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2 Patterns and local pile-up probability


2.1 General concepts

  The way to identify an event in a frame is to look for a local maximum above some statistical threshold, then to perform a proximity analysis around this maximum. The charge from an X-ray may either concentrate in a single pixel (monopixel events) or appear over several adjacent pixels (split events) forming a specific pattern or grade. The normal practise is to accept only isolated patterns (i.e. all pixels around the pattern, touching it even by a corner, must be below threshold).

\resizebox {\hsize}{!}{\includegraphics{patterns.eps}}\end{figure} Figure 1: Possible charge patterns for X-ray events. The black pixel is the local maximum. The grey pixels are other pixels above threshold. The white pixels must be below threshold (they form the exclusion zone for secondary monopixels). The crosses denote pixels whose value does not influence the classification of the main event. The third pattern (called mono +) is not necessarily isolated (it may include single events touching another one by a corner). The diagonal and mono + patterns are not produced directly by X-rays, but can occur as a consequence of pile-up

True X-rays may appear in a finite number of charge patterns (Fig. 1):

Monopixels, with no other pixel above threshold around.
Bipixels, with two adjacent pixels above threshold.
Tripixels, with two pixels above threshold beside the maximum. They are necessarily L-shaped (aligned tripixels are not produced by true X-rays) and the corner pixel always receives the largest charge.
Quadripixels, forming a closed square (elongated four-pixel patterns are not produced by true X-rays).
Bipixels can be horizontal or vertical. Tripixels can have any of four orientations. One usually defines the position of a split event by that of the pixel with highest charge, allowing the definition of four orientations for bi- and quadripixels also. I call $\alpha_i$ the proportion of X-rays creating a charge pattern with i pixels (this is also the pattern distribution of measured events in the ideal situation with no pile-up). The $\alpha_i$ usually depend on energy. The proportion of split events increases with energy for front illuminated CCDs such as XMM/EPIC-MOS. For back illuminated CCDs it is minimum at medium energy. Examples of $\alpha_i$ are [0.778, 0.195, 0.014, 0.013] at the energy of AlK (1.49 keV), [0.459, 0.303, 0.046, 0.192] at the energy of Ni$_{\rm K\alpha}$ (7.48 keV), measured with the XMM/EPIC-MOS (EPIC MOS team, 1994). The situation where a sizable fraction of events is split over more than four pixels is discussed in Sect. 4.6.

For a relatively large incoming flux, two X-rays may materialize as monopixels in adjacent pixels and be counted as an apparent split event. This comes in addition to the true pile-up (two X-rays in the same pixel). In the following I compute the apparent pattern distribution (after pile-up), neglecting the possibility that small amounts of charge (below threshold) in peripheral pixels of two close-by charge patterns may combine to exceed threshold.

I call $\lambda$ the incoming X-ray flux/pixel/frame (assumed uniform and random) and $\mu_j$ the expected (expectation value of the) count rate/pixel/frame in pattern j (to avoid ambiguities, I note i charge patterns of incoming photons and j patterns of detected events). Except in Appendix C the paper deals only with ideal (as opposed to noisy) quantities. The actual measured count rate would be $\mu_j$with Poisson fluctuations.

I call $\mu_j^{\rm t}$ the expected count rate/pixel/frame in pattern j of clean (not piled-up) events. $\mu_j^{\rm t}$ does not correspond to a measurable quantity, since one has no way to tell a clean event from a piled-up one a posteriori, but it is useful to know what it is. $1 - \mu_j/(\alpha_j \lambda)$ is called the flux loss. It is the loss in detection efficiency due to pattern overlap. $1 - \mu_j^{\rm t}/\mu_j$ is called the pile-up fraction. It is the fraction of measured events whose energy will be wrong. Note that in this paper I always refer to the flux loss and pile-up fraction per pattern. This is appropriate since the pattern is known in the data. Global quantities would be obtained by summing over the patterns.

In the next subsection and 6 I derive the relationship between $\mu_j$, $\mu_j^{\rm t}$, $\lambda$ and $\alpha_i$.The formulae appear as a product of the probability to get such a pattern times the probability not to destroy it by receiving another photon in the exclusion zone around the "target'' event. For $\mu_j^{\rm t}$, probability to get clean (not piled-up) events, the first (production) term is simply $\alpha_j \lambda$.

Since the probability not to receive a charge pattern of type i in a pixel is always of the form $\exp(-\alpha_i \lambda)$, the second (suppression) term is always of the form $\exp(-\gamma_j \lambda)$, where $\gamma_j$is a sum over all pixels in the exclusion zone (including the target event). I note $\gamma_j$ here the coefficient associated to $\mu_j^{\rm t}$ (clean events). For $\mu_j$ the exclusion zone would not include the target event, since additional charge patterns there would preserve the event's geometry. The exclusion zone is larger for larger patterns (because not one of their i pixels must be close to the original pattern), so that $\gamma_j$ will appear as $\sum n_{ji} \alpha_i$, where nji is the area of the exclusion zone for incoming pattern i and measured pattern j. Mathematically, one may start by convolving the area which must be below threshold around the measured pattern j, plus the pattern itself (black, grey and white pixels in Fig. 1), with the area of the incoming pattern i for a given orientation (black and grey pixels in Fig. 1). The exclusion zone (Fig. 2) may then be obtained by setting all pixels above 0 in the convolution to 1, multiplying by the fraction of patterns with that orientation, and summing over all orientations.

For $\mu_j$, the production term must include all possibilities of pile-up preserving the geometrical shape of the pattern, and the exclusion zone does not include the target event.

The formulae have been checked against simulations (Ferrando et al. 1996) for various pattern distributions and incoming flux. Except for $\mu_3$ which is known to be approximate anyway (6.3) I have seen no evidence of discrepancy. Early comparison to calibration data is presented in Appendix C.

2.2 Monopixels

\resizebox {\hsize}{!}{\includegraphics{exclude.eps}}\end{figure} Figure 2: Exclusion zones for keeping a single event clean (not piled-up) as a function of the charge pattern of the secondary photon. R (L, U, D) stands for the fraction of patterns extending rightward (left, up, down) from the central pixel. For tri- and quadripixels the individual patterns are DR, DL, UR, UL which stand for the fraction of patterns extending down and rightward, etc. If the pattern distribution is symmetrical, R = L = U = D = 0.25 for bipixels, 0.5 for tri- and quadripixels. DR = DL = UR = UL = 0.25 for tri- and quadripixels. The total exclusion area (sum of all coefficients) is the same (9, 12, 15, 16 for mono-, bi-, tri- and quadripixels) even if the distribution is not symmetrical

I first consider the simplest case of monopixel events. For such an event to remain clean (not piled-up) there must be no other pixel above threshold in the exclusion zone (Fig. 1, top left), nor in the target pixel. Therefore there must be no other monopixel pattern in the 9 corresponding pixels, whereas bipixels must be avoided over a larger area: the same 9 pixels, plus 3 pixels outside (on the left, top, right or bottom depending on the bipixel's orientation) from which the bipixel would spill over into a pixel immediately next to the event. The same reasoning leads to 15 pixels to be avoided for tripixels and 16 for quadripixels. Fig. 2 shows graphically the exclusion zone for all orientations of the secondary charge patterns. Its area is the same for all orientations because the target (monopixel) is symmetrical.

To get the production term for $\mu_1$, one can first say that monopixels can appear only from one or more superposed monopixels. The probability to have zero monopixels in a given pixel is $\mathrm e^{-\alpha_1 \lambda}$. Therefore the probability to have at least one is $1 - \mathrm e^{-\alpha_1 \lambda}$.Finally the target pixel must be removed from the exclusion area for monopixels.

Finally, using $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 1$,one obtains:
\gamma_1 & = & 9 \: + \: 3\,\alpha_2 \: + \: 6\,\alpha_3 \: + \...
 ...{\rm t} & = & (\alpha_1 \lambda) \; \mathrm e^{-\gamma_1 \lambda}.\end{eqnarray} (1)
The probabilities to get other patterns are detailed in Appendix A.

\includegraphics [width=8.8cm]{flux_empil_flat.eps}\end{figure} Figure 3: Expectation value of the monopixel count rate (solid line, scale on the left axis) as a function of input photon flux for a flat spatial distribution. The dashed line is the ratio of single events in diagonal events to truly isolated monopixels (right axis). The dotted line is the fraction of piled-up single events (right axis). Two pattern distributions are superposed. The upper lines correspond to the ideal case of all monopixels. The lower lines correspond to the pattern distribution [0.459, 0.303, 0.046, 0.192] which includes larger events. For this case the monopixel count rate (solid line) was divided by $\alpha_1 = 0.459$ to allow easier comparison with the case of all monopixels. It is apparent that the presence of large events further suppresses the count rate, but reduces the proportion of diagonal events and of piled-up events

2.3 Measured count rate versus input photon flux

  From Eq. (2) (and those in 6) it is easy to plot, for a given pattern, the expected count rate as a function of incoming flux. Figure 3 illustrates this for the ideal case of a pure monopixel distribution. The maximum count rate is achieved for an incoming flux of  
\lambda_0 = \frac{\left\vert \ln(1-\alpha_1 / \gamma_1) \right\vert}{\alpha_1}\cdot\end{displaymath} (4)
For a pure monopixel distribution $\alpha_1 = 1$ (so $\gamma_1 = 9$), $\lambda_0 = 0.118$and the maximum count rate is $4.33\ 10^{-2}$.This corresponds to the maximum of the radial curve (Fig. 4). At this value of $\lambda$ the pile-up fraction (dotted curve) is still no larger than 5.77%.

Because the expected count rate for a flat spatial distribution is not strictly increasing as a function of incoming photon flux, there is an ambiguity in reconstructing the incoming photon flux. This ambiguity can be overcome by considering at the same time the fraction of diagonal events over truly isolated monopixels (dashed line in Fig. 3). This is also by itself ambiguous, but matching both is not. At this kind of count rate the statistics (even in a single frame) is enough to distinguish solutions unambiguously, except at very high incoming flux, which is not a very interesting case.

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