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Transformation coefficients, NGC7790 & Steve Howell's Growth Curve Method

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WBY's picture
Transformation coefficients, NGC7790 & Steve Howell's Growth Curve Method

 Although used less frequently than M 67, NGC 7790 is another standard star field for calculating transformation coefficients. As part of the XZ Cet "stellar experiment" I imaged NGC 7790 a few times in the early winter to use in transformation coefficient calculations. I spent much too much time measuring through different apertures sizes with different annulus sizes and comparing them to determine which seemed to give the best results on average ( lowest zero point standard deviation). After all that, I still had a lot of scatter – significantly more than I thought I should be getting from good photometry data. See the attached Tv coefficient plot of v-v0 vs. B-V as an example.  

Conditions were far from ideal and the uncorrected 3000 mm focal length Dahl Kirkham telescope I used in conjunction with a fairly large format camera ~ 2000 x 3000 pixels at 0.9 microns (yielding ~ 30 x 20 min FOV) suffers significant distortion across the FOV. Further, there are a number of stars in the field that could suffer from crowding. That suggested that I should apply the growth curve method from Steve Howell's June 1989 PASP paper "Two-dimensional Aperture Photometry: Signal-To-Noise Ratio of Point-Source Observations and Optimal Data Extraction Techniques" to my instrumental magnitude measurements. 

Before I recalculate everything all over again for all of the images, I would like to confirm that I am applying Steve Howell's method correctly. Please see the attached spread sheet. 

1. I calculated the growth curves (delta magnitude vs measurement aperture) for the first 10 stars in the standard star list (from the AAVSO NGC 7790 standard field) in the image NGC - except for #7 which had an obvious crowding problem and an exceptionally large deviation from the average zero point residual - and plotted them in tab "Growth Curves (1-10)."

2. From those growth curves I selected those that seemed reasonably close to the "normal" shape that a growth curve should have, although you might argue about the curve of star 6. The selected growth curves are plotted in tab "Growth Curves Select (1-10)." 

3. I plotted growth curves for all of the remaining stars in tabs "Growth Curves (11-20)" and "Growth curves (21-31)" and selected ones close to the "normal" growth curve shape as shown on tabs "Growth Curves Select (11-20)" and "Growth Curves Select (21-31)."

4. I averaged the growth values for the stars selected to have "normal" growth curves at each aperture from 2 to 20 pixels, on tab "Select Avgs" and plotted the result on tabs "Growth Avg Select" and “Growth Avg Select Zoom.” The resulting "normal" growth curve looked reasonable to my eye. There was a bit of wobble starting at 18 pixel aperture which could result from slight crowding in a number of the stars selected for calculation of the “normal” growth curve. Because of this slight “wobble” I decided to use the magnitudes at 17 pixel aperture radius as my stellar magnitudes on the corrected magnitude plots described below.

5. I plotted the error values (from the CCD equation) at each aperture and selected 6 pixels as the aperture to use as the base for growth. The growth values used for each star from their 6 pixel value are shown in column "CU" on the "Select Avgs" tab. The values in the “CV” column accomplish the same thing but starting with radius 7 they use an iterative method from the corrected instrumental magnitude at the next smaller aperture radius to facilitate pasting into column “AB” on tab “Photometry_NGC7790_B001_all-30-”  in the rows applicable to image NGC

6. The “corrected” (perhaps it would be better to call them “adjusted”) instrumental magnitudes for all stars are calculated in column “AC” on tab “Photometry_NGC7790_B001_all-30-”  in the rows applicable to image NGC and are plotted on tabs “corr magns 1-10”, “corr magns 11-20” and “corr magns 21-31.”

7. The corrected magnitudes at 17 pixel aperture radius are summarized in Column “AC of tab “Corrected photometry & Residuals.”The zero point for the magnitude of each star the average zero point and the average zero point and the standard deviation of the residuals are calculated for the corrected and uncorrected magnitudes in columns “AE” through “AH” of the same tab.  The average zero point for the corrected and uncorrected magnitudes are 19.762 and 19.806 respectively, a separation of only about 0.04 magnitudes. However the standard deviation of residuals for the corrected magnitudes  only 0.050 magnitudes compared to 0.309 magnitudes. The uncorrected residuals are about 6 times as large as for the corrected magnitudes.

These results seem to indicate that using Howell’s method should reduce the scatter in my transformation plots significantly. However, before I go through the considerable effort of applying this method to all of the images from several nights, I want to be sure that I am doing it correctly. If anyone is experienced in using Howell’s growth curve method and can confirm if I have done it correctly or correct any errors I have made, I would appreciate it.

Brad Walter

WGR's picture
Transformation Tool

Hello Brad

I woiuld like to look at your spreadsheet, but I don't have any way to read xlsx files.  Can you save them as regulary xls files?  If so, I will go thru them in detail and then comment.  

It will be interesting to see if this helps your transformation coef scatter.  My knowledge is that growth curves are mostly used to get the best SNR from a single star.  However, when looking to get the accurate magnitude measurement, that PSF or using apertures that include the same percentage of the wings will work better.  BTW:  What ratio of FWHM to Aperture radius looks best based on the growth curves?




PVEA's picture
Transformation Tool

Hi Gary,

You can convert xlsx to xlc files using some online services as this one:

I am not familiar to Steve Howell's Growth Curve Method and I am also curious of it's applicability.


WBY's picture
Howell Growth Curve Method

The referenced paper by Steve Howell, "Two Dimensional Aperture Photometry: Signal-To-Noise Ratio of Point-Source Observations And Optimal Data-Extraction Techniques"

If I understand correctly, the basic Idea is that "correct" growth curves (flux ratio or magnitude difference vs. aperture size) in an image should have the same shape because stellar profiles in a perfect, flat and undistorted image with consistent background should be the same shape. This is the theoretical situation that won't be exactly true because of distortions from seeing, variations in focus across the field of view and background variations. As you increase your measurement aperture size to try to get all of the flux from stars above the background level, the shape of individual growth curves may vary from the "correct" shape for that image because measured background levels under-subtract or over-subtract the actual background, crowding may occur (even if it is from very dim background stars you might not notice by visual examination of the imaages), defects may be included in the measurement aperture, etc. So you have a problem. You want to use a large aperture so that you collect all of the light from stars of different brightness above the sky background level but beyond a relatively small aperture radius close to the FWHM, you increase the error (reduce the S/N). 

Pick some bright isolated stars in the image and measure them using a whole bunch of different apertures from 1 pixel radius up to about 3X FWHM. I used 1 pixel increments except near the FWHM where I used 0.5 pixel increments. Calculate and plot growth curves. Select ones that approximate the theoretical shape and average them to construct a "correct"growth curve to use on other stars. 

For each star you want to measure in the image, there will be an aperture radius that provides maximum signal to noise ratio (essentially equivalent to the smallest CCD equation error). This aperture will be at approximately (but not exactly) at an aperture RADIUS equal to the FWHM of the stellar profile and the aperture that minimizes error (or maximizes S/N) will vary a bit from star to star but should fall within a narrow range. In my images that seemed to be in the range of 6.0 to 6.5 pixel aperture radius.

Apply the "correct" growth curve your created to the magnitudes of the stars of interest at a radius that is at or near the min error (max S/N). for may data I picked the base magnitudes at 6.0 pixel radius so I am adding the difference in magnitude of my "correct" growth curve from the 6.0 pixel radius out to a radius where the growth curve flattened. I actually constructed corrected magnitude vs. aperture curves for each star rather than just adding the magnitude difference of my "correct" growth curve at 6 pixels and 17 pixels because it helps you see if your corrections make sense. The Magnitude vs aperture curves should now all be very close to the theoreticallt 'correct shape. Theoretically that should give you more precise instrumental magnitudes and should imrove the accuracy of measurement for stars that suffer crowding or other deffects introduced with increased aperture size. Since I am doing photometry of a standard star field, i can tell if I have greater precision by comparing the standard deviation of the zero point residuals of the growth curve corrected magnitudes vs the uncorrected magnitudes. For the image of NGC 7790 that I did this applying growth curve corrections decreased the standard deviation of zero point residuals from 0.309 magnitudes to 0.050 magnitudes. The precision of the uncorrected instrumental magnitudes is pretty bad. The precsion of the corrected magnitudes is much better. 

So now I have magnitudes that should include all of the light from each star based on the magnitues at or very near the aperture giving the highest signal to noise ratio which eliminates a large portion of the extraneous variation that occurs from "stuff" in my image that has nothing to do with the stars themselves.  

Brad Walter

WBY's picture
Howell's growth curve method

Thank's Gary. I know slogging through someone elses spread sheet is a chore. An xls version of the spreadsheet is attached.

There are a lot of tabs in this spreadsheet but most of them are graphs. I don't like to work only with the numbers. I like to see how they look on a graph. It has saved me numerous times from making bad mistakes or using bad data. All calcualtions are on tabs "Photometry_NGC7790_B001_all", "Select Avgs" and "Corrected Photom." Most of the columns on these pages are simply the output from Mira Pro. I have tried to indicate below columns where the growth curve correction calculations are actually made in this mass of numbers. 

Excel gave me a warning that a function may not have converted from xlsx to xls correctly. The only function that I can think of that may be a problem is the STDEV.S function used on the "Corrected Photometry" tab. Older versions do the same standard deviation of a sample calculation but the function name is different.  So if you see any spaces with NAME instead of a result, this is probably the issue. Even though I saved it as an xls, I can't see the error because the Excel version I use knows how to interpret the function. Older versions may not. 

Since I only did the Growth calculations for image 001B1H180, the first of 4 images in the set, there are a lot of blank rows in columns Z through AC on tab "Photometry_NGC7790_B001_all" where I calculated the corrected magnitudes using the corrections from column "CV" on tab "Select Avgs."

It may seem odd that I did the correction two ways - Columns "CU" and "CV" on the "Select Avgs" tab. After I had caclulated column "CU" I realized a step wise correction calculation would cut and paste better than one that referenced a specific cell in each group of data. 

The other thing you may notice is that aperture 16.3 is out of order. For some reason Excel would not sort that value in order in the third sort level when initially structuring the data on tab "Photometry_NGC7790_B001_all" That's why the formulas for apertures 17 and 16.3 break the pattern in the column "AC", "Corr Magn", on tab "Photometry_NGC7790_B001_all" and column "CV", Correction from next smaller aperture", on tab "Select Avgs" break the calculation pattern. 

Thanks again for your offer to check my methodology.

Brad Walter

WBY's picture
FWHM in the image

The Average FWHM of stars 1, 8, 16 and 18 (from AAVSO NGC 7790 std star field PDF) is 6.77 with a standard devisation of 0.26. the best SNR is a toss-up between radius of 6.0 and 6.5 based on the error graphs in my spreadsheet. So, splitting the difference,  the aperture RADIUS for best SNR is about 92% of the average FWHM That seems to agree well with the results Howell expresses in section 4 of his paper:

"The S/N is a maximum at fairly small radii (approximately the FWHM of the stellar profile but not in general equal to it) and decreases in both directions away from that point, the maximum is not necessarily at the same radius for all sources, and the decrease in S/N away from the maximum is dramitic for objects of all flux levels."  

You can see that the lowest (maximum S/N) in my data is not at exactly the same for all stars, but falls in a relatively narrow band of apertures. 


HQA's picture
curve of growth analysis

Hi Brad,

I was away for the weekend.  70's in New Hampshire - winter is over!

There are some details missing from your email.  Your CCD has 9 micron pixels, not 0.9 micron.  That yields about 0.9arcsec/pixel.  Your selected aperture for your analysis (17 pixel radius) then has a diameter of about 30arcsec.  Assuming a kinda typical seeing of 4arcsec, this means the diameter is about 7 FWHM, which doesn't surprise me as having the highest signal/noise because you are primarily dealing with bright stars, well above the sky background (which is the primary limiting factor).

You then used 6 pixel radius (about 11arcsec, or 2-3fwhm) as your base aperture size, did your photometry at that aperture size, then corrected your photometry as if it were taken at 17 pixel radius.

If I understand this correctly, then you've been making a couple of basic mistakes.

First, forget about the aperture correction - just do all of your measurements at 6 pixel radius.  My guess is that your standard deviation will be even lower than the results you got corrected.  The reason for this is that you are adding a second layer of assumptions - that the curve of growth is well determined and the same for all of your stars, which won't be the case since you've already stated that you have optical distortion.

Where curve of growth analysis is useful is for the case where you have a faint star in a crowded field, where you have to use a small aperture, and a comp star which is brighter and which you want to get higher signal/noise by using a larger aperture.  Then you need to adjust either star to match the same aperture as used for the other star.  Otherwise, you are comparing measures taken at two different points in the star profile and your differential photometry will be wrong.

For transformation calculations, you want to use apertures small enough so that you don't have blends, which is what you were getting with your 30arcsec diameter apertures and what was causing the 0.3mag scatter.  Getting maximum signal/noise is less important, as your errors will be dominated by crowding when dealing with clusters.  To be even more accurate, you should use the same diameter aperture that the original source used.  For example, Landolt uses a 14arcsec aperture, so I use a 14arcsec digital aperture on my photometry to match what passed through Arlo's photometer - any close companions would be included in both data sets.

So my advice to you - forget about aperture corrections and curve of growth analysis - it is unnecessarily complicating your analysis.


WBY's picture
Growth curve analysis

Arne you are correct my camera is 9.0 microns per pixel not 0.9. Sorry that was a typo. However Unless I am doing the math wrong that should be 0.62 arcsec per pixel. Using the small angle formula 9 micron pixel and 3 000 mm (3E6 micron) focal length  and converting from radians to arcsec gives the following calculation 9/3.0E6*3600*180/PI() = 0.62. This agrees with the plate scale that Maxim DL calculates for my setup.  Then 6 pixels aperture radius equals radius of 3.7 arcsec and diameter of 7.4 arcsec and 17 pixels yields 10.5 arcsec aperture radius and 21 arcsec diameter.

You estimated my typical seeing with uncanny accuracy. I measured 4 stars scattered around the FOV  in each of the 4 B filter images in this set (IDs 1, 8, 16, & 18 in your 10/14/1999 sequence). The average FWHM of these stars in this image is 6.77 pixels or 4.2 arcsec with a standard deviation of 0.26 pixels  

This doesn't alter your basic point of using smaller aperture and forgetting about the growth curve. That is great news since this adds a whole bunch of calculation. If I understand you correctly you recommend using an aperture diameter (not radius) of between 2x and 3x the FWHM in a cluster. Is that correct?

I don't know what aperture diameter was used to establish the sequence for this cluster. I suspect that you might have that information provided you still have records going back to October 14, 1999. 

For non-cluster stars that have S/N of more than 100, I have typically used an aperture RADIUS of between 2.0x and 2.5x the average FWHM of a few stars selected around the FOV. That seems to be in the range that is most frequently recommended, and when doing aperture photometry on a series of images taken relative close in time on a given night through the same filter, I find that I get the most consistent results with a fixed aperture radius of 2.3x (diameter 4.6x) the average FWHM for the series provided the scatter of the FWHM is not more than a fraction of a pixel. This seems to be about 2X the aperture size you are recommending for photometry of a cluster. Should I be using smaller apertures for non-cluster (uncrowded) stars as well?

As an aside, I have tried using different apertures for different images scaled to the FWHM, but unless the variation in FWHM among images is large, generally a constant aperture has given more consistent results (lower standard deviation of magnitudes). 

Thanks for the comments. This will save me a whole bunch of calculation time. It would be helpful if you could respond to the two questions in this e-mail:

1. Do I understand correctly that as a rule of thumb, I should be using an aperture DIAMETER in the range of 2x to 3x FWHM (apetture radius = 1.0x to 1.5x FWHM) when measuring stars in clusters? (except for standard star fields for which it is best to use the same aperture as was used to establish the field sequence)

2. As a rule of thumb, should I be using a smaller aperture DIAMETER than the 4x to 5x FWHM (aperture radius = 2.0x to 2.5X FWHM) I currently use for reasonably bright (S/N >100 ) uncrowded stars?


Brad Walter

WGR's picture
Radius, not Diameter

Hello Brad

You wrote:

1. Do I understand correctly that as a rule of thumb, I should be using an aperture DIAMETER in the range of 2x to 3x FWHM (apetture radius = 1.0x to 1.5x FWHM) when measuring stars in clusters? (except for standard star fields for which it is best to use the same aperture as was used to establish the field sequence)

This is not correct.  If you think about it, DIAMETER of 2x FWHM puts the measuring aperture right at half max, and avoids the wings completely.  This is wrong.  This is less that 1 sigma of the PSF.  One clearly wants most of the wings, within reason.


So the rule of thumb is to use 2x to 3x for the RADIUS!






WBY's picture
Radius Not Diameter


Thanks for the reply. I would not normally use an aperture that small but  Arne's comment relating to aperture photometry in clusters seems to be advising use of much smaller than normal apertures: 

"You then used 6 pixel radius (about 11arcsec, or 2-3fwhm) as your base aperture size, did your photometry at that aperture size, then corrected your photometry as if it were taken at 17 pixel radius.

If I understand this correctly, then you've been making a couple of basic mistakes.

First, forget about the aperture correction - just do all of your measurements at 6 pixel radius.  My guess is that your standard deviation will be even lower than the results you got corrected.  The reason for this is that you are adding a second layer of assumptions - that the curve of growth is well determined and the same for all of your stars, which won't be the case since you've already stated that you have optical distortion."

At Arne's value of 0.9 arcsec per pixel a 6 pixel radius is not 11 arcec. at 0.9 arcec per pixel the diameter is 11 arcsec and the diameter is in the range of 2x - 3x the typical FWHM of about 4 arcsec (FWHM that Arne mentions in his post (the image FWHM is very close to that at 4.2 arcsec).  

That is why I am asking Arne to reply to those questions because this is about half the aperture size I would normally use. Crowding is much worse in clusters. Therefore different limiting conditions may apply compared to stars in relatively uncrowded conditions outside of clusters.

Brad Walter

HQA's picture
diameter, not radius

What I meant is what Brad assumes: diameter.  If you look at Steve Howell's paper, he finds the optimal signal/noise to occur at a radius equal to the fwhm, or in other words, a diameter equal to 2x fwhm.  Remember that "fwhm" refers to the diameter of the seeing disk of a star, not its radius; it is just a convenient scale factor for these calculations.

I personally use diameter because I started my career using photoelectric photometers, which were based on the diameter of the entrance hole.  Many software packages plot using radius, and either method is ok as long as you carefully describe which one you are using.  For the majority of my photometry, I prefer either 14arcsec diameter apertures (when calibrating) or sometimes 4-5fwhm diameter apertures (when doing generic variable-star work).  This is bigger than the 2x fwhm listed in the first paragraph, because it is my empirical experience that small apertures suffer from other defects, such as the methodology used to "round" apertures formed with square pixels, or the method of centroiding.

However, for clusters like NGC7790, I may use a smaller-than-normal aperture because of crowding.  Say that my seeing is 4arcsec, so that my normal aperture would have 16-20arcsec (4-5fwhm) diameter.  If I reduced my aperture size to 8arcsec diameter (2x fwhm), I'd get rid of many cases of a contaminating star in my measuring aperture.  However, if what I am doing is comparing my results to some standard photometry where that researcher used a 14arcsec diameter aperture, then I may be avoiding a contaminating star that the other researcher included, and my results will differ.  I'm right and they're wrong, but the values in the literature include the contamination, so if I want to compare, I also have to include the contamination.  So always look at the original source for calibration data, and find out if you can how the measurement was made, and try to duplicate it as closely as possible.


WBY's picture
Diameter Not Radius

Thanks, Arne.

I thought I understood correctly but I wanted to be sure since it is easy to confuse between diameter and radius and your suggested aperture for a cluster is about half of what I would normally use and have seen in many sources regarding normal "uncrowded" Photometry. 

I get the point about the added assumption of the growth curve. I am going to compare the zero point residual spread at 6 pixel aperture to those from the growth curve adjusted magnitudes. After some further thought, the entire growth exercise is silly in this application as opposed to just using the small aperture raw magnitudes (no comparison star). The spread of zero point residuals from the six pixel aperture should be the same as from the growth curve method since, in essence, it does nothing but add a fixed magnitude difference obtained by averaging the magnitude growth of the "selected stars"  from 6 pixels to 17 pixels. Similarly, the zero points of the individual stars between the 6 pixel magnitudes and the growth curve adjusted magnitudes should be different by a fixed amount, provided I have actually made the growth adjustments as I think I made them. The proof will be in the pudding when I compare the results. 

It would make sense, however if you were using different apertures for target and comparison stars, and needed to adjust the photometry for the difference in apertures, as you described in your original post. 

Brad Walter

WBY's picture
Using the 6.0 pixel Aperture Radius

Well, I calulated the zero point for the image (average of zero points of the individual stars) and the standard deviation of the zero point residuals for the individual stars. The zero point shifted by exactly the growth correction I used to go from the 0.6 pixel aperture to 17 pixels and the standard deviation of the zero point residuals for the stars remained exactly the same as for the growth curve adjusted magnitudes.  That is the expected result which does nothing more than show that I applied the correction as I thought I had and nothing is gained by calculating growth curves compared to using the aperture that gives the highest S/N (lowest uncertainty), or even more simply, an aperture RADIUS equal to the FWHM when doing photometry in a cluster. I assume the same would apply to other crowded areas such as dense star fields in the Milky Way (keeping in mind Arne's caveat about using the same aperture used to establish the sequence stars in the cluster if you know it).

Thanks Arne, that saves many hours of pointless calculation. 

Brad Walter

WBY's picture
Transformation Redux

OK, I measured three nights of BVRI photometry of NGC 7790 Using a 7 pixel aperture radius (14 Pixel diameter) which at my pre corrector plate scale is 4.3 arsec radius (8.6 arcsec diameter). The small aperture is per Arne's suggestion above. I adjusted the aperture upward slightly after looking at FWHM information from all of the images I would be using for the photometry. 

I measured all with the same aperture. I don't think that is actually necessary but it was easier and faster than trying to pick the lowest noise aperture in each image. I had two sets of images for each night with IRVBBVRI or IRVBBBBVRI sordering in each set. The sets were at significantly different airmasses. I did this to see if airmass would have a significant effect and also to provide data that would allow me to calculate second order extinction coefficients. The lower airmass set for each night provided lower scatter of linear regression residuals in the plots. They did not always produce Tv closer to zero or T for color indexes closer to one. Than regressions of the data from both sets of measurements for a given night. 

I used an ensemble photometry approach rather than using a particular standard star as a comp star for all of the others. I did this to avoid any "color bias" that might be introduced because of deviations in color of the other stars from the comp and because the particular software I use allows me to easily calculate Raw instrumental magnitude -2.5*Log(netcounts*gain/exptime). The zero point for each image was established by averaging the residuals from stadard mag for all 31 standard stars in each image (averages of Std mag - Raw mag) and adding the image zero point to the raw mag for each star. Then for each set of images I averaged the two (or sometimes four, for the B filter) magnitudes for each filter in each set of images so that the resulting magnitude values for a set would be at as close to the same airmass as possible. Then I plotted V-v vs. B-V, (B-V)-(b-v) vs. (B-V), (V-R)-(v-r) vs (V-R), etc. for the lowest airmass data set only and both data sets  for each night and ran linear regressions on the plots and added the regression lines. I could have simply regressed the data, but I like to see the plots to get an idea of the fit and, if I am rejecting and discrepant data points it allows me to see the difference (and whether I actually eliminated the correct point).  I ran two regressions for each of these groups of data. The first used all of the data points. I used a t value  for the residuals from the regression line of 2.5 x the standard deviation as the criteria for rejecting discrepant points. This amounts to approximately a 98% probability that the point is discrepant for a 31 point sample (30 degrees of freedom).Then I ran the regression again after eliminating any discrepant points to get the slope. The slope, M, = Tv for V-v vs. (B-V). However, for the color indexes, Tbv for example, T = 1/(1-M) because (B-V)-(b-v) = (1-1/Tbv)(B-V) -Zbv/Tvb, Where Zbv is the zero point for that data set. *

The results for each of the three nights  and their average are the following:































If anyone is interested examples of the detailed spreadsheets are attached. 

*The reason for plotting the differences between the standard color index and the instrumental color index vs. the standard color indexis that it gives much bigger variations in slope for small variations from the standard system The ratio of   Tbv= 1.1 to T bv= 1.01  plotting (b-v) vs (B-V)is only about 1.09, so the relative change in slope is small for a big change in transformation coefficient. The corresponding ratio of slopes plotting (B-V) - (b-v) vs. (B-V) is about 9.1 giving a much larger relative change in slope. This allows you to see your variation from the standard system more clearly. Also, If your data exactly matches the standard system, M = 0 in all cases, not just for V-v vs. B-V. If you just plot instrumental color index vs. standard color index, you want to be close to zero for Tv but close to 1.0  the color index Transformantions and a slope of 1.0 can be at essentially any angle on the plot depending on the scaling of the axes. (See Astronomical Photometry by Henden and Kaitchuck section 4.6 as a reference).  If anyone is interested samples of the calculation spreadsheets are attached. 

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