F Xu

N C Siersch

S Gréaux

A Rivoldini

H Kuwahara

N Kondo

N Wehr

N Menguy

Y Kono

Y Higo

A.-C Plesa

J Badro

Daniele Antonangeli
email: daniele.antonangeli@upmc.fr

Low Velocity Zones in the Martian Upper Mantle Highlighted by Sound Velocity Measurements

Sound wave velocities and density were measured for pressures, temperatures and mineralogical assemblages representative of Mars' mantle • Experimental results support the existence of a low shear velocity layer in the shallow Martian mantle 2 • Phase equilibria experiments suggest the existence of ferrian minerals in the oxidized regions of the uppermost Martian mantle 3

Introduction

The NASA's Discovery program mission InSight is the first mission, almost 40 years after the end of Viking 2 lander operations [START_REF] Anderson | Seismology on Mars[END_REF], aiming at investigating the seismology of a rocky planetary body other than the Earth and the Moon. The spacecraft successfully landed on Mars in November 2018 [START_REF] Banerdt | Initial results from the InSight mission on Mars[END_REF] and since early winter 2019, its seismometer SEIS is continuously recording global seismic activity [START_REF] Giardini | The seismicity of Mars[END_REF]. In the first year of operations, three quakes were strong enough to perform single station seismic analyses of Mars' subsurface and upper crust [START_REF] Lognonné | Constraints on the shallow elastic and anelastic structure of Mars from InSight seismic data[END_REF][START_REF] Knapmeyer-Endrun | Thickness and structure of the Martian crust from InSight seismic data[END_REF]. Observations of direct and reflected body-wave phases from these and few other marsquakes have been used to provide first constraints on seismic structure to a depth of 800 km [START_REF] Khan | Upper mantle structure of Mars from InSight seismic data[END_REF] and the size of the core [START_REF] Stähler | Seismic detection of the Martian core[END_REF]. However, in order to invert seismic models for information on the composition and mineralogy of Mars' interior, knowledge of the sound velocities and density of relevant minerals and mineralogical assemblages at pertinent thermodynamic conditions are critically needed. Furthermore, most of the Martian seismic data available so far do not allow for independent source localization [START_REF] Giardini | The seismicity of Mars[END_REF] and a-priori knowledge of sound velocity variations as a function of pressure and temperature would play an essential role in interpreting the nature and depths of the observed seismic events. Self-consistent thermodynamic mineralogy and seismic wave velocity models of the Martian mantle have been computed for a series of pressure-temperature profiles and for a selection of bulk chemical compositions (Smrekar et al., 2019 and references therein) along self-consistent adiabats by Gibbs free-energy minimization [START_REF] Connolly | The geodynamic equation of state: What and how[END_REF] using the thermodynamic formulation of [START_REF] Stixrude | Thermodynamics of mantle minerals -I. Physical properties[END_REF], 2011). While internally consistent, these calculations rely on available thermodynamic databases (e.g. [START_REF] Stixrude | Thermodynamics of mantle minerals -II. Phase equilibria[END_REF][START_REF] Holland | Newthermodynamic models and calculated phase equilibria in NCFMAS for basic and ultrabasic compositions through thetransition zone into the uppermost lower mantle[END_REF], which have been largely constructed on the basis of minerals with compositions relevant to Earth's mantle. Experiments and calculations have not yet sufficiently constrained the thermoelastic properties of Fe-rich end members, especially Fe 3+ -rich, minerals across the full range of pressure and temperature of Mars' mantle. Since Mars' mantle is believed to be especially enriched in Fe (Fe# = [Fe/(Fe+Mg)]x100 = 25, e.g. [START_REF] Dreibus | Mars, a volatile-rich planet[END_REF] compared to Earth's mantle (Fe# = 10, e.g. [START_REF] Takahashi | Melting of a dry peridotite KLB-1 up to 14 GPa: Implications on the Origin of peridotitic upper mantle[END_REF], potentially with oxidized regions in its uppermost part [START_REF] Tuff | Volcanism on Mars controlled by early oxidation of the upper mantle[END_REF], current estimations do not encompass the full range of possibilities, and proposed velocity models are based on, and possibly biased by, an incomplete dataset.

The only experimental study so far investigating the phase assemblages of Mars' mantle has been performed by [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF]. The synthetized aggregates at Mars' upper mantle conditions were predominantly made of olivine and pyroxene down to ~200 km depth corresponding to ~2 GPa. From this depth, pyroxene dissolves into garnet through the entire upper mantle, until olivine transforms into wadsleyite at ~1100 km depth (13.5 GPa), marking the beginning of the Martian counterpart of the transition zone as defined for the Earth's mantle [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF]. The compositions of the olivine and pyroxene phases measured by [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF] were found to be characterized by 25 mol.% of the respective Fe end-member. Despite the reducing conditions in the experiments (both of the starting material and synthesis setup), the garnets present in the assemblages representative of the upper mantle contain not only 21 mol.% almandine (Fe 2+ 3Al2Si3O12) component, but also 5 mol.% of the skiagite (Fe 2+ 3Fe 3+ 2Si3O12) component, the latter being a Fe 3+ -rich end-member of garnet whose thermoelastic properties have not been studied in much detail, especially lacking high-temperature data [START_REF] Woodland | Equations of state for Fe 3 2+ Fe 2 3+ Si 3 O 12 "skiagite" garnet and Fe 2 SiO 4 -Fe 3 O 4 spinel solid solutions[END_REF][START_REF] Ismailova | Effect of composition on compressibility of skiagite-Fe-majorite garnet[END_REF][START_REF] Vasiukov | Sound velocities of skiagite-iron-majorite solid solution to 56 GPa probed by nuclear inelastic scattering[END_REF]. Since Mars is believed to have oxidized regions at least in the uppermost mantle [START_REF] Tuff | Volcanism on Mars controlled by early oxidation of the upper mantle[END_REF], studying more oxidizing conditions during phase equilibria experiments in order to obtain mineralogical assemblages which are potentially more representative of the Martian mantle is essential.

Here we measured the bulk compressional (vP) and shear (vS) wave velocities as well as densities of representative Martian upper mantle mineral aggregates synthesized at various redox conditions by means of in situ ultrasonic experiments combined with synchrotron radiation at pressures and temperatures relevant to the Martian upper mantle. Our results provide direct guidance to the interpretation of seismic observations and extend on available datasets for thermodynamic models.

Materials and Methods

Homogeneous spherical glass samples (diameter ~ 2.5-3 mm) were produced in an aerodynamic levitation laser furnace, by fusing a powder mixture with bulk composition close to the one proposed by Taylor (2013) (oxides expected to be present at a level below 1 wt.% were not included in the starting material), employing different gas mixtures (Ar, CO2, H2), and quenching the melt. The glass spheres were then polished to cylinder shapes and re-equilibrated in a multi-anvil or a piston-cylinder press, to crystallize phase assemblages analogous of the Martian mantle at high pressure and temperature (UM1 at 3 GPa and 1373 K; UM2 at 3 GPa and 1473 K; MM1 at 8 GPa and 1473 K, see Table S1). All recovered samples were well sintered. Assemblages were further analyzed using a scanning electron microscope (SEM) and electron microprobe (EMP)/ transmission electron microscope (TEM) to determine grain size, mineral phases and compositions (Table 1 andS1). For more detailed information please refer to the Supporting Information.

Sound wave velocity and density measurements were carried out at the beamline BL04B1, SPring-8, Japan, in three in situ experiments using combined ultrasonic interferometry, X-ray radiography and X-ray diffraction under high-pressure and high-temperature. Measurements have been conducted up to 6.8(1) GPa and 900 K for the upper mantle assemblages, and up to 10.6(1) GPa and 1200 K for the mid-mantle assemblage (see Supporting Information and Figure S1). The two-way travel time (t) was measured by the pulse echo method, analyzing the time delay between the buffer and sample echoes (Figure S2). At each pressure and temperature, the sample length (L) was measured by X-ray radiography, imaging two gold foils placed at both ends of the sample (Figure S3), and the compressional and shear wave velocities were calculated using the relation Energy-dispersive diffraction patterns were collected for both the samples (Figure S5) and pressure marker using a solid-state Ge-detector at a fixed 2θ angle of ~ 6°. The unit-cell volumes of the Au pressure marker were used to determine the pressure at each temperature point according to the P-V-T equation of state of gold (Tsuchiya, 2003). The unit-cell parameters of olivine, orthopyroxene, garnet and magnetite, if detected (Table S2), were determined using full profile LeBail refinements within the GSAS software package in the EXPGUI interface (Toby, 2001;Larson & Von Dreele, 2004).

Results and Discussion

The three aggregates show distinct mineralogies, mainly differing in proportions of olivine, pyroxene, garnet, and, when present, magnetite (Figure S6). The phase assemblages equilibrated at Martian upper mantle P-T conditions (UM1 and UM2) are comparable in terms of phase proportions and compositions (Table 1), with 50-54 vol.% olivine, 43-44 vol.% orthopyroxene and 3-6 vol.% magnetite, as determined from analysis of SEM and TEM images (see Supporting Information). Differently from the results of [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF], garnet is absent in these samples due to the stabilization of magnetite (Table S3). Clinopyroxene shows a small grain size, i.e. <500 nm, making it only observed in TEM composition mapping but not detectable by SEM and EPMA measurements (Table 1, Table S3). The Fe# of olivine in both assemblages is identical (Fe# = 23), whereas in orthopyroxene, the Fe# of UM1 is 20 compared to a Fe# = 22 in UM2, which can be attributed to the fact that 3 vol.% more Al-Mg-rich magnetite is present in UM1 than in UM2 incorporating some amount of Fe into its structure. Magnetite is observed in these upper mantle assemblages irrespective of the gas flux used in the glass synthesis and the capsule material used for the high P-T equilibrations (Table S1), and this is another major difference with respect to the previous study by [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF] (Table S3). One possible explanation is the use of Re and Pt foils and the Fe 3+ -rich oxide mixtures as starting material in this study, in contrast to the more reducing conditions imposed by the capsule materials (graphite-lined Pt capsules) containing the previously reduced starting material used by [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF]. Noteworthy, a recent study reported the stability of magnetite next to majoritic garnet and pyroxene when starting from a bulk composition similar to the one investigated here [START_REF] Tao | Experimental investigation of Fe3+rich majoritic garnet and its effect on majorite geobarometer[END_REF]. More importantly, the presence of Fe 3+ bearing phases, and possibly magnetite, in the Martian mantle assemblage is reasonable under the assumption of an Fe-rich and oxidized uppermost mantle [START_REF] Tuff | Volcanism on Mars controlled by early oxidation of the upper mantle[END_REF]. Still, ferrian minerals have not been taken into account in any of the mineralogy and velocity models considered so far (e.g., Smrekar et al., 2019 and references therein).

The phase assemblage of the run under mid-mantle P-T condition (MM1) contains 23 vol.% of garnet, 30 vol.% of orthopyroxene and 47 vol.% of olivine (Table 1). The Fe# in olivine and orthopyroxene are 23 and 18, respectively, with the garnet being rich in Fe 3+ , containing 23 mol.% skiagite (Fe 2+ 3Fe 3+ 2Si3O12) and 18 mol.% andradite (Ca3Fe 3+ 2Si3O12) components. Such an enrichment in Fe 3+ is significant, especially if compared to the previous phase equilibrium study [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF].

The compressional and shear wave velocities for all three runs increase monotonically with pressure and decrease with increasing temperatures (e.g. UM2 in Figures S4). Both vP and vS for the upper mantle experiments UM1 and UM2 are very similar, within mutual uncertainties (Table S4). This observation can be readily explained in view of the similar phase assemblages and mineralogy of the two samples. The wave velocities vP and vS for the mid-mantle experiment MM1 are higher than those measured for the upper mantle experiments (Table S4) and with a larger increase with increasing pressure, presumably related to the presence of garnet in this phase assemblage (Table S1).

Using the derivatives of vP and vS with respect to T and P from the experimental dataset of this study (Table S5), vP and vS of each phase assemblage are calculated as a function of depth along two different Martian mantle temperature profiles (areotherms) (Figure 1). These areotherms are chosen as representative cases out of a larger series established by recent 3-D thermal evolution models (Plesa et al., 2018). Both areotherms have been calculated assuming a crustal heat production rate compatible with the surface abundance of heat producing elements measured by the gamma ray spectrometer instrument onboard Mars Odyssey [START_REF] Taylor | Bulk composition and early differentiation of Mars[END_REF][START_REF] Hahn | Martian surface heat production and crustal heat flow from Mars Odyssey gamma-ray spectrometry[END_REF]. Case 12 is representative for a relatively thin crust (45 km) that contains about 44% of the total bulk amount of heat producing elements and is characterized by a steep temperature increase down to depths of ~350 km, from where the temperature profile flattens out towards the lower mantle (Figure S7). Case 56 considers a thick crust (87.1 km) that contains most of the heat producing elements, i.e., 94%, leading to an overall less pronounced temperature increase at shallow depths. In this latter case, the areotherm starts flattening out at a depth of 800 km reaching a temperature approximately 420 K lower compared to case 12 (Figure S7). From our experimental data we predict a low-velocity layer, i.e., a decrease in wave velocities with increasing depth, at shallow mantle depths (~150-350 km depths range) for both vP and vS for the thin-crust areotherm (case 12). For the thick-crust areotherm (case 56), a low-velocity layer is present for vS, whereas variation of vP with depth (initial small reduction followed by a slight increase) are too small to derive firm conclusions. Results along more areotherms for thin and thick crust and different thermal evolution models (Plesa et al., 2018) are shown in the Supporting Information (Figure S8). Different temperature profiles affect absolute values of velocities, but not the overall behavior with depth for the two classes of areotherms (i.e. thin and thick crust). The occurrence of extended lowshear-velocity zones is predicted for all phase assemblages considered here and can be explained by the temperature-induced velocity decrease prevailing over the pressure-induced velocity increase at depths shallower than 400 km.

The obtained experimental results along the case 12 areotherm are compared to wave velocities modelled along the same areotherm for aggregates having the same bulk composition of the experiments but assuming all Fe to be present as Fe 2+ due to the absence of Fe 3+ in end-member phases in the Stixrude & Lithgow-Bertelloni ( 2011) database (Figure 2). Specifically, equilibrium phase assemblages are determined by Gibbs free energy minimization using Perple_X [START_REF] Connolly | The geodynamic equation of state: What and how[END_REF]) (Table S3), and the elastic properties of the aggregates are computed with the thermodynamic formulation and thermodynamic database of [START_REF] Stixrude | Thermodynamics of mantle minerals -I. Physical properties[END_REF], 2011). The overall agreement between experiment-based velocity profiles and models is good, in particular for the mid-mantle assembly, where the two are within uncertainties. For uppermantle assemblies modeled vP is below (UM1) or above (UM2) experiment-based velocity profiles by ~1%, while vS is below by ~2% in the case of UM1 or close for UM2. Accordingly, neglecting the presence of Fe 3+ bearing minerals leads to less than 2% difference between the results obtained in our experiments and self-consistent thermodynamic models.

The experimental results along the areotherm case 12 are also compared to mantle velocity models [START_REF] Smrekar | Pre-mission InSights on the Interior of Mars[END_REF] computed along the same areotherm using the thermodynamic approach detailed above (Figure 3a). Several bulk chemistry compositions proposed for the mantle of Mars are considered here -MA: [START_REF] Morgan | Chemical composition of Mars[END_REF]LF: Lodders & Fegley, 1997;EH45 and EH70: Sanloup et al., 1999;MM: Mohapatra & Murty, 2003;TA: Taylor, 2013;and YO: Yoshizaki & McDonough, 2020; together with the bulk chemistry of our starting material (Table 1). It is important to note that the experiment-based results are computed for constant mineralogy, corresponding to equilibrium mineralogy at depth ~250 km (UM1 and UM2) and ~650-680 km (MM1). Thermodynamic models use Gibbs free-energy minimization and thus have a mineralogy that evolves with depth. Both vP and vS obtained from the ultrasonic experiments and modeled for our starting composition plot on the lower end of the velocities calculated with thermodynamic models, close to estimations for the composition EH70 suggested by [START_REF] Sanloup | A simple chondritic model of Mars[END_REF], despite the significant difference in major element concentration (+7.3 wt.% SiO2; -0.53 wt.% Al2O3, -6.7 wt.% FeO, -3.2 wt.% MgO and -0.43 wt.% CaO of EH70 with respect to our composition).

Assuming the areotherm case 56, both the experimental and calculated wave velocities increase slightly, without changes to the relative difference. Overall, we observe generally faster wave velocities from the previously published models compared to our experiments and model. The TA model, which has a bulk chemistry very close to that of our experiments, shows a vP/vS ratio very close to that modeled for our starting composition (Figure 3b) but absolute velocities systematically above our measurements (difference up to ~0.23 km/s and up to ~0.1 km/s for vP and vS, respectively). This offset can be explained by the larger amounts of clinopyroxene in the TA model (up to 19.1 %) and less orthopyroxene (13.3 %) compared to the phase proportion modeled for the starting material composition of this study, resulting in the stability of only 5.6 % of clinopyroxene and a larger amount of orthopyroxene (up to 28.2 %) (Table S3). Indeed, at pressure and temperature conditions of the case 12 areotherm clinopyroxene is faster compared to orthopyroxene [START_REF] Flesch | Sound velocities of polycrystalline MgSiO3-orthopyroxene to 10 GPa at room temperature[END_REF][START_REF] Kung | In-situ elasticity measurement for the unquenchable high-pressure clinopyroxene phase: Implication for the upper mantle[END_REF][START_REF] Li | Elasticity of diopside to 8GPa and 1073K and implications for the upper mantle[END_REF][START_REF] Kung | High-temperature elasticity of polycrystalline orthoenstatite (MgSiO 3)[END_REF].

In comparison to a typical mineralogy of the Earth's mantle [START_REF] Ringwood | Composition of the crust and upper mantle. The Earth's Crust and Upper Mantle[END_REF], calculated along the same areotherms (Figure 3a), the wave velocities through Mars' upper and mid-mantle are significantly slower than those of the Earth, ~0.5 km/s for vP and ~0.3 km/s for vS. This can be attributed not only to the higher garnet and olivine over pyroxene content in the Earth, but also to the lower Fe# of ~10 assumed for the Earth versus a Fe# of ~25 expected for Mars, which has a significant effect on the bulk wave velocities [START_REF] Stixrude | Thermodynamics of mantle minerals -II. Phase equilibria[END_REF].

Irrespective of absolute velocity values, a low-S-wave-velocity layer at shallow depths is expected according both experiments and for all self-consistent thermodynamic models, further supporting the independence of this finding from mineralogy. Available Martian seismic records suggest a possible S-wave shadow zone at an epicentral distance range of ~40-60° from the InSight landing site [START_REF] Giardini | The seismicity of Mars[END_REF][START_REF] Khan | Upper mantle structure of Mars from InSight seismic data[END_REF], which might result from the presence of a low-velocity layer in the upper mantle. Our experimental results and the above-presented considerations provide a direct explanation for this observation and support the existence of a low-S-wave-velocity zone in the upper mantle of Mars [START_REF] Khan | Upper mantle structure of Mars from InSight seismic data[END_REF] as a direct consequence of the dominant temperature-induced effects on seismic velocities along all areotherms considered in this study (Plesa et al., 2018). Whether or not a low-P-wave-velocity layer will be observed on Mars would provide constraints on the thermal structure of the upper mantle, with a low-P-wavevelocity zone being a strong indication of hot temperatures in the shallow mantle, such as the temperature profile case 12 of Plesa et al. (2018).

The vP/vS ratio between 100 and 800 km depth, which directly affects the difference in the arrivals between P-waves and S-waves, is shown in Figure 3b. This ratio shows a smooth monotonic increase for all three Martian compositions investigated in this study. Within experimental uncertainties, shown as shaded areas, vP/vS is the same for the two upper mantle assemblages. Furthermore, the experimental data clearly falls into the same range as calculated for different Martian compositions using thermodynamic end-member data by [START_REF] Stixrude | Thermodynamics of mantle minerals -II. Phase equilibria[END_REF]. Compared to a typical vP/vS for Earth's mantle composition [START_REF] Ringwood | Composition of the crust and upper mantle. The Earth's Crust and Upper Mantle[END_REF] along a Martian temperature profile, the bulk vP/vS for Mars is larger, especially for depths > 400 km. Another interesting observation is that the dependence of vP/vS on pressure in our assemblages does not significantly vary along isotherms, for temperatures in the range of 300 -2500 K tested by extrapolation of results in this study. Thus, the measured vP/vS ratio provides important constraints for the estimation of the epicentral distance of marsquakes recorded by InSight.

Conclusions

In this study, we report the stability of magnetite, although to a small fraction, in Martian mantle phase assemblages at conditions directly pertinent to its upper mantle, especially in an oxidized environment. Additionally, the presence of Fe 3+ -bearing garnets was inferred in the midmantle phase assemblage. We thus argue for a Martian mantle mineralogy more complex compared to what has been considered so far (e.g. [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF]Smrekar et al., 2019 and references therein). Sound velocity measurements have been carried out at high pressure and high temperature and suggest seismic wave velocities possibly slower compared to the majority of the velocity models that have been previously proposed for Mars. Whether the difference between experiments and thermodynamic models is the result of the presence of iron as Fe 3+ or of the different phase proportions in our assemblages compare to those predicted by existing thermodynamic equilibrium calculations remains to be assessed. Nevertheless, the presence of low-S-wave-velocity layer extending between 150 and 350 km depth, and possibly up to ~500 km depth, is independent of the considered mineralogy and thermal structure and is therefore a robust conclusion that is compatible with the findings from recent InSight seismic observations [START_REF] Giardini | The seismicity of Mars[END_REF][START_REF] Khan | Upper mantle structure of Mars from InSight seismic data[END_REF]. The vP/vS ratio depends only weakly on mantle composition and mineralogy and is largely independent of temperature. Hence the vP/vS ratio that has been measured in this study provides direct constraints for the estimation of the epicentral distance of quakes and guidance for the interpretation of seismic data collected on Mars. S3).

Tables

b The TEM or EMP in the parathesis indicate the composition was obtained by TEM or EMP analysis, respectively. Pressure, temperature and duration of the synthesis run are also indicated.

Introduction

In section S1 to S4 we provide more detailed information on sample synthesis, sample characterization, in situ synchrotron experiments and data analysis. In section S5 we provide additional discussion about sample's density, bulk and shear moduli. Figure S1 shows the P-T path followed in the three in-situ ultrasonic interferometry and X-ray diffraction experiments. Figure S2 and S3 show a representative ultrasonic signal and a typical X-ray radiography image, respectively, of sample MM1 at 7.9 GPa and 1000 K.

Figure S4 shows the wave velocities and densities versus pressure of sample UM2 collected along isotherms. Figure S5 shows typical energy-dispersive X-ray diffraction pattern collected for the upper mantle experiment UM1. Figure S6 shows representative backscattered electron images for the three mineralogical assemblages here investigated.

Figure S7 shows two representative areotherms used in this study selected from the larger set in Plesa et al. (2018). Figure S8 provides wave velocities as a function of depth along six areotherms representative for Mars. Figure S9 shows the bulk density of studied mineralogical assemblages versus pressure at 300 K. Figure S10 shows the bulk density of the three phase assemblages investigated in this study along the case 12 areotherm. Figure S11 shows the adiabatic bulk modulus KS and shear modulus G for the three bulk assemblages studied versus depth. Table S1 provides the experimental conditions and run products of all synthesis experiments. Table S2 reports unit-cell lattice parameters and densities of all phases present in the three mineralogical assemblages as determined from energy-dispersive X-ray diffraction. Table S3 outlines the phase proportion obtained in this study by experimental observations and modeling in comparison with the previous report by [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF]. Université, France or JEOL JSM-7000F at the Geodynamics Research Center (GRC), Ehime University, Japan. All samples showed a homogenous grain distribution with a typical grain size of ~1-5 µm, and clear grain boundaries, typical of the so-called equilibrium texture (Figure S6). et al., 2016) using TEM/EMP data (Table 1). This approach has been proven to be a suitable method to determine Fe 3+ /ΣFe when Mössbauer spectroscopy or other comparable techniques are not practical, with specific applications to garnets (Quinn et al., 2016). The main uncertainties of this method derive from the composition determination, especially from Si, as Si 4+ is the most abundant cation and has the highest charge. Accordingly small uncertainties are expected for magnetite and garnet.

Major element composition of each sample was measured on a Zeiss Auriga 40 FEG-SEM equipped with a Bruker EDXS detector. Quantification was done using the Klines of Fe, Mg, Al, Si, Ca, and O was obtained by stoichiometry, using the PhiRhoZ routine with pre-loaded standards. Large window scans (200 x 200 µm) were performed over multiple areas of the sample to show homogeneity and estimate analytical uncertainties (Table 1, method 1). The bulk composition of each sample was also assessed from the phase proportions and compositions of each phase (Table 1, method 2). Mineral compositions and phase proportions were checked before and/or after synchrotron measurements (see below). All compositions and phase proportions are reported in Table 1.

S3. In situ synchrotron experiments

Sound wave velocity and density measurements were performed at BL04B1, SPring-8, Japan. Three in situ experiments using combined ultrasonic interferometry, Xray radiography and X-ray diffraction were carried out under high-pressure and hightemperature conditions in the DIA-type 1500 tons multi-anvil apparatus (SPEED1500), equipped with eight cubic tungsten carbide (WC) anvils. The high-quality samples were carefully polished at GRC to a thickness of 1.0 mm maintaining parallel and mirror-quality surfaces using ~0.5 µm diamond-coated polishing tools. The samples were surrounded by a MgO sleeve and placed in a modified 14/8 Co-doped MgO octahedron serving as pressure medium, equipped with a cylindrical Re foil heater. Two 2 µm thick Au foils were placed at the top and bottom of the sample, which acted as sample length indicator as well as to enhance the bonding between the individual layers. An Al2O3 corundum buffer rod was placed on top of the sample being in direct contact with the WC anvil. A pre-pressed pellet of MgO + Au + BN was placed at the back of the sample to enhance hydrostatic conditions at high pressure and to serve as pressure sensor during the experiments. The temperature was monitored with a D-type (W97Re3-W75Re25) thermocouple, whose hot-junction was placed at the bottom of the pressure marker. At first, the press load was increased to reach the target pressure of the first heating cycle, then temperature was increased to either 1000 or 1200 K (MM1) or 900 K (UM1 and UM2) (see Table S4 for details) and kept stable for 30 minutes to 1 hour to release deviatoric stresses that might have developed during cold compression. Data were collected for decreasing temperature, from 1000 K or 1200 K (MM1) or 900 K (UM1 and UM2) down to 300 K in steps of 200-300 K. The press load was then further increased and the overall procedure was repeated for three subsequent cycles (Figure S1). For ultrasonic measurements, a piezoelectric LiNbO3 transducer was attached with a small amount of epoxy on a truncation of one WC anvil that was polished to mirror-quality. By applying alternating current, S-and P-waves were produced at the resonance frequencies of 40 and 60 MHz, respectively. A typical P-wave signal at high pressure and temperature can be seen in Figure S2. The variation in sample length at different pressure and temperature conditions was measured by imaging the two Au foils placed at both ends of the sample using X-ray radiography (Figure S3). Radiographs were recorded using a high-resolution CCD camera (C11440, Hamamatsu Photonics, Iwata, Japan) with 1 pixel resolution corresponding to ~1.47 µm. Energy-dispersive diffraction patterns between 0 and 145 keV were collected both for the sample and pressure marker on an SSD detector at a fixed 2θ angle of ~ 6°, which was calibrated with the unit-cell volume of Au in the pressure marker before compressing each experiment. An example of a representative XRD pattern for UM1 collected at 5.5(1) GPa and 900 K is shown in Figure S5.

S4. Data analysis

The unit-cell volumes of the Au pressure marker were used to determine the pressure at each temperature point according to the P-V-T equation of state (Tsuchiya, 2003). The unit-cell parameters of olivine, orthopyroxene, garnet and, when possible, magnetite (Table S2) were determined using full profile LeBail refinements within the GSAS software package in the EXPGUI interface (Toby, 2001;Larson & Von Dreele, 2004). The densities (ρ) of the phase assemblage were then calculated according to the phase proportions obtained from SEM/TEM images and the volumes measured by XRD.

The unit-cell volumes of magnetite could be clearly determined by XRD only for sample UM1 and not for sample UM2 due to the low amount of magnetite present in the phase assemblages (3 vol.%). In this latter case, the volumes at specific pressures and temperatures were calculated using the thermal equation of state proposed for MgFe2O4 by Levy et al., (2004), due to the missing thermal expansion information of the end-member magnetite (Fe3O4) at high pressures. in order to get the pressure and temperature dependence of each mineralogical assemblage (see Figure S4).

S5. Bulk density, bulk and shear moduli

The room temperature bulk densities for each experiment versus pressure are shown in Figure S9. For the upper mantle experiments UM1 and UM2, the larger density of UM1 reflects the higher abundance of magnetite (6 vol.%) in this phase assemblage relative to UM2 (3 vol.%). Similar to the wave velocities, the bulk density at ambient temperature of the mid-mantle experiment MM1 follows a smooth transition from the bulk densities of the upper mantle assemblages. The evolution of the bulk density at constant mineralogy along the areotherm case 12 is shown in Figure S10. Similar to the wave velocities, an initial decrease in densities can be observed due to the larger temperature effect relative to the pressure effect. Compared to thermodynamic end-member models calculated with Perple_X, which however are not for a constant mineralogy but for a mineralogy evolving along the areotherm, the bulk densities are comparable to the experimentally determined densities, although their phase assemblages containing olivine, pyroxene and garnet differ quite significantly depending on the bulk starting composition.

With knowledge of both bulk density and wave velocities, it is possible to calculate the bulk and shear moduli KS and G following: Table S2. Unit-cell lattice parameters and densities of all phases present in the three mineralogical assemblages studied using energydispersive X-ray diffraction.

𝑣 𝑃 = √ 𝐾 𝑆 + 4
P (GPa) T (K) Ol Py Gt/Mag a (Å) b (Å) c(Å) V (Å 3 ) ρ (g/cm 3 ) a (Å) b (Å) c (Å) V (Å 3
) ρ (g/cm 3 ) a (Å) V (Å 3 ) ρ (g/cm 3 ) Table S3. Experimentally observed phase proportion (in weight percent) in comparison with results obtained for the bulk composition of the starting material by Gibbs free energy minimization at the same pressure and temperature conditions. For completeness we also report the phase proportion at the same pressure experimentally obtained by [START_REF] Bertka | Mineralogy of the Martian interior up to core-mantle boundary pressures[END_REF] for the [START_REF] Dreibus | Mars, a volatile-rich planet[END_REF] bulk chemistry. Phases are olivine (Ol), orthopyroxene (Opx), clinopyroxene (Cpx), garnet (Gt) and magnetite (Mag). 1 and the density of each phase at room pressure and 300 K in Table S2. b Clinopyroxene, expected in our samples according to estimated bulk chemistry (Bulk (1)) and the bulk chemistry of the starting material (see Table 1), can't be detected in the SEM image and EMP composition point measurement due to its small grain size and amount. c The phase proportion from Bertka & Fei (1997) (their Figure 8).

  and temperature dependences of each mineralogical assemblage were derived by global fitting the respective P-T-vP-vS datasets to a linear function following: 𝑣 𝑃,𝑆 = 𝑣 0𝑃,𝑆 + 𝜕(𝑣 𝑃,𝑆 ) 𝜕𝑃 * 𝑃 + 𝜕(𝑣 𝑃,𝑆 ) 𝜕𝑇 * (𝑇 -300𝐾) ⁄ ⁄(see also FigureS4).

  the DLR Management Board Young Research Group Leader Program and the Executive Board Member for Space Research and Technology. The Focused Ion Beam (FIB) and Scanning Electron Microscope (SEM) facility at IMPMC is supported by Région Ile de France grant SESAME 2006 N°I-07-593/R, INSU-CNRS, Institute de Physique (INP)-CNRS, University Pierre et Marie Curie-Paris 6, and by the French National Research Agency (ANR) grant ANR-07-BLAN-0124-01. Infrastructures and services at the Geodynamics Research Center (GRC), Ehime University, were made available through the PRIUS program. The synchrotron radiation experiments were performed under SPring-8 Non-Proprietary Research Proposals Numbers 2019A1473 and 2019B1305, in collaboration with the partner user program (2019A0069 and 2019B0069). This paper is InSight Contribution 199. Datasets for this research (main paper and Supporting Information) are available at https://doi.org/10.5281/zenodo.4709036 (doi: 10.5281/zenodo.4709036).

Figure 1 .

 1 Figure 1. Experiment-based compressional vP and shear wave velocities vS of three different mineralogical assemblages versus depth along two different areotherms. The black and orange lines represent the two areotherms assuming a thin crust (case12, 45 km) or thick crust (case 56, 87.1 km), respectively. Extended low-velocity zones are visible for vS in both cases. VP shows a low-velocity zone only for the thin crust areotherm (case 12). The shaded areas represent the experimental uncertainties. Results are for constant mineralogy, corresponding to equilibrium mineralogy at depth ~250 km (UM1 and UM2) and ~650-680 km (MM1), indicated as stars in the figure.

Figure 2 .

 2 Figure 2. Experiment-based compressional vP and shear wave velocities vS of three different mineralogical assemblages versus depth along a thin crust areotherm (case 12, 45 km average crustal thickness, black) in comparison to velocities according to

Figure 3 .

 3 Figure 3. Experiment-based compressional vP and shear wave velocities vS (a) and vP/vS (b) of three different mineralogical assemblages studied here versus depth along the thin crust areotherm case 12 (45 km average crustal thickness). The grey shaded areas represent the experimental uncertainties. Self-consistent thermodynamic models (colored lines) calculated for different bulk starting compositions proposed in literature show wave velocities faster than the experimental ones. We note, however, that experimental results are for constant mineralogy, corresponding to equilibrium mineralogy at depth ~250 km (UM1 and UM2) and ~650-680 km (MM1), indicated as stars in the figure, while selfconsistent thermodynamic models have been computed for mineralogy evolving with depth.

  In order to analyze the compositions of all phases present in the individual experiments, samples UM2 and MM1 have been studied by electron microprobe (EMP), performing focused beam analyses at 15 kV, 10 nA using the Cameca SX-FIVE at Centre Camparis, Sorbonne Université. UM1 sample, showing grain sizes below ~1 µm, was thinned at IPGP by a Zeiss Auriga 40 focused ion beam (FIB) instrument (~50-100 nm thick slices) and analyzed at IMPMC using a JEOL 2100F transmission electron microscope (TEM) operating at 200 kV. Phase proportions of each sample were determined from the SEM and TEM images analyzing the greyscale histograms using the software ImageJ (individual greyscale maxima represent the different phases). Due to the technical difficulties of measuring Fe 3+ vs. Fe 2+ content of each of the very small phases present in our assembly by more direct methods (e.g., Mössbauer spectroscopy), the valence states of iron in Py, Gt and Mag were estimated by charge balance method (Quinn

Vp

  and vS were calculated according 𝑣 = 2𝐿 𝑡 with L representing the sample length obtained from X-ray radiography and t being the two-way travel time taken from ultrasonic data at each pressure and temperature condition measured during the experiments. The P-𝑆 = 𝑣 0𝑃,𝑆 + 𝜕(𝑣 𝑃,𝑆 ) 𝜕𝑃 * 𝑃 + 𝜕(𝑣 𝑃,𝑆 ) 𝜕𝑇 * (𝑇 -300𝐾) ⁄ ⁄

  and G are shown in FigureS11versus depth along the thin crust areotherm case 12 and are compared to self-consistent thermodynamic models. The elastic moduli clearly reflect the presence of magnetite and Fe 3+ -rich garnet in the experimental phase assemblages, leading to lower bulk and shear moduli compared to the self-consistent thermodynamic models, and hence indicate slower wave velocities (Figure3a). The availability of only Mg-or Fe 2+ end-member data does not allow the direct comparison of thermodynamic models and experiments for the same mineralogy.

Figure S1 .

 S1 Figure S1. P-T path of the three in situ ultrasonic interferometry and diffraction experiments at BL04B1, SPring-8, Japan. The upper mantle experiments UM1 and UM2 (a) were first compressed to target pressure before increasing the temperature to 900 K in

Figure S2 .

 S2 Figure S2.Representative ultrasonic signal collected for MM1 at 7.9 GPa and 1000 K.

Figure S3 .

 S3 Figure S3. X-ray radiography image collected on MM1 at 7.9 GPa and 1000 K. By measuring the distance between the two gold foils, the sample length at each individual pressure and temperature point can be determined.

Figure S4 .

 S4 Figure S4. Wave velocities vP (a), vS (b) and bulk density (c) versus pressure collected along isotherms for the upper mantle experiment UM2. When not visible, propagated uncertainties are smaller than the symbols. A linear fitting procedure allows the determination of the pressure and temperature dependence of vP, vS and the bulk density.

Figure S5 .

 S5 Figure S5. Representative energy-dispersive X-ray diffraction pattern collected for the upper mantle experiment UM1 at 5.5(1) GPa and 900 K. Blue thick marks correspond to indexed reflections of magnetite, red thick marks to orthopyroxene and black thick marks to olivine.

Figure S6 .

 S6 Figure S6. Backscattered electron images for the three mineralogical assemblages investigated in this study collected on the samples recovered after the ultrasonic experiments. The upper mantle experiment UM2 (a) and UM1 (b) contain a mixture of olivine (ol), orthopyroxene (opx) and magnetite (mag). The mid-mantle phase assemblage MM1 contains olivine, orthopyroxene and garnet (gar). The black areas in a) and c) are voids in the samples formed during sample preparation for SEM analyses and were not present during the high pressure and temperature ultrasonic experiments.

Figure S7 .

 S7 Figure S7. Two representative areotherms, case 12 (black) and case 56 (orange) for the Martian mantle used in this study. Both areotherms are part of a set of temperature profiles that were obtained from 3-D thermal evolution models as discussed in Plesa et al. (2018).

Figure S8 .

 S8 Figure S8. Wave velocities vP and vS as a function of depth along six areotherms representative for Mars. The areotherms consider different mantle rheologies, core radii and crustal thicknesses. The present-day crustal heat production rate is similar to the average value derived from the surface abundances of heat producing elements. Cases 11, 12, 26 and 109 are for thin crust areotherms (45 km average crustal thickness) and cases 55 and 56 are for thick crust areotherms (87.1 km average crustal thickness)(Plesa et al., 2018). Thin crust areotherms show low-velocity zones for both vP and vS whereas the thick crust areotherms only show low velocity zones for vS.

Figure S10 .

 S10 Figure S10. Bulk density of the three phase assemblages investigated in this study along the case 12 areotherm. For comparison, models based on thermodynamic end-member databases are plotted. N.B. experimental densities are for constant mineralogy (equilibrated at depth corresponding to either ~250 km or ~650-680 km), while models are for mineralogy changing with depth.

Figure S11 .

 S11 Figure S11. Bulk modulus KS and shear modulus G for the three bulk assemblages studied versus depth. For comparison, the elastic moduli calculated from the self-consistent thermodynamic models are shown.

Table 1 .

 1 Phase proportions of measured samples and compositions of constituent phases.

	The bulk composition of each sample is estimated in two different ways: (1) by SEM
	analysis performed over large window scans (200 x 200 μm) and averaged over multiple
	areas of the sample; and (2) from the phase proportions determined from the SEM/TEM
	images and compositions of each phase. In case (1) all Fe is counted as FeO. The starting
	chemical composition of oxide mixture is in wt.% SiO2: 44.6; Al2O3: 3.1; FeO: 18.5; MgO:
	31.2; CaO: 2.5. Phases are olivine (Ol), orthopyroxene (Opx), garnet (Gt) and magnetite
	(Mag).
	Run NO.	Ol	Opx a	Gt	Mag	Bulk (1)	Bulk (2)
	UM1 (TEM) 3 GPa, 1373K, 1 hour b
	SiO2	39.0	53.2		1.4	45.7	43.2
	Al2O3	-	3.7		10.0	3.2	2.2
	FeO c	21.1	11.8		24.1	19.3	17.1
	Fe2O3 c	-	1.9		59.1	-	4.1
	MgO	39.8	27.4		5.3	29.7	32.4
	CaO	0.1	2.0		0.1	2.0	1.0
	UM2 (EMP) 3 GPa, 1473K, 24 hours
	SiO2	39.4	50.9		1.6	46.4	42.4
	Al2O3	0.5	4.3		14.3	3.3	3.0
	FeO	20.6	7.7		21.2	18.1	14.9
	Fe2O3	-	6.9		55.2	-	6.1
	MgO	39.0	28.3		7.5	29.5	32.5
	CaO	0.5	1.9		0.1	2.8	1.1
	MM1 d (EMP) 8 GPa, 1473K, 23 hours
	SiO2	39.1	55.7	40.3		45.8	44.3
	Al2O3	0.2	0.5	10.9		3.3	2.9
	FeO	21.1	11.3	8.8		18.6	15.3
	Fe2O3	-	1.3	17.1		-	4.4
	MgO	39.2	30.0	15.9		29.8	31.0
	CaO	0.4	1.2	6.9		2.5	2.2
	Phase proportion
	UM1	44	50		6
	UM2	43	54		3
	MM1	47	30	23
	a Only orthopyroxene was counted here due to the non-detection of clinopyroxene in SEM and EMP
	measurement. (see also table

  Table S4 shows the experimental pressure and temperature, compressional (vP) and shear (vS) wave velocities, and densities of measured samples and Table S5 reports the results of 2-Dimensional global linear fitting for velocity and density according to the equation 𝑀 = 𝑀 0 + 𝜕𝑀 𝜕𝑃 * 𝑃 + 𝜕𝑀 𝜕𝑇 * (𝑇 -300 𝐾)

	⁄	⁄	with M=
	{vP, vS, ρ}.

Table S1 .

 S1 Experimental conditions and run products of all synthesis experiments. The gas flux used in aerodynamic levitation laser furnace for making glass. b Run products are Ol, olivine; Py, pyroxene; Mag, magnetite; Gt, garnet.

	Run	Gas a	Capsule P (GPa)	T (K) Time (hr)	Product b	Grain
							size (µm)
	UM1	Ar	Re	3	1373 1	Ol, Py, Mag 1-2
	UM2	Ar+CO2+H2	Re+Pt	3	1473 24	Ol, Py, Mag 2-5
	MM1	Ar	Re	8	1473 23	Ol, Py, Gt	1-3

a

  The weight percentage is derived from the volume percentage in Table

	Condition	Ol	Opx	Cpx	Gt	Mag
		This study a
	3 GPa, 1373K 43.1 48.9	-b		8.0	UM1
	3 GPa, 1473K 43.3 52.8	-		3.9	UM2
	8 GPa, 1473K 46.5 28.6	-	24.9		MM1
			Modelling
	3 GPa, 1373K 51.7 32.0	4.7 11.6
	3 GPa, 1473K 51.7 34.1	3.4 10.8
	8 GPa, 1473K 51.8 28.2	5.6 14.4
		Bertka & Fei (1997) c
	3 GPa	58.4 18.9 12.8 9.9
	8 GPa	58.4	8.2	23.5 9.9

a

Table S4 .

 S4 Experimental pressure and temperature, compressional (vP) and shear (vS) wave velocities, and densities of measured samples.

	P (GPa)	T (K)	vP (km/s)	vS (km/s)	ρ (g/cm 3 )
			UM1
	0.0001(1)	300			3.5244(3)
	2.76(2)	900	7.88(1)	4.44(1)	3.5356(6)
	2.23(3)	700	7.94(1)	4.48(1)	3.5451(6)
	1.68(4)	500	7.99(1)	4.52(1)	3.5566(6)
	1.19(8)	300	8.04(1)	4.55(1)	3.5655(6)
	3.79(5)	900	7.97(1)	4.44(1)	3.5596(6)
	3.10(7)	700	8.02(1)	4.49(1)	3.5704(5)

Table S5 .

 S5 Results of 2-Dimensional global linear fitting for velocity and density 𝑴 = 𝑴 𝟎 + 𝝏𝑴 𝝏𝑷 * 𝑷 + 𝝏𝑴 𝝏𝑻 * (𝑻 -𝟑𝟎𝟎 𝑲) ⁄ ⁄ with M= {vP, vS, ρ}.

	Sample	vP0	∂vP/∂P	∂vP/∂T	vS0	∂vS/∂P	∂vS/∂T	ρ0	∂ρ/∂P	∂ρ/∂T
		(km•s -1 )	(km•s -1 •GPa -1 )	(km•s -1 •K -	(km•s -1 )	(km•s -1 •GPa -1 )	(km•s -1 •K -	(g •cm -3 )	(g •cm -3 •GPa -1 )	(g •cm -3 •K -1 )
				1 )			1 )
	UM1	7.94(1)	0.080(3)	-0.00045(2)	4.53(1)	0.017(2)	-0.00024(1)	3.532(2)	0.0282(7)	-0.000128(4)
	UM2	7.89(1)	0.083(2)	-0.00048(1)	4.50(1)	0.017(2)	-0.00024(1)	3.488(2)	0.0276(6)	-0.000120(4)
	MM1	8.01(1)	0.098(5)	-0.00056(2)	4.54(2)	0.039(3)	-0.00037(2)	3.550(4)	0.0258(7)	-0.000124(5)

c The valence states iron in Opx, Gt and Mag was calculated according to the charge balance method(Quinn et al., 2016). See supplement S2 for more details.

2.64(10)

Acknowledgments

We thank two anonymous reviewers for their thoughtful and constructive comments that have improved an earlier version of this paper. We also thank the editor A. J. Dombard for handling the revision process of the manuscript. The authors wish to thank Imène Estève and Stéphanie Delbrel for their help with ion-milling and sample analysis by SEM at the Institut de Minéralogie de Physique des Matériaux et de Cosmochimie (IMPMC, Paris) and Michel Fialin and Nicolas Rividi for their help during microprobe analysis at the Centre Camparis, Sorbonne Université (Paris, France). We would like to thank Stephan Borensztajn for his help with the FIB cutting at IPGP (Paris, France). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation Programme (Grant agreement 724690). This work was supported by the French Space Agency (CNES), focused on SEIS instrument of the InSight mission. AR is financially supported by the Belgian PRODEX program managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office. ACP gratefully acknowledges the financial support and endorsement from

S1. Samples' synthesis

Starting material with composition in weight percent of oxides close to the one proposed by Taylor (2013) (oxides expected to be present at a level below 1 wt.% were not included in the starting material) was prepared from synthetic SiO2, Al2O3, Fe2O3, MgO and CaCO3 powders that were ground in an agate mortar under ethanol for 1 h to obtain a homogeneous mixture. Homogeneous glass spheres (diameter ~ 2.5-3 mm) were then produced at the Institut de Physique du Globe de Paris (IPGP), France, by fusing and quenching powder mixtures in an aerodynamic levitation laser furnace under different gas fluxes, employing gas mixes of Ar or Ar+CO2+H2 (Table S1). The resulting glass spheres were further polished into a cylindrical shape of 2 mm diameter and 1.5 mm length. Two glass cylinders (UM1, MM1) were encapsulated by a Re foil of 25 µm thickness (Table S1). An additional glass cylinder (UM2) was prepared with a double capsule design using Re foil of 25 µm thickness as inner capsule and Pt foil of 20 µm thickness as outer capsule material (Table S1). Two high pressure and high temperature syntheses (UM1, MM1) were performed in the Walker-type multi-anvil apparatus installed at IPGP at the conditions summarized in Table S1. 14/8 COMPRES assemblies were used, equipped with a graphite furnace, and with the capsule being inserted into a MgO sleeve and placed into the center of the assembly. The temperature was monitored with a D-type thermocouple inserted into a 4-hole alumina tube with the hot junction placed close to the capsule. One experiment (UM2) was conducted in the piston-cylinder apparatus at the IPGP using a ½'' assembly at 3 GPa and 1473 K (heating duration 24 h). The capsule surrounded by a MgO sleeve and MgO spacers was inserted into a cylindrical graphite furnace and talc sleeve which acted as pressure medium. The temperature was recorded with a S-type (Pt90Rh10-Pt100) thermocouple with its hot junction placed near the capsule. All synthesis details are reported in Table S1.

S2. Samples' characterization

All recovered samples were well sintered and further analyzed by a set of electron microscopy techniques. Initial analyses have been conducted either using the Zeiss Ultra55 field emission scanning electron microscope (FE-SEM) at the IMPMC, Sorbonne


Rodrigo Barra
email: rodrigo.barra.novoa@gmail.com

MBA, DBA Novoa Economist

State of the Art and Future Applications of Digital Health in Chile

Keywords: -digital medicine, healthcare, medtech, public health, technology

This article explores how new digital health technologies can transform health care and disease management in Chile and other Latin American economies. Digital Medicine has the potential to democratize healthcare by allowing patients to become more involved in their wellbeing. This can provide instant guidance and information on a healthy lifestyle, which can be tailored to individuals. Constant health monitoring through digital technology can help solve easily diagnosable health problems or provide direct support to patients who are in control of their own health. These transformative technologies are paving the way for the next generation of healthcare. In the article you will discover the implications digital health has on various stakeholders, for example, how digital health benefits patients and healthcare professionals. It will also consider the opportunities and challenges surrounding the technology involved.

Introduction

Technological advances have contributed to the transformation of the healthcare system in many ways. For example, robotic-assisted surgeries and digital records and databases. Change is constant within the healthcare system. It is a growing network that is always looking for ways to improve. Recent technological advances have led to a push in the healthcare system to improve patient care. Digital medicine is a wide range of technological systems. These systems aid in diagnosis and treatment without the constant assistance and guidance of a healthcare professional, such as a physician, surgeon, and nurse. In doing so, they provide independence to patients and allow them to actively monitor the health of citizens [START_REF] Barello | Burnout and somatic symptoms among frontline healthcare professionals at the peak of the Italian COVID-19 pandemic[END_REF][START_REF] Simonetti | The impact of survey nonresponse on estimates of healthcare employee burnout[END_REF].

Methodology

The methodology proposed for the development of the article is based on a rigorous literature review that responds to the new research needs in the field of digital medicine and the use of technological systems in the health sector. We have chosen to analyze international literature and a series of published articles, to lead the reader to a broad scenario that allows to understand at a qualitative level the implications of technological and intellectual evolution in the way clinical information are stored, recorded, and generated. This has made possible the development of digital medicine, potentially more accurate, more efficient, more experimental, and more distributed than current medical practice. The critical steps in the creation of digital medicine involve a careful analysis of the impact that new technologies may have on the development towards a new paradigm of medical care in Chile [START_REF] Day | Digital PCR strategies in the development and analysis of molecular biomarkers for personalized medicine[END_REF][START_REF] Hood | A personal view on systems medicine and the emergence of proactive P4 medicine: predictive, preventive, personalized and participatory[END_REF].

Literature review

Digital health is the convergence of digital technologies with healthcare, lifestyles, and society to improve the efficiency of healthcare delivery and make medications more personalized and accurate. "Digital medicine can be defined as digital technologies and products that directly impact the diagnosis, prevention, monitoring and treatment of a disease, condition or syndrome. This includes telemedicine, digital diagnostic tests, biotechnology, digital therapeutics, and immersive technologies such as virtual and augmented reality" [START_REF] Topol | The patient will see you now: the future of medicine is in your hands[END_REF].

Digital medicine

Objectives of digital medicine to use in the 21st century to transform the public and private healthcare system. Digital medicine uses wearable devices and applications that make it easy to track a patient's condition. This data can then be collected and communicated with physicians from a variety of specialties. Previously, this would have been difficult to do. The goal of digital medicine is to improve patient care delivery by tracking diseases to create personalized disease management and care processes. This can be achieved through various types of technology and devices, such as implantable and wearable technology. The use of digital technology enables constant monitoring of patients to identify the early onset of disease, as well as the patient's daily behavior. Digital medicine can be used for preventive purposes, showing problems that will develop unless corrective action is taken. This can be achieved by using DNA sequencing to identify and diagnose diseases in humans. Big data can also be used to aid in diagnosis and treatment [START_REF] Barra Novoa | Current overview of assistance bioethics committees in Chile[END_REF][START_REF] Susilo | Understanding digital technology trends in healthcare and preventive strategy[END_REF].

As technology brings the world closer together, it can be used to actively engage patients with physicians. Patients and physicians can now communicate with other healthcare professionals around the world to get a second opinion. The use of digital technology also means that physicians can gather a substantial amount of information to detect disease patterns. These could lead to early treatment and prevention of outbreaks. All the above can be made possible by transforming the healthcare system into a digital network. Then, technology can be used to initiate a much-needed change in the delivery of care and the practice of medicine. It is important that physicians, patients, and engineers work closely together when designing digital health systems. This ensures that all stakeholders understand the complex challenges related to healthcare [START_REF] Steinhubl | The emerging field of mobile health[END_REF][START_REF] Becker | mHealth 2.0: experiences, possibilities, and perspectives[END_REF]. End users, such as physicians and patients, must also be involved in the design process to create an efficient solution that meets their needs. Digital tools not only record clinical data, but they also can transform the information and generate medical information. In doing so, digital technology potentially creates a more accurate, more efficient, more experimental, and more distributed medical practice than ever before. As with any new technology, there are critical steps to ensure that the impact of the technology is carefully analyzed and evaluated. This is necessary to ensure that it will create a new way of working in healthcare with minimal disruption [START_REF] Blevins | How Good Ideas Die: Understanding Common Pitfalls of Medtech Innovation[END_REF][START_REF] Purcell | Switzerland's medtech prowess[END_REF].

Traditional medicine

The clinical practice involves an evaluation of a patient by a physician for the purpose of diagnosing, prognosticating, treating, and preventing a disease, condition or syndrome using clinical judgment. This judgment is established by an examination of the patient's medical history, followed by a medical interview and a physical evaluation to identify signs and symptoms. Several diagnostic methods are used to help rule out diseases. These are based on information provided by the patient and diagnostic tests. Throughout the diagnostic process, the patient is required to develop a level of trust in the physician's judgment and proposed treatment. The outcome is based on the patient reporting in detail to the physician about his or her symptoms and the physician's interpretation. Second judgments are rare as they depend on the complexity of the patient's condition and the availability of a specialist. It can also be a costly and time-consuming process. This means that the diagnostic process is mainly between the patient and the physician [START_REF] Hagedoorn | Climate change and human health: impacts, vulnerability and public health[END_REF][START_REF] Vance | Social internet sites as a source of public health information[END_REF].

Key factors of digital medicine

The international literature, and especially that developed in the UK, states that to improve patient care, the key factors that digital medicine should address the following key factors:

1) Disease prevention

This can be achieved using mobile technology, such as wearable devices, implantable devices and smartphones that collect and analyze data to prevent future illness and disease. Preventive action can help people stay well for longer and use fewer hospital services. Digital medicine can help people proactively monitor and manage their health before a condition worsens [START_REF] Kersey | A review of recent developments in fiber optic sensor technology[END_REF][START_REF] Hagedoorn | Climate change and human health: impacts, vulnerability and public health[END_REF]. 2) Early diagnosis This is possible by using technology to monitor a patient's health. As an example, wearable technology can be used to track conditions prior to a diagnosis. In addition, a heart rate monitor can inform the user of changes in their heart rate, prompting them to speak to a medical professional. This leads to treatment of a previously undiagnosed condition. Another example of how digital medicine can enable early diagnosis is software that can be used to digitally observe a tissue sample. This can then be shared online with practitioners around the world, providing a second opinion. Both examples can result in quick and efficient diagnoses. Diagnosing a patient earlier means that the necessary steps to improve the outcome of their condition and modify its progression can be taken sooner. This can help people self-manage more effectively, reducing the need for hospital admission.

3) Efficient treatment

Not all consultations are necessary in a clinic or hospital. Rather than being limited to surgery hours and GP availability, technology can be used to provide remote consultations at a more convenient time for the patient. This has the potential to reduce the number of ER visits and admissions by monitoring patients remotely. As such, pressure on overburdened hospitals, physicians and medical professionals is reduced.

4) Rehabilitation

Smartphone apps and remote technologies can be used to track patient rehabilitation and recovery. This allows the patient and the healthcare professional to monitor the patient's recovery, ensuring that the patient is performing the correct rehabilitation exercises. These processes can improve the patient's recovery rate.

Patient expectation and need

Technology can be used as a support mechanism to help patients stay well and manage their illnesses. It can also improve care for people with chronic or complex conditions.

1) Time and Cost

The introduction of digital medicine into the healthcare system can generate time-and cost-saving benefits for patients and healthcare professionals. Some of these benefits include: a) Reduced hospital admissions b) Fewer adverse drug reactions and prescription drugs c) Reduced medical errors 2) Remote diagnosis and treatment

Recent advances in the way information and images are stored and shared digitally allow users to access and store substantial amounts of data remotely. They provide real-time information through image-sharing platforms. Instant data exchange and access to data can generate innovative ideas and discussions in different parts of the world. It can help get a second opinion when there is uncertainty and provide support. All of this can be accomplished from the comfort of an individual's own home. Online platforms that host physicians and medical social networks give patients access to experts and communities that can help empower them. This can help them address their conditions and concerns at an early stage [START_REF] Liddell | Technology in the NHS. Transforming the patient's experience of care[END_REF][START_REF] Lupton | Digital health now and in the future: Findings from a participatory design stakeholder workshop[END_REF].

The impact of digital medicine on healthcare professionals

Digital medicine can impact the roles of healthcare professionals in different ways:

1) Doctors Digital technology can be used to constantly monitor patients and provide remote rehabilitation. This has the potential to give doctors more time to interact with their patients. As such, this can lead to improved quality of care and better patient outcomes.

2) Nurses

Digital technologies can perform repetitive tasks, for example, taking blood pressure. They can also perform administrative functions such as scheduling appointments and referrals. This can free up nurses' time, allowing them to care for patients in a more meaningful way and provide greater attention and emotional support.

3) Care assistants

Technology-based care has the potential to support face-to-face and remote care. This can be done by using technology to enable constant patient monitoring and communication, which helps care assistants provide a personal touch to those in need.

4) Physicians

Digital pathology is used to analyze tissue samples to diagnose diseases such as cancer. Instead of using traditional microscopes and glass slides, the slides are scanned into a computer and are available for review wherever the physician is located. The pathology team at St. James's in England has reported that more than 99% of cases are assessed using digital pathology, with microscopes being used only in exceptional cases.

The use of such technology allows physicians to share opinions and collaborate on complex cases quickly, without having to be in the same place as a glass slide. This provides time-efficient internal and external consultations. Physicians can provide answers immediately because they have access to the images [START_REF] Morley | Collaboration in health care[END_REF][START_REF] England | Improving outcomes through personalised medicine[END_REF].

Challenges and opportunities in medical care

Interdisciplinary work creates opportunities to improve products and services by gathering a variety of perspectives and inputs. The healthcare system in Chile and Latin America currently faces several key challenges, including:

1) An aging population 2) A growing population due to increasing immigration rates 3) Evolving health care needs 4) Closure of services due to centralization

As a result of these factors, the healthcare system in Chile requires efficient methods and processes to improve the quality of healthcare while keeping costs to a minimum. Technology is helping to achieve this. The combination of healthcare and technology allows healthcare professionals to access premium tools and resources. This can help them make properly informed diagnoses and deliver efficient outcomes. An interdisciplinary approach to innovation can help develop an effective product. The teams involved contain members from different backgrounds, medical fields, and professions. Typically, they will include the users, the designer, the engineer, and a representative of the funding agency. They can share their experience, knowledge, and skills to develop an innovative product that benefits diagnosis, treatment, and patient care [START_REF] Ebert | Digital Transformation[END_REF][START_REF] Imison | Delivering the benefits of digital health care[END_REF]. Patient and physician involvement is key when developing a product for use in a healthcare setting. Both stakeholders will determine what they require, and their needs will create the usability requirements for the product. Then, the design team will develop a product based on the patient's and physician's needs. The design team must also ensure that the product meets the requirements of the funding source. Therefore, the involvement of an end user in this process is beneficial. Without this, there is the potential for designers to overlook basic elements that could have great benefit to the physician and patient (Taylor, 2015;[START_REF] Kanan | Energy harvesting for wearable wireless health care systems[END_REF].

Digital medicine opportunities

Digital medicine has created opportunities where the convergence of interests, opinions and ideas has resulted in a product readily available in the marketplace. The combination of different fields merging, for example, medicine and informatics, has created many opportunities for knowledge sharing. By doing so, specialists can develop innovative devices that would not be possible without interdisciplinary collaboration. As such, as the technology and healthcare sectors come together more easily, patients are sure to benefit from more of these advances in health and medicine.

Internet of things

The Internet of Things is a network of sensor-enabled physical devices connected to each other via the Internet. This connection makes it possible for devices to collectively collect and analyze data. The data obtained provides opportunities for patients to self-manage their care. The fitness watch is an example of a wearable device that combines software, hardware, and an Internet connection. It can use an accelerometer, gyroscope, compass, ambient light sensor, and an optical heartbeat sensor. The sensors on the wearable device allow the user to observe feedback on their daily activity. They allow tracking of: 1) Heartbeat 2) Number of calories burned throughout the day 3) Hours of quality sleep 4) Number of steps taken during a day

The fitness watch could also use Wi-Fi to wirelessly sync with wearable devices, allowing the user to view their health information on a larger screen. This allows the user to monitor their daily activity and observe how their interactions affect bodily functions. As a result, users can actively monitor their health for fitness purposes.

Big data

Big Data is the process of collecting data from numerous sources to identify patterns and trends to develop products and services tailored to customer needs. Big Data in healthcare can be used to revolutionize healthcare. Big Databased big data can be defined as a large set of computer-analyzed data obtained from multiple sources. These could include transactional, machine-related, clinical, or social sources. Big Data can reveal patterns and trends related to specific aspects of the data. This allows those with access to tailor their products and services to meet customer needs and optimize operations and infrastructure (Taylor, 2015;[START_REF] Sadiku | Internet of Things in Healthcare[END_REF].

Benefits of big data for the healthcare industry

In the context of healthcare, big data refers to health data collected from numerous sources, including: 1) Electronic medical records 2) Medical imaging 3) Pharmaceutical research 4) Medical devices Big data is distinct from traditional electronic medical and human health data used for decision making in that it is available in extraordinarily high volume. In addition, it is derived from many sources and is highly varied in structure and nature. IBM Watson a supercomputer that combines artificial intelligence with analytical software, developed a breakdown of Big Data into the following four dimensions [START_REF] Data | The four V's of big data[END_REF]. 1) Volume: the size of the data 2) Velocity: the speed at which the data is generated 3) Variety: the different types of data 4) Veracity: the reliability of the data in terms of accuracy

The healthcare industry can benefit significantly from big data, as it is an organization that generates large amounts of data daily. Four areas where big data can help the healthcare system are as follows:

1) Internet of Things Any device that generates and shares data about a patient's health is part of the Internet of Things. 2) Care management Big data can be used to identify patterns of care and provide a broader view of evidence-based clinical practice. As such, obtaining big data from patient analytics can enable analysis of patient behavior that may have gone unnoticed before, e.g., hospital readmissions.

3) Predictive analytics

This process analyses the probability of a future outcome based on historical data patterns. Predictive analytics can help improve care delivery by making informed decisions based on data about events before they happen.

4) Genomic sequencing

Human genome sequencing uses a patient's genomic information to develop an individualized strategy for diagnosis and therapy. Big Data infrastructure provides an efficient and effective approach to discover patterns and process Big Data sets from numerous genomes.

Finally, Big Data can also be used outside the healthcare sector, helping other industries to achieve substantial gains. These benefits could be the development of new products, intelligent decision making, and a reduction in costs and time.

Conclusions

The rising cost of health care in Chile and Latin America has created a need to make the health care system more efficient and cost-effective. Appointments, referrals, missed appointments, misdiagnoses, and ambulance calls cost the health care system a considerable amount of time and money. Value-based medicine aims to provide costeffective, science-based medical care that incorporates patient values that improve patients' life expectancy and quality of life. This approach could help monitor people remotely, prevent the likelihood of disease, and provide accurate medications and treatments. This not only benefits patients, but also physicians, nurses and other stakeholders within the healthcare system. Access to physicians working remotely could be an answer to the current unsustainable demand on the Chilean healthcare system. It could also help the implementation of more personalized patient care by giving physicians the time to support longer appointments through telephone and video consultations with patients. In turn, this could reduce waiting times in public hospitals and private clinics.


Christelle Baunez
email: christelle.baunez@univ-amu.fr

Mickael Degoulet
email: mickael.degoulet@univ-amu.fr

Stéphane Luchini
email: stephane.luchini@univ-amu.fr

Patrick A Pintus
email: patrick.pintus@univ-amu.fr

Miriam Teschl
email: miriam.teschl@ehess.fr

COVID-19 Acceleration and Vaccine Status in France -Summer 2021 *

Keywords: JEL Classification Numbers: I18, H12 COVID-19, Vaccine Status, Viral Acceleration, Acceleration Index, France

Objectives: This note provides an assessment of COVID-19 acceleration among groups with different vaccine status in France.

Methods

: We assess viral acceleration using a novel indicator introduced in Baunez et al. (2021). The acceleration index relates the percentage change of tests that have been performed on a given day to the percentage change in the associated positive cases that same day. We compare viral acceleration among vaccinated and unvaccinated individuals in France over the period May 31st -August 29, 2021.

Results: Once the state of the epidemic within each groups is accounted for, it turns out that viral acceleration has since mid-July converged to similar levels among vaccinated and unvaccinated individuals in France, even though viral speed is larger for the latter group compared to the former.

Conclusion:

Our results call for an increasing testing effort for both vaccinated and unvaccinated individuals, in view of the fact that viral circulation is currently accelerating at similar levels for both groups in France.

Introduction

Existing vaccines against SARS-CoV-2 currently on the market in Western societies are said to have a high efficacity in reducing the number of symptomatic cases, and also in cutting transmission (e.g. Disease Control and Prevention (CDC) indicates for example that fully vaccinated people with the Delta variant can spread the virus, although it appears for a shorter period of time. [START_REF] Baunez | Tracking the Dynamics and Allocating Tests for COVID-19 in Real-Time: an Acceleration Index with an Application to French Age Groups and Départements[END_REF] It is therefore of great importance for any public health authority to be able to measure and understand correctly the viral spread, including among the vaccinated population.

In France, we observe that at the end of August 2021, about 6 times more new positive cases are found among the unvaccinated individuals, compared to the vaccinated. This indicates that new infections are much larger for the unvaccinated but those absolute levels do not correct for the fact that unvaccinated people are tested much more each day (about 3 times more, as of August 29, 2021). The usual way to account for tests is to compute the ratio of the number of new positive cases generated on a given day to the number of new tests performed on that same day, the positivity rate.

At the end of August, the positivity rate for the unvaccinated about is twice as large compared to that of the vaccinated. [START_REF] Blanquart | Characterization of vaccine-breakthrough infections of SARS-CoV-2 Delta and Alpha variants and within-host viral load dynamics in the community in France[END_REF] This suggests that the virus is spreading faster among the non-vaccinated than the vaccinated population and one may conclude that vaccines have succesfully contributed in slowing down viral transmission among the vaccinated group.

The challenge with this conclusion however is that a direct comparison between the positivity rate of different groups is not as straightforward. One general difficulty is that it relies on the implicit assumption that those positivity rates do not change with sample size. If, for instance, the number of tests of vaccinated would be increased to match those of non-vaccinated, their positivity rate is likely to differ. One reason for this is the question of who is getting tested. If among vaccinated people primarily those with symptoms are getting tested, nothing can be inferred about whether there may be asymptomatic cases among vaccinated people. Current research suggests that asymptomatic vaccinated may have a lower viral charge than asymptomatic unvaccinated individuals, and would thus be contributing less to transmission, but the full verdict is not yet known (Blanquart et al. 2021, Chau et al. 2021, Riemersma et al. 2021). Another important difficulty is that the positivity rate is level-dependent. A more meaningful comparison however puts the new share of tests that return a positive result in relation to the cumulated level of tests and cases in order to see how the daily positivity rate evolves with respect to its history in the respective group. This makes the measure level-independent in a way that indicates whether the viral spread is accelerating or decelerating, which is, as we argue below, more important information for public health than any variations of viral speed over time.

In Baunez et al. [1], we propose an acceleration index that monitors, in real-time, whether the pandemic is decelerating or accelerating, that is, whether harm (here represented as the number of cases) is accelerating or decelerating (Taleb 2012).The advantage of this index is that it measures the sensitivity of cases with respect to a change in tests. The intuition is that if we increase the amount of tests by a certain percentage and we find a greater positive percentage change of cases, then the pandemic is accelerating. If we find less than that percent change of cases, the pandemic is decelerating. The index is thus an elasticity, which measures the responsiveness or sensitivity of one variable (here cases) in response to another variable (here tests). It gives a number that takes account of the level of tests and cases and is, consequently, more informative and precise than the positivity rate about viral propagation. As long as acceleration is greater than one, the pandemic is not under control, whether the daily positivity rate is small or not. For a succesful vaccination campaign, we want to see low speed and deceleration of viral spread. If the viral transmissions are low, but with a tendency to accelerate, vaccine campaigns alone will not be able to curb the pandemic. This translates directly into an important policy goal: to curb the pandemic, it is important to test ever more, even alongside a vaccine campaign, and to find ever fewer cases. That is, the policy goal should be deceleration, and not only lower speed.

In this paper, we calculate the acceleration index for the vaccinated and unvaccinated population since the start of the vaccination campaign in France until the end of current available data, which is August, 29 2021. What we find is that despite vaccination, the pandemic is still accelerating in both population groups, although we see a decrease in acceleration in both groups since about the beginning of August. Moreover, despite lower absolute numbers among the vaccinated, viral spread accelerates similarly among the vaccinated and non-vaccinated populations. This finding suggests that vaccines are not fully protective against infection and cannot, by themselves alone, curb the pandemic especially if even vaccinated people continue to transmit the virus. Testing remains an important instrument to observe viral spread together with additional health policies such as, for example, contact tracing, quarantine, mask-wearing and social distancing, if necessary.

Methods

In Baunez et al. [1], we have proposed a real-time indicator, the acceleration index, that measures whether the pandemic is accelerating or decelerating in a given population. The acceleration index is defined as the elasticity of cumulated positive cases to cumulated tests. It is thus a level-free measure of responsiveness of cases to tests in relative terms. Suppose that data is available about the number of tested and positive persons, up to end date T . Denote {p 1 , . . . , p T } the historical times series of the new (per period) number of positive persons from date t = 1 to end date t = T .

Similarly, {d 1 , . . . , d T } is the historical times series of new (per period) diagnosed/tested persons.

Denote P t = t τ =1 p τ and D t = t τ =1 d τ
the cumulative numbers of positive and diagnosed persons up to date t. Importantly, variables P t and D t therefore define the state of the epidemic at date t, in terms of the current stocks of positives and tests, cumulated since date 1. In fact, those states reflect the history of each variable and it might differ across the groups of individuals that we consider below, due to vaccine status.

The acceleration index, denoted ε T at date T , thus gives the percentage change of cases divided by the percentage change of testing as follows:

ε T = P T -P T -1 P T ÷ D T -D T -1 D T = p T P T ÷ d T D T (1)
where the latter equality follows from the very definition of cumulated variables, the variations of which are just daily flows. When ε T > 1 we say that the epidemic is accelerating (it is on the loose), since a given growth rate of cumulated tests produces a larger growth rate of cumulated positive cases, while it is decelerating (the pandemic is under control) when ε T < 1. As a consequence, our indicator can tightly be linked to an arguably desirable objective of public health policy, which is to get proportionally less infected people when tests are increasing.

Rearranging the terms of the latter equality, we see that the acceleration index relates to the daily and average positivity rates, in the following way:

p T d T daily positivity rate = P T D T average positivity rate × ε T acceleration index (2)
where the average positivity rate is here defined as the ratio of stocks at end date T . In sum, the acceleration index is the elasticity of cumulated positive cases to cumulated tests (that is, the ratio of their growth rates), which may also be decomposed as the ratio between daily and average positivity rates. It then follows that if the daily positivity rate is always constant, then ε T = 1 We also estimate a Pearson correlation coefficient on a rolling window of 7 days between the acceleration index for the fully vaccinated and that of the recent incomplete vaccination, efficient incomplete vaccination, not vaccinated. This provides a simple measure of moment-by-moment local synchrony of viral propagation across groups with different vaccine status.

Results and Discussion

Our main results are depicted in Figure 1. Panel (a) depicts the amount of daily tests that are performed for the 4 groups. It reveals that the amount of testing is both time-varying and dependent on vaccine status, with unvaccinated individuals getting tested about 3 to 5 more times over the considered period. Panel (b) in Figure 1 depicts the daily positivity rates, defined as p T /d T at each end date T , for the four groups. Looking at daily positivity rates alone, one concludes that viral speed is greater among unvaccinated people, particularly when compared with fully vaccinated individuals. On August 29, the daily positivity rate for unvaccinated people is about twice as large compared to that of the fully vaccinated. However comparing levels of the daily positivity rate across groups does not give any indication about whether the pandemic worsens faster for the unvaccinated or improves quicker for the vaccinated. For this to know, one needs to look at the acceleration of viral spread for both groups. This means to look at the change of speed or viral spread, which can be understood by putting the daily positivity rate in relation to the average positivity rate. Doing so, one obtains a level-independent measure of the dynamic of the pandemic because it puts the change in viral spread expressed through the daily positivity rate in relation with the "past" viral spread or average positivity rate, that is the cumulated number of cases over cumulated number of tests. Intuitively this means that if today we find more cases per tests than we did on average on all previous days, then the pandemic is accelerating, if we find less cases per tests today than we did on average in the past, the pandemic decelerates. This measure thus is a more informative way to compare between groups because it tells us that given current testing strategies, if of two groups one has a higher elasticity, this group has a higher acceleration/lower deceleration of viral spread in comparison to the other one.

This is exactly what the acceleration index does and in panel (c) we report its level for the 4 groups.

Our acceleration index is lvel-free since, as we see in Equation ( 2), the daily positivity rate, i.e.

the daily flows of positives and tests for a particular day, respectively p and d, are corrected by the average positivity rate, i.e. their stock values, respectively P and D for that particular day.

We therefore see in panel (c) of Figure 1 that viral acceleration is essentially similar among fully vaccinated and unvaccinated individuals. This is because even though the unvaccinated have a larger daily positivity rate than the vaccinated, they also have a larger average rate: unvaccinated people are less protected against direct SARS-CoV-2 infection and therefore have a larger P/D over the epidemic past. But vaccinated people, who should be protected against direct SARS-CoV-2 infection have a lower daily positivity rate but with respect to a lower average positivity rate, which means that with respect to their epistemic past, viral spread is accelerating as quickly as among the non-vaccinated. This is insofar suprising as one may think that a succesful vaccination campaign would contribute to the deceleration of viral spread among the vaccinated. Yet this is not what we observe.

What is also striking in panel (c) is how close the acceleration indices stay over the entire period.

More precisely, the resurgence of the acceleration regime, at the beginning of the summer break in early July, starts a little bit earlier for fully vaccinated individuals compared to unvaccinated people in France. The level of acceleration is about 1.2 for the two groups on August 29, 2021.

This means with 1% of additional tests (the total number of test, or its stock, is increased by 1%), we find 1.2% additional cases (the total number of cases, or stock of positive cases, increases by 1.2%) in both groups. Acceleration for both groups reached its peak during the second half of July ( a factor of about 3) and is steadily decreasing since the beginning of August. But the pandemic remains in an acceleration regime for both groups. The acceleration indices have been very much synchronized since the resurgence of the acceleration regime, starting early July. This is confirmed in panel (d), where we report the pairwise Pearson correlation coefficients on a rolling window of 7 days. In particular, the correlation between acceleration indices for fully vaccinated and unvaccinated individuals has reached its highest level mid-July and is still close to 1 at the end of the sample. In sum, our acceleration index shows no large difference across vaccinated and unvaccinated individuals since early/mid July.

Conclusion

The main result that we document in Figure 1 to make access Antigen-testing difficult and costly for individuals later this year, to focus on PCRtesting by prescription only and to impose a vaccine passport ("pass sanitaire") that gives certain advantages to vaccinated people, notably to travel and to visit social gatherings without prior testing is questionable on the grounds that it does not guarantee to be effective to help curbing sustainably the COVID-19 acceleration that is at this date still observed in France.

  Layan et al. 2021, Olliaro et al. 2021, Prunas et al. 2021, Salo et al. 2021). But the Delta variant of SARS-CoV-2 has created new uncertainties notably with respect to transmission and vaccin breakthrough, which refers to the observation that even vaccinated individuals may get infected and also transmit the virus to other individuals (e.g. Gazit et al. 2021, Riemersma et al. 2021, Brown et al. 2021). A certain number of studies are starting to suggest that the Delta variant "[...] is believed to spread faster than other variants" (Planas et al. 2021) and the US-based Centers for

  because current and average speeds are equal. If ε T > 1, daily positivity rate exceeds average rate, indicating acceleration, while deceleration prevails when ε T < 1. We compute the acceleration index for France over the period May 31st -August 29, 2021, for individuals who are grouped according to their vaccine status. We use data about vaccine status in France that exploit two French databases SI-DEP and VAC-SI. SI-DEP contains, for each day, the total number of PCR tests realised as well as the number of positive tests. VAC-SI contains the vaccine status of the tested patients. Vaccine status is divided into four categories: (1) not vaccinated, (2) one vaccine jab for less than 7 days for Pfizer, Moderna and Astra Zeneca vaccines or 14 days for Janssen, (3) one vaccine jab for more than 7 days or two jabs for less than 7 days (Pfizer, Moderna, Astra Zeneca) and (4) full vaccination (2 jabs for more than 7 days for Pfizer, Moderna and Astra Zeneca and jab for more than 14 days for Janssen. The matching has been done by the DREES (Direction de la recherche, des études, de l'évaluation et des statistiques) from the Ministry of Health (DREES, 2021) and data are open source 3 .

Figure 1 :

 1 Figure 1: Epidemic dynamics in France across 4 groups with different vaccine status -May, 31 to August 29, 2021; panel (a) depicts the number of SARS-CoV-2 tests; panel (b) depicts positivity rates (defined as ratio of daily positive cases over daily tests); panel (c) depicts the acceleration indices (defined as the elasticity of cumulated cases to cumulated tests); panel (d) depicts Pearson correlation coefficients on a rolling window of 7 days between the acceleration index for the fully vaccinated and that of the other 3 groups; data obtained from DREES (a) RT-PCR Testing

  is that in France, COVID-19 acceleration has been of similar magnitude among both the vaccinated and the unvaccinated population since early July 2021, despite the daily positivity rate being larger among the latter group. Positivity rate is a leveldependent indicator that roughly informs us about viral spread or speed, but what is important to know is whether viral spread is accelerating or decelerating. Our acceleration index is a levelindependent measure that indicates the percentage increase in cases following a percentage increase in tests and thus captures better whether the pandemic is under control or not. Our indicator can also serve as a basis for public health policy to curb COVID-19, that, if successful, would be one that sees ever fewer positive cases as testing efforts increase, i.e. our indicator would indicate deceleration. Currently, our indicator does not indicate deceleration of viral spread among the vaccinated, which may indicate that vaccination campaign is not fully achieving its aim. Riemersma et al, 2021; Public Health England Technical Briefing 20). Chia et al (2021) report that if vaccine breakthrough occurs, the viral load is the same for vaccinated and non-vaccinated individuals for the first few days, but that the viral load is decreasing more quickly for vaccinated individuals thereafter. A study from June 2021 in Vietnam has shown that the viral load with the delta variant in cases of vaccine breakthrough was over 250 times higher than in older strains (Chau et al. 2021). This means that while vaccination offers in general effective biological protection against severe cases and mortality (see e.g. Public Health England COVID-19 vaccination report 31; CDC Science Brief of September 15, 2021), it does not mean that vaccinated people cannot get infected with the virus and that they do not contribute to viral spread. Moreover, as it is known that even asymptomatic individuals may transmit the virus (e.g. Byambasuren, O. et al. 2020), asymptomatic vaccinated as well as non-vaccinated individuals may often go unperceived with no systematic testing effort. Testing for both, non-vaccinated and vaccinated individuals thus remains an important public health strategy to control the pandemic and to bring it into a deceleration regime. Indeed, research shows that test sensitivity is secondary to frequency for efficient COVID-19 screening (Larremore et al., 2021) and that there should not be a "one size fits all" testing strategy, primarily based on PCR-screening (Mina and Andersen 2021). The decision of the French authorities for example to concentrate mainly on vaccination to combat COVID-19,

	The fact that acceleration persists even among the vaccinated population is broadly consistent
	with recent biological evidence that vaccinated and unvaccinated individuals have similar viral

loads where the SARS-CoV-2 delta variant is most prevalent and that vaccine breakthroughs are seen more often in comparison with previous variants (see

Brown et al, 2021; Musser et al, 2021;

https://www.cdc.gov/coronavirus/2019-ncov/variants/delta-variant.html

See press releases by the DREES, in French, and notably the release on August 20, 2021 about differences in cases, tests and positivity rate between vaccinated and unvaccinated individuals, available at https://drees.solidarites-sante.gouv.fr/communique-de-presse/ debut-aout-huit-fois-moins-de-tests-positifs-et-onze-fois-moins-dentrees-en, last accessed September 16, 2021.

Data used in this note are available at https://data.drees.solidarites-sante.gouv.fr/explore/dataset/ covid-19-resultats-regionaux-issus-des-appariements-entre-si-vic-si-dep-et-vac-s/information/, accessed September 16, 2021.

* This work was supported by the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -A*MIDEX.