de Naurois (2006) - Analysis methods for Atmospheric Cerenkov Telescopes

tags: IACT proceeding translating annotating

大氣契忍可夫望遠鏡的分析方法

Information 資訊

• URLs
  • arXiv:astro-ph/0607247
  • [INSPIRE],[ADS] Proceedings of the 7th Workshop on Towards a Network of Atmospheric Cherenkov Detectors, pp.149–161 (2006)
• BibLaTeX
  ​​​​@inproceedings{denaurois2006,
  ​​​​    author        = {de Naurois, Mathieu},
  ​​​​    title         = {Analysis methods for Atmospheric Cerenkov Telescopes},
  ​​​​    booktitle     = {7th Workshop on Towards a Network of Atmospheric Cherenkov Detectors 2005},
  ​​​​    pages         = {149--161},
  ​​​​    year          = {2006},
  ​​​​    month         = {7},
  ​​​​    doi           = {10.48550/arXiv.astro-ph/0607247},
  ​​​​    archivePrefix = {arXiv},
  ​​​​    eprint        = {astro-ph/0607247},
  ​​​​}
  

Highlights 重點整理

• Sec. 1 第 1 節

  • The data analysis techniques include: 資料分析技術包括：
    1. Hillas-parameter based anaylis 基於 Hillas 參數的分析
       the earliest adopted; based on elliptical images 最早採用；基於橢圓圖像
    2. Model Analysis 模型分析
       first adopted by CAT; compared to pre-calculated model/template CAT 首先採用；跟預先計算好的模型／範本比較
    3. 3D Model Analysis 3D 模型分析
       based on ellipsoidal model of photosphere 基於光球的橢球模型
  • The three techniques are complementary in many senses; and there is also much space for them to improve. 這三種技術在很多方面都是互補的；它們也都還有很大的改進空間。

• Sec. 2 第 2 節

  • The Hillas-parameter based analysis models an image by a two-dimensional ellipse. 基於 Hillas 參數的分析是利用二維橢圓對圖像建模。

Table of Contents 目錄

• de Naurois (2006) - Analysis methods for Atmospheric Cerenkov Telescopes
  • Information 資訊
  • Highlights 重點整理
  • Table of Contents 目錄
  • 0. Abstract 摘要
  • 1. Introduction 概論
  • 2. Hillas-parameter based analysis 基於 Hillas 參數的分析
    • 2.1 Introduction 簡介
    • 2.2 Single telescope reconstruction 單一望遠鏡重建
    • 2.3 Stereoscopic reconstruction 立體成像重建
    • 2.4 Gamma-hadron separation 伽瑪－強子鑑別
  • 3. Model analysis 模型分析
    • 3.1 Introduction 簡介
    • 3.2 Gamma-hadron separation 伽瑪－強子鑑別
  • 4. 3D Model analysis 3D 模型分析
    • 4.1 Introduction 簡介
    • 4.2 Gamma-hadron separation 伽瑪－強子鑑別
    • 5. Comparison 比較

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0. Abstract 摘要

Three different analysis techniques for Atmospheric Imaging System are presented. The classical Hillas parameters based technique is shown to be robust and efficient, but more elaborate techniques can improve the sensitivity of the analysis. A comparison of the different analysis techniques shows that they use different information for gamma-hadron separation, and that it is possible to combine their qualities.

本文介紹了大氣成像系統的三種不同分析技術。基於經典 Hillas 參數的技術被證明是穩健和有效的，但更精細的技術可以提高分析的靈敏度。對不同分析技術的比較表明，它們使用不同的訊息進行伽馬－強子鑑別，而且它們的特性是可以結合的。

1. Introduction 概論

From the beginning of ground based gamma ray astronomy, data analysis techniques were mostly based on the “Hillas parametrisation” ^[1] of the shower images, relying on the fact that the gamma-ray images in the camera focal plane are, to a good approximation, elliptical in shape. More elaborate analysis techniques were pioneered by the work of the CAT collaboration on a model analysis technique, where the shower images are compared to a more realistic pre-calculated model of image. Other analysis techniques, such as the 3D Model analysis were developed more recently with the start of the third-generation telescopes. The 3D Model analysis is, for instance, based on the assumption of a 3 dimensional elliptical shape of the photosphere. These analysis techniques are complementary in many senses. We will show that they are sensitive to different properties of the shower, and can therefore be used to cross-check the analysis results or be combined together to improve the sensitivity. No analysis is currently really winning the race, and there is much space for further improvements.

從地面伽馬射線天文學開始，數據分析技術的基礎主要是簇射圖像的「Hillas 參數化」^[1:1]，而這參數化又仰賴以下事實，即，在良好的近似下，相機焦平面中的伽馬射線圖像呈橢圓形。 CAT 合作團隊在模型分析技術方面的工作開創了更精細的分析技術，其中將簇射圖像與更逼真的預計算圖像模型進行比較。 隨著第三代望遠鏡的開始，最近開發了其他分析技術，例如 3D 模型分析。例如，3D 模型分析基於光球為三維橢球狀的假設。這些分析技術在許多方面是互補的。我們將證明它們對簇射的不同特性很敏感，因此可以用它們來交叉檢查分析結果，或者將它們組合在一起，以便提高靈敏度。目前還沒有一種分析方式脫穎而出，而且還有很大的改進空間。

2. Hillas-parameter based analysis 基於 Hillas 參數的分析

2.1 Introduction 簡介

In a famous paper of 1985^[1:2], M. Hillas proposed to reduce the image properties to a few numbers, reflecting the modelling of the image by a two-dimensional ellipse. These parameters, shown on figure 1, are usually:

• length
  $L$ and width
  $w$ of the ellipse
• size (total image amplitude)
• nominal distance
  $d$ (angular distance between the centre of the camera and the image centre of gravity)
• azimuthal angle of the image main axis
  $\phi$
• orientation angle
  $\alpha$

在 1985 年的一篇著名論文中^[1:3]，M. Hillas 提議將圖像的屬性降為幾個數字，反映了用二維橢圓對圖像進行建模。這些參數如圖 1 所示，通常是：

• 橢圓的 長度
  $L$ 和 寬度
  $w$
• 規模（總圖像振幅）
• 標稱距離
  $d$（相機中心與圖像重心之間的角距離）
• 圖像主軸的方位角
  $\phi$
• 位向角
  $\alpha$
  

2.2 Single telescope reconstruction 單一望遠鏡重建

In single telescope observations, the shower direction was estimated from the Hillas parameters themselves (and in particular from the image length and size), either with lookup tables or with ad-hoc analytical functions. But the choice of a symmetrical parametrisation of the shower led to degenerate solutions, on each side of the image centre of gravity along the main axis.
單台望遠鏡的觀測是根據 Hillas 參數本身（特別是圖像長度和規模）來估計簇射方向，估計時使用查找表，或使用特製的解析函數。但是既然簇射參數的選擇是對稱的，就會導致在圖像重心沿主軸的兩側出現退化的解。^[2]

In order to break this degeneracy, other parameters — based in particular on the third order moments — were added later.
為了打破這種退化，後來添加了其他參數——特別是基於三階動差的參數。^[3]

The shower energy is usually estimated with a similar technique, from the image size and nominal distance.
簇射能量通常是使用類似的技術，根據圖像規模和標稱距離進行估算。

2.3 Stereoscopic reconstruction 立體成像重建

The stereoscopic imaging technique, pioneered by HEGRA^[4], provides a simple geometric reconstruction of the shower: the source direction is given by the intersection of the shower image main axes in the camera, and the shower impact is obtained in a similar manner. The energy is then estimated from a weighted average of each single telescope energy reconstruction.

由 HEGRA ^[4:1] 首創的立體成像技術可對簇射進行簡單的幾何重建：光源方向由相機中簇射圖像主軸的交點給出，並用類似的方法獲得簇射的撞擊參數。然後對每個單一望遠鏡重建出的能量進行加權平均，以估算能量。

2.4 Gamma-hadron separation 伽瑪－強子鑑別

The Hillas parameters not only allow to reconstruct the shower parameters, but also can provide some discrimination between γ candidates and the much more numerous hadrons. Several technique were developed, exploiting to an increased extent the existing correlation between the different parameters (e.g. Supercuts^[5], Scaled Cuts^[4:2] and Extended Supercuts^[6]). We will use here the Scaled Cuts technique, in which the actual image width (

$w$) and length (

$l$) are compared to the expectation value and variance obtained from simulation as a function of the image charge

$q$ and reconstructed impact distance

$\rho$, expressed by two normalised parameters Scaled Width (

$\text{SW}$) and Scaled Length (

$\text{SL}$):

$$\begin{array}{}\text{(1)}& \text{SW}=\frac{w(q,\rho )-⟨w(q,\rho )⟩}{{\sigma }_w(q,\rho )},\text{SL}=\frac{l(q,\rho )-⟨l(q,\rho )⟩}{{\sigma }_l(q,\rho )}\end{array}$$

Hillas 參數不僅可以重建簇射參數，還可以在候選的伽馬（事件）和數量更多的強子（背景）之間提供一些區分。有好幾種技術已經被開發出來，它們用了更多不同參數之間的現有相關性（例如 Supercuts^[5:1]、Scaled Cuts^[4:3] 和 Extended Supercuts^[6:1]）。我們將在這裡使用 Scaled Cuts 技術，它是將實際圖像的寬度 (

$w$) 和長度 (

$l$) 與模擬獲得的期望值和變異數進行比較，並且是圖像電荷量

$q$ 和重建撞擊距離

$\rho$ 的函數，由兩個歸一化參數——縮放寬度（Scaled Width，SW）和縮放長度（Scaled Length，SL）——來表示：

$$\begin{array}{}\text{(1)}& \text{SW}=\frac{w(q,\rho )-⟨w(q,\rho )⟩}{{\sigma }_w(q,\rho )},\text{SL}=\frac{l(q,\rho )-⟨l(q,\rho )⟩}{{\sigma }_l(q,\rho )}\end{array}$$

These parameters have the noticeable advantage of being easily combined in stereoscopic observations in Mean Scaled Width and Mean Scaled Length:
這些參數具有明顯的優勢，可以立體成像觀察中輕鬆組合成平均縮放寬度（MSW）和平均縮放長度（MSL）：^[7]

$$\begin{array}{}\text{(2)}& \text{MSW}=\frac{\sum _{\text{tel}}\text{SW}}{\sqrt{n_{\text{tel}}}},\text{MSL}=\frac{\sum _{\text{tel}}\text{SL}}{\sqrt{n_{\text{tel}}}}\end{array}$$

From simulations, one can show that the Mean Scaled Width and Mean Scaled Length are almost uncorrelated for γ candidates (

$\rho =0.15\pm 0.01$) and can therefore be combined in a single variable Mean Scaled Sum (MSS) :

$\mathrm{M}\mathrm{S}\mathrm{S}=(\mathrm{M}\mathrm{S}\mathrm{W}+\mathrm{M}\mathrm{S}\mathrm{L})/\sqrt{2}$.
從模擬中可以看出，平均縮放寬度和平均縮放長度對於 γ 射線的候選事件幾乎不相關（相關係數

$\rho =0.15\pm 0.01$），因此可以組合成一個單一變量——平均縮放和（MSS）：

$\mathrm{M}\mathrm{S}\mathrm{S}=(\mathrm{M}\mathrm{S}\mathrm{W}+\mathrm{M}\mathrm{S}\mathrm{L})/\sqrt{2}$。

3. Model analysis 模型分析

3.1 Introduction 簡介

The Model Analysis, introduced by the CAT collaboration^[8] (with a single telescope) and further developed in the H.E.S.S. collaboration^[9], is based on the pixel-per-pixel comparison of the shower image with a template generated by a semi-analytical shower development model. The event reconstruction is based on a maximum likelihood method which uses all available pixels in the camera, without the requirement for an image cleaning. The probability density function of observing a signal

$S$ in a given pixel, given an expected amplitude

$\mu$, a fluctuation of the pedestal

${\sigma }_{\text{p}}$ (due to night sky background and electronics) and a fluctuation of the single photoelectron signal (p.e.)

${\sigma }_{\text{s}}\approx 0.4$ (PMT resolution) is given by the formula:

The log-likelihood function

$\mathcal{L}=2\sum _{\text{pixel}}\log [P_i(S_i\mid \mu ,{\sigma }_{\text{p}},{\sigma }_{\text{s}})]$ is then maximised to obtain the primary energy, direction and impact. In contrast to the Hillas Analysis technique, the shower reconstruction works in an identical way for a single telescope or for a stereoscopic array.

模型分析（Model Analysis）是由 CAT 團隊 ^[8:1]（使用單個望遠鏡）引入，並在 H.E.S.S. 團隊中進一步發展。^[9:1]，它是讓簇射圖像與「半解析式簇射發展模型生成的模板」逐像素比較而進行的。事件重建是基於最大概似法，該方法使用相機中的所有可用像素，無需圖像清理。對於給定的預期的圖像幅度

$\mu$、背景的波動

${\sigma }_p$（由於夜光背景和電子電路）和單一光電子（p.e.）的信號波動

${\sigma }_s\approx 0.4$（PMT 解析度）下，在給定像素中觀察到信號

$S$ 的機率密度函數由以下公式給出：

然後求對數概似函數

$\mathcal{L}=2\sum _{\text{pixel}}\log [P_i(S_i\mid \mu ,{\sigma }_{\text{p}},{\sigma }_{\text{s}})]$ 的最大值，以得到初級能量、方向和衝擊參數。與 Hillas 分析技術相比，這種模型分析技術在重建簇射時，對於單一望遠鏡或立體成像陣列的工作方式都是相同的。

3.2 Gamma-hadron separation 伽瑪－強子鑑別

4. 3D Model analysis 3D 模型分析

4.1 Introduction 簡介

4.2 Gamma-hadron separation 伽瑪－強子鑑別

5. Comparison 比較

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1. [1] A. Hillas, “Cerenkov light images of EAS produced by primary gamma”, Proc. 19nd I.C.R.C. (La Jolla), Vol 3, 445 (1985). my annotation. ↩︎ ↩︎ ↩︎ ↩︎

2. Annotation: As far as I understand, this may infer that the real Cherenkov image is not a perfect ellipse symmetrical about the main axis. 譯注：就我的理解，這可能意味著，真實的契忍可夫圖像並非完美的、關於主軸兩側對稱的橢圓形。 ↩︎

3. Related source?

   ↩︎

4. [3] A. Daum et al., “First results on the performance of the HEGRA IACT array”, Astropart. Phys. 8, 1 (1997). ↩︎ ↩︎ ↩︎ ↩︎

5. [7] P. T. Reynolds et al., “Survey of candidate gamma-ray sources at TeV energies using a high-resolution Cerenkov imaging system: 1988–1991”, ApJ 404, pp.206–218 (1993). ↩︎ ↩︎

6. [8] G. Mohanty et al., “Measurement of TeV gamma-ray spectra with the Cherenkov imaging technique”, Astropart. Phys. 9 1, pp.15–43 (1998). ↩︎ ↩︎

7. 譯注：求和中的指標

   $\mathrm{t}\mathrm{e}\mathrm{l}$ 是指立體成像望遠鏡的編號，
   $n_{\text{tel}}$ 應該是指望遠鏡的數量。 ↩︎

8. [6] ↩︎ ↩︎

9. [4] ↩︎ ↩︎