Data Selection and Reduction
4.4 Shower Reconstruction
The calibrated air shower images from the selected data set were cleaned of noise, parametrised withHillasparameters, and then the shower direction and impact point were determined using a fast and simple reconstruction algorithm. The energy of the shower could only be estimated by comparison with simulated air shower images. The reconstruction methods used are based on the techniques developed in the first stereoscopic IACT project HEGRA [see e.g. Hofmann et al., 1999].
4.4.1 Image Parametrisation
Image Cleaning Using Tail Cuts
The intensity distribution of an air shower image in the telescope camera is superimposed on night sky background fluctuations in each pixel. A simple cleaning is performed with the fol-lowing procedure: For each pixel, it was required that its recorded intensity exceeded a threshold ofh p.e. or at leastlp.e. in case it has a neighbouring pixel with an intensity exceeding hp.e.
The parameters h,l (thetail cuts) were chosen such that the signal to noise ratio forγ-rays is optimal and at the same time insensitive to variations in the NSB level. Figure 4.12 shows a single shower image from observations on PSR B1259−63 before and after calibration (left and middle respectively), and after image cleaning (right).
(Sun Mar 28 22:56:49 2004 369745us)
(Sun Mar 28 22:56:49 2004 369745us) Event 28561
(Sun Mar 28 22:56:49 2004 369745us) CT4 Cleaned Intensity
Figure 4.12: Air shower image in a camera at different analysis stages. Left: Raw ADC counts in the high gain channel. Middle: Calibrated image with pixel intensities in photo-electron units. Right:
Cleaned image after tail cuts have been applied together with the image parameters. The computed ellipse is artificially enlarged by a factor of 5.
Hillas Parameters
The cleaned images were parametrised using their first and second moments according to Hillas [1985]. The image is described as an ellipse with the parameters:
. Width,Length,
. Φ,LocalDistance (dlocal) determining the orientation andcentre of gravity of the image with respect to the camera centre,
. andSize, the total image intensity in p.e.,
These parameters are illustrated in Fig. 4.12, right, for a γ-ray candidate image taken from PSR B1259−63 data. The ellipse is artificially enlarged by a factor of 5.
Image Quality Cuts
In order to use only images of good quality for which the analysis methods below can be applied, two image quality cuts were applied:
. dlocal < dmaxlocal was required, ensuring that the image centre of gravity was not close to the camera edge to avoid truncated images.dmaxlocalwas set to 80% of the camera radius (0.035 rad).
. Images were required to have a minimum image amplitude, Size > Sizemin. The actual cut value was derived in the cut optimisation procedure described in Sec. 4.5.3 which optimised the signal to noise ratio forγ-rays.
4.4.2 Geometric Shower Reconstruction
The geometric air shower properties relevant for this analysis are the shower direction and the position where the shower axis intersects the ground plane – the shower core position. Both quantities were calculated using a method similar to the standard reconstruction technique for a system of stereoscopic IACTs [Hofmann et al., 1999].
Shower Direction
CT1 CT4
θy
θx
Figure 4.13: Reconstruction of the shower di-rection using the image parametrisation of two telescopes for a measuredγ-ray like event.
The major axis of each recorded shower im-age represents a projection of the shower axis into the camera system and therefore the initial direction of the primary particle lies somewhere close to this axis. Using more than one shower image from a single event, the shower direction was determined by in-tersecting the major axis of the image el-lipses within the camera coordinate system as shown in Fig. 4.13. If more than two tele-scope images were available for a particular event, the different quality of the images was considered to further increase the reconstruc-tion accuracy. To each pair i of images, a weight
Wi = sin(Φ1−Φ2) (s−11 +s−12 )(w1/l1+w2/l2)
was assigned, where s, w, and l are the Size, Width, and Length of the image respectively.
The estimated direction is found from the weighted average of all intersection positions. The weightWi implies that images which had a high intensity and a high aspect ratio were favoured against fainter and circular images. Pairs of images with major axes tilted towards each other have greater weights because the intersection point is much more accurately determined than for nearly parallel images. Furthermore, if one of the images did not pass the image quality selection cuts, the corresponding pair was discarded.
Shower Core Position
Figure 4.14: Tilted ground system.
A similar approach was used for the determi-nation of the shower core position. Within the ground coordinate system, the major axis of the telescope image points towards the core position. Again, using the stereoscopic method with more than one image, the same algorithm as for the determination of the shower direction was applied. Since the cam-era plane of a telescope is only parallel if it points to zenith, a coordinate system was used which is tilted with respect to the ground and perpendicular to the telescope pointing axis (see Fig. 4.14).
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Figure 4.15: Reconstructed shower parameters for simulatedγ-rays. Left: Shower direction within the camera system.Right: Shower core position on the ground level.
4.4.3 Accuracy of the Shower Reconstruction Methods
In order to quantify the accuracy of the reconstruction, the sample of simulated γ-rays at the zenith angleΘ = 40◦(typical for the PSR B1259−63 dataset) was used to determine the resolu-tion of the reconstructed direcresolu-tion and core posiresolu-tion. Figure 4.15 shows the distriburesolu-tion of the difference between the true and reconstructed shower parameters.
Angular Resolution
For each simulated event passing the shape selection cuts (see Sec. 4.5), the angular distanceθ between the true and the reconstructed shower direction was computed. Figure 4.16 (left) shows the resulting distribution ofθ2. The angular resolution can be quantified by the value θ68%, for which 68% of the events have a smaller value of θ. The resulting resolution for the simulation sample isθ68% =0.09◦. The distribution is significantly non-Gaussian with a long tail resulting from telescope images for which the estimation of shower orientation was bad. This distribution represents the H.E.S.S. point spread function for γ-rays, and can be parametrised by a sum of four Gaussians (see Fig 4.16, left):
Nγ(θ)=
Figure 4.16, right, shows the energy dependence of the angular resolution using equidistant energy bins on a logarithmic scale between 200 GeV and 50 TeV. It is obvious that the resolu-tion improves dramatically with increasing energy. This improvement can be exploited in the determination of the position of aγ-ray excess (see Sec. 5.2).
θ2
Figure 4.16: Angular resolution for simulatedγ-rays after image shape cuts. Left: Distribution of the squared angular distanceθ2between the true and reconstructed shower direction for simulatedγ-rays.
The vertical line shows the 68% containment radius, θ68%. Right: Angular resolution as a function of the true energy of the simulated shower.
Impact Parameter Resolution
Figure 4.17: Distribution of the difference between the true and reconstructed impact parameter ∆β = β−βtrue
for simulatedγ-rays after allγ-ray selection cuts.
The information of the shower core position is used in the reconstruc-tion of the energy of primaryγ-rays (as described below). For this pur-pose, the distance from the shower core to each telescope position, the so called impact parameter β is re-quired. Since this shower parameter is only important for energy recon-struction, the cut on shower direc-tion was applied as well on the sim-ulation sample, as it was done in the spectral analysis in Sec. 5.3, in or-der to quantify the impact parameter resolution.
In Figure 4.17, the distribution of the difference between the true and the reconstructed impact parameter
∆β = β − βtrue is shown. The mean deviates only 2 m from the nominal value, the RMS is
∼ 15 m, and decreases with increasing shower energy in a similar way to the angular resolution (see Fig. 4.18, left). However, the value is biased towards negative values which is an arti-fact of the arrangement of the telescopes in the array. The bias remains below 4 m within the considered energy range (see Fig. 4.18, right).
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20 Impact Parameter Resolution vs. True Energy
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4 Impact Parameter Bias vs. True Energy
Figure 4.18: Impact parameter resolution for simulatedγ-rays. The RMS (left) and bias (right) of the
∆βdistribution as a function of the true shower energy.
4.4.4 Energy Reconstruction
Figure 4.19:Shower energy lookup table. The mean true energy of simulatedγ-rays Etrueis shown as a function of true shower impact distance and expected image ampli-tudeSize. This distribution reflects the radial distribution of the Cherenkov light density on the ground for a given image Size.
Assuming that a constant fraction of the energy of a primary γ-ray is con-verted into Cherenkov light, the in-tensity collected by a single telescope depends only on the extinction of this light and the distribution on the ground, the Cherenkov “light pool”.
Therefore, the shower energy was es-timated from the image Size in each telescope using the expectation value hEiof simulatedγ-rays from a lookup table such as that shown in Fig. 4.19.
The variation of the atmospheric ex-tinction depending on the telescope pointing is accounted for by the de-pendence on the zenith angle Θ, and the Cherenkov light distribution on the ground is considered via the impact parameterβ assuming rotational sym-metry (see Fig. 3.3):
hEiMC=hEiMC(Size,Θ, β).
Reconstruction Algorithm
The shower energyEof eachγ-ray candidate event was reconstructed via:
E= 1
whereNtelsis the number of andtthe index for the individual telescopes which passed theγ-ray selection cuts, Sizet andβt are the image amplitude and the impact parameter of the event for telescopet, respectively, andΘis the zenith angle of the array pointing. Sinceγ-rays were only simulated for discrete zenith angle valuesΘ(see Sec. 3.3.3),hEiMCwas linearly interpolated in cosΘ.
Figure 4.20:Distribution of the relative difference be-tween the simulated (true) and reconstructed energy (Ereco−Etrue)/Etruefor simulatedγ-rays.
Figure 4.20 shows the distribution of the difference between the recon-structed and true energy ∆E = Ereco − Etrue relative to the true energy Etrue, for the complete sample of simulated γ-rays withΘ = 40◦. The energy res-olution was defined as the RMS of this distribution and found to be∼ 20%. A Gaussian approximation of∆E/Etrue is not valid due to the increasing bias at low energies.
Figure 4.21 shows the energy reso-lution and bias as a function of the true energy. Within the range considered, the energy resolution improves with in-creasing shower energy. The bias in the reconstructed energy becomes greater
than 10% below≈400 GeV of primaryγ-ray energy, indicating that the reconstruction method is not efficient below this energy for observations at this zenith angle.
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0.22 Energy Resolution vs. True Energy
[TeV]
Energy Bias vs. True Energy
Figure 4.21: Energy resolution and bias for simulatedγ-rays afterγ-ray selection cuts. Left, Right:
RMS and mean of the distribution∆E/Etrueas a function of the true shower energy Etrue, respectively.