Bernard Le Gall
email: blegall@univ-brest.fr

Christine Authemayou

Axel Ehrhold

Jean-Louis Paquette

Denise Bussien

Gilles Chazot

Arthur Aouizerat

Yves Pastol

LiDAR offshore structural mapping and U/Pb zircon/monazite dating of Variscan strain in the Leon metamorphic domain, NW Brittany

Keywords: Armorican massif, Leon metamorphic domain, offshore structural mapping, LiDAR records, foliation trajectory map, dextral ductile shearing, U/Pb zircon/monazite dating

come

Introduction

Trajectory maps of ductile fabrics provide useful insights into the finite strain pattern and the relative chronology of deformation in pervasively deformed terrains. Drawing trajectory maps of planar ductile fabrics is generally an easy task in continuously exposed deformed terrains. This issue is much more problematical in ancient orogenic belts where basement rocks are usually highly eroded and scarcely exposed. In such unfavourable conditions, trajectory maps can be elaborated either by drawing the envelope curve of foliation trends for each discrete field measurement point, or by using geostatistical approaches. The latter method has been recently applied to specific structural domains of the Armorican Variscides [START_REF] Gumiaux | Tectonic history of the Hercynian Armorican shear belt (Brittany, France)[END_REF]. Where ancient deformed terrains extend laterally offshore into marine areas devoid of any sedimentary cover, remote sensing imagery could be a powerful potential method for imaging submerged basement structures. Such geophysical surveys have recently been performed in the Molene archipelago, off NW Britanny (Fig. 1a), by combining LiDAR (Light airborne detecting and ranging) and multibeam swath records. The resulting composite high-resolution dataset provides an exceptional structural picture of the Variscan bedrock over a ~12 x 20 km area, in the western offshore extent of the Leon metamorphic domain (LMD in the text) (Fig. 1b). It has been known for a few decades that the LMD forms an exotic terrane within the Armorican Variscides. Its overall structure is dominated by large-scale metamorphic thrust

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M A N U S C R I P T sheets, further dissected by both sinistral (Guisseny-Porspoder; GPSZ), and more numerous dextral (North Armorican, Elorn, and Pierres Noires; NASZ, ESZ, PNSZ) ductile transcurent shear zones. In both cases, transcurent shearing is accompanied by synkinematic granitic intrusions (see [START_REF] Ballèvre | Paleozoic history of the Armorican Massif. Models for the tectonic evolution of the suture zones[END_REF], for a review). However, major uncertainties still exist with regards to the internal architecture and map trace of the entire shear zone pattern. The specific role of the ESZ and PNSZ, which separate the LMD from Paleozoic terrains of the Central Armorican domain to the south, is also debatable. These tectonic boundaries have been regarded as either dextral transform zones [START_REF] Balé | Les complexes métamorphiques du Léon (NW Brittany) : un segment du domaine éohercynien sud-armoricain translaté au Dévonien[END_REF] or large-scale thrusts related to an oceanic suture zone [START_REF] Rolet | L'évolution géodynamique dévono-carbonifère de l'extrémité occidentale de la chaîne hercynienne d'Europe sur le transect Armorique-Cornwall[END_REF][START_REF] Faure | Paleozoic tectonic evolution of Medio-Europa from the example of the French Massif Central and Massif Armoricain[END_REF][START_REF] Ballèvre | Paleozoic history of the Armorican Massif. Models for the tectonic evolution of the suture zones[END_REF].

Possible solutions are proposed in the present paper, based on a high-resolution

LiDAR/echosounder digital elevation model (DEM) that supplies, for the first time, an accurate structural map of part of the offshore Variscan basement in Brittany. Various types of planar structures, extracted from the elaborated DEM, are interpreted in terms of ductile fabrics from seafloor rock sample analyses, and correlations with strained rocks continuously exposed inland along a 10 km-long coastal section. Our offshore/onshore results, that combine a foliation trajectory map (offshore) with kinematic data (onshore), ultimately allow us to propose an improved kinematic model for part of the Variscan history of the LMD. An intricate dextral shear zone network, arranged into a regional-scale strike-slip relay zone, post-dated the earlier regional foliation associated with top-to-the-NNE shearing. The timing framework of Variscan ductile strain in the LMD is further calibrated by new U/Pb zircon dating of synkinematic intrusions.

Geological setting

The LMD is a 30 x 80 km granitic and metamorphic domain, NW of the Armorican massif (Fig. 1a). It is part of a mosaic of structural domains that have been juxtaposed during

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the Middle/Late Carboniferous along large-scale ductile strike-slip discontinuities, including the North-and South-Armorican shear zone [START_REF] Gapais | Is the Hercynian belt of Brittany a major shear zone[END_REF]. The strongly strained and metamorphosed rocks in the LMD differ markedly from the much less deformed terrains in the surrounding North-and Central-Armorican domains that nearly escaped

Variscan shortening (Fig. 1a).

The overall structural arrangement of the LMD is generally thought to be a result of the combined effects of early large-scale thrusts, later intruded by various synkinematic granitoids during transcurrent shearing. Though not totally cylindrical, the structure of the LMD is synthetized on the 30 km-long cross-section of Fig. 1c. It shows a broad N70°Etrending upright antiform, involving a pile of thrust slices with metamorphic grades increasing northwestwards, i.e. downsection, from greenschist facies to partial-melting conditions. The upper allochthonous units in the southern flank of the antiform comprise, from top to bottom, the Brest orthogneiss, the protolith of which yielded ages of 504 ± 15 Ma [START_REF] Marcoux | Géochronologie revisée du dôme du Léon[END_REF], the Le Conquet amphibolite-facies micaschists and metagabbros, dated at 478 ± 4 Ma [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF] and the Lesneven gneisses that pass northwards into sillimanite-rich facies with pockets of anatectic granites. The basal unit is thrust over the para-autochthonous

Treglonou orthogneiss in the core of the eroded antiform. Its protolith has been dated at 385-391 Ma by U/Pb method on zircons [START_REF] Marcoux | Géochronologie revisée du dôme du Léon[END_REF]. Published U-Th/Pb monazite ages of synthrusting metamorphism range from 340-335 Ma [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF] to 315 Ma [START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF]. Relics of an earlier (439 ±12 Ma) HP eclogitic event [START_REF] Paquette | Géochronologie et géochimie des éclogites du Léon: nouvelles contraintes sur l'évolution géodynamique du Nord-Ouest du Massif armoricain[END_REF] are found locally in mafic lenses within the high-grade paragneiss exposed on the northern flank of the antiform [START_REF] Cabanis | Les éclogites du Pays de Leon (NO du Massif armoricain) : étude petrologique et géochimique ; implications géodynamiques[END_REF]).

The regional antiformal stack is cross-cut by two distinct types of ductile strike-slip shear zones that differ in age, kinematics, and types of related granitic intrusions (Figs. 1b andc). Early dextral shearing occurred to the south along (1) the PNSZ that extends as a 10-15

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km-long corridor throughout a narrow band of undated (Pierres Noires) granites and part of the Brest gneisses [START_REF] Chauris | Notice explicative de la feuille Plouarzel/île d'Ouessant à 1/50000[END_REF] (2) the NASZ that cuts though the southern margin of the 321±5 Ma St-Renan-Kersaint synkinematic granite (U-Th/Pb monazite ages; [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], in turn, secant to the south with respect to the strained Lesneven metamorphic terrains [START_REF] Goré | Cinématique hercynienne du cisaillement nord-armoricain à la bordure du granite syntectonique de Saint-Renan[END_REF]. Younger sinistral shearing took place further north, along the N70°E-trending PGSZ, which also acted as the root zone of the 301± 3 Ma Aber-Ildut southerly-directed granitic sheet [START_REF] Le Corre | Le Léon : un domaine exotique et géophysique au NO de la chaîne varisque armoricaine (France)[END_REF][START_REF] Marcoux | Notice explicative de la feuille Plouguerneau à 1/50000[END_REF].

Prior to our work, the offshore extent of the LMD Variscan framework was poorly documented. Published geological maps of the Molène archipelago only show a regular pattern of N70E-oriented granitic and metamorphic belts, with local structural disturbances, marked by the map curvature of the foliation in the Beniguet and Quemenes southern islands (Fig. 1b) [START_REF] Chauris | Notice explicative de la feuille Plouarzel/île d'Ouessant à 1/50000[END_REF]. Very little was known about the offshore map trace of the LMD shear zone pattern, except about those of the PNSZ which was assumed to continue westwards as a 15 km-long discontinuous range of islets [START_REF] Chauris | Notice explicative de la feuille Plouarzel/île d'Ouessant à 1/50000[END_REF], whereas the PGSZ was correlated with the dextrally offset (15 km) North Ouessant shear zone [START_REF] Andreieff | Les terrains antécambriens et paléozoïques au large du Léon et leur relation avec la couverture secondaire et tertiaire de la Manche[END_REF] on both sides of the La Helle NW/SE-trending fault.

Offshore/onshore dataset

Between 2010 and 2011, an integrated airbone laser altimetry (LiDAR) and swath bathymetry survey, funded by IFREMER and the National Marine Park of Iroise, was carried out over a ~320 km 2 offshore area in the Molene archipelago (Fig. 2 and Table 1).

LiDAR techniques alone are commonly used in a variety of geoscience applications, including, river network analysis, structural mapping, and more generally speaking, geomorphic studies [START_REF] Webster | The application of LiDAR-derived digital elevation model analysis to geological mapping: an example from the Fundy Basin, Nova Scotia, Canada[END_REF]. However, combining aerial laser bathymetry and

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technique that has been successfully applied in coastal and river environmental studies [START_REF] Poppe | Integrated multibeam and LiDAR bathymetry data offshore of New London and Niantic[END_REF].

LiDAR dataset

Airborne LiDAR records can provide high-resolution topographic data based on the two-way travel time of laser pulses. The resulting DEM's have proven to be an accurate tool for either mapping surficial features, such as faults [START_REF] Kondo | A fault scarp in an urban area identified by LiDAR survey: A Case study on the Itoigawa-Shizuoka Tectonic Line, central Japan[END_REF], or quantifying volumetric changes in beach budget contributions [START_REF] Young | Application of airborne LiDAR for seacliff volumetric change and beach-sediment budget contributions[END_REF]. The use of airborne lasers to measure bathymetry and seabed topography, however, is considerably more complex and challenging than topographic mapping because of a variety of technical and environmental problems. The readers are referred to [START_REF] Flood | Commercial implications of topographic terrain mapping using scanning airborne laser radar[END_REF] for a general overview of airborne laser scanning technology and principles. Water heavily absorbs the near-infrared energy of terrestrial LiDAR, making it ineffective over streams. Consequently, airborne LiDAR bathymetry (ALB) must operate in the blue-green regions of the spectrum, and traditionally these systems have been designed for maximum water penetration using a relatively high-power green laser fired at fairly low repetition rates. ALB has been successfully applied to coastal areas for bathymetric programs [START_REF] Irish | Scanning laser mapping of the coastal zone: the SHOALS system[END_REF], and to river environmental studies [START_REF] Mckean | Remote sensing of channels and riparian zones with a narrow-beam aquatic-terrestrial LiDAR[END_REF].

For the Molene archipelago under study, ALB data were recorded in April/May 2010 over a 170 km 2 area, i.e. 60% of the total offshore studied area, encompassing emerged lands and the shallowest offshore areas, with an elevation/depth range between +32m down to -27m (Fig. 2). The LiDAR measurements were georeferenced into the RGF93 datum used by the Global Positioning System. After correction for the influence of tides, LiDAR data were

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exported into the Fledermaus software to produce ASCII grid files in 1 x1 km tiles, using a weighted moving average algorithm at 5m resolution with a weight diameter of 3.

Swath bathymetry surveys

In the present study, acoustic swath-mapping surveys were conducted in the Molene archipelago to complete the ALB dataset, using a multibeam echosounder in the deepest parts (-10 m down to -100 m), as well as an interferometric sidescan sonar in shallow water (Fig.

2).

A surface of 54 km 2 in water depth of <20 m was investigated using the hydrographic survey boat V/O Haliotis (Ifremer), equipped with a hull-mounted GeoSwath interferometric sidescan sonar (Table 1). The GeoSwath is a fully integrated sonar tool designed for high-resolution surveys. It is based on phase measuring (interferometry)

technology, which provides the advantages of a wide swath and high resolution in a compact and robust system, and is suitable for deployment in shallow waters where towed sidescan sonars have particular problems. A full description of GeoSwath system performance is given by [START_REF] Hiller | The Use of Phase Measuring (Interferometric) Sonars: Choosing Appropriate Data Processing Methodologies[END_REF].

Acoustic swath-mapping surveys were also conducted over the 97 km 2 outer part of the archipelago by the costal vessel N/O Thalia, equipped with a EM1000 hull-mounted multibeam echosounder (Fig. 2). The multibeam data acquired insonified 100% of the seafloor with at least a 20% overlap of the echosounding corridors. The Simrad EM1000 is an ultra-wide Swath sounder, suitable for surveying swaths of up to 7.5 times the water depth.

Since the 90's, it has been used routinely for seafloor geological mapping of the continental shelf [START_REF] Acosta | Sea floor morphology and Plio-Quaternary sedimentary cover of the Mallorca Channel, Balearic Islands, Western Mediterranean[END_REF].

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Bathymetric data from 2500 acoustic profiles were first processed for corrections (filtering, sound speed velocity, navigation, tidal) with the Caraibes software (© Ifremer), in order to elaborate a DEM displaying the same vertical referential as the LiDAR dataset, and to standardize the chart datum to lowest astronomical tides.

Seafloor rock sampling

A total of 52 seafloor rock samples have been collected in selected offshore areas (Figs. 2 and3), in order to constrain the structural interpretation of fabrics extracted from the bathymetric map of Fig. 3. Divers directly collected 15 rock samples at depths of < 20 m. The other rock samples (37) were acquired by gravity Cnexo-ville rock cores in deeper waters.

Though not oriented, in situ rock samples provide useful informations about strain conditions.

One major new insight is the recognition of a large-scale gabbro/diorite intrusive complex whose emplacement was constrained by U/Pb zircon dating.

Digital elevation model

For our study, each acquisition system results in a specific DEM with its own spatial resolution in the range 0.5 m (GeoSwath) -5m (LiDAR) (Table 1). Data were processed using the SonarScope software (© Ifremer), by the nearest-neighbour interpolation method, and then merged to provide a unique georeferenced raster grid with a spacing of 5 x 5 m (horizontal resolution), and a maximum vertical resolution of 0.5 m.

Focused views with the highest horizontal resolution (0.5 m) were locally acquired around specific structures. The resulting gridded DEM was imported into a geographic information system (GIS) for geomorphic analyses. Improved hillshaded imagery was produced in ESRI to enhance structural mapping. Hillshading is known to be subject to directional bias. To address this issue, two shaded relief images were first obtained by using illumination sources with two perpendicular azimuths (45° and 270°) and a constant tilt of 30°, and then superposed by transparency in order to enhance the trace of the total population of linear fabrics. The latter were then digitized manually from the bathymetric map.

Analysis of bathymetric and geological data

Geomorphic offshore patterns

With respect to the -10 m isobath, the Molene archipelago extends as a 10 x 20 km elongated plateau, with a NW-SE-trending elevated axial zone, outlined by six major islands (Fig. 3). It is dissected by an intricate network of NE-SW-, NW-SE-, and to a lesser extent, N-S-trending depressions, inferred to be the surficial expression of post-Variscan faults. These young brittle structures are not discussed in the present paper, with the exception of the La Helle fault used to restore the initial map-geometry of laterally offset basement structures.

Offshore geological mapping

The geological map of the Molene archipelago in Fig. 4 has been obtained by merging exposed geology with lithological data from seafloor rock samples (52), along with the analysis of bedrock surface roughness observed on the improved hillshaded offshore DEM (Fig. 3). Most of the metamorphic and plutonic seafloor rock samples collected offshore have onshore counterparts exposed in the islands and on the Leon mainland (Fig. 1b). Several interesting points arise from the examination of the Molene geological map. Those dealing with structural issues are discussed in the next section. Because of the intrusion of the St-

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the LMD antiform do not extend laterally offshore into the studied area.

-To the south, the N70°-trending Brest gneiss unit dies out rapidly westwards as the result of its obliquity with the EW map-trace of the 1-2 km-wide mylonitic zone following the PNSZ.

-To the north, the St-Renan granite is interdigitized with migmatites of the Plouarzel complex, west of the Molene island.

-Further north, both the leucogranitic (west) and the red/porphyric (east) facies of the Aber-Ildut granite overlap the highly-strained rocks of the St-Renan-Plouarzel granitoid complex.

-The Groac'h Zu intrusive complex, identified SW of Trielen island, also includes the diorite/gabbro rocks, locally exposed, in small islets around the intrusion (Chauris and Hallegouet, 1989). New radiometric dating has been performed on collected gabbro core-rock samples.

In situ U/Pb geochronology

Available U-Th/Pb dating of monazites yielded ages of 321 ± 5 Ma for the syntectonic emplacement of the St-Renan granite along the NASZ [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], and ages of 338 ± 5 Ma [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], and 315-305 Ma [START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF] for the amphibolite-type metamorphism recorded by the Le Conquet micaschists. In order to achieve a more complete time framework of the Variscan strain in the LMD, U-Th/Pb zircon/monazite dating has been performed, both for the offshore Groac'h Zu intrusion, and for coherency purposes, for the onshore part of the St-Renan granite.

Zircon and monazite U-Th/Pb geochronology was conducted in the Laboratoire Magmas & Volcans in Clermont-Ferrand (France) using LA-ICP-MS. The method used for dating is

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similar to those developed by [START_REF] Paquette | High resolution (5 µm) U-Th/Pb isotopes dating of monazite with excimer laser ablation (ELA)-ICPMS[END_REF]. Detailed analytical procedures and conditions used in this study can be found in [START_REF] Hurai | Age of deep crustal magmatic chambers in the intra-Carpathian back-arc basin inferred from LA-ICPMS U-Th/Pb dating of zircon and monazite from igneous xenoliths in alkali basalts[END_REF][START_REF] Hurai | U-Pb geochronology of zircons from fossiliferous sediments 1 of the Hajnáčka I maar (Slovakia)type locality of the MN16a biostratigraphic subzone[END_REF]. U, Th and Pb concentrations (available on-line as supplementary material) were calibrated relative to the certified GJ-1 zircon standard [START_REF] Jackson | The application of laser ablation-inductively coupled plasma-mass spectrometry to in situ U/Pb zircon geochronology[END_REF], and the Moacyr monazite standard [START_REF] Gasquet | Miocene to Messinian deformation and hydrothermalism in the Lauzière Massif (French Western Alps): New U-Th-Pb and Argon ages[END_REF]. Concordia ages and diagrams were generated using Isoplot/Ex v. 2.49 [START_REF] Ludwig | User's manual for Isoplot/Ex Version 2.49, a geochronological toolkit for Microsoft Excel[END_REF]. The zircon and monazite analytical results were projected on 207 Pb/ 206 Pb versus 238 U/ 206 Pb diagrams [START_REF] Tera | U-Th-Pb systematics in three Apollo 14 basalts and the problem of initial Pb in lunar rocks[END_REF]) and 206 Pb/ 238 U versus 208 Pb/ 232 Th diagrams, respectively.

The dated sample from the Groac'h Zu intrusion is a highly altered but unstrained gabbrodiorite (cv10 in Fig. 3). It is composed of large plagioclase and amphibole grains, as well as oxides and apatite. Biotites or pyroxenes are pseudomorphosed into chlorite. Small, rare quartz grains are also present. The zircon crystals are translucent, yellow, and subhedral, but appear partly broken. This hypidiomorphic shape of the zircon grains has already been observed in Variscan gabbros [START_REF] Gebauer | Isotopic systems-Geochronology of eclogites. In : Eclogite Facies Rocks[END_REF][START_REF] Paquette | Geochemical and isotopic study of a noriteeclogite transition in the European Variscan belt: implications for U/Pb zircon systematics in metabasic rocks[END_REF], and is merely due to the morphology of the free areas between the surrounding minerals, when zircon crystallizes as a late phase in the magma.

Two fresh samples from the St-Renan granite have been collected in a quarry near St Renan (Location in Fig. 1). Sample 13DB10 is a fine-grained granite with quartz, microcline, feldspars, and small grains of biotite and muscovite. Sample 13DB11 is more porphyric. It contains less quartz, a greater amount of biotite, and some large feldspars. Neither magmatic nor tectonic fabrics are present in these samples. Zircon crystals are translucent, colourless to light brown, euhedral, and short prismatic. Monazites are yellow to brown and also euhedral.

Morphologies of both mineral types are very similar and may be confused.

A set of thirty-four 33 µm-wide spots were analyzed by laser ablation coupled to ICP-MS on cv10 single zircon crystals (Fig. 5d). All the analyses are concordant, and yield an age of

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317.9 ± 2.0 Ma (Fig. 5) which is the crystallization age of the zircons during the emplacement of the gabbro-diorite. U content generally ranges between 100 to 300 ppm and Pb content is close to 10 ppm. The Th/U ratios of 0.3 are characteristic of magmatic zircons.

Twenty-seven 20 µm-wide laser spots were performed on the 13DB-10 and on eleven zircon crystals from the granite. Ten spots measured in zircon cores and one whole crystal display a large range of ages, mostly from 380-410 Ma (Fig. 5b) up to 980 Ma and 1.8 Ga (Fig. 5a and supplementary material). These inherited ages are related to the crust recycling during the granite petrogenesis. The remaining seventeen spots measured on whole-grains and outer rims are concordant in the U/Pb diagram, and yield an age of 316.0 ± 2.0 Ma (Fig. 5b). A set of twenty-seven 10 µm-wide laser spots was performed on the monazite crystals from the same St-Renan granitic samples. Twenty-one analyses, concordant in the U/Th/Pb diagram, define an age of 316.7 ± 1.5 Ma (Fig. 5c), i.e. similar to that obtained on the zircons, and precise the crystallisation age of the rock. The six remaining analyses are scattering between 330 and 360 Ma (Fig. 5c) and, similar to the zircons, recorded previous geological events of the protoliths.

These new results undoubtedly demonstrate that both gabbro and granite samples are contemporaneous at 316-318 Ma. Our dating of the St-Renan granite is compatible within error limits with the 321 ± 5 Ma age on monazites previously obtained by the EMPA technique [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF].

Magmatic/tectonic ductile 2D-pattern

Lineaments extracted from the high-resolution bathymetric map of The first type of linear lineaments (fabric 'a') is marked by a submeridian network of fault-like discontinuities, spatially confined within granitic intrusions. To the north, in the Alber-Idult granite (Fig. 3), they delineate a pattern of <100 m-wide elongated substratum blocks and intervening flat-bottomed sedimentary depressions (Fig. 6a1). They correlate onshore, in the Lampaul-Plouarzel area, with vertical cooling joints that likely formed at a late stage of granite emplacement (Figs. 3and 6a2).

More closely-spaced lineaments extend with a N120°E orientation NE of the Aber-Ildut granite (fabric 'b' on Fig. 6b1). Their onshore continuation to the SE corresponds to a composite planar structure, in a vertical position, involving a magmatic flow pattern overprinted by a later network of (1m-spaced) tectonic joints (Fig. 6b2). These late magmatic features are not discussed here. Conversely, emphasis is placed on more widelydistributed lineaments, occurring preferentially in the southern half of the offshore studied area, and among them three types of fabrics are recognized.

-Fabric 'c' is marked by a closely-spaced network of lineaments, extending with a dominant N70°E orientation, over a length of > 1 km (Figs. 4 and6c1). It represents the regional foliation, as indicated by the in situ micaschist rock sample cv8, and by the microstructure of rock sample cv29 that shows a typical gneissic fabric (Lesneven series) expressed by bands of elongated quartz-FK-plagioclase and biotite-muscovite assemblages (Fig. 3). At a larger scale, fabric 'c' lineaments pass laterally onshore into the foliated Le Conquet micaschists (Fig. 6c2).

-Fabric 'd' corresponds to sigmoidal lineaments, spatially restricted within narrow (1-2 km-wide) and elongated (>10 km-long) corridors that typically resemble ductile shear zones [START_REF] Ramsay | Strain variations in shear belts[END_REF] -Fabric 'e' is chiefly expressed to the southwest by sinuous and more widely-spaced lineaments, drawing a 5 km-wide fold-like structure, parallel to the metamorphic host-rock foliation, in the so-called Graoc'h Zu intrusive complex (Figs. 4 and6e1). The thin section of the corresponding granite seafloor rock sample (p11) shows the preferred linear orientation of most porphyroclasts (quartz, FK, plagioclase, micas), with no evidence of tectonic strain (Fig. 3).

Onshore kinematic dataset along the St-Mathieu-Corsen coastal section

Structural analysis of the metamorphic units continuously exposed along the St-Mathieu-Corsen coastal section (Fig. 7) has been performed in order to further constrain offshore/onshore structural correlations, and also to provide accurate kinematic criteria. The exposed metamorphic rocks, of Paleozoic age [START_REF] Cabanis | Remise en cause de l'existence d'un socle orthogneissique antécambrien dans le pays de Léon : étude géochronologique par les méthodes Rb/Sr et U/Pb des othogneiss de Tréglonou et Plounevez-Lochrist[END_REF], form a S/SE-dipping monoclinal structure, intruded to the north by unstrained facies of the St-Renan granitic

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massif (Fig. 7). These rocks have been deformed under amphibolite facies conditions that increase northwards, i.e. towards the structurally lower parts of the crustal pile, from the gneiss of Brest, the Le Conquet micaschists, down to the gneiss of Lesneven (the reader may refer to [START_REF] Jones | Phase relations in Al 2 SiO 5 polymorphs; Le Conquet region, north-western Brittany, France[END_REF][START_REF] Jones | Progressive metamorphism in a crustal-scale shear zone : An example from the Leon region, NW Brittany, France[END_REF][START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF], for a detailed description of the metamorphic history). Anomalous superposition of either metamorphic units (inverted grades, high gradeover lower grade-rocks), or structural units (abrupt strain gradients), which should have provided arguments for major tectonic contacts as postulated by [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], are missing along the overall transect under study [START_REF] Schulz | Cadomian and Variscan metamorphic events in the Leon domain (Armorican massif, France) : P-T data and EMP monazite dating[END_REF]. The presence of an oceanic suture zone, outlined by the metagabbro-amphibolite lenses within the Le Conquet micaschists [START_REF] Faure | Paleozoic tectonic evolution of Medio-Europa from the example of the French Massif Central and Massif Armoricain[END_REF]2008), is also unlikely since the ophiolitic nature of the metabasites is yet to be demonstrated [START_REF] Ballèvre | Paleozoic history of the Armorican Massif. Models for the tectonic evolution of the suture zones[END_REF][START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF].

Evidence for superimposed structures exists more frequently along the southern part of the section. An analysis of porphyroblast inclusions has been used to argue for a polyphased ductile strain history (5 distinct phases reported by [START_REF] Jones | Progressive metamorphism in a crustal-scale shear zone : An example from the Leon region, NW Brittany, France[END_REF]. However, most exposed strained rocks display a single foliation (S), generally associated with a shear plane (C) in porphyric/coarse-grained facies. The map-trace of the regional foliation in Fig. 7 strictly follows the offshore lineament pattern labeled 'c' in Fig. 6. The main goal of our structural field study is only to discriminate various styles of structures and strain within spatially distributed structural domains that might correlate laterally with the ductile fabrics identified on the offshore DEM of Fig. 3. Defining the precise strain/stress history of the LMD metamorphic rocks is beyond the scope of this paper.

Three distinct structural domains, with specific ductile deformation styles, are distinguished on the 10 km-long transect (Figs. 7 and8). Two of them roughly correspond to those previously described, involving top-to-the-NNE thrust-related shear [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], and dextral transcurrent shear [START_REF] Balé | Les complexes métamorphiques du Léon (NW Brittany) : un segment du domaine éohercynien sud-armoricain translaté au Dévonien[END_REF][START_REF] Goré | Cinématique hercynienne du cisaillement nord-armoricain à la bordure du granite syntectonique de Saint-Renan[END_REF]. Additional

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kinematic data are supplied by our detailed structural study that further enable us to identify a third type of structural domain, evoked by [START_REF] Jones | Progressive metamorphism in a crustal-scale shear zone : An example from the Leon region, NW Brittany, France[END_REF] as being dominated by transtensional shear in a spatially restricted zone.

The most relevant deformational features of each structural domain are briefly described, below.

-Domain (a) : Dominant top-to-the-NE thrust-related shear Domain (a) is here referred to as a 5 km-wide zone that nearly coincides with the Lesneven gneissic unit, exposed on the northernmost part of the transect (Figs. 7 and8b). The dominant outcrop-scale structure is a foliation plane (S), striking northeasterly, and dipping steeply to the southeast. An anomalous zone exists in the Brenterc'h area, where a pronounced shallowing of the foliation dip (down to ~25°) is accompanied by the counter-clockwise rotation of its trend toward the north (Fig. 7). That map-scale deflection evokes a postfoliation fold compatible with sinistral shearing. A stretching lineation (L) locally lies within S as elongated quartz-feldspar assemblages. It displays a wide range of steeply plunging attitudes, generally >45°, and consistently toward the SW (Fig. 8a). On the XZ planes, orthogonal to S and parallel to L, S is locally associated with a shear plane (C) (Fig. 8b). The resulting composite S/C planar fabric systematically shows a top-to-the-NE sense of shear. If assuming, to a first approximation, that L indicates the transport or shear direction of deformation, it follows that the NE-directed displacement is oblique, and combines a prominent top-to-the-N reverse component (Fig. 8c1), in addition to sinistral ductile shearing (Fig. 8b). A similar sense of (post-foliation) shear is deduced from the above-mentioned mapscale fold.

-Domain (b) : dextral ductile shearing

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Two discrete zones typically show structural features of pure strike-slip ductile shear at the two extremities of the transect (Fig. 8b). To the north, tourmaline-bearing facies of the St-Renan granite are highly deformed along a 1.5 km-wide zone, occurring 1-2 km north of its magmatic contact with the foliated Lesneven gneiss (see above). That is the coastal section of the NASZ in which the main fabric corresponds to subvertical anastomosing foliation (S) and shear (C) planes, striking E-W on average (Fig. 7). On the S-surface, a well-developed stretching lineation is outlined by elongated K-feldspars in a sub-horizontal position (e.g.

Goré and Le Corre, 1987, for details). When observed in XZ sections, the S/C fabric indicates a rotational deformation, as well as a dextral sense of shear (Fig. 8b).

To the south, orthogneisses and a subordinate metasedimentary sequence of quartzitic layers and metapelites of the Brest unit both occur from a vertical to a southerly-dipping position within a ~1 km-wide zone extending from St Mathieu to Penzer (Figs. 7 and8b). Within the so-called PNSZ, the orthogneisses are penetratively deformed by a composite S/C planar fabric, in a vertical position, striking N80°E (Figs. 7 and8b). The preferred elongation of quartz-feldspar assemblages on the S-planes outlines a sub-horizontal to shallowly-plunging stretching lineation (Fig. 8c6). Sigmoid-shaped K-feldspar and quartz phenocrysts observed in XZ sections indicate dextral shear (Fig. 8c5). A similar sense of shear is also recorded by dextrally folded mylonitic bands. Ductile dextral shearing is expressed in another way in the vertical quartzite-metapelite sequences by asymmetrical and steeply-plunging isoclinal folds displaying an axial plane parallel to the N80°-oriented schistosity of the metapelites. Further north, criteria for dextral shearing still exist, but in moderately-inclined gneisses (Brest unit) (Figs. 8a,b).

-Domain (c) : polyphase and extensional ductile strain

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The third structural domain involves most of the Le Conquet micaschists unit exposed from Penzer to Portez beach (Fig. 8b). With respect to the two other structural domains, this 2.5 km-long intermediate domain is demarcated by a more shallow (southerly) dip of the foliation, generally <45°S, and a conspicuous stretching lineation (L), with variable plunging azimuths (Fig. 8a). L is defined by various markers, including quartz pressure-shadows on garnet porphyroblasts, and corrugated and/or boudinaged quartz layers (Fig. 8c4). In many places, the strong development of the lineation results in a S-L-type tectonic fabric in the Le Conquet micaschists (Porz Liogan in Fig. 8b). The diagram of Fig. 8a shows its wide range of plunge within S, but with an asymmetrical distribution to the SW. The prominent southwesterly-plunging lineation, generally outlined by either sigmoidal K-feldspars in gneissic facies, or by tails and pressure-shadows on rotated garnets in micaschists (Fig. 8c3), is systematically associated with a component of dextral shear. That resulted in a bulk SWdirected trans-tensional displacement on the S surfaces (Fig. 8b), already evoked by [START_REF] Jones | Progressive metamorphism in a crustal-scale shear zone : An example from the Leon region, NW Brittany, France[END_REF]. In the northern part of domain (c), from Porz Liogan to Portez (Fig. 7), SW-directed extensional shearing coexists with top-to-the-N reverse shear which is recorded by rotated inclusions (S1) in garnets [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF].

Since the two opposite shear strains both display a nearly parallel NE-SW transport lineation, evidence for superimposed patterns are not easily documented within domain (c).

Nevertheless, a relative timing framework can be reasonably deduced from the following evidence. More convincing structures are supplied by the interference pattern in Fig. 8c2. It shows an early foliation, affected by N-verging asymmetrical folds, both being dissected by a dense network of top-to-the-S extensional shear bands. Porphyroblast/foliation relationships in garnet-bearing micaschists also provide additional supportive evidence (Fig. 8c3). The early foliation, preserved as inclusions in rotated garnets, is tentatively regarded as remnants

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of top-NE reverse shear-related structures [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], later overprinted by the regional foliation and its spatially associated SW-directed extensional shear planes.

Regional ductile strain framework

The structural sketch map in Fig. 9 combines both the offshore lineament trajectory map of Fig. 4, obtained by extrapolating the trace of various ductile fabrics, and structural and kinematic data from the SW Leon coastal section. Three major types of magmatic/tectonic ductile structures are clearly identified : the internal fabric of the Groac'h Zu intrusion, the regional foliation, and a number of discrete shear zones. Their mutual interference patterns, in addition to newly acquired radiometric ages of intrusions, supply accurate temporal constraints for the kinematic model proposed in the last paragraph.

The more widely distributed fabric is the regional foliation which coincides accordingly with the offshore continuation of the LMD metamorphic terrains to the SE (Fig. 9). The offshore foliation pattern is dissected by six shear zones, with C/S internal fabrics that indicate a dextral sense of shear for five of them, the so-called Fromveur, North Molene, Groach Zu, Beniguet, and Pierres Noires shear zones (FSZ, NMSZ, GZSZ, BSZ and PNSZ, respectively). The only sinistral shear zone, i.e. the South Molène shear zone (SMSZ), is cut to the east, and post-dated by the St-Renan granite (Fig. 4). The SMSZ is thus assumed to be linked to top-to-the-NE shear strain that also comprises a sinistral component onshore, whilst being post-dated by the St-Renan granite.

Concerning the prominent dextral shear zone pattern, offshore/onshore correlations can be readily constructed about the EW-trending PNSZ that extends continuously throughout the coastline (Figs. 4 and9). On the contrary, the offshore extent of the NASZ is more problematic, being erased by Quaternary deposits, and further dextrally offset by the La Helle fault (Fig. 4). Among the two possible candidates, i.e. the N70°E-oriented FSZ and NMSZ,

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the latter is the more appropriate structure, given (1) its ~ 4 km apparent dextral offset with regards to the onshore NASZ, i.e. similar to the offset of the reference Aber Ildut granite on both sides of the La Helle fault (Fig. 4), and (2) the nearly similar 10 km map-distance between the NMSZ and NOSZ (offshore), and between the NASZ and PGSZ (onshore), keeping in mind that the NOSZ is the western counterpart of the GPSZ (Fig. 1b).

Further south, complex interactions exist between the GZSZ and BSZ dextral shear zones, the regional foliation, and the Groac'h Zu intrusion (Figs. 9 and 10). The BSZ appears to merge southwards onto the EW-oriented PNSZ, whereas to the NE it transects the EW trace of the regional foliation (Figs. 10b,e). To the north, the GZSZ forms the southern tectonic boundary of the Groac'h Zu intrusive complex, cutting through its concentric internal fabric that however tends to merge into the map-trace of the shear zone (Fig. 10b). Given the unstrained character of the cv10 (gabbro) and p11 (granite) in situ seafloor rock samples (see above), the internal fabric of the Groac'h Zu intrusive complex is assumed to be a magmatic feature (compositional layering or magmatic flow).

Between the two shear zones, the foliation draws large-scale asymmetrical folds, kinematically consistent with dextral shearing (Fig. 10d). Beyond the NE tip zone of the two shear zones, the foliation is severely disrupted and folded along a submeridian network of sinistral faults (Fig. 10c). To the NW, the foliation shows a map-scale deflection around the northern elliptic trace of the intrusion boundary.

Discussion and conclusions

Tectonic event chronology

The two successive episodes of ductile strain previously identified on the southern flank of the regional-scale LMD antiform [START_REF] Balé | Les complexes métamorphiques du Léon (NW Brittany) : un segment du domaine éohercynien sud-armoricain translaté au Dévonien[END_REF][START_REF] Rolet | L'évolution géodynamique dévono-carbonifère de l'extrémité occidentale de la chaîne hercynienne d'Europe sur le transect Armorique-Cornwall[END_REF] are

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confirmed in the present work. However, our onshore/offshore structural approach allows us to accurately detail both the kinematics (onshore) and spatial extent (offshore) of their respective deformations.

A first episode of top-to-the-NE transpressional ductile shear, associated with a sinistral shear component, resulted in the development of the N70-80°E regional foliation. To a first approximation, this early strain pattern is kinematically compatible with a maximal principal stress axis in the NE quadrant. The N70°E sinistral SMSZ identified in the offshore lineament trajectory map is assumed to be a map-scale expression of the sinistral shear component documented onshore. Both the foliation (onshore) and the SMSZ are cut and thus post-dated by the 316 ± 2 Ma-old St-Renan granite.

Evidence for a second episode of transtensional ductile tectonics is found, both onshore, in spatially restricted areas, and offshore in five dextral shear zones. Onshore, the deformation is partitioned into discrete dextral strike-slip shear zones, and top-to-the-SW transtensional shear zones. In terms of regional stress conditions, the dextral transtensional strain pattern likely developed in response to the counter-clockwise rotation of the maximal principal stress axis in the NW quadrant. During this episode, magmatic intrusions took place in close connection with dextral shear zones. The synchroneity of the St-Renan granite (dated here at ~316 Ma) and the NASZ has long been advocated onshore [START_REF] Goré | Cinématique hercynienne du cisaillement nord-armoricain à la bordure du granite syntectonique de Saint-Renan[END_REF].

Syntectonic emplacement of the coeval (~318 Ma) Groac'h Zu intrusion along the dextral

GZSZ is also supported here by the fact that its internal magmatic fabric tends to merge into the southern sheared boundary of the intrusion. Additional supportive evidence is observed immediately east of the intrusion, where the foliated structure of the country-rocks is highly disturbed by a submeridian network of sinistral faults. By analogy with results of experimental modeling of a rigid rotated body within a viscous matrix during simple shear Along with our onshore structural data (Figs. 7 and 8), the mutual relationships between the Groac'h Zu intrusion, the shear zones, and the regional foliation supply additional constraints for the temporal development of the two successive strain events. Indeed, the emplacement of the Groac'h Zu intrusion by progressive ballooning in previously strained terrains (first shear episode) might have caused, concomitantly with dextral ductile shearing, later deformation (folds) in the foliated country-rocks around the southern intrusion boundary (Fig. 10).

The third shearing event that occurred at ~300 Ma further north along the N70°E sinistral PGSZ [START_REF] Le Corre | Le Léon : un domaine exotique et géophysique au NO de la chaîne varisque armoricaine (France)[END_REF] has no structural expression in our study area. Both its kinematics and trend likely resulted from the back-rotation of the stress axes.

With regards to the numerous published structural, metamorphic, and radiometric data (e.g, [START_REF] Paquette | Géochronologie et géochimie des éclogites du Léon: nouvelles contraintes sur l'évolution géodynamique du Nord-Ouest du Massif armoricain[END_REF][START_REF] Le Corre | Le Léon : un domaine exotique et géophysique au NO de la chaîne varisque armoricaine (France)[END_REF][START_REF] Jones | Progressive metamorphism in a crustal-scale shear zone : An example from the Leon region, NW Brittany, France[END_REF][START_REF] Godard | Peraluminous sapphirine formed during retrogression of a kyanite-bearing eclogite from Pays de Léon, Armorican Massif, France[END_REF][START_REF] Schulz | Cadomian and Variscan metamorphic events in the Leon domain (Armorican massif, France) : P-T data and EMP monazite dating[END_REF][START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF][START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF], we find it difficult to correlate the tectonic events argued here with the metamorphic episodes reported by previous authors.

According to [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF], an early HP metamorphic event and a second MP-MT event are coeval with top-to-the-N thrusting at <340 Ma, whereas a third MP-HT event operated synchronously with younger dextral shearing at 335-327 Ma. The two metamorphic episodes of [START_REF] Schulz | Cadomian and Variscan metamorphic events in the Leon domain (Armorican massif, France) : P-T data and EMP monazite dating[END_REF] and [START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF] consist of a first HP event related to crustal thrusting at 360-340Ma, and a second MP-MT to -HT event at 340-300 Ma within a dextral transtensional setting.

Consequently, the N-verging thrust episode of [START_REF] Faure | The Léon domain (French Massif armoricain): a westward extension of the Mid-German Crystalline Rise? Structural and geochronological insights[END_REF] roughly correlates with our first top-to-the-NE shear event, and its associated age, around 340 Ma, is in accordance with our timing (older than 320 Ma). The second top-to-the-SW extensional shear

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event emphasized here has not been observed by these authors. On the other hand, transtensional tectonics is suggested by [START_REF] Schulz | Cadomian and Variscan metamorphic events in the Leon domain (Armorican massif, France) : P-T data and EMP monazite dating[END_REF] and [START_REF] Schulz | Monazite EMP-Th-U/Pb age pattern in Variscan metamorphic units in the Armorican Massif (Brittany, France)[END_REF], but his corresponding time-range (340-300 Ma) is much larger than the robust U/Pb ages acquired in the present study.

Dextral shear zone pattern

One of the new insights from our structural work is the identification, in the offshore extent of the LMD, of a much more intricate dextral shear zone pattern than those previously reported from onshore data that includes only part of the NASZ and PNSZ (Fig. 1b). The overall map arrangement of the offshore/onshore dextral shear zone pattern, once completed to the SE by the Elorn Shear Zone (ESZ), shows a linked system composed of two major EWstriking segments, i.e. the NASZ and PNSZ, that connect via a 10 x 80 km overlapping zone (Fig. 11). This relay zone comprises an array of second-order right-lateral shear zones, striking N70°E, i.e. the ESZ, BSZ, GSSZ and FSZ. According to the terminology of Woodcock and Fisher (1986), the overall system mimics a strike-slip duplex compatible with a relay zone in an EW-trending dextral system. The resulting stretching direction should be NE-SW, i.e. nearly parallel to the mean mineral lineation related to our transtensional tectonic event onshore. The 80 km-long relay zone identified here is thus a new major structural feature in the dextral transcurrent network that dissected the Armorican Variscides during the Late Carboniferous (Fig. 11). One implication of our findings is that it rules out the 'transform' model of [START_REF] Balé | Les complexes métamorphiques du Léon (NW Brittany) : un segment du domaine éohercynien sud-armoricain translaté au Dévonien[END_REF] that implied the extent of the ESZ beyond the PNSZ, as far south as the South Armorican domain.

The structural sketch in Fig. 11 along the NASZ and PNSZ, with their long axis oblique to the shear zones (Fig. 11). Similar geometries have long been considered as criteria for the synchronism of granitisation and dextral shearing along, for example, the eastern course of the NASZ [START_REF] Guillet | Le complexe granitique de Plouaret (Bretagne) : mise en évidence structurale et gravimétrique de diapirs emboîtés[END_REF], or the SASZ (Hanmer and Vigneresse, 1980). Furthermore, the close spatial association of the intrusions and dextral shear zones suggests strong interactions between pluton-induced thermal weakening of the crust and the location of discrete crustal-scale shearing.

One other important issue concerns the structural significance of the transtensional tectonic regime documented onshore. According to [START_REF] Jones | Progressive metamorphism in a crustal-scale shear zone : An example from the Leon region, NW Brittany, France[END_REF] and Schultz ( 2007), it might have accompanied the progressive exhumation of the entire LMD metamorphic terrains during a HT event. However, given the restricted spatial distribution of these structures, further fieldwork is really needed to assess whether the transtensional tectonic regime strictly resulted from either the relay zone kinematics, or a crustal-sale exhumation process during late orogenic collapse. From this point of view, the origin of the Leon antiform as a compressive structure [START_REF] Rolet | L'évolution géodynamique dévono-carbonifère de l'extrémité occidentale de la chaîne hercynienne d'Europe sur le transect Armorique-Cornwall[END_REF]) can be also questioned, and possibly substituted by a 'metamorphic core complex' hypothesis. In this way, fruitful comparisons could be attempted with the nearly contemporaneous extensional systems that also characterized the South Armorican Variscan belt [START_REF] Gapais | La zone de cisaillement de Quiberon : témoin d'extension de la chaîne varisque en Bretagne méridionale au Carbonifère[END_REF][START_REF] Cagnard | Late pervasive crustal-scale extension in the south Armorican Hercynian belt (Vendée, France)[END_REF][START_REF] Turrillot | The top-to-the-southeast Sarzeau shear zone and its place in the late-orogenic extensional tectonics of southern Armorica[END_REF][START_REF] Turrillot | Late orogenic exhumation of the Variscan high-grade units (South Armorican Domain, western France), combined structural and 40Ar/39Ar constraints[END_REF]. zone. S, regional schistosity. d. Asymmetric folds and coeval shear planes post-dating the regional schistosity. e. Cross-cutting relationships between the regional foliation (early) and later shear surfaces in the BSZ. and Le Gall, 1986, modified). Abbreviations as in Figs. 1 and9.

  Fig. 3 are assumed to represent the trace of regional-scale structures in the basement seafloor. Five distinct types of linear fabrics (labeled 'a to e') are identified from their specific physical attributes (length, spacing, spatial distribution, sinuosity, and cross-cutting relationships). Their structural interpretation is based on evidence from seafloor rock samples, structural map features, lateral A C C E P T E D M A N U S C R I P T onshore correlations, and kinematic criteria supplied by the St-Mathieu-Corsen reference coastal section (next paragraph). Most of the offshore area under study is dissected by a dense network of closely-spaced and short lineaments, confidently regarded as the map-trace ofVariscan penetrative ductile structures (Fig.4). These key features, extensively discussed in the paper, are cut by a discrete network of fault-like structures, probably late-to post-Variscan in age, and are not addressed here.

  (Figs. 4 and 6d). The obliquity of the internal C/S-type patterns indicates dextral or sinistral shearing(Figs. 6d1a, d1b). C/S-type fabrics are also observed as micro-scale structures in offshore in-situ samples showing thin bands of asymmetrical quartz and K-feldspar porphyroclasts (S), further dissected by oblique shear planes (C) (samples cv2, Pierres Noires granite; cv24, Plouarzel migmatite; cv29, Lesneven gneiss; Fig.3). Shear criteria are also documented onshore, as dextral C/S ductile fabrics in high-strained St-Renan granitic facies involved in the NASZ (Figs.3 and 6d2b), or as sinistral C/S fabrics in Lesneven orthogneisses (Figs.3 and 6d2a).

(

  [START_REF] Van Den Driessche | Tectonic evolution of the Montagne Noire (French Massif Central) : a model of extensional gneiss dome[END_REF], these structures are assigned to the clockwise rotation of the pluton at a cooling stage, during progressive dextral shearing.

  also emphasizes the spatial association of the ~ coeval St-Renan and Groac'h Zu intrusions with the NASZ and GZSZ/PNSZ, respectively. In detail, we also observe the asymmetrical drop-shaped map-trace of the various magmatic intrusions A C C E P T E D M A N U S C R I P T

Figure 1 .

 1 Figure 1. Geological features of the Leon metamorphic domain (LMD). a. Location of the

Figure 2 .

 2 Figure 2. Location map of geophysical records and seafloor rock sampling in the Molene

Figure 3 .Figure 4 .

 34 Figure 3. 5 m-resolution bathymetric map of the Molene archipelago obtained from merging

Figure 5 .

 5 Figure 5. Concordia diagrams of the dated igneous rocks: a. and b. Tera-Wasserburg plots of

Figure 6 .

 6 Figure 6. Shaded relief images of fabrics extracted from the digital elevation model of the

Figure 7 .Figure 8 .

 78 Figure 7. Detailed structural and geological map of the SW Léon coastal zone from the

Figure 9 .

 9 Figure 9. Foliation trajectory map (including magmatic fabrics and ductile shear zones)

Figure 10 .

 10 Figure 10. Focused views of the 5m-resolution alti-bathymetric map at the eastern extremity

Figure 11 .

 11 Figure 11. Kinematic model of the southern part of the LMD during the ~320 Ma dextral

Table 1 :

 1 Characteristics of the remote sensing dataset used to produce a single DEM of the studied offshore area.

ACCEPTED MANUSCRIPT

Acknowledgments

The Parc Naturel Marin d'Iroise (Philippe Le Nilliot) and the SHOM are acknowledged for permitting publication of the LiDAR and echosounder offshore data.

Petrological correlations with onshore granites has greatly benefitted from the expertise of Martial Caroff (Brest university). Many thanks to Jean-Pierre Oldra for making the thin sections. This work benefited from a State Grant (A.A.) from the French Agence Nationale de la Recherche in the Program ANR-10-LABX-19-01, Labex Mer, and from Swiss National Science Foundation (SNSF) grant no. PA00P2-145309 (DB). We thank Romain Augier and Denis Gapais who helped improve this manuscript.


Fang Feng

Philippe Morin

Yanne K Chembo

Alexej Sysoliatin

Stefan Wabnitz

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Experimental demonstration of spectral sideband splitting in strongly dispersion oscillating fibers

Keywords: mixing, (190.4410) Nonlinear optics, parametric process

By using a highly nonlinear, dispersion oscillating optical fiber operating in the telecom C band, we experimentally demonstrate the splitting experienced by quasi-phase matched gain sidebands in the strongly dispersion managed regime of a dispersion oscillating fiber as the power of a continuous-wave pump laser is increased over a certain threshold value. Excellent agreement is found between the theoretical predictions and our experimental measurements.

Modulation instability (MI) is an ubiquitous nonlinear process that has been widely investigated in various fields of physics and applications including plasmas, hydrodynamics and optics, to cite a few. In the presence of a high power continuous wave (CW), MI leads to the emergence and amplification of gain sidebands in the wave spectrum. In nonlinear fiber optics, such a process has been demonstrated in fibers with anomalous, constant group velocity dispersion (GVD) [1], as well as in normal GVD fibers by enabling the fulfillment of the nonlinear phase-matching condition through either fourth-order dispersion [2], birefringence or a multimodal structure [3,4]. More recently, a renewed experimental and theoretical interest in MI studies has been stimulated by the availability of fibers with a longitudinal and periodic modulation of their dispersion properties [5]. Indeed, thanks to the periodic dispersion landscape, which leads to quasiphase-matching (QPM) of the nonlinear four-wave mixing (FWM) process, MI sidebands can be observed even in the regime of normal average GVD of a dispersion-oscillating optical fiber (DOF) [6][7][8]. Recent experimental studies have indeed confirmed the QPM-induced MI process in the normal GVD regime of a microstructure DOF around 1 m [5], as well as of conventional highly nonlinear DOF at telecom wavelengths [9,10].

MI induced by the longitudinal variations of chromatic dispersion has been theoretically investigated earlier for a wide range of configurations, ranging from fibers with a sinusoidal dispersion profile and a spatial period of a few tens of meters [5,9], up to dispersion-managed optical transmission fiber spans with periods of several kilometers [6,11,12]. Whenever the dispersion fluctuations can be considered as a perturbation to nonlinear wave propagation, that is when their amplitude is smaller or comparable with the value of the average dispersion, QPM-induced MI leads to the emergence of well-separated, and unequally spaced gain sidebands, symmetrically placed around the pump. On the other hand, it has been recently theoretically shown that when the amplitude of dispersion fluctuation grows much larger than the average GVD (strong dispersion management regime), a spectral splitting process may occur in the sideband spectrum [13].

In this letter we report the first experimental validation of this prediction, by using a highly nonlinear DOF pumped in the telecom C band. As a basis for our discussion, let us first recall the expected evolution of QPM sidebands as the pump power is increased. For that purpose, we consider a continuous wave (CW) pump and make use of both a Floquet linear stability analysis (LSA) as well as of the existing analytical laws. Next, we will describe the implemented experimental setup and the corresponding experimental results which demonstrate the transition from the regime of usual QPM-induced MI to the novel regime of MI which exhibits spectral sideband splitting. Finally, we compare our experimental results with the full numerical simulations of the nonlinear Schrödinger equation (NLSE).

The evolution of the optical field in an optical fiber can be described by the NLSE :

2 2 2 | | 0 2 () i z t z            (1)
where  is the complex electrical field, z is the propagation distance and t the retarded time. The NLSE includes both the Kerr nonlinearity  and the second-order dispersion 2, which is supposed to periodically evolve in the longitudinal direction according to the sinusoidal law

  2 22 ( ) sin 2 / av amp zz       ( 2
)
where 2av is the average dispersion of the fiber, 2amp is half of the peak-to-peak amplitude of the dispersion variation, and  is the spatial period of the fluctuations.

In the presence of relatively weak amplitude sinusoidal longitudinal GVD variations, QPM of the FWM (or MI) process leads to the appearance of resonant gain sidebands, whose angular frequency shift relative to the pump can be analytically predicted as follows [6] :

where Jp is the Bessel function of order p.

In order to illustrate the process of spectral splitting, we consider here the same optical fiber that was previously employed in refs. [9,10]. This 400-m long highly nonlinear fiber has a nonlinear coefficient of 10 W -1 km -1 . The amplitude of dispersion fluctuations 2amp is estimated to be 2 ps 2 /km, with a spatial period =20 m. By changing the pumping wavelength, it is possible to access different values of the average dispersion 2av. Therefore, we have compared spectra which are obtained for the pumping wavelengths of 1536 and 1554 nm. The corresponding average dispersion values 2av are 0.7 ps 2 /km and 0.13 ps 2 /km, respectively, so that the ratio 2amp / 2av is equal to 2.8 and 15.4 in the two cases. Let us consider the resulting MI gain spectra for a CW pump power ranging in the interval between 0 and 11 W.

Equations ( 3) and ( 4) do provide accurate predictions of the value of the sideband gain which is expected at the resonant frequencies p. However these equations do not give us an insight on the actual shape of the gain sideband around those frequencies. Using the LSA of the CW pump by means of the Floquet's approach has been shown to be a very powerful method to calculate the details of MI gain sidebands for optical fibers with spatially periodic parameters [13,15,16]. In particular the Floquet LSA enables to correctly describe the spectral sideband splitting effect. Therefore we may take advantage of this method to plot the theoretically expected gain sideband for the two previously mentioned pumping wavelengths. The corresponding results, showing the impact of the pump power on the MI gain spectrum, are summarized in Fig. 1. For the pump wavelength of 1536 nm (panel a), the two main sidebands are provided by the 1 st and 2 nd QPM conditions. They experience a continuous growth and their position is in excellent agreement with Eq. ( 3).

The behavior of the MI spectrum is radically different for a high ratios 2amp/2av>>1, as illustrated by panel b of Fig. 1.

Indeed, in this configuration the MI gain which is predicted by Eq. ( 4) exhibits a non-monotonic evolution with pump power.

More importantly, for some values of pump power the gain gp vanishes. Consequently, several branches of the 1 st QPM sideband can be distinguished (four in our case, with frequency offsets around 6.8, 5.8, 4.7 and 3 THz, respectively) in panel b of Fig. 1: hence the spectral splitting phenomenon is observed. By comparing panels a and b of Fig. 1 we may note that the observation of relatively sideband gains (i.e., above 150 dBs) requires higher pump power values in the strongly dispersion oscillating regime. Our experimental setup for the observation of MI sideband splitting is depicted on Fig. 2(a). A CW pump delivered by a wavelength tunable external cavity laser is phase modulated at a low frequency (typically 100 MHz), in order to avoid any detrimental Brillouin backscattering upon propagation in the fiber. Next an intensity modulator is used to temporally slice into the CW pump. This modulator is driven by a deterministic non-return-to-zero sequence of a single one followed by 31 zeros repeating at a frequency of 4 GHz. This enables to achieve, for a given average power, a more than twenty-fold increase of the peak power of the resulting long pulse. The resulting pulses repeat every 8 ns, have a 250 ps duration and an intensity profile close to a super-Gaussian shape as outlined on Fig. 2(b). A high power erbiumdoped fiber amplifier (EDFA) is then used to obtain an average power up to 33 dBm. A narrow optical bandpass filter (OBPF) is inserted to limit the build-up of amplified spontaneous emission (ASE), and to adjust the power level at the DOF input.

The highly nonlinear DOF under test was designed and fabricated to have a zero dispersion wavelength (ZDW) in the telecom C band (ZDW = 1556 nm). The fiber preform was prepared by the MCVD process using mixtures of The values estimated at wavelengths 1536 and 1554 nm corresponds to the values used in the previous numerical simulations. At the output of the fiber, a 90/10 coupler is used to simultaneously monitor the output spectrum on an optical spectrum analyzer (OSA) and the output power on a power meter (Pow).

Let us first consider the MI which is induced by a pump centered at 1536 nm. The corresponding results are plotted on Fig. 3(a) for two different power levels. For a power of 3.5 W, we clearly observe the emergence of the 1 st QPM sideband (grey curve). The small side-lobe in the vicinity of the pump at lower frequencies is due to residual ASE. For relatively high pump powers (i.e., 7 W, see the black solid curve), the intensity of the 1 st QPM sideband is largely increased, and its central frequency is shifted towards lower frequencies, in agreement with Eq. (3). In this case we also observe in Fig. 3(a) the emergence of the 2 nd QPM sideband, as well as the generation of a set of equally spaced, widely shifted sidebands that originate from the cascaded or multiple four wave mixing (MFWM) process involving the pump and the 1 st QPM sideband [9].

A more systematic study of the influence of the input pump power on the output MI spectrum is presented on Fig. 4(a). The experimental results obtained by the averaging of 36 measurements confirm the generation and growth of the 1 st and 2 nd QPM sidebands. Their evolution as a function of pump power is in qualitative agreement with the Floquet LSA results presented Fig. 1(a), except for the emergence of MFWM sidebands. For pump powers higher than 10 W, the different sidebands merge and a continuum is formed, with a noticeable spectral asymmetry owing to the Raman effect The overall spectral behavior is consistent with our previous measurements carried out with a ns Q-switched laser emitting at 1534 nm [9].

The picture is very different when dealing with a pump whose wavelength is located in the vicinity of the ZDW (see Fig. 3(b)), hence a more careful analysis is required in this case. In agreement with the theoretical prediction of Fig. 1, higher pump powers are required to reach a level that can be detected on the OSA. The results obtained for a pump power of 7W reveal the emergence of three distinct sidebands, which result from the splitting of the 1 st QPM sideband. The most powerful branch 3 is shifted by 5 THz from the pump. When increasing the pump power to 9 W, the branch 4 which is shifted by nearly 3 THz grows the largest, in agreement with the Floquet analysis. Once again, FWM-induced sidebands are also observed and they overlap branch 2 (shifted by 6 THz) of the 1 st QPM sideband. The experimental signature of MI sideband splitting was clearly confirmed by systematic measurements of the MI spectrum evolution (24 measurements are summarized on Fig. 4(b)). Here the emergence and growth of the 2 nd QPM sideband is also be observed, along with a broad Raman peak at -13 THz. For pump powers higher than 10 W, a continuum is formed.

In order to further validate our experimental results as well as the analysis of the sideband splitting process, we carried out simulations of the propagation of the pulsed pump in the DOF. For this purpose, we used the extended NLSE including the third order dispersion term (4.7 10 -5 ps 3 /m) and the delayed Raman response of silica. Given their potential impact in the process of spectral splitting, we also included linear fiber loss (0.7 dB/km) [13] and the pump pulse shape [17]. Simulation results are summarized in Fig. 5, and reproduce nearly all of experimentally observed features which were not predicted by the LSA of Fig. 1.

In conclusion, we presented the first experimental demonstration of the sideband spectral splitting effect in strongly dispersion oscillating fibers. The Floquet-based LSA was completed by pulse propagation simulations.

  1, 2, 3 … More recently, it has been shown that the gain gp experienced by the p th sidebands may also be analytically predicted by the formula[14] :

Fig. 1 .

 1 Fig. 1. Evolution of the output gain spectrum according to the input power for an average dispersion of 0.7 ps 2 /km (a) and 0.13 ps 2 /km (b). The corresponding pumping wavelengths are 1536 and 1554 nm. Results are obtained from the linear stability analysis based on the Floquet theory. The predictions from Eq. (3) are plotted with blue line.

Fig. 2 .

 2 Fig.2. (a) Experimental setup. Details of the pump pulse (b) and of the fiber dispersive properties (c).

4 SiF

 4 in order to obtain the necessary refractive index profile.

Fig. 2 (

 2 Fig.2(c) shows the measured values of the average dispersion of the fiber as a function of wavelength. The values estimated at wavelengths 1536 and 1554 nm corresponds to the values used in the previous numerical simulations. At the output of the fiber, a 90/10 coupler is used to simultaneously monitor the output spectrum on an optical spectrum analyzer (OSA) and the output power on a power meter (Pow).Let us first consider the MI which is induced by a pump centered at 1536 nm. The corresponding results are plotted on Fig.3(a) for two different power levels. For a power of 3.5 W, we clearly observe the emergence of the 1 st QPM sideband (grey curve). The small side-lobe in the vicinity of the pump at lower frequencies is due to residual ASE. For relatively high pump powers (i.e., 7 W, see the black solid curve), the intensity of the 1 st QPM sideband is largely increased, and its central frequency is shifted towards lower frequencies, in agreement with Eq. (3). In this case we also observe in Fig.3(a) the emergence of the 2 nd QPM sideband, as well as the generation of a set of equally spaced, widely shifted sidebands that originate from the cascaded or multiple four wave mixing (MFWM) process involving the pump and the 1 st QPM sideband[9].A more systematic study of the influence of the input pump power on the output MI spectrum is presented on Fig.4(a).

Fig. 3 .

 3 Fig. 3. (a) Experimental output spectrum for pump wavelength of 1536 nm for pump powers of 3.5 and 7 W (grey and solid black lines respectively). (b) Experimental output spectrum for pump wavelength of 1554 nm for pump powers of 7 and 9 W (grey and solid black lines respectively).

Fig. 4 .

 4 Fig. 4. Evolution of the output spectrum according to the input power recorded experimentally for a pump wavelength of (a) 1536 nm and (b) 1554 nm. Results are plotted using a 40 dB dynamics.

Fig. 5 .

 5 Fig. 5. Evolution of the output spectrum according to the power for a pump wavelength of (a) 1536 nm and (b) 1554 nm.. Results obtained from the numerical integration of the extended NLSE. Results are averaged over 36 simulations.

We thank Julien Fatome and Kamal Hammani for stimulating discussions. We acknowledge financial support of the Conseil Regional de Bourgogne (Pari Photcom), funding of the Labex ACTION program (ANR-11-LABX-01-01), the Italian Ministry of University and Research (grant no. 2012BFNWZ2), the Landau Network-Cariplo Foundation and the Russian Science Foundation (RSCF 14-12-01338).


Nadjib Achir
email: nadjib.achir@univ-paris13.fr

Mérouane Debbah
email: merouane.debbah@supelec.fr

Paul Muhlethaler
email: paul.muhlethaler@inria.fr

Massive MIMO Cooperative Communications for Wireless Sensor Networks: Throughput and Energy Efficiency Analysis

The objective of this study is to analyze a new disruptive deployment of wireless sensors in order to cope with the explosive demand for bandwidth while taking into account energy consumption considerations. The work is grounded on the idea of massive network densification by drastically increasing the number of sensors in a given area in a Time Division Duplex (TDD) mode. Using ideas from the recent Massive MIMO technology (more than 400 antennas, without any modification of the network infrastructure), we transpose the idea to a massive deployment of sensors and show the benefits of such an infrastructure. This research is expected to provide the optimal deployment of massive wireless sensor networks in terms of cost/performance/complexity/energy efficiency trade-off and define the next generation wireless Machine-to-Machine (M2M) communication.

I. INTRODUCTION

In our modern world, more than half of the world's population lives in urban or sub-urban areas. Last year, the world's population reached 7 billion and current projections predict that it will exceed 9 billion by 2050, 70% of which will live in cities. This continuous expansion of urban populations, increases the need to efficiently design and control complex and interdependent urban infrastructures related to urban living, such as public safety, public transportation, traffic management, resource/energy utilization, interpersonal communications, social activities, and entertainment, etc. Providing such an efficient control of urban infrastructures greatly depends on the availability of and access to large amounts of information about urban environments and infrastructures. Data gathering is the key issue in order to efficiently monitor existing infrastructures or provide new urban services. Basically, we need to acquire data from a number of sources, process this huge amount of data, and then push or make these data available to city infrastructure managers and citizens in a user-friendly manner.

Currently, among all the available technologies, we believe that Wireless Sensors Networks (WSNs) are undoubtedly best suited to efficiently acquire and disseminate data on a large scale. Recent advances in Micro-Electro-Mechanical Systems (MEMS) technology, wireless communications, and digital electronics have enabled the development of low-cost, low-power nodes (also called sensor nodes) that are able to communicate over short distances. These sensor nodes consist of sensing, data processing units, and communication components. Consequently, a large number of companies are proposing new small devices which can monitor different phenomena such as temperature, humidity, vibration, pressure, and several other factors. These sensors can be deployed to build a Wireless Sensor Network (WSN). Thus, WSNs could be used to monitor the urban environment in real time, to facilitate automated control and to collect information for decision making. Nowadays, sensors are incorporated in most of our modern facilities, such as mobile phones, vehicles, buses, bus stops, bikes, etc. For example, mobile phones, with their increasing capabilities are used as voice communication devices but also as a sensing device able to collect and transfer data such as images, audio, GPS position, speed, etc. All these sensors could play an important role in providing a vast amount of dynamic information about their environment. Therefore wireless sensor networks could be a valid solution to urban monitoring problems by bringing new services for the city and for the citizens.

Over the last decade, WSNs have generated considerable enthusiasm within the networking research community. Many studies have been conducted in order to apply wireless sensor networks to a wide range of applications. The majority have focused on several networking issues, such as routing, MAC, data gathering and dissemination mechanisms. Unfortunately, few studies have been conducted on large-scale WSNs, where the number of nodes can reach thousands or even more, which is the case if we consider the context of smart cities. In fact, the few studies such as [START_REF] Vicaire | Achieving long-term surveillance in vigilnet[END_REF], [START_REF] Xu | A wireless sensor network for structural monitoring[END_REF], [START_REF] Werner-Allen | Motelab: a wireless sensor network testbed[END_REF], [START_REF] Barrenetxea | Sensorscope: Out-of-the-box environmental monitoring[END_REF], [START_REF] Dutta | Trio: enabling sustainable and scalable outdoor wireless sensor network deployments[END_REF] that explicitly considered large-scale wireless sensor network are often optimized to meet the specific needs of the application and do not fully leverage the general network behaviors. Moreover, due to the intrinsic sensors characteristics including limited energy, limited communication range and a relatively large area of interest (urban areas), full connectivity could not be achieved.

To the best of our knowledge, the mainstream methods that could be used to solve the problem of information gathering in large-scale WSNs are the hierarchical routing protocols [START_REF] Chen | A distributed and effective cluster routing protocol of sensor networks[END_REF], [START_REF] Quang | Adaptive routing protocol with energy efficiency and event clustering for wireless sensor networks[END_REF], [START_REF] Yuan | A data gathering algorithm based on mobile agent and emergent event-driven in cluster-based wsn[END_REF], [START_REF] Ge | Joint clustering and optimal cooperative routing in wireless sensor networks[END_REF], [START_REF] Dimokas | Energyefficient distributed clustering in wireless sensor networks[END_REF]. Indeed, in a hierarchical routing protocol, the sensor network is partitioned into several groups with different assignment levels. The nodes of the low levels are responsible for sensing and collecting the information from their environment, whereas the high level sensors are responsible for gathering the information from their low level sensors and then forwarding this information to the next high level. This hierarchical routing approach has proved to be more energy-efficient than a complete flat network. Unfortunately, this approach requires the deployment of higher energy nodes acting as cluster heads in order to gather, process and send the information from lower level nodes to higher level nodes. Since the cluster heads are acting as relays they will see their energy consumption fast increasing, unless they are given a high energy capability. In addition, the malfunctioning of the cluster head can lead to a strong system degradation in terms of performance. To overcome these limitations, we propose to use, in this paper, a distributed Massive MIMObased collaboration between the sensors [START_REF] Hoydis | Massive mimo in the ul/dl of cellular networks: How many antennas do we need?" Selected Areas in Communications[END_REF], [START_REF] Marzetta | Noncooperative cellular wireless with unlimited numbers of base station antennas[END_REF] in order to improve the performances of large-scale wireless sensor networks.

Conventional Multiple-Input Multiple-Output (MIMO) wireless communication, through the use of multiple antennas both at the transmitter and at the receiver sides, has the potential of multiplying the capacity of a single channel of bandwidth W by the rank of the channel matrix. For sufficiently rich scattering, this rank is r =m i n ( M, N), where M and N denote the number of transmitting and receiving antennas, respectively. MIMO has the capability of multiplexing r independent channels, and therefore achieve an r-fold increase in system capacity. Recently cooperative MIMO has gained substantial research interest [START_REF] Gesbert | Outdoor mimo wireless channels: models and performance prediction[END_REF], [START_REF] Telatar | Capacity of multi-antenna gaussian channels[END_REF] due to its capability to exploit conventional MIMO techniques for physically constrained mobile devices. In this case, the spectral efficiency for users is greatly improved when users share their antennas to jointly transmit and decode the data. In this paper, we focus on massive MIMO cooperative communications in the case of wireless sensor networks. The main idea is to characterize the performances of the system in terms of throughput/complexity/energy efficiency trade-off.

The reminder of this paper is organized as following: Section II describes the system model including the training and the transmission periods. At the end of this section we derive both the overall system throughput and energy consumption. Section III describes the simulation and results. Finally, section IV concludes the paper and discuss directions for our future works.

II. SYSTEM MODEL

Let N be a set of sensors randomly distributed in a two dimensional field. We consider that the N sensors are identical and equipped with one transmit antenna. In addition to the detection capability, each sensor is able to communicate in order to notify the sink of the event that has been detected. Basically, when an event is detected, each sensor can send its message directly to the sink. In this paper, we assume that a set of K sinks are deployed within the area. In addition, we consider that the sensors are able to cooperate with each other in order to form a virtual antenna array to achieve a virtual MIMO communication system. This Virtual MIMO system is completely distributed since all the sensors are in different locations. Finally, we focus our study on a reporting-based application, where the sensors have to periodically report the same data related to their environment to the sink(s). This reporting period needs to be divided into two phases. The first phase is devoted to the channel learning/estimation in the TDD scenario. The channel estimation phase enables the sensors to optimize the distributed beamforming transmission. Basically, to estimate the channel, each receiver (sink) has to send a learning sequence. The second period is dedicated to the transmission phase. Within this period, the transmitters send the data to the sink(s), by adding, if necessary, a preprocessing of the information sent, such as a beamforming with the Hermitian transpose of the noisy estimate of the channel matrix acquired during the learning period.

It is clear that the efficiency of the system is related to the time spent in each period. To increase system performance, we need to increase the duration of the first period in order to have a good estimation of the channel matrix. Likewise, we also have to increase the duration of the second period in order to increase the global system bandwidth. We must obviously find a trade-off between the duration of each period. Another important issue is the number of sensors involved in the collaboration. It is clear that increasing the number of sensors will increase the bandwidth. However, this will also increase the amount of energy consumed by the network.

C =[c 1 c 2 ••• c K ], such that : C H C = M •Id.
During the training period the signal received at the transmitters can be formulated as follows:

Y = 2 6 6 4 y 1 (1) ••• y 1 (M ) y 2 (1) ••• y 2 (M ) . . . . . . . . . y N (1) ••• y 1 (M ) 3 7 7 5 Y = 2 6 6 4 h 1,1 ••• h 1,K h 2,1 ••• h 2,K . . . . . . . . . h N,1 ••• h N,K 3
n 1,1 ••• n 1,M n 2,1 ••• n 2,M . . . . . . . . . n N,1 ••• h N,M 3 7 7 5
We note

H T = 2 6 6 4 h 1,1 ••• h 1,K h 2,1 ••• h 2,K . . . . . . . . . h N,1 ••• h N,K 3
n 1,1 ••• n 1,M n 2,1 ••• n 2,M . . . . . . . . . n N,1 ••• h N,M 3 7 7 5
as the transfer matrix and the the noise matrix respectively.

For a given sensor (say sensor 1), we can write the received signal from K sinks as follows

[ y 1 (1), ••• ,y 1 (M )]=[h 1,1 , ••• ,h 1,K ] • C T + [ n 1,1 , ••• ,n 1,M ]
Note that each sensor needs to know the whole code matrix C, since each receiver has to do the same processing. We assume that the matrix C is a known matrix, thus there is no need for any cooperation between the sensors. The receivers can then estimate the channel as follows

[ b h 1,1 , ••• , b h 1,K ]= 1 M • [ y 1 (1), ••• ,y 1 (M )]• C ⇤ = 1 M • [ h 1,1 , ••• ,h 1,K ] • C T • C ⇤ + 1 M • [ n 1,1 , ••• ,n 1,M ] • C ⇤ =[ h 1,1 , ••• ,h 1,K ] +[ b 1,1 , ••• ,b 1,K ]
where

[ b 1,1 , ••• ,b 1,K
]=b is the error vector. It is a Gaussian vector of K random variables of mean zero. We can compute the second moment of this error:

E[b H b]= 1 M 2 • E[C T n H n C ⇤ ] = 1 M 2 • C T σ 2 • Id • C ⇤ = σ 2 M • Id
Therefore, the estimated channel can be formulated as follows

8i 2{1, ••• ,N}8 j 2{1, ••• ,K} b h i,j = h i,j + b i,j ,
where b i,j ⇠N(0, σ 2 M ). In the following we note c H the K lines and N columns matrix and c H b h i,j ].

B. Transmission Period

we suppose that all the sensors measure the same data s. They also do not cooperate and each sensor knows only its channel to the K sinks (i.e. not the channels of the other sensors to the K sinks). The sensor will exploit the TDD protocol and channel reciprocity to beam-form the sensed data. The received signal at the sinks can be formulated as follows: where x i is the data sent by sensor i. In our case, the transmitted data x i are computed as follows:

2 6 4 r 1 . . . r K 3 7 5 = 2 6 4 h 1,1 ••• h N,1 . . . . . . . . . h 1,K ••• h N,K 3
2 6 4
x 1 . . .

x N 3 7 5 = 1 p K c H ⇤ 2 6 4 s . . . s 3 7 5
Thus we have for instance : Asymptotically, if N !1and K is fixed and h i,j ⇠N (0, 1).

x 1 = 1 p K K X i=1 b h ⇤ 1,i s Therefore, 2 6 4 r 1 . . . r K 3 7 5 = 1 p K H T c H ⇤ 2
1 N H T H ⇤ ! 1 N 2 6 4 N ••• 0 . . . . . . . . . 0 ••• N 3 7 5 = Id
and therefore in the asymptotic setting, we have for

j 2{ 1, ••• ,K}: r j = N p K s + z j
Thus, each sink is able to get the data without any interferences from the other receivers. This supposes of course that the channel is perfectly known but provides a guideline as to how the sensors should perform distributed beam-forming. Unfortunately, in practice, we have for m 2{ 1, ••• ,K}:

r m = 1 p K ([ h 1,m , ••• ,h N,m ]) c H ⇤ 2 6 4 s . . . s 3 7 5 + z m = 1 p K ([ h 1,m , ••• ,h N,m ]) c H ⇤ 2 6 6 4 P K j=1 b h ⇤ 1,j . . . P K j=1 b h ⇤ N,j 3 7 7 5 + z m Therefore: r m = 1 p K [h 1,m P K j=1 b h ⇤ 1,j + •••+ h N,m P K j=1 b h ⇤ N,j ] • s + z m
This results in:

r m = 1 p K (|h 1,m | 2 + •••+ |h N,m | 2 ) • s + v m + z m where: v m =[ N X i=1 h i,m K X j=1 b ⇤ i,k + N X i=1 h i,m K X j=1 j6 =m h ⇤ i,j ]s
Using Shannon's formula, the rate is then equal to:

(1 -α) log 1+ P K ( P N i=1 |h i,m | 2 ) 2 E[|v m | 2 ]+σ 2 ! .
In the following, we assume that the fading is Rayleigh and thus h i,k ⇠N(0, 1). We have:

E[|v m | 2 ]=E h ⇣ N X i=1 h i,m K X j=1 b ⇤ i,j + N X i=1 h i,m K X j=1 j6 =m h ⇤ i,j ⌘ s 2 i E[|v m | 2 ] = E[|s| 2 ] h | N P h i,m K P j=1 b ⇤ i,j | 2 +2< ⇣ ( N P i=1 h i,m K P j=1 b ⇤ i,j )( N P i=1 h ⇤ i,m K P j=1 j6 =m h i,j ) ⌘ + | N P i=1 h i,m K P j=1 j6 =m h ⇤ i,j | 2 i E[|v m | 2 ] = P Kσ 2 M N P i=1 |h i,m | 2 + P | N P i=1 h i,m K P j=1 j6 =m h ⇤ i,j | 2
Thus the total rate is equal to:

R =(1-α) K X m=1 log 1+ P K N P i=1 |h i,m | 2 2 Kσ 2 Ts αT N P i=1 |h i,m | 2 + σ 2 + P | N P i=1 h i,m K P j=1 j6 =m h ⇤ i,j | 2 !
In addition to the total rate, we compute the total energy consumed by the system which corresponds to the amount of energy consumed by the K sinks during the training period (i.e. αT ) plus the amount of energy consumed by the sensors during the transmission period (i.e. (1α)T )). We can formulate the total energy consumed E as follows:

E = αKP +(1-α)P N X i=1 ⇣ Kσ 2 + E(| K X j=1 h i,j | 2 ⌘
The problem we address is, given a maximum energy E consumed, to find α, K and N in order to maximize the total rate R.

III. RESULTS

We start our evaluation by analyzing the performances of the system with one sink (i.e. K =1 ). We fix the number of collaborating sensors, N , to 1000 and we study both the global rate and the energy consumption of the system. In Figure 2, we plot the global rate versus the period of time dedicated to the training period. As expected, the global rate is maximized for a given fraction of time dedicated to the training period (8% in the figure). This is due to the fact that having α<0.08 the sensors can not accurately estimate the channel, which results in a low throughput. On the other hand, increasing the training period (α>0.08), results in an accurate estimation of the channel, but also leads to a shorter transmission period, which also results in a low throughput. The optimal value is obtained for α ⇤ =0.08. In Figure 3, we plot the maximal throughput that can be obtained when fixing the number of sinks to 1 and varying the number of sensors from 1 to 1000. We can notice that the throughput increases as the number of sensors increases. However the more we increase N , the more marginal the potential gain becomes. However, increasing the number of collaborating sensors greatly increases the energy consumption. Indeed the energy consumption increases linearly with the number of sensors. In Figure 4, we plot the K ⇤ i.e. the number of sinks required to maximizes the total rate with a number of sensors equal to 200, 400, 600, 800 1000, respectively, and for a given maximum amount of energy available for each sensor. As we can observe, from a given number of sinks the global rate is maximized. It is not necessary to increase further the number of sinks.

IV. CONCLUSION

In this paper, we present a framework for massive MIMO cooperative communications for Wireless Sensor Networks. Our main objective is to analyze the performances of the deployment of a large number of sensors. This deployment should cope with a high demand for real time monitoring and should also take into account energy consumption. We have assumed a communications protocol with two phases: an initial training period followed by a second transmit period. The first period allows the sensors to estimate the channel state and the objective of the second period is to transmit the data sensed. In this paper, we analyzed the impact of the proportion of time devoted to each period. We study the throughput obtained with respect to the number of sensors when there is one sink. We also compute the optimal number of sinks with respect to the energy spent for different values of sensors. This work is a first step to establish a complete framework to study energy efficient Wireless Sensor Networks where the sensors collaborate to send information to a sink.

Fig. 1 .

 1 Fig. 1. Massive MIMO protocol.

Fig. 2 .

 2 Fig. 2. Throughput versus fraction of training α.

Fig. 3 .

 3 Fig. 3. Throughput versus Number of sensors.

Fig. 4 .

 4 Fig. 4. Optimal number of sinks maximizing the total rate.