Bingzhi Liu

Shijun Dong

Janney Debleza

Weiye Chen

Qiang Xu

Hong Wang
email: zhdwang@ustc.edu.cn

Jérémy Bourgalais

Olivier Herbinet

Henry J Curran

Frédérique Battin-Leclerc

Zhandong Wang

Experimental and Updated Kinetic Modeling Study of Neopentane Low Temperature Oxidation

Keywords: neopentane oxidation, kinetic modeling, Korcek mechanism, carboxylic acids, synchrotron vacuum ultraviolet photoionization mass spectrometry

Neopentane is an ideal fuel model to study low temperature oxidation chemistry. The significant discrepancies between experimental data and simulations using the existing neopentane models indicate that an updated study of neopentane oxidation is needed. In this work, neopentane oxidation experiments are carried out using two jet-stirred reactors (JSRs) at 1 atm, at a residence time of 3 s, and at three different equivalence ratios of 0.5, 0.9, and 1.62. Two different analytical methods (synchrotron vacuum ultraviolet photoionization mass spectrometry and gas chromatography) were used to investigate the species distributions. Numerous oxidation intermediates were detected and quantified, including acetone, 3,3-dimethyloxetane, methacrolein, isobutene, 2-methylpropanal, isobutyric acid, and the peroxides, which are valuable for validating the kinetic model describing neopentane oxidation. In the model development, the pressure dependence of the reaction classes of Q ̇OOH + O2 and Q ̇OOH decompositions are considered, which improves the prediction of the low temperature oxidation reactivity of neopentane. Another focus of model development is to improve the prediction of carboxylic acids during the low temperature oxidation of neopentane. The detection and identification of iso-butyric acid indicates the existence of the Korcek mechanism during neopentane

Introduction

Considering energy resources, combustion efficiency, and the strict control of pollutant emissions, further exploration of combustion chemistry and the development of advanced combustion technology is of contemporary significance. [START_REF] Sun | The effect of ozone addition on combustion: Kinetics and dynamics[END_REF] To achieve cleaner combustion, new engine strategies [START_REF] Manente | Gasoline partially premixed combustion, the future of internal combustion engines?[END_REF][START_REF] Reitz | Review of high efficiency and clean reactivity controlled compression ignition (RCCI) combustion in internal combustion engines[END_REF][START_REF] Yao | Progress and recent trends in homogeneous charge compression ignition (HCCI) engines[END_REF][START_REF] Agarwal | challenges and path forward for low temperature combustion engines[END_REF] are developing rapidly. In particular, low temperature combustion (LTC) has recently attracted great interest. [START_REF] Reitz | Review of high efficiency and clean reactivity controlled compression ignition (RCCI) combustion in internal combustion engines[END_REF][START_REF] Yao | Progress and recent trends in homogeneous charge compression ignition (HCCI) engines[END_REF][START_REF] Agarwal | challenges and path forward for low temperature combustion engines[END_REF] LTC is a combustion strategy for advanced internal combustion engines (ICEs), including homogeneous charge compression ignition (HCCI), [START_REF] Yao | Progress and recent trends in homogeneous charge compression ignition (HCCI) engines[END_REF] reactivity-controlled compression ignition (RCCI), [START_REF] Reitz | Review of high efficiency and clean reactivity controlled compression ignition (RCCI) combustion in internal combustion engines[END_REF] and premixed charge compression ignition (PCCI), [START_REF] Jia | The effect of injection timing and intake valve close timing on performance and emissions of diesel PCCI engine with a full engine cycle CFD simulation[END_REF][START_REF] Jia | Numerical evaluation of the potential of late intake valve closing strategy for diesel PCCI (premixed charge compression ignition) engine in a wide speed and load range[END_REF] which has great advantages in simultaneously reducing pollutant emissions and improving combustion efficiency. However, LTC is currently facing many design and research challenges that require the support of numerical calculations and computational fluid dynamics (CFD) simulations. [START_REF] Agarwal | challenges and path forward for low temperature combustion engines[END_REF] Consequently, further studies on fuel chemistry and establishing precise detailed combustion kinetic models is one of the prerequisites.

The low temperature oxidation of large alkanes is governed by successive radical additions to O2 and intramolecular H-atom migration reactions. The reaction scheme leads to many oxygenated species and makes the low temperature oxidation reaction networks particularly complex, especially when the mechanism of the 3 rd addition to O2 is explored and added. [START_REF] Wang | Unraveling the structure and chemical mechanisms of highly oxygenated intermediates in oxidation of organic compounds[END_REF][START_REF] Wang | Exploring hydroperoxides in combustion: History, recent advances and perspectives[END_REF][START_REF] Wang | New insights into the low-temperature oxidation of 2-methylhexane[END_REF][START_REF] Wang | Additional chain-branching pathways in the low-temperature oxidation of branched alkanes[END_REF][START_REF] Wang | Third O2 addition reactions promote the low-temperature auto-ignition of n-alkanes[END_REF] Neopentane (2,2-dimethyl propane) is one of the unique representatives of branched alkanes with a symmetrical molecular structure and only identical H-atoms and thus gives a single alkyl radical by H-atom abstraction reactions. This leads to a more concise low temperature oxidation network. Accordingly, neopentane is an ideal model fuel to study low temperature oxidation chemistry and to develop detailed kinetic models of alkanes. Some early work [START_REF] Baker | Addition of neopentane to slowly reacting mixtures of H2 + O2 at 480°C. Part II. The addition of the primary products from neopentane, and the rate constants for H and OH attack on neopentane[END_REF][START_REF] Curran | An intermediate temperature modeling study of the combustion of neopentane[END_REF][START_REF] Wang | A flow reactor study of neopentane oxidation at 8 atmospheres: Experiments and modeling[END_REF][START_REF] Dagaut | Oxidation of neopentane in a jet-stirred reactor from 1 to 10 atm: an experimental and detailed kinetic modeling study[END_REF] conducted preliminary explorations on the reaction mechanism of neopentane. Recently, Bugler et al. [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF][START_REF] Bugler | Revisiting the Kinetics and Thermodynamics of the Low-Temperature Oxidation Pathways of Alkanes: A Case Study of the Three Pentane Isomers[END_REF] provided the latest ignition delay time (IDT) results of the pentane isomers and proposed a detailed model in which the thermodynamic data were systematically updated and consistent rate rules were used for the important reaction classes. Eskola et al. [START_REF] Eskola | Time-resolved measurements of product formation in the low-temperature (550-675 K) oxidation of neopentane: a probe to investigate chain-branching mechanism[END_REF] used synchrotron vacuum ultraviolet photoionization mass spectrometry (SVUV-PIMS) combined with a flow reactor to explore the oxidation products of neopentane in the temperature range of 550-675 K; the potential intermediates from keto-hydroperoxide (KHP) decomposition were discussed. Hansen et al. [START_REF] Hansen | The impact of the third O2 addition reaction network on ignition delay times of neo-pentane[END_REF] conducted low temperature oxidation experiments of neopentane using a jet-stirred reactor (JSR) and a time-resolved flow reactor. KHP and keto-dihydroperoxide (KDHP) species were detected. They developed a kinetic model including the 3 rd O2 addition reaction network based on the model of Bugler et al. [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF][START_REF] Bugler | Revisiting the Kinetics and Thermodynamics of the Low-Temperature Oxidation Pathways of Alkanes: A Case Study of the Three Pentane Isomers[END_REF] to explore the effect of the 3 rd O2 addition reactions on IDTs. Very recently, neopentane oxidation in JSRs was studied at 1 atm in the temperature range of 500-850 K and φ = 0.5 by Bourgalais et al., [START_REF] Bourgalais | Jet-Stirred Reactor Study of Low-Temperature Neopentane Oxidation: A Combined Theoretical, Chromatographic, Mass Spectrometric, and PEPICO Analysis[END_REF] and the products were analyzed via a combined theoretical, chromatographic, mass spectrometric, and photoelectron photoion coincidence spectroscopy (PEPICO) study. Although the current neopentane kinetic models in the literature have been validated extensively, discrepancies between the measured and predicted neopentane reactivity, as well as mole fractions of many important species still exist, such as 2-methylpropanal ((CH3)2CHCH=O), 3,3-dimethyloxetane (CH3C(CH2OCH2)CH3), acetone (CH3COCH3), isobutene ((CH3)2C=CH2), and methacrolein (CH3C(=CH2)CH=O). The new JSR experimental data and the corresponding analyses in the work of Bourgalais et al. [START_REF] Bourgalais | Jet-Stirred Reactor Study of Low-Temperature Neopentane Oxidation: A Combined Theoretical, Chromatographic, Mass Spectrometric, and PEPICO Analysis[END_REF] show that the existing neopentane models need to be improved. These latest works 17- 21 provide a starting point and valuable references for the further extension and improvement of the neopentane kinetic model.

In this work, SVUV-PIMS combined with gas chromatography (GC) was used to detect and quantify the key intermediates and final products of neopentane oxidation in JSRs at 1 atm and at equivalence ratios of 0.5, 0.9, and 1.62. Important oxidation intermediates, e.g., acetone, 3,3dimethyloxetane, methacrolein, isobutene, 2-methylpropanal, and iso-butyric acid ((CH3)2CHC(=O)OH), were identified and quantified, which are crucial to clarify the low temperature oxidation mechanism of neopentane and to refine and develop the kinetic model. Next, the rate constants of key reaction classes in the neopentane submechanism, e.g., the reaction classes of Q ̇OOH + O2 and Q ̇OOH decompositions, were re-examined and evaluated based on new JSR experimental data and available studies in the literature. Furthermore, carboxylic acid formation chemistry was revisited. The Korcek mechanism was added, and the rate constants for the reactions of ȮH radical addition to acetaldehyde were updated. The reaction pathways of ȮH radical addition to acetone and subsequent decomposition were proposed to better predict the acetic acid mole fraction distribution.

This updated model satisfactorily predicts current experimental data and various experimental data available in the literature.

Experimental method

In this work, we carried out neopentane oxidation experiments in two JSRs using the analytical methods of SVUV-PIMS and GC. The results obtained by different analytic methods are mutually verified and supplemented. The diluent gas used in the JSR-SVUV-PIMS experiment was argon (Ar), while the diluent gas used in the JSR-GC experiment was helium (He). The influence of different dilution gases on the experimental results is negligible. The initial concentration of neopentane in all experiments was 0.015, and the residence time was 3 s. The detailed JSR experimental conditions are summarized in Table 1.

The JSR experiments based on SVUV-PIMS

The first experimental work was carried out in NSRL (Hefei, China). The specific description of this experimental platform can be found in recent work. [START_REF] Xu | Comprehensive study of the lowtemperature oxidation chemistry by synchrotron photoionization mass spectrometry and gas chromatography[END_REF] Gaseous neopentane (purity 99.8%) is mixed with diluent gas (Ar) and oxygen and enters the JSR. There are four nozzles with a diameter of 0.3 mm in the sphere of the JSR. The gas mixture forms jets through the four nozzles and is evenly mixed in the reactor. The mixture is sampled by the tapered quartz nozzle that is welded on the JSR sphere to form a molecular beam, which is ionized by the SVUV-PIMS. The mass resolution (m/Δm) of SVUV-PIMS was ~4000 at m/z 72 in this experiment. Methods of species identification and quantification can be found in previous work. [START_REF] Herbinet | Detailed product analysis during the low temperature oxidation of n-butane[END_REF] The photoionization cross-sections (PICSs) are obtained from an online database, [START_REF]Photonionization cross section database (Version 2.0[END_REF] available literature data, [START_REF] Dodson | VUV photoionization cross sections of HO2, H2O2, and H2CO[END_REF][START_REF] Moshammer | Quantification of the Keto-Hydroperoxide (HOOCH2OCHO) and Other Elusive Intermediates during Low-Temperature Oxidation of Dimethyl Ether[END_REF][START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF] and experimental measurements in this work. The uncertainties of the measured mole fractions are within ± 10% for neopentane and oxygen, ± 20% for species with known PICSs, and a factor of two for those with estimated PICSs.

The JSR experiments based on GC

JSR experiments under the same conditions were also carried out in LRGP (Nancy, France) using an analytical method based on online GC-FID/GC-MS. A detailed description of the experimental setup can be found in previous publications. [START_REF] Herbinet | Detailed product analysis during the low temperature oxidation of n-butane[END_REF][START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF][START_REF] Wang | Experimental investigation of the low temperature oxidation of the five isomers of hexane[END_REF] The outlet of the JSR is connected to two GCs through a heated transfer line. [START_REF] Herbinet | Detailed product analysis during the low temperature oxidation of n-butane[END_REF][START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF] The first GC is equipped with a Carbosphere packed column and a thermal conductivity detector (TCD) that allows the acquisition of the mole fraction distributions of H2, O2, CO, and CO2. The second GC is equipped with a PlotQ capillary column and a flame ionization detector (FID) to quantify molecules ranging from methane to reaction products containing up to five carbon atoms and one or two oxygen atoms. [START_REF] Herbinet | Detailed product analysis during the low temperature oxidation of n-butane[END_REF][START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF] The mass spectrometer (MS) analyser was adopted for the identification of reaction products according to the NIST Mass Spectrometry database. For O2, CH4, CO, CO2, and some common hydrocarbons, standard gases were used for TCD and FID calibration. For other species detected by FID, the effective carbon number (ECN) method was used for quantification. The uncertainties of the species mole fractions calculated using the standard gas are within ± 5%, while those obtained using the ECN method are ± 10%.

Kinetic model development

The current model is developed based on NUIGMech1.2 [START_REF] Dong | A comprehensive experimental and kinetic modeling study of 1-hexene[END_REF] and the previously published pentane mechanism. [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF] The submechanisms of the three pentane isomers were developed using consistent rate rules and thermochemistry. [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF] The neopentane subchemistry was well validated against various experimental data, including IDTs measured in STs and RCMs and some JSR species data measured at high temperatures. Since the new JSR data of this study mainly focused on neopentane oxidation in the low temperature regime, the model updates mainly focused on the low temperature chemistry of neopentane submechanism.

In this study, one of the major updates in the neopentane submechanism is adding pressuredependent rate constants for the reaction classes of Q ̇OOH + O2 and Q ̇OOH decompositions. These reaction classes are critically important for alkane low temperature chemistry, as the rate constants directly controlled the branching ratios of chain propagation and chain branching pathways. In the previous pentane mechanism, [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF] only high-pressure limits were used for these reaction classes.

However, the high-pressure limits are slightly too fast to simulate JSR oxidation at 1 atm. Goldsmith et al. [START_REF] Goldsmith | Role of O2 + QOOH in low-temperature ignition of propane. 1. Temperature and pressure dependent rate coefficients[END_REF] theoretically investigated the kinetics of the O2 + propyl system, highlighting the pressuredependent reaction rate coefficients of the Q ̇OOH radicals. Recently, Duan et al. [START_REF] Duan | A theoretical kinetics study on low-temperature oxidation of n-C4H9 radicals[END_REF][START_REF] Duan | Theoretical calculation of low-temperature oxidation of heptyl radicals and O2[END_REF] also performed theoretical studies of the rate constants of O2 addition to n-butyl and four heptyl radicals, with significant pressure dependence observed above ca. 500 K. In the current model, the rate constants at 1 atm for both Q ̇OOH + O2 and Q ̇OOH decompositions are included and are a factor of two slower than the high-pressure limit values across the temperature range. The rate constants of these reactions are assumed to reach the high-pressure limits at pressures of 10 atm and higher.

Other major updates of the current mechanism include the new formation reaction pathways of carboxylic acids. In this work, formic acid, acetic acid, and iso-butyric acid were measured. In previous studies, [START_REF] Jalan | New pathways for formation of acids and carbonyl products in low-temperature oxidation: the Korcek decomposition of γ-ketohydroperoxides[END_REF][START_REF] Ranzi | New reaction classes in the kinetic modeling of low temperature oxidation of n-alkanes[END_REF][START_REF] Zhang | An updated experimental and kinetic modeling study of n-heptane oxidation[END_REF] it was suggested that carboxylic acids are mainly formed via the Korcek mechanism, i.e., the decomposition of γ-keto hydropeoxide (γ-KHP) species. Scheme 1 shows the Korcek mechanism for neopentane proposed by Eskola et al. [START_REF] Eskola | Time-resolved measurements of product formation in the low-temperature (550-675 K) oxidation of neopentane: a probe to investigate chain-branching mechanism[END_REF] γ-KHP undergoes cyclic peroxide isomerization that leads to two aldehydes (formaldehyde and 2-methylpropanal) and two acids (formic acid and iso-butyric acid). The rate constants for the Korcek mechanism of the C5 KHP are adopted to be the same as the Korcek reaction of C3 KHP calculated by Goldsmith et al. [START_REF] Goldsmith | Effect of non-thermal product energy distributions on ketohydroperoxide decomposition kinetics[END_REF] , with an assumption that the extra methyl groups in the C5 KHP (3-hydroperoxy-2,2-dimethylpropanal) are spectators. In addition, the reactions of ȮH radical additions to formaldehyde/acetaldehyde and subsequent reactions can also lead to the formation of formic acid, as shown in Scheme 2. In NUIGMech1.2, [START_REF] Dong | A comprehensive experimental and kinetic modeling study of 1-hexene[END_REF] the rate constant of ȮH radical addition to formaldehyde and the subsequent decomposition of the adduct to formic acid were adopted from the work of Xu et al. [START_REF] Xu | Ab initio study of the OH + CH2O reaction: The effect of the OH••OCH2 complex on the H-abstraction kinetics[END_REF] In this work, the rate constant of ȮH radical addition to acetaldehyde is determined to be the same as the reaction of ȮH radical addition to formaldehyde. For the decomposition of the adduct to formic acid via β-C-C bond scission, the rate constant was estimated based on the decomposition of the isopropoxy radical. [START_REF] Zádor | Unimolecular dissociation of hydroxypropyl and propoxy radicals[END_REF] Scheme 1 The γ-KHP decomposition of neopentane via the Korcek mechanism. [START_REF] Jalan | New pathways for formation of acids and carbonyl products in low-temperature oxidation: the Korcek decomposition of γ-ketohydroperoxides[END_REF] ȮH radical addition to acetaldehyde and the subsequent scission of the C-H bond could lead to the formation of acetic acid, as shown in Scheme 2. In addition, we propose another pathway that occurs via ȮH radical addition to acetone and the subsequent decomposition, producing a methyl radical and acetic acid. The branching ratio of ȮH addition to acetone has been discussed in a number of studies. [START_REF] Yamada | The reaction of OH with acetone and acetone-d6 from 298 to 832 K: Rate coefficients and mechanism[END_REF][START_REF] Henon | Features of the potential energy surface for the reaction of OH radical with acetone[END_REF][START_REF] Vasvári | Reaction and complex formation between OH radical and acetone[END_REF] It is suggested that this reaction pathway is slower than H-atom abstraction from acetone.

However, as acetone is one of the major products of neopentane oxidation, the reaction pathways of ȮH radical addition to acetone and subsequent decomposition may be important in the prediction of acetic acid. In this study, the rate constant of ȮH addition to acetone was estimated by analogy with ȮH addition to formaldehyde, [START_REF] Xu | Ab initio study of the OH + CH2O reaction: The effect of the OH••OCH2 complex on the H-abstraction kinetics[END_REF] with the activation energy decreasing by 2 kcal/mol, considering the structural difference between formaldehyde and acetone. Furthermore, the rate constants of the adduct decomposition of ȮH addition to acetone were estimated by analogy with the β-C-C bond scission of the isopropoxy radical. 38

Scheme 2

The formation reaction pathways of formic acid and acetic acid by the ȮH radical addition to aldehydes and ketone in the model.

Results and discussion

Species measurement and quantification

Some important intermediates (acetone, formaldehyde, 3,3-dimethyloxetane, methacrolein, isobutene) were detected by both SVUV-PIMS and GC. Carbon monoxide and methane were measured by GC, and water was measured by SVUV-PIMS. Additionally, peroxides (hydrogen peroxide and methyl peroxide), carboxylic acids (formic acid, acetic acid, and iso-butyric acid), and alcohols (methanol) were detected, identified, and quantified by SVUV-PIMS. The PICSs of formaldehyde, hydrogen peroxide, acetic acid, and methyl peroxide were adopted from the literature studies. [START_REF] Dodson | VUV photoionization cross sections of HO2, H2O2, and H2CO[END_REF][START_REF] Moshammer | Quantification of the Keto-Hydroperoxide (HOOCH2OCHO) and Other Elusive Intermediates during Low-Temperature Oxidation of Dimethyl Ether[END_REF][START_REF] Herbinet | Experimental and modeling investigation of the low-temperature oxidation of n-heptane[END_REF][START_REF] Dong | Improving quantification of hydrogen peroxide by synchrotron vacuum ultraviolet photoionization mass spectrometry[END_REF] The PICS of iso-butyric acid was measured in this work. Details of the PICS measurement method can be found in previous work, [START_REF] Zhou | Determination of absolute photoionization cross-sections of aromatics and aromatic derivatives[END_REF] and the table of PICS for iso-butyric is provided in the Supplementary material. The PICSs used for other species are from the online database. [START_REF]Photonionization cross section database (Version 2.0[END_REF] 3,3-Dimethyloxetane is a critically important intermediate in the chain-propagation process.

Eskola et al. [START_REF] Eskola | Time-resolved measurements of product formation in the low-temperature (550-675 K) oxidation of neopentane: a probe to investigate chain-branching mechanism[END_REF] carried out time-resolved measurement and analysis on this intermediate. The PICS of 3,3-dimethyloxetane at the parent ion m/z 86.07 is very small (the maximum PICS at 9.5-11.0 eV is ~0.007 Mb), but it has a very large PICS at its daughter ion m/z 56.06. [START_REF] Eskola | Time-resolved measurements of product formation in the low-temperature (550-675 K) oxidation of neopentane: a probe to investigate chain-branching mechanism[END_REF] For SVUV-PIMS, 3,3dimethyloxetane was quantified by the PICS of the daughter ion m/z 56.06 at 9.8 eV. We note that isobutene also contributed a signal at m/z 56.06, so the signal of isobutene needed to be subtracted to obtain the mole fraction of 3,3-dimethyloxetane. In this subtraction, we first obtain the mole fraction of isobutene at 9.3 eV. We note that the appearance energy for the daughter ion m/z 56.06 of 3,3dimethyloxetane is ~9.45 eV, while the ionization energy for isobutene is ~9.2 eV. The photon energy of 9.3 eV could exclude the strong interference of the daughter ion of 3,3-dimethyloxetane. Here, we used 1,3-butadiene as a reference to obtain the mole fraction of isobutene at 9.3 eV. Considering the low PICS of isobutene at 9.3 eV, the uncertainty in its mole fraction is estimated to be ± 50%. Then, the mole fraction of 3,3-dimethyloxetane was quantified at 9.8 eV after subtracting the isobutene signal.

Considering the impact of error propagation, the mole fraction uncertainty of 3,3-dimethyloxetane is also estimated to be ± 50%. Under the three experimental conditions presented in Fig. 1, both the experimental and modelpredicted results show significant low temperature reactivity and a notable negative temperature coefficient (NTC) zone. The reactivity of neopentane decreases as the equivalent ratio increases.

NUIGMech1.2 [START_REF] Dong | A comprehensive experimental and kinetic modeling study of 1-hexene[END_REF] overpredicts neopentane reactivity in the temperature range 550-675 K, while the model updated in this work can satisfactorily predict the reactivity. However, both models overpredict fuel reactivity in the temperature range of 750-850 K for φ = 0.5 and φ = 0.9. The overprediction is more evident at φ = 0.5. These major products formed during oxidation are also overpredicted at φ = 0.9, as seen in Figs. S1-S4 in the temperature range of 750-900 K. In contrast to the observation in this work, the models predicted the high temperature reactivity of neopentane well, as shown in Figs.

S5-S7, with a much lower fuel initial mole fraction of 0.001 that was measured by Dagaut et al. [START_REF] Dagaut | Oxidation of neopentane in a jet-stirred reactor from 1 to 10 atm: an experimental and detailed kinetic modeling study[END_REF] The sensitivity analyses in Fig. S8 indicate that some of the most sensitive reactions are related to the base chemistry, and it is believed that future updates of base chemistry will improve the model predictions in the intermediate range (above 750 K).

The final products of the low temperature oxidation of neopentane are carbon monoxide and water. The GC method failed to detect carbon dioxide at 500-800 K, although the maximum carbon dioxide concentration predicted by the present model is ca. 500 ppm at 500-800 K and at φ = 0.5. The comparisons of the experimental and simulated results for oxygen, carbon monoxide, and water at φ = 0.5, φ = 0.9, and φ = 1.62 are also presented in Fig. 1. Considering the experimental error and model uncertainty, the updated model satisfactorily predicts the reactants and final products of the low temperature oxidation of neopentane.

In the following discussion on the oxidation intermediates, we will focus on the results at the experimental condition of φ = 0.5, and the results at other conditions of φ = 0.9 and φ = 1.62 are presented in the Supplementary material.

The first and second additions to O2

Neopentane oxidation is initiated by H-atom abstraction initially by O2 and subsequently by ȮH radicals to form neopentyl radicals (Ṙ). These radicals add to O2 to form the neopentyl peroxy radical (RȮ2). Thereafter, a dominant reaction channel is the isomerization of RȮ2 radicals to hydroperoxy neopentyl radicals (Q ̇OOH) via a 6-membered ring transition state. The fate of Q ̇OOH radicals is critical for the low temperature chemistry of alkanes because they directly control the branching ratios of the chain-propagation and chain-branching pathways. In NUIGMech1.2, 29 the high-pressure limit rate constants of the first (Ṙ) addition to the O2 reaction calculated by Miyoshi et al. [START_REF] Miyoshi | Systematic computational study on the unimolecular reactions of alkylperoxy (RO2), hydroperoxyalkyl (QOOH), and hydroperoxyalkylperoxy (O2QOOH) radicals[END_REF] were used. The rate constants of the second (Q ̇OOH) addition to the O2 reaction were based on the first Ṙ addition to the O2 rate constants, with the A-factor decreasing by a factor of two based on the suggestions of Goldsmith et al. [START_REF] Goldsmith | Role of O2 + QOOH in low-temperature ignition of propane. 1. Temperature and pressure dependent rate coefficients[END_REF][START_REF] Goldsmith | Theoretical rate coefficients for allyl+HO2 and allyloxy decomposition[END_REF] The rate constant of the Q ̇OOH decomposition is derived from the rate rule of the LLNL model. [START_REF] Sarathy | Comprehensive chemical kinetic modeling of the oxidation of 2-methylalkanes from C7 to C20[END_REF] The pressure dependence of the reaction classes of Q ̇ OOH + O2 and Q ̇ OOH decompositions, as described in Section 3, is considered by this model to satisfactorily capture the neopentane low temperature oxidation reactivity at 550-675 K.

Isobutene and 3,3-dimethyloxetane are the predominant species during neopentane low temperature oxidation. This is because the RȮ2 radical does not have any β-hydrogen and lacks the reaction pathway of concerted elimination that leads to olefins and HȮ2 radicals. Instead, the RȮ2 radicals dominantly isomerize to Q ̇ OOH radicals, which can either decompose to form 3,3dimethyloxetane and an ȮH radical or undergo β-scission to form isobutene, ȮH and formaldehyde;

formaldehyde is the byproduct in the formation of isobutene. These two reaction channels compete with each other, and the formation of isobutene and 3,3-dimethyloxetane inhibits the chain-branching process, which is directly related to the low temperature reactivity of neopentane. Furthermore, the RȮ2 radical derived from the first addition to O2 reacts with HȮ2 radicals and leads to the formation neopentyl hydroperoxide (ROOH). The second additions of Q ̇OOH radicals to O2 lead to peroxy neopentyl hydroperoxide radicals (ȮOQOOH), which leads to C5 KHP and ȮH radicals via internal H-migration reactions. ROOH and C5 KHP are the key intermediates formed by the first and second additions to O2 reactions, and their decomposition is crucially important for the low temperature reactivity of neopentane. The dotted and solid lines represent the results predicted by NUIGMech1.2 [START_REF] Dong | A comprehensive experimental and kinetic modeling study of 1-hexene[END_REF] and the present model, respectively. The error bar for the experimental data is given.

C5 KHP decomposition

The dominant KHP consumption pathway is the formation of ȮH radicals and keto alkoxy radicals through O-OH bond scission. The keto alkoxy radical leads to the formation of acetone via CH2O loss and the following reactions in Scheme 3. Acetone is one of the dominant products during the low temperature oxidation of neopentane. These reaction channels are one of the main sources for the formation of acetone and formaldehyde. The kinetic model updated in this work predicts the formation of these two intermediates well, as shown in Fig. 3 and Fig. S1.

Scheme 3

The formation reaction pathways of acetone via C5 KHP decomposition.

In addition to the above pathways of KHP unimolecular ȮH-loss and subsequent reactions, another potential pathway of KHP is decomposition via the Korcek mechanism, [START_REF] Jalan | New pathways for formation of acids and carbonyl products in low-temperature oxidation: the Korcek decomposition of γ-ketohydroperoxides[END_REF] which is known to lead to a carbonyl compound and a carboxylic acid. The first decomposition pathway, Scheme 1 (1), of the Korcek reaction leads to the formation of 2-methylpropanal and formic acid. 2-Methylpropanal was detected and identified using GC and SVUV-PIMS, and formic acid was detected and identified using SVUV-PIMS. The second decomposition pathway, Scheme 1 (2), leads to the formation of formaldehyde and iso-butyric acid. Eskola et al. [START_REF] Eskola | Time-resolved measurements of product formation in the low-temperature (550-675 K) oxidation of neopentane: a probe to investigate chain-branching mechanism[END_REF] observed a very weak product signal at m/z 88.05 during neopentane oxidation but could not identify its structure. Bourgalais et al. [START_REF] Bourgalais | Jet-Stirred Reactor Study of Low-Temperature Neopentane Oxidation: A Combined Theoretical, Chromatographic, Mass Spectrometric, and PEPICO Analysis[END_REF] obtained photoionization efficiency (PIE) spectra for m/z 88.05 during neopentane oxidation at 600 K and φ = 0.5. Their results suggested that the signal of m/z 88.05 came from 3-hydroperoxybut-1-ene. In this work, a PIE experiment was performed at 675 K and at φ = 0.9, and the results support the existence of iso-butyric acid. The signal of m/z 88.05 can be attributed to the combined contribution of isobutyric acid and 3-hydroperoxybut-1-ene, as shown in Fig. S9. The comparison of the measured PIE curve with that of iso-butyric acid indicates that iso-butyric acid is the dominant contributor for m/z 88.05 at 675 K. The measured absolute PICS of iso-butyric acid is presented in Fig. S10.

The detection and identification of 2-methylpropanal, formic acid, formaldehyde, and isobutyric acid indicate the existence of the Korcek reaction during neopentane oxidation. Their mole fraction profiles are presented in Fig. 3. We note that for 2-methylpropanal, the SVUV-PIMS data are significantly higher than the GC measurements. The reason for these discrepancies has been discussed in detail by Bourgalais et al, [START_REF] Bourgalais | Jet-Stirred Reactor Study of Low-Temperature Neopentane Oxidation: A Combined Theoretical, Chromatographic, Mass Spectrometric, and PEPICO Analysis[END_REF] i.e., SVUV-PIMS results were affected by the additional contributions from other isomers at m/z 72.06. These isomers can be well separated by the GC method; only GC data are shown here for better comparison with model-predicted results. For iso-butyric acid, the calculated mole fraction should have the contribution of 3-hydroperoxybut-1-ene, but the PIE comparison indicates that the contribution is not significant. The error bar for the experimental data is given.

It can be seen that their relative ratio is not in accord with the ratio from the Korcek decomposition in Scheme 1. The mole fraction of formaldehyde is much higher than that of iso-butyric acid. As discussed above, formaldehyde comes from other pathways, such as the pathway in Scheme 3. For 2-methylpropanal and formic acid, their mole fractions are close to each other. However, apart from formaldehyde, the model underpredicts the mole fraction of 2-methylpropanal, iso-butyric acid, and especially formic acid. The reaction pathway analysis for these three intermediates at φ = 0.5 and 650 K is shown in Scheme 4. For 2-methylpropanal, the main formation pathway is ȮH radical addition to isobutene and the subsequent reactions, with only a small contribution (~15%) from Korcek decomposition of the C5 γ-KHP. For the formation of formic acid, the Korcek reaction channel of C5 γ-KHP contributed ~52%, followed by the subsequent reaction of isobutene (~36%) and formaldehyde (~12%). For iso-butyric acid, the Korcek decomposition of C5 γ-KHP is the only channel for its formation in the model. The above discussion indicates that Korcek decomposition is a source of carboxylic acid, but they could not fully explain their formation, such as formic acid and iso-butyric acid. The underestimation of the rate constant for the Korcek decomposition may also be a potential explanation for the discrepancy.

Scheme 4

The reaction pathway analysis for 2-methylpropanal, formic acid, and iso-butyric acid at φ = 0.5 and 650 K. The numbers represent the contribution of each reaction channel to the formation of 2-methylpropanal, formic acid, and iso-butyric acid, respectively. Only the main reaction pathways are shown here.

In addition, acetic acid was measured in this work. This carboxylic acid could not be formed by the Korcek decomposition of the C5 γ-KHP in neopentane oxidation. Carboxylic acids are often observed in combustion experiments, [START_REF] Herbinet | Detailed product analysis during the low temperature oxidation of n-butane[END_REF][START_REF] Wang | n-Heptane cool flame chemistry: Unraveling intermediate species measured in a stirred reactor and motored engine[END_REF] and there have been some exploratory studies on their formation during low temperature oxidation. [START_REF] Jalan | New pathways for formation of acids and carbonyl products in low-temperature oxidation: the Korcek decomposition of γ-ketohydroperoxides[END_REF][START_REF] Goldsmith | Effect of non-thermal product energy distributions on ketohydroperoxide decomposition kinetics[END_REF][START_REF] Battin-Leclerc | To Better Understand the Formation of Short-Chain Acids in Combustion Systems[END_REF] Battin-Leclerc et al. [START_REF] Battin-Leclerc | To Better Understand the Formation of Short-Chain Acids in Combustion Systems[END_REF] proposed possible channels for the formation of short-chain monocarboxylic acids, e.g., C1-C3 carboxylic acids that are mainly formed from aldehydes with the same structure. In this work, the reaction of ȮH radical addition to aldehydes and the subsequent decomposition via C-H bond scission--Scheme 2 (3) was added to the model. Furthermore, an additional formation pathway of acetic acid was proposed in this work, i.e., the reaction of ȮH radical addition to acetone and the subsequent decomposition--Scheme 2 (4).

Experimental measured and model-predicted mole fraction profiles of acetic acid are shown in Fig. 3f.

Compared to NUIGMech1.2, [START_REF] Dong | A comprehensive experimental and kinetic modeling study of 1-hexene[END_REF] the current model has significantly improved the mole fractions of acetic acid.

Other products from neopentane oxidation

Other key products of neopentane oxidation were also identified and quantified in this work. The measured and model-predicted mole fraction profiles of ethylene, propene, methanol, ketene, acetaldehyde, methacrolein, hydrogen peroxide, and methyl hydroperoxide at φ = 0.5 are shown in Fig. 4. For these key species, the predictions from the updated model improved significantly, especially under the conditions of φ = 0.9 and φ = 1.62, as shown in

Summary and conclusions

In this work, two different analytical methods (SVUV-PIMS and GC) were used to investigate neopentane oxidation in JSRs at 1 atm and three different equivalence ratios of 0.5, 0.9, and 1.62. The experimental data obtained by different analytical methods are in good agreement; these different analytical methods complement each other, making the experimental results more comprehensive.

Numerous important intermediates were identified and quantified, e.g., acetone, 3,3-dimethyloxetane, methacrolein, isobutene, 2-methylpropanal, formic acid, iso-butyric acid, acetic acid, methyl hydroperoxide, and hydrogen peroxide. These key intermediates are crucial for clarifying the low temperature oxidation mechanism of neopentane and developing the kinetic model.

An updated kinetic model is proposed based on the current experimental data and previous neopentane studies. The pressure dependence of the reaction classes of Q ̇ OOH + O2 and Q ̇ OOH decompositions was considered, which is important for improving the predictions under atmospheric conditions. The identification of iso-butyric acid supports the existence of the Korcek mechanism during neopentane oxidation, which is consistent with the point proposed by Bourgalais et al. [START_REF] Bourgalais | Jet-Stirred Reactor Study of Low-Temperature Neopentane Oxidation: A Combined Theoretical, Chromatographic, Mass Spectrometric, and PEPICO Analysis[END_REF] and Eskola et al. [START_REF] Eskola | Time-resolved measurements of product formation in the low-temperature (550-675 K) oxidation of neopentane: a probe to investigate chain-branching mechanism[END_REF] The Korcek reactions were added to the updated kinetic model. However, the results

show that the Korcek reactions could not fully explain the experimental observations. Regarding the formation of acetic acid, the reaction pathways of ȮH radical addition to acetone/acetaldehyde and subsequent reactions were considered. The model analysis shows that the reaction pathway of ȮH radical addition to acetone and the subsequent decomposition is the most dominant formation channel of acetic acid.

The updated kinetic model was well validated against the speciation data measured in this work and various experimental data available in the literature, including IDTs of both STs and RCMs and JSR species data measured at high temperatures. We note that further improvement of the neo-pentane model is still needed to reduce model uncertainty, including quantum calculations of the newly added reaction pathways discussed in the current study. This model could provide a valuable reference for further updating alkane kinetic models or exploring the low temperature oxidation chemistry of highly branched alkanes.

20 4. 2

 202 Two C5 peroxides were measured in this work, i.e., the C5 ROOH (C5H12O2) and the C5 KHP (C5H10O3). They could be determined to be neopentyl hydroperoxide and 3-hydroperoxy-2,2dimethylpropanal due to the highly symmetrical structure of neopentane. The C5 KHP was also identified as 3-hydroperoxy-2,2-dimethylpropanal ((CH3)2C(CH2O2H)CH=O) by the ionization energy calculated by Hansen et al. Low temperature chemistry of neopentane CHEMKIN PRO software 44 was used to simulate these JSR data. The measured and modelpredicted mole fraction distributions of neopentane oxidation products are shown in Figs. 1-4 and Figs. S1-S4. Figure1shows the mole fraction profiles of neopentane, oxygen, carbon monoxide and water at φ = 0.5, φ = 0.9 and φ = 1.62, respectively. The neopentane model updated in this work can satisfactorily capture the profiles of reactants and final products at different equivalence ratios and different temperatures, especially under fuel-rich conditions.

Figure 1

 1 Figure 1 Measured (symbols) and model-predicted (lines) mole fraction profiles of neopentane, oxygen, carbon monoxide and water at φ = 0.5, φ = 0.9 and φ = 1.62. Dotted and solid lines represent the results predicted by NUIGMech1.2 29 and the present model, respectively. The error bar for the experimental data is given.

Figure 2

 2 Figure 2 presents the measured and model-predicted mole fraction profiles of 3,3-

Figure 2

 2 Figure 2 Measured (symbols) and model-predicted (lines) mole fraction profiles of 3,3dimethyloxetane and isobutene at φ = 0.5 in neopentane low temperature oxidation. The signal profiles ( * ) of ROOH and KHP are also presented with a comparison of the profile shape from model prediction.

Figure 3

 3 Figure 3 Measured (symbols) and model-predicted (lines) mole fraction profiles of acetone, formaldehyde, 2-methylpropanal, iso-butyric acid, formic acid, and acetic acid at φ = 0.5. The dotted and solid lines represent the results predicted by NUIGMech1.2 29 and the present model, respectively.

  Fig.S2. The model predicts well the formation of ethylene and propene under this condition. The simulations show that both ethylene and propene are mainly derived from the H-atom abstraction reaction and subsequent reactions of isobutene. In addition, ketene and methacrolein are also mainly derived from isobutene. Methoxy radicals are the main precursors of methanol, which are dominantly formed from O-O bond scission of methyl hydroperoxide. For methyl hydroperoxide, methyl radical addition to O2 and subsequent reaction of CH3Ȯ2 with HȮ2 radical is responsible for its formation. The model overpredicts the mole fraction of methanol by a factor of two; this overprediction may be caused by the overprediction of methyl hydroperoxide by the model. We note that the model predicts the mole fraction profiles of methanol and methyl hydroperoxide well at φ = 0.9 and φ = 1.62. Acetaldehyde is an important intermediate with a wide range of sources, in which C5 KHP decomposition and subsequent reactions are one of the dominant formation pathways. For acetaldehyde, SVUV-PIMS data are in good agreement with the prediction by the updated model, but the GC data are lower than SVUV-PIMS data by a factor of two at φ = 0.5, φ = 0.9, and φ = 1.62. Hydrogen peroxide plays an important role in low temperature oxidation. The model results show that hydrogen peroxide is mainly produced by the disproportionation reaction of two HȮ2 radicals and the H-atom abstraction reaction from neopentane by HȮ2 radicals and is mainly consumed by decomposing into two ȮH radicals. The model overpredicts the mole fraction of hydrogen peroxide by a factor of 1.5 at φ = 0.5, but underpredicts its mole fraction profiles by a factor of 2 at φ = 0.9 and well predicts its mole fraction profiles at φ = 1.62.

Figure 4

 4 Figure 4 Measured (symbols) and model-predicted (lines) mole fraction profiles of ethylene, propene, methanol, ketene, acetaldehyde, methacrolein, hydrogen peroxide, and methyl hydroperoxide at φ = 0.5. The dotted and solid lines represent the results predicted by NUIGMech1.2 29 and the present model, respectively. The error bar for the experimental data is given.

Table 1 .

 1 The JSR experimental conditions in this work

	Location	T (K)	p (bar) τ (s)	φ	neopentane	O2	He/Ar
	NSRL	500-825	1.03	3	0.5	0.015	0.24	0.745
	NSRL	500-825	1.03	3	0.9	0.015	0.133	0.852
	NSRL	500-825	1.03	3	1.62	0.015	0.0741	0.9109
	LRGP	500-825	1.07	3	0.5	0.015	0.24	0.745
	LRGP	500-1100	1.07	3	0.9	0.015	0.133	0.852
	LRGP	500-1100	1.07	3	1.62	0.015	0.0741	0.9109

Acknowledgements

This work was supported by National Natural Science Foundation of China (51976208) and the Hefei Science Center, CAS (2020HSC-KPRD001, 2021HSC-UE005).

Supporting Information

Additional validation results and an updated kinetic model of neopentane oxidation are provided.


Elise Colin
email: elise.colin@onera.fr

PolTimeSAR: Link between the measurement of a time series of N Jones vectors, and the measurement of a single Stokes vector integrating all these measurements. Application to (HH,HV) or (VH,VV) SAR time series

Keywords: change detection, permanent scatterer, coefficient of variation

This paper addresses radar time series acquired with two polarization channels, one copolar and the other contrapolar. We show that the Eigenvalue decomposition of the coherence-covariance matrix of the complex Jones vectors contains the same information as the Stokes vector that would have been obtained by integrating all measurements into a single Stokes intensity measurement. We use two main parameters of this equivalent Stokes vector: its degree of polarization and its principal orientation. We then proposed two polarimetric decompositions: the "Equivalent-Stokes" representation, totally independent of the total intensity, and the "Polarimetric with main orientation" representation. We test these decompositions on the (HH,HV) Tandem-X, and (VV,VH) Sentinel-one time series of the same area.

Keywords time-series •

Polarimetry refers to the ability to transmit and receive signals in various polarizations. We can distinguish between the theoretical tools that describe the polarimetric state of a wave, both for the transmitted and the received wave, and the formalism that describes the polarimetric behavior of the object, i.e. its ability to transform a given polarization wave into a different one.

Most studies have focused on target analysis, which is possible with full polarimetry, i.e. with the acquisition of four polarimetric channels. They distinguish between the calculation of first order statistical parameters (such as Pauli parameters) and second order parameters, i.e. those requiring the calculation of a covariance matrix, involving second order moments, such as the well known polarimetric entropy parameter Cloude and Pottier [1997], and all physical decompositions such as [START_REF] Freeman | Three-component scattering model to describe polarimetric sar data[END_REF] or Yamaguchi decompositions Yamaguchi et al. [2011].

There are two main issues with the use of these theories:

• Fully polarimetric information is not often available, due to its technological cost. Thus, Sentinel-1 images are most often acquired in VV and VH polarization. Tandem-X images are most often acquired in dual polarimetric mode.

• The estimation of the second order parameters is mostly done by choosing samples in a spatial neighborhood of the considered pixel. This estimation is done at the expense of the resolution. Moreover, we have already seen that for some resolution ranges, this estimation method is not satisfactory, because the ergodicity hypothesis is not respected. Thus, for X-band data of metric type, it has already been seen that the calculation of the entropy in a spatial neighborhood leads to very bad results.

Here, we propose to study a formalism that allows to circumvent these two difficulties.

1

First, we restrict our study to the use of radar time series containing two polarimetric channels: a measurement in copolar polarizations, and a measurement in contrapolar polarizations. This includes Sentinel-1 images in (VH,VV) mode.

Next, we consider the time series as a statistical series containing N samples of the same measurement: this configuration is equivalent to considering a fixed emission polarization, and to observe N polarimetric states of the re-emitted wave.

The originality of the approach is threefold:

• We use the tools for analyzing the behavior of a wave, instead of considering the analysis of a target.

• The different statistical samples used in the study of this re-emitted polarimetric state are the different temporal samples. Thus, it is possible to distinguish random from deterministic behavior, without any spatial averaging. • Considering the second order statistical parameters for these N measurements, we consider the formalism of the measurement of a single Stokes vector. This allows us to consider the modeling of the traditional optical measurement of a Stokes vector, as the integration of N deterministic Jones vectors.

In section 2, we explain the formalism of the covariance-coherence matrix of the re-emitted wave. We then propose to derive two main parameters: the randomness of the polarization designated by "polarization diversity" and the main orientation of the wave. We make these calculations explicit in a literal way, which allows a fast implementation from raw signals.

In a third section, we demonstrate the mathematical properties that allow us to link the measurement of a single Stokes vector, and that of a deterministic Jones vector computed as an eigenstate of the time series.

In section 4, we illustrate the previously introduced concepts, on Tandem-X and Sentinel-1 time series, by proposing two representations of the information in false colors. We conclude in section 5.

1 Polarimetric coherence-covariance matrix of the emitted wave

In this section, we consider the case of a time series for which we have two polarimetric channels, copolar and crosspolar. This is notably the case for the continental Sentinel-1 stripmap images (VV and VH), or for a time series of Tandem-X images in the (HH,HV) dual-polarimetric mode. In this case, the time series can be viewed as a succession of Jones vector of the scattered wave, for a wave emitted with a fixed polarization X. In each pixel, we consider observation vector of 2 rows and N columns that we call P. Each column k of P can be seen as the Jones vector acquired in response to an emitted wave, for a fixed polarization X, where E x will be the component of the backscattered field parallel to X, and E y will be the component of the backscattered field perpendicular to X:

P = p 1 , ... p k ... p N = E 1 x E 2 x ... E k x ... E N x E 1 y E 2 y ... E k y ... E N y , (1)
In the case of Tandem-X data in HH and HV polarizations, we have X=H, and E x = S hh , E y = S hv . In the case of Sentinel-1 data, in VV and VH polarizations, we have X=V, and

E x = S vv , E y = S vh .
Recall that each of the vectors (E x , E y ) is a Jones vector, which describes a polarization ellipse of ellipticity ϵ and orientation ψ, by:

tan(2ψ) = 2|E x ||E y | |E x | 2 -|E y | 2 cos δ, sin(2ϵ) = 2|E x ||E y | |E x | 2 + |E y | 2 sin δ, δ = arg(E x E * y ) (2)
This polarization ellipse can be represented in a cartesian plane, it is a deterministic polarization. We can therefore, for a given pixel, consider the N polarization states received, as in the example given in 1: The main idea is to consider our N acquisitions of these polarimetric states as a single but random polarization state, of which we will measure several properties.

For this, we will consider two different formalisms:

• the coherence-covariance matrix of the N Jones vectors

• the associated Stokes vector

The coherence-covariance matrix of the Jones vectors

In order to study the polarimetric properties of this wave, we can measure the coherence-covariance matrix of the wave, sometimes also called the polarization matrix in the Jones formalism [START_REF] Mandel | Optical coherence and quantum optics[END_REF]. This matrix is written:

C = 1 N N k=1 p k p k † = < |E x | 2 > < E x E * y > < E * x E y > < |E y | 2 > = c 11 c 12 c 21 c 22 (3)
where † denotes the conjugate transpose, c 11 and c 11 are real numbers, and c 12 = c * 21 . By construction, this matrix C is Hermitian. When the two receptions E x and E y are totally decorrelated, then the extra diagonal coefficients are zero. On the contrary, if they are very correlated, then the two extra diagonal coefficients are very high.

Expressed in the eigenvector basis, the matrix C will have zero extradiagonal coefficients: the eigenvector basis is therefore the polarization basis where the signals are the most temporally decorrelated.

Because the matrix C is a 2x2 hermitian matrix, it is possible to express literally the eigenvectors and the eigenvalues: if we let:

δ = (c 11 -c 22 ) 2 + 4|c 12 | 2
Then the two eigenvalues are:

λ -= (c 11 + c 22 -δ)/2 λ + = (c 11 + c 22 + δ)/2
with the corresponding relative unitary eigenvectors:

v -= v 0- λ --c 22 c * 12 , v 0-= 1 (λ --c 22 ) 2 + |c 12 | 2 v + = v 0+ λ + -c 22 c * 12 , v 0+ = 1 (λ + -c 22 ) 2 + |c 12 | 2 such that: C = λ -. (v -v - † ) + λ + . (v + v + † ) (4)
The two eigenstates describe the two depolarization axes that are most temporally decorrelated. v + , associated with the largest eigenvalue, is the main state, the one that best describes the main polarization of the ensemble. They are represented for our starting example, on the figure 2.

Main eigenstate orientation

It is interesting to consider then the orientation of the maximum eigenstate, given by:

tan(2ψ) = 2|E x ||E y | |E x | 2 -|E y | 2 cos δ (5)
with:

|E x | = λ + -c 22 , |E y | = |c 12 |, δ = arg(c 12 ) (6)
Figure 2: Superposition of the different eigenstates in gray, and representation of the two eigenstates: the most important in red, the second one in blue.

Scattering diversity of the covariance coherence matrix

We also propose to look at the distribution of the two eigenvalues, thanks to the coefficient ∆, calculated in the same way as the scattering diversity coefficient in [START_REF] Praks | Alternatives to target entropy and alpha angle in sar polarimetry[END_REF], either from the relative powers of the eigenvalues, or from the Froebenius norm of the coherence matrix:

∆ = 2 -2(p 2 1 + p 2 2 ) = 2 -2∥ C∥ 2 F ( 7
)
where p i = λi λ1+λ2 , λ 1 and λ 2 are the two eigenvalues of C, and ∥.∥ F denotes the Froebenius norm, and C = C/trace(C).

This polarimetric depolarization coefficient will have to be compared later with the notions of interferometric coherence, but note that no spatial averaging is required here, and that this parameter is independant from the total intensity.

The Stokes representation

The Stokes representation is particularly well suited to physical measurements, as it allows polarimetric states to be described directly from sums or differences of integrated intensities, easily measured in the optical domain [START_REF] Rasheed | Ellipsometry and polarized light[END_REF]. Contrary to Jones vectors, Stokes vectors allow describing non-polarized or partially polarized states, in addition to fully polarized states.

And it is possible to express the Stokes vector as a function of the total intensity of the beam, its polarization rate, and parameters related to the shape of the polarization ellipse.

From the previous notations, it is written :

S =    < |E x | 2 + |E y | 2 > < |E x | 2 -|E y | 2 > < 2ℜ(E x E * y ) > < 2ℑ(E x E * y ) >    =    s 0 s 1 s 2 s 3    (8)
where < . > denotes the temporal average. We can see that these coefficients are directly related to the covariancecoherence matrix of the wave previously considered. Specifically, we have the following relationships:

       s 0 = c 11 + c 22 s 1 = c 11 -c 22 s 2 = c 12 + c 21 s 3 = -j(c 12 -c 21 ) ,        2c 11 = s 0 + s 1 2c 12 = s 2 + js 3 2c 21 = s 2 -js 3 2c 22 = s 0 -s 1 (9)
A crucial parameter of the Stokes vector is the degree of polarization, which is defined as follows:

doP = s 2 1 + s 2 2 + s 2 3 s 0 (10)
by writing:

(1 -∆) = 2∥ C∥ 2 F -1 = 2 c 2 11 + c 2 22 + |c 12 | 2 + |c 21 | 2 c 2 11 + c 2 22 -1 (11)
which can be transformed, thanks to 9, into:

s 2 0 (1 -∆) = s 2 1 + s 2 2 + s 2 3 (12)
This leads to our first important result:

(1 -∆) = doP 2 . ( 13
)
If we want to express the doP as a function of the initial data, this gives:

doP = 2 < |E x E * y | > < |E x | 2 > 2 + < |E y | 2 > 2 (14)
The degree of polarization of the equivalent Stokes vector of our Jones vector time-series can be simply linked to the so-called scattering diversity defined from the coherence-covariance matrix of the Jones vectors.

We focus now on the ellipticity and orientation properties of this Stokes equivalent vector, to relate them to that of the principal eigenvector v + .

If we reparameterize the unit eigenvector v + using two complexes a and b, so that v + = (a, b) t , then the vector v -can be written v + = (b * , -a * ) t , and the matrix C can be reparameterized thanks to the equation 4 by:

C = λ + |a| 2 ab * a * b |b| 2 + λ -|b| 2 -ab * -a * b |a| 2 (15)
This means that the equivalent Stokes vector can be reparameterized thanks to Eq.9, by :

s =    λ + |a| 2 + λ -|b| 2 (λ + -λ -)(|a 2 | -|b 2 |) (λ + -λ -)2ℜ(ab * ) (λ + -λ -)2ℑ(ab * )    (16)
We recognize in the last three components, a part proportional to the last 3 components of the Stokes vector associated to the eigenvector v + , with a proportionality factor (λ + -λ -) which is positive. This constitutes the proof of our second property:

The Stokes vector of the principal axis or eigenvector of the coherence-covariance matrix of the time series is collinear to the global Stokes vector. It therefore has the same orientation and ellipticity properties.

Fig. 3 represents different points on the Poincare sphere [START_REF] Deschamps | Poincaré sphere representation of partially polarized fields[END_REF]: the different Jones vectors converted into measured Stokes vector appears in grey, as well as the Stokes vectors corresponding to both eigenvectors of the coherence-covariance matrix in blue and red. As the two eigenstates are orthogonal, we can see that the blue and To conclude this part, the eigenvector and eigenvalue decomposition of the coherence-covariance matrix of the time series contains the same information as the equivalent Stokes vector which would have consisted in integrating in time all the radar observations.

Proposition of two colored PolTimeSAR representations

In this section, we propose two colored representations exploiting the main features we found in the previous sections, namely the degree of polarization and the ellipse parameters of the main polarization.

A pure polarimetric representation: the "Equivalent-Stokes" time series representation

The first representation is a representation that exploits only the purely polarimetric information, making it independent of the total intensity notion.

For this one, we use the three parameters ∆, ϕ, and ϵ to make a false color representation of the stability of the temporal polarimetric information: we use the degree of polarization doP to code the brightness of the image (for a strongly temporally polarized pixel, the value will be high), we use the ellipticity to code the saturation (a saturated color corresponds to a linear polarization), and we use the orientation of the polarization to code the hue.

In figure 4, we show the result of this representation on Tandem-X images acquired in HH and HV polarizations for ten dates. These data have a resolution of 2 meters.

All the parameters of this representation are independent of the collected power, even though the deterministic targets appear with very low depolarizations, well contrasted with the natural areas. To better convince us, we show the channel corresponding to the polarization degree parameter at the top right of this figure. In the second representation, we use the average intensity to code the brightness. We use the degree of polarization to code color saturation. The hue always codes the same main orientation parameter.

Figure 5: The "Main-orientation polarimetric" representation over Tandem-X data (Toulouse, France)

In figure 5 we show the result of this representation on previous Tandem-X data. We find the same trends as in figure 4, except for the areas of very low noise ratios where the polarimetric properties are almost no longer visible. A typical example is the Canal du Midi, which has a high polarizing power aligned on the trajectory direction. In 5, we can no longer see this feature because the signal-to-noise ratio is too low.

We represent an extract of the global image footprint in figure 6. This one allows us to see the diversity of the orientations estimated on the deterministic targets.

Figure 6: The "Main-orientation polarimetric" representation: Extract on Tandem-X data (Toulouse, France)

We also applied our representation to a Sentinel-1 time series, for the same period and coverage, i.e. about 40 images in VV and VH polarizations. The result is shown in Figure 7. In the Sentinel-1 data, the contrast between deterministic and nondeterministic targets is less binary, much more nuanced, and the estimation of orientations appears narrower around the main acquisition orientations. This result suggests that, for this wavelength and resolution, the polarimetric behavior could be very different: several levels of depolarizations are involved, but there are fewer purely deterministic targets. We need to confirm this result with other data.

Figure 7: The "Main-orientation polarimetric" representation: comparison between Sentinel-1 ascending mode and Sentinel-1 descending, and Tandem-X data (Toulouse, France)

In figure 8, we zoomed in on an emblematic university building in Toulouse, called a tripod, because of the geometric distribution in three directions of the building. The image on the right is the optical view. In the colored composition, we find the azimuthal orientations of these three buildings, with patterns on the roof that suggest that the backscatter has focused on some details with particular directions.

Figure 8: The "Main-orientation polarimetric" representation: Zoom on a particular building (Toulouse, France)

Conclusion

This paper concerns radar time series acquired with two polarization channels, one copolar and the other contrapolar. It proposes to consider these signals as a succession of Jones vector measurements to fully describe the polarization of the backscattered wave.

We show that the eigenvalue decomposition of the coherence-covariance matrix of these complex Jones vectors contains the same information as the Stokes vector that would have been obtained by integrating all measurements into a single Stokes intensity measurement. We use two main parameters of this equivalent Stokes vector: its degree of polarization, and its principal orientation.

We then proposed two polarimetric decompositions: the first relies only on purely polarimetric descriptors, made independent of the total intensity: ellipticity, orientation, and degree of polarization. We called this representation the "Equivalent-Stokes" time series representation.

A second color representation includes the degree of polarization, orientation, and total intensity. We called this decomposition the "Main-orientation polarimetric" representation.

Applied to the Sentinel-1 and Tandem-X time series over the same area, they show very different results for the two cases. The degree of polarization allows us to see strongly distinct behaviors on Tandem-X between deterministic and non-deterministic targets without using intensity information. The main orientations estimated seem to correspond to the azimuthal directions of the buildings or the deterministic targets.

Further investigations will be necessary to understand the role of wavelength and resolutions and the potential differences in behavior between (HH, HV) and (VV, VH) modes.

Figure 1 :

 1 Figure 1: Examples of 10 elliptic representations corresponding to N=10 (HH,HV) values in a given pixel.

Figure 3 :

 3 Figure 3: Example of reprensentation on the Poincare sphere: In gray, the N individual different Stokes vectors of the time-series; Red Point: Stokes vector of the main eigenvector, Blue Point: Stokes vector of the second eigenvector, Green point: Stokes vector for the whole time-series

Figure 4 :

 4 Figure 4: Pure polarimetric representation: the "Equivalent-Stokes" time series representation

Acknowledgements

The TanDEM-X data were provided by DLR under the scientific proposal OTHER0103. This work would not have been possible without the collaboration of Paola Rizzoli and Jose Luis Bueso Bello in DLR, whom we thank very sincerely. This fruitful collaboration was initiated in the framework of the ONERA-DLR virtual research center for cooperation in "AI and applications in aerospace engineering.

On a personal note, I would like to thank Professor Eric Pottier for inviting me to address the polarization case of Sentinel-1 and for encouraging me to pursue the study of polarimetric time series under the acronym PolTimeSAR.