Transformation for an image series

Science Photometry
1

I have a question about transformation.

Following the older CCDPhotometryGuide v1.1 2015, the transformation for an two color image series is as follows:

Without transformation
Vvar=Δv+Vcomp 1)

With transformation
Vvar=Δv + Tv_bv * Δ(B-V) + Vcomp 2)

Δ(B-V) is the difference in the standard color of the variable versus the standard color of the comparison star and is equal to Tbv * Δ(b-v). In other words, you can derive Δ(B-V) by multiplying your color transform by the measured color difference between the variable and comparison star, Δ(b-v). Then formula 2) can be written as:

Vvar=Δv + Tv_bv * Tbv * Δ(b-v) + Vcomp 3)

Vvar=Δv + Tv_bv * Tbv *((b-v)var - (b-v)comp) +Vcomp 4)

• (b−v)var−(b−v)comp: how different the instrumental colors of the variable and comparison star are.
• Tbv: converts instrumental color difference → standard color difference.
• Tv_bv: tells how much a difference in B ⁣− ⁣V shifts the V magnitude in the system.

Assume you try to correct the V value using above formula 4), you have to measure (b-v)var and (b-v)comp . The B and V filter are selected intermittent for the image series. The question is how to process the b-v values for an image series. There are two options:

A) For transformation correction for V-image, take the v and select the b magnitude of the B-image before or after the V-image.
B) Average the b and v values for all images and use the median, mean or sigma clipped value.

For A) there will be noise in the calculated b-v values.
For B) Does the changing air mass has an influence on the b-v value?

Which method is better, A) or B)?

Han

2

Now I think A) is better and do the following for an comparison ensemble(comp1, comp2, …):

  1. Pairing: For a V image, find a B image with the nearest timestamp.

  2. Instrumental Δv := -2.5 * log10(Flux_v_var/∑Flux_v_comp)

  3. Instrumental (b−v)var​ := -2.5 * log10(Flux_b_var / Flux_v_var)

  4. Instrumental (b-v)comp := -2.5 * log10(∑Flux_b_comp / ∑Flux_v_comp)

  5. Catalog Vcomp:= -2.5 * log10(∑(10^-0.4Vcatalog_comp))

Final transformation equation:

Vvar​=Δv+(Tv_bv​⋅Tbv​⋅[(b−v)var​−(b−v)comp​])+Vcomp​

You can combine Δv+ Vcomp and write it as:

Vvar​= -2.5 * log10( Flux_v_var*∑(10^-0.4Vcatalog_comp)/∑Flux_v_comp) +(Tv_bv​⋅Tbv​⋅[(b−v)var​−(b−v)comp​]) 5)

Where

Flux_v_var is the measured flux of the variable star.
Flux_v_comp are the measured flux values of the comparison stars.
Vcatalog_comp are the catalog magnitudes of the comparison stars.

Assuming the exposure duration time for B and V images is the same. Else the measured flux should be corrected linear with the exposure time.

Any comments?

3

I agree with your final transformation equation.

I can’t comment on the earlier parts of your post because I am not familiar with the conventions usually employed.

4

Going back to your post starting “Now I think A) is better…”, consider formula 2.

Δv = -2.5 * log10(Flux_v_var/ΣFlux_v_comp)

I question the last term ΣFlux_v_comp which sums the v fluxes of all comps.

Instead, surely it is necessary to calculate Δv for the var and each comp in turn, then sum the Δv values and (say) average them.

Similar logic should apply to formula 4. for the instrumental (b-v)comp, the necessity being to obtain the b-v for each comp prior to averaging the result (i.e.), summing the b-v values for averaging, rather than summing all the b’s and all the v’s.

5

No. What I wrote is not correct.

Surely the transformed magnitude must be calculated individually for each var / comp pair, and therefore no summing of fluxes should occur.

6

Thanks for the reply.

Yes using the individually var-pairs and comp-pairs seems the way to go, so A). A next step could be to smooth the individually calculated b-v values with something like a moving average to reduce noise. But maybe field testing will prove that that is not required. Something for the future.

My other problem I faced, was how to calculate the ensemble magnitude and colour. That is not so easy using magnitudes. My first setup was to calculate a weighted mean of the comp star magnitude values and weighted mean colour values. This using the SNR value as weight factor.

However now I assume the combining of comparison star flux values" into a single star makes it mathematical simpler and more rigorous. I will use the formula 5) in my software and field test it.

7

Yes that should reduce noise, but my understanding is that if you ever needed to carry out statistical analysis it is valid to do so only with averages, not moving averages.

By “weighted mean of the comp star magnitude values” are you saying that you would actually take the mean of the magnitudes of the comp stars? And do you imply a similar thing for the comp star colour indices, taking the mean of the colour indices of all of the comps? If that is the intention, it is not valid. It has no astrophysical meaning. The magnitude of each comp star is unique, as is its colour index. But perhaps I misunderstand.

Is the intention to sum the fluxes of all the comp stars? If that is so, I cannot see how that can be valid. Each comp has its own magnitude and colour index. Each comp must be paired, in turn, with the var to calculate a set of transformed magnitudes, one for each comp star. A mean or weighted mean of the transformed magnitudes could then be calculated. If there are 3 comp stars, you will be taking the mean of 3 transformed magnitudes.

Note that I have no knowledge of coding. I am just an amateur variable star observer who has some familiarity with the formulae involved in determining transformation coefficients and transformed magnitudes.

8

Yes that is a third approach. I haven’t thought about it but which weight factors do you apply? Looks complicate to implement. Combing the flux of the comparison stars to one imaginary star seems to me much simpler and sound.

Noise in star colour measurement

The colour of a comparison star is, in principle, steady, except that air mass and light pollution can affect it. However, both of these effects change slowly, so smoothing the comparison colour (B–V) values of an image series could theoretically reduce transformation noise.

For the (B–V) value of the variable star, you may be dealing with much faster colour changes, such as those seen in red dwarf stars. Therefore, smoothing the (B–V) value of a variable star does not seem appropriate.

Note that the program Transformer (used to calculate transformation coefficients) applies the same principle. You can feed it a single B and V image, but if you provide many B and V images, it will average the measurements of the standard stars, thereby improving the accuracy.

Ensemble colour and magnitude

You have to calculate the magnitude and colour of the comparison ensemble. My first approach was to use the weighted mean magnitude and weighted mean colour of the comparison stars (the ensemble). However, the combined flux method seems to be a better approach and effectively achieves the same goal. For example, if you have a bluish and a white comparison star, the ensemble colour should lie somewhere between bluish and white. Likewise, if you have bright and faint comparison stars, the resulting magnitude will fall somewhere between bright and faint.

By combining the fluxes of the comparison stars in the equations, the calculation becomes simpler and more robust.

I am not suggesting that you should select very different comparison stars. They should be similar in magnitude and colour to the variable star. However, they will never be exactly the same, so it is better to account for these differences.

9

I am not aware of any published scientific basis for the above ideas.

10

It is just math. There are two options:

(1) Combining the ensemble comparison stars in the beginning

or

(2) Combining the individual results of each comparison star at the end

Both should give the same result. However for (2) the calculation of the weighted mean of the individual results at the end is not so trivial if you want to do it exact. The mean of two magnitudes can not be calculated by the simple average since they are logarithmic values. You have to convert the magnitudes to a linear flux, apply the weight factors (SNR & colour ?) and then take the logarithmic again:

The mean magnitude of two stars using an equal weight factor is:

m_combined = -2.5 log₁₀(10^(-m₁/2.5) + 10^(-m₂/2.5))

This is not the same as (m1+m2) / 2

So starting by method (1) by combining the ensemble comparison star fluxes in the beginning makes the calculation much simpler. For method (2) it is not clear to me how to calculate the weight factors based on SNR and colour. If the comparison stars are very similar it probably doesn’t matter much, but it is better to have an exact calculation.

Han

11

Hello! Regarding the transformation, I’d like to share a few points.

The transformation relies on calculations that depend on each other, so an iterative approach is used. To transform V, you need a suitable B frame with the closest possible timestamp. But to transform B, you need a transformed V. That’s why we run 3–4 iterations until the result stabilizes. If you want to transform R, you need V; but to transform V, you need B; and to transform B, you need V, and so on.

As for transforming ensemble data, I’ve been using it, but the transformation largely loses its meaning because different color indices get mixed together in the ensemble. The very idea of transforming results that are already combined and blended somewhat undermines the goal of accuracy and precision in data correction. It’s something worth keeping in mind.

Transforming rapidly varying stars - flares or cataclysmic variables where both brightness and color change quickly - is a very risky task. You need a very fast cadence. For example, to transform flare stars, you need B and V frames taken within no more than 30 seconds together. Otherwise, you risk ending up with a distorted color result.

There are many considerations, and I believe transformations should be applied with care. Sometimes, in trying to get “more accurate” results, we can actually degrade the data quality. A separate issue is obtaining reliable coefficients.

Nikola

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12

Hi Nikola,

Thanks for your points. See below.

That is the latest approach. I still have a preference for the classic method. What is gained with the new iterative method?

Any star light is already a mix of light with different wavelengths which you process in one step in the calculations. So blending comparison stars at different image positions into one imaginary star should not make a difference. The x,y position of the comparison stars does not occur in the transformation calculation, so should not make a difference.

For a quality check, I found it useful to report to the user the untransformed and transformed magnitudes of the check star and the error compared to the documented magnitude. If the transformation of the check star magnitude gives an improvement then it is assumed the transformed variable magnitudes are sound. The AAVSO report contains only the untransformed check magnitude as specified by the standard.

Han

Later, the only minor problem I see in combining the comparison star fluxes is the possibility that the imaginary star intensity versus wavelength curve doesn’t follow exactly a black body curve.

13

Hi Han,

Thanks a lot for sharing your approach.

After quite a lot of testing on real data, I’ve come to a slightly different, more “conservative” conclusion regarding transforming ensemble photometry. In my experience, transforming ensemble-based results usually doesn’t bring the expected benefit and it’s often better to leave the ensemble data as it is. Of course, that’s my personal view, based on my own system and filters.

Perhaps it makes sense to transform only in the special case where the comparison stars in the ensemble have very similar B–V. Once you start mixing very different colours, I feel that the ensemble itself already acts as a kind of transformation, and applying an additional formal transformation on top of that introduces a risk of “over-transforming” the data. On top of this, validating and controlling transformed results from ensemble photometry is quite difficult in practice – I would even say close to impossible to do rigorously on a routine basis.

I have tried ensemble transformations, so this is not just a theoretical opinion. From experience with my own system and with the colour response of the filters I use, I’ve noticed the following:

  1. Transforming V is essentially pointless in my case – the correction is usually negligible.
  2. Transforming B can have a bit more impact, but still not something I would call critical for most of the work I’m doing.
  3. Transforming R gives the largest differences, but here another problem appears: the standards. R-band standards often come with relatively large uncertainties, and it’s not easy to find enough genuinely red stars to derive good, stable coefficients.

There is also a point that is sometimes forgotten: when using an ensemble, you are combining the uncertainties of all comparison stars. This naturally increases the final formal error of the transformed result. So even if in principle the transformation should make the data “more correct”, in practice it can end up increasing the overall uncertainty.

So my bottom line is very close to what you say at the end: transformation is a good and useful tool, but it shouldn’t be absolutized. It has to be applied with logic, with an eye on the actual data quality, the available standards, the colour distribution of the comparison stars, and with a clear idea of what is really gained (or lost) in the process.

Best regards,
Nikola

14

Thanks for your insights.

Yes with a proper V filter (having the correct curve) I also noticed the correction is negligible and could make it worse. For a B interference filter is helps. I assume when the filter passband is rectangle shaped transformation is required.

Transformation experiments with my RGB interference filters, so normally reported as TB, TG, TR shows that transformation brings a huge improvement for TB and TR but less for TG.

Yes an ensemble instead of a single comparison star doesn’t always bring an improvement. I can see that directly in the check star transformed values for user use only.

About black body difference when combing comparison star fluxes, Claude.ai has produced for me a public tools showing how the combined star spectrum deviates from a single blackbody. The error seems small. Link:

https://claude.ai/public/artifacts/f9c4c2b8-9b49-44e5-8c71-51679ab83705

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15

Has anyone actually tested this (using summation of fluxes of comparison stars) on real data with (say) three or more comps?

I’ve done one trial and the result makes no sense at all, but of course there may be undetected errors in the calculations.

16

The above is equation 4. from a few days ago (December 24). In my opinion it makes no sense at all. It purports to be some sort of combining of comp star colours, taking all the b fluxes and summing them, taking all the v fluxes and summing those, then inserting those values into the above equation. I can think of no algebraic or scientific basis for this procedure.

17

The colour of an ensemble:

Instrumental (b-v)_comp := -2.5 * log10( sum(Flux_b_comp) / sum(Flux_v_comp) ) (4)

Let me explain with an example. A close pair of stars generates the following flux in a camera with a specific exposure time:

star1: b_flux = 1000, v_flux = 1200
star2: b_flux = 800, v_flux = 900

If the telescope is not well focused, the stars are not separated and look like one star. They then generate the following total flux:

star_sum: b_flux = 1800, v_flux = 2100

The star_sum instrumental magnitudes are:

b = -2.5 * log10(1800)
v = -2.5 * log10(2100)

So:

b - v = -2.5 * log10(1800) - ( -2.5 * log10(2100) )

which equals:

b - v = -2.5 * log10(1800 / 2100)

or:

b - v = -2.5 * log10( sum(b_flux) / sum(v_flux) )

Now assume the telescope is well focused and the two stars are measured separately.

For star1:

b = -2.5 * log10(1000)
v = -2.5 * log10(1200)

For star2:

b = -2.5 * log10(800)
v = -2.5 * log10(900)

From these two measurements you want to calculate the b-v color of the ensemble. This is only possible if you convert the magnitudes back to flux.

To do this, apply:

flux = 10^( -mag / 2.5 )

This recovers the original fluxes. The summed fluxes are then:

sum(b_flux) = 1000 + 800 = 1800
sum(v_flux) = 1200 + 900 = 2100

The ensemble color index is therefore:

b - v = -2.5 * log10(1800) - ( -2.5 * log10(2100) )

which is the same result as for the defocused stars

18

If the exposure times are different then they should be included in the equation since the measured flux is linear with the exposure time:

Instrumental (b-v) := -2.5 log10(∑(Flux_b/exp_b )/ ∑(Flux_v / exp_v) )

19

Hi, Han,

Correct me if I’m missing something, but it sounds like you’re proposing to treat the entire comparison ensemble as a single “pseudo-star” based on summed fluxes.

Basically, you’re calculating the differential magnitude as dv = -2.5*log(F_var / sum(F_comp)), defining V_comp from the total catalog flux, and then applying a color correction based on (b-v)_var - (b-v)_comp (using tight B/V pairing by timestamp).

My take: It’s internally consistent and works well as an “engineering approximation,” provided you strictly define the ensemble as a physical sum of fluxes and stay consistent in flux space.

That said, I wouldn’t call it a universal replacement for the standard AAVSO iterative workflow. The weak point is the ensemble color (b-v)_comp = -2.5*log(sum(Fb) / sum(Fv)) when computed from measured B and V frames. It’s susceptible to pairing cadence and the atmosphere (airmass shifts, transparency changes, or color-dependent extinction). Unless your B and V frames are taken very close together under steady conditions, you risk baking systematic color errors right into your transformation.

If you go this route, there are two practical requirements:

  1. Keep B/V pairing very tight in time, especially at higher airmass.
  2. Use comparison stars with similar colors (and ideally close to the target’s color). Otherwise, the brightest comps will dominate the summed flux, and the “single color” approximation becomes biased.

Additionally, the product Tv, bv * Tbv only works if Tbv is explicitly the mapping from instrumental (b-v) differences to standard (B-V) differences. Otherwise, you might end up double-counting, depending on how those coefficients were derived.

In short, It’s an elegant shortcut if you’re disciplined about your observing and selection, but the iteration-free gain is bought by assuming away exactly the atmospheric and sync effects that the iterative approach is meant to control.

Regards,
Nikola

20

Nikola , thanks for the feedback.

My original intention was to use weighting factors so that a brighter star with less noise would have a higher weight in the ensemble than a fainter star. By working in the flux domain rather than the magnitude domain, the weighting is effectively built in, so brighter stars naturally have more influence which is good. That said, selecting stars with similar magnitudes and colours to the variable star is still beneficial for other reasons.

Han