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XMM-Newton Science Analysis System


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Detection sensitivity

Suppose one has a random variate $c$ with a probability distribution $p(c)$. The nett probability that a given sample of $c$ is due to chance is the integral of $p$ from $c$ to $\infty$:

$$P(c) = \int_c^\infty dx \ p(x).$$

In the case of a single pixel of a single image, $p(c)$ is the Poisson distribution about the expectation value of the counts due to background, $\langle b\rangle$. As described in section 3.2, we say there is a source present in this pixel if $L = -\ln{P}$ is larger than a cutoff value $L_{\rm {cutoff}}$. In principle it is possible to invert the relationship between detected counts $c$ and likelihood $L$, to calculate that value of $c$ which would give $L = L_{\rm {cutoff}}$. This value of $c_{\rm {cutoff}}$, minus the expectation due to background $\langle b\rangle$, is defined here as the detection sensitivity.

Note that this does not mean that a source with an expectation value of counts $\langle s\rangle$ which is greater than $c_{\rm {cutoff}} - \langle b\rangle$ will always be detected. There are always statistical fluctuations to consider. The probability that a source with $\langle s\rangle = c_{\rm {cutoff}} - \langle b\rangle$ will be detected is the integral from $a$ to infinity of the Poisson distribution with expectation value $a$ equal to $c_{\rm {cutoff}} - \langle b\rangle$. This is equal to 0.5 in the limit of large $a$, but becomes significantly less than 0.5 for $a <$ about 1. Also, the detection cutoff is naturally not sharp: fainter sources have still some non-zero probability of detection, and sources brighter than cutoff have always some non-zero probability of non-detection.

Where one is performing source detection in parallel on $N>1$ images, there are $N$ inputs to the calculation of nett likelihood at any given pixel. In this circumstance it is no longer possible to invert this calculation to obtain a single detection sensitivity, since there may be more than one combination of counts which yield the same nett $L$. Here a definition of sensitivity only makes sense in connection with a fixed source spectrum, as follows. Suppose that at the pixel in question the expected count values $\langle c_i\rangle$ are made up from background $\langle b_i\rangle$ plus source $\langle s_i\rangle$. (The expectation value of two summed Poisson variates is equal to the sum of the two expectation values.) Suppose also that we know the source spectrum and are thus able to express the source counts as a product between this spectrum and some nett intensity $S$:

$$\langle s_i\rangle = \langle S\rangle \frac{\langle s_i\rangle}{\sum_i^N \langle s_i\rangle}.$$

Regardless of the precise algorithm employed, likelihood $L$ is some function of the counts $c_i$, ie

$$L = f(c_i).$$

The detection sensitivity $S_{\rm {cutoff}}$ can therefore be defined implicitly as follows:

$$L_{\rm {cutoff}} = f\bigg(\langle b_i\rangle + S_{\rm {cutoff}} \frac{\langle s_i\rangle}{\sum_i^N \langle s_i\rangle}\bigg).$$ (2)