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SourceDetection

This is a description of some code tailored for an X-ray detector, but it is applicable to any photon-counting detector.

Define the counts from a source as $X$. When searching for sources, one defines a region on the detector (i.e., the expected physical size of the object, such as the radius of a galaxy cluster, or some large fraction of the point spread function) and asks how likely it is that the number of counts observed in that region ($S$) would exceed the number one would have expected from the background ($B$). Here, $S=X+B$. In the following, we assume that the uncertainty on the expected background rate is negligible. This is generally reasonable for an imaging detector, because the background can be measured from the larger part of the detector that is not covered by the small region of interest, or it can be modeled based on other information (e.g., CCD read noise for the optical, sky brightness in the infrared, or position on the orbit and event rates in anti-coincidence layers for X-ray detectors).

For large numbers of expected background counts ($B$$>$120 for our code), one can approximate a detection significance by considering the signal to noise in the Gaussian regime:
\begin{equation}
\sigma = S/B^{1/2}.
\end{equation}
Here, $B^{1/2}$ is the 1$\sigma$ uncertainty on the background counts in the limit of large $B$ (i.e., $\ge$100), and assuming that the uncertainty on the background rate is negligible. The signal-to-noise $\sigma$ is the number of standard deviations in a Gaussian distribution that enclose the probability corresponding to the confidence limit, $C$ (where $1-C << 1$). Optical and infrared detectors always operate in this regime.

For small background rates, such as those from an imaging X-ray telescope, Equation~\ref{eq:sn} is not accurate. In photon-counting experiments, the probability that the observed source number of counts ($S$) are produced by background with an expected number of counts $B$ is given by the Poisson distribution:
\begin{equation}
{\rm Prob}(S|B) = \frac{e^{-B} S^B}{S!}.
\end{equation}
We are interested in computing $C$, which represents the cumulative probability that at fewer than $S$ counts would be detected given $B$:
\begin{equation}
C(<S|B) = \sum_{x = 0}^{S-1} \frac{e^{-B} x^B}{x!}.
\end{equation}
Our code inverts this equation in the a straightforward manner: we increment $x$ until $C$ exceeds our specified value, and that defines $S$. This works as long as $C$ can be distinguished from 1 in a double-precision number (this is true for $\sigma$$<$8). We use logarigthms of the factorial ($x!$ can be written as a gamma function, $x! = \Gamma(x+1)$, and IDL has a built-in function lngamma) and exponential terms, to avoid numerical overflows.




Created by: MikeMuno. Last Modification: Monday 06 of October, 2008 20:05:14 CDT by MikeMuno.