Yves Moussallam

Clive Oppenheimer

Bruno Scaillet

Fabrice Gaillard

Philip R Kyle

Nial Peters

Margaret Hartley

Kim Berlo

Amy Donovan

Tracking the changing oxidation state of Erebus magmas, from mantle to surface, driven by magma ascent and degassing

Keywords: oxygen fugacity, sulfur, degassing, XANES, Melt inclusions, CO 2

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The oxidation state of an igneous system is often expressed in terms of oxygen fugacity (fO 2 ), an intensive thermodynamic variable. Oxygen fugacity varies widely in magmas [START_REF] Frost | The Redox State of Earth's Mantle[END_REF] and controls phase equilibria and the composition of minerals, melt and gas. Here we examine a suite of olivine-and anorthoclase feldspar-hosted melt inclusions (MI) from basanites, tephriphonolites and phonolites from Erebus volcano and nearby Hut Point Peninsula, Antarctica [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]. Erebus is an alkaline volcano that sits in an intraplate continental rift, and its uppermost phonolite magma is exposed as a persistent lava lake. We focused on these samples since they represent a near-continuous differentiation series [START_REF] Kyle | Petrologic Evolution of Anorthoclase Phonolite Lavas at Mount Erebus, Ross Island, Antarctica[END_REF] and record depths spanning the entire crust and extending into the upper mantle (Moho depth <20 km). In addition, measurements of sustained gas emissions from the Erebus lava lake show they are more reduced (fO 2 around NNO-1.6, i.e., 1.6 log units below the Ni-NiO solid buffer) than gases sourced from greater depth during Strombolian eruptions (fO 2 around NNO) [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF][START_REF] Burgisser | Backward tracking of gas chemistry measurements at Erebus volcano[END_REF].

METHODOLOGY

Samples used in this study and their preparation are described by [START_REF] Eschenbacher | Evidence from Antarctic mantle peridotite xenoliths for changes in mineralogy, geochemistry and geothermal gradients beneath a developing rift[END_REF] and given in [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF] (see also Table S1). To ensure good preservation of MI, rapidly quenched samples from subaqueous pillow and palagonite breccias and a phonolite sample from an Erebus volcanic bomb erupted in 1997 were used. Fig. 1 shows the typical morphology of melt inclusions hosted in olivine (for basanite to tephriphonolite composition) and anorthoclase (for phonolite composition) crystals. Olivine-hosted MI reach up to 350 µm across and consist of brown-coloured transparent glass of ovoid and negative crystal shape located throughout the crystal. Anorthoclase-hosted MI are of irregular and negative crystal shape, reach up to 600 µm across, and tend to be elongated parallel to the crystal face.

Anorthoclase crystals can reach up to 10 cm in length. Olivine-hosted MI may contain one vapour bubble (many have none) whereas they are common in anorthoclase-hosted MI.

Magnetite, apatite and pyroxene crystals are found in some MI. Pyrrhotite globules are present in MI and as mineral inclusions in all compositions. Only doubly polished wafers in which each side of the MI was polished, and at least a 2020 m 2 area through the inclusion was free of obstructions were examined. The MI were analysed for major and volatile elements by EMP, SIMS and FTIR spectroscopy by [START_REF] Eschenbacher | Evidence from Antarctic mantle peridotite xenoliths for changes in mineralogy, geochemistry and geothermal gradients beneath a developing rift[END_REF] and further discussed by [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]. Details are given in the supplementary information.

XANES analytical Methods

We used synchrotron-based Fe and S K-edge X-ray absorption near-edge structure (XANES) spectroscopy to explore the redox state of the MI using the iron Fe 3+ /ΣFe ratio and sulphur speciation. Glass standards for the XANES analyses were prepared in a controlled atmosphere furnace at the Department of Earth Sciences, University of Cambridge. The furnace was operated at atmospheric pressure (0.1 MPa), and was heated by a silicon carbide resistance wire wrapped around an aluminium cylinder. A controlled mixture of CO and CO 2 gases was circulated through the furnace and monitored by calibrated flow meters. The current was controlled by a Eurotherm 2404 series regulator able to maintain a constant temperature (±2°C) within the furnace. The cooling system consists of a constant flow of water circulating within the furnace. For each experiment, 50 mg of sample was placed in a platinum crucible. The sample was then introduced in the furnace and heated at 1400°C and left to equilibrate for two days. At the end of each experiment, rapid quenching preserved chemical equilibrium as the sample dropped in a container of deionised water. The drop was initiated by short-circuiting the thin platinum wire by passing a current through the thick platinum wires. Four basanite standards were equilibrated at QFM-1, QFM, NNO, NNO+1, and, while tephriphonolite standards were equilibrated at QFM-1, NNO and NNO+1. All standard glasses were analysed by electron microprobe. A 15-point-transect was made across each standard to ensure all samples were homogeneous and to monitor iron and sodium loss.

Micro X-ray Absorption Near-Edge Structure (-XANES) spectroscopy is a synchrotronbased method able to determine the valence state of elements in a range of materials including glass and crystals. XANES analyses at the Fe K-edge were conducted from 6987 to 7350 eV, using a 3 × 2 µm 2 x-ray beam whereas analyses at the S K-edge were conducted from 2400 to 2675 eV using a 5 × 5 µm 2 X-ray beam (Table S2). Both sets of analyses were acquired during a single week at beamline I18 of the Diamond Light Source (in the UK).

The storage ring was operated at an energy of 3 GeV with electron currents of 300 mA. The X-ray beam was monochromatised by a pair of Si (111) crystals (for measurements at both the S and Fe K-edges). Measurements were performed in fluorescence mode. The incident monochromatic X-ray intensity (I0) was measured with a 1.5 cm-long ionization chamber.

The fluorescence intensity (IF) was measured with a four-element silicon drift detector positioned perpendicularly to the beam direction. The sample was oriented at 45° to the incident beam. For analyses at the S K-edge, the sample chamber was flushed with helium to minimise the attenuation of the fluorescence radiation. The ionisation chamber was flushed with helium for both S and Fe K-edge measurements. Energy resolution is estimated at 0.1 eV. The pre-edge feature (7110-7118 eV) on a XANES spectrum at the Fe K-edge corresponds to the 1s→3d electronic transition. The pre-edge region was fitted using a combination of a linear function and a damped harmonic oscillator function (DHO) to fit the baseline, on to which were superimposed two Gaussians to fit the two pre-edge peaks [START_REF] Cottrell | High-precision determination of iron oxidation state in silicate glasses using XANES[END_REF] (Fig. S2). This method has the advantage of reproducing the spectrum extremely well (with a very low residual). We used the centroid (area-weighted average) of the two Gaussians to parameterise against the Fe valence state. We constructed calibration curves using the basanite, and tephriphonolite standards (Fig. S3 andS4). Instrumental drift was small since all Fe K-edge spectra were acquired in a 24 h period and corrected by aligning the pre-edge region of all spectra to that of the corresponding glass standard. Spectral alignment was achieved by matching the first peak on the derivative spectra (at 7111 eV), equivalent to matching the initial rise of the pre-edge region in normalised intensity (Fig. S5). All XANES spectra were examined for contamination from the olivine host. Spectra showing any structure in the edge and post edge region (see Fig S6 for an example) were not processed.

Beam damage

The focus beam of X-rays can cause photo-reduction of the sulphur [START_REF] Wilke | The origin of S4 detected in silicate glasses by XANES[END_REF][START_REF] Métrich | The oxidation state of sulfur in synthetic and natural glasses determined by X-ray absorption spectroscopy[END_REF]. To assess this effect we subjected a phonolitic anorthoclase-hosted melt inclusion to long beam exposure by acquiring four successive spectra. This sample had not been subjected to EMPA prior to XANES analysis. Figure S7 shows that the sulphur was, in fact, oxidized under the X-ray beam. We therefore expect all S K-edge spectra of glasses analyzed in this study to have undergone some amount of oxidation under the beam. We note however that the exposure time to the beam has been similar for all MI and the large differences in the spectra between each sample group cannot be solely attributed to beam damage. The transition at ~2478.4eV typical of S 4+ shown by sample DVDP 3-295 is likely to be an analytical artefact as discussed in [START_REF] Wilke | The origin of S4 detected in silicate glasses by XANES[END_REF].

Determination of the Fe 3+ /∑Fe ratio

The standard equation to relate fO 2 and temperature to Fe 3+ /∑Fe is given by [START_REF] Kress | The compressibility of silicate liquids containing Fe2O3 and the effect of composition, temperature, oxygen fugacity and pressure on their redox states[END_REF]: [START_REF] Sack | Ferric-ferrous equilibria in natural silicate liquids at 1 bar[END_REF], samples 141 and 142 from [START_REF] Kress | Stoichiometry of the iron oxidation reaction in silicate melts[END_REF] and sample U-50 from [START_REF] Kilinc | The ferric-ferrous ratio of natural silicate liquids equilibrated in air[END_REF]. We found that the calculated Fe 3+ /∑Fe ratio was consistently overestimated by the calculation compared to the measured value. We therefore made a slight correction to equation (1) adjusting parameters b and c in order to minimise the difference between calculated and measured data. We thereby found that the experimental data were best reproduced using b=11357.9 and c=-6.7542 (the other constants were left unchanged). The original parameterisation of Equation (1) was not modified for the tephriphonolite composition.

( ) ( ) ∑ (1)
Standards were analysed by wet chemistry at the ISTO-CNRS laboratory in Orléans (Table S3); the difference between calculated and measured values is shown in Fig. S4. The resulting calibration curves are slightly different and not within error of each other. The discrepancy between the two calibration methods reflects an important uncertainty on the absolute Fe 3+ /ΣFe ratio reported. We chose here to use the calibration curve given from calculated values as they are obtained directly from the imposed fO 2 under which the standards were created whereas wet chemistry represents a bulk method that can be affected by uneven amounts of iron loss in the glass standards. Wet chemistry analyses, therefore, do not provide an accurate estimate of the Fe 3+ /ΣFe ratio of the synthetic glasses used as standard. The effect of using the calibration curve from wet chemistry in subsequent figures is illustrated in Fig S8.

Gas-melt equilibrium modelling

The model used in this study [START_REF] Gaillard | The sulfur content of volcanic gases on Mars[END_REF][START_REF] Gaillard | Atmospheric oxygenation caused by a change in volcanic degassing pressure[END_REF]) solves, at each pressure step, the following reactions in the C-H-S-O system, at magmatic pressure and temperature:

CO (gas) + ½O 2 = CO 2(gas) (2) H 2(gas) + ½O 2 = H 2 O (gas) (3) CH 4(gas) + O 2 = CO 2(gas) + 2H 2 O (gas) (4) ½S 2(gas) + O 2 = SO 2(gas) (5)
½S 2(gas) + H 2 O (gas) = H 2 S (gas) + ½O 2 (6)

The gas-melt equilibria (using thermodynamic data for water and carbon dioxide (Iacono- , 2002) and hydrogen [START_REF] Gaillard | Rate of hydrogen-iron redox exchange in silicate melts and glasses[END_REF]) are dictated by the following reactions:

Marziano et al., 2012), sulphur (O'Neill & Mavrogenes
H 2 O (gas) + O 2- (melt) = 2OH - (melt) (7) CO 2(gas) + O 2- (melt) = CO 3 2- (melt) (8) S 2(gas) + 2O 2- (melt) = O 2 +2S 2- (melt) (9) H 2(gas) = H 2(melt) (10)
The iron redox ratio in the melt is computed using Equation (1) for the tephriphonolite and phonolite, and its slightly modified form for the basanite. Note that the increment of pressure steps in the calculations has no effect on the results since thermodynamic equilibrium depends only on the pressure and temperature considered.

RESULTS

XANES at the Fe K-edge

The XANES results of iron at its K-edge are reported in Tables 2 and3; example spectra are presented in Fig. 2; and the results are shown in Fig. 3. The results indicate pronounced reduction of iron (lower Fe 3+ /ΣFe ratio) with decreasing pressure. This pattern holds for MI of similar composition but different entrapment pressure, suggesting that the identified redox change with depth is not simply a reflection of differentiation, crystallisation or crustal assimilation. Furthermore, for samples in which it was analysed, the quenched matrix glass records a Fe 3+ /ΣFe ratio systematically lower than that of the coexisting MI.

XANES at the S K-edge

The XANES spectra of sulphur at its K-edge remain difficult to interpret quantitatively [START_REF] Rowe | Determination of sulfur speciation and oxidation state of olivine hosted melt inclusions[END_REF][START_REF] Wilke | The origin of S4 detected in silicate glasses by XANES[END_REF][START_REF] Métrich | The oxidation state of sulfur in synthetic and natural glasses determined by X-ray absorption spectroscopy[END_REF] as discussed above, but provide useful qualitative information on the nature of dissolved S species. Another transition occurs at ~2467 eV in samples 97009, 97010, 97011 and most strongly in 97018. We could not match this transition with any sulphur-bearing mineral standards but note that it is similar to a peak observed in reduced, iron-free glass spectra from [START_REF] Métrich | The oxidation state of sulfur in synthetic and natural glasses determined by X-ray absorption spectroscopy[END_REF]. While these MI samples all show some oxidation has occurred under the X-ray beam, samples DVDP3-295 and 97009 are, nevertheless, significantly more oxidised (higher sulphate signals) than samples 97010, 97011, AW82033 and 97018 (which have a stronger S 2-signal). Since most MI in DVDP3-295 and 97009 were entrapped at higher pressure compared with the other samples, the S K-edge analyses are consistent with the reducing trend associated with magma decompression evident in the Fe K-edge spectra.

DISCUSSION AND MODELLING

Several mechanisms have been proposed to explain changing redox state in magmas during ascent and/or differentiation. In an earlier study of Erebus [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF], a linear decrease of FeO/Fe 2 O 3 with decreasing pressure (from NNO+0.4 at 800 MPa to NNO-0.6 at 0.1 MPa) was modelled and attributed to the precipitation of olivine, pyroxene and magnetite. However, this presupposes that magmatic differentiation occurs continuously with decreasing pressure, which cannot explain the strong variation of Fe 3+ /Fe in MI of the same composition. We can exclude magnetite precipitation as the driving force behind the observed reducing trend since there is no correlation between the melt iron content and its redox state (Fig. S9). The absence of correlation between the melt Mg number and its Fe 3+ /Fe ratios (Fig. S10) also shows that there is no obvious change in redox with differentiation. This is evident when considering that the tephriphonolites, which are differentiated from the basanite, have very similar oxidation state and Fe 3+ /ΣFe ratio to the undegassed basanites. It is therefore clear that precipitation of Fe-bearing minerals only exerts a second order influence on the melt redox state. Importantly, our dataset counters the idea that MI will requilibrate with redox changes in the host magma on such rapid timescales that they cannot reflect the source (e.g. [START_REF] Gaetani | Rapid reequilibration of H2O and oxygen fugacity in olivine-hosted melt inclusions[END_REF][START_REF] Bucholz | Post-entrapment modification of volatiles and oxygen fugacity in olivine-hosted melt inclusions[END_REF]. It is clear from our data that olivine-hosted MI retain their signature even when the crystals have been in contact with melt and gases at a different redox state.

We propose that fO 2 is controlled by coupled redox reactions between sulphur and iron in the silicate melt and coexisting gas phase [START_REF] Carmichael | Oxidation-reduction relations in basic magma: a case for homogeneous equilibria[END_REF]. Sulphur dissolves in the melt as sulphate (S 6 + ) or sulphide (S 2 -) but is present in the gas phase as SO 2 (S 4+ ) or H 2 S (S 2-) [START_REF] Métrich | The oxidation state of sulfur in synthetic and natural glasses determined by X-ray absorption spectroscopy[END_REF]. The exsolution of sulphur can therefore be accompanied by reduction (e.g., six electrons are liberated with S 2-→ S 4+ ) or oxidation (two electrons are captured with S 6+ → S 4+ ) of dissolved iron, which is the only other major multiple valence state element in the melt, following reactions such as [START_REF] Métrich | The oxidation state of sulfur in synthetic and natural glasses determined by X-ray absorption spectroscopy[END_REF]:

S 2- (melt) + Fe 3+ 2 O 3(melt) = SO 2(gas) + 6Fe 2+ O (melt) + O 2- (melt) (11)
and

SO 4 2- (melt) + 2Fe 2+ O (melt) = SO 2(gas) + Fe 3+ 2 O 3(melt) + O 2- (melt) (12)
In addition, SO 2 degassing is increasingly favoured over H 2 S during magma ascent, since the following equilibrium shifts to the left with reducing pressure [START_REF] Gaillard | Atmospheric oxygenation caused by a change in volcanic degassing pressure[END_REF]:

SO 2 + 3H 2 = 2H 2 O + H 2 S ( 13
)
The pronounced increase of Fe 3+ /Fe 2+ with pressure suggests that reaction (11) is favoured over reaction (12) during decompression.

To test this interpretation, we modified a gas-melt equilibrium model [START_REF] Gaillard | The sulfur content of volcanic gases on Mars[END_REF] to simulate the effect of sulphur degassing on iron speciation. To ensure internal consistency of the model, the entrapment pressures of the MI were recalculated using the same solubility laws on which the model is based. It is also more appropriate for alkaline composition than that used in [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF], which substantially underestimates CO 2 solubility for these compositions. This results in slightly shallower estimates of entrapment depths for evolved compositions than previously reported [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]) (Figure 3C). . The model starting conditions are reported in Table 4 and degassing is simulated from the less degassed magma without adjustment of the model for lower pressure calculations. Figs. 5, 6 and 7 show the results of this gas-melt equilibrium model for basanitic, tephriphonolitic and phonolitic compositions. To first order, the Fe 3+ /Fe vs.

pressure trend is fitted well by the model showing that, during ascent and degassing, the redox state of the magma can evolve by two log units (from NNO+1 to NNO-1.2) for the basanite model. We note also that the model reproduces the reverse L-shaped pattern defined by the MI, with most of the reduction taking place at shallow pressure, as expected for a degassing-driven process but also reflecting that SO 2 exsolution is favoured over H 2 S at shallow depths.

Comparing the sulphur content of MI and matrix glasses relative to their Fe 3+ /Fe ratios, both the basanite and phonolite models reproduce the observations quite well. The fit to the tephriphonolite data is poorer, likely reflecting the limited experimental data on sulphur solubility in magmas of this composition, and possibly secondary processes that influence either parameter (sulphur content or redox state). To a first order, however, it is clear that the matrix glass for this composition shows both lower sulphur content and a more reduced character compared to the corresponding melt inclusions, as predicted by the model. The models suggest that decompression and degassing alone can account for the observed evolution of the Erebus magmas' redox state, throughout the compositional lineage. At

Erebus, this process is promoted by the relatively oxidising conditions of the primitive magmas (hence the high abundance of Fe 2 O 3 , which is the source of oxygen needed to oxidise S 2-in the melt to S 4+ in the gas (Equation 11), and the abundance of sulphur.

The oxidised character of the most primitive magma (~NNO+1) suggests a minimum for the oxidation state of the upper mantle beneath Erebus. This estimate is at the upper end of the range obtained from oxythermobarometry measurements of spinel-bearing peridotites [START_REF] Frost | The Redox State of Earth's Mantle[END_REF], and is more typically associated with subduction zone environments.

Such high fO 2 might, in part, be explained by coupling of the oxidation state of spinel peridotite in rift environments to melt depletion and possibly CO 2 content, with increasing degrees of melt extraction and higher CO 2 content associated with more oxidised mantle [START_REF] Eschenbacher | Evidence from Antarctic mantle peridotite xenoliths for changes in mineralogy, geochemistry and geothermal gradients beneath a developing rift[END_REF]. This is consistent with the very high abundance of CO 2 in the most primitive basanitic MI (>8000 ppm).

At Erebus, measurements of redox conditions recorded in MI and the average low oxidation state of volcanic gas emissions at the surface, independently point to a magmatic system that becomes progressively reduced towards the surface. Our model (Fig. 7) helps reconcile the puzzling observation that gases fuelling sporadic Strombolian eruptions through the lava lake are more oxidised (CO 2 /CO up to 100) than the lake itself (NNO-1.6; CO 2 /CO = 15 [START_REF] Burgisser | Backward tracking of gas chemistry measurements at Erebus volcano[END_REF]). These bursts are thought to arise from sudden ascent of gas that has accumulated at localised asperities in the plumbing system. Their rapid ascent and size hinders re-equilibration with the melt (i.e., Equation 11), thereby preserving their deeper, oxidised character. From our model, such oxidised compositions would be in equilibrium at a pressure of up to ~35 MPa (~1.3 km depth; Fig. 7). However, if the gas slugs cool by adiabatic expansion [START_REF] Burgisser | Backward tracking of gas chemistry measurements at Erebus volcano[END_REF] then depths could be as shallow as ~80 m. This range brackets the depth of the seismic source associated with Strombolian eruptions at

Erebus, of about 400 m [START_REF] Aster | Moment tensor inversion of very long period seismic signals from Strombolian eruptions of Erebus Volcano[END_REF].

CONCLUSIONS

Although this study has focused on an alkaline system with a lava lake, the mechanism proposed here should apply to any iron-bearing magma rich in sulphur and other volatiles, and in particular CO 2 , which will drive fluid exsolution during decompression [START_REF] Wallace | Volcanic SO2 emissions and the abundance and distribution of exsolved gas in magma bodies[END_REF][START_REF] Metrich | Volatile Abundances in Basaltic Magmas and Their Degassing Paths Tracked by Melt Inclusions[END_REF]. We thus expect a similar trend to be found not only in other rift and intraplate volcanic settings but also in volcanic arc settings [START_REF] Kelley | The influence of magmatic differentiation on the oxidation state of Fe in a basaltic arc magma[END_REF]. At a global scale, changes in volcanic degassing pressure could therefore have the potential to affect the atmosphere's redox state (e.g. [START_REF] Gaillard | Atmospheric oxygenation caused by a change in volcanic degassing pressure[END_REF]. Many prior studies have used the redox state of erupted lavas of known age to model the evolution of the mantle's oxidation state and the composition of volcanic gases expelled into the atmosphere over geological time (e.g. [START_REF] Kasting | Mantle Redox Evolution and the Oxidation State of the Archean Atmosphere[END_REF][START_REF] Canil | Vanadium partitioning and the oxidation state of Archaean komatiite magmas[END_REF][START_REF] Delano | Redox history of the Earth's interior since approximately 3900 Ma: implications for prebiotic molecules[END_REF]. The estimated volcanic inputs have, in turn, been used to model the evolution of the Earth's early atmosphere and ocean chemistry, and thereby to infer aspects of biotic evolution (e.g. [START_REF] Canfield | The early history of atmospheric oxygen: Homage to Robert M. Garrels[END_REF][START_REF] Shaw | Earth's atmosphere -Hadean to early Proterozoic[END_REF][START_REF] Sleep | Evolutionary ecology during the rise of dioxygen in the Earth's atmosphere[END_REF]. Our findings suggest that the fundamental premise underlying these interpretations may not be valid, and that degassing conditions may be as influential as mantle processes in determining the oxidation state of volcanic gases.

We do stress, however, that systems which are not degassing sulphur have the potential to remain unaltered in terms of their redox during magma ascent. Indeed, cases where redox change with ascent is limited have been documented [START_REF] Crabtree | An evaluation of the effect of degassing on the oxidation state of hydrous andesite and dacite magmas: a comparison of pre-and posteruptive Fe2+ concentrations[END_REF]. On the other hand, recognising that the redox state of a magma can be a function of depth is a key in relating gas emission measurements to their source region. This connection presents new opportunities for geochemists engaged in measurement and interpretation of redox-sensitive gas couples at active volcanoes [START_REF] Allard | Spectroscopic evidence for a lava fountain driven by previously accumulated magmatic gas[END_REF][START_REF] Aiuppa | Forecasting Etna eruptions by real-time observation of volcanic gas composition[END_REF], and for the application of gas geochemistry to volcano monitoring and hazard assessment. Plot of Fe 3+ /Fe ratio determined by Fe K-edge XANES spectroscopy compared with calculated entrapment pressures [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF] for each melt inclusion. The lava lake Fe 3+ /Fe ratio is computed from measurements of the CO/CO 2 and H 2 /H 2 O ratios in the gases emitted from Erebus' lava lake [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF][START_REF] Moussallam | Hydrogen emissions from Erebus volcano, Antarctica[END_REF]. ). Anhydrite (S 6+ ) [START_REF] Scheinost | Geochemical behavior of uranium in mine tailings at Freital, Germany: A mu-XRF, mu-XAFS and mu-XRD study[END_REF]), sodium sulfite (S 4+ ) [START_REF] Jugo | Sulfur K-edge XANES analysis of natural and synthetic basaltic glasses: Implications for S speciation and S content as function of oxygen fugacity[END_REF]) and pyrrhotite (S 2-) spectra are given for reference (available from the ESRF database http://www.esrf.eu/UsersAndScience/Experiments/Imaging/ID21/php).

FIGURE CAPTION

Figure 5: A. Plot of Fe 3+ /Fe ratios determined by Fe K-edge XANES spectroscopy compared with calculated entrapment pressures (this study) for each melt inclusion. See Figure 3 for derivation of the lava lake Fe 3+ /Fe ratio. The results of gas-melt equilibrium modelling are shown for basanite (blue curve), tephriphonolite (yellow curve) and phonolite (red curve) compositions. B. Plot of ∆NNO value determined by Fe K-edge XANES spectroscopy compared with calculated entrapment pressures (this study) for each melt inclusion. Error bars on the entrapment depth are based on uncertainties in CO 2 measurements. A more realistic error on the calculated pressure would also take account of the error on H 2 O, S, fO 2 and solubility laws and cannot readily be computed. Errors in Fe 3+ /Fe ratios are discussed in the SI.

Figure 6: Plot of Fe 3+ /Fe ratio determined by Fe K-edge XANES compared to measured sulphur contents [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF] for each melt inclusion. See Figure 4 for derivation of the lava lake Fe 3+ /Fe ratio. The results of gas-melt equilibrium modelling are shown for basanite (blue curve), tephriphonolite (yellow curve) and phonolite (red curve) compositions. Errors in sulphur measurements are from [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]; errors in Fe 3+ /Fe ratios are discussed in the SI. Starting conditions for each run are detailed in Table 4.

Figure 7: Calculated fO 2 (as ∆NNO) and volcanic gas redox-couple ratio as a function of pressure, shown in the context of a schematic representation of the Erebus plumbing system. The results of gas-melt equilibrium modelling are shown for basanite (blue curve), tephriphonolite (yellow curve) and phonolite (red curve) compositions. The red star represents the lava lake conditions as determined by surface gas measurements [START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF][START_REF] Moussallam | Hydrogen emissions from Erebus volcano, Antarctica[END_REF].

Moussallam et al.,

  Fig. 4 shows S spectra for each sample group. All spectra show an absorption edge crest at ~2477 eV and a poorlyresolved shoulder at ∼2474 eV, typical of the presence of S 2-species in the glass and in agreement with the presence of pyrrhotite globules, < 3 μm in diameter, in MI from all samples This structure is most pronounced in AW82033 (basanite) and least in DVDP3-295 (basanite). Basanite sample DVDP3-295 shows an intense transition at ~2478.4 eV characteristic of the sulphite anion (SO 3 2-). Both DVDP 3-295 and tephriphonolite sample 97009 show a very strong transition at (~2482.6 eV) typical of sulphate anions (SO 4 2-) while other samples show a less pronounced transition at this energy level and AW82033 (basanite) from more shallow depths shows almost no sulphate signal. Samples 97018 (phonolite), 97010 (tephriphonolite), 97011 (phonolite) and to a lesser extent 97009 and AW82033 show a transition at 2471.2 eV representative of S 2-species in the glass.

Figure 1 :

 1 Figure 1: A. Transmitted light photomicrograph of an olivine hosted melt inclusion. B. X-ray potassium chemical map of anorthoclase hosted melt inclusions.

Figure 2 :

 2 Figure 2: Example of edge-step normalised XANES spectra. A. Example spectra from two matrix glass and three melt inclusions from a basanite sample (DVDP-3-295). B. Example spectra from four tephra-phonolite and four phonolite melt inclusions (samples 97009 and 97018 respectively). C. Example spectra from DVDP-3-295 basanitic melt inclusions compared to a basanite standard equilibrated at conditions near NNO+1. D. Example spectra from 97018 phonolitic melt inclusions compared to a tephra-phonolite standard equilibrated at conditions near QFM-1.

Figure 3 :

 3 Figure 3: A. Plot of CO 2 concentration (in ppm; determined by FTIR[START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF])) compared to Fe 3+ /Fe ratio determined by Fe K-edge XANES. B. Plot of H 2 O concentration (in wt%; determined by FTIR[START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF])) compared to Fe 3+ /Fe ratio determined by Fe K-edge XANES. Error bars on CO 2 and H 2 O measurements are from[START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]; errors in the Fe 3+ /Fe ratio are discussed in the SI. C.

Figure 4 :

 4 Figure4: S K-edge XANES spectra for each sample studied (averages for four or five individual MI spectra are shown). The dashed lines are at 2482.6 eV (S 6+ ), 2478.4 eV (S 4+ ), 2477.0 eV (S 2-) and 2471.2 eV (S 2-). Anhydrite (S 6+ )[START_REF] Scheinost | Geochemical behavior of uranium in mine tailings at Freital, Germany: A mu-XRF, mu-XAFS and mu-XRD study[END_REF]), sodium sulfite (S 4+ )[START_REF] Jugo | Sulfur K-edge XANES analysis of natural and synthetic basaltic glasses: Implications for S speciation and S content as function of oxygen fugacity[END_REF]) and pyrrhotite (S 2-) spectra are given for reference (available from the ESRF database http://www.esrf.eu/UsersAndScience/Experiments/Imaging/ID21/php).

Table 1 :

 1 Fe 3+ /Fe of glass standards determined by: * wet chemistry and †calculated from imposed fO 2 by controlled CO 2 /CO gas flux in furnace using equation from[START_REF] Kress | The compressibility of silicate liquids containing Fe2O3 and the effect of composition, temperature, oxygen fugacity and pressure on their redox states[END_REF] for the tephriphonolite and our modified equation (see method) for the basanite.

	Sample	Fe 3+ /Fe* Error Fe 3+ /Fe †
	XANSTD_Ba_01_QFM-1	0.07	0.01	0.10
	XANSTD_Ba_02_QFM	0.17	0.02	0.15
	XANSTD_Ba_03_NNO	0.26	0.02	0.21
	XANSTD_Ba_04_NNO+1	0.37	0.03	0.30
	XANSTD_TP_01_QFM-1	0.16	0.01	0.13
	XANSTD_TP_03_NNO	0.37	0.03	0.27
	XANSTD_TP_04_NNO+1	0.39	0.03	0.39

Table 2 .

 2 Volatile content centroid position, Fe 3+ /Fe and calculated pressure for all melt inclusion and glasses analysed. Pressure estimate from[START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]. †Pressure recalculated by this study. ** data from Eschenbacher, (1998) ;[START_REF] Oppenheimer | Mantle to surface degassing of alkalic magmas at Erebus volcano, Antarctica[END_REF]; ***error assumed to be 10%; ND not detected

	Melt	Host	Centroid Fe	3+ /Fe Error on	Pressure*	Pressure †	Error on	**S	***Error	**CO 2	**Error	**H 2 O	**Error
	inclusion/Glass					Fe	3+ /Fe	(bar)	(bar)	pressure (bar)	(ppm)	on S (ppm)	(ppm)	on CO 2 (ppm)	(wt%)	on H 2 O (wt%)
	DVDP-3-295j	olivine	7112.73	0.297	0.027	5136.2	4350	400	2448	245	4890	790	1.30	0.14
	DVDP-3-295c	olivine	7112.77	0.340	0.027	8033.9	7350	500	2488	249	5950	1000	1.46	0.17
	DVDP-3-295q	olivine	7112.76	0.327	0.027	5772.2	5570	470	1927	193	5280	880	1.25	0.14
	DVDP-3-295b	olivine	7112.76	0.332	0.027	4753.8	5030	380	2062	206	4000	660	1.67	0.18
	DVDP-3-295R	olivine	7112.75	0.320	0.027	7823.2	6090	560	2007	201	7270	1200	1.39	0.15
	DVDP-3-295g	olivine	7112.79	0.359	0.027	7577.3	8600	500	2187	219	6750	1090	1.75	0.19
	DVDP-3-295-G3		7112.57	0.150	0.027				465	47	ND		0.23	0.03
	DVDP-3-295-G2		7112.56	0.141	0.027				770	77	ND		0.34	0.04
	DVDP-3-295-G1		7112.58	0.162	0.027				325	32	ND		0.26	0.03
	AW82033h	olivine	7112.66	0.239	0.027	1393.4	630	280	1575	157	930	470	0.34	0.05
	AW82033c	olivine	7112.63	0.211	0.027	2101.5	1000	210	1525	152	1650	350	0.11	0.02
	AW82033a	olivine	7112.58	0.163	0.027	876.74	330	70	1405	140	530	110	0.18	0.02
	AW82033d	olivine	7112.60	0.186	0.027	1594.2	750	150	1390	139	1240	240	0.11	0.02
	AW82033g	olivine	7112.61	0.187	0.027	361.6	180	65	1085	108	280	110	0.17	0.02
	AW82033_G1		7112.64	0.223	0.027				1030	103	110	50	0.10	0.01
	AW82033_G3		7112.69	0.260	0.027				1150	115	10	5	0.01	0.001
	AW82033_G2		7112.67	0.246	0.027				990	99	60	30	0.10	0.01
	97009a	olivine	7113.14	0.388	0.027	6417.9	4350	130	665	66	3120	450	0.59	0.09
	97009d	olivine	7113.10	0.354	0.027	3771.5	2200	80	915	91	1490	260	0.45	0.06
	97009g	olivine	7113.16	0.400	0.027	4309	3690	90	545	54	2040	310	0.52	0.07
	97009j	olivine	7113.11	0.357	0.027	7359.6	2980	170	690	69	3590	600	0.49	0.07
	97009_G1		7113.03	0.286	0.027				520	52	170	90	0.15	0.02
	97009_G2		7112.96	0.228	0.027				330	33	190	100	0.12	0.01
	97009_G3		7113.00	0.263	0.027				510	51	220	120	0.12	0.01
	97010f	olivine	7113.12	0.372	0.027	2173.1	1380	90	345	34	740	270	0.13	0.02

*

Table 3 .

 3 Fe 3+ /Fe from wet chemistry calibration and ∆NNO determined by calibrating the centroid position using the standard's ∆NNO values (as imposed by controlled CO 2 /CO gas flux in furnace).

	Melt inclusion/Glass	∆ NNO	Error on ∆ NNO	Fe	3+ /fe	Error on Fe 3+ /Fe
	DVDP-3-295j	0.316	0.355	0.297	0.039
	DVDP-3-295c	0.874	0.355	0.359	0.039
	DVDP-3-295q	0.702	0.355	0.340	0.039
	DVDP-3-295b	0.767	0.355	0.347	0.039
	DVDP-3-295R	0.615	0.355	0.330	0.039
	DVDP-3-295g	1.110	0.355	0.385	0.039
	DVDP-3-295-G3	-1.596	0.355	0.085	0.039
	DVDP-3-295-G2	-1.712	0.355	0.072	0.039
	DVDP-3-295-G1	-1.440	0.355	0.102	0.039
	AW82033h	-0.441	0.355	0.213	0.039
	AW82033c	-0.810	0.355	0.172	0.039
	AW82033a	-1.436	0.355	0.103	0.039
	AW82033d	-1.136	0.355	0.136	0.039
	AW82033g	-1.123	0.355	0.137	0.039
	AW82033_G1	-0.654	0.355	0.189	0.039
	AW82033_G3	-0.173	0.355	0.243	0.039
	AW82033_G2	-0.355	0.355	0.223	0.039
	97009a	1.177	0.285	0.416	0.028
	97009d	0.799	0.285	0.379	0.028
	97009g	1.303	0.285	0.429	0.028
	97009j	0.832	0.285	0.382	0.028
	97009_G1	0.057	0.285	0.306	0.028
	97009_G2	-0.584	0.285	0.243	0.028
	97009_G3	-0.196	0.285	0.281	0.028
	97010f	0.996	0.285	0.398	0.028
	97010g	0.755	0.285	0.375	0.028
	97010d	0.137	0.285	0.314	0.028
	97010b	1.048	0.285	0.404	0.028
	97010c	-0.010	0.285	0.299	0.028
	97011b	-0.666	0.285	0.235	0.028
	97011c	-0.573	0.285	0.244	0.028
	97011a	-0.343	0.285	0.266	0.028
	97011f	-0.126	0.285	0.288	0.028
	97018f	-0.857	0.285	0.216	0.028
	97018a	-0.584	0.285	0.243	0.028
	97018c	-1.023	0.285	0.199	0.028
	97018e	-0.595	0.285	0.242	0.028

Table 4 .

 4 Starting conditions used in for each model run. *Sulphur capacity calculated using[START_REF] O'neill | The Sulfide Capacity and the Sulfur Content at Sulfide Saturation of Silicate Melts at 1400°C and 1 bar[END_REF]

	Model	Pressure (bar)	Temperature (°K)	FeO (wt%)	lnCs*	S (ppm)	fH 2 O (bar)	fH 2 (bar)
	Basanite	8600	1473.15	10.26	-1.42	2400	220	0.4
	Tephriphonolite	4350	1373.15	7.5	-3.94	915	70	0.12
	Phonolite	400	1273.15	5.43	-5.39	670	10	0.06

ACKNOWLEDGEMENTS

This work would not have been possible without the meticulous preparation of the MI samples by Al Eschenbacher while a student at NM Tech. We thank the Diamond Light Source for access to beamline I18 (proposal number SP8219-1) that contributed to the results presented here and Tina Geraki and Konstantin Ignatyev for their help. YM acknowledges support from the University of Cambridge Home and EU Scholarship Scheme and from the Philip Lake and William Vaughan Lewis funds of the Department of Geography, University of Cambridge. CO and PK acknowledge support from the NSF and United States Antarctic Program (grant ANT1142083). F.G. acknowledges the European Research Council for funding support (ERC grant number 279790). We are most grateful to the two reviewers of the manuscript, Nicole Metrich and Elizabeth Cottrell, for their insightful comments on the manuscript.

AUTHOR COMTRIBUTIONS

Y.M. designed and performed the synchrotron experiments, analysed data, performed the modelling and wrote the paper. P.K. provided all the samples and MI maps. C.O., B.S. and P.K. supervised the project and wrote the paper. F.G. designed the modelling and wrote the paper. N.P., M.H., K.B. and A.D. performed the synchrotron experiments. N.P. developed the fitting algorithm for XANES spectra.


Yangyang Chen
email: yangyang.chen@telecom-bretagne.eu

Xavier Lagrange
email: xavier.lagrange@telecom-bretagne.eu

Downlink Capacity Gain Analysis of Mobile Relay in LTE-Advanced Network

Keywords: Mobile Relay, Capacity Gain, Spatial Poisson Process, LTE

In this paper, we analyze whether mobile relay could bring capacity gain to a LTE-Advanced network. The analysis is based on the Shannon formula modified for LTE. The Spatial Poisson Process is used to model the distribution of terminals and mobile relays in the cell. We consider two relay modes in the study: pure-relay mode and femto-like mode. Through the analysis we get the conclusion that penetration loss is a key factor to decide whether mobile relay can bring considerable capacity gain into the system. When penetration loss is 25 dB, pure-relay mode can bring about +30% capacity gain compared with traditional mode, femto-like mode can bring almost +100% capacity gain compared with traditional mode.

I. INTRODUCTION

Relay has been considered as a key technology for 3GPP LTE-Advanced, as it can increase the network capacity, and provide coverage in new area [START_REF]Further advancements for E-UTRA physical layer aspects[END_REF]. There exist two types of relays, fixed relay and mobile relay [START_REF] Wang | Cooperative MIMO channel models: A survey[END_REF]. Fixed relays are usually deployed at locations that are shrewdly chosen, such as shadow area or hot spot. They have already been standardized by 3GPP in release 10. Now in the context of ITS (Intelligent Transport System), the 3GPP is working on enhancing the relay functionality with mobile relays [3]. Mobile relays are mounted in a moving vehicle that provides wireless connectivity to users inside the vehicle via an indoor antenna. The backhaul connection to donor eNodeB is provided via an outdoor antenna.

One example use case is buses, as shown in Figure 1. The link between donor eNodeB and the relay is called the backhaul link, the link between the relay and the embedded UE is called the access link, and the link between the eNodeB and the UE is called the direct link. The buses are well shielded with coated windows, which leads to a rather high penetration loss between outdoor and in-vehicle. Furthermore, the vehicle can be anywhere in the cell without dedicated network planning like fixed relay. Therefore providing the required services to the massive users that are moving with the bus is much more challenging than typical mobile wireless environments. Numerous studies have been done on performance evaluation of fixed relays. However, to the best of our knowledge very few studies have been done towards the scenario that deploying mobile relay in public transportation systems. In the previous related work about mobile relay [START_REF] Lin | Capacity analysis of dedicated fixed and mobile relay in LTE-Advanced cellular networks[END_REF][5] [START_REF] Chen | Downlink data rate gain provided by a mobile relay for LTE-advanced[END_REF]. [START_REF] Lin | Capacity analysis of dedicated fixed and mobile relay in LTE-Advanced cellular networks[END_REF] presented an simplified and ideal capacity analysis of fixed relay and mobile relay with SISO antenna configuration and without shadowing. [START_REF] Sui | The Potential of Moving Relays -A Performance Analysis[END_REF] investigated the performance of dual-hop moving relay assisted transmission, but the scenario they considered was more related to point-to-point transmission, not a network. [START_REF] Chen | Downlink data rate gain provided by a mobile relay for LTE-advanced[END_REF] investigated the data rate gain provided by one mobile relay deployed in a network with only one terminal, but how much capacity gain can be attained through mobile relays in a LTE network is still unknown.

Therefore in this paper, we focus on the capacity gain in a network deployed with mobile relays. The relay in this work is layer 3 relay specified by LTE [START_REF] Iwamura | Relay technology in LTE-Advanced[END_REF]. The sections are organized as follows. Firstly, the system model is given in Section II, including some assumptions and analysis methods. Secondly, some analysis and calculations are given in Section III. Then the performance results are shown in Section IV. A conclusion is drawn in Section V.

II. SYSTEM MODEL

A. Spatial Model

In this study we focus on a downlink LTE network with multi-user and multi-relay. We consider three types of nodes existing in the mobile relay scenario.

1) Mobile relays: which are mounted on public buses. Each relay has an attenuation outside the bus for the backhaul link, and an invehicle attenuation for the access link. 2) Ordinary UEs: which are outside the bus.

They always connect to the eNodeB directly. 3) Embedded UEs: which are inside the bus. When mobile relays are deployed, we assume all embedded UEs are connected to the LTE network through mobile relays. And we assume ordinary UEs and buses are uniformly distributed in the cell. The locations of mobile relays can be represented by the locations of buses. Embedded UEs are uniformly distributed inside the bus. But from the view of the cell, the UEs (ordinary and embedded) are not uniformly distributed in the cell. We use Spatial Poisson Process to model the distribution of mobile relays and UEs [START_REF] Andrews | A primer on spatial modeling and analysis in wireless networks[END_REF]. The related variables are listed in the Table 1.

Based on the Spatial Poisson Distribution, the probability to have n u UEs in area A is:

P (N U = n u ) = (ρ U A) nu n u ! e -ρ U A . ( 1
)
The probability to have n v vehicles in area A is:

P (N V = n v ) = (ρ V A) nv n v ! e -ρ V A . (2)

B. Interference Model

In LTE network, the cell-edge terminals suffer from severe interferences from neighbor eNodeBs. Neighbor eNodeBs are the source of interferences. In 3GPP specified relay network, the relay utilizes the same frequency band for the access link and the backhaul link, so the access link and the backhaul link are separated in time domain. Due to this mechanism, two new types of interferences are brought in the mobile relay scenario. One is Relayto-Relay interference, the other one is Relay-to-UE interference, as shown in Figure 2.

1) Relay-to-Relay Interference:

In FDD LTE, eNodeBs do not need to be synchronized in time domain [START_REF] Holma | LTE for UMTS -Evolution to LTE-Advanced[END_REF]. All mobile relays under the same donor eNodeB are aligned, but mobile relays under different eNodeBs may be misaligned. For example, one mobile relay A under eNodeB 1 is receiving data on the backhaul link, another mobile relay B under

TABLE I THE TABLE OF THE VARIABLES FOR SPATIAL MODEL

Variable

Define A

The area of the cell (km 2 ) N U the random variable stands for the number of UEs in the cell nu the value of the random variable N U N V the random variable stands for the number of buses in the cell nv the value of the random variable N V ρ V the density of buses in the cell ρ U the density of UEs per km 2 in the cell eNodeB 2 is sending data on the access link, they are both in cell-edges and close to each other. In this case, mobile relay A might suffer interference from mobile relay B. The Relay-to-Relay interference is shown as solid line in Figure 2. In our model, we assume all eNodeBs are synchronized, so this type of interference is not considered.

2) Relay-to-UE Interference: Another kind of interference is Relay-to-UE interference. When two mobile relays are transmitting to their embedded UEs, frequency reuse is applied on the access link. If they are close to each other, which is possible for two buses, the embedded UEs can be interfered by the neighbor mobile relay. It's shown as dotted line in Figure 2. This kind of interference is unique in mobile relay scenario. But since the transmitted power of mobile relay is relatively small, when two mobile relays are far away, this kind of interference is generally small. This interference is considered in our work.

C. Propagation Model

In mobile relay scenario the propagation loss can be divided into three parts: outdoor path loss, penetration loss L trans between outdoor and invehicle and in-vehicle path loss. Outdoor path loss is modeled by Okumura-Hata formula. Since the terminal inside the bus is close to the in-vehicle antenna of mobile relay. We consider free space propagation inside the bus. Path loss model is shown in Table II, d (in km) is the distance between two nodes.

The signal power at the receiver can be written with the general formula: where P S is the transmitted power of the transmitter, k and α are constant parameters deduced from propagation loss, ξ is a standard normal random variable that models shadow fading and parameter σ s is the standard deviation of shadow fading.

P R = P S k 10 σs 10 ξ d α (3)

D. System Mode

In this article we compare three modes: traditional mode, pure-relay mode and femto-like mode. The traditional mode is a baseline mode. There are no mobile relays, all terminals connect directly to eNodeB. In the pure-relay mode, when mobile relays transmit to embedded UEs in the access time slot, the eNodeB stops transmitting to ordinary UEs. But in the femto-like mode, frequency reuse is applied: in the access time slot eNodeB and relay transmit to ordinary UEs and embedded UEs respectively at the same time, as shown in Figure 3. So for embedded UEs in the femto-like mode the donor eNodeB is one source of the interferences.

III. SYSTEM ANALYSIS

A. The calculation of SINR

In this section, we compute the SINR of the terminal.

1) SINR in the traditional mode: In this mode, the UE connects to the eNodeB directly. We consider outdoor path loss, penetration loss on the direct link, and the interferences from neighbor

δ i U E = P S k 1 10 σ s1 10 ξ d α1 ( j P j + N ) (4)
where P S is the transmitted power of donor eN-odeB, P j is the received power from other eNodeB j, d is the distance between UE and donor eNodeB, k 1 = 10 -14.178 × 10 -Ltrans/10 , α 1 = 3.5 and N is UE's thermal noise.

2) SINR in two relay modes: In relay network, the backhaul link and the access link are timedivision multiplexed, so the interferences are different in the backhaul time slot and in the access time slot. For ordinary UE, the SINR in the backhaul time slot remains the same as the traditional mode. For mobile relay, the signal that the mobile relay receives from donor eNodeB mainly suffers outdoor path loss. The SINR of mobile relay k in the backhaul time slot is:

δ k relay = P S k 2 10 σ s1 10 ξ d α1 ( j P j + N Relay ) (5)
where P S is the transmitted power of donor eN-odeB, P j is the received power from neighbor eNodeB d is the distance between relay and donor eNodeB, k 2 = 10 -14.178 , α 1 = 3.5 and N Relay is mobile relay's thermal noise.

In the access time slot, the signal that embedded UEs receives mainly suffers in-vehicle path loss. The interferences are from neighbor eNodeBs and other mobile relays. The SINR of embedded UE i in the pure-relay mode is:

δ i U E = P relay k 3 10 σ s2 10 ξ d α2 ( j P j + k P relay,k + N ) (6)
where P relay is the transmitted power of relay, P j is the received power from neighbor eNodeB j, P relay,k is the received power from neighbor relay k, d is the distance between relay indoor antenna and embedded UE, k 3 = 10 -9.84 , α 2 = 2 and N is UE's thermal noise. Note that, in femto-like mode donor eNodeB is one source of interferences for embedded UE. The received power from donor eNodeB for embedded UE i should be added as interference in equation ( 6). For ordinary UEs in femto-like mode, in the access time slot relay is one source of interferences. For them the received power from relays should be added as interference in equation ( 4).

B. Scheduling

We consider equal bit rate scheduling, which means all UEs are served with the same data rate. We define the served data rate as the unified bit rate per second for each UE. The served date rate depends on the number of UEs and channel condition. The data rate on each link can reflect channel condition, which is calculated by Shannon formula modified for LTE and is limited by Modulation Coding Scheme (MCS) [START_REF] Mogensen | LTE Capacity Compared to the Shannon Bound[END_REF]. Note that in relay modes the data that mobile relay transmits to embedded UEs is, at most, the one that relay receives on the backhaul link. The related variables are listed in the Table 3 in the sequence of alphabet.

First we calculate the served data rate in traditional mode. The idea is to calculate how many bits can be transmitted per second for each UE. The inverse of the data rate can be seen as the time to transmit 1 bit. In traditional mode, eNodeB can serve the UEs all the time. So the served data rate R served is:

R served = 1 No i=1 1
Ro,i + N b i=1 Ni,a j=1 1 Re,j (7)
Then in the pure-relay mode, eNodeB can only serve the UEs in the backhaul time slot, so R served

R served = 1 No i=1 1
R1o,i + N b i=1 Ni,a j=1 ( 1 R b,i + 1 Ra,j ) (8)
In the femto-like mode, eNodeB can serve ordinary UEs all the time, so R served is:

R served = 1 No i=1 1 R1o,i+R2o,i + N b i=1 Ni,a j=1 ( 1 R b,i + 1 Ra,j ) (9)

C. Resource Allocation Model

To simplify the resource allocation process, we allocate the resource in the manner of time, as is shown in Figure 4. In this model each mobile relay or UE take the whole system bandwidth, but because of different channel conditions, they consume different time resource.

First eNodeB allocates resources to mobile relays in the backhaul time slot. After that, eNodeB can keep allocating resources to ordinary UEs in the backhaul time slot. Then in the access time slot, mobile relays allocate resources to their embedded UEs. Also, in the femto-like mode donor eNodeB can allocate resources to unserved ordinary UEs simultaneously. The resource allocated to each node i is calculated according to its channel condition:

α x,i = R served R x,i (10)
where x = o1, o2, b, a stands for ordinary UE in the backhaul time slot, ordinary UE in the access time slot, mobile relay and embedded UE respectively.

IV. PERFORMANCE RESULTS

The performance of mobile relay is evaluated in this section. The evaluation is performed by Monte Carlo simulations. The total number of UEs and the number of vehicles (mobile relays) in the cell in each snapshot is a Poisson random parameter. Mobility and handover are not taken into account. Random shadowing is fixed in each snapshot.

The vehicles (per km of road) in France was 37 in 2010 [START_REF]Vehicles (per km of road)[END_REF]. This data include cars, buses, and freight vehicles. We only consider public buses in mobile relay scenario, the density of buses is set to 4 per km 2 .

In the traditional mode, all UEs connect directly to the donor eNodeB. In the pure-relay mode and the femto-like mode the distance between the mobile relay and an embedded UE is assumed to be 20 meters (the bus is usually 40 meters long at most), and the transmitted power of mobile relay is set to -10 dBm [START_REF] Claussen | Selfoptimization of coverage for femtocell deployments[END_REF]. The number of backhaul subframes in one radio frame is set to 6. Some main parameters are listed in Table IV.

A. The Served Data Rate

First of all, we compare the served data rate in three modes under different ratio of embedded UEs in the cell. The simulation result is shown in Figure 5.

As shown in Figure 5, two relay modes have higher served data rate, this is because penetration loss is eliminated in relay modes. With the increase of the proportion of UEs inside the vehicle, more UEs suffer from penetration loss in the traditional mode, so the served data rate decreases. In the relay modes with the increase of UEs inside the vehicle, backhaul link increasingly becomes the bottleneck, so the served data rate also decreases.

B. The Penetration Loss

The Penetration loss affects the performance significantly. We compare system capacity in different modes under different penetration loss when the number of embedded UEs in a bus is fixed to 5 [START_REF] Sui | Moving cells: a promising solution to boost performance for vehicular users[END_REF]. The System capacity is the sum of the achievable data rate of all served UEs. The results are shown in Figure 6. From the figure we can see, the larger penetration loss is, the higher capacity relay modes could achieve, then more capacity gain can be brought by mobile relays. This is because mobile relay eliminates the affect of penetration loss. In traditional mode, penetration loss deteriorates the signal. According to [START_REF] Holma | LTE for UMTS -Evolution to LTE-Advanced[END_REF], penetration loss in vehicles is normally between 20 to 30 dB. When penetration loss is 25 dB, pure-relay mode can bring about +30% capacity gain, femto-like mode can bring almost +100% capacity gain compared with traditional mode, so capacity gain can be attained by deploying mobile relay in public transportation systems.

Also we can see femto-like mode can achieve higher system capacity than pure-relay mode. This is because frequency reuse is applied in femtolike mode, eNodeB and mobile relay can send data simultaneously, which improves the bandwidth efficiency.

V. CONCLUSION

In this paper, we investigate the capacity gain that mobile relay could bring into a public transportation system. The Spatial Poisson Process is used to model the distribution of terminals and buses (mobile relays) in the cell. We compare the performance of traditional mode, pure-relay mode and femto-like mode. The results show that penetration loss is a key factor to decide whether mobile relay could bring capacity gain into system. Normally higher system capacity can be achieved in the femto-like mode due to frequency reuse. When penetration loss is 25 dB, pure-relay mode can bring about +30% capacity gain compared with traditional mode, femtolike mode can bring almost +100% capacity gain compared with traditional mode. Furthermore, in our study we assume mobile relay has the same receiver as the UE, which means the performance of mobile relay can be improved by using better receivers.

Fig. 1 .

 1 Fig. 1. Mobile relay mounted on a bus

Fig. 2 .

 2 Fig. 2. Interferences in mobile relay scenario

Fig. 3 .

 3 Fig. 3. The pure-relay mode vs The femto-like mode

Fig. 4 .

 4 Fig. 4. Resource allocation in time domain

Fig. 5 .

 5 Fig. 5. the served data rate vs the fraction of embedded UEs (with 95% confidence interval)

Fig. 6 .

 6 Fig. 6. System Capacity vs Penetration Loss

TABLE III VARIABLES

 III FOR EQUAL BIT SCHEDULING Variable Define N b the number of buses (mobile relays) in the cell N i,a the number of UEs inside the bus associated with bus (mobile relay) i No the number of ordinary UEs in the cell R a,i the data rate for embedded UE i on the access link R b,i the data rate for mobile relay i on the backhaul link R e,j the data rate for UE j inside the bus in traditional mode R o,i the data rate for ordinary UE i in traditional mode R1 o,i the data rate for ordinary UE i in the backhaul time slot R2 o,i the data rate for ordinary UE i in the access time slot R served the served date rate for all UEs

is:

ACKNOWLEDGMENT

This work was performed within project SYSTUF, which is subsidized by the french ministry of Industry in the framework of the AMI ITS program.