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8.2.1.1 Photon counting noise

With the very accurate electronics used in today’s detectors, photon counting noise206 is the most significant source of uncertainty in most datasets. To understand this noise (error in counting) and its effect on the images of astronomical targets, let’s start by reviewing how a distribution produced by counting can be modeled as a parametric function.

Counting is an inherently discrete operation, which can only produce positive integer outputs (including zero). for example, we cannot 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 book207.

Let’s take \(\lambda\) to represent the expected mean count of something. Furthermore, let’s take \(k\) to represent the output of a counting attempt (hence \(k\) is a positive integer). 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 integer 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 for smaller values near zero, there simply are not enough integers smaller than the mean, than integers that are larger. Therefore to accommodate all possibilities/counts, it has to be strongly skewed to the positive when the mean is small. For more on Skewness, see Skewness caused by signal and its measurement.

As \(\lambda\) becomes larger, the distribution becomes more and more symmetric, and the variance of that distribution is equal to its mean. In other words, the standard deviation is the square root of the mean. It can also be proved that when the mean is large, say \(\lambda>1000\), the Poisson distribution approaches the Normal (Gaussian) 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 has one free parameter (\(\mu=\lambda\) and \(\sigma=\sqrt{\lambda}\)), instead of the two parameters that the Gaussian distribution originally has (independent \(\mu\) and \(\sigma\)).

In real situations, the photons/flux from our targets are combined with photons from a certain background (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, 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 known208. 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 summary, the value in each element of the dataset (pixel in an image) is the sum of contributions from various galaxies and stars (after convolution by the PSF, see Point spread function). Let’s name the convolved sum of possibly overlapping objects in each pixel as \(I_{nn}\). \(nn\) represents ‘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 of that pixel, after adding noise, is a random value taken from a Gaussian distribution with the following mean (\(\mu\)) and standard deviation (\(\sigma\)):

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

In astronomical instruments, \(B\) is enhanced by adding a “bias” level to each pixel before the shutter is even opened (for the exposure to start). As the exposure is ongoing and photo-electrons are accumulating from the astronomical objects, a “dark” current (due to thermal radiation of the instrument) also builds up in the pixels. The “dark” current will accumulate even when the shutter is closed, but the CCD electronics are working (hence the name “dark”). This added dark level further enhances the mean value in a real observation compared to the raw background value (from the atmosphere for example).

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 Brightness, Flux, Magnitude and Surface brightness. It is then internally converted to the flux scale for further processing.

The equations above clearly show the importance of the background value and its effect on the final signal to noise ratio in each pixel of a science image. It is therefore, one of the most important factors in understanding the noise (and properly simulating observations where necessary). An inappropriately bright background value can hide the signal of the mock profile hide behind the noise. In other words, a brighter background has larger standard deviation and vice versa. As a result, the only necessary parameter to define photon-counting noise over a mock image of simulated profiles is the background. For a complete example, see Sufi simulates a detection.

To better understand the correlation between the mean (or background) value and the noise standard deviation, let’s use an analogy. Consider the profile of your galaxy to be analogous to the profile of a ship that is sailing in the sea. The height of the ship would therefore be analogous to the maximum flux difference between your galaxy’s minimum and maximum values. Furthermore, let’s take the depth of the sea to represent the background value: a deeper sea, corresponds to a brighter background. In this analogy, the “noise” would be the height of the waves that surround the ship: in deeper waters, the waves would also be taller (the square root of the mean depth at the ship’s position).

If the ship is in deep waters, the height of waves are greater than when the ship is near to the beach (at lower depths). Therefore, when the ship is in the middle of the sea, there are high waves that are capable of hiding a significant part of the ship from our perspective. This corresponds to a brighter background value in astronomical images: the resulting noise from that brighter background can completely wash out the signal from a fainter galaxy, star or solar system object.


Footnotes

(206)

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

(207)

[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.

(208)

In a real image, a relatively large number of very faint objects can be 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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