The Stereoscopic Reconstruction
5.3 Geometrical Reconstruction of Showers
The concept of stereo imaging is based on the simultaneous detection of air showers in different projections by at least two telescopes separated by a distance, which is comparable with the radius of the Cherenkov light pool.
In the H.E.S.S. experiment the telescopes are built at the corners of a square with side length of 120 m. The advantages of stereoscopic observations are: (i) better quality of reconstruction of shower parameters, as compared to a single telescope; (ii) very powerful rejection of background showers, (iii) effective suppression of night-sky as well as muon background [10]. Depending on the size of the array, the dynamic energy range in registration of γ rays may be extended towards multi-TeV showers.
Using the coordinate transformations given in Appendix B, the shower direction, the shower core position, the height of the shower maximum, and
work, the weighting factor for the intersecting ith and jth telescopes (i, j = 1,2,3,4) is defined as follows
Abbildung 5.7: The shower direction is calculated as the weighted mean of the intersection points of the image major axes of triggered telescope pairs in the nominal system.
wij = |sin(φi − φj)|
1/sizei + 1/sizej
, (5.8)
whereφi(j) is the angle between the image major axis and the x-axis in the nominal system for the i(j)th telescope. The weight wij is derived
empirically from the fact that the telescope pairs with larger angles bet-ween their intersecting image major axes and telescopes having brighter images provide a more precise determination of the shower axis. There-fore, with the help of this weight, telescope pairs with their major axes being approximately parallel to each other and telescopes with very faint images are suppressed to improve the reconstruction quality.
More information on different methods of the reconstruction of the shower direction can be found in [94].
Abbildung 5.8: The shower core position is calculated by intersecting the directional lines formed by the telescope coordinates and image orientation angle (Φ) in the z-projected plane of the tilted system. The intersection points are weighted with the angle of intersection, and the mean of these intersection points gives the shower core position.
• Determination of Shower Core Position: The core position is cal-culated in the tilted system. First the x,y coordinates of the telescopes are transformed from the ground system into the tilted system (Appen-dix B). Using the telescope coordinates and the orientation angle φ, a directional line is calculated for each triggered telescope. For each pair of triggered telescopes an intersection between the directional lines (Fi-gure 5.8) attached to the actual telescopes positions in the z-projected
the convention is used to express the final core position in the x-y plane corresponding to z = 0. So, the shower axis, which is defined by ~p and the shower directional vector ~r are located in the plane of z = 0. The position of the shower core in the x-y plane is given by C, and it is~ found as follows
C~ = ~p − pz
rz ~r , (5.10)
where pz and rz are the components of ~p and ~r in z-direction, respec-tively.
5.3.1 Angular Resolution and Accuracy of Shower Co-re Localization
In Figure 5.9 (top-left) (or (top-right)), the difference∆θx (or∆θy), between the x-(or y-)components of the reconstructed shower direction and the actual simulated shower direction is shown. The radial angular residual,∆θ, is found by
∆θ = q∆θ2x + ∆θ2y . (5.11)
Higher values of∆θ would imply a bad reconstruction of the shower direction basically due to the erroneous reconstruction of the major axis of the simula-ted/recorded images. This problem arises mainly for very faint images, which are produced by showers at rather large impact distances from the telescope, as well as by showers of very low energies.
The images detected at relatively large impact distances have an elonga-ted angular shape and enable an accurate determination of the image orien-tation ([112]), as long as the images are not truncated by the camera edges.
sdir_diffX_12
Shower Core Position Residuals score_diffY_12
RMS 22.73
Abbildung 5.9: Distributions of angular residuals (∆θx (top-left), ∆θy (top-right)) and residuals of shower-core position (∆Rx (middle-left), ∆Ry
(middle-right)) of the reconstructed γ-ray candidates are shown. The sho-wers are simulated for zenith angles of 45◦ and quality cuts are applied (see text). Radial angular resolution, σ∆θ , is the value of ∆θ (on the x-axis), left of which the number of entries is 68% of the total number of entries in the
∆θ distribution (bottom-left). Similarly, the shower-core position resolution, σ∆R, can be found from the ∆R distribution (bottom-right).
The distribution of ∆θ after the quality cuts is shown in Figure 5.9 (bottom-left), which still has a long tail towards higher values of∆θ. In the distribution of angular residuals, the value, for which 68% of the whole entries of the ∆θ distribution are within the error circle around the source position, is taken as the angular resolution, σ∆θ, of the reconstruction. Similarly, the accuracy of localizing the shower core position (shower-core resolution),σ∆R, is found from the distribution of shower-core residuals given in Figure 5.9 (bottom-right).
It was found that the angular resolution of the IACT strongly depends on the telescope multiplicity ([112]), where the higher multiplicity of triggered telescopes results in a better angular resolution. To observe telescope multi-plicity dependence of angular and shower-core resolutions at various energies, the values ofσ∆θ andσ∆R are calculated for each energy bin between 0.1 TeV and 20 TeV and for telescope multiplicities of 2, 3, 4, and≥2 (Figure 5.10).
From Figure 5.11, it is apparent that the showers with higher energies located close to the center of the telescope array yield the best angular and shower-core resolutions. For example, the average angular resolution is found to be of ∼0.16◦ and the average shower core resolution is found to be of
∼33 m at zenith angle of 45◦. The accuracy of localization of the shower core position correlates with the accuracy in the angular resolution.
It is found that the dependence of angular resolution on the source po-sition in the FoV (or the pointing modus of the telescopes) can be ignored within a reasonable angular distance from the center of the camera. Therefo-re, so-called wobble-mode observations up to 1.5◦ do not significantly affect the angular resolution [112].
Energy [TeV]
Angular Resolution for Zenith 45o
2 telescopes
Core Resolution for Zenith 45o
2 telescopes 3 telescopes 4 telescopes telescopes >= 2
Abbildung 5.10: The angular- (top) and shower core (bottom) resolutions at a zenith angle of 45◦ are calculated per energy bin for telescope multiplicities of 2, 3, 4, and ≥ 2 (blue, red, green, and black markers, respectively).
Energy [TeV]
1 10
0.04 0.06 0.08 0.1 0.12
Energy [TeV]
1 10
Resolution [m]
10 20 30 40 50
Core Resolution for Zenith 45o
core <= 300m , telescopes >= 2 core <= 200m , telescopes >= 2 core <= 100m , telescopes >= 2
Abbildung 5.11: The angular- and shower core resolutions at a zenith angle of 45◦ are calculated per energy bin for telescope multiplicities≥2. and after various cut values applied on the shower core: 100, 200, and 300 (shown by green, red, and black markers, respectively).
Energy [TeV]
Abbildung 5.12: The relation between the energy of the shower and the mea-sured image amplitude is shown for different zenith angles (0◦ (top), 45◦ (middle), 60◦ (bottom)) and for two different true impact distance ranges:
the range of 60 - 80 m shown with black markers and the range of 140 - 240 m shown with red markers.
True Impact Distance [m]
0 100 200 300 400 500 600 700
log( True Energy ) [GeV]
1.5
Mean Amplitude for Zenith 00
True Impact Distance [m]
0 100 200 300 400 500 600 700
log( True Energy ) [GeV]
1.5
Mean Amplitude for Zenith 450
True Impact Distance [m]
0 100 200 300 400 500 600 700
log( True Energy ) [GeV]
1.5
Mean Amplitude for Zenith 600
Abbildung 5.13: The look-up tables containing mean amplitude values for 50 true energy bins and 50 true impact distance bins shown for zenith angles of 0◦ (top), 45◦ (middle), and 60◦ (bottom), where the color code indicates the mean amplitude value in each true energy and impact distance bin.
The relationship between the amplitude of the image at a specific true impact distance and energy of the shower can be tabulated using Monte Carlo
simulations. The tables are produced for zenith angles of 0◦, 20◦, 30◦, 40◦, 45◦, 50◦, 55◦, 60◦. Each of these tables is subdivided into 50 logarithmic bins of true energy in the range of 0.02 - 20 TeV and 50 bins of true impact distance within a range of 0 - 750 meters. The range of impact distance is taken large enough to include the higher energy events at larger zenith angles 55◦ - 60◦. In Figure 5.13 the tables for zenith angles 0◦, 45◦, and 60◦ are shown. The x-axis and y-axis represent the true impact distance and the true energy, respectively. The color code indicates the value of the mean amplitude for each bin. In observations the basic information that can be obtained for each recorded event consists of the image amplitude for each of triggered telescope, the reconstructed impact distance, and the zenith angle. The table corresponding to the zenith angle of observation (z0) is selected. For the true impact distance bin including the measured impact distance value (ID0) the tabulated true energy and mean amplitude values are plotted and a straight line is fitted to the data. The resulting fit parameters and the measured image amplitude (A0) are used to reconstruct the energy (EReco(z0)) for this specific impact distance bin. In case, there is no table found corresponding to the measured zenith angle, the closest two tables produced for zenith angles zlow and zhighare read. For ID0, energies EReco(zlow)and EReco(zhigh)are calculated from these tables. EReco(z0)is obtained from the linear interpolation incos(z) between EReco(zlow) and EReco(zhigh) as follows: where EReco(z0) is the reconstructed energy for the ith triggered telescope (i = 1 , ... , Ntel) and Ntel is the total number of triggered telescopes). The reconstructed energy for the system (ERecosys ) is then the mean of reconstructed energies for all triggered telescopes. ERecosys is the final measure of the shower energy used for the spectrum evaluation.