[Aavso-photometry] Standard deviation calculation

[arne]

Brad Walter wrote:
> 1. I don't understand why the STDEV of K-C is added in quadrature to the
> Poisson noise of the target star. When you measure the target star in
> differential photometry you are actually measuring K-T. When you measure K-C
> you are including all of the stochastic error elements that are in the K-T
> measurements if C and T are the same magnitude. 
> 2. How do you correct the error estimate for differences in magnitudes
> between C and T? It seems to me you have to do a calculation something like
> the following:
> STDEV(k-t) = SQRT((STDEV(k-c)^2 - SE(k)^2)*SE(t)^2/SE(c)^2 + SE(k)^2)
> 
> Where SE(x) = -2.5*Log(1+1/SNR(x))
> 
------------------------
> Usually you work with three stars for time series: the target (V),
> the comparison star (C), and the check star (K).  Usually you
> obtain measures for all three stars on every image.  If you
> form the difference between the check and the comparison star (K-C),
> this represents the major part of the error in estimating the
> target, since typically these two stars are similar in magnitude
> to the target and their difference roughly matches the error of
> a constant star with the target's brightness.
> 
> So the usual way of handling a time series is to calculate the
> mean and standard deviation of (K-C), and report the standard
> deviation of (K-C), added in quadrature to the Poisson noise
> of the target, as the error for the target star.
> 
> If your sky conditions are varying, you may want to break up the
> calculation of the standard deviation of (K-C) into smaller time
> intervals, and modify the reported error for each of those intervals.
> Arne
> 
I walk a fine line between giving a full complex description of the
error calculation process, and giving something that gets most of
the way there so that the average report can be improved beyond
a pure reporting of Poisson error.

The full error equation has been given by Michael Newberry on this list if
I remember correctly, and can also be found in Steve Howell's book.
Remember that you need to work in flux space and not magnitude space
for a careful calculation.

What Brad is saying is that you are working with (at least) three stars
in this process: target (T), comparison (C) and check (K).  Each of these
stars will be of different brightness than the others, and therefore will
have different Poisson error.  I think that what needs to be remembered
is that, for most amateur situations, Poisson error is actually nearly
negligible in comparison with the other error sources.  That is why the
(K-C) estimate is suggested in reporting errors to the AAVSO, as it
includes implicitly many of the possible error souces and is simple to
calculate.  I included the Poisson error of the target in quadrature to
guard against those cases where the comparison/check are much brighter
than the target (such as estimating the brightness of a cataclysmic variable
in quiescence).  If the target/check are fainter, then including the
target Poisson error will be a very minor overestimate.  So the quadrature
summation presented in my posting above is an approximation and should have
been described that way.

There is no simple mechanism that gives you a perfect estimate of the error.
The (K-C) process gets you most of the way there and is one that we might
get most observers to use (rather than the very optimistic Poisson error
estimate that many use now).  If you want to try something more accurate,
then by all means look at the real error equation and see what results
you obtain.
Arne