HRC Degapping Corrections

Introduction

The HRC uses two one-dimensional grids of wires to determine the locations of x-ray events. The charge cloud emitted from the back of the bottom MCP is collected on these crossed grids, the Crossed-Grid Charge Detector (CGCD). The charges on the grid wires are combined and amplified in a set of charge amplifiers and the signals from the three amplifiers at the center of the charge distribution on each axis are included in telemetry stream from the HRC. The principle of operation of the CGCD can be found at this link.

The "3-tap" position algorithm is used to convert the three amplifier (or tap) signals from an axis into a position relative to the central tap. If Q_A, Q_B, and Q_C are the signals from the three taps, then event position can be approximated by

X_fine = (Q_C - Q_A)/(Q_A + Q_B + Q_C).

Unfortunately, the size of the charge cloud can span a range of wires larger than can be collected by the three taps; as a result some information can be lost. The loss of information, in turn, results in the mis-location of the event position when using the simple algorithm given above. The worst case occurs when the event occurs half-way between two taps. In this case two of the three signals will be about equal and the third signal will not be balanced by the signal from a "fourth" tap. The event position will be biased away from the mid-point between the two nearly equal signals toward the center tap. This generates a gap in the image. It should be stressed that these gaps are not caused by a lack of detector response but by the inadequacy of the simple position determination algorithm. Murray and Chappell (1989) have shown that the accuracy of the event position reconstruction can be improved by applying linear or quadratic corrections to the relative position:

X_corrected = a×X_fine + b×sign(X_fine)×X_fine²

For a more complete description of modeling the HRC CGCD see Murray and Chappell (1989) or "Simulating the Crossed-Grid Charge Detector Response to an Event Charge Cloud".

Determining Degap Corrections

Flat field illumination of the entire detector provides a good data set for the determination of the linear or quadratic degapping corrections. If the illumination and the detector response are uniform, then the mean number of counts expected in each pixel of the image should be the same. Since the position determination algorithm assumes that the two axes are not coupled, the two-dimensional image can be projected into two one-dimensional histograms for separate analysis on each axis. Below is an example of a section of one histogram of undegapped data from a flat-field test of the HRC-I.

In this projection the gaps at the 1/2 coarse position points are easily seen. A linear degap correction would close the gap by shifting all the calculated positions away from the center of the coarse position by a constant fraction. To the extent that the distribution of events (in the region were there are events) is constant, the linear correction is good. In the figure below the same region as in the previous figure has been degapped using a linear correction (a = 1.055, b = 0.0).

Most of the evidence for gaps has disappeared from this projection. There is a small residual gap between 30 and 31 but this could be removed by choosing a different value for the linear correction around this gap. The linear correction is determined by how far the edge of the distribution must be moved to reach the half-way point between amplifiers.

A more difficult situation arises when the distribution of events is not flat. In the above figures there is some suggestion of this across coarse position 30, which peaks toward the center. A more extreme example is shown below in a projection of undegapped data from HRC-S flat field data.

Here the event distribution is not flat over the region in which events are located. The events near the center of the coarse position must be moved a larger fraction away from center than those which are farther out. This can be accomplished using quadratic degapping corrections. The figure below shows this same HRC-S section as previously but this time using a set of fixed quadratic coefficients to degap the data (a = 1.18, b = -0.16).

There are some residual problems at the gaps, both over- and under-correction, but these can be handled by adjusting the coefficients for each coarse position.

Rather than looking for how much "stretch" is required to fill the gap (a linear term), the quadratic coefficients can be found by minimizing the deviation between the positions of the edge of the event distribution and the points which include 1/4, 1/2, and 3/4 of the events from the coarse position center and their expected locations (at 0.5, 0.125, 0.25 and 0.375 respectively). This idea is shown schematically in the following figure. This is similar to the linear "stretch" except that the coefficients are found that do the best job of shifting each of the four points toward the appropriate location.

The point labeled X_1/8 in this figure should correspond to a value of 1/8 relative to the coarse position. Similarly X_1/4, X_3/8, and X_1/2 should correspond to values of 1/4, 3/8, and 1/2. The minimization can be performed for the "minus" and "plus" side of each coarse position providing a set of position dependent degapping parameters.

Details of the derivation of a specific set of degap parameters are given at the following links.

Linear-Quadratic Comparison

The quadratic coefficients will remove central peaking in the data as shown in the figures from the HRC-S above. But an interesting question is how much different are the event locations between linear and quadratic degapped data. The figure below shows the difference in the event location between quadratic degapping (using a = 1.18 and b = -0.16, as for the HRC-S U-axis above) and linear degapping (using a = 1.108 and b = 0.0).

These values reflect the the more extreme values for quadratic degapping coefficients needed for the HRC. The maximum difference between the locations using the two techniques is about 2 pixels. The differences will be even less on the HRC-S V-axis and the HRC-I since the gaps are smaller and the undeppaged histograms show less peaking toward the center.

Difficulties and Cautions

There is one major assumption that goes into the derivation of the degapping coefficients discussed above: the mean number of events per unit detector area is constant. There are two ways in which this assumption is invalid. First, the illumination of the detector may not be flat. Second, the response of the detector may not be constant. This second assumption is obviously not true for the HRC-S, since the UV/ion-shield transmission causes detector response variations. The first assumption was also not true for the HRC-S flat field data because the High-Energy Suppression Filters had to be occulted during the flat field measurements to prevent reflections. Both of these problems appear in the HRC-S U-axis projection shown above; the tap at coarse position 6 is where the illumination is occulted and coarse position 10 contains the transition between the thin and thick Al coatings that make the top of the "T" shape.

It may be possible to account for the "non-flatness" of the histograms in the derivation of degapping parameters by using a template for the expected variations rather than assuming a "flat" distribution. This is an area for future work.

References

Dr. Michael Juda
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Mail Stop 70
Cambridge, MA 02138, USA
Ph.: (617) 495-7062
Fax: (617) 495-7356
E-mail: mjuda@cfa.harvard.edu