GNU Astronomy Utilities


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8.2.1.1 Photon counting noise

Thanks to the very accurate electronics used in today’s detectors, this type of noise is the main cause of concern for extra galactic studies. It can generally be associate with the counting error that is known to have a Poisson distribution. The Poisson distribution is about counting. But counting is a discrete operation with only positive values, for example we can’t count \(3.2\) or \(-2\) of anything. We only count \(0\), \(1\), \(2\), \(3\) and so on. Therefore the Poisson distribution is also a discrete distribution, only applying to whole positive integers.

Let’s assume the mean value of counting something is known. In this case, we are counting the number of electrons that are produced by photons in a detector (for example CCD) pixel. Let’s call this mean \(\lambda\). Furthermore, let’s take \(k\) to represent the result of one particular counting attempt. The probability density function of \(k\) can be written as:

$$f(k)={\lambda^k \over k!} e^{-\lambda},\quad k\in {0, 1, 2, 3, \dots }$$

Because the Poisson distribution is only applicable to positive values, naturally it is very skewed when \(\lambda\) is near zero. One qualitative way to explain it is that there simply aren’t enough integers smaller than \(\lambda\), than integers that are larger than it. Therefore to accommodate all possibilities, it has to be skewed when \(\lambda\) is small.

But as \(\lambda\) becomes larger and larger, the distribution becomes more and more symmetric. One very useful property of the Poisson distribution is that the mean value is also its variance. When \(\lambda\) is very large, say \(\lambda>1000\), then the normal (Gaussian) distribution, see Point Spread Function, is an excellent approximation of the Poisson distribution with mean \(\mu=\lambda\) and standard deviation \(\sigma=\sqrt{\lambda}\).

We see that the variance or dispersion of the distribution depends on the mean value, and when it is large it can be approximated with a Gaussian that only has one free parameter (\(\mu=\lambda\) and \(\sigma=\sqrt{\lambda}\)) instead of two that it originally has.

The astronomical objects after convolution with the PSF of the instrument, lie above a certain background flux. This background flux is defined to be the average flux of a region in the image that has absolutely no objects. The physical origin of this background value is the brightness of the atmosphere or possible stray light within the imaging instrument. It is thus an ideal definition, because in practice, what lies deep in the noise far lower than the detection limit is never known114. However, in a real image, a relatively large number of very faint objects can been fully buried in the noise. These undetected objects will bias the background measurement to slightly larger values. The sky value is therefore defined to be the average of the undetected regions in the image, so in an ideal case where all the objects have been detected, the sky value and background value are the same.

As longer wavelengths are used, the background value becomes more significant and also varies over a wide image field. Such variations are not currently implemented in MakeProfiles, but will be in the future. In a mock image, we have the luxury of setting the background value.

In each pixel of the canvas of pixels, the flux is the sum of contributions from various sources after convolution. Let’s name this flux of the convolved sum of possibly overlapping objects, \(I_{nn}\). \(nn\) representing ‘no noise’. For now, let’s assume the background is constant and represented by \(B\). In practice the background values are larger than \(\sim1,000\) counts. Then the flux after adding noise is a random value taken from a Gaussian distribution with the following mean (\(\mu\)) and standard deviation (\(\sigma\)):

$$\mu=B+I_{nn}, \quad \sigma=\sqrt{B+I_{nn}}$$

Since this type of noise is inherent in the objects we study, it is usually measured on the same scale as the astronomical objects, namely the magnitude system, see Flux Brightness and magnitude. It is then internally converted to the flux scale for further processing.


Footnotes

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See the section on sky in Akhlaghi M., Ichikawa. T. 2015. Astrophysical Journal Supplement Series.


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