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

[Wolfgang Renz]

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.

>> CCD observers are asked to use exposure times such that you can
>> obtain a signal-to-noise of 50 per observation.
> 
> For CCD observing, what constitutes a 'fainter-than' observation?

I think that its generally accepted that a 3 sigma detection above the
noise is sufficient certain to report such a measurment with its high
error.

If its a less than a 3 sigma detection, it should be probably reported
as a 'fainter than' with the value of the background + 3 sigma as mag
But this mag value is unfortunatly nothing that is available with usual
photometry software. So maybe someone of the experts can recom-
mend how it can be fast and easily determined.

> If I can see the variable on the image but only 10:1 S/N (50 mini-
> mum needed for observation, below), is that a 'fainter-than'?

IMO no, as its a better than 3 sigma detection. Its just a measure-
ment with a less than recommended SNR.


The exact same question with the exact same SNR came up already
before. So I've attached the answers that came up below.


Clear skies
 Wolfgang

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



----- Original Message ----- 
From: "Michael Newberry" <mnewberry at mirametrics.com>
To: "Laurent Corp" <laucorp at wanadoo.fr>; <aavso-photometry at mira.aavso.org>
Sent: Saturday, January 27, 2007 9:25 PM
Subject: Re: [Aavso-photometry] magnitude limit

> I did not write the AAVSO alert notice, so I cannot say why it wants
> S/N >= 50. The AAVSO is saying they want the observation to be
> within about 0.02 magnitudes of what it should be based entirely on
> random errors. The value of "sigma" is the random error of the mea-
> surement. Call it the standard deviation, if you will. If you measured
> two stars in the same field and your measurements were quoted with
> 0.02 magnitude uncertainty and the two stars differed by 0.02 mag-
> nitudes, then you could say they differ in magnitude by (difference /
> uncertainty) = 1-sigma. That would be like a 1-sigma detection.
> There's only a 68% chance that they really are different values 
> when the difference is only 1-sigma.
> 
> Here's generally how all this works.
> 
> In earlier posts, the value S/N = 3 was given for the definition of a 
> "detection" above the sky background, not for a "good measure-
> ment". Detection means that you are comparing the significance
> of a spot against it being just a random fluctuation that is really
> just random noise. The scientific goals of the project determine
> how "good" the measurement must be. Before I given an example,
> consider these definitions which relate Signal to Noise Ratio ("S/N"
> or "SNR") to flux and magnitude:
>    M = Magnitude
>    dM = Magnitude difference
>    F = Flux
>    dF = Flux difference
>    S/N = Signal to Noise Ratio
>    dM = -1.0857 dF / F
> Since dF / F = 1 / S/N, you get this:
>    dM = -1.0857 / S/N   or equivalently,   S/N = -1.0857 / dM
> 
> We'll drop the negative sign, since they just tell you which direction
> the quantities change; in other words, increasing flux means a
> numerically decreasing magnitude.
> 
> As I said, the magnitude uncertainty is based on how good the
> measurement needs to be. For example, suppose there are two
> competing models for a star's brightness and they make different
> predictions that are separated by 0.06 magnitudes. To distinguish
> one model from the other using photometric observations, how
> good do the observations need to be? Goodness, here, is defined
> as the S/N, or ratio of the measurement to its uncertainty. To be
> "minimally convincing", you need a 3-sigma detection. The required
> magnitude uncertainty is calculated as follows: 0.06 magnitude / 3
> sigmas = 0.02 magnitudes. That corresponds to a signal to noise
> ratio of S/N = 1.0857 / 0.02 which is about 55. We could roughly
> call this a S/N of about 50. This result assumes that all the photo-
> metric errors are properly included in the value of dM. An even
> smaller dM would give a more significant measurement but in this
> example, 0.02 magnitudes was required as a minimum.
> 
> Michael Newberry
> 
> 
> 
>> ----- Original Message ----- 
>> From: "Walt Cooney"
>> To: "'Laurent Corp'"; aavso-photometry
>> Sent: Saturday, January 27, 2007 8:56 PM
>> Subject: Re: [Aavso-photometry] magnitude limit
>> 
>>> At a S/N level of 3 the accuracy of your photometry is approximately
>>> the reciprocal of 3 or 1/3 of a magnitude.  3 is normally taken as the
>>> minimum sigma level for successful detection (for astrometry for in-
>>> stance) but not what you want to use to do a brightness measurement.
>>>  The AAVSO is in effect asking for photometry measurements with
>>> accuracy of approximately 1/50 or 0.02 mags. Real accuracy would
>>> actually be somewhat less than that due to many factors.
>>> 
>>> Clearest skies,
>>> Walt
>>> 
>>> 
>>> 
>>> -----Original Message-----
>>> From: aavso-photometry On Behalf Of Laurent Corp
>>> Sent: Saturday, January 27, 2007 12:44 PM
>>> To: aavso-photometry at mira.aavso.org
>>> Subject: Re: [Aavso-photometry] magnitude limit
>>> 
>>>> I assume you are talking about CCD data and not visual observa-
>>>> tions. In my paper, I give 2 formulae for S/N (Signal to Noise ratio).
>>>> The magnitude limit is usually taken to be a photometric measure-
>>>> ment with S/N = 3.  So you just solve the equation for exposure time
>>>> when S/N = 3. That estimate will include all the random noise sources
>>>> in the data image, but will not include processing noise that result
>>>> from correction bias, dark, and flat field. However, you should be
>>>> developing/using a data reduction technique that reduces the pro-
>>>> cessing noise to negligble levels.
>>> 
>>> Ok, but another question, i am agree that if the S/N = 3 , the measure
>>> will be good.
>>> But, why the AAVSO Alert Notice require a S/N = 50 ?
>>> 
>>> Laurent



----- Original Message ----- 
From: "Michael Newberry"
To: "Todd Schell"; aavso-photometry
Sent: Tuesday, January 30, 2007 12:38 AM
Subject: Re: [Aavso-photometry] Differential Photometry Questions


> Hi Todd,
> 
> Regarding the S/N (or SNR) value you mentioned, that is the signal
> to noise ratio for all the photons inside the star measuring aperture -
> including those from the sky background. That value includes the sky
> background so you get an idea of how much information you are
> collecting. The reason it does not calculate the S/N for the star
> *above* background is because that quantity is already provided in
> the table in the form of the magnitude random errors. Remember
> that S/N = 1.0857 / sigma(m), where sigma(m) is the random error.
> As a rule, the value in the "S/N" column will be higher than the S/N
> of the star measurement because the sky photons add noise but
> no information to the star measurement. The ratio of the value in the
> "S/N" column to the S/N calculated form the magnitude errors indi-
> cates how much of the magnitude measurement is being degraded
> by the sky brightness.
> 
> Another point to consider is that Mira gives you 2 random error esti-
> mates labeled "Error" and "Error(T)" (in older version of Mira they
> were "Error(E)" and "Error(T)").  So, along with the "S//N" column,
> you have 3 different S/N values to consider.
> 
> The column labeled "Error" is the random error based on the noise
> in the background annulus around the star, as well as instrumental
> parameters (Gain, Readnoise, and pixels in the star and sky sam-
> ples). The Error(T) column is a theoretical error based on the star
> and sky signal and instrumental parameters and does not depend
> upon the sky noise. In other words, "Error(T)" is "ideal" and "Error"
> measures what you actually got. Since there can be noise in the sky
> (actually, anywhere in the image) that is not accounted for by the
> instrumental and exposure parameters, the Error(T) is a smooth
> function of magnitude whereas the Error value has statistical fluc-
> tuations from star to star at the same magnitude. If you plot the
> values of Error and Error(T) against magnitude, you will see a shot-
> gun scattering of points that should track along the curve formed
> by Error(T). If the actual errors are much above the Error(T) values,
> then there is a significant source of noise in the image that is not
> accounted for by photon statistics, readout noise, and gain.
> 
> Michael Newberry
> Mirametrics, Inc.