Pileup for PSF concentrated measurements

When more than one X-ray photon interaction with the CCD places charge in the same or neighboring pixels, the ACIS event processing software has difficulties reconstructing the energy, grade, or even number of photons. The effect of these multiple photon charge cloud combinations is called `pileup'. As discussed in Chapter 4, the case of uniform illumination pileup removal is reasonably well modelled for moderate incident flux levels.

In this section we examine the XRCF data to develop empirical measures of the proper correction for pileup in the case of a focussed illumination emerging from the HRMA and striking ACIS. Data for this situation were collected as part of the Count-Rate Linearity tests (cf. Table ).

If the HRMA Point Spread Function (PSF) were infinitely narrow, then a simplified treatment of pileup would consist of the following simple algorithm. The spectrum of a mono-energetic incident flux would appear in the CCD as a series of peaks, each separated by the energy of the individual photons. The number of events found at the incident energy, E$_{\rm L}$,would be the number of frames containing a single photon. The number of events seen at the apparent energy of $2 \times {\rm E}_{\rm L}$ would be the number of frames with two piled-up photons. The number of events at $3 \times {\rm E}_{\rm L}$is the number of frames with three piled-up photons, and so on. Thus to extract the true number of incident photons one could integrate over each peak in the observed CCD spectrum and sum them, after weighting by how many photons occur in each peak. Expressed as an equation:  

$$n ~=~ \sum_{i=1}^{\infty} i \cdot c_i$$ (65)

where

$$c_i ~=~ \int_{i \cdot {\rm E}_{\rm L} - \Delta E}^{i \cdot {\rm E}_{\rm L} + \Delta E} I(E) dE$$ (66)

I(E) is the observed CCD spectrum and $\Delta E$ is the energy halfwidth of the line.

  

Figure 6.19: Plot of CCD detected flux vs. BND count rate; circles are CCD flux in single photon peak alone - diamonds are CCD flux corrected for higher order pileup peaks.

Unfortunately this simple algorithm is insufficient for complete correction. Fig. 6.19 shows the result of fitting Gaussian profiles to each peak in the countrate linearity measurements at 1.486 keV (Al-K$\alpha$). The circles are the number of counts per second in the single photon peak alone, while the diamonds are the inferred rate based on equation 6.11. Only events falling into the ASCA grades 02346 were included.

  

Figure 6.20: Plot of CCD detected flux vs. BND count rate using ROI; circles are CCD flux in single photon peak alone - diamonds are CCD flux corrected for higher order pileup peaks.

If the same algorithm is applied, but replacing the number of events by the result of a simple region of interest (ROI) starting from the top of the lower order pileup peak, going up to the top edge of the given pileup peak, then the pileup correction for the same data looks like Fig. 6.20. The pileup correction is clearly still not correcting for all events (otherwise the diamonds would fall on a straight line, indicating direct proportionality between the CCD inferred rate and the BND rate), but using ROI detects significantly more events than Gaussian fits to the peaks. The implication is that the interaction between multiple charge clouds produce event spatial distributions which cause a loss of charge (to the event reconstruction algorithm). This is not impausible if we consider that the HRMA PSF is not perfect, but causes photons to be distributed with a $\sim$0.5 arc second FWHM width over the CCD (roughly a pixel). Thus succeeding photons do not always strike the same pixel, but frequently neighboring pixels. When this occurs further charge splitting results in some charge outside the 3x3 event reconstruction neighborhood, and thus to loss of charge. For the succeeding plots we use the ROI method when reconstructing events.

  

Figure 6.21: Plot of Pile Up Fraction vs. number of counts per frame; Circles: CCD frame time is held constant at 0.11 seconds, while beam intensity is varied; Stars: CCD frame time varies, while beam intensity is constant.

In the Count-Rate Linearity tests the number of photons per frame was regulated in two ways: the X-ray beam intensity was increased at a single frame time (0.11 second); and the X-ray beam intensity was held constant while the CCD frame times were increased (0.11, 0.22, 0.33, 0.66 second). In principle the relevant quantity describing the pileup behavior should be the number of photons per frame (which is the product of the frame time times the rate of photons per frame). To check this we plot the `Pile Up Fraction' versus counts per frame with constant frame time (filled circles) and with constant incident X-ray flux (stars; Fig. 6.21).

The `Pile Up Fraction' is defined as the ratio of the number of events inferred in the n=2 and higher peaks divided by the total number of events, including the n=1 peak. The two sets of points are in good agreement, leading us to conclude that the pileup effect can be treated as a function of the total counts per frame (within the PSF), independent of the frametime.

  

Figure 6.22: Plot of pileup corrected CCD flux with all grades (with and without ACIS grade 255) - Open symbols: all grades, including ACIS 255; Filled symbols: all grades, excluding ACIS 255.

If, instead of using the standard grade selection (g02346), we accept all events regardless of grade, then the pileup correction of equation 6.11 becomes much better. Figure 6.22 shows the correlation of the total ACIS rate (all grades) after pileup correction versus the incident beam (as determined by the BND counting rate). Note that the circles form a nearly straight line, indicating that the pileup corrected CCD rate is proportional to the BND rate, and hence the incident flux. Even if some X-rays are not being counted, the linearity and proportionality shows that we will be able to calibrate a conversion factor to correct piled-up photons into incident X-ray flux.

Unfortunately the total rate expected from background events in orbit will saturate the telemetry if no grade selection is applied. A significant reduction in charged particle events can be achieved by merely excluding the ACIS grade 255 events (i.e. all eight neighbors of the central pixel exceed the split event threshold). The proportionality of the CCD corrected rate to the BND rate remains, indicating that exclusion of grade 255 still allows flux pile-up correction.

The success of the pileup correction in this monochromatic case does not mean that the pileup problem is solved in general. In astrophysical spectra the usual case is a distribution of many photon energies. When multiple photons are combined we lose the ability to individually recognize them. Moreover as the incident energy changes so to does the event spreading, which means that the monochromatic case will need to be explored at differing energies.

Finally we will need to explore the effect of pileup using simulation to check the efficiency of our correction to that predicted by the model. In this way we can validate the model for application to more complex, and realistic, situations.