 and 0.793. I'll see if there is any correlation between the measured magnitude and the B-V of the reference stars.
"I'll see if there is any correlation between the measured magnitude and the B-V of the reference stars."
A while ago I found it instructive to choose several pairs of phòtometric standards, or pairs of good comp stars, set one in each pair as the target and the other as the comp. After determining the magnitudes of the targets, plot the difference between measured and catalogue mag (y axis) against target - comp B-V difference (x axis).
For the various values of B-V difference, the corresponding y values will be the error due to that difference.