Measuring the channel-energy relationship of the SVOM / ECLAIRs camera

Space Engineering, master's level

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Measuring the channel-energy relationship

of the SVOM / ECLAIRs camera

Arvind Vairavan

Master’s Thesis

and Improvement of their Modelling

Author:

Supervisor:

Mini Gupta

Yanez Carlos

A thesis undertaken within:

Centre National d’études Spatiales, Toulouse, France

Submitted in partial fulfillment of the requirements for the degree of:

Masters Techniques Spatiales et Instrumentation

Faculté Sciences et Ingénierie

Université Paul Sabatier – Toulouse III

as part of the

Joint European Master in Space Science and Technology (SpaceMaster)

Measuring the channel-energy relationship

of the SVOM / ECLAIRs camera

Arvind Vairavan

_{Laurent Bouchet}

Olivier Godet

Institut de Recherche en Astrophysique et Planétologie, Toulouse, France

Analysis of the Representation of Orbital Errors and Improvement of their modelling

DISCLAIMER

ACKNOWLEDGEMENTS

I express my sincere and deepest gratitude to my supervisors Laurent Bouchet and Olivier Godet for their guidance throughout the thesis project. Their expertise, invaluable guidance and

constant encouragement added considerably to my knowledge and experience in space

Contents

1 Introduction 6

1.1 The SVOM Mission . . . 7

1.2 The ECLAIRs camera . . . 8

1.3 Working Principle of the detector and the electronics . . . 9

1.4 ECLAIRs Background . . . 10

1.4.1 Simulated ECLAIRs Background . . . 11

1.5 Problem Definition . . . 11

2 Method 12 2.1 Input Data . . . 12

2.2 Spectral Fitting . . . 13

2.2.1 Spectral Fitting . . . 14

2.3 Chi-square statistics and best fit parameters . . . 14

2.4 Error fitting and confidence level . . . 15

2.4.1 Estimating the guess values for spectral fitting . . . 15

2.4.2 Continuum functions . . . 16

2.5 Spectral fitting for peaks . . . 17

2.6 Computation of gain and offset . . . 17

3 Energy to Channel Scale Conversion 18 3.1 Estimation of the gain and offset offset coefficients . . . 18

3.2 Optimize for robustness and CPU time efficiency . . . 18

3.2.1 The 8 keV line . . . 19

3.2.2 The 23 keV lines . . . 20

3.2.3 The 75 keV lines . . . 20

3.3 Channel-Energy Relationship . . . 22

4 Discussion 24

5 Conclusion 25

6 Future Works 25

A Sigma Error Distribution 28

B Peak Position table 28

List of Figures

1 Illustration of the GRB firemodel. . . 6

2 Distribution of the various instruments aboard the SVOM platform and telescopes of the ground component of the project. . . 7

3 View of ECLAIRs main components. . . 8

4 241_{Am spectrum of one pixel of the prototype . . . 10}

5 Input spectrum of the in-flight simulated data . . . 12

6 Flowchart explaining the procedure used to verify and optimize the method . . . 13

7 Linear and Quadratic regressions. . . 16

8 Spectrum fitting for the peak with blue line depicting the linear fit for the continuum and orange curve depicting the peak fit . . . 17

9 Spectrum fitting for the peak with blue line depicting the quadratic fit for the continuum and orange curve depicting the peak fit . . . 17

10 Spectral fitting for peak 1 at 8 keV. . . 19

11 Spectral fitting for peak 2 at 23 keV. . . 20

12 Spectral fitting for peak 3 at 75 keV. . . 21

13 Evolution of peak centroids in channel and their errors with data acquisition time. . . 22

14 Evolution of gain and offset with DA time. . . 22

15 Evolution of gain and offset when certain peaks are ignored while calibrating the channel-energy relationship. . . 23

16 Illustrating the Gaussian distribution distribution with mean µ . . . . 28

List of Abbreviations

CdTe - Cadmium Telluride

CXB - Cosmic X-Ray Background DoF - Degrees of Freedom FoV - Field of View

FWHM - Full Width at Half Maximum GRB - Gamma Ray Bursts

GRD - Gamma Ray Detectors GRM - Gamma Ray Monitor

MXT - Micro-channel plate X-ray Telescope

Abstract

The French-Chinese Space based multi-band Variable astronomical Object Monitor(SVOM) mission will be dedicated to the study of the high-energy transient sky with a particular interest for Gamma-ray bursts (GRBs). The mission is due to be launched in 2021. These cosmological and very powerful stellar explosions signal the catastrophic formation of a compact object (a black hole or a neutron star). GRBs appear as short-lived, highly variable and intense bursts of Gamma-ray light, followed by a long-lived multi wavelength afterglow emission characterized by a rapid flux decay over time. To study GRBs, SVOM will embark two wide-field instruments and two narrow-field telescopes on-board an agile platform. The wild field instruments will study the GRB prompt emission from 4 keV to 5 MeV and the narrow-field telescopes will study the afterglow emission in X-ray (0.2 - 10 keV) as well as in the visible once the satellite repoints towards the GRB position calculated by the ECLAIRs instrument.

ECLAIRs, the prime instrument on board SVOM, is a 4-150 keV 2-D coded mask telescope in charge of autonomously detecting and providing the first GRB position. ECLAIRs will measure the arrival date, the position on the detection plane (6400 CdTe pixels) and the energy of each detected photon.

The measured energy will be encoded over a linear dynamics (210) as a

num-ber (channel) by the readout electronics. To reconstruct the energy scale in physical units (keV), it is necessary to calibrate the linear channel-energy re-lationship (i.e to measure the gain and offset coefficients) for each of the 6400 ECLAIRs detectors. Experimental on-ground measures on the detection plane irradiated using radioactive sources will be performed to calibrate the relation-ship. In flight, due to the aging of electronics and detectors induced by the radiation environment along the SVOM orbit, the reconstruction of the energy scale may change for some pixels over time, making it necessary to re-calibrate the relationship for these pixels. To do so, we will make use of the measured ECLAIRs background, in particular the instrumental lines present below 90 keV.

My thesis work focuses on investigating a robust and CPU time efficient method to derive updated values of the gain and offset coefficients for the 6400 ECLAIRs pixels using in-flight simulated ECLAIRs background spectra. I introduce the SVOM mission and the problem definition in section 1. In section 2.2, I explain the spectral fitting process of both the most intense instrumental lines and their underlying continuum using various models through an iterative

χ2minimization process. In section 3.1, I describe the method used to estimate

Abstract

La mission spatiale Sino-Française SVOM, dont le lancement est prévu fin 2021, effectuera une étude détaillée du ciel transitoire en synergie avec les dé-tecteurs multi-messager (dédé-tecteurs d’ondes gravitationnelles et de neutrinos) avec un intérêt particulier pour les sursauts gamma. Ce sont des pénomènes dus à des phénomènes cosmiques catastrophiques signalant la formation catas-trophique d’un trou noir ou d’une étoile à neutrons lors de l’effondrement du cœur de certaines étoiles massives ou la coalescence de deux étoiles à

neu-trons. Les sursauts gamma se présentent sous la forme de éclats intenses

de lumière gamma de courte durée suivies d’une émission rémanente multi-longueurs d’onde de plus longue durée. Pour les étudier, SVOM disposera de deux instruments à grand champ et de deux télescopes à champ étroit embar-qués sur une plate-forme agile. Les instruments à grand champ permettront de mesurer l’émission prompte de 4 keV à 5 MeV et les télescopes à champ étroit permettront de mesurer l’émission de rémanence dans les rayons X (0,2 - 10 keV) ainsi que dans le visible une fois que le satellite les repointera dans la direction du sursaut détecté par ECLAIRs.

ECLAIRs est une caméra à masque codé 2D, fonctionnant dans la gamme d’énergie 4–150 keV, en charge de la détection et de la fourniture de la pre-mière localisation des sursauts gamma. ECLAIRs mesurera la date d’arrivée, la position sur le plan de détection et l’énergie de chaque photon détecté en vol. L’énergie mesurée sera codée sur une dynamique linéaire par un nombre (canal). Pour reconstruire l’échelle en énergie en keV, il nous faut mesurer les coefficients gain et offset de la relation canal-énergie. Au sol, nous ef-fectuerons des mesures sur le plan de détection en l’illuminant avec plusieurs sources radioactives pour étalonner cette relation. Cette dernière pourrait être amenée à évoluer dans le temps essais au sol, nous utilisons des sources ra-dioactives émettant des photons à des énergies spécifiques pour étalonner la relation Canal-Energie de chaque détecteur. Cependant, cette relation peut être amenée à évoluer dans le temps du fait du vieillissement des détecteurs et de l’électronique. Il convient donc de surveiller cette évolution pour les 6400 détecteurs d’ECLAIRs une fois en vol. Pour ce faire, une stratégie consiste à utiliser, comme points de référence, les raies instrumentales vues en émission dans le spectre de bruit de fond mesuré par ECLAIRs. Ces raies sont pro-duites par la fluorescence de matériaux composant l’instrument en dessous de 90 keV (cuivre et plomb) faisant suite au bombardement de l’instrument par des particules présentes sur l’orbite de SVOM.

Ma thèse porte sur l’étude d’une méthode robuste et économe en temps permettant de déduire de nouvelles valeurs des coefficients de gain et de offset pour les 6400 pixels d’ECLAIRs à l’aide de spectres (simulés) de bruit de fond d’ECLAIRs. Je présente la mission SVOM et la définition du problème dans la section 1. Dans la section 2.2, j’explique le processus d’ajustement spectral que j’ai utilisée pour modéliser les raies instrumentales et le continu

sous-jacent via un processus itératif de minimisation du χ2. Dans la section 3.1, je

1

Introduction

The Chinese National Space Agency and the French Space Agency (CNES) have decided to jointly implement a satellite mission dedicated to the study of GRBs and other powerful transient events. The mission is expected to be launched by the end of 2021.

The SVOM mission (see Figure 2 [1]) will carry out a detailed study of the high energy transient sky in synergy with multi-messenger detectors (eg. gravitational wave detectors [2]) with a particular interest in GRBs [3]. They appear as flashes of X and Gamma-ray photons due to cosmological explosions signaling the catastrophic formation of a compact object such as black hole or a neutron star and the launch of ultra-relativistic jets directed towards the Earth following the gravitational core

collapse of some massive stars (M > 20 M) [4] or the coalescence/merger of two

compact objects (e.g. NS-NS, NS-BH) [5]. Lasting anywhere from a few milliseconds to several minutes, the prompt emission of GRBs shine hundreds of times brighter

than a typical supernova with an isotropic radiated emission about 1054_{ergs [6]. The}

prompt emission is followed by the afterglow emission of a GRB (see Figure 1).

Figure 1: Illustration of the GRB firemodel. Image: NASA Goddard Space Flight Center

A black hole is a region of spacetime that irrevocably consumes all matter and radiation which comes within its sphere of influence [7]. A neutron star ( M

1.5 M) is one of the possible end points of the evolution of massive stars. The

Figure 2: Distribution of the various instruments aboard the SVOM platform and telescopes of the ground component of the project. The components MXT, VT, ECLAIRs and GRM are on the satellite payload while GFT-1, GFT-2 and GWAC are ground based telescopes.

1.1

The SVOM Mission

afterglow emission in the 0.2 - 10 keV energy range for the Micro-channel plate X-ray Telescope(MXT) [10] and in the 400-1000 nm optical range for the Visible Telescope (VT) [11], respectively. The GRM with a total FoV of 2.6 sr consists of three detectors called GRDs (Gamma-Ray Detectors). The GRM will provide information on the spectral shape and the light curve of GRBs during the prompt phase in the 15 keV – 5 MeV energy range [12].

The secondary objective is to to measure the broadband spectral characteristics and temporal properties of the GRB prompt emission. The mission puts a special emphasis on two categories of GRBs: very distant GRBs at z>5 which constitute exceptional cosmological probes and faint nearby GRBs to probe the nature of bursts and the physics at work in the explosion.

The mission lifetime is expected to last a minimum of 3 years, with a possible extension by 2 years. SVOM will operate in low orbit at an altitude of 625 km

with an inclination of 30◦ [3]. SVOM will use an anti-solar pointing to maximize

the chance of follow-up by large ground facilities of SVOM detected GRBs.

1.2

The ECLAIRs camera

Figure 3: View of ECLAIRs main components. The ECLAIRs Instrument consists of a 2-D Coded Mask, a lateral passive shield made of Pb and Cu layers, a detection plane consisting of 200 XRDPIX detection modules of 32 Schottky CdTe detectors, divided into 8 sectors of 800 detectors each, a readout electronics box and a data processing unit, UGTS, including the GRB trigger software.

ECLAIRs is developed by a consortium of several French laboratories (IRAP1_{, APC}2

and CEA-Saclay3) under the supervision of CNES. Figure 3 shows the ECLAIRs

sub-systems. ECLAIRs is a 2 sr ( 90x90deg2_{) coded mask camera designed to detect}

around 70 GRBs/year [13]. Due to the space and mass allocation constraints on the

1_{L’Institut de Recherche en Astrophysique et Planétologie} 2_{Laboratoire Astroparticules et Cosmologie}

satellite platform, the ECLAIRS detection plane occupies 1024 cm2 in geometrical area [1]. To optimize the GRB detection sensitivity (i.e. the number of GRBs), the energy threshold will be set at 4 keV, improving the capability to detect X-ray flashes (i.e.GRBs for which the prompt emission photons are mostly emitted in X-rays) and high redshift (z >5) GRBs [14].

The ECLAIRs X- and Gamma-ray imaging camera is used for GRB detection and localization. It works in the 4–150 keV energy range with a detection plane and a real-time data-processing electronic system (UGTS). IRAP builds an array

of 80 × 80 semiconductor detectors. Each detector has a 4 × 4 mm2 geometrical

area and a 1 mm thickness. The detectors are arranged on the cold plate into 8 independent sectors, each coupled to a readout electronics.

The coded mask camera for X- and Gamma- rays is responsible for triggering GRB observations and providing their first localization with an accuracy better than 10 arc-minutes in the 4-50 keV imaging band. The passive shielding is designed so that the background photons coming from outside the FoV will be completely absorbed by the shielding compounds.

The photons from a certain direction in the sky project the mask on the detector of a coded mask camera. This projection has the same coding as the mask pattern but is shifted relative to the central position over a distance that uniquely corre-sponds to the direction of the photons. The detector accumulates the sum of the number of shifted mask patterns. The position is encoded in each shift and the in-tensity of the sky at the given position is encoded as the strength of the shifted mask. After a certain illumination period, the accumulated detector image is decoded to a sky image by determining the strength of the shifted mask pattern [15].

1.3

Working Principle of the detector and the electronics

Incident photons interact with CdTe detectors depositing their energy either totally

through the photo-electric effect or partially through Compton scattering. The

photoelectric effectis the dominant effect in the energy range of ECLAIRs [16]. Once a photon deposits energy in the CdTe detector, this creates a charge cloud (a cloud

of electrons [e-] and holes4 _{[h]) at a certain depth in the detector depending on the}

photon energy. The total energy deposited in the crystal is directly proportional to the number of e-/h pairs formed.

To fully deplete the detector of free charge carriers, the detectors are polarized with a reverse bias of -300 V. This induces an electric field which causes the electrons and holes created by the photon interactions to drift from their initial location. The electrons will move towards the anode while the holes will move towards the cathode inducing the creation of a build-up charge over time on the electrodes [17].

The first stage of electronics (pre-amplifier) converts the charge collected on the anode to a voltage impulse signal with a certain amplitude. The amplitude of the analog signal is proportional to the number of collected electrons onto the anode. Once the amplitude of the analog signal is larger than the low-level threshold, the electronics will read the event (i.e. provide a time tag, a position on the detection plane and measure the energy). The amplitude of the analog signal is measured

4_{A hole or an electron hole, in solid state physics, is the effect created by the absence of an}

and encoded by a analog-to-digital converter into a number (channel) over a 10 bits linear dynamics.

1.4

ECLAIRs Background

Figure 4: 241_{Am spectrum of one pixel of the prototype}

The SVOM satellite will be subject to different sources of background in space such as extra-galactic components (X-ray and Gamma-ray diffuse background) and near Earth components (Gamma-ray albedo, neutrons, secondary protons located under the radiation belts) [18]. The extra galactic diffuse background is the dominant background up to 70 keV [13] and internal background starts to become dominant above 70 keV.

The instrumental lines induced by fluorescence of materials placed around the detectors are used to compute the ECLAIRS background. The Pb and Cu layers in the passive shield as well as the CdTe material used on the detectors deposit energies on the energy spectrum range due to their interaction with photons. The energy deposited by Cu is observed at 8 keV, CdTe in the 27 keV region and Pb in the 75 keV region.

Figure 4 shows the spectrum of241Am source emitting photons [19]. The emission

1.4.1 Simulated ECLAIRs Background

The ECLAIRs event count rate is dominated by the photon background due to its large field of view and large effective area [18]. The background comprises of the Cosmic X-Ray Background (CXB) [20], the reflection of CXB on the Earth’s at-mosphere and the radiation from the Earth’s atat-mosphere (albedo).The instrument background expected in orbit, therefore, needs to be estimated to enhance the sci-entific performances of the mission. The background will vary along the orbit given the mission profile and observing strategy (see section 1.1). To compute the back-ground spectrum, a Monte-Carlo approach [21] was used to simulate the primary and secondary interactions between particles from the main components of the space environment that SVOM will encounter along its Low Earth Orbit (LEO).

The Monte Carlo database allows the diffuse background to be estimated at all points of the orbit. This database contains tal the energy deposits in the ECLAIRs

detection plane from simulations made using GEANT45_{. The simulations were}

performed using the release 4.10.1 of the GEANT4 C++ toolkit [22]. An algorithm then makes it possible to estimate the background noise at any position of the satellite in the orbit of SVOM by selecting adequate events from the database.

GEANT4 enables in describing the detailed mass model of the payload and the spacecraft as well as drawing particles (photons, electrons, protons, neutrons, ...) with a specific spatial and energy distribution. It also enables in tracing the paths of the primary particles through the body of the camera as well as any secondary particles generated during the different physical processes (Compton, Rayleigh scat-tering, photo-electric effects, pair annihilation and creation, and nuclear interaction). The output of GEANT4 gives a spectrum of the energy deposits as well as the depth of each interaction for each detector.

1.5

Problem Definition

The measured energy encoded in channel by the electronics needs to be converted into physical units (i.e. keV) using a linear relationship (channel = gain × E + offset). In order to calibrate the channel-energy relationship of each detector, the photons emitted by radioactive sources in ground tests are used as references [19]. In flight, due to the aging of electronics and detectors induced by the radiation environment along the SVOM orbit, the reconstruction of the energy scale may change for some pixels over time, making it necessary to re-calibrate the relationship for these pixels.

Upon observation of significant change from the initial energy expected, new set of values for gain and offset are computed. To calibrate this relationship for the 6400 ECLAIRs detectors on-ground, experimental measures will be performed on the flight mode of the detection plane using a set of radioactive sources emitting photons at known energies over the ECLAIRs energy band. This is done using the instrumental lines from the ECLAIRS background. The aim of the thesis is to investigate a robust and CPU time efficient method to derive updated values of the gain and offset coefficients for the 6400 ECLAIRs pixels using in-flight simulated ECLAIRs background spectra.

2

Method

In the first step I calibrate the free parameters of the model through an iterative

χ2 _{minimizing spectral fitting process. In the second step I establish the}

channel-energy relationship between the positions of the most intense lines in channels and the expected energy values of the intense peak lines by estimating the gain and offset values. And in the final step, I optimize the method for robustness in order to perform the operations in the shortest possible time and efficiently across 6400 detectors.

The ECLAIRs scientific specifications about the accuracy of reconstructing the energy scale states that the energy scale reconstruction below 80 keV shall be per-formed with an accuracy better than ± 0.15 keV (i.e. ± 1 channel) while above 50 keV it shall be better than ± 0.25 keV (± 1.7 channels).

2.1

Input Data

Figure 5 shows the input spectrum of the background expected in flight obtained from Monte-Carlo simulations in the 4 keV-150 keV energy range. The spectrum is degraded to the resolution of the CdTe detectors using a Gaussian with a Full-Width Half-Maximum (FWHM) of 1.6 keV. The particular data studies the scenario when the Earth is in the field of view (FoV) of ECLAIRs and the external diffuse background is low as compared to a scenario when the Earth is not in the FoV [3]. This enables in studying the nominal case scenario where the intense peak lines from the deposited energies are more prominent. The input data used here is taken from the simulations run on GEANT4 [23].

Figure 5: Input spectrum of the in-flight simulated data. The Pb and Cu layers in the passive shield as well as the CdTe material used on the detectors deposit energies on the energy spectrum range due to their interaction with photons. The energy deposited by Cu is observed at 8 keV, CdTe in the 27 keV region and Pb in the 75 keV region.

on which they are superimposed, are variable along the orbit. Hence, it is important to optimize the monitoring of these emission lines to adapt to the changes while estimating the energy-channel relationship. And this monitoring needs to be done for each detector in flight.

The input data describing the deposited energies (see Figure 5), is chosen based on the multiplicity of events that show the peak lines more prominently. For the 8 keV and 75 keV lines, the multiplicity is given as 1 while for the 27 keV, the multiplicity is 2. The input data is then distributed into bins with a bin size of 1 Channel as shown in the figure 5. An event refers to the to the interaction of a photon and detection of the energy in the detector. Simple events are single events detected in a defined coincidence window. Events can be classified as multiple events if they occur on different modules or different pixels within the same module.

2.2

Spectral Fitting

Fig. 6 below explains the procedure that is followed to evaluate the method used to calculate the channel-energy relationship. To do the fitting process, three particular regions are chosen where the peaks are prominent, 5 keV to 14 keV, 14 keV to 40 keV and 70 keV to 80 keV (see section 1.4). The output of the GEANT4 simulations is in keV, but to perform the analysis I need to convert the energy scale to how it appears as outputs of the electronics. Hence, I make use of an arbitrary set of gain and offset parameters derived from experimental data. An initial value of 7 channel/keV for gain and 0.5 Channel for offset are chosen from the prototype of the detection plane is used to convert the energy (in keV) to Channel scale.

curve represents the probability distribution with standard deviation σ relative to the average of a random distribution. The error function is obtained by integrating the normalized Gaussian distribution as shown in equation 1.

erf (x) = √2

π ×

Z x

0

e−t2dt (1)

For a random variable that is normally distributed with mean 0, erf(x) describes the probability of the function falling in the range [0, x]. Gauss error function was defined in python to perform a fit to the the data with a model. The error function is comprised of three parameters, centroid - the energy position at which the peak was observed, amplitude - the number of counts at the peak position, and σ - standard deviation of the observed peak.

The error function at each interval of the Guassian distribution can be defined as:

E(x, µ, σ) = 1/2 × (erf (xi+1√ − µ

2σ ) − erf (

xi− µ

√

2σ )) (2)

where erf is the erf function, x is bins representing channels, µ is the centroid and σ is the width.

2.2.1 Spectral Fitting

Spectral fitting is the process of specifying the model that provides the best fit to the specific curves in a dataset. Scipy is the scientific computing module of Python providing built-in functions for well-known Mathematical functions. Python packages support multiple regression and optimization functions such as polyfit, curvefit, linregress, lmfit, linalg.leastsq, etc.

Curvefit is one such function provided by Scipy package that is chosen because of itse low run time and its simple functionality requiring only a model function, data, and initial guesses to perform the fitting operation. Curvefit uses non-linear least squares to fit a function, f, to the data. Least squares minimization is the process of minimizing the sum of the squares of the residuals in the fitting process. A residual is the difference between an observed value, and the fitted value provided by a model. A linear or quadratic function, used to estimate the background, is then combined with the Gaussian function to model the spectrum data.

Additionally, the curvefit procedure also has the possibility of utilizing multiple parameters and the ability to execute least square minimization while returning both best fit parameters as well as the covariance matrix showing the uncertainty and interdependence of each parameter.

2.3

Chi-square statistics and best fit parameters

The estimation of the best fit parameters is carried out through an iterative mini-mization fitting process between the model and the simulated measurements. To es-timate the goodness of the spectral fitting process, chi square statistics is computed.

The χ2 test indicates whether the divergence of the observed values is reasonable

by the best fit value of the parameters centroid(µ), amp(A), σ, slope (m) & con-stant(C) such that a model defined by function y = f(x:µ, A, σ) best describes the data. The goodness of the fit estimated using chi square statistics, determines the best fit parameters that minimize the difference between the model and the data.

χ2 = ΣN_i=1(f (x : µ, A, σ, m, C) − yi)2) (3)

This is done by systematically choosing input values from within an allowed range till the minimum value of the difference is obtained. The python command used to estimate the best fit parameters and the error of each parameter in performing the fit is scipy.optimize_curvef it. The command fits a linear function to the data set by doing least-square minimization. The command returns the popt and pcov values. popt is a one dimensional array of the best estimates for the parameter values with each entry matching the order in the function definition and pcov is the covariance matrix showing the uncertainty and interdependence of each parameter in popt. Covariance matrix is a matrix whose element in the (i, j) position is the covariance between the i-th and j-th elements of a vector. One σ standard deviation errors on the parameters are computed by taking the square root of the diagonal of the covariance matrix.

2.4

Error fitting and confidence level

Any measured data has uncertainties which need to be taken into consideration to perform the fit. The model follows a Gaussian distribution with mean 0 and

standard deviation as p(data). To determine the uncertainty in a given data, the

deviation of errors in the data is given as input when performing curve fit using the python function. To compute the fit for the data along with error, 1-sigma error values, i.e a statistical calculation that refers to data within one standard deviation from mean value. It is calculated as the square root of the data (refer Appendix A). A confidence interval determines the level of uncertainty there is in any particular statistic. For this thesis, a 90 % confidence level is estimated for the fitting process.

2.4.1 Estimating the guess values for spectral fitting

The basic idea of all non-linear fitting routines is to start with some initial guesses for the parameters to be fitted and then obtain the best fit parameters by minimizing the fitting function. The peaks are estimated using the scipy.signal.f ind_peaks command on python with the sigma estimated at 1.6/2.35 ∼ 0.68keV ∼ 4 channels (FWHM [25].) The python command takes a one-dimension array as input and through comparison of neighbouring values, it estimates the local maxima. This is done by performing a convolution on the data with a function characterized by a width parameter and a height parameter.

2.4.2 Continuum functions

Once the peaks are found using the python command, the continuum is constructed for the region excluding the peak bounded by ±2σ from the peak position. To construct the continuum, both linear and quadratic extrapolation are considered.

The coefficients of a linear function y = mx+c or a quadratic function y = ax2_+bx+c

are calculated from the extrapolated data excluding the peak. In case of linear, all four regressions are considered to extrapolate and construct the continuum, as shown in figure 7. The quadratic regression, shown in figure 7e is estimated by applying a least-squares estimation to fit a polynomial function to the data.

1. linlin: y = m × x + c 2. linlog: y = m × log(x) + c 3. loglin: log(y) = m × x + c 4. loglog: log(y) = m × log(x) + c

(a) (b)

(c) (d)

(e)

2.5

Spectral fitting for peaks

The continuum fit with a linear function consisting of 2 parameters, slope and con-stant, combined with 3 parameters of the Gaussian fit, centroid, Amplitude and σ are then used to perform the peak fitting. The five parameters are fed as input to the function defined by the following equation:

peakf it = √amp

2π × σ × E(x, µ, σ) + m × x + C (4)

where amp is amplitude, x is bins representing channels, mean is the centroid, σ is the width, and m and C are the co-efficients of the linear fit. In case of quadratic regression, six parameters, including two coefficients, are fed as input to the function defined by the following equation:

peakf it = √amp

2π × σ × E(x, µ, σ) + c1 × x

2_{+ c2 × x + C (5)}

where c1, c2 and C are the co-efficients of the quadratic fit.

To perform the spectral fitting, initial guess values including the coefficients de-termined by the continuum extrapolation and the three parameters for the Gaussian function are given (see section 2.4.1).

Figure 8: Spectrum fitting for the

peak with blue line depicting the lin-ear fit for the continuum and orange curve depicting the peak fit

Figure 9: Spectrum fitting for the

peak with blue line depicting the quadratic fit for the continuum and orange curve depicting the peak fit The quadratic polynomial function performs a better fit at the tail ends of the peak.. There were no significant changes in the Gaussian parameters between the linear and quadratic continuum functions. Statistics in flight will be linear(see sec-tion 1.1), hence a linear continuum funcsec-tion with less parameters is better adapted.

2.6

Computation of gain and offset

3

Energy to Channel Scale Conversion

3.1

Estimation of the gain and offset offset coefficients

The measured energy from the read-out electronics will be coded on a linear dy-namics by a number (channel) for each of the 6400 ECLAIRS detector. To convert a given channel in physical units (keV), it is necessary to calibrate for each detec-tor its Channel-energy relation. As discussed in Section 1.5, the channel-energy relationship is defined by parameters gain and offset.

The standard method to measure the gain and the offset to reconstruct the energy scale is to derive the centroid of the peak in Channel and to associate it with the expected line energy by performing a linear fit [6]. From the simulated background data, one peak, three peaks and two peaks are observed at around 8 keV, 27 keV and 75 keV, respectively.

The channel-energy relationship is given by a linear function:

Channel = gain × Energy + offset (6)

I used the centroid values derived for the 6 lines (see Section 2.6) and their corresponding values in keV (see table in the Appendix B) to measure the gain and offset coefficients from Eq. 6 using a linear regression routine. The best fit values obtained for Centroids of each of the 6 peaks and the expected energy values are given as inputs to a linear relationship function to retrieve the coefficients, gain

and offset [26]. The units of gain is keV−1. The obtained best fit parameters and

the an 1-σ errors for gain and offset are 7.33 ± 0.008 Channel/keV and 0.33 ± 0.18 Channel, respectively. The values obtained are close to the initial values used to convert energy to channel (see 5).

3.2

Optimize for robustness and CPU time efficiency

One of the objectives of the thesis is to investigate the amount of data that will be needed to reach a given accuracy on the reconstruction of the energy scale. To do so, I decided to define a series of 9 data acquisition (DA) durations from 533 days to 2 days (see table in Appendix B) to create the input background spectra. The statistics of number of events observed at each energy/channel is reduced to roughly half of that in each iteration. This iterative procedure is stopped if one of the following conditions fails to be met:

1. The measured centroid value of any observed lines has a 1 sigma error within 1 channel

2. When the spectrum statistics make it difficult to perform the line peak(s) search properly.

3.2.1 The 8 keV line

For the peak expected at 8 keV introduced by Cu, the corresponding best fit value in channel is 57.25. Figure 10 shows a comparison between the model representing the data using the guessed values (left column) and the fit using the best-fit parameters (right column).

(a) (b)

(c) (d)

(e) (f)

3.2.2 The 23 keV lines

Figure 11 describes the spectral fitting for the 3 instrumental lines induced by Pb.

(a) (b)

(c) (d)

(e) (f)

Figure 11: Spectral fitting for peak 2 at 23 keV with 193 degrees of freedom. Figures 11a, 11c and 11e depict the comparison between the model and data using guess estimates for data acquisition times of 533 days, 66 days and 4 days, respectively and the figures 11b, 11d and 11f show the corresponding final fit. The ratio on the plot describes the correlation between the model and fit.

3.2.3 The 75 keV lines

(a) (b)

(c) (d)

(e) (f)

Figure 12: Spectral fitting for peak 3 at 75 keV with 75 degrees of freedom. Figures 12a, 12c and 12e depict the comparison between the model and data using guess estimates for data acquisition times of 533 days, 66 days and 4 days, respectively and the figures 12b, 12d and 12f show the corresponding final fit. The ratio on the plot describes the correlation between the model and fit.

suited to be used for estimating the channel-energy relationship.

Figure 13: Evolution of peak centroids in channel and their errors with DA time. The dashed lines represent the expected position of centroid from the instrumental lines.

3.3

Channel-Energy Relationship

Figure 14 shows the evolution of gain and offset while constructing the channel-energy relationship considering all 6 peaks, namely, 1 peak at 8 keV, 3 peaks at 23 keV and 2 peaks at 75 keV. The gain and offset values are not consistent with the expected values. This could be due to the uncertainties introduced by the lines at 23 keV (see figure 13).

(a) (b)

Other scenarios are also considered to calibrate the channel-energy relationship by excluding certain peaks. This is done to understand the role of each observed peak and its uncertainty in calibrating the channel-energy relationship. Figure 15 shows the evolution of gain and offset when channel numbers calibrated at various peaks are excluded from the gain and offset calculation.

(a) (b)

(c) (d)

(e) (f)

Figure 15: Evolution of gain and offset while calibrating the channel-energy rela-tionship. The line at 8 keV is excluded in figures 15a and 15b with 3 DoF. The lines at 23 keV are excluded in figures 15c and 15d with 1 DoF. Only the dominant lines of each region are included in excluded in figures 15e and 15f with 3 DoF.

role i.e indication of higher error rate in calibrating the channel-energy relationship as compared to gain. Additionally, the peaks at 23 keV region play a significantly important role in calibrating the channel-energy relationship. This can be observed in figure 15d where the offset uncertainties are high when the peaks in the 23 keV region are excluded from the data used to obtain the channel-energy relationship.

4

Discussion

I considered different models for describing the continuum such as linear and quadratic functions. Although, the quadratic functions better described the data for high

statistics (χ2 _{at 3.683 with degree of freedom(DoF) 73 for linear and 1.478 with}

DoF 72) for quadratic with a degree of freedom of 66), the linear function was equally reliable for low statistics (see Appendix C).

5

Conclusion

ECLAIRs mission on board the SVOM satellite will detect and locate GRBs. The readout electronics on the detection plane converts the energy deposited by an event into a channel number. The gain and offset coefficients of all the 6400 pixels need to be computed. My thesis evaluates the method to calculate the channel-energy relationship using the in flight background lines. The estimation of the free model

parameters was carried out through an iterative χ2 minimization fitting process.

The χ2 _{value obtained for the modelling of data indicates the goodness of the fit.}

However, despite the good statistical fit, it is observed that some parts of the low-end tail of the energy lines are not properly fitted by the model when very low statistics are considered.

Accurate modelling of data in fitting lines were successfully performed. From the line position in channels with known energies, I estimated the channel-energy relationship. The gain and offset parameters obtained are compliant with ECLAIRs scientific specifications. The reconstructed energy scale had uncertainties less than ± 0.15 keV (i.e. ± 1 channel) and, thus, accurate. Although the modeling of data is tested, the robustness of the method still needs to be evaluated across multiple simulated data. The thesis could be further extended by estimating the channel-energy relationship across all 6400 pixels

Finally, the computation time taken to perform the operation for one pixel was reduced from the initial 5.35 seconds to 1.2 seconds while taking the entire simu-lated data into consideration. When, the statistics are further reduced, the CPU computing time required to perform the operation while being compliant with the ECLAIRs scientific requirements is acceptable to estimate the channel-energy rela-tionship across all 6400 pixels. The computation was performed on a core i7 CPU, 8GB RAM machine.

6

Future Works

To assess the robustness of the method, simulated data of noisy pixels could be introduced. A detector is considered noisy when its count rate suddenly increases. The presence of a noisy pixel may have an impact on locating bursts as they reduce the sensitivity of the affected modules due to the dead time losses where dead time is defined as the fraction of lost events during the readout process.

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[19] A. Bajat, O. Godet, J. L. Atteia, et al. Calibration of the spectral response of the SVOM/ECLAIRs detection plane. In Space Telescopes and Instrumentation 2018: Ultraviolet to Gamma Ray, volume 10699 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, page 106995J, Jul 2018. [20] E. Boldt. The cosmic X-ray background. Physics Reports, 146(4):215–257, 1987. [21] O. Godet, D. Barret, J. Paul, et al. Background simulations of the wide-field coded-mask camera for X-/Gamma-ray of the French-Chinese mission SVOM. In 37th COSPAR Scientific Assembly, volume 37, page 1041, Jan 2008.

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[24] The Minimum Chi-Square Method

https://ned.ipac.caltech.edu/level5/Wall2/Wal3_4.html (accessed: 01.05.2019). [25] Gaussian Function

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ECLAIRs camera on board the mission SVOM. Experimental Astronomy,

46(2):337–356, Nov 2018. [27] Sigma significance

A

Sigma Error Distribution

A 1-sigma error indicates that if one analyzes the same sample many times, 68% of the values will be in the range of µ - 1 - sigma and µ + 1 - sigma. If 2 sigma error is used, then approximately 95% (95.4%) of the measurements will fall on that range. Figure 16 explains the different error confidence that can be chosen[27].

Figure 16: Illustrating the Gaussian distribution distribution with mean µ

B

Peak Position table

The tables below describe the parameter values of Gauassian combined with linear

model and χ2 _{statistics as a function of the DA time. The rows highlighted in}

each of the tables indicate the lowest statistics for which the ECLAIRs scientific specifications criterion are met.

Peak 1 reduced χ2 DA Time

Centroid (channel) Amplitude(counts) σ(channel) (dof=73 ) (days)

57.26 ±0.075 12126 ±323 5.18 ±0.08 3.69 533 57.22 ±0.08 5996 ±236 5.18 ±0.09 2.2 267 57.24 ±0.098 3027 ±132 5.12±0.1 1.61 133 57.34 ±0.13 1470 ±63 5.18 ±0.20 1.47 66 57.10 ±0.18 745 ±40 5.18 ±0.15 1.33 33 57.19 ±0.27 374 ±32 5.20 ±0.3 1.49 16 57.39 ±0.39 179 ±25 5.02 ±0.43 1.46 8 57.19 ±0.43 97 ±17 4.47 ±0.47 1.16 4 56.10 ±1.23 42 ±9 8.15 ±1.72 1.57 2

Peak 2 reduced χ2 DA Time

Centroid (channel) Amplitude(counts) σ(channel) (dof=193 ) (days)

161.925 ±0.04 8320 ± 294 2.13 533 161.952 ±0.048 4159 ±182 4.85 ±0.04 1.53 267 161.967 ±0.06 2084 ±116 4.81 ±0.05 1.17 133 161.917 ±0.08 1028 ±62 4.88 ±0.07 1.12 66 161.765 ±0.12 502 ±43 4.91 ±0.1 1.06 33 161.731 ±0.17 251 ±29 4.69 ±0.15 1.12 17 161.606 ±0.27 118 ±18 5.03 ±0.25 1.44 8 161.89 ±0.32 61 ±13 5.13 ±0.3 1.07 4 161.975 ±0.37 34 ±7 4.83 ±0.34 0.74 2

Table 2: Best fit parameter value and chi square statistics for peak 2 at 23 keV.

Peak 3 reduced χ2 _{DA Time}

Centroid (channel) Amplitude(counts) σ(channel) (dof=75 ) (days)

511.82 ±0.12 6953 ±245 4.62 ±0.13 2.03 533 511.87 ±0.14 3477 ±166 4.76 ±0.16 1.34 267 511.76 ±0.2 1687 ±98 4.78 ±0.22 1.2 133 511.59 ±0.26 856 ±53 4.62 ±0.38 1.11 66 511.61 ±0.34 437 ±39 4.63 ±0.38 1.1 33 511.38 ±0.43 231 ±25 4.32 ±0.47 1.05 17 511.83 ±0.78 147 ±19 4.87 ±0.62 1.55 8 511.72 ±0.9 74 ±14 4.91 ±0.75 1.63 4 512.45 ±1.05 42 ±8 5.02 ±1.03 1.06 2

Table 3: Best fit parameter value and chi square statistics for peak 3 at 75 keV.

C

χ

Statistics comparison

Linear Continuum(73 DoF) Quadratic Continuum(72 DoF)

(a) (b)

(c) (d)

(e) (f)

(g) (h)