Idalina Moubiya Mouélé

Patrick Dudoignon

Abderrazak El Albani

Alain Meunier

Philippe Boulvais

François Gauthier-Lafaye

Jean-Louis Paquette

Hervé Martin

Michel Cuney

Idalina Moubiya Mouélé

.9 Ga paleoalterations ofArchean granitic basement of the Franceville basin (Gabon)

Keywords: Archean, granitoid, hydrothermal, Paleoproterozoic, diagenesis, Francevillian 5

Introduction

The Archean basements have been frequently altered by multiple hydrothermal and/or metamorphic events [START_REF] Card | Metamorphism of the Middle Precambrian (Aphebian) rocks of the eastern southern Province: Metamorphism in the Canadian Shield[END_REF][START_REF] Murakami | Early Proterozoic weathering processes under low O2 conditions reconstructed from a 2.45 Ga paleosol in Pronto[END_REF][START_REF] Nedachi | Geochemistry and mineralogy of the 2.45 Ga Pronto paleosols[END_REF][START_REF] Ohmoto | Evidence in pre-2.2 Ga paleosols for the early evolution of atmospheric oxygen and terrestrial biota[END_REF][START_REF] Ohmoto | Devolatilization equilibria in graphitic systems[END_REF]. Additionally, in some cases, their erosion surface may conserve the traces of weathering processes forming the so-called paleosols which are researched for their paleo-atmospheric signification (Grandstaff et al., 1986;Holland, 1994;Maynard, 1992;Mossman and Farrow, 1992;[START_REF] Ohmoto | Evidence in pre-2.2 Ga paleosols for the early evolution of atmospheric oxygen and terrestrial biota[END_REF][START_REF] Rye | Paleosols and the evolution of atmospheric oxygen; a critical review[END_REF]. At last, when overlain by sedimentary deposits, the basement rocks may have suffered mineral reactions triggered by the diagenetic fluid invasion [START_REF] Mercadier | Migration of brines in the basement rocks of the Athabasca Basin through microfracture networks[END_REF]. Because they are operating successively, all these events may superimpose their effects in some places. Then, the genetic message carried by the secondary minerals is difficult to interpret without an accurate petrographical study. This is the backbone of the present work whose goal is to characterize the alteration events which have affected the Archean basement under the Francevillian sedimentary series in Gabon. It was globally affected by post-magmatic to metamorphic mineral reactions [START_REF] Mathieu | Reconstitution des paléocirculations fluides et des migrations élémentaires dans l'environnement des réacteurs nucléaires naturels d'Oklo (Gabon) et des Argilites de Tournemire[END_REF][START_REF] Sère | Géochimie des minéraux néoformés à Oklo (Gabon): histoire géologique du bassin d'Oklo, une contribution pour les études de stockages géologiques de déchets radioactifs[END_REF] but its alteration state near the unconformity is still poorly documented. Our work, focused on that interface, was made possible thanks to several drill-holes which have been dug for uranium ore exploration. The goal is to determine the origin of the secondary minerals and to date their formation.

Geological setting

The Chaillu massif is located in the South Gabon which is Northeastern part of the Congo craton (Fig. 1). It forms the granitic basement of the Paleoproterozoic sediments deposited in the Franceville basin. It is composed of different granitoid and orthogneiss bodies dated from 3.1 to 2.5 Ga [START_REF] Bouton | Notice explicative de la carte géologique de la République du Gabon à 1/200 000, feuille Franceville-Boumango[END_REF]Caen-Vachette et al., 1988). These bodies are structurally and mineralogically heterogeneous because of two major plutonic events [START_REF] Bouton | Notice explicative de la carte géologique de la République du Gabon à 1/200 000, feuille Franceville-Boumango[END_REF][START_REF] Chevallier | Notice explicative de la Carte géologique de la République Gabonaise à 1/ 1 000 000[END_REF][START_REF] Prian | Contribution à la reconnaissance géologique et pétrographique de l'Archéen du massif du Chaillu et du Francevillien du Bassin des Abeilles (région deKoulamoutou -Lastoursville, Gabon)[END_REF][START_REF] Thiéblemont | Notice explicative de la Carte géologique et des Ressources minérales de la République Gabonaise à 1/1000 000[END_REF][START_REF] Thomas | Carte géologique de la République Gabonaise à 1/ 1 000 000[END_REF].

The first one took place during the Mesoarchean (2928 ± 6 Ma to 2870 ± 5 Ma). It led to the intrusion of gray-coloured calc-alkaline granitoids belonging to typical a Archean Trondhjemite, Tonalite, Granodiorite suite (TTG) associated with migmatized quartz diorites (Thiéblemont et al., 2009). Locally, these rocks are cross-cut by diorite sills. The second period of magmatism belongs to the Neoarchean (2800 to 2550 Ma). It is characterized by the intrusion of alkaline to calc-alkaline pink to red magmatic rocks: syenites, syenogranites, granites and pegmatites [START_REF] Thiéblemont | Notice explicative de la Carte géologique et des Ressources minérales de la République Gabonaise à 1/1000 000[END_REF].

The Franceville basin sedimentary series are composed of five unmetamorphosed formations, 4000 m thick, sequentially labeled from FA at the base up to FE (Weber, 1968). The FA formation (500-1000 m thick) is directly deposited on the eroded surface of the Archean granitoids basement (Chaillu massif). It is essentially formed of conglomerates and sandstones typical of fluvio-deltaic deposits. Its upper part (Fig. 1) hosts the natural nuclear reactors of the Oklo, Okélobondo and Bangombé uranium ore bodies [START_REF] Gauthier-Lafaye | Natural fission reactors in the Franceville basin, Gabon: A review of the conditions and results of a "critical event" in a geologic system[END_REF][START_REF] Naudet | Des Réacteurs Nucléaires Fossiles[END_REF]. The FA formation in the Kiéné area, exhibits a vertical granulometric evolution (conglomerate to sandstone and mudstone). The color of the sediments changes upwards from red to green to black [START_REF] Gauthier-Lafaye | Les gisements d'uranium du Gabon et les reacteurs d'Oklo. Modèle métallogénique de gîtes à fortes teneurs du protérozoique inférieur[END_REF][START_REF] Haubensack | Environnement des grès protérozoïques et des indices uranifères du secteur Kiéné dans le bassin de Franceville (République Gabonaise): aspects sédimentologiques et géochimiques[END_REF]. The fine-grained black levels are composed of detrital quartz, plagioclase and biotite. Plagioclase and biotite are locally replaced by calcite and illite + chlorite assemblage respectively [START_REF] Haubensack | Environnement des grès protérozoïques et des indices uranifères du secteur Kiéné dans le bassin de Franceville (République Gabonaise): aspects sédimentologiques et géochimiques[END_REF]. The green colored intermediate levels are rich in illite (or phengite) and chlorite which replace detrital biotite and locally feldspar. The sandstone is mainly composed of quartz grains cemented by calcite, illite and chlorite. The conglomeratic sandstones at the base of the FA sedimentary series lie directly on the granitoid basement. Their mineralogical composition is dominated by a microcrystalline illitic material filling the porosity and replacing feldspars. When disseminated between quartz crystals, the illitic matrix may contain calcite, anhydrite or gypsum. The sandstones at the very base of the series, are known to be red colored due to the precipitation of hematite in the cleavages of altered biotites and to the staining of detrital quartz grain surfaces. This is not the case in the eight studied drills crosscutting the unconformity we studied here: no trace of red coloration has been observed probably because detrital ferromagnesian minerals are lacking or perhaps, because of a secondary discoloration.

Sampling and analytical methods

Sampling

The eight drill cores studied here are located in the Kiéné region (250 km 2 ), the central part of the Franceville basin, close to the Mikouloungou U-ore deposit and the Kaya-Kaya fault (Fig. 2a). Petrographic investigations were performed on 38 samples collected along the eight drill cores (KA 13, GR1, GR5, GR23, GR31, GR20 GR43 and KA6) down to variable depth (5 to 36 m) below the unconformity between sandstones and granitic basement (Fig. 2b). From the deepest levels to the sediment interface, the granitoids exhibit three dominant "facies" which correspond to an increasing alteration degree: (i) the least altered one, observed at depth, are gray TTG or pink-to-red syenitic-to-granitic rocks; (ii) the altered granitoids in which the initial petrographical texture is preserved; (iii) the intensely altered granitoids which are green colored close to the sandstone-basement unconformity for all the studied drill-cores. The granitoids are largely altered and red colored near the sandstone contact in the GR20 and GR31 drill cores (Fig. 2b).

Analytical methods

The petrographical observations of the granitoids were performed using optical and scanning electron microscopes (OM, SEM). Polished thin sections were carbon coated before examination using the backscattered electron mode imaging (Jeol JSM 5600LV). X-ray spectra were collected with an energy dispersive device according to following analytical conditions: 15 kV accelerating voltage, 1 nA probe current and a 16.5 mm focalization distance. Micro-chemical analyses were obtained using a CAMECA SX50 electron microprobe (CAMPARIS analytical center, University Pierre and Marie Curie, Paris VI) equipped with wavelength dispersive spectrometers (15 kV accelerating voltage, 1 nA probe current, 1µm beam spot diameter). Bulk rock analyses were carried out at the SARM-CRPG (Nancy, France; Table 1). The samples were fused with LiBO 2 , dissolved with HNO 3 , analyzed using ICP-EAS and ICP-MS for major and trace elements respectively. The calibration is based on international standards. The mineralogical composition of each sample has been determined using X-ray diffraction of randomly oriented powders (Bruker D8 ADVANCE A25 diffractometer, CuKα). The accelerating voltage and probe current were of 40KV and 40 mA, respectively. Clay mineral identification was detailed using oriented < 2µ m preparations scanned in the air dried and ethylene glycol saturation states in the 8 to 30° 2θ CuKα angular range. The C and O isotope compositions of calcitic veins were measured in the stable isotope laboratory of the Géosciences department (Rennes, University of Rennes 1).

Calcite was reacted with anhydrous H 3 PO 4 at 50°C and the isotopic compositions measured on a triple collector VG Optima mass spectrometer.

The monazite crystals observed in two samples (KA6 437.85 and GR1 631) near the unconformity level were used to constrain the ages of plutonic rock emplacement and their subsequent alterations. These crystals were examined using SEM, mostly in backscattered electron mode, and analyzed using a CAMECA SX50 electron microprobe (CAMPARIS center, University Pierre and Marie Curie, Paris VI). U-Th-Pb isotopic data were obtained by laser ablation inductively coupled plasma spectrometry (LA-ICPMS) at the Volcanology Laboratory of Clermont-Ferrand University. The analyses involved the ablation of minerals using a Resonetics Resolution M-50 powered by an ultra-short-pulse (<4ns) ATL Atlex Excimer laser system operating at a wavelength of 193 nm [START_REF] Muller | Initial performance metrics of a new custom-designed ArF excimer LA-ICPMS system coupled to a two-volume laser-ablation cell[END_REF]. A 7 µm laser spot diameter and repetition rate of 1 Hz with energy of 8 mJ producing a fluence of 15 J/cm 2 was used for zircon dating. The ablated material was carried into helium flux and then mixed with nitrogen and argon before injection into the plasma source of an Agilent 7500 cs ICP-MS equipped with a dual pumping system to enhance the sensitivity. The alignment of the instrument and mass calibration were performed before each analytical session using the NIST SRM 612 reference glass, by inspecting the signal of 238 U and by minimising the ThO + /Th + ratio (< 1%).The mean sensitivity for 238 U using a spot size of 44 µm is about 25,000 cps/ppm. The analytical method for isotope dating of zircon with laser ablation ICPMS is similar to that developed for monazite and reported by [START_REF] Paquette | High resolution (5 µm) U-Th-Pb isotope dating of monazite with excimer laser ablation (ELA)-ICPMS[END_REF]. The signals of 204 (Pb+Hg), 206 Pb, 207 Pb, 208 Pb, 232 Th and 238 U masses are acquired. The occurrence of common Pb in the sample can be monitored by the evolution of the 204 (Pb+Hg) signal intensity, but no common Pb correction was applied owing to the large isobaric interference from Hg. The 235 U signal is calculated from 238 U on the basis of the ratio 238 U/ 235 U= 137.88. The analyse procedure is based on 30 seconds of background integration with laser off followed by 1 minute integration with the laser firing and a 30 seconds delay to wash out the previous sample and prepare the next analysis. Data are corrected for U-Pb fractionation occurring during laser sampling and for instrumental mass discrimination (mass bias) by standard bracketing with repeated measurements of the Moacyr monazite standard [START_REF] Gasquet | Miocene to Messinian deformation and hydrothermal activity in a pre-Alpine basement massif of the French western Alps: new U-Th-Pb and argon ages from the Lauzière massif[END_REF]. Repeated analyses of Manangoutry monazite standard [START_REF] Paquette | High resolution (5 µm) U-Th-Pb isotope dating of monazite with excimer laser ablation (ELA)-ICPMS[END_REF], treated as unknown, independently control the reproducibility and accuracy of the corrections. Data reduction was carried out with the software package GLITTER ® from Macquarie Research Ltd [START_REF] Achterbergh | Data reduction software for LA-ICP-MS: In: Laser ablation-ICPMS in the earth science[END_REF][START_REF] Jackson | The application of laser ablation-inductively coupled plasma-mass spectrometry to in situ U-Pb zircon geochronology[END_REF]. For each analysis, the time resolved signal of single isotopes and isotope ratios was monitored and carefully inspected to check the presence of perturbations related to inclusions, fractures, mixing of different age domains or common Pb. Calculated ratios were exported and Concordia 206 Pb/ 238 U vs 208 Pb/ 232 Th ages and diagrams were generated using the Isoplot/Ex v. 3.23 software package (Ludwig, 2001).The concentrations in U-Th-Pb were calibrated relative to the certified contents of Moacyr monazite standard [START_REF] Seydoux-Guillaume | Experimental resetting of the U-Th-Pb systems in monazite[END_REF].

Results

The least-altered granitoids

The least altered gray or pink-to-red granitoids are observed in the deepest parts of the different drill cores far from the unconformity (Fig. 2b). The primary minerals are quartz, plagioclase (An 0 -An 20 ), biotite, alkali feldspar and locally amphibole. Zircon, apatite, uranothorite and monazite are the common accessory mineral phases. The gray granitoids present sometimes a gneissic structure.

The magmatic rocks range from diorite to granite. The whole rock major element compositions of the less altered rocks reflect the variety of these rock types (Table 1): i.e. SiO 2 (52.1 -74.5 wt%), FeO T (0.3 -9.5 wt%), . In order to determine their magmatic type, the chemical composition of these rocks has been plotted in the Q-P chemical-mineralogical diagram of Debon and Le fort (1983) (Fig. 3). Because even the best preserved samples from the eight drill cores exhibit petrographic features of a hydrothermal alteration event the rock samples were selected according to their low ignition loss value: 2.5 Wt%. The grey TTGs range from quartz-diorite (dq) to tonalite (to) composition, while the pink and red rocks correspond to granodiorite (gd) and adamellite (ad) (Table 1; Fig. 3). The chondrite-normalized REE patterns show that tonalities present the typical highly fractionated pattern: high LREE content, low HREE content, no significant Eu anomaly. These properties are typical of the Archean TTG [START_REF] Martin | Effect of steeper Archean geothermal gradient on geochemistry of subductionzone magmas[END_REF]. On the other hand, the granodiorite and adamellite have lower REE contents with a positive Eu anomaly; such REE patterns have already been interpreted in Archean rocks as resulting from the fractionation by accessory phases such as allanite and zircon [START_REF] Martin | Petrogenesis of Archaean trondhjemites, tonalites, and granodiorites from eastern Finland: major and trace element geochemistry[END_REF](Fig. 4). Finally the higher REE content of the diorite is similar to that of the late Archean sanukitoids [START_REF] Martin | The sanukitoid series: magmatism at the Archaean-Proterozoic transition[END_REF][START_REF] Stern | Petrogenesis of the Archean sanukitoid suite: State university at Stony Brook[END_REF]. In other words, all the less altered rock samples display the features typical of late Archean magmatic associations.

The altered granitoids

Samples located far from the unconformity

All the Kiéné granitoid samples located far from the unconformity exhibit the presence of chlorite. A pervasive chloritization process has severely affected the primary ferromagnesian minerals, i.e. total replacement of biotite and amphibole. Chlorite is associated to epidote, titanite and allanite (Fig. 5a). According to [START_REF] Lowell | Lateral and vertical alteration-mineralization zoning in porphyry ore deposits[END_REF] and Titley et al. (1986), this paragenesis is typical of a propylitic alteration. Concomitantly, the plagioclases are partially replaced by small sized micas (sericite) forming a microcrystalline matrix in which calcite, pyrite and chalcopryrite crystals are locally disseminated. Some fractures crosscut the propylitzed rocks. They are filled by a zoned deposit: chlorite along the walls and calcite sealing the central part (fig. 5b)

Samples close to the unconformity

The Kiéné granitoids are affected by an intense illitization in the vicinity of unconformity giving them a green color. This alteration is mainly invasive and extends down to 4-5 m depth.

Plagioclase, K-feldspar and previously chloritized biotite are totally or partially replaced by illite (Fig. 6a andb). The feldspars are corroded by a microcrystalline matrix (5-20 µm) while the chloritized biotite are replaced by larger illite crystals (10-50µm). The quartz bodies, themselves, exhibit corrosion features and are locally crosscut by illite veinlets. The illitization intensity decreases with depth becoming nearly undetectable between 5 and 7 m down from the unconformity. Here, only illite veinlets are still observable. Close to the unconformity, the GR20, GR23 and GR31 granitoid samples become red colored. The chloritized biotites are replaced by an illite + hematite assemblage (Fig. 6b). The rocks are crosscut by calcite ± anhydrite veins (Fig. 6c). Some monazites have been locally observed disseminated through the illitic matrix.

Alteration features of FA sandstones

The FA sandstones above the unconformity exhibit detrital quartz and K-feldspar crystals embedded in an illitic micro-crystalline matrix while the detrital grains, whatever their composition, are invaded along their inter-granular joints. Samples GR23 629 and GR20 818 are cemented by anhydrite and/or calcite deposits (Fig. 6d).

Geochemical data of bulk rocks

Because of the predominance of hydrolyzed and carbonated secondary phases in the secondary minerals, the loss on ignition (L.O.I) has been chosen here as an indicator of the alteration degree. In most samples, L.O.I. is significantly greater than that of the corresponding unaltered rocks. Indeed, the measured values range from 0.76 wt% in the freshest granitoids to 4.5 wt% for the most altered. However, the average L.O.I. calculated from a compilation of 60 TTG samples, is much lower: 0.25 wt% [START_REF] Huang | Origin of Archean tonalite-trondhjemite-granodiorite (TTG) suites and granites in the Fiskenaesset region, southern West Greenland: Implications for continental growth[END_REF]. Consequently, the absolute value of L.O.I. cannot be directly used as a reliable marker because of two reasons: (i) for a given rock type, it depends on the quantity of secondary minerals produced by alteration processes; (ii) it varies according to rock types. For instance, diorite L.O.I. typically averages to 1.55 wt% [START_REF] Debon | A chemical-mineralogical classification of common plutonic rocks and associations[END_REF], tonalite for 0.32 and granodiorite for 0.12 [START_REF] Huang | Origin of Archean tonalite-trondhjemite-granodiorite (TTG) suites and granites in the Fiskenaesset region, southern West Greenland: Implications for continental growth[END_REF]. The tonalite value will be used here as the reference in the calculation of relative L.O.I. variations ∆LOI = (LOI sample -LOI ref )/(LOI most altered sample -LOI ref ). According to this definition, the ∆LOI calculated for each basement sample shows a regular increase toward the unconformity (Fig. 9). The most altered levels are found in the illitic zone in agreement with petrographic observations. In order to address the problem of element mobility, it appears suitable to normalize all chemical element values to that one remaining immobile during the alteration processes. As already proposed by several authors [START_REF] Nesbitt | Mobility and fractionation of rare earth elements during weathering of a granodiorite[END_REF][START_REF] Ohmoto | Evidence in pre-2.2 Ga paleosols for the early evolution of atmospheric oxygen and terrestrial biota[END_REF][START_REF] Rye | Paleosols and the evolution of atmospheric oxygen; a critical review[END_REF]White et al., 2001), Ti or Zr are classically considered as reference immobile elements. Here, Ti has been chosen because of its higher content. Then, each element content in the altered samples is corrected by the TiO 2 reference /TiO 2 sample ratio. Propylitic and illitic processes being acceptably discriminated by the increasing loss on ignition (Fig. 7), the ∆LOI parameter has been used to study the variations of the major and some minor element amounts as a function of alteration intensity (Figs. 8 and9). These variations between altered and reference samples (defined by the lowest ∆LOI) are calculated as follows: ∆SiO 2sample = (SiO 2 sample × TiO 2 reference /TiO 2 sample ) -SiO 2 reference (wt%).

Because the piece of rocks sampled in the drill cores cannot be at the size of a representative sample, the experimental error is not calculable. This is why we adapted the variation scales to their abundance in the specimens: 0 to 50 for SiO 2 and Al 2 O 3 , -7 to 2 for all the other components. Reported in the same scale, ∆SiO 2 and ∆Al 2 O 3 show contrasted distribution with ∆LOI: the first one is systematically negative while the second remains roughly constant. Silica loss and aluminium conservation are commonly reported in water rock interactions due to their large difference of solubility. The weight percent variations are more limited for Fe 2 O 3 , MgO and CaO. If they are not significant for the first and second ones, it seems that the losses of the calcium are more important in the illitic alteration zone. The chemical differences between propylitized and illitized samples are clearly evidenced by the alkaline element behaviour. All exhibit increasing losses of Na 2 O with ∆LOI but they more important for the second ones. The K 2 O amount remains roughly constant in the propylitized zone while it significantly increases in the illitic one. The variations for trace elements are expressed in ppm (Fig. 9). The illitized samples are enriched in rubidium and cesium because of their potassium similar chemical affinities for micas. Strontium is depleted in all the samples, the losses being more important in the illitized ones. Cerium and lanthanium amounts are highly variable in the propylitized samples while they are systematically depleted in the illitized ones (Figs. 9 and 10). The variations of uranium are erratic but thorium seems to be systematically depleted. In spite of the uncertainties due to the limited size of the samples, it is possible to distinguish the geochemical characteristics of the two alteration types. Indeed, the propylitic alteration is almost chemically conservative: the losses of SiO 2 and Na 2 O are limited. This is not the case for illitization which is depleted in all the components except aluminium (conserved) and potassium (enriched). The behaviour of trace elements outlines this difference.

Crystallo-chemistry of secondary phases

Chlorite and illite-phengite are present in all the studied samples. It is to be noticed that no dior trioctahedral expandable mixed-layer minerals have been detected in the studied samples (Fig. 11a).

Chlorite

XRD patterns of randomly oriented powders show that chlorites are of the IIb polytype. The (001)/(002) peak intensity ratio decreases from depth to top (0.60-0.50 to 0.20) indicating that the relative proportion of Fe and Mg ions changes from the deep propylitic zone to the upper illitic level (Fig. 11b). This is coherent with the micro-chemical analyses of individual chlorite crystals.

Most of the half-formula units calculated on a 14 oxygen basis, plot inside the brunsvigite composition field defined by Foster (1960). This is the case for all the chlorites in the propylitic zone (Table 2,Table 3, Fig. 12b). The Fe/(Fe+Mg) ratio varies from 0.3-0.5 ( propylitic zone) to 0.6 -0.7 (illitic level).

Illite-phengite

The XRD patterns of randomly oriented powders indicate that the illite-phengite minerals are composed of a mixture of 2M 1 and 1M polytypes. They are small sized crystals when replacing the feldspars or infilling the quartz microcracks (5-20µm) but become larger when replacing chlorites (10-50µm). Their half-formula units calculated from microprobe analyses using a 11 oxygen basis, are coherent with that of dioctahedral micaceous minerals (Tables 4,5). Two populations are clearly separated in the MR 3+ , 2R 3+ , 3R 2+ diagram proposed by Velde (1985).The phengitic crystals are found in the altered chlorites as shown by the mixing line while the illitic ones are found in the altered feldspars (Fig. 12a). Both exhibit a quite similar range of Fe/(Fe+Mg) ratio: 0.46-0.61 and to 0.52-0.76 for phengite and illite respectively. However, phengites are always K-richer than illite: 0.84-0.95 and 0.73-0.84 respectively.

Epidote-allanite

Epidotes are of the pistachite type (Table 6). They form everywhere in the rock, partly replacing primary feldspars or chloritized biotites or precipitating into veinlets. Their Fe/(Fe+Al) ratio varies from 0.30 to 0.42. Allanite crystals, on the contrary, form exclusively in chloritized biotites. They present a rather constant Fe/(Fe+Al) ratio (0.3-0.32). Their main chemical characteristic is the very high Nd, Ce and La contents: 0.16-0.07, 0.38-0.36, 0.25-0.17%respectively.

Monazite

Monazite crystals have been observed in two samples: KA6 437.85 (0.85 m from the unconformity) and GR1 631 (1 m from unconformity). The first one is a strongly illitized rock containing Fe-rich chlorites. Euhedral elongated millimeter size crystals of monazite are scattered in the illitic matrix. They seem to be perfectly unaltered and do not exhibit any chemical zoning in spite of light grey intensity variations observed in the BSE imaging contrast (Table 7). The GR1 631 sample is less illitized since only feldspars are altered. Most of the micro-cracks are filled by calcite, and locally by quartz and pyrite. Here, the monazite crystals are anhedral, often grouped in aggregates and intimately associated with sulfide and illite. Compared to the euhedral crystals of the KA6 437.85 sample, monazite crystals of the GR1 631 one exhibit higher U, La and Ce and lower Th, Ca, Nd, Sm contents. The Th/U ratios are very low (Tables 8.9). They are characterized by lower Th content (Th<12000 ppm) compared to that of the unaltered monazites described by Mathieu et al. (2001) in the FA Sandstone (average Th = 40000 ppm for 94 analyses).

Carbonates

Calcite is observed in many samples from depth to the upper levels in the FA sandstone. Far from the unconformity in the propylitic zone, it appears as xenormorphic crystals located in altered plagioclase and in veinlets where it is associated to chlorite. It is observed as euhedral crystals filling the fissures near the unconformity where it is locally associated to anhydrite. In some of the FA sandstone samples, it constitutes the inter-granular microcrystalline cement.

Regardless of their habit, the calcite crystals do not show significant chemical composition differences.

Geochronological data

The isotopic dating using laser-ICPMS microanalyses were performed on monazite crystals sampled near the FA sandstone-granitoid unconformity in the KA6 437.85 and GR1631 (Tables 8 and9). The first one is Th-rich (11614-2909 ppm). Plotted in the 206 Pb/ 238 U versus 207 Pb/ 235 U diagram (Fig. 13), the U-Pb data of the KA6 437.85 monazite give two groups of discordia ages: from 2998±25Ma (MSWD = 0.72) to 2922±24 Ma (MSWD = 1.4) and 2621±30Ma (MSWD = 1.5). The GR1 631 monazite being common Pb-rich, the dating is obtained using the 207 Pb/ 206 Pb versus 206 Pb/ 238 U. The Tera-Wasserburg diagram (Fig. 14) gives an age of 1870 ± 26Ma (MSWD = 1.04).

C and O isotope compositions of calcite

The calcite isotopic analyses were performed on the propylitized and illitized granitoids far from and closed to the unconformity respectively. The δ 13 C values obtained varies from -5 to -14‰ and the δ 18 O ones from 10.2 and 17.7‰, most of them being close to 10.5 -11‰ (Tab. 10).

The C and O isotope compositions are not inter-correlated, nor with depth, intensity of carbonation, nature of alteration (propylitization, illitization), or element mobility.

The large range of δ 13 C values suggest that carbon comes from two different sources: one having a δ 13 C > -5‰, which might be surface-derived continental carbon including atmospheric CO 2 , and another one having a δ 13 C < -14‰, which could possibly be a signature of the evolved organic matter similar to that identified in the sediments of the Franceville Basin [START_REF] Albani | Large colonial organisms with coordinated growth in oxygenated environments 2.1[thinsp]Gyr ago[END_REF]Weber and Gauthier-Lafaye, 2013). The δ 18 O values centered around 10.5 -11‰ point to formation at low temperature hydrothermal conditions (150-300°C), which is consistent with the common mineralogical association of calcite + chlorite, where the fluids reached isotopic equilibrium with the granitic basement. On the other hand, the KA6462 sample has a much higher δ 18 O value, which probably relates to calcite formation at lower temperatures than for the others, regardless of the nature of the fluid involved in calcite crystallization.

As a whole, the C and O isotope compositions are consistent with the presence of a diagenetic fluid that would have circulated along or close to the unconformity, imposing low temperature hydrothermal conditions to secondary alteration processes as it is the case for the Athabasca basin [START_REF] Richard | From evaporated seawater to uranium-mineralizing brines: Isotopic and trace element study of quartz-dolomite veins in the Athabasca system[END_REF]. Part of the carbon was taken from the organic matter which is particularly abundant in the FB Francevillian sediments. A similar case is given by the Athabasca basin where metamorphic fluids have interacted with graphite bearing metasediments of the basement.

These fluids have been able to precipitate calcite with such isotopic signatures, at greenschists conditions, in equilibrium with chlorite. [START_REF] Sère | Géochimie des minéraux néoformés à Oklo (Gabon): histoire géologique du bassin d'Oklo, une contribution pour les études de stockages géologiques de déchets radioactifs[END_REF] interpreted the presence of chlorite, albite, quartz and calcite in veins crosscutting the Chaillu massif as witnessing a metamorphic episode of the greenschist facies type. The petrographic observation performed on the eight drill cores show that all the biotite crystals have been chloritized regardless of the parent rock type: granitoids or gneiss. This implies that the chloritization took place after deformation. Consequently, it cannot be related to a metamorphic event but rather to a propylitic alteration which preserves the texture of rocks. Indeed, the chlorite-sericite-epidote/calcite assemblage is commonly described in propylitized granitoids [START_REF] Lowell | Lateral and vertical alteration-mineralization zoning in porphyry ore deposits[END_REF]Titley et al., 1986).

Discussion

Propylitic alteration

The ∆Fe 2 O 3 /TiO 2 , ∆MgO/TiO 2 , ∆CaO/TiO 2 , and ∆K 2 O/TiO 2 ratios do not vary with ∆LOI in the propylitic zone (Fig 10). Some samples plot far from general trend due to the local invasion by secondary minerals such as calcite (high CaO/TiO 2 ratio-GR23 631) in fracture and plagioclase.

The ∆SiO 2 /TiO 2 and ∆Na 2 O/TiO 2 ratios decrease slightly while ∆Al 2 O 3 /TiO 2 remains roughly constant. This means that the propylitized rocks do not have experienced significant exchanges of chemical component with the external reservoir. It is a nearly isochemical set of mineral reactions in which the potassium released by biotite during chloritization is consumed by the crystallization of sericite. This is typical of propylitic alteration in which the water/rock ratio is limited [START_REF] Berger | Chemical parameters controlling the propylitic and argillic alteration process[END_REF]. The temperature conditions can be roughly estimated using mineral indicators.

The allanite is known to be found at rather high temperature: 400 °C [START_REF] Berger | Chemical parameters controlling the propylitic and argillic alteration process[END_REF][START_REF] Smith | Monazite U-Pb dating of staurolite grade metamorphism in pelitic schists[END_REF]Wing et al., 2003) while the chlorite-epidote-paragenesis is formed in the range 350 to 200 °C [START_REF] Norman | Petrology and geochemistry of propylitic alteration at Southwest Tintic[END_REF]. These conditions are compatible with that prevailing during the cooling stage following the intrusion of plutonic rocks.

Illitic alteration

Petrographic observations suggest that the propylitic alteration is overprinted by an illitic one in the upper part of the basement within 4-5 m below the unconformity. It is typically characterized by a massive replacement of all the primary and secondary minerals except the sericite zones in feldspars which seem to have been preserved. This could explain the coexistence of the 2M 1 and 1M polytypes of the white micas fraction. The degree of replacement decreases with depth down to 7 m where illite is observed only in veins. Epidote and allanite have totally disappeared while chlorite seems to remains up to the unconformity. However, its composition changes becoming iron-richer composition. This indicates that it has recrystallized in a more stable version considering the local chemical condition imposed by the fluid responsible of the illitic alteration. As shown by the strong decrease of the ∆CaO and ∆Na 2 O ratios on one hand and the increase of the ∆K 2 O ones on the other hand, it appears that the illitic alteration induces more important chemical changes of the rocks that the propylitic one did (Fig. 8). It is obvious that it operates in an open system in which the fluids have flowed. The mineral similarities between illite in the most altered basement samples and the upper sandstones militate for extensive chemical exchanges driven by the diagenetic fluids along the unconformity. The very low thorium content of the small euhedral monazite crystals indicates that they have crystallized at low temperature [START_REF] Overstreet | Metamorphic grade and the abundance of ThO2 in monazite[END_REF] and are probably synchronous with the illitic alteration.

Could a paleosol have been present along the unconformity?

Based on the study of fifty pre-Devonian rock units that have been described as paleosols, [START_REF] Rye | Paleosols and the evolution of atmospheric oxygen; a critical review[END_REF] proposed that five criteria must be respected to evidence the weathering effects: (i) the paleosol is developed on a homogeneous parent rock and has been preserved in place; (ii) textural; (iii) mineralogical; (iiii) chemical; (iiiii) soft sediments features. A geological formation that meets all these criteria is classified as definite paleosol; when all criteria are met but one it is likely a paleosol; all but two is a possible paleosol; not more than three criteria is unlikely a paleosol.

Considering the eight drill cores studied here, no petrographic features typical of a weathering process have been observed, even in the reddish samples from the GR31 or the GR20 drill cores.

None of them contain any evidence of typical weathering processes such as primary phase recrystallization inside microsytems, clay translocation deposits (cutans) or increasing porosity toward the surface (Velde and Meunier, 2008). On the contrary, the rocks, irrespective of their petrographic properties, are all invaded by a single clay mineral phase: illite. If any physical traces of a weathering event have been preserved after the sediment deposition, they have been totally erased by the diagenetic alteration processes.

As it is the case for any water/rock interactions induced by pure water or diluted solution invasion, the most soluble components are leached out from the altered rock (Ca, Na, Mg) while the less soluble (Al, Ti) ones are concentrated and K is enriched (Fig. 8). These chemical characteristics do not meet the criteria proposed by [START_REF] Rye | Paleosols and the evolution of atmospheric oxygen; a critical review[END_REF]. Consequently, the probability for a paleosol to have been formed on the unconformity surface before being buried under the sediments of the Franceville basin is very weak. The superimposition of an illitic alteration upon a previous propylitic one is the most probable scenario here. It is supported by the distinct chemical trends evidenced by plotting the bulk rock compositions in the Al 2 O 3 -CaO, Na 2 O, K 2 O-FeO, MgO system (Fig. 15). Two trends are clearly separated: (i) a shift toward the epidote-chlorite assemblage typical of the propylitic alteration or magmatic differentiation; (ii) a shift toward the illite pole. The two different chemical pathways are coherent with the progressive upward variation of Rb, Cs and Sr amounts which are increasing and decreasing respectively (Fig. 9). The same contrasted chemical behavior is also observed for the REE distribution: the amounts of the light ones (LREE) do not vary during the propylitic alteration while they are leached out during the illitic event (Fig. 9 and 10). This is also coherent with the disappearance of the allanite crystals which could be related to the percolation of oxidizing diagenetic fluids invading the basement from the overlying sedimentary basin through the unconformity [START_REF] Cuney | Extreme light rare earth element mobilization by diagenetic fluids in the geological environment of the Oklo natural reactor zones, Franceville basin[END_REF].

The negative δ 13 C values of calcite crystals in veins indicate that the diagenetic fluids involved in the illitization process were mixed with organic compounds deriving from oil maturation in the FB sediments (Gauthier-Lafaye and Weber, 1989; Weber and Gauthier-Lafaye, 2013).

Timing of alteration events.

The isotopic U-Pb data gathers into three major dating groups. The oldest one, from 2998 ± 25 Ma to 2922 ± 24 Ma is consistent with the age of plutonic intrusions forming the basement. The 2621 ± 30 Ma intermediate group might correspond to the hydrothermal propylitic event related to the intrusion of the neoarchean plutonic bodies. It is to be noticed that the dating obtained here from monazite crystals is coherent with that measured in altered biotites using the K-Ar method: 2696 ± 60 Ma [START_REF] Bonhomme | Etudes radiochronologique complémentaire de la série du Francevillien et de son environnement[END_REF]. The youngest group, i.e. 1870 ± 26 Ma, has been measured on newly formed monazites from the illitized zone. Their extremely low Th contents reflect a crystallization process under low temperature conditions. It is consistent with the diagenetic event of the Franceville basin [START_REF] Bonhomme | An example of lower proterozoic sediments: The Francevillian in Gabon[END_REF].

Conclusion

The mineralogical and geochemical studies of the basement of the Franceville Basin along a nearly E-W profile show that the basement is composed of different plutonic rocks. Dating obtained from 8 drill holes give Archean ages (3.0 -2.9 Ga) similar to the ones obtained on rocks outcropping in the Chaillu Massif. No metasedimentary formations have been intersected. Most of these rocks present evidences of a propylitic alteration which postdate any ductile deformation and which may be associated to the intrusion of some Neoarchean plutonic rocks at about 2.6 to 2.7 Ga [START_REF] Thiéblemont | Notice explicative de la Carte géologique et des Ressources minérales de la République Gabonaise à 1/1000 000[END_REF]. The mineral reactions produced a Mg chlorite + epidote + allanite + titanite paragenesis typical of nearly close systems: no significant mobility of the major elements suggesting weak Water/Rocks ratios. Then, a massive illitic alteration associated with hematization, has transformed the previously propylitized rocks over a thickness of 2 to 5 m below the unconformity in the Franceville basin. It has been triggered by the percolation of oxidizing diagenetic fluids from the FA conglomeratic-sandstone formation at the base of the sedimentary basin through the unconformity. Similar fluid invasions have been described in other basins: Southern Norway [START_REF] Munz | Deep penetration of sedimentary fluids in basement rocks from southern Norway: Evidence from hydrocarbon and brine inclusions in quartz veins[END_REF][START_REF] Oliver | 100th Anniversary Special Paper: Numerical Models of Extensional Deformation, Heat Transfer, and Fluid Flow across Basement-Cover Interfaces during Basin-Related Mineralization[END_REF], Spain [START_REF] Bouch | Direct evidence of fluid mixing in the formation of stratabound Pb-Zn-Ba-F mineralisation in the Alston Block[END_REF], France, Northwestern Massif Central [START_REF] Munoz | Continental basinal origin of ore fluids from southwestern Massif central fluorite veins (Albigeois, France): evidence from fluid inclusion and stable isotope analyses[END_REF], Athabasca, Canada [START_REF] Richard | From evaporated seawater to uranium-mineralizing brines: Isotopic and trace element study of quartz-dolomite veins in the Athabasca system[END_REF] and Kombolgie, Australia [START_REF] Derome | A detailed fluid inclusion study in silicified breccias from the Kombolgie sandstones (Northern Territory, Australia): inferences for the genesis of middle-Proterozoic unconformity-type uranium deposits[END_REF]. Most of the major and light Rare Earth elements have been heavily depleted. This is particularly the case for Ca, Na and Mg while Al Adamellite(KA6 472), Granodiorite (KA6 445,GR31 873 and GR31 866),Tonalite (KA6 467,KA6 454,KA6 449,KA6 444,KA6 438,GR23 640), Diorite (KA6 462). The structural formulae are calculated on the 12.5 oxygen basis.

Gabon -granite GR1-631

Figure 14 ( GR23 640,50 GR23 631,50 GR23 630,90 GR23 629,90 GR23 629,25 GR23 629,18 GR20 851 GR20 845 GR20 833 GR20 825 GR20 821 GR20 819 GR20 818 SiO 2 (Wt %) 64 GR31 873 GR31 866 GR31 863 GR31 859 GR31 858 GR31 857 KA6 472 KA6 467,90 KA6 462 KA6 454 KA6 449 KA6 445,60 KA6 444 KA6 438,7 KA6 437,85 KA6 437 0,02 0,02 0,02 0,01 0,00 0,01 0,01 0,02 0,01 0,01 0,00 0,02 0,01 K 0,01 0,01 0,00 0,00 0,00 0,01 0,00 0,01 0,01 0,00 0,00 0,01 0,00 Nd 0,00 0,00 0,01 0,00 0,01 0,16 0,09 0,06 0,07 0,17 0,15 0,08 0,13 Ce 0,01 0,00 0,01 0,03 0,00 0,38 0,36 0,36 0,37 0,42 0,35 0,38 0,31 La 0,01 0,00 0,01 0,00 0,00 0,25 0,22 0,22 0,17 0,23 0,19 0,21 0,18 Fe/(Fe+Al) 0,32 0,30 0,33 0,30 0,30 0,42 0,42 0,40 0,41 0,35 0,31 0,30 0,30

  amounts are roughly preserved. No paleosol chemical or petrographical traces have been detectedhere. The calcite δ 13 C values confirm the diagenetic origin for the fluids which have percolated into the basement. They were also responsible of the uranium transfer and deposition in the vicinity of the organic-rich sediments of the FB formation which gave the natural nuclear reactors[START_REF] Gauthier-Lafaye | Natural nuclear fission reactors: time constraints for occurrence, and their relation to uranium and manganese deposits and to the evolution of the atmosphere[END_REF][START_REF] Mathieu | Geochemistry of palaeofluids circulation in the Franceville basin and around Oklo natural nuclear reaction zones (Gabon)[END_REF]. AKNOWLEDGMENT Titley, S.R., Thompson, R.C., Haynes, F.M., Manske, S.L., Robison, L.C., and White, J.L., 1986, Evolution of fractures and alteration in the Sierrita-Esperanza hydrothermal system, Pima County, Arizona: Economic Geology, v. 81, p. 343-370. Velde, B., and Meunier, A., 2008, The Origin of Clay Minerals in Soils and Weathered Rocks: With 23 Tables, Springer. Weber, F., 1968, Une série précambrienne du Gabon, le Francevillien: sédimentologie, géochimie, relations avec lkes gîtes minéraux associés, Université de Strasbourg. Weber, F., and Gauthier-Lafaye, F., 2013, No proof from carbon isotopes in the Francevillian (Gabon) and Onega (Fennoscandian shield) basins of a global oxidation event at 1980-2090 Ma following the Great Oxidation Event (GOE): Comptes Rendus Geoscience, v. 345, p. 28-35. White, A.F., Bullen, T.D., Schulz, M.S., Blum, A.E., Huntington, T.G., and Peters, N.E., 2001, Differential rates of feldspar weathering in granitic regoliths: Geochimica et Cosmochimica Acta, v. 65, p. 847-869. Wing, B., Ferry, J., and Harrison, T.M., 2003, Prograde destruction and formation of monazite and allanite during contact and regional metamorphism of pelites: petrology and geochronology: Contributions to Mineralogy and Petrology, v. 145, p. 228-250.

Figure captions Fig. 1 :

 captions1 Figure captions

Fig. 2 :

 2 Fig. 2: (a) Location of the studied drill cores (KA13, GR1, GR5, GR20, GR23, GR31, GR 43 and KA6) close to Kaya-Kaya fault in the Franceville basin; (b) Simplified geological logs of the studied drill cores.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Bulk rock chemical compositions in the Debon and Le Fort diagram (1983). In brackets (L.O.I.), Q = normative quartz, P = normative feldspars. (a) GR20 and GR31 drill cores, (b) GR23, KA6 drill cores. Dot arrow = magmatic differentiation trend. The white triangles represent the average compositions of the diorite quartzite (dq), tonalite (to), granodiorites (gd), and adamellite (ad)

Fig. 5 :

 5 Fig. 5: Photomicrographs of the granitoids affected by propylitic alteration: (a) biotite is completely replaced by the chlorite associated to rod shaped allanite and titanite, backscattered electron SEM image; (b) sericitization of the feldspar, chloritization of biotite and chlorite calcite sequence of fissure infilling. chloritized biotite = chl biot, chlorite = chl, allanite = All, titanite = tin, feldspar = Felds, calcite = Cal, QZ = quartz.

Fig. 6 :

 6 Fig. 6: Photomicrographs of illitization features in granitoids; (a) feldspars are completely replaced by illite while quartz (Qz) are cracked and enveloped by illitic matrix; (b) The chloritized biotite is partially to totally replaced by the Fe-oxide + phengite (Phg) assemblage; (c) fracture filled by calcite (cal); (d) Cementation in the FA sandstone by anhydrite.

Fig. 7 :

 7 Fig. 7: Evolution of ∆ LOI with depth of tonalite.

Fig. 8 :

 8 Fig. 8: Plot of ∆SiO 2 , ∆Al 2 O 3 , ∆CaO, ∆Na 2 O, ∆Fe 2 O 3 , ∆MgO, ∆K 2 O versus ∆LOI of propylitized (black diamond) and illitized (open circle) tonalite.

Fig. 9 :

 9 Fig. 9: Plot of ∆Rb, ∆Sr, ∆La, ∆Ce, ∆Cs, ∆Th, ∆U versus ∆LOI of propylitized (black diamond) and illitized (open circle) tonalites.

Fig. 10 :

 10 Fig. 10: Evolution of the chondrite-normalized rare earth element patterns of the granitoids, from the least altered to the most altered (illitized) granitoids. Open square = illitized granitoids, black losange = superimposed illitic on propylitic alteration, open circle = least altered and propylitized granitoids.

Fig. 11 :

 11 Fig. 11: (a) Evolution of the XRD patterns of the < 2 µm fractions from KA6 467 m (granitoids) to KA6 436 m (sandstone); (b) Evolution of the XRD patterns of chlorite from KA6 467 m tonalite), to KA6 438 drill core < 2µm fraction in oriented glycol saturation preparations. Chl = chlorite, ill = illite.

Fig. 12 :Fig. 13 :

 1213 Fig. 12: (a) Representation of the four clay mineral domains in the MR 3+ , 2R 3+ , 3R 2+ triangle (Velde, 1985): illite, phengite (Ph), chlorite upper zone (chlup) and chlorite lower zone (chl low ), phengite + chlorite mixture (Ph + chl up ) domain. (b) Evolution of chemical compositions of the chlorites analyzed from the depth (lower zone) to the unconformity (upper zone) in the Fe/(Fe + Mg) versus Si diagram. Open diamond = chlorite of lower zone, open circles = chlorite of upper zone.

Fig. 14 :

 14 Fig. 14: Representation of Pb and U isotopic data 207 Pb/ 206 Pb versus 206 Pb/ 238 U diagram. MSWD = Mean Square Weighted Deviation. Monazites of GR1 631 sample.

Fig. 15 :

 15 Fig. 15: (a) Representation of the alteration effects in the Al 2 O 3 -CaO* + Na 2 O + K 2 O -FeO + MgO diagrams (Rainbird et al., 1990). CaO* = CaO of silicates. L.alt = least altered rocks, Pro + dif = propylitized rocks + magmatic differentiation, Ill. = analyzed illites, phg = analyzed phengites. Illite = theorical illite, Ka = kaolinite, Gi = Gibbsite. Black arrows indicate the least altered rock to-propylitized rock and the propylitized rock-to-illitized rock. (1) 2.6 Ga propylitic event, (2) 1.87 Ga late diagenetic (illitic) event.

  Figure 1

  Figure 15

Table captions Table 1 .

 captions1 Whole-rock geochemical data for the Archean granitoids at Kiéné area.

Table 2 :

 2 Representative microprobe analyses of chlorites from the propylitic zone. The structural formulae are calculated on the 14 oxygen basis.

Table 3 :

 3 Representative microprobe analyses of chlorites from the illitic zone. The structural 796 formulae are calculated on the 14 oxygen basis.

Table 4 :

 4 Representative microprobe analyses of phengites from 797 the illitic zone. The structural formulae are calculated on the 11 oxygen basis.

Table 5 :

 5 Representative microprobe analyses of illites from the illitic zone. The structural 800 formulae are calculated on the 11 oxygen basis.

Table 6 :

 6 Representative microprobe analyses of epidotes and allanites from the propylitic zone.

Table 7 :

 7 Representative microprobe analyses of monazites.

Table 8 :

 8 Representative Laser-ICPMS analyses and age calculations of monazites (sample 805 KA6 437.85).

Table 9 :

 9 Representative Laser-ICPMS analyses and age calculations of monazites (sample 807 GR1 631).

Table 10 :

 10 δ 13 C values (‰ PDB) and δ 18 O (‰ SMOW) measured on six samples affected by carbonated alteration.

Table 1 (

 1 Continued)

	Table
		,74	63,91	61,72	63,42	57,00	70,17	69,05	68,08	56,74	60,99	62,10	55,55	75,20
	TiO 2	0,45	0,50	0,39	0,44	0,56	0,09	0,30	0,19	0,32	0,52	0,38	0,70	0,07
	Al 2 O 3	16,15	15,07	15,81	15,58	22,83	5,17	15,35	16,49	18,45	17,66	18,62	23,36	5,04
	FeOt	4,23	4,01	3,93	4,07	4,08	0,87	2,65	1,86	8,16	4,71	4,33	6,86	0,62
	MnO	0,05	0,06	0,07	0,02	0,02	0,00	0,03	0,02	0,06	0,07	0,04	0,01	0,00
	MgO	3,15	2,05	1,88	1,22	1,46	0,31	1,79	0,89	1,61	2,06	2,01	0,73	0,26
	CaO	2,24	4,19	5,07	2,67	0,58	8,17	2,03	3,07	6,68	5,32	1,12	0,69	6,50
	Na 2 O	5,13	4,66	5,49	2,42	0,35	0,07	5,21	5,84	4,66	5,46	2,30	0,14	0,03
	K 2 O	1,89	1,80	1,42	4,50	8,34	2,32	1,23	1,43	0,59	0,71	5,01	7,38	1,68
	P 2 O 3	0,23	0,26	0,23	0,27	0,03	0,00	0,10	0,08	0,07	0,14	0,08	0,19	0,04
	LOI	1,95	3,53	4,17	4,52	4,03	12,20	1,65	1,55	2,09	2,03	3,26	3,94	9,10
	Total	100,20	100,03	100,18	99,12	99,25	99,37	99,38	99,50	99,41	99,67	99,24	99,55	98,55
	Ba (ppm)	1060,00	1147,00	476,40	4229,00	1700,00	830,40	401,20	622,10	243,70	388,30	2552,00	831,90	469,50
	Rb	40,77	70,11	61,46	261,50	433,60	107,80	30,21	32,20	15,80	17,39	247,00	259,20	62,30
	Sn	1,12	1,18	1,07	0,97	1,91	0,32	0,59	0,00	0,91	1,16	0,87	1,93	0,31
	Sr	802,90	695,90	533,40	775,50	115,50	780,00	490,10	738,60	1293,00	573,20	204,50	15,23	385,50
	Ga	21,62	21,71	20,30	20,46	33,50	6,43	17,50	17,85	26,20	22,31	23,63	29,91	6,09
	Y	9,72	11,63	11,09	12,01	25,19	7,12	3,76	3,66	9,05	14,35	7,46	13,02	3,71
	Zr	141,80	179,50	137,30	155,50	221,00	337,00	59,26	100,80	368,40	158,80	32,17	286,50	124,20
	Nb	4,07	3,61	2,83	3,54	7,49	2,29	2,10	1,32	1,04	3,62	2,73	7,48	1,83
	Cs	0,50	1,89	1,43	4,49	16,31	1,52	0,42	0,33	0,33	0,27	4,32	17,79	3,04
	Cu	0,00	47,83	93,11	20,02	8,18	10,75	7,91	23,41	5,25	15,86	39,96	4,06	22,26
	Zn	67,14	66,51	99,82	93,88	81,95	19,63	43,66	33,26	76,97	66,83	110,70	39,94	15,07
	Hf	3,59	4,34	3,35	3,82	5,39	7,78	1,63	2,87	8,81	3,82	1,00	6,99	3,17
	Ta	0,26	0,23	0,20	0,44	0,45	0,25	0,14	0,08	0,11	0,35	0,20	0,42	0,17
	Pb	7,20	42,63	381,32	23,23	19,27	5,44	6,62	10,64	8,72	10,26	3,24	8,25	1,93
	Th	5,68	7,55	2,20	41,43	12,45	11,86	2,85	0,62	7,52	1,16	4,56	20,04	6,45
	U	0,50	1,05	1,31	4,33	4,78	3,59	0,57	0,52	2,50	0,34	0,95	5,69	1,22
	V	64,36	74,24	70,28	62,46	256,10	146,40	40,54	14,51	80,11	73,27	51,95	84,92	19,92
	La	33,46	65,07	58,12	39,61	114,90	14,42	11,47	12,42	22,37	11,01	17,16	20,31	3,40
	Ce	59,83	119,40	105,20	75,67	180,30	24,56	22,37	25,32	41,44	25,44	29,71	41,50	5,93
	Pr	6,76	12,87	11,33	9,08	19,94	3,15	2,71	3,01	4,23	3,42	3,42	4,77	0,75
	Nd	24,99	46,31	41,11	32,99	74,04	12,13	10,30	11,32	13,56	14,44	2,73	17,84	2,85
	Sm	4,28	7,29	6,41	5,78	12,69	2,40	1,72	1,92	2,00	3,18	2,85	2,94	0,60
	Eu	1,36	1,68	1,35	1,87	2,98	0,66	0,56	0,53	0,61	0,73	0,90	0,64	0,21
	Gd	3,02	4,46	3,94	4,15	9,36	1,84	1,23	1,11	1,61	2,77	2,15	2,62	0,70
	Tb	0,37	0,52	0,48	0,54	1,21	0,24	0,16	0,14	0,22	0,43	0,30	0,37	0,10
	Dy	1,80	2,44	2,25	2,65	5,90	1,19	0,75	0,66	1,24	2,57	1,64	2,14	0,63
	Ho	0,31	0,39	0,37	0,42	0,89	0,22	0,13	0,12	0,26	0,49	0,29	0,45	0,12
	Er	0,80	1,02	0,99	1,05	2,07	0,67	0,35	0,32	0,78	1,41	0,77	1,41	0,34
	Tm	0,11	0,14	0,14	0,14	0,26	0,10	0,05	0,04	0,12	0,21	0,11	0,22	0,05
	Yb	0,72	0,93	0,89	0,87	1,53	0,78	0,16	0,14	0,22	1,36	0,72	0,37	0,34
	Lu	0,11	0,14	0,14	0,13	0,19	0,14	0,06	0,05	0,15	0,21	0,10	0,26	0,06

Table 2

 2

	Chlorite in the propylitic zone

Table 3

 3

				Chlorite in the illitic zone
	SiO 2 (Wt %)	27,65	26,27	26,40	26,18	25,47	26,38	27,97
	TiO 2	2,05	2,10	1,83	1,35	0,03	0,04	0,19
	Al 2 O 3	15,62	15,58	15,85	15,51	17,79	18,77	21,39
	FeOt	31,22	31,48	29,90	32,43	31,55	29,70	34,02
	MnO	0,34	0,21	0,46	0,40	0,42	0,37	0,16
	MgO	10,27	10,70	10,44	10,18	10,57	10,79	5,82
	Ca O	0,08	0,07	0,09	0,13	0,06	0,03	0,02
	Na 2 O	0,05	0,02	0,07	0,04	0,06	0,01	0,00
	K 2 O	0,20	0,13	0,15	0,44	0,33	0,64	0,91
	Tota l	87,48	86,55	85,18	86,65	86,29	86,73	90,47
	Si	3,02	2,92	2,96	2,93	2,85	2,89	2,87
	Al IV	0,98	1,08	1,04	1,07	1,15	1,11	1,13
	Al VI	1,04	0,96	1,05	0,98	1,19	1,32	1,47
	Ti	0,17	0,18	0,15	0,11	0,00	0,00	0,02
	Fe 2+	2,86	2,93	2,80	3,04	2,95	2,72	3,22
	Mg	1,67	1,77	1,74	1,70	1,76	1,76	1,03
	Mn	0,03	0,02	0,04	0,04	0,04	0,03	0,03
	Ca	0,01	0,01	0,01	0,01	0,01	0,00	0,00
	Na	0,01	0,00	0,02	0,01	0,01	0,00	0,06
	K	0,03	0,02	0,02	0,06	0,05	0,09	0,06
	∑ oct	5,57	5,66	5,60	5,72	5,90	5,80	5,72
	Fe/(Fe+Mg)	0,63	0,62	0,62	0,64	0,63	0,61	0,76

Table 4

 4

	Phengite

Table 5

 5

	Illite

Table 6

 6

				Epidote						Allanite
	Na 2 O (Wt %) 0,10	0,15	0,10	0,04	0,00	0,06	0,04	0,10	0,04	0,03	0,02	0,08	0,04
	MgO	0,00	0,01	0,11	0,01	0,18	0,22	0,16	0,12	0,44	0,28	0,80	0,12	0,25
	Al 2 O 3	19,84	21,31	19,26	21,06	22,08	13,59 13,69 13,65	13,19	14,56	17,23	15,82	18,27
	SiO 2	34,15	36,96	32,96	35,67	37,94	28,98 30,00 29,06	28,79	28,03	32,56	28,11	33,05
	K 2 O	0,05	0,11	0,03	0,04	0,01	0,08	0,04	0,06	0,10	0,03	0,03	0,06	0,03
	CaO	20,28	21,00	20,58	21,65	21,46	10,15 10,74 10,64	12,00	10,45	11,03	11,11	11,30
	TiO 2	0,11	0,07	0,04	0,04	0,45	0,23	0,44	0,47	1,70	0,22	0,64	0,24	0,13
	MnO	0,00	0,06	0,02	0,12	14,74	0,20	0,33	0,13	0,14	0,22	0,24	0,48	0,62
	FeOt	14,34	14,39	14,68	14,23	0,01	15,25 15,48 14,41	14,45	12,26	12,40	10,76	12,01
	Nd 2 O 3	0,00	0,00	0,16	0,00	0,36	3,07	1,79	1,03	1,34	3,15	3,25	1,48	2,83
	Ce 2 O 3	0,32	0,15	0,20	0,90	0,00	10,11	9,82	9,55	9,87	10,91	10,37	9,84	9,23
	La 2 O 3	0,38	0,13	0,27	0,00	0,09	6,67	6,04	5,80	4,41	5,92	5,70	5,52	5,40
	Total	89,57	94,33	88,42	93,76	97,32	88,61 88,56 85,03	86,46	86,06	94,26	83,64	93,16
	Si	2,99	3,05	2,95	2,99	3,03	2,95	3,00	3,01	2,94	2,93	3,01	2,95	3,05
	Al	2,05	2,07	2,03	2,08	2,08	1,63	1,62	1,67	1,59	1,79	1,88	1,96	1,99
	Ti	0,01	0,00	0,00	0,00	0,03	0,02	0,03	0,04	0,13	0,02	0,04	0,02	0,01
	Mn	0,00	0,00	0,00	0,01	0,00	0,02	0,03	0,01	0,01	0,02	0,02	0,04	0,05
	Fe 2+	0,94	0,89	0,99	0,90	0,89	1,17	1,17	1,12	1,11	0,97	0,86	0,85	0,83
	Mg	0,00	0,00	0,01	0,00	0,02	0,03	0,02	0,02	0,07	0,04	0,11	0,02	0,03
	Ca	1,90	1,86	1,98	1,94	1,84	1,11	1,15	1,18	1,31	1,17	1,09	1,25	1,12
	Na

Table 7

 7

			Monazite (KA6 437,85)			Mona zite (GR1 631)
	SiO 2 (Wt %)	0,37	1,22	0,39	0,43	0,88	0,22	0,72	0,42
	P 2 O 5	30,73	29,76	30,99	30,76	30,13	28,66	29,55	29,66
	Ca O	1,14	1,37	1,12	1,18	0,28	0,25	0,28	0,20
	Y 2 O 3	0,97	0,90	0,99	1,04	0,74	0,79	0,68	0,59
	La 2 O 3	15,05	14,98	15,61	15,63	18,74	18,17	17,30	20,10
	Ce 2 O 3	30,47	29,80	30,31	30,07	33,50	33,26	33,76	33,16
	Pr 2 O 3	2,99	2,92	2,95	3,04	3,54	3,88	3,33	3,21
	Nd 2 O 3	13,10	11,93	12,42	12,01	8,20	8,76	8,30	7,32
	Sm 2 O 3	1,40	1,65	1,51	1,65	0,59	0,67	0,37	0,59
	Gd 2 O 3	0,72	0,69	0,63	0,79	0,37	0,79	0,47	0,08
	PbO	0,05	0,13	0,08	0,09	0,02	0,02	0,04	0,00
	ThO 2	0,87	1,33	0,72	0,96	0,03	0,00	0,01	0,01
	UO 2	0,01	0,01	0,00	0,02	0,01	0,09	0,04	0,01
	Tota l	97,46	96,27	97,29	97,22	97,04	95,57	94,85	95,36
	Th/La	0,06	0,09	0,05	0,06	0,00	0,00	0,00	0,00
	ΣREE	64,71	62,86	64,42	64,23	65,68	66,32	64,20	65,07

Table 8

 8

	2 sigma	2 sigma		2 sigma
	error	error	Age (Ma)	error

Table 9

 9

							2 sigma		2 sigma
	GR1-631						error		error		Age (Ma) 2 sigma error
	Analysis_	Pb ppm	Th ppm	U ppm	Th/U	Pb207/U235 Pb207/U235 Pb206/U238 Pb206/U238	Rho	Pb207/Pb206 Pb207/Pb206
	05120912c	231	1429	337	4,2	5,78	0,24	0,35	0,01	0,85	1976
	06120912c	143	582	316	1,8	5,38	0,24	0,33	0,01	0,78	1931
	07120912c	265	1799	399	4,5	5,51	0,22	0,33	0,01	0,87	1948
	08120912c	191	1387	199	7,0	7,69	0,35	0,36	0,01	0,78	2401
	17120912c	72	31	224	0,1	6,03	0,31	0,35	0,01	0,68	2037
	18120912c	125	442	264	1,7	6,49	0,36	0,35	0,01	0,65	2157
	19120912c	96	77	303	0,3	5,87	0,32	0,34	0,01	0,66	2057
	20120912c	158	234	466	0,5	5,78	0,31	0,34	0,01	0,66	1993
	21120912c	133	25	301	0,1	10,97	0,60	0,38	0,01	0,66	2917
	22120912c	322	2263	294	7,7	10,03	0,56	0,38	0,01	0,64	2760
	23120912c	363	1934	359	5,4	14,02	0,79	0,41	0,01	0,64

Ba (ppm) 1014,00 740,50 577,10 874,60 538,40 1800,00 3861,00 976,40 1355,00 2100,00 1551,00 5313,00 1109,00 990,40 2007,00 2237,00 543,70 Rb 61,78 60,31 194,40 182,50 168,60 74,69 95,06 25,94 70,29 62,86 54,99 59,19 63,20 71,04 246,80 252,40 59,84 Sn 0,62 0,72 1,54 0,58 0,75 0,00 0,00 1,05 2,07 1,20 1,33 0,00 3,23 0,89 3,45 1,89 0,00 Sr 222,40 202,30 121,80 59,10 15,89 131,80 553,80 658,20 643,50 659,70 587,60 634,60 370,10 586,30 57,23 52,65 32,24 Ga 17,28 17,18 24,91 20,08 26,15 8,77 13,20 16,31 24,81 22,77 23,01 14,97 23,95 18,14 31,57 28,22 4,31 Y 4,62 13,04 8,68 12,32 11,28 3,97 0,80 9,95 35,31 11,16 11,25 2,02 9,67 8,55 11,46 15,69 2,11 Zr 57,04 165,80 110,60 97,47 167,30 47,66 16,28 120,30 234,30 154,00 146,90 50,36 160,50 150,50 218,10 262,70 59,63 Nb 2,91 5,71 3,92 2,97 5,50 1,34 0,21 3,61 8,33 5,07 5,08 0,23 5,39 4,86 6,21 10,41 1,31 Cs 0,55 0,71 3,40 2,72 8,01 4,47 0,55 0,15 1,85 0,79 0,91 0,39 1,25 2,14 15,50 7,05 1,10 Cu 6,54 11,37 34,02 4,29 0,00 12,30 8,30 17,85 65,30 42,13 18,92 123,80 41,43 0,00 0,00 0,00 5,55 Zn 30,30 41,49 268,60 74,10 59,29 29,28 0,00 46,25 130,00 50,28 62,36 0,00 58,56 32,64 39,11 73,37 0,00 Hf 1,61 4,71 3,05 3,43 5,39 1,33 0,38 2,96 5,47 3,79 3,65 1,60 3,98 3,63 5,42 6,92 1,63 Ta 0,15 0,35 0,27 0,08 0,24 0,12 0,01 0,21 0,47 0,33 0,34 0,02 0,34 0,43 0,67 1,00 0,13 Pb 7,25 6,66 5,70 10,27 3,43 2,07 8,99 4,04 29,46 45,59 16,15 10,68 45,39 88,14 5,57 10,07 4,50 Th 1,38 8,79 2,09 4,37 9,97 4,13 0,77 15,08 17,55 10,46 6,38 0,67 12,69 8,79 7,52 24,00 3,45 U 0,51 0,89 1,49 1,99 2,78 1,35 0,23 2,78 4,98 3,97 2,78 0,61 3,97 4,83 2,83 12,94 0,60 V 10,49 18,94 79,74 67,65 14,88 14,27 4,92 86,66 199,90 116,20 128,40 4,08 88,59 37,92 54,50 75,21 2,50 La 6,82 11,55 14,41 25,61 4,92 17,36 1,45 34,16 103,70 67,71 49,74 10,40 45,73 25,89 55,00 11,44 11,30 Ce 12,31 23,10 28,22 46,72 10,56 28,32 2,25 63,12 213,90 124,50 103,90 17,91 88,34 50,81 101,30 22,80 16,67 Pr 1,26 2,72 3,44 5,22 1,17 3,49 0,23 7,55 24,13 14,27 11,01 1,85 9,58 5,97 10,45 2,88 1,99 Nd 2,91 5,71 3,92 2,97 4,60 11,81 0,85 27,66 96,64 51,54 40,18 6,34 5,39 22,82 45,21 12,33 6,61 Sm 0,94 2,21 2,96 3,66 1,33 2,07 0,17 4,79 16,82 7,69 6,53 0,85 6,65 3,99 8,11 3,55 0,90 Eu 0,61 0,76 0,89 0,95 0,45 0,52 0,21 3,15 11,24 4,50 3,93 0,56 1,21 2,83 1,57 1,33 0,21 Gd 0,91 2,30 2,69 2,92 1,56 1,65 0,15 0,41 1,42 0,49 0,47 0,06 4,01 0,34 5,05 3,41 0,55 Tb 0,15 0,38 0,35 0,41 0,25 0,20 0,02 2,10 7,20 2,27 2,32 0,32 0,46 1,70 0,58 0,50 0,07 Dy 0,83 2,24 1,78 2,23 1,62 0,86 0,11 2,10 7,20 2,27 2,32 0,32 2,14 1,70 2,67 2,73 0,38 Ho 0,15 0,43 0,27 0,40 0,35 0,13 0,02 0,35 1,25 0,37 0,39 0,06 0,34 0,29 0,43 0,48 0,06 Er 0,40 1,14 0,62 1,04 0,96 0,30 0,07 0,96 3,40 0,97 1,08 0,18 0,91 0,81 1,15 1,27 0,19 Tm 0,06 0,17 0,08 0,15 0,14 0,04 0,01 0,13 0,48 0,14 0,15 0,03 0,13 0,11 0,17 0,19 0,03 Yb 0,36 1,16 0,47 0,99 1,01 0,24 0,07 0,87 3,06 0,94 1,01 0,20 0,83 0,78 1,14 1,23 0,17 Lu 0,06 0,19 0,07 0,16 0,17 0,04 0,01 0,13 0,48 0,15 0,15 0,03 0,12 0,13 0,18 0,19 0,03

We thank the Gabonese Government, the Sylvia Bongo Foundation, Région Poitou-Charentes, FEDER program, CNRS-INSU and University of Poitiers for their financial supports.

Wealsothank the « Direction Générale des Mines et de la Géologie du Gabon » (DGMG) for itstechnical support.


Pietro Gori

Olivier Colliot

Linda Marrakchi-Kacem

Yulia Worbe

Fabrizio De

Vico Fallani

Mario Chavez

Sophie Lecomte

Cyril Poupon

Andreas Hartmann

Nicholas Ayache

Stanley Durrleman

A Prototype Representation to Approximate White Matter Bundles with Weighted Currents

Quantitative and qualitative analysis of white matter fibers resulting from tractography algorithms is made difficult by their huge number. To this end, we propose an approximation scheme which gives as result a more concise but at the same time exhaustive representation of a fiber bundle. It is based on a novel computational model for fibers, called weighted currents, characterised by a metric that considers both the pathway and the anatomical locations of the endpoints of the fibers. Similarity has therefore a twofold connotation: geometrical and related to the connectivity. The core idea is to use this metric for approximating a fiber bundle with a set of weighted prototypes, chosen among the fibers, which represent ensembles of similar fibers. The weights are related to the number of fibers represented by the prototypes. The algorithm is divided into two steps. First, the main modes of the fiber bundle are detected using a modularity based clustering algorithm. Second, a prototype fiber selection process is carried on in each cluster separately. This permits to explain the main patterns of the fiber bundle in a fast and accurate way.

Introduction

Tractography from diffusion-weighted magnetic resonance imaging (DW-MRI) is a technique capable to virtually map the neural architecture of the human brain white matter (WM). This method is very useful for a better characterization of neurological diseases, for surgical planning or for the study of anatomofunctional relationships, for example. Tractography methods result in a set of 3D tracts that are commonly referred to as "fibers" which represent an estimate of the trajectory of large groups of neural fibers. WM fibers are traced starting from points, called seeds, inside one or more voxels and they are constituted by segments connecting different voxels. Tractography algorithms can be divided into two classes: deterministic and probabilistic. Deterministic methods reconstruct the fiber following the principal direction given by the diffusion model inside each voxel while the probabilistic ones use a randomly perturbed version of the main fiber direction. The idea behind probabilistic methods is to start many fibers from every seed in order to obtain maps of connectivity that are related to the probability that a certain voxel is connected to the starting seed. The choice of the starting voxels depends on the application and it ranges from the whole-brain to particular areas of the gray matter (ROI). One can also choose a second ROI as ending area in order to retrieve only the fibers connecting two precise areas of the gray matter. This paper focuses on such sets of fibers, called fiber bundles, which connect the cortical surface to the basal ganglia. A fiber is therefore characterised by its pathway between voxels and by the anatomical locations of its starting and ending point. Tractography algorithms usually result in a considerable amount of fibers. This makes difficult the development of efficient computational methods and the visualisation and interpretation of the brain connections that have been captured. To this end, we propose a novel approximation scheme for fiber bundles which reduces the number of tracts to analyse conserving almost all the information related to the fibers pathway (geometry) and to the distribution of the fibers endpoints (connectivity).

The core idea is to approximate a fiber bundle B with a set of weighted prototypes {τ i M i }, chosen among the fibers, which represent ensembles of similar fibers. The weights {τ i } are linked to the number of fibers approximated by the prototypes {M i }. In order to do that, we need an appropriate dissimilarity measure based on both the geometry and the connectivity of a fiber. Usual dissimilarity measures present in the literature consider only the geometry of the fibers, like the Hausdorff distance [START_REF] Gerig | Analysis of Brain White Matter via Fiber Tract Modeling[END_REF], the Chamfer distance (or modified versions of it) [START_REF] Gerig | Analysis of Brain White Matter via Fiber Tract Modeling[END_REF][START_REF] O'donnel | Automatic Tractography Segmentation Using a High-Dimensional White Matter Atlas[END_REF][START_REF] Zhang | Identifying White-Matter Fiber Bundles in DTI Data Using an Automated Proximity-Based Fiber Clustering Method[END_REF][START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF], the total square loss between two fibers represented as Gaussian mixture model [START_REF] Liu | Unsupervised Automatic White Matter Fiber Clustering Using a Gaussian Mixture Model[END_REF] or Fourier descriptors [START_REF] Batchelor | Quantification of the shape of fiber tracts[END_REF]. Other dissimilarity measures consider only the ending points of the fibers [START_REF] Brun | Coloring of DT-MRI Fiber Traces Using Laplacian Eigenmaps[END_REF] or are based on the voxel space [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF][START_REF] Tunç | Multinomial probabilistic fiber representation for connectivity driven clustering[END_REF]. Here we propose a novel dissimilarity measure which considers both the pathway and the anatomical locations of the termini of the fibers. It can be seen as an extension of the framework of currents [START_REF] Durrleman | Registration, Atlas Estimation and Variability Analysis of White Matter Fiber Bundles Modeled as Currents[END_REF] and we have called it: weighted currents. As usual currents, it does not need point-to-point correspondence between fibers, it has a closed form and easy to compute expression and it can be used to compare fiber bundles. In addition, two fibers modelled as weighted currents are considered similar if and only if their pathways are alike and their endpoints are close to each other. Fibers are considered as vectors in the space of weighted currents. This permits to see a fiber bundle B as the sum of all its fibers F j : B = j F j and its approximation as a weighted sum:

K i=1 τ i M i .
The proposed approximation scheme is based on the minimization of the residual error between B and {τ i M i } which can be easily written as: ||B -

K i=1 τ i M i || 2
W * in the framework of weighted currents. The final goal is to find the smallest set of prototypes which minimizes this approximation error. An exhaustive analysis of all the possible combinations of prototypes is not feasible and therefore we propose a greedy algorithm. Our strategy is first to decompose a fiber bundle into modes, and then approximate every mode assuming that its fibers follow a gaussian distribution. In order to do that, the fiber bundle is first divided into independent clusters, which can be seen as different modes, using a clustering algorithm. Afterwards, the prototype fiber selection process is carried on in each cluster separately. This permits a fast (parallel computing), concise and at the same time exhaustive approximation of the fiber bundle.

Usual clustering approaches for WM tracts are based on hierarchical [START_REF] Zhang | Identifying White-Matter Fiber Bundles in DTI Data Using an Automated Proximity-Based Fiber Clustering Method[END_REF][START_REF] Gerig | Analysis of Brain White Matter via Fiber Tract Modeling[END_REF][START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF][START_REF] Liu | Unsupervised Automatic White Matter Fiber Clustering Using a Gaussian Mixture Model[END_REF], spectral [START_REF] O'donnel | Automatic Tractography Segmentation Using a High-Dimensional White Matter Atlas[END_REF][START_REF] Brun | Coloring of DT-MRI Fiber Traces Using Laplacian Eigenmaps[END_REF] or EM [START_REF] Maddah | A unified framework for clustering and quantitative analysis of white matter fiber tracts[END_REF] techniques. In many cases one needs to fix in advance parameters like the number or the size of the clusters. A solution, which does not require parameter setting, is to employ one of the quality functions in the field of networks community detection: the modularity [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF]. It determines how good a cluster division is by comparing the similarity of fibers inside clusters with respect to the similarity of fibers between clusters. Optimization methods based on the maximization of this quality function permit to automatically find the number of clusters and how to divide the fiber bundle into clusters.

Eventually, the prototypes are chosen by selecting the fibers that minimize the residual error between B and {τ i M i } in the spirit of PCA (Principal Component Analysis) exploiting the fact that every mode is assumed to be gaussian (unimodal). This permits to explain and approximate almost the entire variability of the fiber bundle in a fast and accurate way.

Approximation scheme for WM fiber bundles

In the first paragraph we introduce the concept of weighted currents from a formal mathematical point of view. Afterwards, we present the two steps of the approximation scheme based on weighted currents: the modularity optimization clustering algorithm and the prototype fiber selection.

Weighted Currents can be seen as an adaptation of the functional currents presented in [START_REF] Charon | Functional Currents : A New Mathematical Tool to Model and Analyse Functional Shapes[END_REF] where the "functional signal" attached to each fiber is the anatomical location of its starting and ending point. Let X be a WM tract (an oriented and rectifiable curve in R 3 ) which can be modelled as a polygonal line of N segments and f c and f b two 3D vectors in M =R 3 xR 3 containing the coordinates of the two extremities. For example, f c might represent the extremity of the fiber on the cortical surface and f b on the basal ganglia. In practice, f c and f b could be the mean values of the last n points, i.e. n=3, if the extremities are supposed to be in an area with a low SNR. The tract X, exactly as for currents, can be seen as a discrete weighted current via: C X (w) = N i=1 w(x i , f c , f b ) T α i where x i and α i are the centres and the tangent vectors of the N segments respectively and w is a vector field belonging to a reproducing kernel Hilbert space (RKHS) W on R 3 xM . A natural way to build the kernel associated to W is as the tensor product of kernels defined separately in the geometrical space R 3 and in the "functional" space M . This means that the inner product in the framework of weighted currents between two tracts

X (C X (w) = N i=1 w(x i , f c , f b ) T α i ) and Y (C Y (w) = M j=1 w(y j , t c , t b ) T β j ) can be defined as: C X , C Y W * = K c (f c , t c )K b (f b , t b ) N i=1 M j=1 α T i K g (x i , y j )β j
where K c , K b and K g are Gaussian kernels parametrized by their scale parameters: λ c , λ b and λ g respectively. The underlined part is the inner product between usual currents which measures overall differences in the pathway of the two fibers. λ g defines the range of interactions between the points of X and Y . The multiplication by the new terms K c and K b means that two fibers are considered similar if the pathways are similar, like in usual currents, but also if the endpoints of the two fibers are at a distance smaller than λ c on the cortical surface and than λ b on the basal ganglia (see Fig. 1). The space of weighted currents is a vector space which implies that a fiber bundle B is seen as the sum of its fibers

F i : C B = N i C
Fi and that it is also possible to compute the mean of a fiber bundle:

C F = 1 N N i C Fi .
In the following, we will assume that each fiber is modelled as weighted current writing simply F instead than C F . As we will see, our processing will be calculated using only simple computations of the Gram matrix Γ = { F i , F j W * } i,j=1,...,N .

Fig. 1. Tracts that have an angle smaller than 45 degrees with the red one using currents (green, #118) and weighted currents (blue, #8). Fibers belong to a thalamocortico bundle of the right hemisphere resulting from a probabilistic tractography (see Sec.3). Green tracts are more than the blue ones and they are also more spread, connecting anatomical locations far from the ones of the red fiber. This shows why weighted currents are more suitable for clustering and approximating fiber bundles.

Modularity Optimization Clustering A fiber bundle is seen as a sum of vectors in the space of weighted currents, each vector representing a fiber. Every vector can then be considered as a vertex of a weighted graph where the weighted edges are the inner products between every couple of fibers. The first step of the proposed algorithm consists in finding clusters in this set of vectors representing the main modes of the fibers distribution. We use a clustering algorithm based on the maximization of a quality function Q called Modularity [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF]:

Q = N C c=1    i∈c F i 2 W * j / ∈c F j 2 W * -( i∈c j / ∈c F i , F j W * ) 2    (1)
where N C is the number of clusters. This equation can be easily rewritten in terms of the Gram matrix Γ as: Q=

N C c=1 (s T c Γ s c )((1-s c ) T Γ (1-s c ))-(s T c Γ (1-s c )) 2
where s c (k)=1 if F k belongs to cluster c and 0 otherwise. In the simple case of N C = 2 this equation can be rewritten as:

Q = F1 2 W * F2 2 W * -F1 , F2 2 W *
where F1 and F2 are the means of the two clusters. Maximizing Q means therefore looking for two clusters whose means are as orthogonal as possible and at the same time their norms should be as close as possible. This can be generalized to N C clusters by saying that the goal of modularity is to create clusters with balanced norms characterised by fibers orthogonal to the fibers in the other clusters and parallel to the fibers in their own cluster. Unfortunately, exact modularity optimization is a NP-complete problem. The "Louvain" algorithm [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] is a greedy solution divided into two steps which are repeated iteratively. At the beginning every WM tract forms a different cluster. The first part consists of associating every WM tract to all its neighbour clusters finding the one that leads to the largest increase in Q. This step is repeated until no change would produce an increase in Q. The second part consists in merging all the WM tracts of one cluster in one single supervertex. Two supervertices have a weighted edge equal to the sum of all the inner products between the fibers of the initial clusters. The two steps are repeated until no change would produce an increase in Q. At the end of this process the fiber bundle is separated into different clusters without fixing in advance neither the number of clusters nor their size.

Prototype Fiber Selection The goal of the Prototype Fiber Selection (PFS) process is to concisely represent the fiber bundle B with a set of weighted prototypes {τ i M i } chosen among the fibers. If we wanted only one weighted prototype

τ 1 M 1 which minimizes ||B-τ 1 M 1 || 2 W * , it would be: M 1 = argmax Fz B, Fz ||Fz|| 2 W * = argmax Fz N 2 F , Fz ||Fz|| 2 W * with τ 1 = B,M1 ||M1|| 2 .
This means that we would look for the fiber most similar to the average of the bundle. This scheme works fine only in a uni-modal setting (i.e. gaussian) but not in a multi-modal one since the fiber chosen would be the one closest to the center of the different modes. If the modes are far from each other the fiber chosen could be also considered as an outlier. This is why it is fundamental finding the main modes of the bundle through the previous clustering step. Once defined the main modes, a PFS is performed independently on each one of them. One prototype is not sufficient to explain the whole cluster. So, as for instance in PCA, we remove from each fiber (F i ) its orthogonal projection onto the prototype (π(F i ) = Fi,M1 M1

||M1|| 2 ) and we select, in this new representation (r(F i ) = F iπ(F i)), the fiber most similar to the new average as second prototype (M 2 = argmax r(Fz) N 2 r( F ), r(Fz) ||r(Fz)|| 2 ). We iterate this process for each cluster C j until:

||C j - Kj i=1 τ i M i || W * ≤ γ||C j || W * .
At iteration k, the k weights {τ i } are computed by the orthogonal projection of the cluster C j to the space spanned by the selected k prototypes {M i }. It is important to notice that all these computations are based on the Gram matrix Γ of the fiber bundle, also when computing a new prototype:

r(F i ), r(F j ) = F i , F j - Fi,M Fj ,M ||M || 2 =Γ ij - Γ iM Γ jM ||Γ M M || 2 .
After selecting the prototypes in each cluster, the weights are recomputed by the orthogonal projection of the whole bundle B to the entire set of prototypes in order to retrieve the correct values also for the prototypes close to the boundary between two different clusters.

Experiments and Discussion

We illustrate the algorithm on 36 fiber bundles from 6 subjects connecting the cortex to thalamus [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF], putamen [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF] and caudate [START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF] of the right hemisphere using both deterministic and probabilistic tractography estimated from HARDI data ( [START_REF] Marrakchi-Kacem | Mapping Cortico-Striatal Connectivity onto the Cortical Surface: A New Tractography-Based Approach to Study Huntington Disease[END_REF] and references therein). The fiber bundles include the commisural fibers which have been truncated at the interhemispheric fissure [START_REF] Marrakchi-Kacem | Mapping Cortico-Striatal Connectivity onto the Cortical Surface: A New Tractography-Based Approach to Study Huntington Disease[END_REF]. The segmentation of the sub-cortical structures and of the cortex is done using FSL and FreeSurfer respectively [START_REF] Worbe | Distinct structural changes underpin clinical phenotypes in patients with Gilles de la Tourette syndrome[END_REF]. The result of the clustering and the prototype representation for both deterministic and probabilistic tractography on a cortico-thalamus fiber bundle are shown in Fig. 2. Other results are presented in the supplementary material. In the deterministic case only 491 fibers out of 9525 have been chosen as prototypes (∼5%) in order to explain 90% of ||B|| W * while with the probabilistic tractography only 2652 fibers out 10410 have been used as prototypes (∼25%) to explain 85% of ||B|| W * . The average reduction in the deterministic case is 4.7% explaining 90% of ||B|| W * and 24.6% in the probabilistic case explaining 84% of ||B|| W * . Once calculated the Gram matrix (∼120 min), the computation of the whole approximation scheme is very fast: about 40 sec for the clustering and 30 sec for the PFS in a PC, Intel Xeon, 4 cores, 3.20GHz using 10410 fibers. Fig. 3 shows the same bundle of Fig. 2 but its goal is to point out the differences between our approximation scheme and a uniform downsampling resulting in the same number of tracts. The probability densities of the endpoints of the original fibers on both cortex and thalamus are more similar to the ones of our scheme than with a uniform downsampling. In fact, some small modes are lost with the uniform downsampling. Probability densities have been computed using gaussian kernels, taking into account the weights τ i of the prototypes for our approximation. The Kolmogorov-Smirnov test fails to show statistically significant differences between the densities of the original fibers and of the weighted prototypes and in most of the cases it finds significant differences at the 5% level between densities from original fibers and uniform downsampling.

Conclusions

We have presented here a new approximation scheme for white matter fiber bundles which results in a concise representation maintaining almost all the information related to the pathways of the fibers and to the locations of the fiber endpoints. It is based on a new computational model for fiber bundles, called weighted currents, which permits to compare fibers considering both their pathways and the locations of their endpoints. Moreover, it allows to treat fibers as vectors. We have tested this method on fiber bundles resulting from both deterministic and probabilistic tractography showing that the number of tracts to analyse can be reduced up to 3% of the initial number of fibers in the deterministic case and up to 17% in the probabilistic case, explaining in both cases more than 84% of the norm of the fiber bundles. We have also shown that the connectivity information, seen as the probability density of the fibers endpoints onto the gray matter, is more preserved with the proposed method than using a uniform downsampling. This information is important in many neuro-anatomical studies since it could be used, for instance, to characterise neurodevelopmental disorders [START_REF] Worbe | Distinct structural changes underpin clinical phenotypes in patients with Gilles de la Tourette syndrome[END_REF]. Future works will include this new representation into registration and atlas construction methods in order to find changes in white matter organization across subjects. Another possible development could be to use this method for the white matter segmentation problem [START_REF] O'donnel | Automatic Tractography Segmentation Using a High-Dimensional White Matter Atlas[END_REF][START_REF] Guevara | Robust clustering of massive tractography datasets[END_REF]. Fig. 3. Probability densities of the endpoints of the fibers using the right thalamocortico fiber bundle, prototypes P and an equal number of fibers coming from a uniform downsampling U of the original fiber bundle. U is a poorer approximation of the original density than P since it reveals fewer modes, as shown by the arrows.

Fig. 2 .

 2 Fig. 2. First row: clusters of the right thalamo-cortico bundle highlighted in different colors. Second row: resulting prototypes visualized as tubes whose radii are proportional to their weights τi. The two columns on the right show the same fiber bundle from both lateral views.

Acknowledgements The research leading to these results has received funding from the program "Investissements d'avenir" ANR-10-IAIHU-06.