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Simulations Of Alternative Designs For The Glast Calorimeter

Heather Arrighi, Jay Norris, and Steve Ritz
December 5, 1998

  1. Summary
  2. Introduction
  3. Implementation
  4. Analysis
  5. Results
  6. Discussion

1. Summary

We performed simulations using Glastsim to study alternative designs for the GLAST calorimeter. We simulated designs with a pair of SSD planes following a pre-calorimeter zone of 1 to 4 layers (1.25 to 5. radiation lengths), and a 10-cm gap following the SSD planes. The SSDs afford a precise measurement of the shower vertex. Since at least two points are needed to reconstruct the trajectory, to estimate the achievable angular resolution we utilize the shower barycenter — the next best-measured point — as determined in the lower calorimeter layers.

Two normally incident illumination patterns were studied with a monoenergetic beam of 10-GeV gammas: an infinitesimal pencil beam impinging on the centers of crossed crystals in the hodoscopic arrangement near the center of GLAST; and uniform illumination of the center module. We compare results for embedding SSD planes below the first CsI layer to those obtained for the Baseline Calorimeter geometry (8 layers, 9.9 radiation lengths) with uniform illumination of all 25 modules, for which the 68% containment radius (R68) is » 2.2°:

  • Pencil beam illumination of the split calorimeter with a pair of SSDs positioned below the first CsI layer, with no 10-cm gap, resulted in R68 » 1.9°, improved by ~ 10 - 15% compared to the Baseline R68 statistic.
  • Inclusion of a 10-cm gap after the first layer and SSD planes yielded an R68 » 1.0°, an improvement by a factor of two compared the Baseline, and attributable to a larger level arm across which the shower is measured, combined with a precise measurement of the shower vertex.
  • However, the second, more realistic illumination pattern of the entire central module, with 10-cm gap included, yielded R68 » 1.4°. The difference is reminiscent of results obtained in previous simulations of normal incidence, pencil beam versus uniform illumination of the coarse-dimensioned crystals.
  • Only » 63% of gammas convert in the top layer; hence the R68 » 1.4° applies only to this fraction. Trajectories of the remaining » 37% would be measured without benefit of the 10-cm gap, using the moments analysis. The weighted estimate (0.63 x 1.4°) + (0.37 x 2.2°) » 1.7°, is similar to that obtained for the two error distributions are combined their respective fractions and the R68 statistic computed, yielding » 1.8°.

2. Introduction

GLAST (Gamma-ray Large Area Space Telescope) is designed with three major components: tracker, calorimeter, and anti-coincidence shield. Given GLAST's current design, an incoming gamma ray will convert (i.e. pair produce) within the tracker approximately forty percent of the time. The remaining sixty percent will continue into calorimeter and pair produce there. In addition, downward-moving events which enter the side of the calorimeter have no opportunity to be detected by the tracker. It is desirable to recover these "lost" events and extract as much information about them as possible. To that end, work is being done to use the calorimeter as a coarse tracking device for those events which have no tracker information. This will be referred to as CAL-only mode.

Use of the calorimeter as a tracking device presents some challenges. Specifically, the calorimeter is composed of large Cesium Iodide (CsI) crystals with dimensions 32 cm (height) x 3.2 cm (width) x 2.3 cm (height). A calorimeter module (one of 25 such modules in the whole instrument) consists of eight layers, with ten crystals per layer. The layers are arranged in a hodoscopic pattern: alternating layers have their longitudinal coordinate oriented in either the X or Y direction. Thus, hits within a CsI crystal have a corresponding coordinate in three-space, where two of the coordinates are simply the center of the crystal, and the third (longitudinal) coordinate is determined by the difference in light collected by photo-diodes at the two ends of a crystal. Since the logs have relatively coarse dimensions, limited positional accuracy is obtained for CAL-only events.

When a gamma ray enters the calorimeter, an electromagnetic shower ensues. The shower vertex, or initiation point, will provide constraining information for determining the direction of the incoming gamma ray in the absence of tracker information. Hence, an important task in using the calorimeter as a tracking device involves measuring the shower vertex. To improve upon the shower vertex precision obtained solely with the CsI crystals, Silicon Strip Detectors (SSDs) may be used. These are the same detectors used in the tracker. Currently, the GLAST design uses SSDs with a strip pitch of 200 μm. Again, SSDs are set up in pairs, in order to obtain measurements in two orthogonal directions. Using SSDs in the calorimeter would require modifying the original design. The modified design envisages a pre-calorimeter consisting of one or more layers of CsI logs, followed by a pair of SSDs and possibly a 10-cm gap; and finally the remaining layers of the calorimeter. There are two clear reasons for including the gap. It will increase the lever-arm and decrease the amount backsplash from the main calorimeter that can impinge on the anti-coincidence shield. Figure 1 illustrates the instrument with SSDs included in the calorimeter along with a 10-cm gap. The configuration with an optional 10-cm gap will be referred to as Pre-CAL.

Instrument with SSDs included in the calorimeter along with a 10-cm gap

This document describes the results of simulating a hodoscopic CsI calorimeter where SSDs are inserted, with and without a gap, between the pre-calorimeter and main calorimeter. The following sections describe the simulation design, the resulting measurements, and conclusions.

3. Implementation

Simulations were done using Glastsim 5.0 released in June, 1998. Glastsim includes a hodoscopic calorimeter configuration. To incorporate the Pre-CAL configuration required modifying the existing code. The number of layers in the pre-calorimeter is set in the initialization file for the simulation. The pre-calorimeter is followed by a 1-cm gap and then a pair of SSDs with a 3-mm gap between them. None of the other infrastructure from the tracker trays is incorporated into the simulation. These changes required modifying the geometry of the calorimeter, as well as reading out the SSD hits from the calorimeter. The classes that were modified were: Calorimeter, CsIData, GlastData, Tray, and CalController.

The modifications to Calorimeter involved introducing an additional option for the configuration of the calorimeter. Originally the two options were vertical logs and the hodoscopic horizontal logs. Each option is represented by its own member function in the Calorimeter class. A new member function called PreCal inserts a pair of Trays in the calorimeter between the main calorimeter and the pre-calorimeter. The Tray objects are Silicon layers which correspond to SiDetectors. The location of the SSDs and the presence of the 10-cm gap are specified on the input from the initialization file. Glastsim denotes which materials in the instrument description behave as detectors. Each detector has a score function which is called for every particle that traverses a detector material. For example, the CsIDetector score function keeps track of the amount of energy deposited in a CsI crystal. At the end of each event, a list of hit CsI crystals and SSDs has been generated. The CalController class handles the readout of the data from the detectors within the calorimeter. Previously, only CsIDetectors existed within the calorimeter. The Pre-CAL configuration includes SiDetectors as well. It was necessary to modify CalController to recognize SiDetectors. An additional member function visit was overloaded to accept SiDetectors. This function maintains a list of the hit detectors. The list is later stored through the call to the CsIData::load. This function was also modified to accept SiDetectors and a new class was created called Hit. The Hit class stores the vital information about each hit SSD. The class contains the coordinates of the center position, the tower id, the strip id number, and the amount of charge deposited in this detector in fC. A list of Hit objects is created to store parameter values for hit SSDs. No further processing of the SSD hits is performed within Glastsim. Instead, all hit information is output to the DAT file, which contains the data from all hit detectors for each event.

After the modifications to Glastsim, sets of runs were performed. The runs varied the number layers in the pre-calorimeter, 1 - 4 (1.25 - 5. rad lengths), with or without the 10-cm gap between the SSDs and the main calorimeter (eight runs total). The beam was monoenergetic, with E = 10 GeV. We were interested in CAL-only events - those events where the gamma did not convert in the tracker. To insure that all events were CAL-only, the gammas were shot just above the calorimeter at normal incidence, bypassing the tracker completely. Originally, to understand a simple configuration, all events were shot into the centers of crossed crystals, near the center of the central module of the calorimeter using a pencil beam. Later, an additional set of eight runs was performed with a single difference: the beam at normal incidence uniformly illuminated the center tower of GLAST.

4. Analysis

An IDL procedure was written to analyze the performance of the Pre-CAL configuration. The measure of interest is the two-dimensional point spread function (PSF), which is the angular distribution of differences between actual and reconstructed particle directions within the calorimeter. The distribution includes all particles for which energy deposition greater than a given threshold occurred in the SSDs. A useful statistic is derived from the angular distribution: the radius which contains 68% of the events.

A first-order estimate of the best achievable PSF can be calculated from an event's reconstructed trajectory using only two points: the shower vertex as measured by the SSDs near the bottom of the pre-calorimeter, and the shower centroid which develops deeper in the main calorimeter. The best measure of the centroid - for a shower completely contained laterally within the calorimeter - is formed from the energy-weighted coordinates of crystals with any energy deposition, the barycenter:

bCenter = å xi Ε i / å Εi      (1)

The information from the SSDs comprises the discretized coordinates (200 μm spacing) of the hit strips. These hits were used to estimate the entrance point of the incoming particle. Naively, one may calculate the mean hit coordinate (X or Y), or find the median hit coordinate. However, this simplistic approach produces poor results since hits that are far from the actual origination point of the incoming particle are included in the determination of mean or median. These "fringe" hits are due to backsplash from the shower within the calorimeter as well as production of daughter particles with large angles in the SSDs.

Therefore, rather than calculate a simple mean or median coordinate for all hits in an SSD plane, we should eliminate the fringe hits as intelligently as possible. To reconstruct a better estimate of the true shower center, we made use of an algorithm presented by Y.H. Chang et al. (1995). The method is expedient for our purposes - considering the desires for simplicity of both instrument design and reconstruction algorithm - in that it does not require a measurement of the deposited charge to determine which hits to include in calculating the shower vertex. Only a record of hit or no hit is required, above or below threshold, Ethres. The Chang et al. algorithm proceeds in three steps:

  1. for hit strips with E > Ethres, find the median (m1) hit and compute rms;
  2. find the second median (m2), the middle strip within the range m1 ± rms;
  3. compute the center of gravity (COG) in the window m2 ± nw strips,

where nw is adjustable. The X and Y COG statistics from the respective SSD planes then provide estimates of the X-Y position of the shower vertex.

This algorithm was coded in IDL, and the DAT files from the runs were analyzed. The appended tables summarize the results. Chang et al. used nw = 4 for SSDs with a strip pitch of 50 μm. Since the SSDs in our simulations have a coarser strip pitch of 200 μm, our initial choices were also nw = 2 and 4 (in fact, larger ranges for nw tended to yield higher dispersions for the vertex position). We computed the COG statistics for two SSD thresholds, 1.29 fC and 5.74 fC. The first is the current value used in the tracker design and corresponds to ~ _ MIP (~ 45 keV). The second threshold, corresponding to ~ 1 MIP, was applied to see what effect, if any, removing lower energy hits would have on the vertex measurement.

We then used the vertex measurement and the energy-weighted centroid of the CsI hits (bCenter) to form a two-point reconstructed trajectory of the incoming gamma ray. From the ensemble of reconstructed trajectories we computed the PSF and 68% containment radius. Let the shower vertex be denoted by Γ and bCenter by Β. The vector V connecting vertex and barycenter is then

Vi = Γi - Βi; Vj = Γj - Βj; Vk = Γk - Βk      (2)

and the direction cosines are

cos α = Vi / ½ ; cos β = Vj / ½ ; cos γ = Vk / ½      (3)

From the direction cosines, θ and φ were computed:

θ = γ; φ = Arctan (cos α / cos β)      (4)

The PSF was generated from the distribution of differences between reconstructed trajectories (θ, φ) and the Monte Carlo trajectories (θmc, φmc):

ε = Arccos[ cos(θmc) cos(θ) + sin(θmc) sin(θ) cos(φ - φmc) ]      (5)

For the simulations, all events have θmc = 0°; hence ε = Arccos[ cos(θmc) cos(θ) ].

5. Results

Since the vertex measurements were insignificantly different for nw = 2 and 4, and only marginally different for the two thresholds, we describe results for the case nw = 2 and Ethres = 45 keV (~ _ MIP). The three pairs of tables present results for embedding the pair of SSDs below the 1st, 2nd, 3rd, and 4th layers which form the pre-calorimeter, in each case with and without the 10-cm gap before the remaining portion of the calorimeter. A pair of tables contains results for pencil beam illumination (A) and uniform illumination of the central module (B). The numbers of reconstructed events, out of 1000 incident on the calorimeter, are ~ 625, 860, 950, 975 for SSD embedding below layers 1 through 4, respectively. Inclusion of more layers in the pre-calorimeter results in more gammas converted and reconstructed. The statistics listed in Tables 1 - 3 are necessarily only for the gammas which convert and whose trajectories were reconstructed using the two-point method.

In Table 1 the one-dimensional 68% containment ranges in the SSD planes are listed for X and Y coordinates separately. The X and Y measurements are separated vertically by only 3 mm; any differences in the 68% containment range are basically statistical. Progressing from embedding below layer 1 to layer 4, the shower vertex is measured to an accuracy of ~ 0.55, 1.0, 1.4, and 1.8 mm, respectively, for pencil beam illumination (Table 1A). For center module illumination (Table 1B), the vertex measurements are ~ 0.6, 1.2, 1.7, and 2.1 mm. On average, runs with center module illumination yield vertex measurements with approximately 20% larger errors than for pencil beam illumination; the increase is probably attributable to dead space on the SSD planes. The vertex determinations appear to be independent of the presence of 10-cm gap, whose influence on the SSD measurements would be to decrease the fringe hits resulting from backsplash; the (hypothetical) decrease is negligible since fringe hits are efficiently eliminated by the positional reconstruction algorithm. As we will see from Tables 2 and 3, an improvement is afforded by the SSDs only when embedded below layer 1 or layer 2, in which cases the reconstruction is better than that obtained for the baseline calorimeter. Essentially, the improvement from embedding at the higher layers results because the measurement accuracy for the shower vertex then exceeds that of the barycenter, by a factor of 2 - 2.5. For embedding below layers 3 and 4, the vertex position is not measured any better than the barycenter.

TABLE 1A - 68% Containment of SSD Shower Vertex (Pencil Beam Illumination)

Layer 1

  X (mm) Y (mm)
With 10-cm Gap 0.65 ± 0.16 0.42 ±> 0.10
     
Without 10-cm Gap 0.57 ± 0.16 0.57 ± 0.15

Layer 2

  X (mm) Y (mm)
With 10-cm Gap 0.88 ± 0.10 0.96 ± 0.08
     
Without 10-cm Gap 0.89 ± 0.12 1.16 ± 0.09

Layer 3

  X (mm) Y (mm)
With 10-cm Gap 1.38 ± 0.11 1.37 ± 0.08
     
Without 10-cm Gap 1.41 ± 0.09 1.45 ± 0.10

Layer 4

  X (mm) Y (mm)
With 10-cm Gap 1.77 ± 0.09 1.74 ± 0.13
     
Without 10-cm Gap 1.71 ± 0.10 1.88 ± 0.11

TABLE 1B - 68% Containment of SSD Shower Vertex (Uniform Illumination of Center Module)

Layer 1

  X (mm) Y (mm)
With 10-cm Gap 0.49 ± 0.13 0.55 ± 0.14
     
Without 10-cm Gap 0.54 ± 0.15 0.91 ± 0.13

Layer 2

  X (mm) Y (mm)
With 10-cm Gap 1.36 ± 0.17 1.13 ± 0.14
     
Without 10-cm Gap 1.18 ± 0.13 1.29 ± 0.14

Layer 3

  X (mm) Y (mm)
With 10-cm Gap 1.72 ± 0.12 1.59 ± 0.09
     
Without 10-cm Gap 1.73 ± 0.12 1.90 ± 0.13

Layer 4

  X (mm) Y (mm)
With 10-cm Gap 2.09 ± 0.11 1.97 ± 0.13
     
Without 10-cm Gap 2.05 ± 0.11 2.14 ± 0.14

In Table 2 the one-dimensional 68% containment ranges for the barycenter are listed for the X and Y coordinates. For pencil beam illumination (Table 2A) the Y coordinate errors are systematically ~ 0.3 -0.4 mm larger than the X coordinate errors, which range from ~ 1.6 to ~ 2.0 mm as the embedding layer increases from 1 to 4. The X-Y difference may be attributable to an inescapable side effect of the hodoscopic geometry: Since one layer must come first, and since for a monoenergetic beam, the shower will tend to be defined in a certain vertical range spanning a small number of layers, then either the X, or Y, layers will tend to have more longitudinal - and better measured - coordinates falling within the shower range. Also, the X-Y difference should be independent of the presence of the 10-cm gap because the gap does nothing to redefine the vertical shower zone relative to the Z coordinates of the CsI layers.

TABLE 2A - 68% Containment of Barycenter (Pencil Beam Illumination)

Layer 1

  X (mm) Y (mm)
With 10-cm Gap 1.55 ± 0.09 2.08 ± 0.10
     
Without 10-cm Gap 1.60 ± 0.05 1.99 ± 0.05

Layer 2

  X (mm) Y (mm)
With 10-cm Gap 1.66 ± 0.09 2.04 ± 0.08
     
Without 10-cm Gap 1.53 ± 0.10 1.93 ± 0.10

Layer 3

  X (mm) Y (mm)
With 10-cm Gap 1.90 ± 0.10 2.25 ± 0.09
     
Without 10-cm Gap 1.53 ± 0.10 1.98 ± 0.08

Layer 4

  X (mm) Y (mm)
With 10-cm Gap 2.50 ± 0.13 2.79 ± 0.13
     
Without 10-cm Gap 1.75 ± 0.09 1.97 ± 0.07

For center module illumination (Table 2B) the sense of the difference in X-Y errors is reversed, with the X coordinate errors being similarly larger. The explanation must lie in the fact that averaging over a module produces a difference in the measured shower depth, due to the discrete coordinates of the crystal. Or perhaps these apparent differences are attributable to statistical fluctuations (1 σ ~ 0.1 mm). Regardless, independent of embedding layer or coordinate measured, center module illumination yields barycenter errors which are ~ 35% larger (average of X and Y coordinates) than those obtained with pencil beam illumination.

TABLE 2B - 68% Containment of Barycenter (Uniform Illumination of Center Module)

Layer 1

  X (mm) Y (mm)
With 10-cm Gap 2.52 ± 0.10 2.52 ± 0.11
     
Without 10-cm Gap 2.51 ± 0.10 2.28 ± 0.10

Layer 2

  X (mm) Y (mm)
With 10-cm Gap 2.72 ± 0.10 2.25 ± 0.09
     
Without 10-cm Gap 2.57 ± 0.09 2.33 ± 0.09

Layer 3

  X (mm) Y (mm)
With 10-cm Gap 2.68 ± 0.10 2.61 ± 0.06
     
Without 10-cm Gap 2.55 ± 0.09 2.35 ± 0.09

Layer 4

  X (mm) Y (mm)
With 10-cm Gap 2.91 ± 0.11 2.61 ± 0.08
     
Without 10-cm Gap 2.63 ± 0.09 2.30 ± 0.10

By combining the shower vertex and barycenter point, we arrive at an estimate of the reconstructed trajectory. While better estimates which make more use of the available information may exist, the two-point definition - vertex to barycenter - appears nearly optimal: The vertex has been measured with high precision using the SSD planes and reconstructed with a finely tuned algorithm designed to eliminate outlying hits, while the barycenter averages all information available about energy deposition in the crystals as a function of position. (While developing the moments analysis algorithm, we tried variations on the barycenter calculation, including different weightings and energy thresholds. The simplest approach of linearly weighting all positions with any energy deposition yielded the most accurate reconstruction. The moments are computed with respect to the barycenter.)

Tables 3A and 3B list the resulting 68% containment radii derived from the two-point reconstruction for the several cases, with and without 10-cm gap, and the four levels of SSD embedding. The 68% containment radii (R68) with no gap for embedding levels 1 - 4 are » 1.9° (3.2°), 1.95° (3.9°), 2.2° (5.6°), and 3.1° (12.0°), respectively, for pencil beam (center module) illumination.

TABLE 3A - 68% Containment Radius of PSF (Pencil Beam Illumination)

Layer 1

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 601 1.04 ± 0.09
     
Without 10-cm Gap 651 1.93 ± 0.11

Layer 2

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 857 1.12 >± 0.06
     
Without 10-cm Gap 864 1.95 ± 0.09

Layer 3

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 953 1.30 ± 0.04
     
Without 10-cm Gap 940 2.19 ± 0.07

Layer 4

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 978 1.74 ± 0.05
     
Without 10-cm Gap 973 3.11 ± 0.07

TABLE 3B - 68% Containment Radius of PSF (Uniform Illumination of Center Module)

Layer 1

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 639 1.37 ± 0.05
     
Without 10-cm Gap 614 3.21 ± 0.23

Layer 2

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 844 1.65 ± 0.06
     
Without 10-cm Gap 835 3.86 ± 0.12

Layer 3

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 905 1.93 ± 0.06
     
Without 10-cm Gap 936 5.56 ± 0.18

Layer 4

  68% containment radius (degrees)
  Events nw = 2
With 10-cm Gap 960 2.96 ± 0.09
     
Without 10-cm Gap 966 11.97 ± 0.47

Inclusion of the 10-cm gap improves the R68 statistic to » 1.0° (1.4°), 1.1° (1.65°), 1.3° (1.9°), and 1.7° (3.0°), for embedding levels 1 - 4, respectively.

Figure 2 summarizes these results for embedding SSD planes below layers 1 and 2, and compares with the performances of the Baseline Hodoscopic Calorimeter and the Tracker. It is clear that the combination of (1) an increased lever arm across which the shower measurement is made, and (2) a precise measurement of the vertex, are necessary to realize an significant improvement in the CAL-only angular resolution. However, the improvement for the realistic case of uniform illumination of a module is only ~ 1.4° compared to ~ 2.2°, for the ~ 65% of photons which convert in the first layer; the balance - not to be discarded - must be measured using the moments analysis.

Results for embedding SSD planes below layer 1

Results for embedding SSD planes below layer 2

6. Discussion

The purpose of this study was two-fold: To learn what improvements can be had in the angular resolution for CAL-only mode events by (1) inclusion of a pair of SSD planes following a pre-calorimeter zone of 1 to 4 layers, and (2) inclusion of a 10-cm gap following the SSD planes. The SSDs afford a precise measurement of the shower vertex. But since at least two points are needed to reconstruct the trajectory and the next best-measured point is the barycenter - with its larger error dominating - the angular resolution appears to be improvable, at best, by ~ 30%, from 2.2° to 1.4° for 10 GeV gammas, uniformally normally incident on a module. Inclusion of both the pair of SSD planes and 10-cm gap are necessary to realize this improvement, since with the gap, the barycenter error continues to dominate the overall measurement. The estimate of angular resolution should include those photons which do not materialize in the first one or two CsI layers of the pre-calorimeter:

  • Only 63% of gammas convert in the top layer; hence the R68 » 1.4° applies only to this fraction. Trajectories of the remaining 37% would be measured without benefit of the 10-cm gap, using the moments analysis. The weighted estimate (0.63 x 1.4°) + (0.37 x 2.2°) = 1.69°, is similar to that obtained when the two error distributions are combined with respective fractions and the R68 statistic computed, yielding 1.80° ± 0.09.
  • For embedding SSDs below the second CsI layer, the overall R68 obtained is similar. Approximately 84% of gammas convert in the top two layers. The weighted estimate is then (0.84 x 1.65°) + (0.16 x 2.2°) = 1.74°; combining the actual distributions and computing R68 yields 1.73° ± 0.07.

Photons converting in the lower CsI layers could be eliminated to improve angular resolution, but this measure would appear to be wasteful, given the sources for which CAL-only photons provide a real benefit: (1) gamma-ray bursts, whose rates (even for dim bursts) completely dominate the residual (CAL-only) proton background rate within the Baseline Calorimeter PSF error circle (Bonnell and Norris 1999); and (2) the brightest ~ 100 (EGRET-detected) sources, on which longer integrations may be performed. These CAL-only source sensitivities are based on preliminary estimates from neural-network plus rule-based rejection methods and comparison with the GeV sources catalog of Lamb and Macomb (1997). Preliminary science simulations utilizing the CAL-only PSF as a function of polar angle and energy appear to confirm this conclusion (Digel et al. 1999).

References

Norris, J.P., and Bonnell, J.T. 1999, "Simulations of Gamma-Ray Bursts Observed by GLAST", Proc. of 19th Texas Symposium

Chang, Y.H., et al. 1995, "A study of spatial resolution for a preshower detector with aluminum absorber and silicon strip sampler", NIM-A, 374, 156-163

Lamb, R.C., and Macomb, D.J. 1997, "Point Sources of GeV Gamma Rays", ApJ, 488, 872-880

Digel, S., et al. 1999, "Simulations of GLAST Observations", in preparation.