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8.2.1.1 Photon counting noise

With the very accurate electronics used in today’s detectors, photon counting noise153 is the most significant source of uncertainty in most datasets. To understand this noise (error in counting), we need to take a closer look at how a distribution produced by counting can be modeled as a parametric function.

Counting is an inherently discrete operation, which can only produce positive (including zero) integer outputs. For example we can’t count \(3.2\) or \(-2\) of anything. We only count \(0\), \(1\), \(2\), \(3\) and so on. The distribution of values, as a result of counting efforts is formally known as the Poisson distribution. It is associated to Siméon Denis Poisson, because he discussed it while working on the number of wrongful convictions in court cases in his 1837 book154.

Let’s take \(\lambda\) to represent the expected mean count of something. Furthermore, let’s take \(k\) to represent the result of one particular counting attempt. The probability density function of getting \(k\) counts (in each attempt, given the expected/mean count of \(\lambda\)) 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 (note the factorial operator, which only applies to non-negative integers), naturally it is very skewed when \(\lambda\) is near zero. One qualitative way to understand this behavior is that there simply aren’t enough integers smaller than \(\lambda\), than integers that are larger than it. Therefore to accommodate all possibilities/counts, it has to be strongly skewed when \(\lambda\) is small.

As \(\lambda\) becomes larger, the distribution becomes more and more symmetric. A 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, is an excellent approximation of the Poisson distribution with mean \(\mu=\lambda\) and standard deviation \(\sigma=\sqrt{\lambda}\). In other words, a Poisson distribution (with a sufficiently large \(\lambda\)) is simply a Gaussian that only has one free parameter (\(\mu=\lambda\) and \(\sigma=\sqrt{\lambda}\)), instead of the two parameters (independent \(\mu\) and \(\sigma\)) that it originally has.

In real situations, the photons/flux from our targets are added to a certain background flux (observationally, the Sky value). The Sky value is defined to be the average flux of a region in the dataset with no targets. Its physical origin can be the brightness of the atmosphere (for ground-based instruments), possible stray light within the imaging instrument, the average flux of undetected targets, or etc. The Sky value is thus an ideal definition, because in real datasets, what lies deep in the noise (far lower than the detection limit) is never known155. To account for all of these, the sky value is defined to be the average count/value of the undetected regions in the image. In a mock image/dataset, we have the luxury of setting the background (Sky) value.

In each element of the dataset (pixel in an image), the flux is the sum of contributions from various sources (after convolution by the PSF, see Point spread function). Let’s name the convolved sum of possibly overlapping objects, \(I_{nn}\). \(nn\) representing ‘no noise’. For now, let’s assume the background (\(B\)) is constant and sufficiently high for the Poisson distribution to be approximated by a Gaussian. 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

(153)

In practice, we are actually counting the electrons that are produced by each photon, not the actual photons.

(154)

[From Wikipedia] Poisson’s result was also derived in a previous study by Abraham de Moivre in 1711. Therefore some people suggest it should rightly be called the de Moivre distribution.

(155)

In a real image, a relatively large number of very faint objects can been fully buried in the noise and never detected. These undetected objects will bias the background measurement to slightly larger values. Our best approximation is thus to simply assume they are uniform, and consider their average effect. See Figure 1 (a.1 and a.2) and Section 2.2 in Akhlaghi and Ichikawa [2015].


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