ACKNOWLEDGMENTS

The life of being a PhD student is about to finish, a little bit more than three years brought me many laughter and tears. I experienced many new things, living alone in a foreign country, trying to learn how to cook, trying to learn French, working with a group of people who speak different mother tongues but very passionate about their works… I want to say thanks to many people, first of all, of course, professor Ludovic Duponchel. You gave me this opportunity, lead me to the world of chemometrics. I am extremely grateful for your patience, your enthusiasm and immense knowledge. This endeavor would not have been possible without your continuous support during these three years, especially the period of writing.

Then, many thanks to associate professor Raffaele Vitale for your help in the work, I am very thankful for all suggestions you provided.

Also, I would like to thank the China Scholarship Council for the opportunity and financial support of doing this PhD.

Really big thanks to my dear friend Hui, I couldn't be able to finish this journey without your support and help, I couldn't imagine my life without you. Thank you very much for all the nice food you prepared for me, and many nice times we spent together. For me, you are the family, may we have many more happy times in the future. Special thanks to Ali Eftekhari and Dario Cevoil, for your wise advice and the help. I would like to extend my sincere thanks to the students in the lab, thank all of you for the support and many happy times.

Great thanks to my family, especially my mon for the many long phone calls we had during this PhD to help me relaxing and the belief in me. Thanks should also go to my dear friends (张乃丹, 汪雅静, 佟彤) in China, for all those emotional support you provided thousands of miles away.

Qicheng Wu

吴祺琤

vi

GENERAL REMARK

This thesis has been particularly impacted by the COVID health crisis that we have all endured.

It is undeniable that it had repercussions on the research period of this thesis and particularly because it has not been possible to obtain additional months of funding to limit this impact. This does not detract from the quality of the work presented in this manuscript even if we would all have liked to have more time to go further in this very interesting research with strong expectations from the LIBS community.

Pr. L. Duponchel.

LIST OF FIGURES

LIST OF TABLES

Table I-1 Examples of basic expressions characterizing the laser beam, its intensity distribution, its propagation, and its focusing behavior [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part I: Review of Basic Diagnostics and Plasma-Particle Interactions: Still-Challenging Issues Within the Analytical Plasma Community[END_REF]

ABSTRACT

Today, Laser-Induced Breakdown Spectroscopy (LIBS) imaging is in full change. Indeed, always more stable instrumentations are developed, which significantly increases the signal quality and naturally the analytical potential of the technique for the characterization of complex and heterogeneous samples at the micro-scale level. Obviously, other intrinsic features such as a limit of detection in the order of ppm, a high field of view and high acquisition rate make it one of the most complete chemical imaging techniques to date. It is thus possible in these conditions to acquire several million spectra from one single sample in just hours. For sure, different chemometric approaches allow us today to propose a valorization of the acquired spectral data at both qualitative and quantitative levels. From a fundamental point of view, the main characteristics of the plasma, such as its temperature and electron density, have a significant impact on the intensities of the emission lines and even on the presence of some of them over a wide range of wavelengths. Of course, theoretical models could allow us, in some way and under very strict conditions, to correct the emission line intensities but this would remain a complex and very time-consuming task. In fact, we could only consider this strategy for a few spectra but not for hundreds of thousands or even millions of spectra of a LIBS hyperspectral imaging dataset that is available today. The first problem we have been working on in this thesis is to develop a robust predictive model to quantify an element from a LIBS spectrum independently of the plasma temperature and electron density. The second problem we addressed during this thesis is related to the intrinsic structure of a hyperspectral imaging dataset. Indeed, we always want to extract information on both major and minor compounds and even traces potentially contained in the hyperspectral data cube we have just acquired. However, minor compounds and traces are often present on a small number of pixels representing a very small variance in the spectral dataset.

Unfortunately, the majority of chemometric algorithms exploiting the concept of expressed variance do not (or hardly) allow the detection of these compounds, especially when the signal-to-noise ratio is limited. It is therefore the second purpose of this thesis to introduce an algorithm which can give all compounds a chance to be detected regardless of their concentration.

xv

RÉSUMÉ

Aujourd'hui, l'imagerie LIBS est en pleine évolution. En effet, des instruments toujours plus stables sont développés, ce qui augmente significativement la qualité du signal et naturellement le potentiel analytique de la technique pour la caractérisation d'échantillons complexes et hétérogènes à l'échelle du micron. Bien évidemment, d'autres caractéristiques intrinsèques telles qu'une limite de détection de l'ordre du ppm, un champ d'analyse élevé et une vitesse d'acquisition importante en font l'une des techniques d'imagerie chimique les plus complètes à ce jour. Il est ainsi possible dans ces conditions d'acquérir plusieurs millions de spectres d'un seul échantillon en quelques heures. Certes, différentes approches chimiométriques nous permettent aujourd'hui de proposer une valorisation des données spectrales acquises tant au niveau qualitatif que quantitatif. D'un point de vue fondamental, les principales caractéristiques du plasma, telles que sa température et sa densité électronique, ont un impact significatif sur les intensités des raies d'émission et même sur la présence de certaines d'entre elles sur une large gamme de longueurs d'onde. Bien sûr, des modèles théoriques pourraient nous permettre, d'une certaine manière et sous des conditions très strictes, de corriger les intensités des raies d'émission mais cela resterait une tâche complexe et très longue. En fait, nous ne pourrions envisager cette stratégie que pour quelques spectres, mais pas pour des centaines de milliers, voire des millions de spectres d'un ensemble de données d'imagerie hyperspectrale LIBS disponibles aujourd'hui. C'est le premier problème sur lequel nous avons travaillé dans cette thèse dans le but de développer un modèle prédictif robuste pour quantifier un élément à partir d'un spectre LIBS indépendamment de la température du plasma et de la densité électronique. Le deuxième problème que nous avons abordé au cours de cette thèse est lié à la structure intrinsèque d'un jeu de données d'imagerie hyperspectrale. En effet, nous souhaitons toujours extraire des informations sur les composés majeurs et mineurs et même sur les traces potentiellement contenues dans le cube de données hyperspectrales que nous venons d'acquérir. Cependant, les composés mineurs et les traces sont souvent présents sur un petit nombre de pixels représentant une très faible variance dans le jeu de données spectrales. Malheureusement, la majorité des algorithmes chimiométriques exploitant le concept de variance exprimée ne permettent pas (ou peu) la détection de ces composés, surtout lorsque le rapport signal/bruit est limité. C'est donc le second objectif de cette thèse que d'introduire un algorithme permettant de donner une chance à tous les composés d'être détectés quelle que soit leur concentration.

INTRODUCTION

As a spectroscopic technique, LIBS has numerous applications in many areas and this technique has showed its ability to the world. Compared with some other commonly used spectroscopy, LIBS has many advantages, such as the fast data acquisition, no sample preparation, capable of analysing sample in gas, liquid and solid form and the amazing ability to analysis almost all the elements. LIBS showed great power in qualitative analysis, however, the quantitative analysis is perhaps its Achilles's heel if we are very strict about the origin of the signal. The reason for making LIBS quantitative analysis a bit complex is the varying nature of the plasma, with different temperature and electron density coming from matrix effects. Although temperature and electron number density information are important, the vast majority of studies either do not consider them or simplify the effects they might have on the analysis due to the difficulty of acquiring them. Of course, there are different solutions for this issue, from using standard reference to complex physical calibrations, the coupling of other analytical instruments or the use of chemometric strategies. In reality, it is first not always possible to have references, the calibration is way too complex and not very practical for large datasets, and coupling with other instrument is not always as easy. Compared with previous solutions, chemometrics has significant advantages, for instance, no extra measurement needed, friendly to big data, and once the analysis model has been established, it is reusable. Among the many chemometric methods, artificial neural network (ANN) is an approach that has been quite quickly exploited in the field of LIBS and is suitable for large spectral datasets. Needless to mention the rise and popularity of ANN in many fields in recent years, the astonishing ability of dealing with non-linear complex problems makes it a celebrity in many scientific domains. The first objective of this thesis will be to develop a data processing strategy for LIBS spectra based on neural networks in order to propose a quantitative analysis that is robust to the potential variations of temperature and electron density of the plasma observed in complex and heterogeneous samples.

With the development of analytical instrument, the datasets we acquired become bigger and bigger, especially in LIBS hyperspectral imaging. For example, the datasets acquired in the 2000s contained a thousand spectra, whereas today they can contain more than 10 million spectra acquired in a few hours on a single sample. The instrumental developments have of course allowed this growth without compromising the quality of the spectra. During all these years, we could also observe a real increase in the skills of the LIBS community which opened to the concepts of chemometrics. From this very positive observation concerning LIBS imaging, we could then ask ourselves where is the problem? It comes in fact from the natural will of a researcher to want to discover everything of his complex sample, i.e. to detect both major, minor and even trace elements. Unfortunately, the majority of chemometrics algorithms based on the notion of variance do not allow to detect all of them and it is therefore the interest of this work to propose an alternative chemometrics strategy for a real exhaustive exploration of large LIBS imaging datasets. This thesis manuscript will therefore be organized in three chapters. The first chapter will cover the principles of LIBS spectroscopy and chemometrics with a state of the art. The second chapter will report on the first experimental part of this thesis with the development of a neural network strategy for robust quantification in the face of temperature and electron density variations in the plasma. The third chapter, the second experimental part of this thesis, will develop a chemometric strategy for the detection of all chemical contributions of a complex sample independently of their explained variance within the considered imaging spectral dataset. Laser-induced breakdown spectroscopy (libs)

Chapter

I.1.1 Introduction of LIBS

Laser-Induced Breakdown Spectroscopy, known as LIBS, is an emission spectroscopy capable of analysis of multi-elements in basically any kinds of samples both qualitative and quantitative [START_REF] Owolabi | Development of hybrid extreme learning machine based chemo-metrics for precise quantitative analysis of LIBS spectra using internal reference pre-processing method[END_REF].

LIBS is a rather powerful spectroscopy for it is rapid, very little or no sample preparation needed, small quantity of sample required, analyzed in-situ and remotely. LIBS uses a high energy laser pulse on the sample to excite and ionize it to generate plasma, which will later produce spectra of the sample for analysis. Nd: YAG (Neodymium doped Yttrium Aluminum Garnet) lasers is one of the most commonly used lasers for LIBS. In general, the process of LIBS starts with the ablation of a small amount of sample from laser to generate plasma, the light from the plasma will then be directed and collected through an optical system, then the spectrum will be generated and detected by spectrometer for further analysis. The LIBS spectrum for the existence of different elements is distinguished by the characteristic wavelength of each element with the help of NIST (National Institute of Standards and Technology), and the quantity of each element can be told by measuring the intensities of spectrum.

Further quantification of each element can be done with either calibration curves or through a calibration free LIBS (CF-LIBS). For the many advantages of

I.1.2 History

After the invention of pulsed ruby laser in 1960, and the development of Q-switched laser in 1963, laser was capable of breakdown and generate laser plasma that can be used analytically, LIBS was born. In the 60s and 70s, LIBS have developed the abilities to detect samples in gas and liquid form, and theory and modeling were also developed to assist the better understanding of the laserinduced plasma. During the 70s and the early 80s, some applications for LIBS were appeared with the work from Los Alamos National Laboratory [36][37]. But it was not until in the 1980s with the huge improvement in laser, the introduction of Nd: YAG laser, and detector technology that made LIBS popular in different areas [START_REF] Casini | Detection of pollutants in air by Time-Resolved LIBS System[END_REF].

In the 90s, with the huge development of computer, laser and spectrometers, there were further development in instruments of LIBS. With the invention of calibration-free LIBS(CF-LIBS) [START_REF] Ciucci | New Procedure for Quantitative Elemental Analysis by Laser-Induced Plasma Spectroscopy[END_REF] and the use of LIBS on NASA's Mars rover Curiosity around 2000, LIBS again caught people's attention and became one of the major techniques for qualitative and quantitative analysis [START_REF] Whitehouse | Remote material analysis of nuclear power station steam generator tubes by laser-induced breakdown spectroscopy[END_REF].

I.1.3 Characterization of the laser parameters and the ablation process

(Principle of LIBS)

I.1.3.1.

Laser light

The description of laser light is frequently determined by terms like intensity, irradiance, fluence, radiant exposure, power of energy, density, volumetric energy density, and photo flux. All these terms can be decided with equations in

𝑡 & = ∫ 𝑡𝑃(𝑡)𝑑𝑡 !"#$% ∫ 𝑃(𝑡)𝑑𝑡 !"#$%
Time of the pulse center (i.e. when

𝐼(𝑡) = 𝐼 '()

). This formula is valid for any pulse shape T1.4

𝐼(𝑡) = 𝐼 & exp (- 𝑡 * 𝜏 & * )
Gaussian temporal profile (Ref [START_REF] Radziemski | Laser-induced plasmas and applications[END_REF], p.

26)

T1.5

𝜏 ! = 2𝜏 & √𝑙𝑛2 FWHM (full width at half-maximum) (Ref[36]) T1.6 𝐼 ̅ = + ! , ! =∫ -(/)1/ , ! = 1.06𝐼 & 23 43
Average pulse irradiance(Ref [START_REF] Radziemski | Laser-induced plasmas and applications[END_REF]) T1.

I.1.3.2. Ablation

Ablation process is an important process for LIBS, and ablation efficiency is a parameter related to it. By definition, ablation process is the result of a complex interaction, involving laser parameters, sample properties, and plasma chemistry [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part I: Review of Basic Diagnostics and Plasma-Particle Interactions: Still-Challenging Issues Within the Analytical Plasma Community[END_REF]. Table I-2 from review [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part I: Review of Basic Diagnostics and Plasma-Particle Interactions: Still-Challenging Issues Within the Analytical Plasma Community[END_REF] listed the equations related to ablation efficiency. A simpler explanation is given with the example of surgery for medical application, as removing clots in the blood to restore the normal flow and at the same time maintain the integrity of blood vascular. As stated by its name, this parameter is an efficiency, like "quantum efficiency", has no units. It is defined by the ratio of number of atoms removed from the sample to the number of laser photons impinging on the surface [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part I: Review of Basic Diagnostics and Plasma-Particle Interactions: Still-Challenging Issues Within the Analytical Plasma Community[END_REF]. However, in order to quantity this important parameter, another parameter, "ablation sensitivity" was introduced [START_REF] Amponsah-Manager | Microchip laser ablation of metals: investigation of the ablation process in view of its application to laser-induced breakdown spectroscopy[END_REF]. This parameter can be defined by the graphs in plotting the signal measured versus laser pulse energy, and has the unit of Volt/J, count/J and others if needed. Altogether ablation efficiency and ablation sensitivity are usually used to characterize the ablation process. Another thing that worth mentioning is that a similar definition used in LA-ICP-MS (Laser Ablation Inductively Coupled Plasma Mass Spectrometer) is called detection efficiency, which means the ratio of ions reaching the detector and the number of atoms released during laser ablation and takes into account aerosol losses during transportation as well as incomplete vaporization in the ICP (Inductively Coupled Plasma) [START_REF] Wälle | Detection efficiencies in nano-and femtosecond laser ablation inductively coupled plasma mass spectrometry[END_REF].

𝜂 (5# = 𝜌 $ 𝑆 # ℎ $ 𝑁 < ℎ𝑣 𝑀 $ 𝑄 # = 𝜌 $ ℎ $ 𝐹 # ] 𝑁 < ℎ𝑣 𝑀 $ ^ Atom
E F ⁄ (𝐼) A * ⁄ (𝐴) A B ⁄ (𝜆) A * ⁄ (𝜏 > ) A B ⁄
Mass removal rate (Ref. [START_REF] Phipps | Impulse coupling to targets in vacuum by KrF, HF, and CO 2 single-pulse lasers[END_REF]) T2.17

𝜓 ≡ 𝐴 2[𝑍 * (𝑍 + 1)] A * ⁄
Parameter entering Eq. T2.17; A=atomic mass; Z=ionic charge(for singly charged ions Z=1) T2. [START_REF] Martin | Analysis of preservative-treated wood by multivariate analysis of laser-induced breakdown spectroscopy spectra[END_REF] Ablation rate is another useful parameter in ablation process, it is characterized by the total mass ablated per unit time per unit area (gcm -2 s -1 ). Since the laser sample interaction in LIBS is a rather complex process, the ablation mass and the useful ablation mass can be different, and the chemical properties of ablation in plasma can be different from sample in solid form. To conclude, a unified approach is still needed though the problems mention above.

I.1.4

Local thermodynamic equilibrium, theoretical equilibrium, plasma parameters and their evaluation

In this section, two of the most important parameters in LIBS will be introduced, electron number density (Ne) and plasma temperature(T), and also the concept of local thermodynamic equilibrium (LTE) will be discussed.

I.1.4.1. Local thermodynamic equilibrium (LTE)

Local thermodynamic equilibrium is defined as the plasma state reaching a local equilibrium.

The best option would always be for the plasma to reach a complete local equilibrium. However, with all the other many processes to reach the perfect balance they all need to reach a single temperature, which is rather difficult criteria to meet. When the plasma is in a theoretical complete local equilibrium, the process of excitation of atoms by collision with electron is equal to the reverse deactivation process, and collisional ionization equal to three-body collisional recombination, and radiation emitted is equal to the radiation absorbed [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part I: Review of Basic Diagnostics and Plasma-Particle Interactions: Still-Challenging Issues Within the Analytical Plasma Community[END_REF]. When the laws that used to describe the full thermodynamic equilibrium can apply, and at the same time the collisions dominate, a complete local thermodynamic equilibrium is reached. Since radiative transitions between low-lying levels (resonance transitions) are defined by high values of the Einstein coefficient of spontaneous emission, and the former are depopulated faster than the high levels, a radiative equilibrium is easier to reach. If these levels are not considered, a partial local thermodynamic equilibrium is reached [START_REF] Huddlestone | Plasma diagnostic techniques[END_REF].

Table I-3 shows equation related to LTE, among many equations, the first condition to be reached is the number of electron density should be less than a value to meet electron energy distribution function (EEDF), a Maxwell-Boltzmann distribution. Equation I-1tells us that for T = 6000 K, Ne should be less than 10 21 cm -3 , and for T = 20 000 K, Ne should be less than one 6.7 ´ 10 21 , and in most LIBS lab this can be acquired.

𝛬 = ℎ $2𝜋𝑚𝑘𝑇 ! I-1 𝑛 ! (𝑐𝑚 "# ) ≪ Λ "$ = ( 2𝜋𝑘𝑚𝑇 ! ℎ % ) $ % ⁄ I-2
An ideal plasma source is of course preferred, however, compared with ICP, laser-induced plasma is not really an ideal plasma source for spectroscopy, to solve this problem, equation I-2is needed. When an energy is suddenly applied to a system that already reach equilibrium, an extra period of time will be needed for the new equilibrium to reach, and this period of time is called relaxation time 𝜏 '!()* , and defined by equation I-

I-3

Kinetic model were also brought into the LIBS for cases [START_REF] Capitelli | Laser-induced plasma expansion: theoretical and experimental aspects[END_REF] to study the distribution of electron energy. These consideration are all list in Table I-3, and among them all, the first equation is

McWhirter criterion [START_REF] Huddlestone | Plasma diagnostic techniques[END_REF], perhaps one if the most important equation to determine the LTE in LIBS.

This criterion is derived from the form [START_REF] Griem | Validity of Local Thermal Equilibrium in Plasma Spectroscopy[END_REF] which is originally used to minimum Ne necessary to ensure partial and complete thermodynamic equilibrium, and also requires the collisional rate larger than the radiative one at least 10 times [START_REF] Huddlestone | Plasma diagnostic techniques[END_REF]. There are many different applications that derived from McWhirter criterion and discussions, a conclusion could reach is that the only use of it is not enough to determine LTE.

𝜎 #" = x 2𝜋 * √3 z { 𝑓 #" 𝑔̅ 𝑒 B 1 2 𝑚 % 𝑣 6 * Δ𝐸 "# |
Cross-section of inelastic collisions (cm 2 ); Δ𝐸 (erg); e (statC); 𝑣 6 = electron velocity (cm s -1 ) (Ref. [START_REF] Cristoforetti | Local Thermodynamic Equilibrium in Laser-Induced Breakdown Spectroscopy: Beyond the McWhirter criterion[END_REF])

T3.6 𝑋 #" (𝑇 % ) = 𝑛 % (𝜎 #" 𝑣) = 4𝜋 𝑓 #" 𝑒 B 𝑛 % (𝑔̅ ) Δ𝐸 "# ] 2𝜋 3𝑚𝑘𝑇 % ^A * ⁄ 𝑒𝑥𝑝 ]- Δ𝐸 "# 𝑘𝑇 % ^
Collisional excitation rate (s -1 ); m = electron mass (g); k (erg K -1 ) (Ref. [START_REF] Cristoforetti | Local Thermodynamic Equilibrium in Laser-Induced Breakdown Spectroscopy: Beyond the McWhirter criterion[END_REF]) T3.7

𝜏 G%# ≈ 1 𝑛 % (𝜎 #" 𝑣 % ) = 6.3 × 10 B 𝑛 % 𝑓 #" 〈𝑔̅ 〉 Δ𝐸 "# (𝑘𝑇 % ) A * ⁄ 𝑒𝑥𝑝 ] Δ𝐸 "# 𝑘𝑇 % ^
𝛼 N = 𝑛 6,N 𝑛 N = 𝑛 6,N 𝑛 6,N + 𝑛 (,N 𝑛 6,N 𝑛 (,N = 𝛼 N (1 -𝛼 N )
Definition of degree of ionization, 𝑛 6,N = 𝑛 (,N = ions (atoms) number density of species j (Ref. [START_REF] Boumans | Theory of Spectrochemical Excitation[END_REF], p. 157) T3.15 T3.20 Which is difficult to find nevertheless. decrease the destruction of the sample compared to the conventional LIBS, and increased the LOD [START_REF] Measures | Laser interaction based on resonance saturation (LIBORS): an alternative to inverse bremsstrahlung for coupling laser energy into a plasma[END_REF][81] [START_REF] Goueguel | Resonant laser-induced breakdown spectroscopy for analysis of lead traces in copper alloys[END_REF]. However, with the double-pulse laser and hyphenated LIBS, more information (both elemental information and molecular) about the sample can be acquired, or more information about the power. Micro-LIBS is also proposed to deal with the problem of the heterogeneous sample.

(𝐼 "# 2 ) N (𝐼 "# ) N = x 𝑔 " 2 𝐴 "# 2 ℎ𝑣 "# 2 𝑔 " 𝐴 "# ℎ𝑣 "# z [ 𝛼 N (1 -𝛼 N ) ] x 𝑍 (,N 𝑍
𝐼 "# = 𝐹 %)! 𝐴 "# 𝑔 " 2𝑍 6 [ ℎ C (2𝜋𝑚𝑘) C * ⁄ ]𝑛 % 𝑛 6 𝑇 6HI 4C * ⁄ 𝑒𝑥𝑝 ] 𝐸 6HI -Δ𝐸 6HI 𝑘𝑇 6HI - 𝐸 " 𝑘𝑇 %)O ^ Line ''
𝐼 "# 𝜀 O (𝜆) = x ℎ B 3 C * ⁄ 𝑐 C 256𝜋 C 𝑒 P 𝑘 z 𝐴 "# 𝑔 " 𝑍 6 1 𝑇 % 𝑒𝑥𝑝 •- 𝐸 6 -Δ𝐸 6 𝑘𝑇 % • 𝑒𝑥𝑝 • -𝐸 " 𝑘𝑇 %)O • P𝜉 ]1 -𝑒𝑥𝑝 ℎ𝑐 𝜆𝑘𝑇 % ^+ 𝐺𝑒𝑥𝑝 ]- ℎ𝑐 𝜆𝑘𝑇 % ^Q ] 𝜆 Δ𝜆 '%($ ^ Line-
The stand-off LIBS for space exploration [START_REF] Wiens | Combined remote mineralogical and elemental identification from rovers: Field and laboratory tests using reflectance and laser-induced breakdown spectroscopy: IDENTIFICATION USING REFLECTANCE AND LIBS[END_REF][84] and organic explosives [START_REF] Dong | Experimental study on the characteristics of molecular emission spectroscopy for the analysis of solid materials containing C and N[END_REF][86] has also been developed. In the archaeological area, remote control LIBS is very handy [87][88]. Based on the information above, considerations about the set-up of LIBS will be discussed below in different sections and trying to cover as many aspects as possible with details.

I.1.5.1. Laser and focusing optics

In the area of description of laser light, many different terms were used to describe the properties of which, as it is more physics related, many details will not be discussed here in detail,

Table I-1 listed equations for many parameters to characterize the laser itself. In the parts where it's more "chemical" related, the selection of ablation wavelength and pulse duration are popped out. As stated in article [START_REF] Radziemski | Laser-induced plasmas and applications[END_REF], the choice of wavelength should be based on the material, UV is better for ceramics, stones, and metal. In water analysis, due to the huge absorption of UV, IR would be a better choice. As for the pulse duration is concerned, a cleaner crater can be achieved with a ultrashort pulse, along with intense neutral atom emission [START_REF] Radziemski | Laser-induced plasmas and applications[END_REF]. The types of laser are dealt with for certain cases in these articles [START_REF] Gatti | Spherical shock waves in laser produced plasmas in gas[END_REF][90] [START_REF] Rodolfa | Capabilities of Surface Composition Analysis Using a Long Laser-Induced Breakdown Spectroscopy Spark[END_REF] for detailed study of interest. There isn't a single guideline for choice of laser light, but a general guideline could be given as: a single-mode profile should be used for singleshot analysis, and a multimode profile performs better for single-spot sampling. Based on one article [START_REF] Lednev | Laser beam profile influence on LIBS analytical capabilities: single vs. multimode beam[END_REF], a single-mode Gaussian beam profile has the best performance in lateral resolution, depth profile, chemical mapping, and multimode profile should be used for trace analysis. Also, something not to be forgotten is the rise of fiber laser also showing good potential in LIBS, and providing higher power and higher repetition rate.

I.1.5.2. Optical transfer of plasma radiation, spectrometer and detection

Optical train is the term used for describing the radiation from plasma to detector, it has important role in signal collection efficiency of the entire system. In general, two types of optical train exist, one is the simplest when a commercial instrument is involved, which leaves one to trust the instrument for its optical quality and spectral characteristics. Another one is, of course, custom made, which gives one great opportunity to tune parameters to achieve better results. Detailed discussions about the merits of optical train are in books written by Gallou and Sirven [START_REF] Gallou | Plasma Emission Collection for Characterization and Analytical Optimization[END_REF], as this is not the purpose of this section.

I.1.6 Modeling LIBS spectra

It is important to model LIBS spectra, as the meaning of model here can be interpreted as the capability to give the description of the whole process from ablation to emission. Of course, it means having a detailed description of plasma-sample interaction, its plasma formation, its expansion, processes happened in plasma, collection of radiation, the signal generation, and the process from signal to concentration [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF]. One could imagine a perfect model would be able to provide the estimation of the spectrum, all its characteristics, the detailed information of the sample composition and maybe many more deeper understandings of the whole system. Unfortunately, this ideal model is far to be real but almost perfect. Just take one tiny aspect of the whole system to clarify this, for instance, the interaction between laser and sample, according to the review [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF]. The possible variations of conditions that such model has to consider are:

1. The sample can be in solid, liquid or gas state;

2. The sample can either absorb the laser or be transparent to it;

3. A solid sample can be a metal, a semiconductor or a dielectric with wide band gap;

4. The laser in use is just one pulse or a series of pulses;

5. Different laser wavelength can be used;

6. The duration of the laser pulse can be in nanosecond, picosecond or femtosecond time scale;

7. Is the laser characterized by its power or its beam shape;

8. Is the measurement under vacuum, atmosphere of certain gas, or a regular room atmosphere?

9. If the measurement uses two consecutive laser pulses, then the time delay between them can be adjusted for certain purposes;

The previous list shows that LIBS has a series of very complex situations to consider in experiment setup, and this is exactly because of the physical and chemical plasma processes are considered. It's true that a complete system maybe probably can be determined in details but the balance of a perfect model and a model that is good enough for understanding should be reached. Thus, the model should try to use as few parameters as possible to simplify such process. Besides, those parameters should also be able to be observed in experiment. According to Bogaerts [START_REF] Van Dijk | Plasma modelling and numerical simulation[END_REF], all present plasma models are comprised, judiciously reduced problems, which nevertheless try capturing all the physical phenomena one wishes to understand. Putting this consideration in mind, the LIBS models being published are numerous. However, this is not a topic for this work, detailed studies having already published in reviews [START_REF] Van Dijk | Plasma modelling and numerical simulation[END_REF][96] [START_REF] Gornushkin | Radiative models of laser-induced plasma and pump-probe diagnostics relevant to laser-induced breakdown spectroscopy[END_REF].

General considerations

The quantitative analysis of LIBS is a strong aspect of this method. However, it is not an easy task because of the very complex interactions between not only the laser and the sample, but also plasma and particles. The former interactions rely on the characteristics of laser and the properties of sample, the latter depends on time and space. All these combined together generate matrix effect, one thing that makes quantitative analysis difficult for LIBS. Despite probably is only well-known disadvantage of LIBS, LIBS has many advantages in the areas of qualitative chemical analysis. This method should not be seen as a technique with flaws. A statement made by Mermet [START_REF] Mermet | Processing of shot-to-shot raw data to improve precision in laser-induced breakdown spectrometry microprobe[END_REF] like LA-ICP-MS, a method that can be considered as a superstar method in analytical area, also has weaknesses when applied to trace analysis. Not to mention that LIBS can use internal standard calibration or external standard calibration for quantitative analysis. So, the technique cannot be seen as so weak considering the analytical techniques but also the fact improvements we can see in the field.

As stated previously, LIBS has many applications in many fields, being the best advantage of LIBS.

Also, the lack of sample preparation is yet another strong aspect of this method. The many publications of quantitative analysis with LIBS also showed that for certain specific cases, LIBS showed the ability of performing adequate analysis. To conclude, quantitation is possible with LIBS, but it is not perfect and this should be the area where focus goes.

I.1.7.2. Calibration and curves of growth

One of the important aspects that has been constantly mentioned in many books about spectrochemical analysis and plasma spectroscopy is the relation between thermal radiance 𝐵 +,!'-)( of emission line as function of number density of atomic species in the plasma. Many equations related to this part are summarized in Table I-4, some of the very important part will be listed as follows.

𝜎 1677 = ˜[𝜎(𝐵) !,L $%%&!'() ] * + [𝜎 $"' ] * -2𝜃 ™𝜎(𝐵) !,L $%%&!'() š [𝜎 $"' ]› A * ⁄
This expression relates the standard deviation of the difference of the two signals to the standard deviation of the sum and the correlation coefficient.

T4.2 𝜃 = (𝜎 $"' ) %)! * + (𝜎 V ) %)! * -(𝜎 1677 ) %)! * 2(𝜎 $"' ) %)! (𝜎 V ) %)!
The correlation coefficient cannot be calculated directly from the standard deviation of the sum of the two signals, but can be evaluated from the standard deviations of the sum of the difference and of the background.

T4.3

(𝑅𝑆𝐷) $"' = 𝜎 $"' (𝑆 + 𝐵) = (𝜎 V * + 𝜎 ; * + 2𝜃𝜎 V 𝜎 ; ) A * ⁄ (𝑆 + 𝐵)
This expression shows how the relative standard deviation of the sum of the analyte and background signals is related to the individual standard deviations and the correlation coefficient.

T4.4

(𝑅𝑆𝐷) $"' = oe 𝜎 K * 𝑃 * + 𝜎 K *+ * 𝑃 6$ * -2𝜃 ] 1 𝑃𝑃 6$ ^𝜎K 𝜎 K *+ • A * ⁄
This expression is relevant when an internal standard is used to normalize the signal fluctuations. P and Pis refer to the signal peaks measured for the analytical line and that of the internal standard chosen.

T4.5

The relation related to the shape of the analytical calibration curves, usually known as the curves of growth. It is written as equations I-4 to I-7, this is the classical form of the emission and the analyte atomic number density in plasma.

𝐵 +,!'-)( = (𝐵) . W /()012 2𝑇 In the equations above, the spectral radiance of the blackbody radiation given by either Planck

or Wien laws at the given temperature is (𝐵) . W 2𝑇 3 3(Wcm -2 sr -1 nm -1 ). 𝑇 3 and 𝑘 . * (𝜆) are the wavelength dependent absorption coefficient with stimulated emission, while 𝑘 . (𝜆) is the wavelength dependent absorption coefficient without stimulated emission. 𝑘(𝜆) = 𝜎(𝜆)𝑛 , and 𝜎(𝜆) is the wavelength dependent absorption cross section (cm 2 ) of the transition and 𝑛 (cm -3 ) is the atom number density.

𝜏 . (𝜆) stands for the optical thickness, 𝑙 is the length of plasma in the direction of observation. The last equation above is called the total absorption factor, 𝐴 + .

When the self-absorption is ignored, such as in the situation where the optical thickness is more than the whole wavelength range of the line profile is ≪1, then there will be the equation I-8

𝐵 +,!'-)( = (𝐵) . W 2𝑇 3 3 4 𝑘 . ( 50
!
(𝜆)𝑙

I-8

When the optical thickness is thin, 𝐵 +,!'-)( will be a linearly growth to the wavelength integrated absorption coefficient, and directly related to the atom number density. This is the linear part of the curves of growth. When the number density becomes bigger, the Doppler core of the line saturates, but the (Lorentzian) wings of the line are still optically thin [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF]. The different values of parameter will result in the shape of the curves showing a plateau for some range of number density.

In the end, the pressure broadening will effect on the wings, and 𝐵 +,!'-)( grows with concentration with a slower rate, and this is the square root part of the curves [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF].

When the condition is extreme, for instance, the optically condition is thicker than the unity, the blackbody value will be calculated in equation I-9, where the analytical dependence of the emission on analyte number density is lost.

𝐵 +,!'0)( = (𝐵) . W (𝑇 3 ) I-9
The self-absorption parameter is an important parameter in deciding the shape of the curves of growth, and it is defined as follows (equation I-10),

𝐾 . ≡ 𝑘 . (𝜆)𝑙 {1 -𝑒𝑥𝑝[-𝑘 . (𝜆)𝑙]} ≡ 𝜏 . (𝜆) 1 -𝑒𝑥𝑝[-𝜏 . (𝜆)] I-10
Error! Reference source not found. is one example of the curves of growth, this figure is s imulated by Jorg Hermann (Aix-Marseille Université, provided by professor Vincent Motto-Ros, Université Claude Bernard Lyon 1).

The effect of self-absorption can be found in literatures [START_REF] Aragón | Characterization of laser induced plasmas by optical emission spectroscopy: A review of experiments and methods[END_REF][99][100] [START_REF] Omenetto | An expression for the atomic fluorescence and thermal-emission intensity under conditions of near saturation and arbitrary self-absorption[END_REF]. The detailed calibration curves part can be found in paper [START_REF] Mermet | Calibration in atomic spectrometry: A tutorial review dealing with quality criteria, weighting procedures and possible curvatures[END_REF].

Analytical sensitivity and detection limits

The definition of detection limit is a subject of many theoretical and standardization processes.

Being an emission spectroscopy, LIBS could benefit from the already existed many literatures. The most useful way for define the detection limits in LIBS would be signal-to -noise ratio(S/N) and signalto-background ratio(S/B). The reasons for choosing these two parameters as the indicator are these are the most measured and reported during the optimization of LIBS measurements. In the LIBS spectrum, the response observed at the given wavelength is the sum of all the emissions of the sample(analyte).

In a simple version, the average response of analyte after the subtraction of background would be taken as the signal. With the consideration of noise being present in both signal of target and background, the follow equations (I-11 to I-13) can be written:

𝑋 7 FFF = 𝑋 789 FFFFFF -𝑋 9 FFFF I-11 𝑁 7 = 𝑋 : -𝑋 : FFF I-12 𝑁 9 = 𝑋 9 -𝑋 9 FFFF I-13
𝑋 : and 𝑋 9 are the responses for each spectrum, while 𝑋 : FFF and 𝑋 9 FFFF are the average responses related to signal and background. 𝑁 7 and 𝑁 9 represent their noises. S/N ratio is used to evaluate the quality of measurement. It is defined as the equation I-14 follows,

𝑆 𝐵 = 𝑋 F $ 𝑁 I % = 𝑋 F J (𝑋 -𝑋 F ) % 𝑛 I-14
This equation shows that the ratio of the signal amplitude to the noise amplitude is the S/N ratio, and the noise amplitude is defined as the root mean square value.

As for LOD, the classical definition [START_REF] Voigtman | Limits of detection and decision. Part 1[END_REF] defines it as the concentration that results in a signalto-background noise ratio of 𝑘(and in most cases, 𝑘 = 3 is used). LOD can be written in the following 𝑋 ; is the smallest dissemble signal, 𝑠 9 stands for the standard deviation of the background, 𝑏 is the magnitude of the slope associated with the linear part of the calibration curve [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF]. The signalto-background and signal-to-background noise can also fit for detection limit as equation I-18 to I-20,

𝑐 ; = 𝑘( 𝑠 9 𝑋 9 FFFF )( 𝑋 9 FFFF 𝑏 ) I-18 𝑐 ; = 𝑘𝑐 < (𝑅𝑆𝐷) 9 S 1 𝑋 : FFF 𝑋 9 FFFF ⁄ U I-19 𝑐 ; = 𝑘( 𝑠 9 𝑋 9 FFFF )(𝐵𝐸𝐶) I-20
(𝑅𝑆𝐷) 9 is the relative standard deviation of the background, 𝑘 = 3, 𝐵𝐸𝐶 stands for the background equivalent concentration. For LIBS, equation I-19 is recommended for LOD.

I.1.8 Matrix effects and normalization approaches

I.1.8.1.

Definition of matrix effects

As a spectroscopy with so many advantages, the only possible withdraw of LIBS could be the quantitative analysis, and that usually caused by matrix effects. Matrix effect is a common phenomenon observed in several analytical spectroscopies, which is usually the significant change in the observed signal at a given spectroscopic transition of the same element in two different samples with the same concentration. This difference can be categorized into the laser sample interactions difference that comes from differences in laser ablation mechanism and efficiency, or the changes in the plasma parameters (mainly the difference from temperature and electron number density), or a combination of both. In one sentence, this response can be affected by chemical, physical, spectral and instrumental reasons.

Different strategies to try to correct this matrix effects can be found in literature [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF]. A short summary will be given here. Firstly, for any emission spectroscopy, the intermediate physical processes are the keys that lead to the understanding of the analytical signals, so is for LIBS. A relationship between the intensity of the spectral line and the number of atoms could be established if the constant excitation conditions can exist. However, the nature of the sample has huge influence on excitation conditions that leads to the famous matrix effects. There are two types of matrix effects: one is related to the entry of the sample into the plasma and the other one is what happened in the plasma [START_REF] Lee | Recent Applications of Laser-Induced Breakdown Spectrometry: A Review of Material Approaches[END_REF]. These two types of effects are in the area of laser ablation and plasma modeling and diagnostics, respectively. Secondly, we can use a spectrochemical buffer, which means a substance that is added to sample in the effort of trying to reduce the influence of the composition and characteristics for the spectral line intensity. This method can be seen as an extension of the internal standard, and the adding part has a conflict with the general concept that LIBS requires little of no sample preparations. Thirdly, one can manage the measurement of spectral intensities with two methods: total energy and stead-state can be linked to LIBS from dc arc emission. Total energy requires a complete volatilization of the sample, and integration of spectral line over the entire evaporation time.

"Stead-state" means that a constant feeding of the sample into the plasma. One paper [START_REF] Thompson | A review of interference effects and their correction in chemical analysis with special reference to uncertainty[END_REF] talked about the matrix effects with special reference to uncertainty and offered three groups of ways to reduce the effects: these strategies are matrix matching, internal calibration(or internal standard) and correction of the effect by either measuring the effects directly or some property of the matrix to have a correction function or a joint analyte calibration.

I.1.8.1.1. Consideration of the concentration units

When the matrix effects are only related to the entry material into the plasma, but not from what started inside the plasma, it can be assumed that the identical plasma conditions are reached, in this one the temperature of the plasma and the electron number density are the same and there is no chemical reaction in the plasma. At the same time, the same condition should also happen for the same detection parameters, which includes the same bandwidth, gain and integration time and many more.

When the above-mentioned conditions are met, the concentration of a certain element in the matrix can be expressed in weight fraction or in mole fraction both in percent. Weight fraction carries the direct information of the sample, while the mole fraction requires the knowledge of the constitute of the sample to supply the information. For an emission spectral line, the intensity is now linked to the number of atoms in the sample.

I.1.8.2. Normalization approaches

Based on the discussions above, the normalization parameters for LIBS would be the mass ablated, the plasma temperature and electron number density. The general normalization of spectra usually takes the intensity of the spectral line of the analyte in the sample to that of another suitable element present in the sample, or the spectral continuum. There are criteria for selection of spectral lines in LIBS made by Barnett[106]. There are nine criteria in total, five being related to the element choice and four from the choice of internal standard line. They require that the analyte and the ones that used as the internal standard should fit the following conditions. First, they should have similar volatilization rate, second, they should have comparable ionization energies, and last, they should have similar atomic weights. The choice of the spectral lines should be driven by certain rules: They should have first the same excitation energies. They should not have self-absorption and they should have similar intensity. The detailed discussions can be found in the paper [START_REF] Barnett | Theoretical principles of internal standardization in analytical emission spectroscopy[END_REF]. Finally, another paper [START_REF] Zorov | A review of normalization techniques in analytical atomic spectrometry with laser sampling: From single to multivariate correction[END_REF] studied acoustic waves, plasma emission, the electrical current and Mie scattering as the reference signal for LIBS in the normalization approaches. And there are detailed discussions about the considerations of each different reference. Paper [START_REF] Leach | Methods for shot-to-shot normalization in laser ablation with an inductively coupled plasma time-of-flight mass spectrometer[END_REF] which is dedicated to the normalization procedure for certain type of samples, such as the stainless steel samples, can also be found. Besides the plasma temperature and electron number density, one paper [START_REF] Aguilera | Study of matrix effects in laser induced breakdown spectroscopy on metallic samples using plasma characterization by emission spectroscopy[END_REF] also mentioned that the volume of the plasma region can be detected along with some other parameter proportional to the total number of atoms present in the former volume should also be considered. Otherwise, when the only line intensity was used for the comparison, matrix effects can be hidden because of the compensation of plasma parameters.

I.1.8.3. An approach to evaluate matrix effects

Based on the nature of the matrix effects, the reason behind this "problem" can be found maybe through complex theoretical models and repeated accurate experiments. However, this is not the optimal way. In a review [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF], the authors introduced a possible way to evaluate matrix effect. For certain selected elements which has the same concentration or in a very narrow range of concentrations, the observed signals of those elements in different matrices can be described as a function of the energy of the excited state from the transitions studied. Two matrices will be used to construct calibration curves for each element separately. From the results they have by then, the existence of matrix effects can be found with the plot of the log of the intensity ratio for the two matrices versus the excitation energy. From this plot, the parameters (ablation mass, temperature) that caused matrix changing can also be found.

I.1.9 Absolute analysis and Calibration Free Laser-Induced Breakdown

Spectroscopy (CF-LIBS)

The definition of absolute analysis is very strict, that is only when the theory can describing a single measurement and is reliable to calculate the concentration directly in absolute unit of a physical parameter [START_REF] Winefordner | Comparing several atomic spectrometric methods to the super stars: special emphasis on laser induced breakdown spectrometry, LIBS, a future super star[END_REF][111],it also means providing quantitative results without any other standard reference materials. Some other chemical analysis methods only use the standard reference infrequently to reach a stage where this method is steady both for time and repetition. It is then called standardless analysis [START_REF] Winefordner | Comparing several atomic spectrometric methods to the super stars: special emphasis on laser induced breakdown spectrometry, LIBS, a future super star[END_REF], there are chemical analysis methods where absolute analysis were discussed, for instance AAS [112][113], AFS(atomic fluorescence spectroscopy) [START_REF] Omenetto | Laser-induced fluorescence in a furnace: A viable approach to absolute analysis?[END_REF], X-ray analysis [START_REF] Pouchou | Standardless X-ray analysis of bulk specimens[END_REF][116],

and laser ionization orthogonal time-of-flight mass spectrometry(TOFMS) [START_REF] Yu | Applicability of Standardless Semiquantitative Analysis of Solids by High-Irradiance Laser Ionization Orthogonal Time-of-Fight Mass Spectrometry[END_REF]. These experiments are close enough to an absolute analysis, however, the difficult of reproduction the same results cannot be neglected. "Progress in this field can result from a better knowledge and more precise description of the theoretical fundamental phenomena responsible for generation of the analytical signal, and from the development of instrumentation with perfectly reproducible and/or theoretically described characteristics.'' quote from IUPAC Technical Report [START_REF] Hulanicki | Absolute methods in analytical chemistry (Technical Report)[END_REF]. The pure pursuit of absolute analysis may seem not that practical.

Since the absolute analysis is in some way "beyond reach", attention should be focus on the standardless approaches. In the area of LIBS, Calibration-Free (CF) LIBS can be referred as one of such approaches. It means a procedure which is capable of giving results without calibration standards.

Based on the original definition, this approach does not model the plasma, it is not part of LIBS modeling. CF uses the relation between emission intensity and concentration of samples in the plasma.

When CF being considered as a model procedure, it would be categorized as a post-breakdown modeling approach. Another thing that worth mentioning is that for CF, matrix is not an interference but being analyzed together with the sample. This method can be considered as quantitative, semiquantitative or qualitative depending on the analyte being a major, minor or trace components in the relatively error in percentage [START_REF] Herrera | Semiquantitative analysis of metal alloys, brass and soil samples by calibration-free laserinduced breakdown spectroscopy: recent results and considerations[END_REF]. To conclude, the importance of just infrequent use of reference standards means that the reproducibility of LIBS has been improved, and the matrix effects being in an acceptable level. The role of CF-LIBS is vital for some cases and that is a guarantee this method will continue in application and improvements. With the development of the modeling, CF-LIBS is excepted to be a much more mature method in the coming years. Further detailed discussions about CF-LIBS can be found in these reviews [120][97].

I.1.10 Signal enhancement

When the advantages of LIBS being mentioned, sensitivity usually is not among them. It is not a surprise that a lot of efforts has been made into improving the sensitivity of LIBS as long with the detection power. The detection powers the reciprocal of the limit of detection [START_REF] West | Reviews of books[END_REF], by doing so the confusion results could be avoided (a decrease for the LOD value is actually an increase for the performance of the method). An approach is that keep the noise at the same and trying to increasing the analyte signal, since the larger signal-to-noise ratio will have lower detection limits. Many ways can improve the signal level in LIBS, double-pulse and multi-pulse approaches are perhaps two of the most popular ones. Double-pulse LIBS is a solution for the problem that during the ablation and excitation of the plume cannot be controlled separately with single-pulse [START_REF] Omenetto | Analytical laser spectroscopy[END_REF]. From the observation, a large proportion of atoms in the neutral state is present even 40 ls after the plasma formation, a second laser fired at this time could reheat the plasma, thus causing further atom excitation and emission, as proposed by Uebbings [START_REF] Uebbing | Reheating of a Laser-Produced Plasma by a Second Pulse Laser[END_REF]. There are increasing number of publications on the double-pulse LIBS in solid samples in recent years, apart from for the liquid samples where this method is originated. Many experimental parameters are involved in this approach, detailed information can be found in a paper by De Giacomo [START_REF] De Giacomo | Experimental and theoretical comparison of single-pulse and double-pulse laser induced breakdown spectroscopy on metallic samples[END_REF]. Like the double-pulse LIBS, multi-pulse LIBS was based on similar consideration: optimize the ablation and excitation independently. A single flash lamp pumped Qswitched Nd: YAG laser was used in order to obtain a sequence of laser emission bursts at a given (low) repetition rate, each burst consisting of several (up to 6) Q-switched short laser pulses emitted during a single flash lamp pulse [125][126]. Applications can be found with steel samples [START_REF] Sattmann | Laser-induced breakdown spectroscopy of steel samples using multiple Q-switch Nd:YAG laser pulses[END_REF][127],

alloys [START_REF] Galbács | Accurate quantitative analysis of gold alloys using multi-pulse laser induced breakdown spectroscopy and a correlation-based calibration method[END_REF][129], metallic elements in air [START_REF] Galbács | Multi-pulse laser-induced plasma spectroscopy using a single laser source and a compact spectrometer[END_REF] and tin sample [START_REF] Jedlinszki | An evaluation of the analytical performance of collinear multi-pulse laser induced breakdown spectroscopy[END_REF]. Apart from these two approaches, there are other methods for improving the signal, such as magnetic field enhancement, resonant laser ablation, and resonantly enhanced laser ablation.

I.1.11 Comparison of LIBS and other spectroscopic methods

I.1.11.1. Atomic absorption spectroscopy (AAS)

Atomic absorption spectroscopy (AAS) is a spectroscopy used for detection of chemical compositions by absorption of lights of atoms. This technique is based on the fact that when an atom is excited, the electrons will only absorb definite amount of energy (light) to reach higher energy states.

Then in the de-excitation process the electron emits photons with specific wavelength related to the elements, which can thus be delivered and qualified. AAS uses dilute solutions as samples, and the quantification is done though calibration curves. The limit of detection (LOD) of AAS is from 0.003 ppm to 20 ppm, the precision is within 1-2 %, and sensitivity can be enhanced with graphite furnace AAS [START_REF] García | Atomic absorption spectrometry (AAS)[END_REF]. AAS can achieve quite high precision but require complex sample preparation, it is not portable, and requires larger quantity of sample.

I.1.11.2. Inductively coupled atomic emission spectroscopy (ICP-AES)

Inductively coupled atomic emission spectroscopy (ICP-AES), or inductively coupled plasma optical emission spectrometry (ICP-OES), is a type of emission spectroscopy for detection of chemical elements. This technique uses inductively coupled plasma to generate excited atoms and ions, which emit electromagnetic radiation which can be used for distinguishing specific element. The concentration can be measured by the use of specific radiation. The samples are analyzed in solution, and multiple elemental detection is available. ICP-AES has negligible self-absorption and autoreversal effects, the LOD being 10 ppb and accuracy of 1-5%. It is really better than AAS, and can detect some elements that are difficult for AAS. However, this technique like AAS, also requires complex sample preparation and cannot be used in remote analysis or spectroscopic imaging. It needs several minutes to hours for analysis. In addition, the vital drawback of this method is the lack of enough authenticated methods for impurities detection and calculation [START_REF] Khan | Application, principle and operation of ICP-OES in pharmaceutical analysis[END_REF].

I.1.11.3. X-ray fluorescence (XRF)

X-ray florescence is a non-destructive technique that can be used to detect the elemental composition of material. This technique analyzes the secondary (fluorescent) emission of the sample that were triggered by the primary X-ray. When an atom in sample is excited by the primary X-ray, the electrons on the inner shell of the atom are ejected, the electrons on the outer shell would fill in the vacancy on the inner shell. This filling process will release the energy in the form of fluorescence, and the energy is equal to the specific difference in energy between two quantum states of the electron.

Each element has unique florescence peaks and the intensities can be used to estimate the concentration, and that is the fundamental basis of the XRF analysis. XRF has been applied in many fields like oil and gas industry [START_REF] Lia | Application of Elemental Analysis via Energy Dispersive X-ray Fluorescence (ED-XRF) for the Authentication of Maltese Extra Virgin Olive Oil[END_REF], metal fabricating [START_REF] Vrielink | Applicability of X-ray fluorescence spectroscopy as method to determine thickness and composition of stacks of metal thin films: A comparison with imaging and profilometry[END_REF], art and archaeology [START_REF] Alfeld | Revealing hidden paint layers in oil paintings by means of scanning macro-XRF: a mock-up study based on Rembrandt's "An old man in military costume[END_REF], and so on. This method requires no sample preparation and provides fast detection, but just for heavy elements. The LOD is in the level of ppm and the accuracy is between 3-23%. The quantitative analysis involves using of calibration curves, and XRF is non-portable and quite expensive.

I.1.11.4. Energy-dispersive X-ray spectroscopy (EDX)

Energy-dispersive X-ray spectroscopy, short for EDX, EDX, EDXS or XEDS, is an analytical technique for elemental analysis or chemical characterization. This method is similar to XRF, when the atom in the sample is focused under an electron beam, the inner electron will be excited and ejected, the electron in the outer shell will fill the hole left. The filling process will release the energy in the form of X-ray, the number and energy of the X-ray will be measured by an energy-dispersive spectrometer. Since the X-rays are characteristic to different element and also to the energy difference between two shells of an atom, this method can be used as a way to analyses the composition of the sample. The difference between XRF and EDX is the source used for excitation, XRF uses X-rays while EDX an electron beam. The LOD of EDX is between 1000 to 3000 ppm [START_REF] Hussain Shah | Laser induced breakdown spectroscopy methods and applications: A comprehensive review[END_REF], the precision is around ±0.1%. EDX does not require complex sample preparation and the experiment itself is not difficult to be carried. Coupled with SEM (Scanning Electron Microscopy), EDX can provide resolved image of sample for spatial information and matrix effects.

EDX requires a high voltage for the electron beam, and when coupled with SEM, EDX-SEM cannot undertake in-situ analysis. And the secondary fluorescence and secondary electron may appear, causing additional spectral lines in the spectrum. Overlap of spectral lines may happen due to the additional rays by secondary emissions.

I.1.11.5. Conclusion of the comparison

Compared with other techniques, LIBS has several advantages:

1. LIBS requires small amount of sample and basically no sample preparations.

2. Multi-element detection is possible, sample can be in gas, liquid and solid form.

3. LIBS is a fast method for analysis, it can be done within a minute. Today, we have a high acquisition, for instance, acquisition of 1000 spectra per second.

4. LIBS in theory can detect all the elements, which is not possible for a technique like AAS.

5. Can be coupled with other techniques, such as MS (Mass Spectroscopy), Raman and etc. for multimethod experiment.

6. LIBS can be used remotely and is portable.

7. Possible for have spectroscopic imaging experiment at the micro-scale spatial resolution.

However, there are also some disadvantages:

1. Compared with AAS, the LOD of which could fall in the range of ppb, LIBS only has LOD of ppm.

2. Self-absorption of LIBS affects the precision of quantitative analysis.

3. The common existence of matrix effects in LIBS reduces the accuracy of quantitative analysis.

As stated above, LIBS is a simple, fast, free of sample preparations, portable method for analysis of chemical composition of specimen, it has some merits and also a few drawbacks. Combined together, as an important analytical method, LIBS has a wide application in material analysis in many areas.

I.1.11.6. Algorithms for chemometric explanation of LIBS spectra

As mentioned before, accuracy of quantitative analysis is not the advantage of LIBS, however, with the help of chemometrics, the situation could change. Chemometrics is using mathematics, statistic and computer science to better understand the chemical information, the use of which in the area of LIBS can be categorized into 2 major branches, qualitative (also classification and clustering)

and quantitative (calibration) analysis. The application of chemometrics can help to improve the stability and reliability of LIBS online analysis [137].

I.1.12 Qualitative analysis

The chemometrics methods used here are mainly two types: supervised and unsupervised approaches. [START_REF] Vítková | Fast identification of biominerals by means of stand-off laser-induced breakdown spectroscopy using linear discriminant analysis and artificial neural networks[END_REF], Random Forest (RF) [START_REF] Sheng | Classification of iron ores by laser-induced breakdown spectroscopy (LIBS) combined with random forest (RF)[END_REF]and etc., and they will be discussed in the next parts.

I
I.1.12.1.1.

Partial least squares discriminant analysis (PLS-DA)

PLS-DA is a widely applied method in chemometrics based on partial least squares regression (PLS). It uses a matrix X with independent variables and a matrix Y for the categorical variables to construct the model for training. Then the unknown sample is being introduced, the model will give the class of the unknown sample by the predicted PLS values.

In the food quality control area, PLS-DA [START_REF] Nespeca | Detection and quantification of adulterants in honey by LIBS[END_REF] was used to build a classification model for distinguish the real honey from honey added with sugar cane syrup, along with other chemometrics methods like variables selection and some data preprocessing methods, the author showed their model performed 100% accuracy in adulterant detection for honey. Combined with PLS, LIBS showed the possibility of a fast, simple, sample preparation free authenticity certification of honey samples.

Another example [START_REF] Kim | Detection of Nutrient Elements and Contamination by Pesticides in Spinach and Rice Samples Using Laser-Induced Breakdown Spectroscopy (LIBS)[END_REF] is the application of PLS-DA in LIBS for detection of the usage of pesticide in spinach and rice. Though similar elements exist in both pesticide contaminated sample and pesticide free sample, the relation between the LIBS intensity and concentrations of each element can be established. With the confirmation of inductively coupled plasma optical emission spectroscopy (ICP-OES), LIBS combined with PLS-DA can be a rapid way to detect pesticide achieving a misclassification rate lower than 2% for spinach samples.

In the field of recycling plastic bottles, LIBS was used combined with SW-PLS-DA (partial least squares discrimination analysis based on spectral windows) to build classification able to classify 20 different types plastic bottles commonly used in supermarket chain in China [START_REF] Liu | Rapid classification of plastic bottles by laser-induced breakdown spectroscopy (LIBS) coupled with partial least squares discrimination analysis based on spectral windows (SW-PLS-DA)[END_REF]. In their work, a basic PLS-DA model was also built for classification with the highest accuracy of 67.50%. In order to improve the results, methods for selecting input variables were applied, including CWT (continuous wavelet transform) and SW (spectral windows) based on CWT, different parameters of CWT were tuned and the best results showed an accuracy of 93.93%. There were other chemometric methods has been applied for comparisons, including a basic PLS-DA, supporter vector machine (SVM) and random forest (RF). SW-PLS-DA showed the best result and also showed the potential of online analysis for plastic recycling. The classification result is better when the differences are larger between the classes than the differences within a class itself. As a consequence, when the differences are not significant enough among different classes, the model built is not optimized good enough to have good results. In an example [START_REF] Pease | Source provenance of carbonate grains in the Wahiba Sand Sea, Oman, using a new LIBS method[END_REF] of using LIBS/SIMCA to classify grains from Wahiba Sand Sea, Oman, the methodology of combining LIBS with SIMCA showed good results of classification carbonated sand grains from different areas in the sand sea, and showed the potential sources of the sand grains to better understand the formation of the dune. This example showed the promise of applying this method in complex mixing patterns in sample. In the area of recycling polymer, LIBS was applied to 6 types of polymers, acrylonitrile-butadiene-styrene(ABS), polystyrene (PS), polyethylene(PE), polycarbonate(PC), polypropylene(PP), and polyamide(PA) [START_REF] Costa | Identification and classification of polymer e-waste using laser-induced breakdown spectroscopy (LIBS) and chemometric tools[END_REF]. Emission lines were selected for some elements and used for classification. SIMCA and (k-nearest neighbors) KNN were utilized to build classification models, and satisfactory results were achieved with 98% accuracy for KNN and 92% for SIMCA. It showed promising analytical capabilities of LIBS with chemometrics methods in identification and classification of plastics.

I.1.12.1.3.

Artificial neural network (ANN)

Artificial neural network (ANN) is a modulization tool containing groups of nodes that was inspired by the biological neurons in animal brains. It is a typical non-linear method, the basic unit called neuron imitates the function of its own original type, a biological neuron, transmitting the signal to the next ones. Briefly, an artificial neuron receives signals and processes them, the output of the neurons is computed by specific non-linear algorithms. The detailed discussion will be in the later part.

Since ANN can learn from known pairs of inputs and outputs to predict unknow input after training.

It has lots of applications in many different areas, such as image recognition [START_REF] Sun | A new method of feature fusion and its application in image recognition[END_REF], voice conversion [START_REF] Desai | Voice conversion using Artificial Neural Networks[END_REF], machine translation [START_REF]Neural Machine Translation using Recurrent Neural Network[END_REF], medical diagnosis [START_REF] Amato | Artificial neural networks in medical diagnosis[END_REF], finance fraud [START_REF] Omar | Predicting fraudulent financial reporting using artificial neural network[END_REF], and etc.

In LIBS, ANN was applied to the classification of wood samples [START_REF] Cui | Laser-induced breakdown spectroscopy (LIBS) for classification of wood species integrated with artificial neural network (ANN)[END_REF]. In this case, loadings of PCA were used to compress data, after the normalization. Two different methods were applied in ANN, multilayer perceptron network and Broyden-Fletcher-Goldfarb-Shanno iterative algorithm, and achieved 100% for the correct classification rate (CCR). This work has also been compared with PLS-DA, KNN and SIMCA, with the CCRs of 82.5%, 95.83% and 51.67%, respectively. The results demonstrated that the combination of ANN and LIBS could be used for analyzing and cataloged different wood samples.

In an example [START_REF] Vítková | Fast identification of biominerals by means of stand-off laser-induced breakdown spectroscopy using linear discriminant analysis and artificial neural networks[END_REF] of apply ANN in LIBS, both LDA and ANN were applied in fast identification of archeological materials in situ. 18 samples in 7 types (include shell, bricks, soil pellets, ceramic, teeth, bones and mortars.) were used to train the ANN, input of the network in this case are the PCA scores of the data, in the consideration of reduce the size of the data. ANN showed a better result than LDA, only one out of eight samples are not correct for ANN, and for LDA 2 samples were not correct.

In the area of chemistry, one [155] of the early attempts of applying ANN into LIBS data utilize eight spectral regions of interest of polymers as the input into a feedforward network for classification.

The network was modularized into two sub-networks, one to distinguish PVC (polyvinyl chloride)

with one neuron as output layer and three neurons for the hidden layer, the other one has three neurons as output and eight neurons for the hidden layer for PE (polyethylene), PP (polypropylene) and PET (polyethylene terephthalate). The results should be considered as very good when the accuracy is all above 93%. Another application [START_REF] Boueri | Identification of Polymer Materials Using Laser-Induced Breakdown Spectroscopy Combined with Artificial Neural Networks[END_REF] of ANN in LIBS applied the similar network into the spectra of eight polymers, 13 spectral lines were used as the inputs after normalization. In this article, a threelayer network was built, the number of neurons of each layer has been modified to give the best results.

The application of this ANN has successfully overcome the influence of the matrix effects from the samples, and achieved a satisfactory accurate classification.

In archaeological area, LIBS was combined with ANN to classify archaeological ceramics samples [START_REF] Ramil | Application of artificial neural networks for the rapid classification of archaeological ceramics by means of laser induced breakdown spectroscopy (LIBS)[END_REF]. Two strategies were used, one using the whole spectra of LIBS, and another one using spectral sub-domains. In both the methods, ANN showed more than 95% of correct classification rate.

LIBS's ability to identify material is useful in the area of extraterrestrial exploration, such as the study of Mars. And in one similar example [START_REF] Koujelev | Laser-induced breakdown spectroscopy with artificial neural network processing for material identification[END_REF] success rate when unidentified samples were included. And real-time rough material identification were also carried out among materials like metal alloys, marble, granite, soil, clay, rock, sediments and silicon oxide [START_REF] Koujelev | Laser-induced breakdown spectroscopy with artificial neural network processing for material identification[END_REF]. The results within the set of aluminum alloys showed that this method has high sensitivity. This article [START_REF] Koujelev | Laser-induced breakdown spectroscopy with artificial neural network processing for material identification[END_REF] demonstrated the possibility of LIBS with ANN for analysis of material on the surface of Mars.

I.1.12.1.4.

Support vector machine (SVM)

Support vector machine is an original binary supervised learning model, it has a learning algorithm based on kernel function, and it is suitable for small dataset. In SVM, a data is regarded as a p-dimensional vector (as the same number of the list), and the algorithm will try to find a hyperplane with a dimension of (p-1) to separate them. And it is called a linear classifier, according to the theory, the hyperplane is more than one, and the one with largest separation between the two classes is the most reasonable choice. If the hyperplane being chosen is the one that the distances from one data point to the nearest one on each side of the plane is maximized, then it is called a maximum-margin hyperplane. As mentioned above, SVM is original binary method, however, by reducing the single multiclass problem into multiple binary problems, multiclass SVM also exists within finite groups.

In pharmaceutical area, LSSVM(least squares support vector machine), PLS-DA and SIMCA were applied for discrimination in LIBS measurements [START_REF] Dingari | Incorporation of Support Vector Machines in the LIBS Toolbox for Sensitive and Robust Classification Amidst Unexpected Sample and System Variability[END_REF], and the results showed that LSSVM (96% in average accuracy) improved the sensitivity in classification compared with PLS-DA (95.67%) and SIMCA (94.03%), and for the robustness test, LSSVM also showed a good result with 92.47% in accuracy compared with SIMCA of 83.53% and PLS-DA of 83.95%. This example showed that combined with SVM, LIBS can provide a sensitive and robust results in classification of pharmaceutical samples.

In an example [START_REF] Yu | Laser-Induced Breakdown Spectroscopy Coupled with Multivariate Chemometrics for Variety Discrimination of Soil[END_REF] of soil analysis using LIBS, PCA was first applied to select 7 emission lines and then SIMCA and LSSVM were both used to classify 6 types of soils. The correct discrimination rates were 90% for SIMCA and 100% for LSSVM. The LSSVM model was further applied to discriminate 8 types of soils for verification. This research indicated that coupled with LSSVM, LIBS could conduct classification for soil samples.

I.1.12.1.5.

K-nearest neighbor (KNN)

K-nearest neighbor is a multiclass method for samples that are difficult to classify in linear method. KNN calculates the nearest distance of a given sample to K number of known samples, and the given unknown sample will be categorized according to the distances of different known classes.

In an example [START_REF] Neiva | Proposition of classification models for the direct evaluation of the quality of cattle and sheep leathers using laser-induced breakdown spectroscopy (LIBS) analysis[END_REF] of leather quality evaluation, KNN, SIMCA and PLS-DA were applied into analysis of LIBS data from both sheep and cattle leather samples. The results showed that models achieved satisfactory classification rate for SIMCA: 75.2%, PLS-DA: 80.5% for calibration data, and for validation data, the classification for KNN: 80.9% and for SIMCA: 71.6%. These results confirmed that LIBS combined with chemometrics can be used in quality classification in leather.

In the medical area, KNN was also used to distinguish fresh soft tissues based on LIBS spectra [START_REF] Li | Discrimination of soft tissues using laserinduced breakdown spectroscopy in combination with k nearest neighbors (kNN) and support vector machine (SVM) classifiers[END_REF]. Chemometrics methods were applied to discriminate fat, skin and muscle tissues, and also among highly similar tissues like ham, loin and tenderloin muscle. The methods included PCA, KNN and SVM. For fat, skin and muscle tissues, KNN and SVM were applied and achieved accuracy more than 99.83% with sensitivity over 99.5% and specificity over 99.8%. For the discrimination of 3 highly similar tissues, SVM gave the best results with an accuracy of 76.84%, a sensitivity over 0.742 and a specificity over 0.869, which is an acceptable performance considering the molecular complexity of such sample. The results showed that assisted with chemometrics LIBS could be a powerful tool for discrimination of soft tissues, even for highly similar samples.

I.1.12.2. Unsupervised methods I.1.12.2.1.

Principal component analysis (PCA)

Unsupervised pattern recognition is based on the distances between samples, similar samples would have shorter distances in multidimensional space and for unknown samples the distances would be longer. One of such common method is PCA. In an example [START_REF] Unnikrishnan | Analytical predictive capabilities of Laser Induced Breakdown Spectroscopy (LIBS) with Principal Component Analysis (PCA) for plastic classification[END_REF] of classification of four types of plastics with LIBS, PCA was applied with success. The results were further confirmed with statistical parameters, Mahalanobis distance and spectral residuals. Another example [START_REF] Akpovo | Regional discrimination of oysters using laser-induced breakdown spectroscopy[END_REF] of discrimination by PCA was done with oyster samples. PCA was applied with factor analysis together with the elemental composition from LIBS, and managed to classify samples from different sites with a result of 91.3%

unselected cases correctly discriminated.

I.1.13 Quantitative analysis

The quantitative analysis in LIBS is managed in two ways, calibration methods and calibration free methods [START_REF] Ciucci | New Procedure for Quantitative Elemental Analysis by Laser-Induced Plasma Spectroscopy[END_REF]. Within the calibration methods, it can be further categorized into univariate and multivariate approaches. The conventional quantitative analysis in spectroscopy usually establishes the relationship between concentration of either an element or a component and the intensity of the spectra line. This univariate method is accurate and widely used in particular spectroscopies, however with LIBS, it often failed to obtain the better results. The complexity of LIBS data makes the traditional regression method no longer available in some cases, so it is interesting to take advantage of the abundant information that LIBS data can provide with the use of multivariate methods.

Univariate calibration

The univariate calibration is one of the most widespread and simplest quantitative analysis methods. It establishes the relationship between the intensity of an emission line of an element in the sample and its concentration. In the area of LIBS, the group of Andrade [START_REF] Andrade | Direct Determination of Contaminants and Major and Minor Nutrients in Solid Fertilizers Using Laser-Induced Breakdown Spectroscopy (LIBS)[END_REF] used this method to determine the concentration of 12 elements in solid fertilizers. Correlation between the predicted values and the reference ones was right. Other examples can also be found in literature, for instance, 11 elements in fingernails were determined by LIBS with the help of calibration curves [START_REF] Rusak | Quantitative determination of calcium, magnesium, and zinc in fingernails by laser-induced breakdown spectroscopy[END_REF]. LIBS was used in inline analysis of liquid slag in steel works, and the major components were in the good agreement of the reference samples from the laboratory [START_REF] Sturm | Laser-Induced Breakdown Spectroscopy for 24/7 Automatic Liquid Slag Analysis at a Steel Works[END_REF]. Univariate method has the advantage of being simple and has good agreement in many cases, however, it may not be able to meet the requirements for heterogeneous samples due to the fluctuation of the laser energy. A common method here is to use an internal standard added for the samples. In a quantitative study [START_REF] Unnikrishnan | Calibration based laser-induced breakdown spectroscopy (LIBS) for quantitative analysis of doped rare earth elements in phosphors[END_REF] of rare-earth elements(La and Nd) in phosphors by LIBS, a standardization method was applied. Another case [START_REF] Lasheras | Quantitative analysis of oxide materials by laser-induced breakdown spectroscopy with argon as an internal standard[END_REF] used argon environment to reduce the influence of the air at atmospheric pressure for quantitative analysis of oxides in minerals. The results showed a real improvement of the signal quality and better linearity of the calibration curves. The addition of the internal standard also improves the accuracy of the univariate calibration. Still this method is not universal and cannot meet the standard of the highaccuracy analysis. ANN [START_REF] Vítková | Fast identification of biominerals by means of stand-off laser-induced breakdown spectroscopy using linear discriminant analysis and artificial neural networks[END_REF][175], SVM regression [START_REF] Zhang | Quantitative and classification analysis of slag samples by laser induced breakdown spectroscopy (LIBS) coupled with support vector machine (SVM) and partial least square (PLS) methods[END_REF][177] and RF [START_REF] Wu | Quantitative analysis of nonmetal elements in steel using laser-induced breakdown spectroscopy combined with random forest[END_REF].

I

Partial least squares (PLS)

PLS is perhaps one of the most commonly used multivariate statistical analysis methods in many areas of analytical chemistry. The number of latent variables is an important parameter in this method PLS is particularly suitable for tasks where the number of independent variables is bigger than the number of samples. In PLS, considering the samples as a matrix X, the orthogonal decomposition of the X and the response matrix Y will be used to build a regression model based on feature variables.

Because of the mathematical theory of PLS, it is an accurate and fast method. In an analysis [START_REF] Bilge | Ash analysis of flour sample by using laser-i nduced breakdown spectroscopy[END_REF] of wheat ash samples via LIBS, PLS was built and showed good results. The ash contents were between 0.48% and 1.39% with a R 2 of 0.992. In another use [START_REF] Bilge | Determination of whey adulteration in milk powder by using laser induced breakdown spectroscopy[END_REF] of PLS in LIBS is a case with detection of adulterated milk powder. The calibration curves from PLS showed that the R 2 and LOD for sweet whey powder, one of the adulterations in this case, was 0.981 and 1.55%, respectively. The R 2 and LOD value for acid whey powder, another material used for adulteration, is 0.985 and 0.55%, respectively.

There are also some other PLS based methods, such as GA-PLS in the quantitative analysis of soil samples in LIBS [START_REF] Anderson | The influence of multivariate analysis methods and target grain size on the accuracy of remote quantitative chemical analysis of rocks using laser induced breakdown spectroscopy[END_REF], a hybrid model that combined PLS with spectrum standardization in coal analysis [START_REF] Hou | A hybrid quantification model and its application for coal analysis using laser induced breakdown spectroscopy[END_REF], a combination of WT(wavelet transform) and PLS in the analysis of Carbon in coal [START_REF] Yuan | A partial least squares and wavelet-transform hybrid model to analyze carbon content in coal using laser-induced breakdown spectroscopy[END_REF].

Multiple linear regression (MLR)

MLR is a regression model for quantitative analysis, using simultaneously several selected wavelengths in the spectral. MLR and simple linear regression were both applied to soil samples for determination of C [START_REF] Ayyalasomayajula | Application of laser-induced breakdown spectroscopy for total carbon quantification in soil samples[END_REF]. The results showed that MLR has better result than simple linear regression.

In another example [START_REF] Xu | Quantitative analysis of cadmium in navel orange by laserinduced breakdown spectroscopy combined with partial least squares[END_REF] of Pb determine in navel oranges, MLR was built and the feasibility of the MLR model was verified with variance analysis and regression statistics. The results showed that MLR had better results. In this article, authors said that MLR method used here can take full information from the LIBS data for quantitative analysis and reduce the influence of the matrix effect.

Artificial neural network(ANN)

In the previous part, we showed that ANN is capable of solving classification problems in LIBS.

In this part, examples showed that ANN is also be able to solve the prediction problems in LIBS due to its self-learning ability. In another example, ANN was used to improve the accuracy and precision of Cr and Ni concentration prediction in steels [START_REF] Li | Analytical-performance improvement of laser-induced breakdown spectroscopy for steel using multi-spectral-line calibration with an artificial neural network[END_REF]. ANN showed better results than the conventional internal calibration method, RMSE of CV for Cr and Ni decreased to 0.010 and 0.023 wt% from 0.018 and 0.067 wt%, respectively. Besides, the relative standard deviation for these two elements also decreased from 11.3% to 6.4% for Cr and from 19.5% to 12.9% for Ni. An on-site analysis [START_REF] Haddad | Artificial neural network for on-site quantitative analysis of soils using laser induced breakdown spectroscopy[END_REF] of soil samples illustrated that combined with ANN, LIBS can be used for quantitative analysis. The ANN showed results of less than 20% of errors for target elements (such as Al, Ca, Cu and Fe), and which means a good efficiency for on-site soil analysis with LIBS.

Support vector machine regression

SVM is one of the machine learning methods based on statistical learning theory. Besides it's would not for classification, SVM can also be used for regression purposes. SVM regression model is based on structural risk minimization, and is capable of manage the balance between complexity and the learning ability according to the sample information. SVM regression is suitable for small size datasets. Combined with advanced kernel functions and prior knowledge, it has many applications in many areas. In an experiment [START_REF] Chun-Long | Comparative analysis of quantitative method on heavy metal detection in water with laser-induced breakdown spectroscopy[END_REF] of quantitative analysis of Ni in water, 3 different regression methods(MLR, ANN and SVM) were applied, and for this case, SVM performed best. There are also other methods such as LSSVM and relevance vector machine (RVM) applied in LIBS. In another analysis [START_REF] Zhang | Quantitative and classification analysis of slag samples by laser induced breakdown spectroscopy (LIBS) coupled with support vector machine (SVM) and partial least square (PLS) methods[END_REF] of major components in slag samples, LSSVM was used to build the model with full spectra as the input. This model decreased the effect of nonlinear factors from self-absorption in plasma and can provide an ideal result. RVM regression method is a sparse probabilistic model for limited samples, it has good fitting and generalization ability, and also can provide the probability distribution of the prediction result, and this method is capable of dealing with noisy samples. In the area of LIBS, a RVM model [START_REF] Yang | A laser induced breakdown spectroscopy quantitative analysis method based on the robust least squares support vector machine regression model[END_REF] was built for Cr in steels samples with the analytical lines who was selected based on the spectrum intensity, wavelength, the widths of half-height. The results showed that in this example, SVM has better performance than SVM, ANN, PLSR.

Random forest (RF)

RF is a regression method based on a decision tree. It contains more than on decision trees generated by bagging integrated learning technology. The final results will be the averaged vote from multiple decision trees. This method can solve the problem of overfitting and has a good balance of noise and outliers. In an analysis [START_REF] Zhang | A novel approach for the quantitative analysis of multiple elements in steel based on laser-induced breakdown spectroscopy (LIBS) and random forest regression (RFR)[END_REF] of 5 elements in steel samples, PLS, SVM and random forest were adopted simultaneously, and RF had the best results than the other two. The existence of S and P in steel samples may have some matrix effect that are difficult to manage quantitative analysis. In a case [START_REF] Wu | Quantitative analysis of nonmetal elements in steel using laser-induced breakdown spectroscopy combined with random forest[END_REF] of LIBS, quantitative analysis of S and P in steel samples, RF showed good prediction compared with PLSR, which showed that this method had a promising potential in the area of steel quality control.

I.1.13.2. Calibration free method

Contrary to the methods based on calibration curves, methods without calibration are called calibration-free method, short as CF. CF-LIBS [START_REF] Ciucci | New Procedure for Quantitative Elemental Analysis by Laser-Induced Plasma Spectroscopy[END_REF] appeared for quantifying elements within unknown samples. This method using plasma emission lines with mathematical models to determine the content of elements of interest from the single spectrum to be considered. It does not require calibration curves or reference samples. Therefore, CF-LIBS can solve the matrix effects that is caused by the different matrices between the unknown sample and the reference sample. Even if the CF method has some improvement mentioned above, it still cannot compensate the variation caused by plasma temperature nor the influence from self-absorption. The importance of plasma temperature has been addressed enough from [], and many different methods has been invented for different cases. A modified version of CF method [START_REF] Gaudiuso | Laser-induced breakdown spectroscopy of archaeological findings with calibration-free inverse method: Comparison with classical laser-induced breakdown spectroscopy and conventional techniques[END_REF] was invented to analyze copper-based alloy samples in archeology, the plasma temperature was determined by a mathematical method related to known samples, and results were compared with the classical LIBS method. There are also CF-LIBS that combines with different methods, such as CF/ANN [START_REF] D'andrea | A hybrid calibration-free/artificial neural networks approach to the quantitative analysis of LIBS spectra[END_REF], CF-LIBS combined with a binary search algorithm [START_REF] Wang | Acidity analysis of iron ore based on calibration-free laser-induced breakdown spectroscopy (CF-LIBS) combined with a binary search algorithm (BSA)[END_REF], CF-LIBS combined with GA to determine the plasma temperature [START_REF] Dong | A method for improving the accuracy of calibration-free laser-induced breakdown spectroscopy (CF-LIBS) using determined plasma temperature by genetic algorithm (GA)[END_REF] and etc..

I.1.14 Conclusion and prospects

The short summary of chemometric methods in LIBS showed the great potential in the data pretreatment and qualitative and quantitative analysis. It proved to be an effective tool for LIBS and improve the accuracy of quantitative analysis in many different areas. LIBS technology has promising application in complex material analysis, combined with chemometrics. The future of this technology without any doubts would further extent its area with better performance. The improvement of instrument brings bigger and bigger datasets, which is exactly the stage for chemometrics. Some theoretical issues of LIBS, like matrix effects and self-absorption has been solved for certain cases, yet still a general solution is on the way, but chemometrics showed a very promising way towards it.

This is what we will show in this thesis.

I.2. A special focus on Artificial Neural Network (ANN)

I.2.1 Introduction

Artificial

I.2.3.2. Training

The learning of an ANN is an important part, during the process, the network compares the differences between output of the network and the known target, then adjust the parameters (weights of neurons, biases, parameters related to the chosen algorithm, etc.) according to the chosen learning rule to make the output more and more close to target. These adjustments are also known as tuning of parameters. After tuning, the output should be as "real" as the target. Then the already trained network will be tested for accuracy, on with new data not used during the training.

One of such examples is image recognition. Loads of manually labeled pictures of "cat" or "not cat" were used as input, then the network will analyze them and generate characteristics from which.

Later, the well-learned network will be used to classify new pictures whether they are cats or not. This type of ANN is considered as supervised learning.

I.2.3.3. Types of ANN

One [186] of the earliest attempts talked about the theories about building the network with and without circles in the structure. Nowadays, there has been a lot of different networks built many ways.

Among them all, they can be categorized into two basic types based on architecture, feedback (recurrent) (Fig. I-6) and feedforward (Fig. I-4).

I.2.3.4. Theoretical features, criticism

The non-linearity behavior is the biggest feature of ANN, and perhaps also the biggest flaw.

Since the exact theory about how a network gain the result through learning is not yet available, artificial neural network sometimes is described as a black box. You know the input, you know the output, but you don't know how the output came out from the input. In the training of ANN, the previous knowledge about the dataset is not necessary, answers have been found themselves as if by magic, almost. However, ANN requires a lot of training, and the size of the dataset could be huge, which would result in enormous computing resources.

Back-propagation (BP) neural network(Fig. I-7) [START_REF] Rumelhart | Learning Internal Representations by Error Propagation[END_REF] is perhaps the most known and used.

Besides, Self-organizing feature mapping (SOM) network [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF], the Hopfield neural network [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF] ,

the radial basis function (RBF) neural network [START_REF] Powell | Radial basis functions for multivariable interpolation : A review[END_REF] and the recurrent neural network [START_REF] Williams | A Learning Algorithm for Continually Running Fully Recurrent Neural Networks[END_REF] are the other commonly used networks.

Theory of back-propagation (BP)

The popularity of ANN rises with its application in voice recognition and visual recognition, where the traditional logical linear methods failed. These applications in ANN introduced a new area later known as deep learning. Its immense capability of dealing with complex models, ANN was applied in many areas in recent years and achieved some promising results. As for chemistry, there are more and more applications these days, but due to the difficulty of having right size of dataset, the lack of straight mechanism, ANN still cannot be considered as a main approach.

I.2.4 ANN in LIBS

The abundant applications of ANN in LIBS has been discussed previously in the previous sections (1.11.1.3 and 1.12.1.2), for both qualitative and quantitative analysis. Many ANNs in LIBS aimed in quantity, or classification, sorting. The other large part of the application of chemometrics in LIBS is calibration, as stated in one review, the calibration in LIBS is an empirical process, the existence of universally good solutions are precluded [START_REF] Hahn | Laser-Induced Breakdown Spectroscopy (LIBS), Part II: Review of Instrumental and Methodological Approaches to Material Analysis and Applications to Different Fields[END_REF]. As mentioned above, lack of enough data is one of the problems for ANN, luckily, we still have simulated data. Then came the idea that why not using the simulated datasets to train ANN, then apply the real data as the test set for the neural network. When the simulation of training dataset is close enough to real data, the problem of lack of data no longer exists. In the following chapters, this idea has been used. Simulated datasets were generated for training ANN in the next chapter, the details of generation and of the ANNs were discussed in details, also the evaluation of results. In chapter 2, the real data was applied as test set to the neural networks that have been trained with simulated data, the results and the problem will also be discussed.

II. ROBUST QUANTITATIVE ANALYSIS OF COMPLEX SAMPLES USING LIBS IMAGING ANF ANN INDEPENDENT OF PLASMA TEMPERATURE AND ELECTRON DENSITY

II.1. Development of an ANN with simulated datasets

II.1.1 Introduction

In the previous chapter, we were able to observe the full potential of LIBS spectroscopy for the characterization of complex materials. We have also seen the different chemometric approaches that allow us today to propose a valorization of the acquired spectral data at both qualitative and quantitative levels. From a fundamental point of view, we explained the importance of the main characteristics of the plasma, such as its temperature and electron density, as they have a significant impact on the intensities of the emission lines and even on the presence of some of them over a wide range of wavelengths. Of course, theoretical models could allow us, in some way and under very strict conditions, to correct the emission line intensities but this would remain a complex and very timeconsuming task. In fact, we could only consider this strategy for a few spectra but not for hundreds of thousands or even millions of spectra of a LIBS hyperspectral imaging dataset available today. This is the problem we have been working on in this chapter. In LIBS imaging, we first acquire spectra systematically over a region of the sample. In a second step, we integrate the emission signal at a wavelength characteristic of the compound of interest in order to generate a spatial distribution map of the latter. This is a basic principle used for many spectroscopic imaging techniques well beyond LIBS spectroscopy. This integration methodology makes sense in LIBS because we often have the possibility to find an isolated emission line, i.e. without interference. Nevertheless, spectroscopic imaging is primarily implemented when one wants to investigate a heterogeneous sample with many variations in matrices and chemical structures, and it is naturally in these conditions that variations in plasma temperature and electron density can vary. These potential variations in plasma parameters between the different spectra of the imaging dataset thus influence the intensity of the selected emission line and ultimately lead to under-or over-estimations of the element concentration and thus a biased chemical image. Researchers using this LIBS imaging technique are for the most part well aware of the problem but are ultimately at a loss.

The aim of this chapter is therefore to develop a predictive model of the quantity of an element in a spectrum independently of the plasma temperature and electron density. In the classical development of such a quantitative model, the calibration or training step requires the availability of experimental data and more precisely a large number of "spectrum/quantity of the element obtained by a reference analytical method" pairs. It is obvious that this is not feasible in LIBS imaging because firstly it is impossible to estimate the concentration of an element by an analytical method on each pixel but secondly, we cannot have the plasma parameters for each of them. The originality of the work in this chapter is therefore to generate synthetic LIBS spectra with varying concentrations of the elements of interest but also including variations in plasma temperatures and electron density. Due to the complexity of the phenomena to be modelled and the potential non-linearities between the inputs and outputs of the model, we have focused on artificial neural networks. The following sections will introduce the spectroscopic data generation, the construction of the ANN models and their optimization. For your information, this chapter accounts for 90% of my research time during the thesis with more than 200 neural networks built.

II.1.2 Generation of a big LIBS synthetic dataset

The originality of our strategy lies in the generation of synthetic spectral data allowing us to control the temperature and electron density variations of the plasma. It is true that this is a risky bet because it is very rare in chemometrics, but the near impossibility of really evaluating these two parameters on real spectra forced us to proceed in this way. We thus started from the Kurucz database [193] which is well known to the LIBS community. It proposes effectively for each element of the periodic table all the emission lines and the associated Einstein coefficients. Based on the equations of Boltzmann and Saha, we were able to simulate the LIBS spectrum of an element for a given temperature and electron density. In order to get closer to the reality of the LIBS spectral data, we also applied to each of the emission lines a Lorentzian profile with a bandwidth of 0.15 nm [START_REF] Motto-Ros | Investigation of signal extraction in the frame of laser induced breakdown spectroscopy imaging[END_REF]. It should be noted that the Stark effect and self-absorption have been neglected in this work, which is above all a first feasibility study and a proof of concept. Table II-1 shows all the elements used in this study as well as the different temperatures and electron densities. At first it might be surprising to see that only 22 elements have been selected in the periodic table but also why these elements in particular. Our goal was first of all to limit our calculation time and the consideration of all the elements of the periodic classification would have undoubtedly made the construction and optimization phases of our predictive models more difficult. However, we have to put this into perspective as it is not trivial to propose a quantification on 22 elements. We must not forget either that all the elements of the periodic table will not be detected in a complex sample because of certain elemental concentrations that are too low but also because of their detection limits in LIBS. For these 22 items in particular, they were chosen because they are potentially contained in a real sample that we will explore later in this chapter, once our models are trained for these quantitative predictions. As we can also see in We see here, by this simple case, that the intensities of the lines are particularly impacted by the variations of these two parameters describing the plasma. This gives us a real understanding of the problem, as the intensity of the lines is usually used to obtain a quantification of an element. So if we do not take into account the plasma parameters we could, for example the decrease in intensity of a line between two spectra linked to different plasmas were observed, but in the incapacity to detect this situation we could interpret it as a decrease in the concentration of this element whereas it could be unchanged. It is obvious that such a situation would lead to the generation of LIBS chemical maps with estimated under-or over-concentrations if we use the usual strategies to generate them. Since real spectra are of course not made up of a single element, we simulated mixture spectra using a linear combination of the pure spectra of the elements with randomly selected concentration ratios for each of them. An in-house Matlab code then allowed us to generate several hundred thousand mixture spectra by varying, if necessary, the plasma temperature, the electron density or even both with the constraint of having the sum of the elementary concentrations equal to 1. We have also taken care to ensure that each element can be considered as a major, minor or even trace element in all generated combinations. Moreover, we also had the possibility to add a more or less important noise on these mixture spectra but also why not to use only some elements from the 22 mentioned above. This strategy gave us the freedom to choose different levels of complexity to better understand our problem. To summarize the procedure of generation of a spectrum, a plasma temperature and an electron density were randomly drawn from the values previously mentioned. With these two parameters being fixed, we considered the pure spectra of the elements associated with these conditions. Then a random set of concentrations of the considered elements were generated, which linked to the previous pure spectra by linear combination. At this point, we obtained a synthetic mixture spectrum to which we could finally add an experimental noise or any other perturbation. This strategy allowed us to generate training and test sets naturally composed of "spectra" / "concentrations" pairs.

II.1.3 General consideration about the development of our artificial neural networks

From the numerous examples in the literature from different scientific domains, it is obvious that neural networks have a strong potential for modeling potentially non-linear phenomena, even without any prior knowledge of the relationship that could link the inputs to the outputs of such a model. Thus, in our case these inputs would correspond to LIBS spectral data and the outputs to the concentrations of the elements of interest. Unfortunately, this methodology is by no means push-button and the predictive potential of a neural network can only be realized after a long optimization phase, which is even longer as the link between inputs and outputs is complex, which is our case in this work.

The first thing to optimize in a network is its structure. It is often by default composed of an input layer of neurons (which accepts LIBS data), a layer of hidden neurons and a layer of output neurons which will predict our concentrations. We can then ask ourselves questions about each of the layers. To begin with, we will have to see if we can use as many input neurons as wavelengths in the spectrum or if we will have to transform our spectral data to limit the input data and thus the number of neurons. We will also have to think about the possibility of using raw data or rather applying pre-processing methods to ease the predictions. We will also have to study the influence of the number of neurons in the hidden layer and also the number of layers themselves. Indeed, too few neurons on the hidden layer would not fulfill the prediction objective, while too many would most likely lead to over-fitting problems [START_REF] Huang | Neural network modeling of salinity variation in Apalachicola River[END_REF].

The balance between the prediction performance of a network and the complexity of its structure is at the stage where the network is just large enough to provide sufficient accuracy and reasonable time in the learning phase. In this work, hidden neurons varied from one to 200. Regarding the number of hidden neuron layers, it is obvious that a single layer should be preferred to limit the total number of parameters to be optimized on the whole network. Nevertheless, we consider sometimes networks with several hidden layers in the framework of complex modelling. It will thus be necessary to remain pragmatic and look at this possibility in our case. As far as the output neuron layer is concerned, we will also have to ask ourselves the right questions. Ideally, it is true that training a single network to simultaneously predict all 22 concentrations and why not the temperature and the electron density would be the ideal situation. We would then have a network with 24 output neurons. This option would undoubtedly be the fastest to develop and then apply to real data. However, it is not clear that such a model is possible and we may have to build several separate networks capable of predicting a limited number of elements, or even just one if the overall network is not predictive enough. In addition to the influence of the neural network structure that we have just seen, we will also have to consider the optimization of parameters intrinsic to the neural network concept. Thus, the weights of a neural network are its memory and thus contain its ability to predict output values based on the input values provided to the input layer of the network. During the training phase, a so-called learning function is used to optimize these weights to best match the input and output data we have provided in our training set. Unfortunately, there are a large number of learning functions that have been developed with highly variable capabilities depending on the problem to be addressed. It is therefore inevitably something that has to be optimized, just like the transfer functions that often transform non-linearly the signals that pass through each neuron of the network. For information purposes, Table II-2 shows all the functions that have been evaluated in this work. It would be naive to think that a simultaneous optimization of all the above-mentioned parameters could be done because it would then correspond to a kind of combinatorial explosion as regards the total number of networks to be optimized. It would not be justified to find the best neural network either, our objective for this thesis being to demonstrate the feasibility of such an approach and therefore a good model could perfectly suffice. The reverting to the impossibility of setting up a systematic study of the parameters will certainly give the reader an impression of linear optimization based on trials and errors but it is difficult to envisage anything else in this particular context. In this work, we will develop more than 15 types of neural network architecture in order to try to predict the concentration of elements. The Table II-3 given as an illustration shows the extent of the work done.

It is not necessary to go into it now as we will develop all these networks in detail in the following sections while trying to justify our choices concerning them. Varied for dataset means that within the datasets, T or Ne are the same for each spectrum. For trainscg, both T and Ne are varied for spectra, for traincgb, Ne is varied for spectra, T is 10000K. Varied for spectra means that T and Ne are different within one dataset.

The development of our ANN approach will thus follow the steps shown in figure Fig. II-2.

Fig. II-2 The scheme of the training

In a first step, we will generate our training and test sets by varying the temperature and electron density. Each of the two sets will then consist of several thousand spectra/concentration pairs. We will then choose the different perturbations to be applied to the spectra as well as the architecture of the network and the parameters intrinsic to the learning. This data will then be used for the learning phase of the network. Finally, the predictive potential of the constructed network will be evaluated using the spectra contained in the test set that did not participate in the training. Four different figures of merit will then be used to estimate the performance of the models and thus compare them:

The Root Mean Square Error (RMSE)

It is defined by equation II-1 and can be used on the training set and on the test set (i.e. RMSEC and RMSEP respectively).

𝑅𝑀𝑆𝐸 = [

∑ (𝑦 ? ^-𝑦 5 )

0 5@# % 𝑛 II-1
with 𝑛 being the number of sample in the considered set, 𝑦 ? ^ the predicted value of the neural network for the sample , and 𝑦 5 the reference value (in our case the real concentration of element)

The Mean Square Error (MSE)

It is given in in the equation II-2. It is by definition the square of the RMSE value.

𝑀𝑆𝐸 = 1 𝑛 _(𝑦 5 -𝑦 `5) % 0 5@#

II-2

The coefficient of determination (R 2 )

It is one of the most commonly used parameters in many areas. Its definition is given in equation II-3

with 𝑛 the number of sample in the dataset, 𝑦 5 the reference value of sample i, 𝑦 F the mean value of the all 𝑦 5 , and 𝑦 ? ^ the predicted value by the network for the sample i. R 2 is also the proportion of the variance in the dependent variable which is predictable from the independent variable(s).

𝑅 % = 1 - ∑ (𝑦 5 -𝑦 F) % 0 5@# ∑ (𝑦 5 -𝑦 ? ^)% 0 5@#

II-3

The calculation time

The calculation time will also be a parameter to be taken into account. Indeed, some networks will take a long time to train and even if they give the best errors on paper, you will have to ask yourself the question of the real added value compared to others which will be slightly less efficient but which will be undeniably faster to train. We will also have to look at the different learning methods that can support GPU computing versus others that can only run on CPU. It will be all the more important to do this study because the most efficient learning algorithms will not necessarily be those that take advantage of parallel computing on GPUs. Before going into the details of this neural network optimization, which will be detailed in the following sections, we thought it would be interesting to show in table 3.4 a synthesis of the 15 types of neural networks that have been developed in this thesis.

The objective here is to better reflect the importance of the work that may have been a bit thankless in the face of some disappointing results, but which are after all an integral part of scientific research

II.1.4 The first ANN

In the very beginning of this project, a simple neural network (ANN No.1) was built based on the example from Matlab neural network tutorial. It has 2048 neurons (corresponding to the 2048 in the spectral domain) as input, one hidden layer with 10 neurons. The transfer function was a sigmoid and the output layer had a linear transfer function. 22 output neurons were also considered (all the concentrations of 22 elements used for the dataset). Raw data was also considered in this first attempt.

The training method was the well-known Levenburg-Marquardt backpropagation (trainlm in Matlab).

It is one of the fastest methods in the Matlab toolbox but takes more memory and only runs on CPU.

The training phase was then force to stop once the network matches one of the following conditions:

1. reaching maximum epoch (1000 by default),

2.

reaching the minimum gradient (1.00 ´ 10 -7 ), process. It first recalls the structure of the network we trained, the algorithms used but also proposes a set of information related to the end of the learning phase. Thus, the training of this network stopped after 75 iterations (i.e. epochs) for a total calculation time of 12 hours which is enormous. It should be noted that the learning algorithm chosen here only runs on a CPU, which explains this slowness. We should also not forget that our network is made of 2048 input neurons and 10 hidden layer neurons which makes more than 20000 weights to be optimized during the training. We will observe MSE value of 33.2 at the end of the learning process. Moreover, the green rectangle indicates that the learning process has been stopped because it has reached the maximum validation checks which is 6.

This type of representation will be used regularly for the other networks built for comparison. In this case, MSE is 37.8937, R is 0.7273 for the whole dataset (on average), the training stopped at the 75 th iteration after 12 hours. These results should be considered poor, since the concentration of one element is between 0 and 1, the MSE is more than 100 times bigger. The possible solutions are: bigger datasets, less output, more neurons, more hidden layers, different learning method and so on. Only the error distribution seemed to improve as we can see on the histogram. This second network was therefore no more acceptable than the previous one and it was necessary to do something completely different. We assumed that neural networks were very sensitive to the range of values used as input. This is indeed all the more important as the transfer functions tend to accentuate this sensitivity. Thus, we decided to work on the normalization of the input data of the network presented in the following network.

II.1.6 Pre-processing for the input, a normalization?

The two networks mentioned above both used raw data, and the results were bad. Hence the pre-processing of data was considered. In the following trials, normalization was applied. These results were encouraging and showed the importance of normalizing the input data of the neural network. At this point in the work, we also wondered if it would be possible to predict the plasma temperature and electron density in addition to the 22 concentrations through the same network. The results of this network are presented in Fig. II-9. Note that we have kept the parameters of the previous network, only the number of neurons of the output layer is increased to 24 to take into account the prediction of these two new parameters. The results were no more satisfactory than the previous network. So far, a few different networks have been built and trained on CPU considering some sets of parameters, but the results were far from good. It was therefore necessary to test more combinations of parameters, but as we saw earlier, the calculation times were far too prohibitive. We therefore decided to prefer learning functions compatible with GPU computing in order to take advantage of parallelization. ANN (No.4) was then built to train on GPU and the default Matlab learning method was chosen (scaled conjugate gradient backpropagation, called trainscg in Matlab). This network was built as the previous one. With this GPU implementation, training time was significant shortened into several seconds. However, this ANN had similar results. Since we now had the ability to train networks very quickly we had the opportunity to test many combinations of parameters but this could not improve the performance of the network for predicting concentrations. We then assumed that a single network might not be able to predict all the concentrations of the elements of interest. The time was therefore to build networks with fewer output neurons, i.e. predicting only a few elements. This inevitably led to the need to train different networks to predict the 22 elements, but we needed to know whether the difficulty of the network to predict was due to its output structure.

II.1.7 Considering less output neurons?

The objective of this part was to simplify the neural networks in order to understand whether the modelling problem concerned this point. As far as the datasets used are concerned we have not changed anything as each spectrum is always a linked combination of the 22 elements. However, this time the network only predicts 2 or 3 elements that have been chosen by chance, namely Ag and Al or Ag, Al and As respectively. A series of networks (No.5) were built with data size varied from 1 000 to 200 000 spectra. It was also important to consider the number of spectra in the training set as it must be large enough in relation to the total number of weights in the network to be optimized. Unfortunately, there is no real criterion for choosing this parameter, hence the need to study the influence of the number of spectra. We then studied 12 different learning methods and the sigmoid and linear transfer function for the hidden and output layers respectively. A 0.1% noise were also added to the spectral data. The prediction performances of all these networks are listed in the Table II From a general point of view, we can already see that the networks have a very different prediction behavior with now much more acceptable MSE and R values. If we first look at the learning functions, we can see that some are worse than others, for example gradient descent backpropagation (traingd) and gradient descent with momentum (traindm). Additionally, the Levenberg-Marquardt backpropagation (trainlm) and Bayesian Regularization (trainbr) can only run with CPU. Therefore, they need much more time to train and make them difficult to use on big datasets. The scaled conjugate gradient backpropagation has then been considered as the learning function to keep in this part.

Regarding the size of the training dataset, we could test values from 1000 to 200000 LIBS spectra In all these networks, the maximum epoch chosen was 1000 and it was not enough for a dataset of 200 000, since most of these ANNs (No.5) stopped by reaching this limit i.e. without reaching the end of the training step. Of course, we could have increased this limit, but such a training already lasted more than 10 minutes. This time obviously seemed much shorter than before, but it was still too long for the many tests we had to do afterwards. In view of these results, it appeared to us that training the networks on a training set of 20,000 spectra allowed the networks to be trained for a time ranging from a few seconds to a few minutes while ensuring acceptable MSE and R values.

II.1.8 Considering different transfer function and/or different number of hidden neurons?

The transfer function of a neuron has a very important place in a network because it transforms the weighted sum of its inputs into a new output value which is then passed on to the next neuron in Although this may seem trivial, these two functions are quite different because the first will present outputs bounded between 0 and 1 while the second will also propose negative values on a domain from -1 to +1. The transfer function of the output layer is a linear one called purelin in Matlab. Table II-6 and Table II-7 give the results of ANN(No.6) trained with tan-sigmoid as transfer function in the hidden layer. We still consider networks built to predict 2 or 3 elements but this time limiting the total number of spectra in the dataset to 20,000. In spite of our discourse, which may seem a little negative at times, the graphs on the righthand side of the figure show that we have a good quality of prediction with a set of points very close to the first bisector and residues close together in the histogram. So, there is no doubt that we are moving in the right direction. It then appeared interesting to see if we could increase the level of prediction by increasing the number of hidden neurons to 20 while keeping the previous conditions of ANNs N°6. So Table II-8 provides the prediction results of this new structure (ANN N°7). As we can quickly see, we cannot observe a real improvement and we can therefore say that we have reached a certain limit in the predictive capacity of the neural network. Nevertheless, we should not lose sight of the fact that ANNs N°6 and N°7 were built on the basis of LIBS spectral data for fixed plasma temperatures and electron densities. Although this strategy might have seemed simplistic at first, it ultimately paid off because we discovered that it was not so simple to build a network capable of simultaneously predicting multiple concentrations even under these conditions.

The observations we made will certainly help us to understand conditions that are even closer to the reality of the experimental data.

II.1.9 Could we manage different temperature and electron density of the plasma?

As said before, the purpose of the series of networks we constructed (ANN N°8) was first to see if it was possible to predict the concentrations of the elements as the temperature and electron density varied. Table II-9 shows the prediction results of ANNs considering a fixed temperature but different electron densities. In general, we could see that the results were quite acceptable and that temperature variations could be taken into account for a given electron density. Nevertheless, we note a difficulty of prediction when the electron densities are different between the training and test sets, which is quite natural (penultimate line of the table ). Table II-10 shows the prediction results when we fix the electron density but consider different plasma temperatures between the training and test sets. These results are not really different and are therefore quite acceptable. We can even add that the network has a certain robustness to temperature variations since these networks predict the concentrations quite well while they have never seen spectra with another temperature than the one chosen for the training batch. In order to improve the predictive power of our models, we wanted to see if increasing the number of neurons on the hidden layer could improve things. Table II-11 shows the result of the predictions considering different temperatures and electron density with an increasing number of neurons ranging from 20 to 30. This variation seems limited, but it should not be forgotten that this number undeniably multiplies the number of weights to be optimized in the network during its training. We quickly see that this increase does not bring much in terms of predictive power and we can therefore stay with this minimum number of 20 neurons on the hidden layer. At this level of optimization, we again asked ourselves whether or not to add a second layer of hidden neurons. This only made the task of training the network more complex, with the loss of predictive potential as we

II.1.10 A time of reflexion

At this point in the thesis, we had already tested a number of networks and two main questions were added. The first was the normalization used for the input data and the second was the representativeness of the 2 or 3 elements we had chosen by default to be predicted. As far as the first point is concerned, we have seen how important normalization is in order to obtain satisfactory predictions. On the other hand, the normalization by the sum did not allow to cover the whole range of output values of the transfer function which is not optimal in the learning process. With this in mind, we decided to normalize the spectra by dividing them by their maximum value, so that the transformed spectral data would be between 0 and 1. Regarding the second point raised, we had decided in the first part of the thesis to work by default on Ag, Al, and As. We could then ask ourselves whether our difficulties might stem from this particular choice of elements to be predicted or whether modelling in general remained difficult. It was therefore decided to develop neural networks with different combinations of 3 elements from the list of 22 considering different temperatures and electron densities but also considering the new normalization procedure. Scaled conjugate gradient backpropagation (trainscg) was the only learning method applied and 10, 20 or 30 hidden neurons were considered in a single layer. The results of these ANNs No.9 are presented in the Table II-12. This table shows that apart from 5 cases (in bold), the R values for training set are all over 0.9, while only three cases (in bold) have R over 0.9 for validation. We thus observe that some networks perform poorly for particular combinations of elements to be predicted for the same network.

Moreover, increasing the number of hidden neurons and the number of iterations did not bring better performance. In these new conditions it was also interesting to know if we could not improve things by considering two layers of 10 neurons each in a new series of ANN No.10 (results in Table II -12). These results are quite different from the previous ones with acceptable performance for almost all element combinations with a few exceptions. As an illustration, the On the one hand, these results were encouraging, but on the other hand the use of two-layer hidden neural networks was not conducive to the robustness of the predictive model or even to the ease of observing the effect of our choices in such networks. It is clear that there was certainly too much weight to optimize in our networks and that we had to find another solution. It was therefore necessary to work on a solution that would limit the number of neurons on the input layer, as the use of spectral data was no longer possible. We did not want to develop a wavelength selection method in the spectral domain and therefore we preferred PCA as a compression method.

II.1.11 Introducing a new spectral representation using PCA

Principal component analysis is the Swiss Army knife of chemometrics. It is applied to a spectral dataset in order to account for all the variances it contains but representing it with a very small number of variables compared to the original. This technique can thus be seen as a filtering method that separates the chemical information in the dataset from the noise. Thus, the transformation of our spectral data and their use for the construction of neural networks will take place as follows: . nopt scores will then be saved for each synthetic sample in the training set. These are the values that will be used as input to the neural network.

6. The spectra of the test set will then be projected onto the nopt principal components nopt giving in turn the nopt scores for each.

We see that such a procedure allows us to drastically reduce the number of input neurons (and thus the total number of weights to be optimized on the network) since we will go from 2048 (i.e. 2048 spectral variables) into nopt. These scores will therefore be used as input to the neural network. On the other hand, we need to optimize the number of principal components. Indeed, a too low number would induce a loss of spectral information from this preprocessing (i.e. compression) step that we would not be able to make up for by learning the network. Too many components would not be welcome either, as only noise could be carried by some of them. As the synthetic spectra consist of 22 elements, it was unlikely that we would be able to build a successful neural network based on less than this number of principal components. We have thus started to build networks that predict only three elements using scaled conjugate gradient backpropagation (trainscg), a sigmoid transfer function and a variable number of neurons in the hidden layer from a few to a few dozen. This first experiments showed that the number of components had to be higher than 24 to obtain good prediction results but also that the number of neurons in the hidden layer had to be higher than the number of components (results not shown here). We then extended this research on the basis of these observations while at the same time attempting to increase the number of concentrations to be predicted to 22 by a single network. These results are presented in Table II-14. We notice that things change fundamentally with this new concept of compression used before the neural network. We obtain excellent MSE and R values, no matter how many elements are to be predicted. Note that a network consisting of 80 input neurons, i.e. using the scores of the first 80 principal components, and 100 hidden neurons is able to successfully predict the concentrations of the 22 elements. Nevertheless, we wanted to try to restrict the number of hidden neurons to 50 (again to limit the total number of weights to be optimized in the network) while looking at the effect of a larger number of components. The results are presented in the Table II-15. It is difficult to speak about the significance of the differences observed here and that is why we will favor the lower number of weights in the network, i.e. 100 input neurons and 50 hidden neurons. Under these conditions, we have a very efficient network capable of predicting the concentrations of the 22 elements, independently of the temperature of the plasma and its electron density. However, we should not forget that such a network will have to be applied to real data which will undoubtedly be tainted by a certain level of noise. As a reminder, the previous networks were trained and tested on data with a noise level of 0.1%. It was therefore logical to estimate the influence of the noise level on the performance of the neural network. We then generated synthetic datasets with 8 noise levels ranging from 0.1% to 20%. For each dataset with a given noise level, we developed a principal component analysis to keep only the first 100 principal components. The 100 scores per sample were then used to train 8 different neural networks with 50 hidden neurons to predict the 22 concentrations.

Table II-16 gives the prediction results obtained these three levels of noise. More precisely, all the MSE and R values in this table is an average of three trails per noise level. It is quite natural to see the predictive capabilities of a model decrease with the level of noise. Nevertheless, we must insist here on a certain robustness of the approach since the MSE errors and correlation coefficients remain more than satisfactory with a rather huge noise. This network capability is particularly important for our imaging problem. Indeed, we always want to increase the number of spectra acquired during a hyperspectral imaging experiment and as the total analysis time must be reasonable we often have to limit the acquisition time for each spectrum. In fact, the signal-to-noise ratio of an imaging dataset is quite often limited, hence the importance of a predictive model capable of withstanding lower data quality, which is our case here. Based on the previous good prediction results, we asked ourselves whether we will be able to predict the plasma temperature from a spectrum based on a well-trained neural network. This approach is quite original since such a network, if it exists, could also allow us to generate plasma temperature maps from a hyperspectral dataset. For this purpose, we used our procedure for generating synthetic spectra by mixing 22 elements in random ratios for different temperatures. The usual data spitting has been used in order to obtain the training set, the validation and the test set. Then, different neural networks have been trained to predict this temperature using a variable number of PCA scores (from 100 to 500) or directly from the 2048 initial spectrum values. The prediction results are given in Table II-17. The first thing that stands out is the fact that a prediction of the temperature seems possible with a certain accuracy. Again, it can be seen that it is better to use the scores coming from PCA than whole the variables in the spectral domain. The best predictions in this case are obtained for a network consisting of 150 input neurons (i.e. using 150 scores of the 150 principal components) and 50 neurons on the hidden layer. Many other attempts were made to reduce the number of hidden neurons and it was finally a network with 10 hidden neurons that allowed us to finalize this task. The results of these last networks are given in Table II-18. Thus, the network containing 100 input neurons and 10 hidden ones gives MSE the value of 9.10 -4 and a R value around 0.98 which is really efficient. We note an excellent linearity of the relationship on the three datasets and homogeneous dispersion for the different temperature levels. It may be surprising to see values between 0 and 1 in these representations but it was necessary to normalize the temperatures before training the network in order to be consistent with the range of the network transfer functions. Temperatures that were initially between 6000K and 10000K were then represented by values between 0.1 and 1 after this normalization. This does not, of course, detract from the quality of the predictions presented in this section. In view of the good results obtained on the temperature prediction, it was logical to try to do the same with the electron density. As before, we generated our synthetic spectral data by keeping the type of neural network and its learning algorithm. The use of PCA has of course been retained. A preliminary normalization of the densities was also necessary since we considered the values 5 × 10 AP , 1 × 10 AS , 5 × 10 AS , which could not be directly considered during the training of the network. Following this normalization, the lowest electron density was thus associated with the value 0.1 and the highest with 1. We then started with a layer of 100 input neurons for the network, i.e. by considering 100 principal components. As usual, we also studied the influence of the number of neurons on the hidden layer. The prediction results obtained are presented in the The first remark we can make is that the quality of the prediction is well below that of the temperature with higher MSE values and much lower R values. It thus seems more difficult to predict the electron density but we cannot really be categorical as we have only used three levels of density here, which does not necessarily help. If we now look at the number of neurons on the hidden layer, we see that the errors are all very close but the lowest are nevertheless obtained with less than 10 hidden neurons.

Assuming that there is no significant difference between the networks using 2, 5 or 10, it is logical to choose the lowest number of hidden neurons, i.e. 2. As an illustration, Fig. II-16 shows the prediction results for a network using 100 input neurons and 1 output neuron.

Fig. II-16 Regression plot of the best results of predicting of Ne

II.1.14 Could we finally predict everything?

In the previous sections, we have seen that the use of principal component analysis has been determinant for the training and prediction of neural networks. We were able to predict first the concentrations of the 22 elements with a single network, and then with two other networks to predict separately the temperature of the plasma and its density on the basis of LIBS spectral information. The objective of this new part was therefore to see if we would be able to predict not only the concentrations but also the temperature and the electron density of the plasma by a single neural network. As usual, the number of input neurons (and therefore the number of principal components) and the number of hidden neurons had to be optimized. As it was necessary to start with a first hypothesis, we decided to fix the number of principal components at 100 since we had seen in the previous sections that this allowed to potentially predict all 22 elements. Thus, Table II-20 gives the prediction results considering a variable number of hidden neurons. As we can see we obtain quite satisfactory results whatever the number of hidden neurons. Even if it is difficult to say if all these differences are significant, we have decided to use 50 hidden neurons which has the lowest MSE value of 1.03E-03. We could then go back to optimize the number of input neurons that we had set a priori in the previous step. The prediction results are given in Table II-21

Table II-21. As we can see, a network using the scores of 120 principal components with 50 hidden neurons allows us to have the lowest error. We can therefore say that with our approach, we are able to predict the concentration of the 22 elements as well as the temperature of the plasma or its electronic density by a single predictive model, which is very interesting. However, during this same discussion, it appeared that we had not seen a much more significant problem. Indeed, the intensities of a LIBS spectrum are naturally dependent on the laser power used.

Thus, the use of two different LIBS instruments or even a single one used to analyze a sample over two different time periods could well present a different overall intensity while from a chemical point of view we still had the same elemental concentrations under these different conditions. This made normalization of the spectra inevitable in order to get rid of the potential global intensity variations in the LIBS spectrum. The only problem was that the good predictive networks we had obtained were based on a neural network using the scores from the PCA that had been done on non-nominalized data.

It was therefore necessary to verify that this PCA/ANN approach was still valid on normalized data, which we did by developing a network with 100 input neurons and 50 hidden neurons to finally obtain an RMSE value of 0.0293 for the validation set and a correlation coefficient of 0.9773. It was obvious that we had lost the predictive potential presented in the previous sections from the PCA on raw data. We have tried to optimize as usual in this work many parameters of the network such as the number of principal components, the global architecture of the network, the transfer functions and the learning function but this did not change anything to the situation (results not presented in the manuscript). Reluctantly we decided to abandon the networks using PCA scores and return to the classical networks exploiting directly the libs spectrum. We had seen before that we could use normalized spectral data to predict the concentrations of some elements even if everything was not perfect. In order to simplify the task of training the neural network as much as possible, we decided to predict only one element at a time. This would undoubtedly require a lot of work as we would have to build 24 different networks for the 24 elements but this option deserves to be tested. As usual, we generated our synthetic dataset based on the pure spectra of the 24 elements considering different temperatures and electron densities. All spectra have been normalized by the maximum value before being used in a network. Of course, we spent a lot of time on the usual optimizations to choose a unique number of hidden neurons equal to 100 for the 24 neural networks. This optimization phase also showed that a much larger number of epochs were needed as well as an increased number of validation checks. Table II-24 gives the predictive capabilities of the 24 neural networks thus optimized. We quickly see that we have good results on all the elements and that this option of trying to simplify the output layer of the network was a worthwhile option. We have thus shown the feasibility of such an approach for the quantification of elements independently of the plasma temperature and its electron density. It is true that you may rightly ask why this option was not considered earlier in this work. This is because we were focused on providing a single network that would be easier to use for prediction purposes. The time spent optimizing these numerous networks was not wasted as it also allowed us to observe different behaviors of the neural networks when faced with LIBS data in different forms, which is equally valuable.

II.2.

Applying our ANN approach for the exploration of a complex mineral sample

II.2.1 Context and description of the considered mineral sample

As explained in the previous section, it is not possible to correct for variations in plasma temperature and electron density for all the spectra in a LIBS imaging dataset, which today often consists of several million spectra. The generation of chemical maps is therefore based on the integration of characteristic emission lines of an element of interest, neglecting these variations so that we cannot know them on each pixel. This is not a fundamental problem if we are studying homogeneous materials, but spectroscopic imaging is primarily used to investigate local heterogeneities in a sample at the micron scale over large areas. The aim of this section is to show how we can use our previous networks trained on synthetic data to predict elemental contributions in a naturally heterogeneous matrix i.e. a mineral for which it is known that variations in plasma parameters are necessarily observed. Since the beginning of the thesis, a rock sample had been selected by our collaborators (Vincent Motto-Ros and Cécile Fabre) and we planned to analyze it with our approach at the end of the modelling phase. Our first aim was to demonstrate the feasibility of the concept and it would have been unreasonable to consider all the elements of the periodic table, hence the short list of 24 elements in line with the supposed presence of these elements in the selected rock. It took only a few hours to acquire 2.38 million LIBS spectra over a spectral range from 280 to 369 nm (and a total of 2048 wavelengths) with a spatial resolution of 20 microns. The cube of LIBS hyperspectral data that we manipulated had the size 1400 × 1700 × 2048.

In the classical implementation of a LIBS imaging experiment, we usually select a rectangular area around the sample, which is then systematically analyzed along lines during the spectroscopic measurement. Of course, the sample studied here is not rectangular and many of the spectra acquired correspond to the resin and not to the rock sample. It is therefore necessary to localize the spectra in the dataset which only contains information about the rock. This may seem trivial but it has been shown many times in spectroscopic imaging that this pixel selection is necessary at the risk of completely biasing the chemometric exploration. We therefore used a Matlab Toolbox (imageSegementer) to generate a so-called mask that allowed us to automatically define which spectra we should keep or which spectra of the resin we should remove from the hyperspectral dataset. This mask is shown inFig.

II-19 , the yellow area representing all the pixels associated with the rock and therefore the spectra that we will consider for the rest of the calculations.

Fig. II-19 Mask generated by the Matlab toolbox, imageSegementer

Comparing this mask with the optical image, it can be seen that the left side of the sample will not be considered because the spectra are in poor quality.

II.2.2 Applying our ANN models to the rock sample

The objective was now to use the latest networks developed in section II-1.15 to generate elemental maps. The latter having been trained on normalized data we had of course to do so for these spectral data. At first glance, it seemed obvious that all that remained was to use each normalized spectrum (included in the mask) as an input to one of the trained networks for a given element to generate the corresponding chemical map. This was a risky bet in the sense that even if our collaborators, experts in LIBS and geology, had proposed a very relevant list of elements, no one could ensure that this list was exhaustive and that the networks had been trained with all the elements likely to be encountered in the sample. In other words, a direct use of the network is not possible at this stage because nobody can say whether each spectrum of the mask contains only elements of this list. So, if we have for example a spectrum that contains an element that is not in the list, its spectrum will be slightly different from the spectra contained in our synthetic spectra database with the consequence of disturbing the prediction of the concentration of the other elements if we force ourselves to use it. We have to find a solution to this problem because we have no choice but to train our network on synthetic spectra with a limited list of elements.

The trick comes from a basic principle in chemometrics. If you have developed a predictive model from a given calibration set of spectra, the future spectra used in the prediction step should be contained within the subspace defined by them. This is exactly the procedure used to detect outliers (i.e. spectra that must be discarded because they are not consistent with the training set). We are going to go even further, because a PCA will be used to filter the rock spectra so that they can be consistent with all the synthetic spectra contained in the training set of the ANNs. In a way, we will remove from the rock spectra all the elemental contributions that are not in the list of 24 elements.

The procedure is therefore as follows:

1) Development of a principal component analysis on the synthetic dataset,

2) Selection of an optimum number of nopt components to best describe the subspace of this first dataset.

3) Projection of the rock sample spectra onto the nopt principal components to obtain nopt scores for each.

4) Reconstruction of each rock spectrum through a linear combination of its nopt scores and the nopt principal components to obtain filtered spectra.

5) Use of the filtered spectra as input of the previously trained ANNs.

An important point was to find the optimum number of principal components nopt. We therefore simply applied this filtering procedure to the entire rock dataset for increasing values of nopt up to 50. For each nopt value, we then compared the corrected spectra to the original rock spectra by simply calculating the difference, which we then squared and summed to obtain sum of squared differences (SSD). The In view of this analysis, 24 principal components seemed to be sufficient to filter the spectra of the rock sample. These corrected spectra could now be used as input to our trained neural networks to predict the quantity of all the element of interest. Before generating elementary distribution maps, we thought it was important to take a closer look at the range of values predicted by the neural networks for each of the elements. The minimum and maximum predicted values obtained over the entire sample area for each of the elements are given in Table II in the interval from 0 to 1 is also given for each element. As a reminder, our neural networks have been trained to predict the relative quantities of one element with respect to another. This value must therefore naturally be between 0 and 1. We quickly notice that all these predictions are not perfect because negative values are present. We should not be too harsh either as the minimums in question are often very close to zero. Furthermore, we must not forget that there is an imprecision in our models, particularly for extremely low or even zero concentrations for a given pixel. This is certainly the case for elements such as Ag, Be, Cu, Ga, Mg, Mn, Zn and Zr for this specific sample. II-27show a comparison of elemental maps obtained through neural networks and the maps of the same elements obtained with the classical method of integration using a specific emission line. It is true that we cannot normally make an absolute comparison of these pairs of maps because we have trained our networks to predict the percentages of an element. Ideally, we would have obtained other concentration maps of each element from a reference method from which we could have obtained relative reference maps. Having said that, we will still try to compare maps on these figures with some caution. In order not to be misled by the images, it is first important to understand how the color scale is generated. For example, for a given image, the dark red color corresponds to the highest value and the dark blue color the lowest. As an example, if we look at the Fe map generated from ANN results (Fig. , we can see that there are many green areas which correspond to a predicted value of about 0.25 (i.e. 25%). If the maximum of the color scale is about 0.48, it means that there are pixels with this value effectively present in the map but they are certainly few in number hence the feeling of not seeing any on this map. Even if nothing can prove our claims at this date, the areas where an element is potentially present seem to be more homogeneous when the ANN approach is used compared to the signal integration method as for example for the elements Al, Fe or Mg. Similarly, we obtain very different maps for the elements Ca, Ce, Cr, Cu, La, Y and Zr. So it seems that we see more with our ANN approach, but all this needs to be verified of course.

II.3. Conclusion and perspectives

The objective of this research was to study the feasibility of a robust quantitative LIBS imaging analysis in the face of variations in plasma temperature and electron density. Indeed, the natural heterogeneity of a sample means that these two parameters are undeniably not constant over its entire surface, with repercussions on the intensities of the emission lines of its constitutive elements. In the context of imaging, we produce astronomical quantities of spectra today and it is impossible for many reasons to know these parameters for each pixel. The correction of the intensities of the acquired spectra on the basis of these parameters is therefore not possible. The originality of the work in this thesis was to propose the construction of a quantitative predictive model based on spectra with controlled and known variations of the plasma electron density and temperature. It is obvious that in these conditions we would have to use simulated spectra to build this model. It is true that this was a very risky gamble as it is not an ideal situation from a chemometric point of view, but we did not really have any alternatives. Based on the hypothesis of strong non-linearity between the spectra and the concentrations to be predicted, we decided to use neural networks which are known for their good ability to model under these difficult conditions. We also made this choice because we knew that we would not have any problem concerning the number of spectra available to build our model since they were simulated. To be honest, we did not expect to spend so much time on neural network optimization at the beginning of the thesis. The complexity of the data structure required a lot of optimizations concerning the architecture of the network, its training and the preprocessing of the spectral data as we have seen. The use of the final networks trained on synthetic data even had to be redesigned to be usable on our real data as we saw in the previous section. In view of these results, we can say that we have demonstrated a certain potential for this analysis concept but there are undeniably things to be done before it becomes a real tool for quantitative characterization in LIBS imaging. It is in this sense that we now present some perspectives of this work in order to converge towards this goal. To begin with, the simulated spectra should eventually take into account the phenomenon of self-absorption which is very often observed in LIBS spectra. It will also be necessary to see the influence of wavelength shifts and why not make the model robust to this perturbation. We will also have to work on the establishment of a criterion which authorizes us or not to use a neural network on a given real spectrum. Indeed, it is obvious that spectra with a low overall intensity (and therefore relatively high noise) are problematic. We also observed in this work that it was very inconvenient not to have a chemical map of the proportions of the elements from a reference method for a final validation of the concept. We still have to find one, although we know that this will be very complicated. Finally, we have seen that the networks used in this thesis have classical architectures that we have known in chemometrics for decades. For the record, these same neural are now called shallow neural networks.

They are so called to differentiate them from deep neural networks which have been developed recently, which demonstrate formidable predictive capacity under very specific conditions. These neural networks are deep because they have a very large number of hidden layers with particular architectures accompanied by learning functions adapted to these new characteristics. It would therefore be interesting to see if we could use such a network to simultaneously predict all the concentrations of the elements of interest from the LIBS spectrum.

III. FINDING EXOTIC PIXELS IN BIG LIBS IMAGING DATASETS

III.1. Introduction

Laser-induced breakdown spectroscopy (LIBS) imaging is actually becoming an essential tool to characterize complex samples in many scientific domains [START_REF] Jolivet | Review of the recent advances and applications of LIBS-based imaging[END_REF][START_REF] Gaudiuso | Laser-induced breakdown spectroscopy for human and animal health: A review[END_REF][START_REF] Fabre | Elemental imaging by laser-induced breakdown spectroscopy for the geological characterization of minerals[END_REF][START_REF] Cáceres | Megapixel multi-elemental imaging by Laser-Induced Breakdown Spectroscopy, a technology with considerable potential for paleoclimate studies[END_REF][START_REF] Trichard | Imaging of alumina supports by laser-induced breakdown spectroscopy: A new tool to understand the diffusion of trace metal impurities[END_REF]. As we have seen in previous sections, in this spectroscopic technique, a pulse laser beam focused on the sample surface generates a plasma from a small amount of vaporized material. Due to the electronic relaxation of excited atoms and ions, an emission spectrum characteristic of the elemental composition of the sample can be acquired using an optical spectrometer. In LIBS imaging experiments, the sample surface is explored in a raster scanning mode (i.e. acquisition of one spectrum for each spatial position of a predefined grid) covering the region of interest. An elemental image can then be generated from the acquired dataset using a simple signal integration of a given emission line. The richness of this imaging approach lies in its many advantages that cannot be observed simultaneously in any other spectroscopic technique. Indeed, LIBS imaging has multi-elemental capabilities, a high acquisition rate (≥100 spectra/s), full compatibility with optical microscopy and ease of use on samples without almost any size restriction (up to several tens of cm²), all under atmospheric conditions. On top of that, this technique has a high field of view and a spatial resolution around 10 μm coupled with a limit of detection in the order of weight ppm. It is thus very convenient to explore a sample at the micronic scale by acquiring several million spectra in just hours. It is also very interesting to look at the evolution of LIBS imaging over time, which is well illustrated in Fig. III-1. We observe here a real explosion of the number of pixels on a very limited time scale, roughly by a factor of 10 every 2 years since the 2010s. More precisely, this figure presents the work of Motto-Ros et al. on the detection of nanoparticles in the kidney [START_REF] Motto-Ros | Mapping nanoparticles injected into a biological tissue using laserinduced breakdown spectroscopy[END_REF]. This evolution is of course explained by regular instrumental developments allowing the acquisition of more and more spectra in a limited time but without compromising their quality. In a way we can say that everything is going well with regard to the acquisition of spectral data in LIBS imaging. Concerning data analysis in LIBS, we see today big differences between the two frameworks of bulk analysis and imaging. Indeed, researchers have quickly learned that multivariate data analysis could bring valuable tools for qualitative and quantitative explorations of samples at the bulk level, for instance by developing regression or classification models as we have seen in chapter one. At the imaging level, there is a relatively limited number of papers dealing with the use of multivariate data analysis in the LIBS community. Indeed, elemental images are, in general, generated from single emission wavelengths, even though the whole spectral domain could be used. The application of chemometric approaches to imaging datasets is in fact more complex, both from a conceptual and practical point of view. Although a large part of the LIBS community is increasingly sensitive to the use of chemometric tools, understanding the concept of hyperspectral imaging, finding appropriate tools for data exploration, and finally interpreting their outputs represent a big task for many non-expert researchers. In addition, it is clear that managing millions of spectra increases the difficulty of this task even if they know the great potential of chemometrics. This is not just about the availability of computational resources, but also, the development of new data exploration tools able to manage such big data structures. Beyond this opinion, which may seem a bit negative, we must consider that there is an evolution of chemometrics within the LIBS community with a real increase of competences, even if the number of publications on the subject is still limited. From this first observation, we could say that everything is going well in the field of LIBS imaging and there are no real obstacles to overcome. So what is the problem? The problem lies in the fact that we want to know everything about the sample we are analyzing, which is quite commendable for a scientist. More precisely, we want to extract information on both major and minor compounds and even traces.

However, minor compounds and traces are often present on a small number of pixels representing a very small variance in the spectral dataset. Unfortunately, the majority of chemometric algorithms exploiting the concept of expressed variance do not (or hardly) allow the detection of these compounds, especially when the signal-to-noise ratio is limited. It is therefore the purpose of this chapter to introduce the IFF algorithm (Interesting features finder) that we have developed in this thesis, the objective of which is to give all compounds a chance to be detected regardless of their concentration [START_REF] Wu | Interesting features finder (IFF): Another way to explore spectroscopic imaging data sets giving minor compounds and traces a chance to express themselves[END_REF]. Table 1S: The list of the 45 selected pixels obtained from the IFF algorithm and their selection frequencies.

III.2. The paper published in SAB

III.3. Conclusion and perspectives

With this chapter, we have demonstrated that the IFF approach allow us to select a short list of interesting pixels (potentially the purest ones) contained in LIBS imaging datasets whatever their explained variance. We thus answer the question posed since we have potentially access to the major and minor constituents and the traces. From an operational point of view, the calculations are done in a very reasonable time (a few minutes for the particle dataset) which is interesting but the calculation is also quite simple. The questions that arise now are how we might value the results of such an approach but also what the prospects might be. There are actually two ways of looking at this. First, if a researcher with little or no experience of chemometrics wishes to exploit the concept, he or she could systematically apply the IFF approach and then look at the spectra that have been selected. Being a specialist in LIBS spectroscopy, he/she will then be able to detect elements that are not expected and perhaps not visible on the mean spectrum of the considered dataset. It would then be possible to select a wavelength of the unsuspected element from which, by integrating the signal, a chemical map could be generated. Second, researchers with chemometrics skills could use this list of selected pixels in different ways but these remain proposals that need to be validated.

This list of pixels should first help to evaluate the rank of the data matrices, i.e. the potential number of pure species present in the system. In the context of signal unmixing (such as MCR-ALS), it could also allow to generate better initial estimates of pure spectral profiles. This selection of pixels could also be seen as a compressed version of the original dataset. As a consequence, this would allow to use algorithms that are difficult to apply to datasets of several hundred thousand or even millions of spectra as other researchers in our team have already done in recent work [START_REF] Nardecchia | Data fusion of LIBS and PIL hyperspectral imaging: Understanding the luminescence phenomenon of a complex mineral sample[END_REF].

GENERAL CONCLUSION

Since its birth, LIBS spectroscopy has come a long way. It is now used as a tool for characterizing complex materials in many industrial fields but also in all scientific fields. It is obvious that the Mars exploration has shed light on this technique, which was not known by all despite its formidable characteristics. Since the 2000s, LIBS imaging has also made a gigantic leap forward as we are now able to generate hyperspectral data cubes of several million spectra acquired in a few hours for a given sample at the micron scale spatial resolution. Over the past decade, LIBS imaging specialists have become aware that chemometrics tools can allow them to explore and exploit their data in different ways than traditional data analysis used on LIBS spectra. The LIBS imaging chemometrics publications now show that we can get a more comprehensive and less biased view of our complex samples. Nevertheless, LIBS imaging has specific characteristics that classical techniques such as vibrational spectroscopy do not have, the phenomenon of photon-matter interaction being from our point of view much more complex. It was clear that some classical chemometrics methods were not fully adapted to these characteristics and it was therefore the objective of this thesis to propose new chemometrics processing methodologies to overcome these obstacles. We therefore addressed two issues of interest to the LIBS community, namely proposing a method capable of making quantitative estimates of elements from the spectra independent of the temperature and electron density of the plasma but also proposing a new data processing strategy giving us a better chance of detecting minor elements and traces in complex samples. In order to answer the first problem, we implemented neural networks trained on simulated LIBS spectral data. This was a very long optimization phase with many pitfalls but we were able to answer the different problems posed throughout this thesis. Of course, this research does not stop here as the proposed neural networks are not yet optimal and their application to real LIBS imaging data needs to be studied further as new constraints have been raised. The second part of this thesis focused on the fact that current chemometric methods are poorly adapted to the detection of minor elements and traces in the LIBS spectra of complex samples. It is important to note that this is one of the strong points of LIBS spectroscopy, which has a very high dynamic range and therefore allows elemental contributions from % to ppm to appear in these spectra. We therefore understand that information on low concentration compounds is present in LIBS spectra but that we do not really have the tools to highlight them in a multivariate analysis approach. We have therefore implemented a strategy based on the mathematical concept of 123 convex hull of the spectral dataset to overcome this problem. This research was also a success as it allowed us to demonstrate that we could detect spectra of some minor particles in an Antarctic sample that had not been detected with conventional chemometrics tools. In conclusion, both aspects of this thesis are undoubtedly hot topics in LIBS imaging. We are therefore convinced that this work will very soon be the beginning of many research projects focused on the improvement of these two concepts or even on their use in more global chemometrics methodologies.

Fig. I- 1

 1 Fig. I-1 The scheme of problems for LIBS model ................................................................... Fig. I-2 Curves of Growth[94] ................................................. Error! Bookmark not defined.

Fig. I- 3

 3 Fig. I-3 Illustration of the different outcomes resulting from choosing the concentration units in weight fraction or mole fraction[94]. ............................................................................................... Fig. I-4 The structure of ANN ................................................................................................. Fig. I-5 The scheme of neuron ................................................................................................. Fig. I-6 Feedback network ....................................................................................................... Fig. I-7 The scheme of backpropagation ................................................................................. Fig. II-1 Influence of the temperature and electron density on the pure elemental spectrum Ca .............................................................................................................................................................. Fig. II-2 The scheme of the training ........................................................................................ Fig. II-3 Neural network No.1 ................................................................................................. Fig. II-4 Results of ANN No.1 ................................................................................................. Fig. II-5 ANN No.2 ..................................................................................................................Fig. II-6 Results of ANN No.2 ................................................................................................. Fig. II-7 ANN No.3 .................................................................................................................. Fig. II-8 Results of ANN No.3 ................................................................................................. Fig. II-9 Another results of ANN No.3 .................................................................................... Fig. II-10 The Sigmoid and tan-sigmoid transfer functions .................................................... Fig. II-11 One example of ANN No.6 ..................................................................................... Fig. II-12 One example of ANN with extra hidden layer ........................................................ Fig. II-13 Another example of ANN with extra hidden layer .................................................. Fig. II-14 Comparison of prediction for ANN No.9 (b) and ANN No.10 (a) .........................Fig. II-15 Regression plot of the best results of predicting only T ..........................................Fig. II-16 Regression plot of the best results of predicting of Ne ............................................

Fig. II- 17

 17 Fig. II-17 Regression plot of the ANN using PCA score with spectral normalization ............

Fig. II- 18

 18 Fig. II-18 The optical image of the mineral sample .................................................................

Fig. II- 19

 19 Fig. II-19 Mask generated by the Matlab toolbox, imageSegementer .....................................

Fig. II- 20

 20 Fig. II-20 Finding the optimal number of components in the filtering process. ...................... Fig. II-21 Comparison of elemental maps generated with ANN and the classical signal integration method (Si and Al) ............................................................................................................

  ABBREVIATIONS AND ACRONYMS

  Fig. I-1 shows the scheme of the problems list above.

Fig. I- 1

 1 Fig. I-1 The scheme of problems for LIBS model

4 𝐵 5 𝑘- 6 4

 456 +,!'-)( = (𝐵) . W 65!0 2𝑇 3 3 4{1 -𝑒𝑥𝑝[-𝑘 . (𝜆)𝑙]} . (𝜆)𝑙 = 𝜏 . (𝜆) I{1 -𝑒𝑥𝑝[-𝑘 . (𝜆)𝑙]}

Fig

  Fig. I-2 Curves of Growth

equations I-15 to I- 17 ,𝑋 ; = 𝑋 9 FFFF + 𝑘𝑠 9 I- 15 𝑋 ; -𝑋 9 FFFF = 𝑘𝑠 9 I- 16 𝑐

 1799159916

  Fig. I-3 shows examples of two metal alloys samples with different concentrations, and equation I-21 can be used to express this plot. 𝑤 = and 𝜒 = are the concentration of element X in the matrix in weight and mole fraction in percentage, respectively. 𝑅 >= stands for the ratio of atomic masses of 𝑀 > and 𝑀 = . 𝜒 = = 𝑤 = 𝑅 >= 1 + 𝑤 = (𝑅 >= -1) I-21 For a single matrix that made up of element A and B with different concentrations, the plot should be look like the one at the up part of Fig. I-3. The dotted vertical line shows the results for different 𝜒 > with the same 𝑤 > . The small inset figure shows the calibration curves of the same two elements with the choice of either the same 𝜒 > or 𝑤 > . The left bottom plot shows the same weight fraction of Cu in Ag and Au will result in different signal intensities and calibration sensitivities. This same weight fraction is different with respect to the mole fractions, while the mole fraction plot simply just shows a single calibration sensitivity. The right bottom plot shows that with the same mole fraction of Cu, we have the same line intensity but only for two specific values of weight fractions. This figureshows that when we talk about the difference of examples as simple as two metal alloys, one should always keep in mind that the differences may not simply come from matrix effects, but also from the unit of the concentration with plays an important role.

Fig. I- 3

 3 Fig. I-3 Illustration of the different outcomes resulting from choosing the concentration units in weight fraction or mole fraction[94].

I. 1 . 12 . 1 . 2 .

 11212 Soft independent modeling of class analog (SIMCA)Soft independent modeling of class analogy (SIMCA) is a pattern recognition method based on PCA. A PCA is calculated on each class of the training set separately. Then statically criteria are used to define the volume containing all the samples of a class. Next, these criteria are used in order to classify unknown samples.

I. 1

 1

  neural network (ANN) was born with the concept of building a network which imitates the behavior of neurons in animal brains to solve problems. Artificial neural networks can learn from dataset to classify, to predict, in a way almost like how a human brain works, learning from experience of means. A typical simple network has an input layer, a hidden layer and an output layer, as shown in Fig. I-4.

Fig. I- 4

 4 Fig. I-4 The structure of ANN

Fig. I- 5

 5 Fig. I-5 The scheme of neuron

Fig. I- 6

 6 Fig. I-6 Feedback network

Fig

  Fig. I-7 The scheme of backpropagation

  Fig. II-1, we have considered in our study 8 levels of temperature and 3 levels of electron density in accordance with experimental conditions likely to be encountered in classical LIBS experiments. For a given plasma temperature and a given electron density, a simulated LIBS spectrum of an element will be referred as a pure spectrum in this work. Fig. II-1 shows the simulated pure spectra of element Ca at varying temperatures and plasma densities.

Fig. II- 1

 1 Fig. II-1 Influence of the temperature and electron density on the pure elemental spectrum Ca

3 .

 3 the performance (mean square error, MSE) reaches zero, 4. reaching the maximum validation checks (6 is the default number, which means the differences in MSE from the last two epochs no longer decreases 6 times in a row, the training no longer improves the MSE), 5. reaching the maximum mu (a parameter of Levenburg-Marquardt backpropagation). The dataset used for training contained 1 000 simulated spectra. It was randomly divided into a training set (75%), a validation set (15%) and a test set (15%) by the toolbox. Fig. II-3 shows an example of the window displayed by our ANN Matlab toolbox at the end of the network learning

Fig. II- 3 1 Fig

 31 Fig. II-3 Neural network No.1

Fig. II- 4

 4 Fig. II-4(d) is a classical representation of the reference values according (target) to the values predicted by the network (output). The dotted line is the first bisector which represents the ideal line where the predictions would be exactly equal to the reference values. We will therefore try to have the points of this figure as close as possible to this line during this work. The colored lines represent the regression lines calculated from the black points of each graph. In other words, they should be as close as possible to the first bisector if the prediction model was perfect. Note that this same figure shows

Fig. II- 4 1 II. 1 . 5

 4115 Fig. II-4 Results of ANN No.1

Fig. II- 5 ANN No. 2 Fig. II- 6

 526 Fig. II-5 ANN No.2

  Each spectrum was divided by the sum of all the values along the 2048 spectral variables. Details about this new network No.3 are given in Fig. II-7. The analysis of the result in Fig. II-8 shows that the data normalization allows to go in the right direction with a MSE significantly smaller (0.00204) although R is merely 0.71.

Fig. II- 7 ANN No. 3 Fig. II- 8

 738 Fig. II-7 ANN No.3

Fig. II- 9

 9 Fig. II-9 Another results of ANN No.3

  another layer. Sigmoid (logsig in Matlab) and tan-sigmoid (tansig in Matlab) are the most common transfer functions used in the hidden layers of a network. Their difference is illustrated in Fig. II-10.

Fig. II- 10 The

 10 Fig. II-10 The Sigmoid and tan-sigmoid transfer functions

  These results show that we are still able to propose satisfactory predictive models. The best learning functions are still the same as previously and the change of transfer function does not allow to obtain a significant improvement in the prediction. Since it is not always clear what lies behind an MSE value or an R, Fig. II-11 shows graphically the results obtained for one of these networks.

Fig. II- 11

 11 Fig. II-11 One example of ANN No.6

  can see on Fig. II-12 and Fig. II-13 and the observation of non-linear effects on the reference values vs. predicted values graphs.

Fig. II- 12

 12 Fig. II-12 One example of ANN with extra hidden layer

  Fig. II-14 provides a visual comparison of the predictive capabilities of three given neural networks (i.e. considered three different combination of elements) when one (b) or two (a) hidden layers are considered in the network.

Fig. II- 14

 14 Fig. II-14 Comparison of prediction for ANN No.9 (b) and ANN No.10 (a)

Fig. II- 15

 15 Fig. II-15 Regression plot of the best results of predicting only T

Fig. II- 17

 17 Fig. II-17 shows how the addition of this normalization has completely disrupted the training of the ANN model.

Fig. II- 17

 17 Fig. II-17 Regression plot of the ANN using PCA score with spectral normalization

  Fig. II-18 provides an optical image of the considered sample. It is approximately 3.4 cm long by 2.8 cm wide and we can easily see the great heterogeneity of the mineral phases. This rock sample was previously embedded in resin and polished prior to LIBS analysis.

Fig. II- 18

 18 Fig. II-18 The optical image of the mineral sample

Fig. II- 20

 20 Fig. II-20 shows the evolution of the SSD value as a function of the number of principal components used during the filtering.

Fig. II- 20

 20 Fig. II-20 Finding the optimal number of components in the filtering process.

  Fig. II-21 to Fig.

Fig. II- 21

 21 Fig. II-21 Comparison of elemental maps generated with ANN and the classical signal integration method (Si and Al)

Fig. III- 1

 1 Fig. III-1 The evolution of LIBS imaging

Figure 1S :

 1S Figure 1S: the seven pure spectra used in the simulated dataset.

Figure 2S :

 2S Figure 2S: an example of a spectrum of a mixture extracted from the dataset containing

Figure 7S :

 7S Figure 7S: Scree plot of eigenvalues.

Figure 8S :

 8S Figure 8S: Localization of selected pixels.

Figure 9S :

 9S Figure 9S: the correlation matrix of the 45 selected spectra from IFF.

Table I

 I

-1 needed.

Table I -1 Examples of basic expressions characterizing the laser beam, its intensity distribution, its propagation, and its focusing behavior[41]

 I

	Expression	Description	Equation number
	𝑃 = # 𝐼 !"#$% (𝑟, 𝜆, 𝜑) 𝑑𝜑𝑑𝜆𝑑𝑟	Light power (Ref[42], p.52)	T1.1
	𝐸 = # 𝐼 !"#$% (𝑟, 𝜆, 𝜑, 𝑡) 𝑑𝜑𝑑𝜆𝑑𝑟𝑑𝑡	Pulse energy (integrated over all distributions)	T1.2
	!"#$% 𝐸 = . 𝑃(𝑡)𝑑𝑡	Power-energy relation of a pulse	T1.3

Table I -2 Definitions related to the ablation process and parameters involved[41]

 I

	Expression	Description	Equation number
	𝜂 (5# = 𝜂 (5# = 𝜂 (5# = 𝜂 (5# = 𝜂 (5# = (𝜂 (5# ) :: = 𝜌𝛿 Φ & 𝜌 𝜇 ( Φ & Φ & -Φ /9 ln ] Φ /9 Φ & ℎ (5# Φ & 𝑉𝑜𝑙𝑢𝑚𝑒 𝑎𝑏𝑙𝑎𝑡𝑒𝑑 ^ 𝑃𝑢𝑙𝑠𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝐶𝑟𝑎𝑡𝑒𝑟 𝑑𝑒𝑝𝑡ℎ 𝑃𝑢𝑙𝑠𝑒 𝑓𝑙𝑢𝑒𝑛𝑐𝑒 = 𝑚 (5# 𝛿 𝑉 (5# Φ & = 𝑚 (5# 𝛿𝑆 # = 𝑆 # ℎ $ 𝑄 # ℎ $ 𝐹 # = ℎ $ 𝑆 # 𝑄 # 𝑆 # 𝛿𝑄 # = 𝑚 (5# 𝑄 # (𝜂 (5# ) ;; = 𝑆 # ℎ $ 𝑄 # = 𝑚 (5# 𝜌𝑄 # 𝜂 (5# = 𝐴𝑡𝑜𝑚𝑠 𝑟𝑒𝑚𝑜𝑣𝑒𝑑 𝑝𝑒𝑟 𝑝𝑢𝑙𝑠𝑒 𝑃ℎ𝑜𝑡𝑜𝑛𝑠 𝑝𝑒𝑟 𝑝𝑢𝑙𝑠𝑒	Ablation efficiency (Ref[48], p. 611) Ablation efficiency (blow-off model) (Ref[48], p.611) Ablation efficiency (steady-state model) (Ref[48], p.611) Ablation efficiency (Ref[49]) Ablation efficiency (Ref[50]) Comparison T2.1 and T2.5 Comparison T2.1 and T2.5 Atom removal efficiency	T2.1 T2.2 T2.3 T2.4 T2.5 T2.6 T2.7 T2.8

  3,

	𝜏 G%#()(/6HI = = (𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑢𝑚𝑛𝑏𝑒𝑟 𝑜𝑓 𝑔𝑎𝑠 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑝𝑒𝑟 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 𝑛𝑒𝑒𝑑𝑒𝑑 𝑡𝑜 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒 𝑒𝑛𝑒𝑟𝑔𝑦) 𝑁 o 𝑣 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑎𝑠 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 𝑝𝑒𝑟 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒)

Table I -3 Most useful and commonly reported equations and definitions in the LIBS plasma diagnostics[41]

 I

			Equation
	Expression	Description	number
	𝑛 T3.1 𝑇(𝑡 + 𝜏 G%# ) -𝑇(𝑡) 𝑇(𝑡) Additional temporal condition for T3.2 ≪ 1 T; 𝜏 G%# =relaxation time 𝑛 % (𝑡 + 𝜏 G%# ) -𝑛 % (𝑡) 𝑛 % (𝑡) Additional temporal condition for T3.3 ≪ 1 𝑛 % 𝑇 % (𝑥) -𝑇 % (𝑥 + 𝜆) 𝑇 % (𝑥) Additional spatial condition for T; T3.4 ≪ 1 𝜆 = diffusion length during 𝜏 G%# 𝑛 % (𝑥) -𝑛 % (𝑥 + 𝜆) 𝑛 % (𝑥) ≪ 1 Additional spatial condition for 𝑛 % T3.5

% ≥ 1.6 × 10 A* (𝑇) A * ⁄ (Δ𝐸) C

McWhirter crierions, 𝑛 % (cm -3 ); T (K); Δ𝐸 (eV) (Refs.

[START_REF] Huddlestone | Plasma diagnostic techniques[END_REF]

[61])

.1.4.2. Equilibrium expressions and diagnostic methods for the evaluation of T and Ne

  𝜋 * 𝜀 & 𝑚 % 𝑐 * z 𝑛

	Δ𝜆 G%$ ≈	3 16	] 𝑔 " 𝑔 #	^A * ⁄	x	𝜆 & C 𝑒 * 𝑓 #"			Resonance of light (ms -1 ); n (m -3 ) (Ref. [77]) interaction broadening, 𝜀 & = vacuum permittivity (C 2 N -1 m -2 ); c = speed	T3.27
	Δ𝜆 ?(I1%GT((#$,Q61/9 = 2.7𝐶 P * U ⁄ 𝑣 C U ⁄ 𝑛 Δ𝜆 ?(I1%GT((#$,$967/ = 0.98𝐶 P * U ⁄ 𝑣 C U ⁄ 𝑛 𝜆 * 𝑐 𝜆 * 𝑐 𝑇 = 𝐸 " # -𝐸 " 𝑘 𝑙𝑛 ] 𝐼 " # # # 𝑔 " 𝐴 "# 𝐼 "# 𝑔 " # 𝐴 " # # # ^Δ𝑇 𝑇 = 𝑘𝑇 ΔE 𝑙𝑛 x 𝐼 "# 2 𝐴 "# 𝑔 " 𝐼 "# 𝐴 "# 2 𝑔 " 2 z = 𝑙𝑛 -[ 2(2𝜋𝑚 % 𝑘) C * ⁄ ℎ C ] x 𝑇 C * ] Δ𝑅 A 𝑅 A ⁄ 𝑛 % z-+ -(𝐸 6HI -Δ𝐸 6HI + 𝐸 " 𝑘𝑇 2 -𝐸 " )	Δ𝑅 < 𝑅 <	^	van der Waals broadening (width),C6 = interaction constant (m 6 s -1 ) (Ref. [77]) van der Waals broadening (shift), v = relative velocity (ms -1 ) (Ref. [77]) Temperature evaluation from the line ratio and associated error (Refs. [65][66][62]) Saha-Boltzmann plot expression 3, p. 132) for evaluating plasma Texc (Refs. [62][78] see also Ref. [67], Chap.	T3.28 T3.29 T3.30 T3.31
	IAn Ideal method could be described as follows[41],
	1. It does not rely upon the assumption of LTE;
	2. It does not require knowledge of the fundamental constants;
	3. It does not necessitate the calibration of the detection system;
	4. It should be relatively easy to implement in ordinary LIBS laboratory.
									to-continuum ratio (note
									that temperatures are kept	T3.21
									different) (Refs. [69][70][71])
	𝑛 % = x Δ𝜆 ;/(G= • 10 E 2.5𝛼 A * ⁄ 𝑛 % = 2(2𝜋𝑚 % 𝑘𝑇) C * z ⁄ ℎ C 𝐼 " 𝐴 "# C * ⁄ = 8.02 • 10 A* x 2 𝑔 " 2 𝐼 " 2 𝐴 "# 𝑔 " 𝑒𝑥𝑝 x-𝐸 6HI + 𝐸 " Δ𝜆 A * ⁄ 𝛼 A * ⁄ C * ⁄ z 2 -𝐸 " 𝑘𝑇 Δ𝜆 Q61/9 = 𝑤 • 𝑛 % 10 AP • [1 + 1.75 • 10 4B 𝑛 % A B ⁄ 𝛼(1 -0.068𝑛 % A P ⁄ 𝑇 4A * ⁄ )] Δ𝜆 $967/ = 𝑤 • 𝑛 % 10 AP • [] 𝑑 𝑤 ^+ 2.0 • 10 4B 𝑛 % A B ⁄ 𝛼(1 -0.068𝑛 % A P ⁄ 𝑇 4A * ⁄ )] 𝛿𝑣 R 𝑣 & = 𝛿𝜆 R 𝜆 & = 7.16 • 10 4S ' 𝑇 𝑀	z	Stark width-𝑛 % relation (linear Stark effect) (Refs. [72][73]) Ion-to-neutral ratio-𝑛 % relation (see Eqs. T3.16, T3.18) (see also Ref. [67], Chap. 3, p. 133) Stark width-𝑛 % relationship (quadratic Stark effect) (Refs. [74][75][76]) (quadratic Stark effect) [Refs. Stark shift-𝑛 % relationship [74](p. 135), [75][76]] Doppler spectral profile, M (Refs. [65][66]) (g/mol) = atomic mass; T (K)	T3.22 T3.23 T3.24 T3.25 T3.26

Table I

 I

	-3 below shows many different methods for LTE
	diagnosis. All these method can be categorized as direct or indirect ones, and also active or passive
	ones by these two papers[79][72], respectively. Direct methods do not require assumptions of type and
	degree of equilibrium within the plasma to know the T and Ne. While the indirect methods require
	certain assumptions for the plasma system. Active methods can use external source to detect the plasma,
	while the passive would only use the plasma itself to measure parameters. Indirect methods require
	certain conditions to be met thus they suffer from the inherent contradiction. However, on the other
	side, these contradictions would allow the system to find certain consistency or a trend. All the
	different equations in Table I-2 and Table I-3 are for different conditions in real LIBS cases, things
	need to be mentioned here are, different temperatures appear in these equations, and extra attention
	should be paid to the unit of the parameters.

I.

1.5 Experiment setup of LIBS

	LIBS showed great potential in many different areas, and has many different sub-categories for
	different purposes. Like Resonant laser ablation (RLA) and resonantly enhanced LIBS (RE-LIBS)

End Simulation: Read concentrations Update Model: One parameter per time Plasma Model : Fluid dynamics Radiative transfer Perturbation theory Simulated Spectrum: Electron number density Ionization states Atomic emission High Correlation?

	Model Parameters:
	Plasma parameters(size and
	velocity), spectral data,
	instrumental parameters
	Yes
	No
	Experimental Input:
	Plasma temperature
	Electron number density
	is perhaps
	very true about this fact, "regardless of the field of application , the acceptability of LIBS is still related
	to the problem of quantitation, involving accuracy, i.e., repeatability and trueness. " . Though LIBS
	lacks of the ability of reproducibility, the great potential of this method cannot be ignored. A technique

Table I -4 Some equations related to the use of the spectral fluctuations approach in LIBS[94]

 I

			Equation
	Expression	Description	number
	𝜎 $"' = ˜[𝜎(𝐵) !,L $%%&!'() ] * + [𝜎(𝑆) I%/,L " ] * + 2𝜃 ™𝜎(𝐵) !,L $%%&!'() š [𝜎(𝑆) I%/,L " ]› A * ⁄	This expression refers to a single isolated contains the correlation coefficient 𝜃 resulting from the sum of the two signals spectral line (signal S) superimposed on a flat background (signal B). The standard deviation	T4.1

Table II -1 Elements, temperature and electron density used in the spectral data generation

 II

	Elements	Temperature (K)	Electron density number (𝑐𝑚 4C )
	Ag, Al, As, Au, Ca,
	Cr, Cu, Fe, Ga, La, Mg, Mn, Mo, Na, Ni, Pb, Si, Sr, Ti, U, Zn,	6000,7000, 8000, 9000, 10 000, 11 000,12000.	5 × 10 AP , 1 × 10 AS , 5 × 10 AS
	Zr.

Table II -2 All the learning function used in this work

 II

	Function name in Matlab	Description	Note
	trainbfg	BFGS quasi-Newton backpropagation.
		Conjugate gradient
	traincgb	backpropagation with Powell-
		Beale restarts.
		Conjugate gradient
	traincgf	backpropagation with Fletcher-
		Reeves updates.
		Conjugate gradient	Backpropagation training
	traincgp	backpropagation with Polak-	functions that use gradient
		Ribiere updates.	derivatives.
	traingd	Gradient descent backpropagation.	Supported on GPU.
	traingda	Gradient descent with adaptive lr backpropagation.
	traingdm	Gradient descent with momentum.
	traingdx	Gradient descent with momentum & adaptive lr backpropagation.
	trainoss	One step secant backpropagation.
	trainrp	RPROP backpropagation.
	trainscg	Scaled conjugate gradient backpropagation.	Default (GPU)
	trainb	Batch training with weight & bias learning rules.
	trainc	Cyclical order weight/bias training.	Supervised weight/bias training
	trainr	Random order weight/bias training.	functions
	trains	Sequential order weight/bias training.
	trainbu	Unsupervised batch training with weight & bias learning rules.	Unsupervised weight/bias training
	trainru	Unsupervised random order weight/bias training.	functions
	trainlm	Levenberg-Marquardt backpropagation.	Default (CPU)
	trainbr	Bayesian Regulation backpropagation

Table II -3 Details about all the ANNs built during the thesis considering the 22 elements to be predicted

 II

	No. Maximum epochs		CPU/GPU	Train	T (K) Validation	Train	Ne	Validation	Elements used in the dataset	Noise	Baseline
	No. 1 1 spectra/sum(spectra) Input layer Pre-processing 1000 2 1000 2 spectra/sum(spectra) 3 spectra/sum(spectra) 3 1000 4 spectra/sum(spectra) 4 1000	Number CPU 2048 CPU 2048 2048 CPU 2048 GPU	Hidden layer Number of neurons Transfer functions varied for dataset 1 -10 tansig varied for dataset -10 tansig 10 tansig varied for dataset -10 tansig varied for dataset -	Output layer Number of neurons varied for dataset Transfer -functions 22 varied for -dataset 22 22 varied for -dataset 24 varied for dataset -	Learning methods 22 trainlm 22 trainbr trainlm 22 trainscg 22	Data size 0.001 0.001 1000 1000 0.001 2000, 5000 5000 0.001	Validation checks none none none none
	5 spectra/sum(spectra) 5 1000 6 5000, 20000 7 1000 8 1000		2048 GPU/CPU GPU GPU GPU	10 10000 10000 10000		logsig ---varied for datasets 2 2,3 1.00E+17 1.00E+17 1.00E+17		---	12: trainscg, traincgb, traincgf, traincgp, traingd, traingda, traingdm, traingdx, trainoss, 2,3(fixed) 2,3(fixed) 2,3(fixed) trainrp, trainlm, trainbr 10: trainscg, traincgb, traincgf, 2,3(fixed)	0.001 1000, 5000, 10000, 0.001 20000,200000 0.001 0.001	none none none none
	6 spectra/sum(spectra) 9 1000 10 1000, 5000		2048 GPU GPU	10		tansig varied for datasets 2,3 varied for datasets			traincgp, traingd, traingda, traingdm, traingdx, trainoss, 2,3(fixed) 3	0.001 5000 0.001	none none
	11 7 spectra/sum(spectra) 1000, 5000 12 5000, 10000		GPU 2048 GPU	20		varied for datasets tansig 2,3 varied for datasets		linear	trainrp 3 traincgf, traincgb, trainscg 3,5,10	0.001 5000, 200000 0.001	none none
	8 spectra/sum(spectra) 13 10000 9 spectra/sum(spectra) 14 10000	2048 CPU PARA 2048 CPU PARA	10,20,25,30 10+10	logsig varied for datasets 2,3 logsig+logsig 2,3 varied for spectra 3			Trainscg (different T and Ne), 3,5,8,10,15,20,22 traincgb (different Ne) trainscg 22	5000, 200000 0.001 0.001,0.005,0.01, 20000 0.02,0.05,0.08,0.1,	none none
	10 spectra/max(spectra)		2048	10,20,30		logsig	3			trainscg	20000 0.2
	11 spectra/max(spectra) 12 spectra/max(spectra) 15 10000	2048 2048 CPU PARA	10+10 10,15,20,30, 40,50	logsig+logsig logsig varied for spectra 3 3,5,10			trainscg trainscg 22	20000 20000 0.001,0.005,0.01, 0.02,0.05,0.08,0.1, 0.2	linear, quadratic curves, mix of both
	13	PCA	24,50,80,100	25,30,35,40, 50,60,80,100	logsig	3,5,8,10,15,2 0,22		trainscg	20000
	14	PCA		100	50		logsig	22			trainscg	20000	20
	15	PCA		100	50		logsig	24			trainscg	20000, 50000	20

Table II -6 Results of ANN No.6 with 2 elements

 II

	Learning methods	Datasize Iterations	MSE	R	Noise	Layer1 FCN	Layer2 FCN	No. elements
	trainscg	5000 20000	73 1000	2.59E-05 0.98984 0.001 tansig purelin 2.87E-06 0.99992
	traincgb	5000 20000	172 356	2.08E-07 0.97693 0.001 tansig purelin 7.11E-06 0.9999
	traincgf	5000 20000	59 423	2.07E-04 0.99309 0.001 tansig purelin 4.13E-06 0.99942
	traincgp	5000 20000	63 270	1.44E-04 0.98851 0.001 tansig purelin 1.35E-05 0.99941
	traingd	5000 20000	1000 1000	2.70E-02 0.81495 0.001 tansig purelin 3.21E-02 0.81868
	traingda	5000 20000	358 209	8.35E-04 0.97229 0.001 tansig purelin 3.64E-03 0.97332
	traingdm	5000 20000	1000 14	2.51E-02 0.83636 0.001 tansig purelin 1.80E-01 0.67192
	traingdx	5000 20000	182 184	3.21E-04 0.98592 0.001 tansig purelin 1.29E-03 0.98981
	trainoss	5000 20000	63 171	3.82E-04 0.98249 0.001 tansig purelin 2.26E-04 0.99712
	trainrp	5000 20000	74 115	3.96E-03 0.93708 0.001 tansig purelin 2.12E-03 0.96161

Table II -7 Results of ANN No.6 with 3 elements

 II

		20000	153	1.54E-02	0.85621
	Learning methods	Datasize	Iterations	MSE	R	Noise	Layer1 FCN	Layer2 FCN	No. elements
	trainscg	5000 20000	82 1000	3.04E-04 4.83E-06	0.98663 0.001 tansig purelin 0.99994
	traincgb	5000 20000	126 470	1.17E-04 2.34E-05	0.98768 0.001 tansig purelin 0.99963
	traincgf	5000 20000	92 505	2.48E-04 1.54E-04	0.99096 0.001 tansig purelin 0.99872
	traincgp	5000 20000	42 139	1.20E-03 1.66E-04	0.98052 0.001 tansig purelin 0.99717
	traingd	5000 20000	1000 1000	1.92E-02 1.92E-02	0.87499 0.001 tansig purelin 0.86751
	traingda	5000 20000	137 138	6.07E-03 6.26E-03	0.94274 0.001 tansig purelin 0.95335
	traingdm	5000 20000	1000 1000	2.71E-02 3.71E-02	0.82758 0.001 tansig purelin 0.77697
	traingdx	5000 20000	177 192	1.22E-03 1.45E-03	0.97709 0.001 tansig purelin 0.98844
	trainoss	5000 20000	68 453	1.78E-03 5.68E-05	0.97685 0.001 tansig purelin 0.99928
	trainrp	5000	90	1.24E-02	0.89319 0.001 tansig purelin

Table II -8 Results of ANN No.7

 II

	trainscg	5000 20000	79 385	1.17E-04 3.47E-05	0.9948 0.9934	0.001 tansig purelin	2
	traincgb	5000 20000	101 173	1.79E-05 0.99645 0.001 tansig purelin 5.82E-05 0.99885	2
	traincgf	5000 20000	141 319	2.11E-05 0.99514 0.001 tansig purelin 2.65E-05 0.99926	2
	trainscg	5000 20000	91 372	4.92E-04 0.99328 0.001 tansig purelin 1.17E-04 0.99818	3
	traincgb	5000 20000	64 395	6.79E-04 0.99177 0.001 tansig purelin 4.75E-05 0.99897	3
	traincgf	5000 20000	127 485	1.27E-04 0.99447 0.001 tansig purelin 5.13E-05 0.99885	3
	Method Datasize Iterations	MSE	R	Noise	Layer1 FCN	Layer2 FCN	No. elements

Table II -9 ANN No.8 with fixed T

 II

		20000	1000	5.44E-06	0.99996	5.00E+16 0.99996 5.00E+16 5000
	traincgb	5000 20000	105 135	1.75E-06 9.56E-05	0.99602 0.99865	5.00E+17 0.97862 1.00E+17 5000 5.00E+17 0.9969 5.00E+17 5000	0.001 logsig purelin	2
	traincgb	5000 20000	92 856	1.80E-04 5.61E-05	0.9924 0.99983	5.00E+16 0.97684 1.00E+17 5000 1.00E+17 0.99953 1.00E+17 5000	0.001 logsig purelin	3
	Learning methods	Datasize Iterations	MSE	R	Ne for calibration	Validation R	Ne for validation	Validation datasize	Noise	Layer1 FCN	Layer2 FCN	No. elements
	trainscg	5000 20000	65 791	1.36E-04 1.65E-05	0.9943 0.9996	5.00E+17 5.00E+17				0.001 logsig purelin	2
	trainscg	5000 20000	72 1000	2.84E-04 5.56E-06	0.99367 0.9999	5.00E+16 5.00E+17 0.93195		5000	0.001 logsig purelin	3

Table II -10 ANN No.8 with fixed Ne

 II

	Learning methods	Datasize Iterations	MSE	R	T for calibration	Validation R	T for Validation	Validation datasize	Noise	Layer1 FCN	Layer2 FCN	No. elements
	trainscg	5000 20000	51 423	2.37E-04 5.21E-06	0.99322 11000 0.97166 12000 0.99941 6000 0.97273 10000	5000 5000	0.001 logsig purelin	2
		5000	46	7.39E-04	0.98946 9000	0.98074	9000	5000
	trainscg	20000	1000	4.32E-06	0.99981 11000 0.58782	6000	5000	0.001 logsig purelin	3
		200000	743	6.71E-06	0.99983 9000	0.63743	6000	5000

Table II -11 Three different hidden neurons values with different T and Ne

 II

		20 neurons	25 neurons	30 neurons
	R Train (T: 11 000, Ne: 5e17)	0.99968	0.99949	0.99907
	R Validation (T: 9 000, Ne: 1e17)	0.9692	0.97049	0.9648
	R Train (T: 9 000, Ne: 5e16)	0.99797	0.99792	0.99863
	R Validation (T: 8 000, Ne: 5e16)	0.93946	0.93865	0.9399

Table II -12 Results of ANN No.9

 II

	447	11000, 5e17 9000, 5e17 2.69E-04	1.40E-02 0.99820 0.91703	30	Au	Al	Sr	5000
	413	11000, 1e17 9000, 1e17 1.86E-02	1.05E-01 0.86662 0.42461	30	Mg	Ag	Mo	5000
	2121 11000, 5e16 9000, 1e17 1.89E-05	2.87E-02 0.99987 0.79994	30	As	Fe	Al	5000
	993	11000, 5e17 9000, 1e17 2.77E-02	1.16E-01 0.79173 0.53309	30	Mg	Na	Fe	5000
	Iterations	Training set plasma parameters	Validation set plasma parameters	MSE training	MSE validation	R training	R validation	Nbr. hydden neurons	Elements considered in the output layer	Epochs
	161	11000, 5e17 9000, 1e17 1.90E-03	8.75E-01 0.98693 0.01840	10	As	Pb	Ga	1000
	1000	8000, 5e17 10000, 5e17 1.41E-04	2.80E-02 0.99905 0.88610	10	Ti	As	Si	1000
	2000	9000, 5e16 6000, 1e17 4.76E-05	1.89E-01 0.99968 0.17017	10	Mn	Ti	Ag	2000
	3597 10000, 1e17 7000, 5e17 1.50E-05	1.27E-01 0.99990 0.79216	10	Ti	Si	Al	5000
	639	6000, 1e17 7000, 5e16 2.20E-04	1.00E-01 0.99852 0.17039	20	Ca	As	Ti	5000
	243	11000, 5e17 6000, 5e16 3.50E-03	2.44E-01 0.97621 0.10943	20	Na	As	Sr	5000
	353	11000, 1e17 6000, 5e16 2.90E-03	1.70E-01 0.97996 0.23552	20	Al	Na	As	5000
	1376 11000, 1e17 9000, 1e17 3.08E-05	5.40E-02 0.99979 0.71034	20	Mo	Al	U	5000
	154	11000, 5e17 9000, 5e16 4.95E-02	1.32E-01 0.57972 0.14083	20	Mg	Na	La	5000
	1005 11000, 1e17 9000, 1e17 7.91E-05	5.21E-02 0.99946 0.73123	20	Mo	La	Cu	5000
	942	11000, 5e16 9000, 5e16 1.59E-04	8.60E-03 0.99892 0.94074	20	Si	Al	Cu	5000
	1306 11000, 1e17 9000, 5e17 3.23E-05	4.74E-02 0.99978 0.61175	20	Fe	La	Si	5000
	884	11000, 5e16 9000, 1e17 6.10E-05	4.56E-02 0.99959 0.66066	20	Ga	La	Fe	5000
	713	11000, 5e16 9000, 5e17 4.89E-04	1.01E-01 0.99667 0.60337	20	Ni	Ag	Ti	5000
	2842 11000, 5e16 9000, 5e16 1.72E-05	8.18E-02 0.99988 0.78043	20	As	Mo	Ag	5000
	649	11000, 5e16 9000, 1e17 8.30E-05	7.12E-02 0.99944 0.52649	20	La	Au	Fe	5000
	182	11000, 5e17 9000, 1e17 6.40E-03	1.74E-01 0.95604 0.53477	20	Mg	Au	Ag	5000
	3854 11000, 5e16 9000, 5e17 5.01E-06	1.87E-01 0.99997 0.21059	20	Cr	Cu	Ca	5000
	387	11000, 5e17 9000, 1e17 1.60E-03	5.21E-02 0.98935 0.83669	20	Na	Ga	Pb	5000
	227	11000, 1e17 9000, 5e16 2.20E-03	2.87E-02 0.98502 0.79473	20	Cu	Pb	Ga	5000
	260	11000, 1e17 9000, 5e17 1.25E-02	2.04E-01 0.91328 0.05646	20	As	Mg	Sr	5000
	575	11000, 1e17 9000, 5e17 2.02E-02	5.86E-01 0.85282 -0.02432	20	La	Mo	Mg	5000
	130	11000, 1e17 9000, 5e17 4.64E-02	5.16E-01 0.61053 0.32034	30	Na	Fe	Mg	5000

Table II -13 Results of ANN No.10

 II

	666	8000, 1e17 12000, 1e17	2.34E-06	3.15E-02	0.99998 0.80501	10+10	Au	La	Sr	1000
	359	12000, 5e17	11000, 5e17	1.68E-02	1.72E-02	0.87937 0.88362	10+10	U	Cr	Na	5000
	1663 8000, 5e16 6000, 5e16	1.28E-04	1.61E-01	0.99914 0.49702	10+10	La	Fe	Mg	5000
	3601	12000, 5e16	6000, 5e16	1.43E-06	2.53E-01	0.99999 0.12979	10+10	U	Al	Ca	5000
	1148 8000, 5e17 8000, 5e17	1.06E-04	1.06E-04	0.99929 0.99927	10+10	Mn Mg	Si	5000
	350	9000, 5e16 6000, 5e16	3.97E-04	5.26E-02	0.99730 0.69429	10+10	Sr	Cu	Ni	5000
	673	9000, 5e16 10000, 5e17	8.55E-06	1.07E-02	0.99994 0.92941	10+10	La	Au	Pb	5000
	910	11000, 1e17	9000, 5e16	1.01E-04	1.32E-02	0.99932 0.96285	10+10	Ti	Au	Pb	5000
	Iterations Training set	Validation set	MSE training	MSE validation	R training	R validation	Hidden neurons	Elements in the output layer considered	Epochs
	1000 7000, 1e17 9000, 5e17	6.88E-06	1.10E-03	0.99995 0.99575	10+10	Fe	U	La
	385	9000, 5e16 10000, 1e17	1.70E-03	1.90E-03	0.98870 0.98724	10+10	La	Mg Ca
	406	9000, 1e17 7000, 5e17	4.80E-03	7.96E-02	0.96679 0.80059	10+10	Sr	Mg	Si
	1000 9000, 5e17 11000, 5e16	5.52E-05	8.43E-02	0.99963 0.59888	10+10	La	As	Na
	1000	11000, 5e17	7000, 5e16	8.18E-06	6.99E-05	0.99994 0.99956	10+10	Ni	Pb	Al
	384	8000, 5e17 9000, 5e16	1.76E-04	5.27E-02	0.99881 0.70382	10+10	Ca	Ag	Pb
	1000 9000, 5e16 10000, 5e16	2.08E-05	1.14E-02	0.99986 0.93495	10+10	Na	Sr	Au
	263	7000, 5e17 11000, 5e16	2.62E-04	1.06E-01	0.99823 0.37254	10+10	Mn	U	Ca
	1000 6000, 1e17 9000, 5e17	7.91E-06	1.45E-02	0.99995 0.90759	10+10	Ga	Au	Al
	378	7000, 5e17 7000, 5e16	1.74E-04	2.09E-02	0.99882 0.85614	10+10	Cr	Mn	La
	1000	10000, 1e17	7000, 5e16	5.09E-06	2.34E-02	0.99997 0.82909	10+10	U	As	Sr
	1000	12000, 5e16	9000, 5e16	7.49E-06	3.29E-02	0.99995 0.79150	10+10	U	La	As
	1000 7000, 5e16 12000, 1e17	2.75E-04	4.12E-02	0.99815 0.77235	10+10	Mg	Sr	Ga
	1000 7000, 5e16 6000, 5e17	5.56E-06	2.53E-02	0.99996 0.87310	10+10	Au	U	Fe
	1000	11000, 5e16	12000, 1e17	1.61E-06	1.02E-04	0.99999 0.99947	10+10	Al	La	Si
	534	6000, 1e17 6000, 1e17	2.54E-05	2.50E-05	0.99983 0.99983	10+10	Ga	Ca	Ti
	597	11000, 5e16	12000, 1e17	1.50E-03	2.62E-02	0.98964 0.91238	10+10	Mn Na	La
	898	7000, 5e17 6000, 5e17	4.22E-06	1.00E-02	0.99997 0.93400	10+10	Cr	Na	Sr

Table II -14 Results of using PCA for inputs

 II

	Iterations	Input neurons	Hidden neurons	Number of elements to be predicted	MSE	Datasize	R	R for test	R for validation	Noise
	4559	24	25	3	6.20E-07	20000	1	1	1
	6960	24	25	8	4.38E-04	20000	0.99532	0.995	0.99504
	6319	24	25	8	1.99E-04	20000	0.99768 0.99778	0.99779
	4754	24	25	10	1.00E-03	20000	0.98727	0.9881	0.98873
	6408	24	30	10	5.22E-04	20000	0.99347 0.99338	0.99355
	10000	24	30	10	3.00E-04	20000	0.99626 0.99818	0.99622
	5570	24	35	10	5.31E-04	20000	0.9934	0.99318	0.9933
	5888	24	35	10	5.43E-04	20000	0.99311 0.99274	0.9927
	5908	24	40	10	4.32E-04	20000	0.99458 0.99434	0.9944
	6137	24	40	10	6.08E-04	20000	0.99237	0.9919	0.9919
	6089	24	50	10	5.93E-04	20000	0.99259 0.99286	0.9921
	9293	24	50	10	2.43E-04	20000	0.99696	0.997	0.9964
	9227	50	50	10	4.75E-05	20000	0.99941	0.9994	0.998
	6394	50	25	10	8.26E-05	20000	0.99896	0.999	0.8329
	4664	50	25	10	1.10E-04	20000	0.99862	0.9986	0.9611(no noise)	0.964
	6221	50	50	10	2.14E-04	20000	0.99732 0.99713	0.9972(no noise)	0.9971
	1792	80	50	10	6.56E-04	20000	0.99178	0.9909	0.9924
	5519	80	50	10	1.05E-04	20000	0.99868	0.9986	0.9988
	9651	80	50	15	6.18E-05	20000	0.99893	0.9988	0.999
	8626	80	50	15	1.26E-04	20000	0.99783	0.9977	0.9977
	9318	80	50	20	2.56E-04	20000	0.9943	0.9939	0.9946
	10000	80	50	22	2.86E-04	20000	0.99305	0.9926	0.9932
	10000	80	50	22	2.40E-04	20000	0.99417	0.9939	0.9946
	3135	80	30	22	8.70E-04	20000	0.97877	0.9784	0.9795
	6243	80	30	22	6.39E-04	20000	0.98439	0.9837	0.9847
	8902	80	60	22	2.17E-04	20000	0.99475	0.9942	0.9949
	10000	80	60	22	2.42E-04	20000	0.99413	0.9934	0.9941
	8284	80	80	22	2.08E-04	20000	0.99495	0.9941	0.9947
	10000	80	100	22	1.82E-04	20000	0.99563	0.9947	0.9959
	7374	100	60	22	2.31E-04	20000	0.99437	0.9931	0.9943
	10000	100	60	22	1.84E-04	20000	0.99556	0.995	0.9956

Table II -15 Results of finding best inputs

 II

	Inputs	Neurons	No. elements	MSE	R training	R for test	R for validation
	100			2.67E-04	0.99353	0.99323	0.99371
	120			2.04E-04	0.99506	0.99324	0.99513
	140			2.31E-04	0.99440	0.99344	0.99488
	150	50	22	2.30E-04	0.99444	0.99303	0.99504
	200			1.48E-04	0.99641	0.99581	0.99753
	250			2.69E-04	0.99348	0.99319	0.99500
	300			2.34E-04	0.99435	0.99328	0.99656

Table II -16 Results of different noise

 II

	Inputs	Neurons	Noise	MSE	R	R for test	R for validation
			0.001	2.72E-04	0.99339	0.99280	0.99284
			0.005	3.01E-04	0.99269	0.99185	0.99003
			0.01	4.82E-04	0.98626	0.98661	0.98519
			0.02	5.13E-04	0.98758	0.98150	0.98473
	100	50
			0.05	5.75E-04	0.98603	0.98415	0.98151
			0.08	8.45E-04	0.97916	0.97869	0.97892
			0.1	8.14E-04	0.98008	0.97570	0.97256
			0.2	1.05E-03	0.97408	0.97003	0.96788
	II.1

.12 Could we predict the temperature from a spectrum?

Table II -17 Predicting only T with 50 hidden neurons

 II

	Input neurons	Noise level	Datasize	Hidden neurons	MSE	R
	100				0.0026	0.95125
	150				0.0016	0.97013
	200				0.0021	0.96261
	250	0.001	20000	50	0.0018	0.96703
	300				0.0034	0.93942
	500				0.0026	0.95212
	2048				0.0070	0.86241

Table II -18 Predicting only T with 10 hidden neurons

 II By way of illustration, Fig. II-15 shows the regression plot (i.e. target vs predicted value) of the best network.

	Input neurons	Noise level	Datasize	Hidden neurons	MSE	R
	50				0.0043	0.91807
	100				0.0009	0.98311
	150	0.001	20000	10	0.0024	0.95685
	250				0.0024	0.95743

Table II -19 Results of predict only Ne with 100 pcs

 II

	Inputs	Neurons	Output	Datasize	MSE	R
		2			0.0325	0.89379
		5			0.0322	0.89603
		10			0.0323	0.89483
	100	20	1	20000	0.0378	0.87617
		50			0.0380	0.87530
		80			0.0418	0.86250
		100			0.0376	0.87656

Table II -20 Influence of the hidden neurons number on the prediction results

 II

	Input neurons	Datasize	Output neurons	Hidden neurons	MSE	R	R for test
				20	4.50E-03	0.95526	0.95569
				30	1.63E-03	0.98412	0.98344
	100	20000	24 (22 elements, T, Ne)	40 50 80 100	1.13E-03 1.03E-03 2.00E-03 1.04E-03	0.98922 0.98991 0.98054 0.98984	0.98686 0.98856 0.98029 0.98814
				120	1.11E-03	0.98929	0.98678
				150	1.92E-03	0.98109	0.98008

Table II -21 Influence of the number of principal components

 II

	Input neurons	Datasize	Output neurons	Hidden neurons	MSE	R	R for test
	100		24		1.04E-03	0.98984	0.98814
	120		24		7.70E-04	0.99254	0.98780
	150	20000	24	50	1.10E-03	0.98912	0.98819
	200		24		9.12E-04	0.99115	0.98960
	II.1.

15 Last but not least, a pitfall but…

  Research on a complex subject is never a smooth river and just when we think we have reached our goal, a new problem often arises that we need to address. A discussion with our collaborators first revealed that the list of 22 elements we had been using from the beginning was not 100% compatible with all the elements of interest that would potentially be present in the mineral sample we were to analyze at the end of the modelling part. We therefore removed the elements As, Au and Sr and added the elements Be, Ce, Eu, Ge and Y. So now, we had a list of 24 elements. Of course, this should not have been a problem for the previous method we had optimized, except for the fact that it was necessary to redo a new PCA analysis and then train a new network with these 24 new elements.

Table II -22 Results of 24 ANNs using normalized spectral data

 II

	Element	RMSE training	R training	RMSE validation	R validation
	Ag	0.0213	0.9900	0.0317	0.9770
	Al	0.0219	0.9896	0.0274	0.9820
	Be	0.0382	0.9625	0.0391	0.9636
	Ca	0.0189	0.9944	0.0221	0.9879
	Ce	0.0400	0.9593	0.0431	0.9499
	Cr	0.0208	0.9910	0.0216	0.9875
	Cu	0.0147	0.9958	0.0184	0.9920
	Eu	0.0358	0.9714	0.0399	0.9563
	Fe	0.0280	0.9819	0.0298	0.9775
	Ga	0.0203	0.9903	0.0248	0.9838
	Ge	0.0393	0.9626	0.0395	0.9526
	La	0.0373	0.9636	0.0411	0.9565
	Mg	0.0194	0.9918	0.0215	0.9892
	Mn	0.0276	0.9812	0.0289	0.9759
	Mo	0.0208	0.9907	0.0255	0.9823
	Na	0.3700	0.9655	0.0400	0.9560
	Ni	0.0191	0.9939	0.0260	0.9814
	Pb	0.0366	0.9727	0.0320	0.9717
	Si	0.0225	0.9880	0.0301	0.9774
	Ti	0.0288	0.9902	0.0297	0.9769
	U	0.0249	0.9860	0.0286	0.9776
	Y	0.0204	0.9914	0.0244	0.9850
	Zn	0.0301	0.9797	0.0336	0.9670
	Zr	0.0168	0.9942	0.0214	0.9889

  -23 Table II-23. The percentage of predicted values

Table II -23 Range of values predicted by neural networks for each element

 II

				Percentage of
	Element	Min. predicted value	Max. predicted value	predicted values between values 0
				and 1
	Ag	-0.0128	0.0160	11.95%
	Al	-0.0384	0.3034	88.67%
	Be	-0.0035	0.0040	40.53%
	Ca	-0.0713	0.2594	46.90%
	Ce	0.1696	0.5080	100.00%
	Cr	0.0058	0.1090	100.00%
	Cu	-0.0081	0.0194	52.11%
	Eu	0.0088	0.2154	100.00%
	Fe	0.0255	0.4856	100.00%
	Ga	-0.0100	0.0258	69.48%
	Ge	-0.0058	0.0582	99.95%
	La	-0.0281	0.4056	99.98%
	Mg	-0.0358	0.0228	70.90%
	Mn	-0.0085	0.0531	54.66%
	Mo	0.0050	0.2302	100.00%
	Na	-0.0143	0.0707	99.94%
	Ni	-0.0320	0.1636	72.68%
	Pb	0.0675	0.2129	100.00%
	Si	0.0034	0.6663	100.00%
	Ti	-0.0096	0.2143	96.73%
	U	-0.0065	0.0698	99.81%
	Y	-0.0014	0.1432	99.72%
	Zn	-0.1682	0.0575	2.04%
	Zr	-0.0277	0.1248	7.21%

Figure 3S: representation of the pixels selected by ESP vs the rest of the dataset in a PCA space considering a 4% noise level.

Figure 4S: representation of the pixels selected by IFF vs the rest of the dataset in a PCA space considering a 4% noise level.

Figure 5S: representation of the pixels selected by ESP vs the rest of the dataset in a PCA space considering a 8% noise level.

Figure 6S: representation of the pixels selected by IFF vs the rest of the dataset in a PCA space considering a 8% noise level.