[Aavso-photometry] CCD 'fainter-than' question

[Wolfgang Renz]

Hi Jim

> Thus, consider a background average of 100 and a sigma of 10. 
> The probability that a pixel in the measuring aperture that measures
> 130 (+ 3 sigma) is just a background outlier is, as stated above,
> 0.0027/2. 
> But how many pixels are in your measuring aperture?
> The math is pi r squared where r is the radius of the aperture in pixels.
> Suppose r = 6 then the number of pixels is 113.  Thus, the probability
> that at least one of the background pixels in the measurement aperture
> is as bright as 130 is only 15%.

The 15% probability of a pixel being 130 or more (>= +3 sigma) is
the same as for being 70 or less (<= - 3 sigma). But as we assumed
a normal distribution, these (usually) even out and leave an average
of (about) 100.

Its always assumed that the CCD and the noise behaves the same
in the sky annulus and the star aperture. This is usually sufficiently
valid. But usually not always in the cases when there exist severe
background gradients like e.g. with aperture photometry in nebulae,
star clusters, galaxies  or when there are strong und non-linear irre-
gularities due to artifical light pollution or internal reflection issues.

To not to introduce biases, one should of course calibrate the raw
frames well enough to get rid of hot/warm/cool/cold pixel/cluster/
columns and and reject frames with uncalibratable bright/dead
pixel/cluster/columns and obvious cosmics in the star aperture.
The effect of these defects is less severe to non-existent in the
sky annulus if one uses software that offers a median/mode or a
min-max/sigma/SDM-rejected mean for the pixel in the sky annulus.

Clear skies
 Wolfgang

-- 
Wolfgang Renz, Karlsruhe, Germany
Rz.BAV = WRe.vsnet = RWG.AAVSO



----- Original Message ----- 
From: "Jim Roe"
Cc: "AAVSO-PHOTOMETRY" <aavso-photometry at mira.aavso.org>
Sent: Wednesday, May 28, 2008 2:23 PM
Subject: Re: [Aavso-photometry] CCD 'fainter-than' question

> Wolfgang Renz wrote:
>> Hi Bob
>> 
>> If one assumes that the noise in the sky annulus is normal distributed,
>> then the following applies for the probability that a star measurement
>> is still just background noise:
>> <http://en.wikipedia.org/wiki/68-95-99.7_rule>
>> <http://en.wikipedia.org/wiki/Standard_deviation#Rules_for_normally_distributed_data>
>> <http://upload.wikimedia.org/wikipedia/commons/8/8c/Standard_deviation_diagram.svg>
>> <http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/400px-Standard_deviation_diagram.svg.png>
>> 
>> So the probability for a measurement to lie outside the following
>> ranges and to be still just a background detection are as follows:
>> +/- 1 sigma   100% - 68.27% = 31.73%
>> +/- 2 sigma   100% - 95.45% = 4.55%
>> +/- 3 sigma   100% - 99.73% = 0.27%
>> +/- 4 sigma   100% - 99.9934% = 0.0066%
>> +/- 5 sigma   100% - 99.99994% = 0.00006%
>> +/- 6 sigma   100% - 99.9999998% = 0,0000002%
>> +/- 7 sigma   100% - 99.99999999974 % = 0.00000000026%
>> But as there are no 'darker than the background' stars, one must
>> half the above probabilities for just 'brighter than the background'
>> stars.
> 
> A precise definition, Wolfgang, but I would try to amplify it with a 
> numerical example.  First off, these results must be applied pixel by
> pixel.
> 
> Thus, consider a background average of 100 and a sigma of 10.  The 
> probability that a pixel in the measuring aperture that measures 130 (+ 
> 3 sigma) is just a background outlier is, as stated above, 0.0027/2. 
> But how many pixels are in your measuring aperture?  The math is pi r 
> squared where r is the radius of the aperture in pixels.  Suppose r = 6 
> then the number of pixels is 113.  Thus, the probability that at least 
> one of the background pixels in the measurement aperture is as bright
> as 130 is only 15%.
> 
> But one should never claim a "detection" based upon one pixel.  There 
> should be several contiguous pixels at or above the threshold before 
> claiming such.  Even then, a bit of caution is warranted as cosmic ray 
> strikes can produce some very star-like images.
> 
> Jim Roe [ROE]