DERIVATION OF S.E. USING SNR

Bruce L. Gary, Hereford, AZ
2003.04.04; 2003.06.21

This web page describes a derivation of standard error for an aperture photometry measurement when signal-to-noise ratio, SNR, is available.  MaxIm DL, for example, displays SNR for whatever star is within the aperture circle.  I recommend using the following equation to represent the stochastic uncertainty for an aperture photometry measurement of magnitude.

SE = -2.5 log10 (1 - 1/SNR)

Given:
   SNR (signal-to-noise ratio)
     Definition of magnitude difference:  dM = -2.5 * log (Y/Yo),
       where log is logarithm to the base 10,
       Y and Yo are two brightness values
      (think of Yo as average magnitude solution)

Find:
     "magnitude SE" (standard error in magnitude units)

Solution:

 SNR = Yo/SE,
    where Yo = average intensity solution and SE is stochastic uncertainty (SE) of Yo solution

 Adding and subtracting SE to and from Yo yields upper and lower Y-values encompassing ~68% of the probability of finding the true solution (assumes Gaussian statistics).

 These two Y-values, Y+ and Y-, are simply Y+ = Yo + SE and Y- = Yo - SE

 Another way of writing these two Y-values is Y+ = Yo + Yo/SNR and Y- = Yo - Yo/SNR

 This can be re-written Y+ = Yo * (1 + 1/SNR) and Y- = Yo * (1 - 1/SNR)

 Consider the ratios of these two Y-values to the average value:
  Y+ / Yo = (1 + 1/SNR), and
  Y- / Yo = (1 - 1/SNR)

 These ratios can be written as magnitude differences:
  dM+ = -2.5 * log (Y+/Yo)
  dM- = -2.5 * log (Y-/Yo)

 Substituting the equations for Y+/Yo and Y-/Yo yields:
  dM+ = -2.5 * log (1 + 1/SNR)
  dM- = -2.5 * log (1 - 1/SNR)        (This is the more conservative SE since it's absolute magnitude is larger)

Comments:

Note 1:   The + and - magnitude errors are different, the more so for smaller SNR.  This is the way things are, and when SNR is large the two dM values can be averaged to give one SE.  Hopefully AAVSO observers will always have large SNR, so that a simple average of the two dM values can be used.  Whenever SNR <= 15 then the inequality of dM errors will exceed 22%.  The value of dM+ will always exceed the absolute value of  dM-.

Note 2:  These errors, dM+ and dM-, are standard errors, or SE - also referred to as 1-sigma errors.  This means that the truth has a 68% probability of existing within the range Yo - SE and Yo + SE (or Mavg - dM- and Mavg + dM+).  Of course, this assumes that calibration uncertainties are small compared to stochastic uncertainties.

Note 3: When using results obtained with the same instrument, the same "seeing," the same quantity of measurements, and the same analysis procedure (same annuli set for the photometry, same reference stars), whenever these same conditions apply, it is acceptable to ignore calibration uncertainties in order to monitor CHANGES in brightness.  Clearly, each observer has different calibration offsets, and this is the reason it's difficult to combine observations from different observers to construct a light curve.  It is much easier to construct a light curve from a single observer (provided that observer has sufficient data and adheres to the same analysis procedure).   When observations are pooled from many observers either empirical offsets must be derived for each observer or each observer must present an estimated calibration uncertainty.  The estimated calibration uncertainty should then be orthogonally added (square-root of the sum of squares) to the stochastic uncertainty, described above as dM- and dM+ before combining with other observations.

Some observers use the approximate equation SE = 1/SNR to calculate magnitude SEs. It is a coincidence that the magnitude scale was defined in such a way that this approximation happens to work - almost.  It is erroneous at all SNR, but especially at low SNR, where it is most important to provide a good estimate of SEs. At high SNR calibration errors dominate accuracy, but at low SNR stochastics are dominate.  The following graph shows how 1/SNR yields magnitude SEs that is always too low.  

Figure 1. Plot of magnitude SEs versus SNR using the correct equation (red) and the approximation 1/SNR (blue dashed).  The ratio (green) is the ratio of the two SEs values.

The 1/SNR version of SEs is 8% low at high SNR and up to 25% low at SNR = 3 (a common criterion for "limiting magnitude.").
 

STOCHASTIC UNCERTAINTY IN RELATION TO CALIBRATION ERROR UNCERTAINTY

It should be kept in mind that every specific situation (hardware, observing strategy, analysis procedure) has its own unique set of uncertainty source components.  The "accuracy" of faint stars will differ from bright ones, not only because faint stars have lower SNR but because components of "calibration error uncertainty" have their own specific dependence upon a star's brightness.  The situation is represented in the next graph, which is explained in more detail on another web page.

Figure 2"Stochastic uncertainty" (green and red) and the several components of "calibration error uncertainty" (dashed blue lines) are orthogonally add to produce "total uncertainty", also called "accuracy."  The 4 components of "calibration error uncertainty" are given letter symbols and are explained in a companion web page.

In this figure the exact location of the various calibration plots are appoximately correct for a specific observing system, observing strategy and analysis procedure, also explained in the companion web page, and will differ for each observer and each observing strategy and analysis procedure. When "stochastic uncertainty" is specified in magnitude units, the "+" and "-" uncertainties bifurcate at low SNR values.  The negative branch is always greater than the positive, as explained in the previous section.  This is illustrated by the separation of the red and green traces in the figure.

Additional description of this graph can be found at SE.

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This site opened:  June 18, 2003 Last Update:  June 21, 2003