Tiffanie Lescure

J Moreau

C Charles

T Ben

Ali Saanda

Hugues Thouin

N Pillas

P Bauda

I Lamy

Fabienne Battaglia-Brunet

Ben Ali

Influence of organic matters on AsIII oxidation by the microflora of polluted soils

Keywords: arsenic, polluted soils, organic matter, microorganisms, AsIII oxidation

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

Introduction

High soil concentrations of the toxic element arsenic (As) may be linked to pollution originating from mining and industrial activities, long-term application of As-containing pesticides [START_REF] Smith | Arsenic in the soil environment: a review[END_REF] or to the geochemical background. On polluted mining sites, characterized by high concentrations of metal(oid)s and poor fertility, organic amendments are often considered as a means of facilitating phyto-stabilization [START_REF] Galende | Field assessment of the effectiveness of organic amendments for aided phytostabilization of a Pb-Zn contaminated mine soil[END_REF]. In addition, agricultural soils affected by diffuse As pollution from pesticides can be fertilized with organic matter [START_REF] Franchini | Rapid transformations of plant water-soluble organic compounds in relation to cation mobilization in an acid oxisol[END_REF][START_REF] Chantigny | Decomposition of de-inking paper sludge in agricultural soils as characterized by carbohydrate analysis[END_REF][START_REF] Chantigny | Fate of carbon and nitrogen from animal manure and crop residues in wet and cold soils[END_REF]. Many studies have described the geochemical interactions between arsenic and organic matter: change of As speciation [START_REF] Redman | Natural organic matter affects arsenic speciation and sorption onto hematite[END_REF], formation of soluble complexes [START_REF] Saada | Adsorption of arsenic(V) on kaolinite and on kaolinite-humic acid complexes. Role of humic acid nitrogen groups[END_REF][START_REF] Redman | Natural organic matter affects arsenic speciation and sorption onto hematite[END_REF], competition for sorption sites [START_REF] Bauer | Mobilization of arsenic by dissolved organic matter from iron oxides, soils and sediments[END_REF]. Conversely, organic matter is also a trophic resource for microorganisms but its influence on microbial activities linked to As is poorly documented, even though the soil microflora plays a major role in geochemical cycles and particularly in As speciation. Bacteria isolated from soils have been shown to oxidize AsIII and/or reduce AsV [START_REF] Macur | Bacterial populations associated with the oxidation and reduction of arsenic in an unsaturated soil[END_REF][START_REF] Inskeep | Detection, diversity and expression of aerobic bacterial arsenite oxidase genes[END_REF][START_REF] Bachate | Oxidation of arsenite by two β-proteobacteria isolated from soils[END_REF], or to methylate this toxic metalloid [START_REF] Huang | Arsenic speciation and volatilization from flooded paddy soils amended with different organic matters[END_REF]. Filamentous fungi isolated from contaminated soils are able to reduce AsV and methylate As [START_REF] Su | Arsenic biotransformation by arsenic-resistant fungi Trichoderma asperellum SM-12F1, Penicillium janthinellum SM-12F4, and Fusarium oxyporum CZ-8F1[END_REF]. Microbial transformations of arsenic in soil have important implications because mobility, toxicity and bioavailability of the element are closely related to its speciation (Smedley and Kinniburgh, 2002).

The global AsIII-oxidizing activity of the microflora should tend to reduce the risk of transfer of the toxic element from soil to surface water or groundwater.

Important bacterial mechanisms involved in AsIII/AsV transformations are the ars system, whose primary function is detoxification with AsV being reduced by an arsenate reductase ArsC, the aio oxidation system through arsenite oxidase, and the arr system of AsV dissimilatory reduction in anaerobic conditions [START_REF] Inskeep | Detection, diversity and expression of aerobic bacterial arsenite oxidase genes[END_REF]. [START_REF] Huang | Arsenic speciation and volatilization from flooded paddy soils amended with different organic matters[END_REF] examined the effect of vegetal organic amendments on arsenic speciation and volatilization in flooded paddy fields. They observed that concentration and type of organic matter had a significant effect on AsIII oxidizing activities and AsV reduction via bacterial arsC genes expression. The organic matter amendments stimulated the methylation and volatilization of As. Yamamura et al. (2009) showed that uncontaminated surface soil microflora exhibit both AsIII oxidizing and AsV reducing activities. However, the influence of organic matter on the behavior of As-transforming bacteria in unsaturated soils was not described. As several studies have suggested that organic substances could lower the efficiency of bacterial arsenic oxidation in the presence of oxygen [START_REF] Challan-Belval | Enhancing pozzolana colonization by As(III)-oxidizing bacteria for bioremediation purposes[END_REF][START_REF] Bachate | Oxidation of arsenite by two β-proteobacteria isolated from soils[END_REF], there is a need to acquire information about the influence of the nature and concentration of organic matter on As oxidizing activities in polluted soils.

3

In this context, the present study aimed to determine the influence of organic matters on the AsIII oxidizing activity of complete microflora of polluted soils in aerobic conditions. Activity measurements made in specific liquid media to assess the global level of target microbial metabolisms are usual: this type of measurement has been proposed to quantify PAH biodegradation rate in soils [START_REF] Kästner | Microbial degradation of polycyclic aromatic hydrocarbons in soils affected by the organic matrix of compost[END_REF], thiosulfate-oxidizing activity in paddy fields [START_REF] Stubner | Sulfur oxidation in rice field soil: activity, enumeration, isolation and characterization of thiosulfate-oxidizing bacteria[END_REF], denitrification [START_REF] Buys | Development and application of a denitrification test based on gas production[END_REF], anaerobic oxidation of ammonium [START_REF] Dapena-Mora | Evaluation of activity and inhibition effects on Anammox process by batch tests based on the nitrogen gas production[END_REF], and FeII oxidizing activity [START_REF] Senko | Characterization of Fe(II) oxidizing bacterial activities and communities at two acidic Appalachian coalmine drainage-impacted sites[END_REF]. For this study, AsIII oxidizing activities were monitored in a previously optimized mineral liquid culture medium [START_REF] Battaglia-Brunet | An arsenic(III)-oxidizing bacterial population: selection, characterisation, and performance in reactors[END_REF] to which two organic substrates differing in their composition, i.e. yeast extract or a synthetic mixture of defined molecules (SMOM) were added at two different concentrations. The composition of the yeast extract is not defined exactly but is close to that of living cells; it is also a source of vitamins that might stimulate the growth of some microorganisms. SMOM is a well-defined complex mixture of molecules displaying some of the main characteristics of natural soil organic matter such as C/N ratio and supply of phenolic and carboxylic groups that may interact with inorganic soil components. Thus, a substrate usually used in microbiology (yeast extract) and an organic amendment designed to present common features with natural organic matter (SMOM) were compared in terms of influence on AsIII biooxidation in conditions of equivalent organic carbon concentrations. The AsIII oxidizing rate constants were correlated with the presence of organic substrates and characteristics of eight polluted soils (physico-chemical, structural and biological features). A study was then made of the combined influence of microbial activities and amendment with complex organic substrates on the speciation and mobilization of intrinsic arsenic in the four most polluted soils.

Material and methods

Soil sampling and characterization

Eight different soils were sampled on three highly polluted sites that could be candidates for the application of aided phyto-stabilization. All materials were surface, non-saturated soils (0-15 cm).

Four different soils were sampled in and around the Cheni disused gold mine site (Limousin; 45°32'59.90"N, 1°09'39.04"E): in a cultivated field, a meadow, a forest and on the site near the arsenic-containing waste dump. Two brownfield soil samples were taken from the Auzon disused industrial site (Haute-Loire; 45°23'13.67"N, 3°21'24.32"E) and two brownfield soil samples were collected on the Salsigne disused gold mine site (Aude; 43°31'89.25"N, 2°38'11.07"E and 43°31'52.75"N, 2°38'73.83"E). The soils were sampled in sterile glass jars, sieved at 2 mm in sterile sieves then stored at 5°C in sterile glass jars. Their water content was determined by drying at 50°C for 24h. Mineral carbon and carbonates were determined by volumetric method per NF ISO 10693.

Total carbon was analyzed per NF ISO 10694. After drying and hand grinding in a mortar, total concentrations of As, Pb, Cd, Zn and Fe were determined in the soils using a portable NITON© X-ray fluorescence field analyzer (XLT999KWY, bulk mode, counting time 60s). Values for the elements Ag, U, Se, Hg, Cu, Ni, Cr and V were below the detection limit. Total As was also analyzed in all soils by Atomic Absorption Spectrophotometry after total digestion, per NF EN 13346. AsIII and AsV were analyzed by HPLC-ICP-MS after extraction with 10 mL of H 3 PO 4 1M added to 0.4 g of sample and microwave heating [START_REF] Vergara Gallardo | Speciation analysis of arsenic in environmental solids reference materials by high-performance liquid-chromatographyhydride generation-atomic fluorescence spectrometry following orthophosphoric acid extraction[END_REF]) in a closed system at 120°C for 20 min. The remaining solution was diluted to 50 mL with ultrapure water and then analyzed with HPLC-ICP-MS, using quantification by standard additions to avoid matrix effects. Arsenic species separation was performed using an anion exchange column (Hamilton PRPX-100) and a mobile phase of ammonium hydrogen phosphate 15 mM at pH 8.5 [START_REF] Thomas | Feasibility of identification and monitoring of arsenic species in soil and sediment samples by coupled high-performance liquid chromatographyinductively coupled plasma mass spectrometry[END_REF]. The percentage of AsIII (%AsIII) was calculated as [AsIII]x100/[total As]. Total nitrogen was analyzed by the Kjeldhal method per NF EN 25663. Granulometry was determined per NF X 31.107, total phosphorus per NF X 31.161, K 2 per NF X 31.108 and pH per NF ISO 10390. The biodegradability of intrinsic organic matter was studied via CO 2 emission (soil respiration), per [START_REF] Rey | Effect of temperature and moisture on rates of carbon mineralization in a Mediterranean oak forest soil under controlled and field conditions[END_REF] Living bacteria were enumerated by microscopy using the Live/Dead® kit (BacLight™ Viability L-13152 Molecular Probes, Invitrogen) per [START_REF] Pascaud | A fluorescence-based assay for measuring the viable cell concentration of mixed microbial communities in soil[END_REF]. The soils (stored at 5°C) were incubated at ambient temperature for 72 h and then suspended at 1% in sodium pyrophosphate and agitated reciprocally for 15 min, sonicated 2 x 20 s at 45 kHz. The suspension was settled for 1 min, 1 mL of the supernatant was then sampled for filtration, application of coloration and observation with a Zeiss Axio Imager Z1 microscope equipped with UV HBO lamp. Living bacteria, exhibiting green fluorescence with the FITC filter, were enumerated on 10 independent fields (each of 5,800 µm 2 ).

Average cell concentrations were calculated from volume of sample used and known filter area.

The characteristics determined for the eight soils samples are detailed in Table 1.

AsIII oxidizing activity tests

The basal culture medium was the CAsO1 medium (pH 6) described in [START_REF] Battaglia-Brunet | An arsenic(III)-oxidizing bacterial population: selection, characterisation, and performance in reactors[END_REF].

This was supplemented with 1 mM AsIII. The soils (stored at 5°C) were incubated at 25°C for 72 h before starting the tests which were performed in 250 mL Erlenmeyer flasks filled with 100 mL medium and plugged with cotton. Each flask was inoculated with a mass of soil equivalent to 0.2 g dry weight. Flasks were incubated at 25°C under reciprocal agitation (150 rpm). The flasks were sampled (3 mL) twice a day; samples were filtered at 45 µm and frozen at -20°C until AsIII/AsV separation was performed. The AsIII oxidation activities were analyzed in (i) simple basal medium, (ii) basal medium amended with 0.2 g/L or 1 g/L of yeast extract (Sigma), and (iii) basal medium amended with two concentrations (0.5 g/L or 2.5 g/L corresponding to equivalent carbon concentrations as 0.2 and 1 g/L of yeast extract, i.e. 0.08 and 0.4 g/L of carbon) of a Synthetic Mixture of Organic Matter (SMOM). The SMOM was designed to mimic the average composition of soil organic matter in terms of C/N ratio, functional groups, amino-acids and sugars, on the basis of data available from the International Humic Substances Society's website (IHSShttp://www.humicsubstances.org). It contained the following molecules: L-arginine (3.2% dry weight), L-aspartic acid monopotassium salt (2.5%), L-glutamic acid monosodium salt monohydrate (3.0%), glycine (1.2%), L-histidine (0.3%), Lisoleucine (0.7%), L-leucine (0.9%), methionine (0.3%), L-phenylalanine (0.6%), L-serine (1.3%), Lthreonine (0.4%), L-tyrosine (0.4%), L-valine (1.5%), succinic acid disodium salt hexahydrate (19.1%), acetic acid (5.6%), propionic acid (2.8%), sodium tartrate dehydrate (8.3%), valeric acid (3.5%), calcium formate (3.7%), citric acid monohydrate dibasic (5.4%), butyric acid (2.1%), oxalic acid (0.8%), palmitic acid (6.7%), gallic acid (3.9%), vanillic acid (3.2%), 4-hydroxybenzoic acid (1.7%), (+)-catechin hydrate (2.9%), protocatechuic acid ethyl (1.2%), trans-ferulic acid (1.0%), pcoumaric acid (0.1%), rutin (5.0%), D-glucose (2.3%), D(+)-galactose (1.1%), D(+)-mannose (1.1%), D(+)-fucose (0.2%), L-rhamnose (0.8%), D(-)-arabinose (0.5%), D(-)-ribose (0.1%), D(+)-xylose (0.8%). The pH of the mixture was adjusted to that of the liquid medium (pH 6). The SMOM concentrations provided the same organic carbon concentration as the 0.2 and 1 g/L of yeast extract, with 1 g/L yeast extract corresponding to 0.4 g/L of organic carbon [START_REF] Holwerda | A defined growth medium with very low background carbon for culturing Clostridium thermocellum[END_REF]. The conditions were therefore 0.08 and 0.4 g/L of added organic carbon for both yeast extract and SMOM.

Incubations were performed in triplicate. The following controls were performed: as all experiments could not be carried simultaneously, the absence of effect of soil storage at 5°C was verified by repeating the test at the beginning and at the end of the study with one of the polluted soils. Abiotic AsIII oxidation controls were performed in the absence of soil inoculation and with sterile soils. For these latter controls, soils were sterilized by autoclaving four times at 120°C, 1h, at 24 h intervals.

AsV was quantified by flame atomic absorption spectrophotometry, after AsV/AsIII separation with the PDC/MIBK method [START_REF] Battaglia-Brunet | An arsenic(III)-oxidizing bacterial population: selection, characterisation, and performance in reactors[END_REF].

First order AsIII oxidizing rate constants were determined by linear least squares regression fitting of the ln[AsV] versus time line, using the following equation:

Ln[AsV] = kt + Constant (1)

Principal Component Analysis

Statistical analyses were carried out using the XLSTAT 2014 software (Addinsoft,version 16.2.01.6189). Pearson correlations were calculated with all soils parameters, and eight independent soil parametersselected as those correlating most strongly with AsIII oxidizing rate constants and their ratios (Electronic Supplementary material ESM1)were subjected to a principal component analysis (PCA). Rate constants and their ratios were integrated as supplementary data.

Soil incubations without addition of AsIII

The four most polluted soils in terms of As concentration were selected for incubation: Auzon 1, Auzon 2, Cheni site and Cheni forest. Experiments were performed in 60 mL flasks with cotton stoppers (aerobic conditions). Slurries were prepared by mixing a mass of soil corresponding to 2.5 g (dry weight) and 25 mL of spring water (Montcalm, Ca 3 mg/L, Mg 0.7 mg/L, Na 2.2 mg/L, K 0.6 mg/L, SO 4 10 mg/L, HCO 3 5.2 mg/L, NO 3 0.7 mg/L, Cl 0.6 mg/L, pH 6.8). Control blanks were prepared with soils sterilized as described in part 2.2. Spring water was autoclaved (120°C, 20 min).

Incubations were performed with and without addition of SMOM at 0.4 g/L of organic carbon. For each condition (blank and experiment, with and without SMOM), six flasks were prepared. One triplicate was sacrificed after 15 min of incubation and the remaining flasks were incubated for 7 days at 25°C, under reciprocal agitation (150 rpm). When the incubations were sacrificed, 10 mL of slurry were filtered at 0.45 µm; 2.5 mL of this filtrate were used immediately for AsIII/AsV separation on resin [START_REF] Ficklin | Separation of arsenic(III) and arsenic(V) in ground waters by ion-exchange[END_REF], the remainder was acidified with 50 µL HCl 37% for flame AAS analysis of total Fe. Separated AsIII and AsV were quantified by graphite furnace AAS (detection limit 20 µg/L).

Samples of biotic soil slurries were taken after 15 min and 7 days of incubation and stored at -20°C.

For biomolecular analyses, genomic DNA was extracted from the -20°C stored soil slurries using the FastDNA® Spin Kit for Soil (Bio101). The community structure and its evolution were monitored by two methods, (1) Capillary electrophoresis-terminal restriction fragment length polymorphism (CE-T-RFLP) diversity analysis of the 16S rRNA gene, as described by [START_REF] Mercier | Biofilm formation vs PCB adsorption on granular activated carbon in a PCB contaminated aquatic sediment[END_REF] and(2) Capillary Electrophoresis Single-Strand Conformation Polymorphism (CE-SSCP, [START_REF] Delbès | Monitoring of Activity Dynamics of an Anaerobic Digester Bacterial Community Using 16S rRNA Polymerase Chain Reaction-Single-strand Conformation Polymorphism Analysis[END_REF]. CE-SSCP analyses were performed with an ABI Prism 310 genetic analyzer using a 47 cm long capillary, a non-denaturating 5.6% CAP polymer (Applied Biosystems). CE-SSCP electrophoregrams were analyzed using the StatFingerprints Version 2 software [START_REF] Michelland | StatFingerprints: A Friendly Graphical Interface Program for Processing and Analysis of Microbial Fingerprint Profiles[END_REF]. For total bacteria enumeration, bacteria were separated from soil particles and from eukaryotes using the methods validated by [START_REF] Lindahl | Evaluation of Methods for Extraction of Bacteria from Soil[END_REF], [START_REF] Lindahl | Improved soil dispersion procedures for total bacterial counts, extraction of indigenous bacteria and cell survival[END_REF] and [START_REF] Bertrand | High molecular weight DNA recovery from soils prerequisite for biotechnological metagenomic library construction[END_REF], including a separation of bacteria from soil particles using the Nycodenz gradient method. The final pellet was suspended in 1 mL of NaCl 0.8%, mixed with 1 mL of absolute ethanol and stored at -20°C.

Details about the methods are given in the Electronic Supplementary Material ESM4. Bacteria were enumerated after fluorescent staining, as described in [START_REF] Kumar | Sulfur and Oxygen Isotope Tracing in Zero Valent Iron Based In situ Remediation System for Metal Contaminants[END_REF].

Results and discussion

AsIII-oxidizing activity measurements

Experiments were performed in order to evaluate the influence of some soil characteristics on the AsIII oxidizing activity of microbes, with and without added organic matter. Controls showed no abiotic oxidation in sterile media nor with autoclaved soils (data not shown). Storage at 5°C did not significantly influence the AsIII oxidation rate (Electronic Supplementary material ESM2). The kinetics of AsIII oxidation differed between soils. Contrasting kinetics according to nature and concentration of added substrates obtained with three soils are shown in Fig. 1. With the Cheni forest soil, high AsIII oxidation rate was obtained without added organic matter (Fig. 1a). With the Cheni site soil (Fig. 1b), results were clearly grouped according to the type of organic matter added and the kinetic was slower without added organic matter. With Cheni meadow soil (Fig. 1c), the higher AsIII oxidizing rates were obtained at 0.08 g/L of added organic matter, the highest being observed with SMOM. The other kinetics are shown in Electronic Supplementary Material ESM3. The time preceding AsIII oxidation varied between 0 and 50 hours, and AsIII was entirely oxidized within 200 hours of incubation.

Many bacteria isolated from soils have been shown to oxidize AsIII [START_REF] Macur | Bacterial populations associated with the oxidation and reduction of arsenic in an unsaturated soil[END_REF][START_REF] Inskeep | Detection, diversity and expression of aerobic bacterial arsenite oxidase genes[END_REF][START_REF] Bachate | Oxidation of arsenite by two β-proteobacteria isolated from soils[END_REF] and AsIII oxidation by complete soil microflora was evidenced by Yamamura et al. (2009). The bacterial AsIII oxidation is linked to the expression of aio genes that were evidenced in soil bacteria [START_REF] Huang | Arsenic speciation and volatilization from flooded paddy soils amended with different organic matters[END_REF][START_REF] Poirel | Arsenite-induced changes in abundance and expression of arsenite transporter and arsenite oxidase genes of a soil microbial community[END_REF].

When no organic substance was added to the medium (C0) and according to the PCA with rate constants as supplementary variables (Fig. 2), the AsIII oxidation rate constant correlated positively with organic carbon and humidity. The highest significant Pearson correlation coefficients were obtained between C0 and the intrinsic soil organic matter parameters (C, N and respiration) (Table 2).

When the culture medium was enriched with organic substrates, the rate constants no longer correlated with intrinsic soil organic matter parameters (Fig. 2 and Table 2), with the exception of a positive significant correlation between C0.4 SMOM and the C/N ratio (Table 2).. Evolution of the AsIII oxidation rate constants when organic matter was introduced in the medium was examined through calculation of the ratio of rate constants, C0/C0.08 and C0.08/C0.4. The evolution of these ratios according to soil respiration enlightens the influence of nature and concentration of organic matter on AsIII oxidation rate constants (Fig. 3). Soil respiration is linked to the intrinsic biodegradable organic matter. The C0/C0.08 ratio is very similar for yeast extract and SMOM (Fig. 3a), and lower than 1 for all soils with the exception of the Cheni forest soil which presents the highest respiration level. Thus, the addition of organic substances at 0.08 g/L of organic carbon exerted a positive effect on AsIII oxidation rate constants, except when the soil contained enough biodegradable organic matter. These results suggest that the AsIII oxidation rate was limited by the availability of organic matter in natural conditions when no organic substrate was added to the liquid medium, and this limitation was removed by supplying 0.08 g/L of available organic carbon, provided by either yeast extract or SMOM. Conversely, the C0.08/C0.4 ratio was higher than 1 in all conditions except when yeast extract was added to media inoculated by the four soils with the lowest respiration levels (Fig. 3b). In all other conditions, increasing the organic substrate concentration from 8 0.08 to 0.4 g/L of organic carbon induced a decrease of the AsIII oxidation rate constant. But the influence of yeast extract on the AsIII oxidizing rate differed from that of SMOM. SMOM exerted a negative effect on AsIII oxidizing rate, more pronounced than that of yeast extract, in particular with the soils with lower respiration levels.

The PCA with ratios as supplementary variables (Fig. 4) showed that the C0/C0.08 ratios obtained with yeast extract and SMOM are located very close to one another on the graph. The positive correlation between C0/C0.08 ratio and intrinsic organic matter concentrations in soils, for both types of added organic substrates, is confirmed by the significant values of the correlation coefficients (Table 2). Conversely, the C0.08/C0.4 ratio no longer correlated with intrinsic soil organic matter, and behaved differently for yeast extract and SMOM (Fig. 4). This ratio correlated negatively with pH for yeast extract and positively with this parameter for SMOM (Tables 2 andFig. 4). However, these correlations, although significant, were weaker than the positive link between C0/C0.08 and the intrinsic organic matter.

Significant correlations between soils parameters are detailed in Table 3. Living bacteria concentration correlated significantly with respiration, C, N and P, indicating that living bacteria concentration was positively correlated with the intrinsic organic matter of the soils. However, the correlation between living bacteria and AsIII-oxidizing rate constants parameters did not reach the significance level (Table 2), even if results have the same pattern as that between intrinsic organic matter and AsIIIoxidizing rate constants parameters. This weakness of significance may be linked to a variable proportion of living As-transforming bacteria between soils.

The only parameters that influenced significantly the AsIII-oxidation rates constants and their ratios were either linked to organic matter, i.e. N, organic C, respiration and C/N, or pH (Table 2). One of the main parameters influencing soil biogeochemistry and microbial activities is pH [START_REF] Rousk | Fungal and bacterial growth response to N fertilization and pH in the 150-year 'Park Grass' UK grassland experiment[END_REF][START_REF] Whittinghill | Effects of pH and calcium on soil organic matter dynamics in Alaskan toundra[END_REF], thus it is not surprising to enlighten an influence of this parameter on microbial As-related activities. Concerning specifically the soil organic matter properties, their significant influence on the AsIII oxidizing rate without added organic substrate (C0) decreased in the following order: total N > org. C > respiration, and the C/N ratio significantly influenced the C0.4 SMOM rate constant. These results suggest that microbial As oxidation may be limited not only by organic C but also by nitrogen. Thus, nitrogen-rich fraction of soil organic matter might influence As-related microbial activities. The C/N ratio in yeast extract is 4.25, whereas it is higher in SMOM (15.51) and in intrinsic organic matter of the eight soils (10 to 23). If the global soil respiration exerted a significant influence on AsIII-oxidizing rate constants, the proportion of biodegradable organic substances in soil organic matter was not related to any AsIII-oxidation related parameter (Table 2).

The soil microbial communities include organisms able to oxidize AsIII and/or reduce AsV [START_REF] Macur | Bacterial populations associated with the oxidation and reduction of arsenic in an unsaturated soil[END_REF], even in unpolluted environments (Yamamura et al., 2009).The AsIII oxidation rate constant is thus linked to the global activity of all microorganisms involved in As speciation. The global rate constant value should be linked to the following parameters, whose relative contributions have not been quantified: the cell density of As-transforming organisms, their physiological state and growth rate, and the kinetic parameters of the reactions (maximum rate and Km) for each organism.

The AsIII-oxidizing activity in the soils also depends on the diversity and density of the different types of microorganisms, parameters that themselves depend on the environmental conditions of soils. The present study was focused on a global activity test that integrates all these variables. Such global activity measurements were developed and applied for other types of soil microbial activities: PAH biodegradation rate [START_REF] Kästner | Microbial degradation of polycyclic aromatic hydrocarbons in soils affected by the organic matrix of compost[END_REF], thiosulfate-oxidizing activity in paddy fields [START_REF] Stubner | Sulfur oxidation in rice field soil: activity, enumeration, isolation and characterization of thiosulfate-oxidizing bacteria[END_REF], and soil respiration, that integrates CO 2 production and consumption [START_REF] Rey | Effect of temperature and moisture on rates of carbon mineralization in a Mediterranean oak forest soil under controlled and field conditions[END_REF] as we integrate here AsIII production and consumption.

To date, all AsIII-oxidizing bacteria isolated from soils have been either heterotrophs [START_REF] Macur | Bacterial populations associated with the oxidation and reduction of arsenic in an unsaturated soil[END_REF][START_REF] Bachate | Oxidation of arsenite by two β-proteobacteria isolated from soils[END_REF][START_REF] Bahar | Kinetics of arsenite oxidation by Variovorax sp. MM-1 isolated from a soil and identification of arsenite oxidase gene[END_REF] or facultative autotrophs [START_REF] Santini | New arseniteoxidizing bacteria isolated from Australian gold mining environments -phylogenetic relationships[END_REF][START_REF] Inskeep | Detection, diversity and expression of aerobic bacterial arsenite oxidase genes[END_REF][START_REF] Garcia-Dominguez | Novel autotrophic arsenite-oxidizing bacteria isolated from soil and sediments[END_REF][START_REF] Dong | Arsenite oxidation by a facultative chemolithoautotrophic Sinorhizobium sp. KGO-5 isolated from arsenic-contaminated soil[END_REF]. The availability of organic substrates should therefore favor increase of cell density of AsIII-oxidizing bacteria. This might almost partly explain the positive effect of organic matters on the AsIII oxidizing rate constants between 0 and 0.08 g/L of carbon, and the positive correlation of the constant with intrinsic organic matter in the without-amendment condition.

The growth with yeast extract or SMOM of two AsIII-oxidizing bacterial strains of contrasting AsIII metabolisms was verified in the conditions of the AsIII-oxidizing tests (data not shown). Both strains grew with both substrates, however they displayed different behaviors with the SMOM: the mixotrophic strain using AsIII as energy source presented a higher growth yield at 0.08 g/L of added C, whereas the heterotrophic organism showed a higher growth yield at 0.4 g/L of added carbon.

Conversely, high concentrations of organic substrates may decrease the specific AsIII-oxidizing activity of bacteria [START_REF] Challan-Belval | Enhancing pozzolana colonization by As(III)-oxidizing bacteria for bioremediation purposes[END_REF][START_REF] Bachate | Oxidation of arsenite by two β-proteobacteria isolated from soils[END_REF] and may also stimulate the aerobic AsV-reducing activity of soil microorganisms (Yamamura et al., 2009). These latter phenomena may explain the decrease in global AsIII-oxidizing rate constants between 0.08 and 0.4 g/L of added organic carbon observed in most of the conditions. In a liquid medium containing 0.5 g/L yeast extract and 1.8 g/L lactate (i.e. 0.92 g/L organic C), Yamamura et al. (2009) observed rapid reduction of 1 mM AsV by unpolluted soil inocula, thus demonstrating the presence of active AsV-reducing microorganisms that may influence As speciation simultaneously with AsIII-oxidizing ones.

Moreover, most AsIII-oxidizing bacteria have both oxidizing and reducing systems. In these 10 experiments, different groups of soil microorganisms may have been favored in each condition (0, 0.08 and 0.4 g/L of added carbon): according to [START_REF] Van Gestel | Microbial biomass responses to soil drying and rewetting: the fate of fast-and slow-growing microorganisms in soils from different climates[END_REF], autochthonous soil microorganisms are adapted to survive in soils containing recalcitrant material, where no abundant supply of easily oxidizable substrate occurs. These organisms exhibit K-selected behavior, with moderate growth rate and moderate nutrient demands, and are able to use diverse, complex materials [START_REF] Odum | The strategy of ecosystem development[END_REF][START_REF] Metting | Soil microbial ecology[END_REF][START_REF] Langer | Classification of soil microorganisms based on growth properties: a critical view of some commonly used terms[END_REF]. In contrast, zymogenous organisms show rapid growth when high energy-containing nutrients are added to soils [START_REF] Paul | Soil microbiology and biochemistry[END_REF], exhibiting rselected behavior, characterized by rapid growth rate and use of simple and readily available substrates. Individual organisms may exhibit both r-and K-selected behavior, however amendment with organic substances should induce a change of behavior in soil microflora. The difference of effects exerted by yeast extract and SMOM at 0.04 g/L carbon may be related to the distinct composition of the two substrates. The r-strategy should be more prevalent in the presence of yeast extract than with the SMOM, whose composition includes some molecules that are not easily biodegradable. Soil incubations without added As

The experiment aimed to evaluate the short term (7 days) influence of amendment with SMOM (0.4 g/L organic carbon) on the mobility of arsenic present in the four most polluted soils, in relation to microbial activity. The incubations remained aerobic throughout the experiment (positive redox potential, Electronic Supplementary Material ESM4). Similar tendencies were observed with the four soils. When microbes were alive and without added SMOM, no arsenic was mobilized in the liquid phase (Fig. 5a). Conversely, in the presence of SMOM, arsenic was solubilized, and this mobilization was significantly higher when the microbes were alive than in the abiotic controls. Regarding the speciation of As in the liquid phase after seven days of incubation (Fig. 5b), AsIII was never detected in the absence of SMOM when microbes were alive. Conversely, AsIII was always detected in biotic conditions with SMOM, in proportion equivalent to that of the abiotic incubations (but with a higher variability between replicates).Present results showed mobilization of arsenic from soils contaminated for decades in presence of active bacteria and added organic matter, in aerobic condition. Iron concentration was increased by addition of SMOM in the soil slurries (Fig. 5c), either in biotic and abiotic conditions. This phenomenon was probably linked to the increase of Fe solubility through complexation with dissolved organic molecules: as a fact, SMOM contains some organic acids, such as citric and oxalic acids, known to chelate iron (Zhang et al., 1985). Whereas Fe concentration increased during incubation in biotic conditions, iron concentration always remained lower in biotic than abiotic conditions, contrary to total arsenic whose release in solution, with SMOM, was clearly higher in biotic than abiotic conditions. Thus, biological reduction of iron does not seem to represent the major mechanism inducing As mobilization. The organic molecules present in the SMOM and their organic degradation products may have mobilized some As by chemical complexation: arsenite and arsenate form aqueous complexes with humic acid and natural organic matter in the presence of bridging metals [START_REF] Redman | Natural organic matter affects arsenic speciation and sorption onto hematite[END_REF][START_REF] Ko | Arsenic speciation and sorption kinetics in the As-hematitehumic acid system[END_REF][START_REF] Kim | Effects of natural organic matter on the coprecipitation of arsenic with iron[END_REF]. In addition, SMOM probably influenced the microbial speciation of As, either decreasing AsIII-oxidizing activity and/or stimulating the AsV-reducing heterotrophs as observed by Yamamura et al. (2009) when they incubated uncontaminated soils spiked with AsV and glucose. Incubating sediments sampled from a disused mine in aerobic conditions, [START_REF] Lee | The effects of different carbon sources on microbial mediation of arsenic in arsenic-contaminated sediment[END_REF] observed bio-stimulation of As mobility by addition of acetate and lactate, but not by glucose, and linked the behavior of As to evolution of pH associated to the metabolism of organic substrates. Here, pH tended to increase in biotic conditions with SMOM (Table ESM4-1), which may have caused desorption of AsV According to [START_REF] Dixit | Comparison of Arsenic(V) and Arsenic(III) sorption onto iron oxide minerals: implications for arsenic mobility[END_REF], sorption of AsV onto iron oxides decreases when pH increases, contrary to AsIII, whose sorption should be favored by pH increase within the pH range of present experiments (pH 4.5 to 7.5). Thus, increase of pH may explain the increase of total As concentration but not that of AsIII.

In addition to influencing microbial speciation of As, presence of SMOM might also have exerted a priming effect [START_REF] Hamer | Priming effects of sugars, amino acids, organic acids and catechol on the mineralization of lignin and peat[END_REF], stimulating the biodegradation of the intrinsic organic matter of the soils and thus mobilizing some arsenic associated with natural organic matter.

The bacterial community structure and diversity of the soils amended or not with SMOM were assessed. The four soils without added organic matter had a constant bacterial diversity profile showing that our incubation procedure alone did not change these characteristics (Fig. ESM4-1 a).

When the soils were amended with SMOM, bacterial diversity changed during incubation and converged at the end of incubation for all soils. (Fig. ESM4-1 b). SMOM did not affect the bacterial diversity in the same way for all soils: it was decreased for the two soils that had the highest initial diversity, and was maintained or slightly increased for the two soils with lower initial diversity. In terms of total bacterial concentration, the mixture induced an increase in the bacterial cell concentration with the Cheni Site soil, however, for the other soils, bacterial counts did not differ with and without addition of the mixture (Fig. ESM4-2). The convergence of the SSCP fingerprint profiles for all amended soils may be related to the selective development of bacteria able to metabolize the most easily biodegradable compounds of the SMOM, as previously observed by [START_REF] Goldfarb | Differential Growth Response of Soil Bacterial Taxa to Carbon Substrates of Varying Chemical Recalcitrance[END_REF], or may have resulted from the buffering effect of metabolized SMOM on pH. Nevertheless, the amendment of soils with the mixture did not result in the emergence of a few dominant species; it allowed the persistence of a diverse bacterial community, and did not significantly modify the total bacterial concentration.

Conclusions

Results suggest that the global AsIII-oxidizing activity of microorganisms in polluted soils is linked to availability of biodegradable organic substrates. AsIII oxidation can be stimulated by some input of organic matter when the natural organic content of the soil is low; the limitation being already removed in the presence of 0.08 g/L organic carbon. Higher intakes of organic substances no longer stimulate AsIII oxidation and, conversely, may induce short term As mobilization through decrease of bacterial AsIII oxidation, stimulation of AsV-reducing organisms and formation of soluble As complexes with organic molecules. In most natural conditions, AsV has lower mobility than AsIII and the speciation of As in soils therefore influences its mobility. These phenomena should be given careful consideration when designing efficient management strategies for highly polluted sites and agricultural lands affected by diffuse As contamination. Yamamura, S., Watanabe, M., Yamamoto, N., Sei, K., Ike, M. (2009). Potential for microbially mediated redox transformations and mobilization of arsenic in uncontaminated soils. Chemosphere 77, 169-174. Zhang, Y., Kallay, N., Matijevic, E. (1985) Interaction of metal hydrous oxides with chelating agents. 7. hematite-oxalic acid and -citric acid systems. Langmuir, 1, 201-206.

Figure Captions

Fig. 1 2 Fig. 2 Figure 3 Fig. 3 4 Fig. 4 Fig. 5

 12233445 Figure 1

Table 1 .

 1 Characteristics of the polluted soils.

	(*) Total As analyzed by NITON; (**) AsIII analyzed by HPLC_ICP-MS; (***)AsIII/(AsIII + AsV) analyzed by HPLC_ICP-MS

Table 3 .

 3 Pearson correlation coefficients between textural, physical, chemical and biogeochemical soil characteristics. Values in bold are significant for a level of significance alpha=0.05. IBCF: Indicator of biodegradable carbon fraction (ratio respiration/organic carbon concentration).

				%						coarse	coarse	CaCO	coarse	humi-							IBCF	living
	Variables	[As]	[AsIII]	AsIII	[Cd]	[Pb]	[Zn]	[Fe]	clay	silt	sand	3	silt	dity	pH	[P]	[N]	org. C	resp.	C/N	h-1	bact.
	[As]	1	0.991	0.899	0.798	0.979	0.832	0.295	-0.523	-0.365	0.807	-0.260	-0.365	-0.099	-0.291	0.322	0.084	0.128	-0.033	0.059	-0.311	-0.195
	[AsIII]	0.991	1	0.874	0.839	0.991	0.779	0.276	-0.480	-0.444	0.824	-0.209	-0.444	-0.207	-0.207	0.253	-0.023	0.015	-0.139	0.043	-0.312	-0.283
	% AsIII	0.899	0.874	1	0.620	0.808	0.696	0.163	-0.534	-0.402	0.733	-0.226	-0.402	0.199	-0.427	0.217	0.007	0.074	-0.124	0.434	-0.471	-0.311
	[Cd]	0.798	0.839	0.620	1	0.851	0.583	0.239	-0.502	-0.356	0.678	-0.296	-0.356	-0.293	-0.175	0.401	-0.087	-0.082	-0.237	-0.161	-0.354	-0.195
	[Pb]	0.979	0.991	0.808	0.851	1	0.796	0.301	-0.470	-0.415	0.815	-0.217	-0.415	-0.287	-0.167	0.280	-0.002	0.030	-0.103	-0.061	-0.245	-0.231
	[Zn]	0.832	0.779	0.696	0.583	0.796	1	0.081	-0.703	-0.165	0.711	-0.368	-0.165	0.102	-0.575	0.664	0.425	0.495	0.417	-0.016	0.085	0.310
	[Fe]	0.295	0.276	0.163	0.239	0.301	0.081	1	0.003	0.545	-0.086	-0.421	0.545	-0.112	0.171	0.034	0.143	0.114	-0.006	-0.373	-0.651	-0.103
	clay	-0.523	-0.480	-0.534	-0.502	-0.470	-0.703	0.003	1	0.057	-0.658	0.774	0.057	-0.268	0.864	-0.793	-0.237	-0.313	-0.233	-0.111	0.165	-0.229
	coarse silt	-0.365	-0.444	-0.402	-0.356	-0.415	-0.165	0.545	0.057	1	-0.594	-0.441	1.000	0.441	-0.151	0.336	0.656	0.619	0.602	-0.439	-0.166	0.604
	coarse sand	0.807	0.824	0.733	0.678	0.815	0.711	-0.086	-0.658	-0.594	1	-0.353	-0.594	-0.213	-0.415	0.303	-0.081	-0.039	-0.106	0.049	-0.055	-0.278
	[CaCO3]	-0.260	-0.209	-0.226	-0.296	-0.217	-0.368	-0.421	0.774	-0.441	-0.353	1	-0.441	-0.273	0.695	-0.697	-0.393	-0.407	-0.326	0.281	0.379	-0.255
	coarse silt	-0.365	-0.444	-0.402	-0.356	-0.415	-0.165	0.545	0.057	1.000	-0.594	-0.441	1	0.441	-0.151	0.336	0.656	0.619	0.602	-0.439	-0.166	0.604
	humidity	-0.099	-0.207	0.199	-0.293	-0.287	0.102	-0.112	-0.268	0.441	-0.213	-0.273	0.441	1	-0.681	0.410	0.599	0.640	0.529	0.398	-0.185	0.463
	pH	-0.291	-0.207	-0.427	-0.175	-0.167	-0.575	0.171	0.864	-0.151	-0.415	0.695	-0.151	-0.681	1	-0.810	-0.530	-0.599	-0.517	-0.208	0.064	-0.461
	[P]	0.322	0.253	0.217	0.401	0.280	0.664	0.034	-0.793	0.336	0.303	-0.697	0.336	0.410	-0.810	1	0.677	0.713	0.652	-0.253	0.066	0.720
	[N]	0.084	-0.023	0.007	-0.087	-0.002	0.425	0.143	-0.237	0.656	-0.081	-0.393	0.656	0.599	-0.530	0.677	1	0.991	0.957	-0.392	0.209	0.842
	org. C	0.128	0.015	0.074	-0.082	0.030	0.495	0.114	-0.313	0.619	-0.039	-0.407	0.619	0.640	-0.599	0.713	0.991	1	0.963	-0.298	0.209	0.851
	respiration	-0.033	-0.139	-0.124	-0.237	-0.103	0.417	-0.006	-0.233	0.602	-0.106	-0.326	0.602	0.529	-0.517	0.652	0.957	0.963	1	-0.368	0.432	0.909
	C/N	0.059	0.043	0.434	-0.161	-0.061	-0.016	-0.373	-0.111	-0.439	0.049	0.281	-0.439	0.398	-0.208	-0.253	-0.392	-0.298	-0.368	1	-0.236	-0.358
	IBCF h-1	-0.311	-0.312	-0.471	-0.354	-0.245	0.085	-0.651	0.165	-0.166	-0.055	0.379	-0.166	-0.185	0.064	0.066	0.209	0.209	0.432	-0.236	1	0.483
	living bact.	-0.195	-0.283	-0.311	-0.195	-0.231	0.310	-0.103	-0.229	0.604	-0.278	-0.255	0.604	0.463	-0.461	0.720	0.842	0.851	0.909	-0.358	0.483	1

Acknowledgements

This work was supported by the French Environment and Energy Management Agency (ADEME), BRGM (convention TEZ 11-16), and the Labex Voltaire (ANR-10-LABX-100-01). The authors gratefully acknowledge Catherine Joulian for expert judgment concerning SSCP profiles and technical support from Dominique Breeze and Pascal Auger.


Anthony Fiche
email: anthony.fiche@ensieta.fr

Arnaud Martin
email: arnaud.martin@ensieta.fr

Jean-Christophe Cexus
email: jean-christophe.cexus@ensieta.fr

Ali Khenchaf
email: ali.khenchaf@ensieta.fr

Continuous belief functions and α-stable distributions

Keywords: Belief functions, pignistic probabilities, plausibility functions, symmetric α-stable distributions

The theory of belief functions has been formalized in continuous domain for pattern recognition. Some applications use assumption of Gaussian models. However, this assumption is reductive. Indeed, some data are not symmetric and present property of heavy tails. It is possible to solve these problems by using a class of distributions called α-stable distributions. Consequently, we present in this paper a way to calculate pignistic probabilities with plausibility functions where the knowledge of the sources of information is represented by symmetric α-stable distributions. To validate our approach, we compare our results in special case of Gaussian distributions with existing methods. To illustrate our work, we generate arbitrary distributions which represents speed of planes and take decisions. A comparison with a Bayesian approach is made to show the interest of the theory of belief functions.

Introduction

Stable distributions have been developed by Paul Lévy in his study of normalized sums of independant and identically distributed terms [START_REF] Lévy | Théorie des erreurs : La loi de Gauss et les lois exceptionelles[END_REF]. It is a class of distributions which includes special cases like Gaussian or Cauchy distributions. Some applications are listed in the literature like in radar [START_REF] Achim | SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modelling[END_REF], engineering [START_REF] Nikias | Signal processing with alpha-stable distributions and applications[END_REF], finance [START_REF] Mcculloch | Financial applications of stable distributions[END_REF], ( etc).

In general, classic statistical problem is dominated by methods based on Gaussian models. However, in many cases, signals are rarely Gaussian and are often complex. Indeed, they can present properties of skewness and heavy tails. A distribution is said to have heavy tails if the tails decays slower than the tail of the Gaussian distribution. Thefore, property of skewness means that it is impossible to find a mode where curve is symmetric. One of avantages of α-stable distributions compared to Gaussian cases is that allows to represent these properties. Consequently, α-stable distributions can be a class of distributions which is used for example to follow texture parameters for pattern recognition.

A Bayesian approach is often used to solve problem of pattern recognition [START_REF] Williams | Bayesian Data Fusion of Multi-View Synthetic Aperture Sonar Imagery for Seabed Classification[END_REF]. However, Bayesian approach requires to know prior probabilities. To avoid to estimate prior probabilities, it is possible to use belief functions.

The theory of belief functions in discret domain has been developed by Dempster [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF], and Shafer [START_REF] Shafer | A mathematical theory of evidence[END_REF]. It can be seen as an extension of theory of probabilities. One of avantages of belief functions is that allows to consider imprecision and uncertainty of measures.

Recently, some works extend theory of belief functions in continuous domain [START_REF] Ph | Belief functions on real numbers[END_REF]. The theory of belief functions has been applied to Gaussian cases in [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF]. Consequently, we try to develop the theory of belief functions to α-stable distributions.

The structure of the paper is as follows: definitions about α-stable distributions are given in section one. Therefore, belief functions on discret and continuous domain are explained in section two. Finally, in section three, a method is proposed to calculate belief functions where information transmitted by sensors are modelized by α-stable distributions and the interest of belief functions is showed compared a Bayesian approach.

α-stable distributions

α-stable distributions allow to represent phenomena of skewness and heavy tails. Consequently, a definition of stable random variable is given, then a way to calculate α-stable distributions is developed.

Stable random variable

The notion of stability has been introduced by P. Lévy in [START_REF] Lévy | Théorie des erreurs : La loi de Gauss et les lois exceptionelles[END_REF]. A random variable X is said stable if ∀(a, b) ∈ (R + ) 2 , it exists c ∈ R + and d ∈ R such as:

aX 1 + bX 2 = cX + d (1)
with X 1 and X 2 independent copies of X. Equation ( 1) define the notion of stability but gives no information how to parametrize α-stable distributions. Consequently, α-stable distributions are defined from characteristic functions.

Characteristic functions

There are in the literature several parameterizations of α-stable distributions, noted S α (β, γ, δ). The most known definition is that used by Samorodnitsky and Taqqu [START_REF] Taqqu | Stable nongaussian random processes[END_REF]. However, the most employed definition is given by Zolotarev [START_REF] Zolotarev | One-dimensional stable distributions[END_REF]. A random variable is said stable if his characteristic function verifies:

if α = 1 φ(t) = exp(itδ-|γt| α [1+iβtan( πα 2 )sign(t)(|t| 1-α -1)]) if α = 1 φ(t) = exp(itδ -|γt|[1 + iβ 2 π sign(t) log |t|]) with α ∈]0, 2], β ∈ [-1, 1], γ ∈ R + * and δ ∈ R.
The four parameters can be interpreted as:

• α is called characteristic exponent
pdf (x) = +∞ -∞ φ(t) exp(-itx)dt (2)
However, it is difficult to represent probability density functions for two reasons. Indeed, the characteristic function is complex and the interval of integration is infinite. Nolan [START_REF] Nolan | Numerical calculation of stable densities and distribution functions[END_REF] suggests to make variable modifications to have a set of integration finite. This method has been programmed in a Matlab code [13].

Each parameter has an influence on plots. Indeed, if α is small, the distribution presents a more important peak (cf. figure 1). When β → 1, distributions is shapeless on left and conversely when β → -1 (cf. figure 2). Moreover, parameter γ allows to dilate and compress distributions (cf. figure 3). Distributions are translated on the abscissa when parameter δ varies (cf. figure 4).

Examples of pdf

Despite the lack of closed formulas, it is possible to describe known distributions. Indeed, when α = 2 and β = 0, Gaussian distribution is defined: with δ corresponding to the mean and σ 2 to the variance. To recognize α-stable distribution, we have σ 2 = 2γ 2 . When α = 1 and β = 0, it is possible to build a Cauchy distribution:

f (x) = 1 √ 2πσ exp - (x -δ) 2 2σ 2 (3)
f (x) = 1 π γ γ 2 + (x -δ) 2 (4)
It is usual to define symmetric α-stable distributions, i.e. a class of distributions where β = 0.

Theory of belief functions

The theory of probabilities is limited because it is difficult to consider imprecisions of measures and it is necessary to know prior probabilities. However, these problems can be solved by using works on belief functions. In this part, the theory of belief functions is developed in discret and continuous domains.

Discret belief functions

Firstly, it is necessary to consider a frame of discernement Θ = {C 1 , C 2 , . . . , C n }, where C j correspond to a class. The belief is represented by a basic belief assignment m defined on 2 Θ and in [0, 1]. m verifies by construction

A∈2 Θ m(A) = 1. When m(A) > 0, A is focal element.
With these definitions, it is possible to construct several functions:

• credibility function, noted bel, we can calculate it by the formula:

bel(A) = B⊆A,B =∅ m(B) (5)
• plausibility function, noted pl, We can define it by:

pl(A) = A∩B =∅ m(B) (6)
We can also calculate pignistic probability used for decision step [START_REF] Ph | Constructing the pignistic probability function in a context of uncertainty[END_REF]. It is an operator which can give probability on every C j ∈ Θ. It is possible to determinate this probability with:

BetP (C j ) = A⊆Θ,Cj ∈A m(A) |A|(1 -m(∅)) (7)
|A| represents the cardinality of A. The combinaison between two mass functions can be realized with orthogonal rule of Dempster. Indeed, m 1 and m 2 and ∀X ∈ 2 Θ :

m(X) = Y1∩Y2=X m 1 (Y 1 )m 2 (Y 2 ) 1 - Y1∩Y2=∅ m 1 (Y 1 )m 2 (Y 2 ) (8)

Continuous belief functions

The theory of belief functions on continuous domain has been developed by Smets

It is possible to define plausibility degree with:

pl([a, b]) = z=b z=-∞ y=+∞ y=max(a,z) f (z, y)dydz (10)
Mass densities can be determinated from probability density functions. However, many mass densities can induce the same probability density function. The transformation is said isopignistic. To avoid this problem, the hypothesis of consonant mass densities is made, ie focal elements are nested. It is possible to create an index u where focal elements are labeled I u , verifying I u ⊆ I u with u > u. It is necessary to apply the principle of least commitment to have this property: Authors [START_REF] Ristic | Belief function theory on the continuous space with an application to model based classification[END_REF] use this method to calculate plausibility functions where probability densities are unimodal Gaussian distributions. Caron et al. [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF] generalize this approach in multidimensional case. The plausibility of a point mass x is given by:

pl(x) = 1 -F n+2 ((x -µ) T Σ -1 (x -µ)) ( 11
)
where µ is the mean, Σ represent matrix of covariance and F n+2 is the cumulative density function of a χ 2 distribution with n + 2 degrees of freedom, defined by:

F n+2 (χ 2 ) = χ 2 0 u n+2 2 -1 2 n+2 2 Γ( n+2 2 ) exp(- u 2 )du (12)
Then, several plausibility functions can be combined by using general Bayesian theorem to obtain mass function at x:

m(x)(A) = Cj ∈A pl j (x) Cj ∈A c (1 -pl j (x)) (13)
The next step consists to convert mass functions into pignistic probabilities by using equation ( 7). This criterion distributes uniformly mass of focal elements on singletons. Finally, the decision is taken by using the maximum of pignistic probability.

Belief functions and α-stable distributions

Belief functions have been applied to Gaussian cases in pattern recognition. However, α-stable distribution have better properties to represent noise [START_REF] Georgiou | Alpha-stable modeling of noise and robust timedelay estimation in the presence of impulsive noise[END_REF]. Consequently, we propose a way to calculate plausibility functions where probability density functions are modelized by symmetric α-stable distributions. Therefore, a comparison is made between method develop in section 3.2 and our method in Gaussian case. Bayesian approach is compared with belief functions in stable cases. Finally, a problem of classification is simulated to show the interest of α-stable distribution compared Gaussian case.

Calculation of plausibility

Smets [START_REF] Ph | Belief functions on real numbers[END_REF] defines plausibility functions in the case of unimodal probability density, noticed Betf , of mode µ. For x > µ, plausibility functions can be calculated by:

pl(x) = t=+∞ t=x (ν(t) -t) dBetf (t) dt dt (14)
where ν(t) verifying Betf (ν(t)) = Betf (t). In symmetric case, equation ( 14) can be simplified [START_REF] Ph | Belief functions on real numbers[END_REF]:

pl(x) = 2(x -µ)Betf (x) + 2 t=+∞ t=x Betf (t)dt (15)
From equation ( 15), we use relation of Chasles to obtain:

t=+∞ t=x pdf (t)dt = t=+∞ t=-∞ pdf (t)dt - t=x t=-∞ pdf (t)dt (
pl(x) = 2(x -µ)pdf (x) + 2(1 -cdf (x)) ( 17
)
It will be useful to compare this result with plausibility functions obtained by Caron et al. [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF], where probability densities are Gaussian, to validate it.

Validation of plausibility functions

We take up the application of identification of flyingobject published in [START_REF] Ristic | Belief function theory on the continuous space with an application to model based classification[END_REF]. Speed can be considered as a feature. These features can be measure by sensors. In general, these measures are imprecise. Distribution of speed can be represented by a Gaussian or α-stable distributions. In our example, three α-stable distributions are chosen arbitrary given by notations defined in section 2.2:

• S 2 (0, 8, 722.5)

• S 2 (0, 7, 690)

• S 2 (0, 10, 730)

These three distributions are plotted in figure 5.

Figure 5: Plot of three Gaussian distributions

As it is explained in section 2.3, these three distributions are Gaussian because α = 2 and β = 0. For each distribution, a plausibility is calculated at every point x between [620; 800] with a step of 1. They are transformed into mass functions by using general Bayesian theorem. Finally, the decision is taken by using maximum of pignistic probability. Plots can be observed at figure 6(a). The same scheme is followed for approach of Caron et al. [START_REF] Caron | Least Committed basic belief density induced by a multivariate Gaussian pdf[END_REF]. We observe in figure 6 x n y * n . Coefficients of correlation are superior than 0.99 in the three distributions. The difference is due to numerical approximations during calculations of α-stable distributions. Consequently, we have shown that approach with α-stable distributions, in the particular case of three Gaussian distributions, is equivalent to current approach of Caron. To conclude, a way to calculate belief functions by supposing probability densities are α-stable distributions has been developed.

Comparison with Bayesian approach

In this section, we take now three distributions that are not Gaussian. The three symmetric α-stable distributions are chosen with parameters:

• S 1.5 (0, 8, 722.5)

• S 1.2 (0, 7, 690)

• S 1.2 (0, 10, 730) Figure 7: Plots of three α-stable distributions.

These distributions can modelize speed feature of different planes. These graphics are represented at figure 7.

It is necessary to explain Bayesian approach in several steps. Firstly, prior probability must be calculated and can be modelized by: The application of Bayes theorem gives posterior probability:

p(x/C j ) = +∞ -∞ φ(t) exp(-itx)dt (18)
p(C j /x) = p(x/C j )p(C j ) n j=1 p(x/C j )p(C j ) (19)
Finally, the decision is chosen by using the maximum of posterior probability. Same steps are followed as them described in section 4.2 to calculate pignistic probabilities. Plots of theses two approach can be observed at figure 8.

When we analyse class probabilities in figure 8(a), decisions are clearly defined at the end of the tails. On the contrary, pignistic class probabilities in figure 8(b) show that it is difficult to take a decision. Indeed, tails are mixed and it is difficult to choose one or the other class. This remark is interesting for the classification when we have a lack of data for learning database.

To show interest of belief functions, we try to classify generated data. For example, 300 samples for each αstable distributions can be simulated by using the same approach given in [START_REF] Chambers | A method for simulating stable random variables[END_REF]. The rate of correct classification are calculated. The two approaches are roughly the same because rates are near 70 %. However, the criterion of decision choosen considers only singletons. In the literature, a criteron of decision, introduced in [START_REF] Appriou | Approche générique de la gestion de l'incertain dans les processus de fusion multisenseur[END_REF] and used in [START_REF] Martin | Decision support belief functions theory for seabed characterization[END_REF] , allows to take decison on union of classes. Consequently, the decision A of 2 Θ is obtained by:

A = argmax(m b (X)(x)pl(X)(x)) (20)
where m b is a basic belief assignment given by:

m b (X) = K b λ X 1 |X| r (21)
r is a parameter in [0, 1]. When r → 0, we have a lack of information and more weight is given to union of classes. On the contrary, when r → 1, more weight is allocated to singletons. K b is a constant of normalization, which respects X∈2 Θ m j b (X) = 1. λ X allows the integration of the lack of knowledge on one of the elements X ∈ 2 Θ . For our application, we choose r = 0.4

Gaussian vs α-stable model

In this section, we try to show α-stable models are more robust in classification compared to Gaussian models. Firstly, 1000 samples of the three symmetric α-stable distributions, defined in section 4.3, are generated. 1/3 is used for the learning database and 2/3 for the test database. Therefore, in Gaussian case, mean and standard-deviation are estimated. In α-stable case, parameters α,γ and δ are estimated by using [START_REF] Mcculloch | Simple consistent estimators of the parameters of stable laws[END_REF]. The next step consists to applicate belief functions, developed in last parts, on test database. A final rate of correct classification is calculated as a mean of 10 successive runs. The obtained results are:

• 58.06 % in Gaussian case • 67.24 % in α-stable distribution The model of probability density functions with αstable distribution gives results roughly better than the hypothesis of Gaussian case. We must take care to the model of estimation.

Conclusions

This paper proposes a way to calculate belief functions by using α-stable distributions in symmetric case. We have confirm this approach with a measure a correlation in Gaussian case. We have showed avantages of belief functions compared a Bayesian approach in a perspective of classification. Indeed, this theory does not need to know prior probability. Furthermore, the theory of belief functions decide not necessary an exclusive class: it is possible to consider a decision as being a union of classes. In our future works, we try to estimate unknown distributions. However, it is necessary to solve several problems. We must generalize belief functions in non symmetric case. Indeed, it is difficult to define ν as Betf (ν(t)) = Betf (t). In symmetric case, the mode is known and equal to δ and the function ν(t) = δ -t. On the contrary, the mode and ν are unknown in non symmetric case. Multimodal distributions can be estimated with a mixture of α-stable distributions as it is realized in [START_REF] Salas-Gonzalez | Finite mixture of α-stable distributions[END_REF]. However, it is difficult to define belief functions in this case. However, it is possible to use works published in [START_REF] Doré | Constructing consonant belief function induced by a multimodal probability[END_REF] where consonant belief functions are build form a multimodal distributions. Finally, our final objective is to develop classification from several SONAR images by using α-stable distributions. Results have been already given in Gaussian case [START_REF] Fiche | Bayesian approach and continuous belief functions for classification[END_REF] and it will be interesting to compare with α-stable distributions.

•

  β is called skewness parameter • γ represents the scale parameter • δ designates location parameter It is possible to obtain probability density function (pdf ) by applying a Fourier transform to characteristic function:

Figure 1 :

 1 Figure 1: Influence of parameter α with β = 0, γ = and δ = 0.

Figure 2 :

 2 Figure 2: Influence of parameter β with α = 1.5, γ = and δ = 0.

Figure 3 :

 3 Figure 3: Influence of parameter γ with α = 1.5, β = and δ = 0.

Figure 4 :

 4 Figure 4: Influence of parameter γ with α = 1.5, β = 0 and δ = 0.

  [8]. Compared to discret domain, mass functions become mass density defined on sets [a, b] of R with m([a, b]) = f (a, b) where f is probability density of {(z, y) ∈ R 2 |z ≤ y} in [0, ∞]. Credibility degree on set [a, b], noted bel([a, b]), is defined with: bel([a, b]) = z=b z=a y=b y=z f (z, y)dydz

  [START_REF] Georgiou | Alpha-stable modeling of noise and robust timedelay estimation in the presence of impulsive noise[END_REF] where pdf designate the probability density functions of α-stable distributions. However, t=x t=-∞ pdf (t)dt correspond to cumulative density function, also written cdf , calculated at point x and by definition, t=+∞ t=-∞ pdf (t)dt = 1. Consequently, equation (15) becomes:

  (b) results of pignistic probabilities. A comparison between two functions can be made by estimating a coefficient of correlation given by the formula: c = N -1 n=0

  (a) pignistic probability of three α-stable distributions (b) pignistic probability by using Caron method

Figure 6 :

 6 Figure 6: Plots of pignistic probability

Figure 8 :

 8 Figure 8: Plots of Bayesian analysis and belief functions analysis.

Figure 9 :

 9 Figure 9: Classification with possible decision on union to consider ambiguity between the classes. The results are illustrated in figure 9. The proportion given to the subset C 2 ∪ C 3 is important compared to the others. It is logical due to the pignistic class probabilities. Consequently, belief functions consider imprecisions of measures whereas it is impossible with a Bayesian approach.