Benoît Dewandel
email: b.dewandel@brgm.fr

Patrick Lachassagne
email: patrick.lachassagne@danone.com.

Faisal K Zaidi
email: fk.zaidi@gmail.com.

Subash Chandra
email: schandra75@gmail.com.

A Conceptual hydrodynamic model of a geological discontinuity in hard rock aquifers: example of a quartz reef in granitic terrain in South India

Keywords: hard rock, discontinuity, granite, weathering, hydrodynamic properties, heterogeneity in hydraulic conductivity

The structure and hydrodynamic properties of geological discontinuities and of a deeply weathered granite aquifer near these structures are described on the basis of geological, geophysical and hydrodynamic investigations in two sites of South India located along a 20-40-metre-wide quartz reef intruding a weathered Archean biotite granite. One of the two sites also comprises a metre-wide dolerite dyke.

Weathering processes appear to be at the origin of fissures development and of a related enhanced local hydraulic conductivity, both in the quartz reef and in the surrounding granite.

The weathering profile in the granite (saprolite and fissured layer) is characterized by an abrupt deepening of the weathered layers in the granite near the contact and in the quartz reef itself. Therefore, the weathering profile shows a ‗U'-shape geometry with, among others, the verticalization of the granite's fissured layer. The hydraulic conductivity of this verticalized layer is on average 5 x 10 -6 m/s and storativity about 10 -3 (-). The hydraulic conductivity of the fissured quartz is 4 to 6 x 10 -6 m/s and its storativity about 3 to 5 x 10 -4 (-). Both media are also characterized by a matrix hydraulic conductivity (10 -7 to 10 -9 m/s) and by a significant heterogeneity in hydrodynamic properties that generates preferential flow paths along the subvertical fissures parallel to the reef axis. A special attention has been paid for characterizing this heterogeneity.

The weathering of the dolerite dyke, however, results in a local low hydraulic conductivity, which consequently does not enhance either the thickness of weathered granite layers or its hydraulic conductivity.

The obtained results complete the conceptual hydrogeological model developed for weathered granite aquifers in characterizing the relationships between weathering processes and hydrodynamic properties near geological discontinuities.

Introduction

Hard rocks (plutonic, metamorphic rocks) form part of the basement of continents and, as such, occupy large areas throughout the World (Africa, South and North America, India, Europe, Asia, Australia, etc.). Groundwater resources in aquifers (HRA) are generally low in terms of available discharge per well (from a few to a few tens of m 3 /h) compared to those from porous, karst, and volcanic aquifers. However, HRA are geographically widespread and thus well-suited for water supply to scattered populations and small to medium-size towns, or suburbs of larger cities. Their groundwater resources largely contribute to economic development, especially in regions exposed to arid and semiarid climatic conditions where surface-water resources are limited as, for example, in Africa and in India [START_REF] Uhl | Results of pumping tests in crystalline-rock aquifers[END_REF][START_REF] Gustafson | Crystalline rock aquifers: their occurrence, use and importance[END_REF]Houston and Lewis, 1998). In such regions, HRA commonly are the only available perennial water resource, supplying the population as well as the agricultural and industrial sectors. However, the knowledge of such aquifers, e.g., their geometry, their hydraulic properties, etc., is currently meagre and needs to be improved.

Significant advances have recently been made in the geological and hydrogeological characterization of such complex aquifers [START_REF] Omorinbola | Verification of some geomorphological implcations of deep weathering in the basement complex of Nigeria[END_REF][START_REF] Owoade | The potential for minimizing drawdowns in groundwater wells in tropical aquifers[END_REF][START_REF] Chilton | Hydrogeological characterization and water-supply potential of basement aquifers in tropical Africa[END_REF][START_REF] Taylor | A tectono-geomorphic model of the hydrogeology of deeply weathered crystalline rock: evidence from Uganda[END_REF][START_REF] Lachassagne | Exploitation of high-yield in hard-rock aquifers: Downscaling methodology combining GIS and multicriteria analysis to delineate field prospecting zones[END_REF]Wyns et al., 2004[START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF][START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF][START_REF] Ayraud | Compartmentalization of physical and chemical properties in hard-rock aquifers deduced from chemical and groundwater age analyses[END_REF][START_REF] Courtois | Country-scale hydrogeological mapping of hard-rock aquifers and its application to Burkina Faso[END_REF][START_REF] Lachassagne | The fracture permeability of Hard Rock Aquifers is due neither to tectonics, nor to unloading, but to weathering processes[END_REF]. These studies show that the geometry and the hydrodynamic properties of HRA mainly result from deep weathering processes of the parent rock. Such processes involve biogeochemical hydrolysis of the minerals constituent of the rocks and their mineral transformation under new surface conditions, mainly into clay-rich materials [START_REF] Tardy | Characterization of the principal weathering types by the geochemistry of waters from some European and African crystalline massifs[END_REF][START_REF] Tardy | Pétrologie des latérites et des sols tropicaux[END_REF][START_REF] Tardy | Dérive des continents, latérites et paléoclimats tropicaux[END_REF][START_REF] Nahon | Introduction to the Petrology of Soils and Chemical Weathering[END_REF]. Additionally, these processes are controlled by the regional geodynamical history, which can lead to multiphase weathering profiles because of alternating erosion and weathering phases [START_REF] Taylor | A tectono-geomorphic model of the hydrogeology of deeply weathered crystalline rock: evidence from Uganda[END_REF][START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF].

In granite-type rocks (e.g. granite, gneiss), a typical weathering profile comprises two main stratiform layers sub-parallel to the paleo-surface contemporaneous of the weathering processes (Fig. 1; [START_REF] Chilton | Characteristics of the weathered basement aquifer in Malawi in relation to rural water supplies[END_REF][START_REF] Chilton | Hydrogeological characterization and water-supply potential of basement aquifers in tropical Africa[END_REF][START_REF] Wyns | Caractérisation multiparamètres des horizons de subsurface (0-100 m) en contexte de socle altéré[END_REF]and 2004[START_REF] Krásný | Hydrogeology of fractured rocks from particular fractures to regional approaches : state-of-the-art and future challenge[END_REF]Maréchal et al., 2007, and others). From the top downward, they are (i) the saprolite layer, a clay-rich material usually characterized by low hydraulic conductivity, and (ii) the fissured layer characterized by dense horizontal fissuring within the first few metres and a depth-decreasing of the sub-horizontal and sub-vertical fissure sets [START_REF] Howard | Hydrogeologic evaluation of fracture permeability in crystalline basement aquifers of Uganda[END_REF]Houston and Lewis, 1998;[START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF]Wyns et al., 2004;[START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF]. This last layer usually assumes the transmissive function of the aquifer. The underlying unfissured and fresh basement is only locally permeable where deep fractures occur, but these conductive zones are much less dense than in the overlying fissured layer and their hydrodynamic properties are highly variable [START_REF] Pickens | Analysis and interpretation of borehole hydraulic tests in deep boreholes; principles model development, and applications[END_REF][START_REF] Blomqvist | Deep groundwater in the cristalline basement in Finland, with implications for waste disposal studies[END_REF][START_REF] Walker | Hydrogeologic studies for nuclear-waste disposal in Sweden[END_REF][START_REF] Kuusela-Lahtinen | Flow dimension as an indicator of hydraulic behaviour in site characterization of fractured rock[END_REF][START_REF] Cho | Relationship between the permeability of hard-rock aquifers and their weathered cover based on geological and hydrogeological observation in South Korea[END_REF]. Flow can be fractionalized, i.e. controlled by the orientation of the fracture network [START_REF] Leveinen | Fractional flow dimensions and hydraulic properties of a fracture-zone aquifer, Leppävirta, Finland[END_REF].

As a consequence, where the formation is not affected by geological discontinuities, such as faults, veins or dykes, and exposed to deep weathering processes, the saprolite and the fissured layers together make up a composite aquifer of up to 100 m thick (Fig. 1).

Hydrogeologically, this medium can be considered as a multilayer system where each layer is characterized by its own specific and quite homogeneous hydrodynamic properties. This relative -homogeneity‖ allows several practical applications [START_REF] Lachassagne | The fracture permeability of Hard Rock Aquifers is due neither to tectonics, nor to unloading, but to weathering processes[END_REF], among which aquifer budgeting [START_REF] Maréchal | Combining the groundwater budget and water table fluctuation methods to estimate specific yield and natural recharge[END_REF] or the development of Decision Support Tools [START_REF] Dewandel | Management of the groundwater resources in semiarid hard rock regions under variable water demand and climatic conditions: development of a Decision Support Tool[END_REF][START_REF] Dewandel | Development of a Tool for managing groundwater resources in semi-arid hard rock regions. Application to a rural watershed in South India[END_REF]. However, where the granite is intruded by quartz, pegmatite or aplite veins, dolerite dykes, or younger granite, or affected by tectonic faults, the layers geometry and their hydrodynamic properties may substantially differ from the above-presented conceptual model. Locating such heterogeneities in hard rocks, commonly resulting from lineament analysis and/or geophysical surveys, constitutes the backbone of borehole siting, as major discontinuities and their intersections are often assumed to be strong indicators of a groundwater resource [START_REF] Mabee | A method of collecting and analyzing lineaments for regional scale fissured-bedrock aquifer studies[END_REF]Minor et al., 1995;[START_REF] Sander | Groundwater exploration based on lineament analysis and reproducibility tests[END_REF][START_REF] Chandra | Integrated studies for characterization of lineaments to locate groundwater potential zones in hard rock region of Karnataka, India[END_REF][START_REF] Sander | Lineaments in groundwater exploration: a review of applications and limitations[END_REF]. However, such structures do not necessarily give impressive results in terms of well productivity [START_REF] Sander | Lineaments in groundwater exploration: a review of applications and limitations[END_REF]; probably because at present their hydrogeological properties in terms of hydrodynamics and geometry are insufficiently understood.

The objective of the present paper is to improve the understanding of the geometry and hydrodynamic properties of such geological discontinuities, through the case-study of a 20 to 40-metre-thick quartz vein intruding granite that was exposed to deep weathering processes. This work has led to a 3-D conceptual hydrogeological model of this structure, on the basis of geological and geophysical surveys, and hydraulic tests carried out on two pilot sites equipped with a total of 21 boreholes (Fig. 2, Maheshwaram area, Ranga Reddy District, Andhra Pradesh, India).

Geological setting

The studied sites are located 40 km south of Hyderabad (Andhra Pradesh, India). The area is characterized by a relatively flat topography with altitudes ranging from 670 to 590 m above mean sea level. The region experiences a semi-arid climate controlled by the periodicity of the monsoon (rainy season: June to October). Mean annual precipitation is about 750 mm, of which more than 90% falls during the monsoon season.

The area is underlain by Archean biotite granite that is locally intruded by leucocratic granite. These granites are affected by deep in-situ multi-phase weathering-and-erosion processes [START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF]. At the regional scale, the weathered horizons more or less follow the current topography and, from top to bottom, are composed of:

-A 1-3-m thick layer of sandy regolith, locally capped by a lateritic crust (<50 cm); -A 10-15-m thick layer of laminated saprolite; -And, a 15-20-m thick layer of fissured granite, where weathered granite and clay minerals can partially fill the fissures.

The granites are locally intruded by up to 40-m-wide quartz veins (Archean in age) and metrewide dolerite dykes (2.5 to 1.6 Ga; G.S.I. 2002; Fig. 2). Quartz reefs systematically occupy positive relief in the area. In order to investigate the impact of such discontinuities on the weathering-profile structure and its hydrodynamic properties, geological and geophysical investigations were carried out on two pilot sites: -IFP 1‖ and -Kothur‖ (Figs. 2 and3). The two sites, separated by 7 km, are located along the same sub-vertical and north-south decametre-wide quartz reef.

Electrical Resistivity Tomography profiles were carried out on the two sites for siting the 21 boreholes used in the study [START_REF] Chandra | Geophysical Characterization of Quartz Reef Intrusive Lineaments in Granite Hard Rock for Groundwater Occurrences[END_REF]. Five ERT profiles were carried on the Kothur site and one on the IFP1 site. All ERT profiles show that the quartz reef, particularly at depth, is characterized by lower resistivities than the surrounding granite, which is attributed to a more developed fissure network at depth (see supplemental material).At Kothur six boreholes were drilled in the central area and in the pinch-out zone of the quartz reef and 5 in the surrounding granite (Fig. 3a and3b). The ten IFP1 boreholes were located within the granite in the pinch-out zone of the quartz reef and near a metre-wide dolerite dyke (Fig. 3a) that was intersected by two boreholes (IFP 1-4 and 1-8).The depth of the 21 boreholes ranges between 35 and 91.4 m (Table 1); detailed geological logs were prepared from drill cuttings (see supplemental material). At IFP1 site, two pre-existing farmer's wells (bw1 and bw2, Fig. 3a) drilled within the granite are also present, but their exact depth is unknown.

The combination of the geophysical and the geological data allowed characterizing both the structure of the geological formations and the geometry of the weathering profile near the intrusion. Geological cross-sections perpendicular and parallel to the reef for the two sites clearly show that the weathering profile is significantly deepened near the quartz reef (Figs 4a, b andc). At the outcrop scale, the orientation of major fissures (metre to decametre wide) in the quartz reef was measured (see supplemental material). The fissures are mostly subvertical and organized in two main sub-orthogonal sets, one sub-parallel (N000 90°) to the quartz vein and the second sub-orthogonal (N100 90°) to it. A few sub-horizontal fissures were also observed, but due to the relatively flatness of the outcrops their number is probably underestimated. Within the first ten metres from surface, the quartz is highly weathered and densely fissured from centimetre to decametre scale. However, near the contact with the granite, i.e. on the sides of the quartz reef and in the pinch-out zones, this highly weathered and fissured material can reach up to 50 m depth (Figs. 4b andc).Near the quartz reef, the granite is also characterized by an important deepening of the weathering profile. On average, saprolite thickness increases by a factor 1.5 to 3 and the fissured layer by a factor 3 to 5, compared to the zone not affected by the intrusion. Locally, the bottom of the fissured layer can reach more than 70 m depth, whereas farther from the quartz reef the weathering profile in the granite stops at few tens of metres from the surface (Figs. 4).

The IFP1 sub-vertical and metre-wide dolerite dyke sub-parallel to the quartz reef is highly weathered within the first 6 m from the surface (clayey material), then poorly fissured up to 15 m, and finally hard and compact. Near this dyke there is no-evidence of a deepening of the weathering front and of the development of a local deeper fissuring.

Previous studies [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF][START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF] showed the relative homogeneity of the hydrodynamic properties of the weathered/fissured hard-rock aquifers where the granite is not affected by such discontinuities. Several questions now arise: are the hydrodynamic properties different in such verticalized weathered-fissured layers? Does the preferential orientation of fissures in the quartz reef play a role? What is the hydrogeological conceptual model of these discontinuities? To try and provide answers to these questions, slug tests and pumping tests were performed on both sites.

Hydrodynamic investigations

Slug tests

Twenty-one slug tests were carried out in the boreholes of both pilot sites and were interpreted using the [START_REF] Bouwer | A slug test for determining hydraulic conductivity of unconfined aquifers with completely or partially penetrating wells[END_REF] technique for unconfined aquifers with completely or partially penetrating wells. The obtained distribution of hydraulic conductivities constitutes a preliminary estimate of the hydraulic conductivity of the weathered layers in the granite and in the quartz aquifers near the wells (Fig. 5).

Near the quartz reef, the hydraulic conductivity of the fissured granite ranges from 8.5 x 10 -8 m/s to 8.8 x 10 -6 m/s with a geometrical mean of 2.4 x 10 -6 m/s and a standard deviation of 0.6 computed on a logarithmic scale (logK slug : -5.6±0.6). In the quartz reef, values are similar and range from 5.0 x 10 -8 m/s up to 1.6 x 10 -5 m/s with a geometrical mean of 2.6 x 10 -6 m/s and a standard deviation on a log scale of 1.0 (logK slug : -5.6±1.0). Within these distributions, a small population of two boreholes (IFP30-3, K=8.5 x 10 -8 m/s and IFP30-6, K=5.1 x 10 -8 m/s) with hydraulic conductivities of less than 10 -7 m/s corresponds to poorly fissured granite and quartz and is possibly representative of the matrix hydraulic conductivity. The hydrodynamic properties of the dolerite dyke were not explored since no water contribution was observed during drilling despite the shallow water-table (4 m from surface). The hydraulic conductivity distributions in the quartz reef and in the nearby granite seem to follow near-log normal distributions, similar to that of the stratiform fissured layer far from discontinuities in the same granite (geometric mean of 4.4 x 10 -6 m/s, [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF]. However, the spatial distribution of hydraulic conductivity in the quartz reef is highly variable and is closely linked with the degree of weathering. Indeed, at the centre of the quartz reef, where a high grade of weathering and fissuring only affects the first metres below surface, the quartz is characterized by quite low hydraulic conductivities (IFP30-1, 30-6; K<5 x 10 -7 m/s), whereas it is significantly more permeable near the contact with granite (e.g., IFP30-9, 30-10, 30-11; K>10 -5 m/s). This shows that most of the conductive fissures develop in the contact zones (i.e. sides and pinch-out zones), whereas the heart of the reef is poorly fissured.

For the granite, no clear relationship between hydraulic conductivity and location relative to the quartz reef is found, suggesting that both near and far from the discontinuity the fissured layer of the granite exhibits similar properties.

Pumping tests

Brief description of the method used

The interpretation of pumping tests in such fissured aquifers commonly is a difficult task, as groundwater flow can be fractionalized [START_REF] Black | Hydrogeology of fractured rocksa question of uncertainty about geometry[END_REF][START_REF] Borgne | Equivalent mean flow models for fractured aquifers: insights from a pumping tests scaling interpretation[END_REF] and controlled by the hydraulic conductivity of fissures, their density and orientation, and also their relationships with low-hydraulic-conductivity blocks or matrix (e.g. [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF][START_REF] Borgne | Assessment of preferential flow path connectivity and hydraulic properties at single-borehole and cross-borehole scales in fractured aquifer[END_REF]. In order to reveal the various aspects of the hydrodynamic properties of such a complex medium and to appropriately model the pumping test, a special attention has been paid to diagnosing of the tests prior to their modelling. Diagnoses were based on the analysis of derivative drawdown curves on log-log plots that allows flow-regime identification [START_REF] Bourdet | A new set of type curves simplifies well test analysis[END_REF][START_REF] Bourdet | Use of pressure derivative in well-test interpretation[END_REF][START_REF] Ehlig-Economides | Use of the Pressure Derivative for Diagnosing Pressure-Transient Behavior[END_REF][START_REF] Spane | DERIV: a computer program for calculating pressures derivatives fir use in hydraulic test analysis[END_REF]Wurstner, 1993, Renard et al., 2009). The tests were then interpreted with WinISAPE (BRGM software) and WTFM [START_REF] Lods | WTFM, software for Well Test analysis in Fractured Media combining fractional flow with double porosity and leakance approaches[END_REF].Table2 presents the main characteristics of the pumping tests.

The observations and models were visually matched on both drawdown and derivative curves.

The sensitivity of the hydrodynamic-parameter estimates was inferred from several fitting tests: it is less than 15% on average for the hydraulic conductivity and less than 20% on average for storativity. Since 34 curves were interpreted for the study, only a few curves are presented here (Fig. 6a, b andc).

Pumping tests at the Kothur site

Three non-simultaneous pumping tests were performed, two within the quartz reef (IFP30-5 and IFP30-10) and one within in the stratiform fissured layer of the granite (IFP30-4) at a quite large distance from the quartz vein (Fig. 3b; Tables 3 to 5).

Pumping in IFP30-5 revealed spherical flow during the early stage of pumping (-1/2 slope of derivative curve; t<100 min; Fig. 6a) that is typical of a borehole partially penetrating the aquifer. As this well intersects about 30 to 40 m of fractured quartz (Fig. 4c), this diagnosis suggests that IFP30-5 is located in a zone where the aquifer is relatively thin compared to where the drawdown progresses. This is corroborated by the geological observations, which show that the thickness of the weathered/fissured zone increases up to 70 m depth north of the site (Fig. 4c). The whole data set from pumping and observation wells has been interpreted with the Hantush model (partially penetrating aquifer with vertical anisotropy in hydraulic conductivity; [START_REF] Hantush | Aquifer tests on partially penetrating wells[END_REF]Fig. 6a). Estimated hydrodynamic parameters are of the same order of magnitude and relatively close from one well to the next with, however, a variability in the estimates and particularly those for hydraulic conductivity and storativity (Table 3); this point will be discussed later. On average, the hydraulic conductivity is 4.0 x 10 -6 m/s, the storativity about 3.2 x 10 -4 (-) and the aquifer thickness is estimated to be about 100 m. The computed aquifer thickness appears slightly overestimated compared to reality (about 70 m), which we explain by the uncertainty of the real productive length in wells (screened thickness on Table 3) but also by the use of the Hantush model that considers a well that partially penetrates the aquifer whereas IFP30-5 is located in a pinch-out zone of the aquifer. The vertical anisotropy ratio in hydraulic conductivity (Kx/Kz) is about 3 and probably reveals a depth-decreasing hydraulic conductivity due to a depth-decreasing density of fissures.

The test performed in IFP30-10 revealed a dual porosity behaviour (Fig. 6b). The overall data set has thus been interpreted with a dual porosity model. Two observation wells (IFP30-6 and 30-7) were interpreted in the matrix, the others being in the fissure network (Table 4; Fig. 6b).

Compared to the previous pumping test, spherical flow is absent showing that IFP30-10 intersected most of the aquifer thickness and thus that the bottom of the aquifer is close to 70 m deep. The estimated hydrodynamic parameters are consistent with the ones obtained in IFP30-5 and of the same order of magnitude (Table 4). This test gives complementary information on the hydraulic parameters of the quartz reef aquifer. On average, the hydraulic conductivity of the aquifer is 6.2 x 10 -6 m/s, the storativity is about 4.6 x 10 -4 (80% in the matrix and 20% in the fissures), and the matrix hydraulic conductivity is about 2.4 x 10 -9 m/s. Variability in hydraulic conductivity and storativity estimates was also noted and will be discussed later.

The third test at Kothur (IFP30-4) characterises the stratiform fissured layer of the granite not affected by the reef. Only two observation wells reacted to the test (IFP30-7 and 30-8). The aquifer behaviour is typical of a dual-porosity medium and the hydrodynamic parameters are very similar (Table 5). The hydraulic conductivity of the fissured granite is on average 6.4 x 10 -6 m/s, the storativity is about 7.2 x 10 -3 (98% in the matrix, 2% in the fissures), and the matrix hydraulic conductivity is about 3.0 x 10 -9 m/s.

Pumping tests at the IFP1 site

At the IFP1 site, wells were drilled in the granite, in the pinch-out zone of the quartz reef (Fig. 3a). A pumping test in IFP1-6 revealed the channelized structure of the aquifer after a few minutes of pumping (1/2 slope of the derivative curve on Fig. 6c). This is consistent with the geological information and particularly the ‗U'-shape geometry of the aquifer (Fig. 4a).

Therefore, a Theis model combined with two parallel no-flow boundaries was used for the interpretation (Fig. 6c). Several hypotheses were tested for the orientation of the two no-flow boundaries. The best result, which was constrained by the modelled drawdowns for the 10 wells, was obtained for a direction sub-parallel to the quartz reef (N195). Aquifer parameters are quite homogeneous (Table 6) with, however, a certain variability in the hydraulic conductivity and storativity values. The average hydraulic conductivity is 4.5 x 10 -6 m/s and the average storativity 7.8 x 10 -4 (-). Distances from IFP1-6 to the two no-flow boundaries are on average 20 and 44 m respectively, which is consistent with the geological information (Fig. 4a). The nearest no-flow boundary, located east of IFP1-6 (d1=19.5±9 m), corresponds to the dolerite dyke while the farthest boundary is the western limit of the verticalized fissured layer of the granite (d2=44.3±2 m).

Nevertheless, some discrepancies to this overall interpretation are found. Interpretation of the observation well IFP1 data gives higher distances to limits, and particularly to the farthest one (138 m) because of the very low storage coefficient computed for this well (S=7 x 10 -5 ; dist.limit=1.5[Tt/S] 1/2 ; [START_REF] Jacob | Drawdown test to determine effective radius of an artesian well[END_REF]. This low value compared to the average found for the site is probably caused by a highly conductive zone connecting this well to the pumping well thus enhancing a rapid pressure transfer between the two wells, quicker than the general one found at the site scale. The re-computation of the distance to limits with the average storage coefficient found at the site scale gives results that are consistent with the ones found in the other wells (Table 6). Compared to other wells, drawdown at IFP 1-3, 1-4 and 1-8 was very low (<0.8 m) which is explained by their location outside the permeable channel (Figs. 3a and4a). The data were interpreted with the Theis model, but the estimated hydraulic parameters, K and S, are non-realistic and differ by several orders of magnitude from the ones obtained in other wells. These wells are thus located in an aquifer that is poorly connected to the permeable channel. These records nevertheless show that the dolerite dyke no-flow boundary is not a completely impervious limit, but rather a low-permeability barrier.

The second test performed in IFP1-8 produced drawdown in only a few observation wells (IFP1-4 and farmer wells BW1 and 2; Fig. 3a). The eight other observation wells did not react to pumping, corroborating the presence of a no-flow boundary between IFP1-8 and IFP1-5, i.e. the dolerite dyke. This test reinforces the scheme deduced from the test at IFP1-6. The interpretations give similar results, with an average hydraulic conductivity of the aquifer at 1.5 x 10 -5 m/s and a storativity, S, of about 2.2 x 10 -4 (-).

Anisotropy and heterogeneity in hydraulic conductivity

Fissure measurements on the quartz-reef outcrops at IFP1 and Kothur sites show that the fissure network is well organized with dominant sets that are sub-parallel and sub-orthogonal to the reef axis (see supplemental material). Moreover, the variability in hydraulic conductivity -particularly within the quartz reef (logK slug : -5.6±1.0) -is high because of the variation in the weathering grade. As a consequence, anisotropy and heterogeneity in hydraulic conductivity are expected and may explain the variability in hydraulic conductivity and storativity deduced from the pumping tests.

Homogeneous and anisotropic aquifer: the stratiform horizontal fissured layer

In an ideal homogeneous and anisotropic aquifer, the hydraulic conductivity on the horizontal plane can be represented by a tensor characterized by two orthogonal major and minor axes.

When a pumping test is performed, the drawdown geometry forms an ellipse that reflects the anisotropy ratio [START_REF] Hantush | Analysis of data from pumping tests in anisotropic aquifers[END_REF][START_REF] Hantush | A method for analyzing a drawdown test in anisotropic aquifer[END_REF][START_REF] Ramey | Interference analysis for anisotropic formationsa case history[END_REF][START_REF] Neuman | Determination of horizontal aquifer anisotropy with three wells[END_REF]. The estimation of the anisotropy ratio and of the major and minor hydraulic conductivity orientations requires several observation wells located at various angle from the two anisotropy axes (e.g. [START_REF] Neuman | Determination of horizontal aquifer anisotropy with three wells[END_REF]. For such a medium and considering an isotropic storativity, drawdown curves from observation wells will all be characterized by the same semi-log slope (Jacob's straight-line), regardless of the distance to the pumping well, leading to equal estimated transmissivity or hydraulic conductivity values for each observation well. Only the intercept of the straight line with the time axis will differ because the anisotropy terms are only incorporated in the well-function (or within the log term for the Jacob's approximation). The analytical solutions used here start (Theis + two no-flow boundaries) or end (Hantush and dual porosity) with radial flow (straight-line on semi-log diagram), suggesting that in the scheme of an ideal anisotropic aquifer the hydraulic conductivity estimates for a particular pumping test should be the same or very close between wells. Such a statement is valid for the test performed at IFP30-4 in the horizontal stratiform fissured granite far from the quartz reef, where observation wells are laid out at right angles (K_30-4K_30-8K_30-3 6.4x10 -6 m/s; Table 5). In addition, the estimated storativity for the two observation wells is very close (7-8x10 -3 ), indicating an anisotropy ratio near 1 when using the earlier mentioned anisotropic model. This suggests that at IFP30-4 the stratiform horizontal fissured layer is neither affected by a significant anisotropy in hydraulic conductivity onto the horizontal plane, nor affected by significant heterogeneities in hydraulic conductivity at the pumping-test scale. However, this information should be confirmed by one or more additional observation wells in other directions, or by additional tests in one of the two observation wells [START_REF] Neuman | Determination of horizontal aquifer anisotropy with three wells[END_REF].

Heterogeneous and anisotropic aquifer: verticalized fissured granite and quartz

In the quartz reef (tests in IFP30-5 and IFP30-10) and in the granite at the pinch-out zone of the reef (IFP1-6), the estimated hydraulic conductivity differs between wells with a factor between 3 and 5 (Tables 3, 4 and6). This implies that the tested aquifers are not ideal anisotropic aquifers or/and are affected by significant heterogeneity in hydraulic conductivity as illustrated by the analysis of slug-test data (Fig. 5).

Based on synthetic aquifers with heterogeneous transmissivity and homogeneous storativity, [START_REF] Meier | An evaluation of Jacob's method for the interpretation of pumping tests in heterogeneous formations[END_REF] and [START_REF] Sánchez-Vila | Pumping tests in heterogeneous aquifers: an analytical study of what can be obtained from their interpretation using Jacob's method[END_REF] showed that for long-duration tests the transmissivity values obtained with the Jacob straight-line method on observation wells at various distances from the pumping well are almost identical to the average transmissivity of the medium, even if the heterogeneity in transmissivity is important. However, storativity estimates vary considerably which, according to these works, is a result of the heterogeneity in transmissivity. Our aquifer hydraulic conductivity and storativity data set can be interpreted similarly: the used methods provide representative hydraulic conductivity values of the aquifers since K estimates are comparable, and the high variability in storativity estimates, which can be about an order of magnitude for the same test, may be the consequence of a heterogeneous hydraulic conductivity field. According to [START_REF] Meier | An evaluation of Jacob's method for the interpretation of pumping tests in heterogeneous formations[END_REF] and [START_REF] Sánchez-Vila | Pumping tests in heterogeneous aquifers: an analytical study of what can be obtained from their interpretation using Jacob's method[END_REF], a well located in a high hydraulic-conductivity zone compared to the estimated one, results in an overestimated storativity value, and vice versa. Furthermore, the obtained transmissivity depends not only on the transmissivity values along the connecting path, but also on the average transmissivity values of the full domain (Sánchez-Vila et al., probably because of the very local estimation assigned to this method (slug tests have an action radius of a few metres around the well). Strong increases of the transmissivity variability of these numerical models, which was introduced by increases in the variance transmissivity fields, produced ellipsoidal shapes of estimated transmissivity around the pumping well, which were even better defined when the highest transmissivity zones were connected by highly permeable fractures. Even though these shapes may be exaggerated due to numerical artefacts, they seem nevertheless related to the transmissivity field. This point was, however, not discussed by the authors.

Proposed methodology for identifying the major and minor axes in hydraulic conductivity and results

In order to understand the variations in hydraulic conductivity estimated from one well to another, we prepared graphs representing the hydraulic conductivity values obtained at each observation well on the X-Y plane (K values from Tables 3, 4 and6). If hydraulic conductivity estimates depend upon the hydraulic conductivity values along the connecting path between the pumping well and the observation well as well as on the average hydraulic conductivity value at the pumping test scale, one may expect variations in estimated values according to the direction between the pumping well and the observation well.

The hydraulic conductivity estimated in a particular observation well (Kobs.well) is plotted as a function of the angle between the geographical north (reference axis) and the pumpingwell / observation-well direction. Each Kobs.well is thus defined by two components, K E representing the projection onto the east-west or X-axis (K E =Kobs.wellxCos) and K N , the projection onto the north-south or Y-axis (K N =Kobs.wellxSin). The pumping well is located at the centre of the graph.

The three tests where the method was applied (Figs. 7a, b andc) clearly show correlations between the estimated hydraulic conductivity and the orientation of the well to the pumping well. Regardless of the distance between pumping well and observation well, the plotted hydraulic conductivities describe ellipses showing that the hydraulic conductivities obtained along the reef axis are systematically larger than the orthogonal estimates. Higher values are explained by the velocity at which the drawdown propagates, which depends upon the hydraulic diffusivity (D=K.b/S; b being the aquifer thickness). Because the variability in hydraulic conductivity is usually much larger than the storativity one, diffusivity is thus large for highly conductive zones and small for low-conductivity zones. Thus, when pumping, a vertical hydraulic gradient from lower to higher conductive zones is induced, which means that the highly conductive fissures or the fissure network that is best connected to the pumping well can be regarded as drainage structures or as lateral extensions of the well [START_REF] Schad | Effects of the investigation scale on pumping test results in heterogeneous porous aquifers[END_REF].

Hydraulic conductivity ellipses are defined by major and minor axes (Kmax and Kmin) and are supposed to characterize an average pseudo or apparent anisotropy introduced by the heterogeneity in hydraulic conductivity of the aquifer, or, in the case of fractured media, the variability in hydraulic connectivity with the different well locations;  representing the angle between the Y-axis and Kmax. The ratios Kmax/Kmin are comparable, between 2.6 and 3.1, and reveal the degree to which the fissures in the two orthogonal orientations are connected; however, the ratio values were not interpreted in a quantitative sense. To agree with previous work -the evaluated hydraulic conductivity at the pumping well should be representative of the whole heterogeneous medium [START_REF] Meier | An evaluation of Jacob's method for the interpretation of pumping tests in heterogeneous formations[END_REF] -but also to validate the analysis, the hydraulic conductivity estimated for the pumping well should represent the average hydraulic conductivity of the aquifer and should be close to the average hydraulic conductivity value deduced from this analysis (K mean =[Kmax . Kmin] 0.5 ). Using such an approach, K mean_30-10 =5.9 x10 -6 m/s is compared to K IFP30-10 = 6.3 x 10 -6 m/s, K mean_30-5 =3.6 x10 -6 m/s to K IFP30-5 = 4.2 x 10 -6 m/s, and K mean_1-6 =5.1 x10 -6 m/s to K IFP1-6 = 5.1 x 10 -6 m/s. Differences are thus small, between 0 to 15%, compared to the ones obtained from pumping wells. However, the ellipses are mismatched in a few points. Some lie outside the ellipse, such as IFP1-1, which suggests that the pressure transfer is easier between this well and the pumping well due to a preferential flow path as previously suggested, or they lie inside the ellipse due to a lower conductive zone (IFP30-8, IF30-7; Fig. 7b).

From this analysis, it results that the major ellipse axes are clearly oriented sub-parallel to the quartz reef. Consequently, for the tests in the quartz reef, it can be concluded that the subvertical fissure set parallel to the reef axis ensures a preferential connection between the pumped and the other wells. However, at right angles to the reef the connection is lower, most probably because of the variation in the degree of fissuring between the heart and the side of the reef.

In the verticalized fissured layer at the quartz vein pinch-out zone (Fig. 7c), we also find variability in the connectivity of the fissure network. The major axis is also sub-parallel to the sub-vertical fissure set and parallel to the reef axis. This result suggests that in the pinch-out zone of the quartz reef, and probably also at the contacts on the eastern and western sides, the verticalized fissured granite is affected by a dominant set of sub-vertical conductive fissures sub-parallel to the reef. This feature differs from the classic stratiform horizontal fissured layer, where horizontal conductive fissures dominate (Fig. 1; [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF].

HYDROGEOLOGICAL CONCEPTUAL MODEL OF THE QUARTZ

REEF

Geological, hydrogeological and geophysical investigations performed along the quartz reef allow completing the understanding of the structure and the hydrogeological properties of hard-rock aquifers exposed to deep weathering processes (Fig. 8).

Aquifer geometry

Near the quartz reef the weathering fronts within the granite (e.g. bottom of saprolite and of the fissured layer) are noticeably deeper compared to the classic stratiform weathering profile.

On average, the thickness of the saprolite increases by a factor of 1.5 to 3 and that of the fissured layer by a factor of 3 to 5. Near the contact, the geometry of the weathering profile in the granite is, as for the classic stratiform weathering profile, characterized by two subparallel layers: the saprolite and the fissured layers. However, they are not sub-parallel to the paleo-surface contemporaneous with the weathering, but sub-parallel to the discontinuity borders. As a result, the weathering profile exhibits a ‗U' shape composed of parallel layers.

The quartz reef is characterized by mainly sub-vertical fissures that are sub-parallel and suborthogonal to the reef axis. The top layer is highly fissured and weathered and the fissure density rapidly decreases with depth. Density in fissuring and the grade of weathering vary also in space and are more pronounced at the contact with granite, i.e. on the sides and in the pinch-out zones, than at the heart of the vein. This ‗U' shape of the weathering layers results from similar processes as those observed in horizontal stratiform weathering profiles:

(i) The constraints resulting from the development of the weathering profile in the granite surrounding the quartz reef induce fissuring of the quartz reef. This explains the numerous metre-to decametre-wide fissures observed in the quartz reef outcrops.

Constraints in the granite are induced by the swelling of some minerals, particularly biotite [START_REF] Wyns | Caractérisation multiparamètres des horizons de subsurface (0-100 m) en contexte de socle altéré[END_REF](Wyns et al., , 2004;;[START_REF] Lachassagne | The fracture permeability of Hard Rock Aquifers is due neither to tectonics, nor to unloading, but to weathering processes[END_REF];

(ii) This weathering-induced fissuring enhances water circulation and in return favours a local deepening of the weathering fronts in the granite near the contact with the quartz reef. The lower strain is thus deviated in the direction of the discontinuity, which favours the development of a verticalized fissured layer in the granite at the contact with the reef. This point is supported by the heterogeneity in hydraulic conductivity, which shows evidence that this layer is characterized by dominant conductive fissures sub-parallel to the intrusion (iii) The induced weathering favours deeper fissures in the quartz vein, and so on.

Thus, during the weathering of granitic formations such a discontinuity favours a local deepening of the weathering fronts in the host rock and thus favours a thickening of the transmissive part of the aquifer (the fissured layer). In addition, the quartz vein is also subject to weathering. It causes small-scale fissures that further enhance the hydraulic conductivity.

This in situ weathering amplifies the above-described process.

Intrusives or veins whose weathering products are of low permeability, such as those of dolerite dykes, do not favour water circulation, neither in their heart, nor around them. The stratiform weathering profile of the neighbouring granite thus remains undisturbed. Moreover, such dykes -as in the case of the IFP1 site-act as low-permeability barriers that compartment the horizontal fissured layer.

Hydrodynamic properties of the aquifer

Granite near the contact with the discontinuity

Far from the quartz reef, the stratiform horizontal fissured layer of the granite is characterized by quite homogenous hydrodynamic parameters (pumping tests in IFP1-8 and IFP30-4): the hydraulic conductivity is about 10 -5 m/s and the storativity ranges between 0.2 and 7 x 10 -3 . These values are similar to the ones found in the same area: K=5 x 10 -5 m/s and S= 6 x 10 -3 , [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF], and K=1.6 x 10 -5 m/s [START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF]. As previously shown by [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF], the granite is characterized by dual-porosity behaviour, being affected by sub-horizontal and sub-vertical fissure networks and low-permeability blocks with a hydraulic conductivity of about 3 x 10 -9 m/s. In addition, the two fissure sets induce a significant horizontal anisotropy in permeability (K horiz. /K vert. : 10) because of the welldeveloped horizontal conductive fissure network. The pumping test performed in IFP1-8 within the horizontal fissured layer did not show such behaviour as a result of the presence of the low-permeability dolerite dyke which became apparent at the early stage of the pumping and masked the dual-porosity behaviour.

The average hydraulic conductivity of the verticalized fissured layer along the quartz reef is about 4.5 x 10 -6 m/s, and the average storage coefficient 7.8 x 10 -4 (-). The similarity between these values and the ones from the ‗classic' stratiform horizontal fissured layer in the area suggests that the conductive zones of both horizontal and verticalized fissured layers have a similar origin, i.e. weathering processes. As in the case of the IFP1-8 test, the two no-flowboundaries that appear at the early stage of pumping in IFP1-6 probably masked the dualporosity behaviour of the verticalized granite. However, this layer may also behave like such a medium, as suggested by the low local hydraulic-conductivity values derived from slug tests, 1 to 4 x 10 -7 m/s (IFP30-3 and IFP1-3), which are typical of the hydraulic conductivity values in the block as given by [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF]. Thus, the verticalized fissured layer of the granite along the contact with the reef is formed by a conductive fissure network (4.5 x 10 - 6 m/s) and possibly by low-permeability blocks (about 10 -7 m/s). In addition, this verticalized layer is characterized by a dominant set of conductive fissures that are sub-vertical and subparallel to the reef axis (Ky>Kx). This result differs from the ‗classic' horizontal fissured layer, where no significant anisotropy was found in the horizontal plane (KxKy at IFP30-4) and where a conductive sub-horizontal fissure network dominates (K x. /K z =10; [START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF].

Quartz aquifer

The fissured quartz is characterized by a dual-porosity behaviour; the average hydraulic conductivity of the fissure network is 4 to 6 x 10 -6 m/s, storativity being 3 to 5 x 10 -4 and mainly ensured by low-permeability blocks (2.4 x 10 -9 m/s). However, hydrodynamic properties in the quartz are highly variable in space and closely depend upon the grade of weathering and fissuring: it is poorly fissured with low permeability in the heart and highly conductive at the contact with granite due to an enhanced fissuring. The fissured quartz is also characterized by anisotropy onto the vertical plane, K x. /K z : 3, which is attributed to a decreasing density with depth of the conductive fissures. As for the verticalized fissured layer of the granite, conductive fissures that are sub-vertical and sub-parallel to the reef dominate in the quartz, inducing a higher hydraulic conductivity parallel to the reef (Ky>Kx); however, the anisotropy in hydraulic conductivity could not be estimated.

Even if at Kothur site the aquifer also has a ‗U' shape, hydraulic tests do not show evidence of channelized groundwater flow. This is explained by the fact that during the tests the groundwater level was very shallow (4 to 5 m deep) whereas it was more than 15 m deep at the IFP1 site. As a consequence, at Kothur the ‗U'-shape aquifer was still well-connected to the surrounding horizontal stratiform fissured layer of the granite aquifer, giving the impression of an infinite aquifer while at the IFP1 site only the ‗U' shape aquifer contributed to pumping.

CONCLUSIONS

The obtained results complement the conceptual hydrogeological model developed for granite aquifers [START_REF] Taylor | A tectono-geomorphic model of the hydrogeology of deeply weathered crystalline rock: evidence from Uganda[END_REF][START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF][START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF] in places where geological discontinuities disturb the weathering profile.

Near the quartz reef, weathering processes appear to be at the origin of an enhanced local hydraulic conductivity both in the vein and in the surrounding granite. The quartz reef and the accompanying verticalized fissured granite layer at the contact constitute a composite aquifer characterized by a ‗U'-shaped geometry that acts as a local drain. This higher permeability is probably at the origin of the sharp deepening of the weathering front in the surrounding granite and in the quartz reef itself. Hydrodynamic properties of the verticalized fissured granite are comparable to those found in the classic horizontal fissured layer. However, the anisotropy in hydraulic conductivity is reversed as sub-vertical fissures control the groundwater flow at the discontinuity, while fluxes in the classic horizontal fissured layer are mostly controlled by a dominant set of sub-horizontal fissures.

Where the weathering can propagate downward, other geological discontinuities such as deep fractures, leucocratic dykes, pegmatite, or contacts between different geological formations, may also constitute local structures with similar aquifer properties as those described here. As a result, the weathering front should also form a ‗U' shape depending upon how vertical the discontinuity is and on the duration of the weathering processes. Such types of structures, searched for and observed by hydrogeologists and geophysicists for borehole siting (e.g. [START_REF] Sander | Lineaments in groundwater exploration: a review of applications and limitations[END_REF] are commonly attributed to -tectonic fracturing‖ (-fault gouges‖). The novelty here is to show that the geometry of the aquifer and its hydrodynamic properties are inherited from deep and local in situ weathering processes, but are unrelated to tectonic activity. Therefore, where this deepening of the weathering front has developed such structures may be valuable targets for borehole siting, particularly because of the enhanced transmissivity due to an increase in aquifer thickness. However, the permeability of such structures is highly variable in space, being low in the heart and high at the contacts. Such contrasting hydrodynamic properties, and also the more or less important development of the weathering profile along these structures, may explain the high variability of well yields using a standard approach based only on the study of lineaments. Thus, lineament studies combined with structural studies of the weathering profile would improve the chances of siting productive wells. At the opposite, discontinuities that produce weathering material of low permeability, such as dolerite dykes, do not locally enhance the thickness of weathered layers in granite or its hydraulic conductivity, making such structures unfavourable targets.

Numerical issues, both for modelling the evolution of the weathering profile along such discontinuities and for modelling of the tests, as well as additional hydraulic tests to assess preferential flow paths (flowmeter tests), to pursue the quantification of heterogeneity in hydrodynamic properties in relation to underground reality (fissures orientation, structure of the weathered layers), are topics of interest for future research. In addition, the development of new analytical solutions for the interpretation of pumping tests in such media would help in improving their characterization, such as including anisotropy, channelized flow and/or the influence of surrounding aquifers. Finally, because of the complex structures and associated properties of such aquifers, special attention has to be paid to water management. The channelized or fractionalized patterns of the flow paths have strong implications in terms of contaminant transport and for delineating appropriate protection perimeters for water-supply wells. Efforts should thus be made for improving the hydrogeological and hydrogeochemical characterization of these aquifers to ensure their safe and sustainable groundwater exploitation. The methods for characterizing the aquifer used here could be used and improved for achieving this goal.

Wyns, R., J. M. Baltassat, P. Lachassagne, A. Legchenko, J. Vairon, and F. Mathieu. 2004. Application of SNMR soundings for groundwater reserves mapping in weathered basement rocks (Brittany, France), Bulletin de la Société Géologique de France, 175

(1), pp. 21-34.

Figure captions

Figure 1. Idealized weathering profile in hard-rock intersected by the current topography. The insert presents the area of interest of the paper: geological discontinuities [START_REF] Wyns | Caractérisation multiparamètres des horizons de subsurface (0-100 m) en contexte de socle altéré[END_REF]. Table 5. Results of the interpretation of the pumping in IFP30-4 (Kothur site; granite aquifer).

Model: dual-porosity. * drawdown at the end of pumping, and ** geometrical mean. Table 6. Results of the interpretations of pumping in IFP1-6 (IFP1 site; granite aquifer).

Model: Theis + two no-flow boundaries. * drawdown at the end of pumping, ** estimated using the average S (7.8E-4), *** the number between comas is estimated using the average S and **** K and S are geometrical mean, do not include data from observation wells IFP 1-3, 1-4 and 1-8 (located in another aquifer); distances to limits are arithmetic mean.

Tables

Figure 2 .

 2 Figure 2. Location of the studied area with main geological features (G.S.I., 2002); Maheshwaram area, Ranga Reddy District, Andhra Pradesh, India.

Figure 3 .

 3 Figure 3. Geological information and borehole locations. (a) IFP1 and (b) Kothur pilot sites. Lines A, B and C materialize the cross sections (see Figs. 6), and the other dotted lines the location of electrical-resistivity profiles (identical to line A for IFP1 site). Grey curves: soil elevation (masl). BW: farmer's wells.

Figure 4 .

 4 Figure 4. Interpreted geological cross sections. (a) perpendicular to the quartz reef at IFP1 site, (b) perpendicular and (c) parallel to the quartz reef at Kothur site. Location of the cross sections is shown on Figure 3 (lines A, B and C).

Figure 5 .

 5 Figure 5. Histogram on a logarithmic scale of the hydraulic conductivity deduced from slug tests analysis. The data from the boreholes located in the quartz reef (average: LogK: -5.58 ±1.0; number of data [n]: 6) and in the granite surrounding the reef (average: LogK: -5.62 ±0.6; n: 15) are compared to the granite far from discontinuities (average: LogK: -5.36 ±1.0, n: 30; Maheshwaram area [same area] Maréchal et al., 2004). LogK: natural logarithm of hydraulic conductivity.

Figure 6 .

 6 Figure 6. Observed and modelled drawdowns and derivatives. (a) Pumping well IFP30-5 and observation well IFP30-2; model: Hantush (WinIsape software). (b) Pumping well IFP30-10, and observation wells IFP30-5 and IFP30-.6; model: dual porosity. IFP 30-6 considers the response of the matrix (WTFM software) (c) Pumping well IFP1-6 and observation well IFP1-9; model: Theis+2 no-flow boundaries (WinIsape software).

Figure 7 .

 7 Figure 7. Heterogeneity in hydraulic conductivity on the horizontal plane. (a) Pumping at IFP30-5; hydraulic conductivity fromTable 3. (b) Pumping at IFP30-10; hydraulic conductivity from Table 4. (c) Pumping at IFP1-6; hydraulic conductivity from Table 6.

Figure 8 .

 8 Figure8. Idealized conceptual hydrodynamic model of the aquifer associated to the quartz reef. Data from the horizontal stratiform fissured layer are from[START_REF] Maréchal | Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fissured layer of a hard rock aquifers[END_REF] and[START_REF] Dewandel | A generalized hydrogeological conceptual model of granite aquifers controlled by single or multiphase weathering[END_REF].

*

  Geology of the aquifer zone; ** drawdown at the end of pumping, and *** number of observation wells that reacted to pumping. The diagnoses are deduced from derivative curve analyses. Table 3. Results of the interpretation of the pumping in IFP30-5 (Kothur site; quartz aquifer). Model: Hantush (partially penetrating well). * drawdown at the end of pumping, ** K and S are geometrical mean; others are arithmetic mean, and *** estimated from geological logs. StDev.: standard deviation. Table 4. Results of the interpretation of pumping in IFP30-10 (Kothur site; quartz aquifer). Model: dual-porosity. * observation well interpreted in the matrix, ** drawdown at the end of pumping, and *** geometrical mean.

  Figures

  Figure6.

Table captions Table 1 .

 captions1 Characteristics of the wells in the IFP1 and Kothur sites and hydraulic conductivity (K) deduced from slug tests.

Table 2 .

 2 Characteristics of pumping tests performed at the IFP1 and Kothur pilot sites.

Table 1 .

 1

					Casing				Saprolite	Highly
					depth				thickness -granite	weathered	Basement	K_slug
	Well ID	site	Depth (m)	(m)	Main Geology		(m)	quartz (m)	depth (m)	(m/s)
	IFP1	IFP1		42.0	8.8 Granite			18.3 -	37.5 7.4E-06
	IFP1-1	IFP1		59.5	5.8 Granite			15.5 -	51 3.2E-06
	IFP1-2	IFP1		35.0	12 Granite			22.5 -	30.2 7.4E-06
	IFP1-3	IFP1		35.0	5.7 Granite			17 -	29 4.4E-07
	IFP1-4	IFP1		68.6	6 Dolerite+granite no data	-	77.7 7.6E-07
	IFP1-5	IFP1		73.2	11.6 Granite			9	32 2.9E-06
										2.5 (overlying
	IFP1-6	IFP1		73.2	22.9 Granite			13.5	the saprolite)	51 3.1E-06
	IFP1-7	IFP1		59.4	27.1 Granite			11.5	44 3.9E-06
									7 m of highly
								weathered
	IFP1-8	IFP1		54.9	11.6 Dolerite+granite	dolerite (clay)	41 6.5E-06
	IFP1-9	IFP1		59.4	18.3 Granite			12.8	49 6.0E-06
	IFP30-1	Kothur		91.4	11.6 Quartz			6.5	91.4 5.6E-07
										4.0 (overlying
	IFP30-2	Kothur		45.7	29.9 Granite			20.6	saprolite)	41 2.8E-06
										2.7 (overlying
	IFP30-3	Kothur		50.3	29.9 Granite			27.1	saprolite)	36.5 8.5E-08
	IFP30-4	Kothur		50.3	29.9 Granite			29.2	42 8.8E-06
	IFP30-5	Kothur		77.7	17.4 Quartz			20.2	77.7 5.1E-06
	IFP30-6	Kothur		50.3	11.6 Quartz			6.8	50.3 5.1E-08
										8.8 (overlying
	IFP30-7	Kothur		64.0	30.2 Granite			16.7	saprolite)	51.5 4.5E-07
	IFP30-8	Kothur		50.0	42.7 Granite			28	38 6.3E-06
	IFP30-9	Kothur		59.4	30.5 Quartz			39.5	50 1.1E-05
	IFP30-10 Kothur		73.0	27.4 Quartz			30	73 1.6E-05
	IFP30-11 Kothur		74.0	12.2 Quartz			25.5	36.5 1.30E-05
	Pumping Wells	Tested aquifer *	Well radius (m)	Pumping rate (m 3 /h)	Duration of the test (min.)		Max. dd. (m)**	Nb. of wells *** obs.	Diagnosis
	IFP1-6	granite		0.11		13.5	1580	21.48	9	Channelized flow
	IFP1-8	granite		0.11		3.5	1430	5.51	3	One-no flow boundary
	IFP30-4	granite		0.11		6.8	1440	10.31	2	Dual porosity
	IFP30-5	quartz		0.11		4.9	1440	8.33	7 Partially penetrating well
	IFP30-10 quartz		0.11		16.3	2890	9.9	8	Dual porosity

Table 2 .

 2

									Aquifer	Screened
							Anisotropy	thickness	thickness
	Well ID		r (m)	s-max* (m) K (m/s)	S (-)	Kx/Kz (-)	(m)	(m)***
	IFP30-5	pump. well	0.11	8.33 4.2E-06 -		3.1	110	35
	IFP30-1	obs.well	30.1	1.12 3.6E-06 1.4E-03	3.5	140	10
	IFP30-2	obs.well	27.3	2.13 2.6E-06 4.0E-04	2.0	130	40
	IFP30-6	obs.well	96.2	0.44 3.4E-06 1.8E-03	3.0	130	20
	IFP30-7	obs.well	52.5	0.45 2.9E-06 7.5E-03	10.0	110	50
	IFP30-9	obs.well	88.0	1.18 5.1E-06 4.5E-05	1.0	100	100
	IFP30-10 obs.well	135.5	1.06 5.5E-06 2.7E-05	1.0	100	100
	IFP30-11 obs.well	180.6	0.85 5.9E-06 3.5E-05	1.0	100	100
	IFP30-3	obs.well	62.5 No reaction -		-	-		-	-
	IFP30-4	obs.well	103.0 No reaction -		-	-		-	-
	IFP30-8	obs.well	118.0 No reaction -		-	-		-	-
				Average**	4.0E-06 3.2E-04	3.1	115.0
				Std.Dev (±) 1.2E-06 2.7E-03	3.0	16.0
	Table 3.
	Well		r (m)	s-max** (m) Kf (m/s) Km (m/s)	Sf (-)	Sm (-)	Stot (-)
	IFP30/10 pump. well	0.11	9.9 6.3E-06	2.4E-09 -	-	-
	IFP30/1	obs.well	123.2	3.38 8.5E-06	4.5E-10 4.9E-05 6.1E-05	1.1E-04
	IFP30/2	obs.well	115.1	3.63 6.5E-06	1.9E-09 4.2E-05 1.8E-04	2.2E-04
	IFP30/5	obs.well	135.7	4.42 6.5E-06	6.0E-10 1.5E-05 4.9E-05	6.4E-05
	IFP30/6* obs.well	222.0	1.27 1.1E-05	1.5E-09 8.5E-05 1.3E-04	2.1E-04
	IFP30/7* obs.well	145.0	1.04 6.0E-06	8.0E-09 2.4E-04 2.2E-03	2.5E-03
	IFP30/8	obs.well	92.0	0.18 2.1E-06	2.6E-08 4.9E-03 6.1E-02	6.6E-02
	IFP30/9	obs.well	49.4	5.5 6.5E-06	1.4E-09 2.6E-05 1.2E-04	1.5E-04
	IFP30/11 obs.well	57.8	4.97 6.2E-06	5.5E-09 5.2E-05 1.9E-04	2.4E-04
	IFP30/3	obs.well	132.8 No reaction	-	-		-	-	-
	IFP30/4	obs.well	144.5 No reaction	-	-		-	-	-
				geomean***	6.2E-06	2.4E-09 8.9E-05 3.5E-04	4.6E-04
				StDev (±)	2.4E-06	8.2E-09 1.7E-03 2.1E-02	2.3E-02

Table 4 .

 4

	Well	WTFM	r (m) s-max* (m) Kf (m/s) Km (m/s)	Sf (-)	Sm (-)	Stot (-)
	IFP30/4 pump. well	0.11	10.31 6.4E-06	4.0E-09 -	-	-
	IFP30/3 obs.well	40.6	1.08 6.5E-06	7.0E-09 9.5E-05 6.7E-03 6.8E-03
	IFP30/8 obs.well	59.8	0.52 6.3E-06	9.0E-10 1.6E-04 7.6E-03 7.8E-03
				geomean** 6.4E-06	2.9E-09 1.2E-04 7.1E-03 7.2E-03
				StDev (±)	1.0E-07	3.0E-09 4.5E-05 6.7E-04 7.2E-04
	Table 5.

Acknowledgements

The authors are grateful to the research-sponsorship from BRGM (France), the Embassy of France in India, ,NGRI (India) and from the French National Research Agency (ANR) under the VMCS2008 program (SHIVA project n°ANR-08-VULN-010-01). Colleagues from NGRI and BRGM are thanked for their fruitful comments, discussions and technical assistance in the field. The three anonymous Journal referees are thanked for their useful remarks and comments that improved the quality of the paper. We are grateful to Dr. H.M. Kluijver for revising the English text.


Romain Tavenard

Mathieu Lagrange

Efficient Cover Song Identification using approximate nearest neighbors

Automatically detecting cover songs imply being robust to several kinds of musical modulations. Timbral variance can be accounted at the feature level, but key and most importantly tempo variations have to be dealt with at the retrieval stage. For that purpose, most state of the art approaches consider exhaustive search based on song to song matching methods that fail at scaling up.

In this paper, we introduce a hybrid technique. It first retrieves the approximate neighbors of each query chroma descriptor. In a second stage, the temporal consistency is exploited to further filter out some matches, thereby filtering irrelevant songs. Our method performs a search in a dataset comprising 80 songs in about 1s, while achieving satisfactory accuracy compared to the best performing techniques of the state of the art.

Introduction

Music retrieval is a wide problem. The typical tasks in the literature range from what is usually coined "music similarity", which typically consists in predicting a more or less loosely defined tag such as the musical genre [START_REF] Logan | A Music Similarity Function Based on Signal Analysis[END_REF]. At the other extreme, fingerprinting [START_REF] Cano | A Review of Audio Fingerprinting[END_REF] aims at detecting the use of a specific audio recording. In between lies a class of alternative tasks, the most widely considered being the detection of cover songs. They distinguish themselves within each other depending on the kind of invariants that are elicited. Are we seeking for different performances of the same musical piece by the same performers, by different performers ? Shall the instrumentation be preserved ?

In this paper, we consider covers to be "different renditions of the same underlying musical piece", as previously used [START_REF] Ellis | The 2007 LabROSA cover song detection system[END_REF][START_REF] Serrà | Cross recurrence quantification for cover song identification[END_REF]. More precisely, two renditions are not constrained to be interpretations of the same musical score. Indeed, a listener usually considers that a song is a cover of another one if they share some kind of harmonic progression, i.e., chord progression and melody line. To reflect this, most algorithms focus on detecting longest sequences of "tonal content" that match within the two songs to be compared. The issue is then to be able to perform this detection in a reliable yet efficient manner within large scale databases.

This challenge is crucially linked to the notion of temporality which is arguably a key aspect in music. In music similarity search, the categories to be considered are so loosely defined that accounting for temporality is often neglected. In audio fingerprinting, the tempo and timing shall not vary. Therefore, short-term temporality is usually encoded for increasing precision.

When considering covers, the tempo and the timing vary from rendition to rendition. There are several way to address this issue. The first is to consider that the tempo is known as a prior knowledge. Assuming that the tempo can reliably be estimated and that the timing variations are negligible, one can compute tonal features that are tempo aligned [START_REF] Ellis | The 2007 LabROSA cover song detection system[END_REF]. The second is to design a matching algorithm that is robust to those changes, by allowing the warping of the time axis [START_REF] Serrà | Cross recurrence quantification for cover song identification[END_REF]. Such approaches lead to the highest accuracy so far, but the matching algorithm is not scalable due to his prohibitive complexity. Another class of methods build a temporal model for each song and evaluate the prediction capability of the model of a query given the tonal sequence of a potential match [START_REF] Serrà | Model-based cover song detection via threshold autoregressive forecast[END_REF].

However, all those methods imply an exhaustive search of the database, which heavily constraints their scope of applicability. In [START_REF] Buccio | A Scalable Cover Identification Engine[END_REF], indexing technique based on inverted index search is considered. The approach is indeed scalable in time as the average query time is linear in function of the database size. However, the accuracy performance is significantly decreased as the database grows in size, most likely because they do not assume any sequentiality of the features, which increases the chance of false positive matches.

In this paper, we propose a method which is geared towards efficiency and allows us to propose a cover detection scheme that is able to deal with large scale databases. The algorithm is composed of two main contributions. First, a fast nearest neighbor strategy allows us to retrieve the most probable matches within the database. Secondly, incoherent matches are filtered out, with respect to some rules that encodes efficiently timing constraints that are robust to tempo changes. The paper is organized as follows: first, previous work is reviewed in Section 2 in order to motivate our approach that is introduced in Section 3. The proposed algorithm is then evaluated in terms of accuracy and scalability in Section 4. Those results are then discussed in Section 5.

Related Work

Cover song detection is an interesting methodological problem as it comes with well defined ground truth and good agreement on the musical concepts that should be encoded, allowing researchers to focus on the design of relevant re-trieval techniques.

Tonal features

In general, one assumes that versions of the same piece preserve the main melodic line and/or the harmonic progression, regardless of its main key. Therefore, tonal or harmonic content is the most employed characteristic in version identification, by means of the so-called Pitch Class Profile (PCP) or chromas. Numerous variations have been presented in order to extract PCP features, and the reader is referred to [START_REF] Gómez | Tonal description of music audio signals[END_REF] for more details on this topic.

Matching of tempo aligned chromagrams

It is proposed in [START_REF] Ellis | The 2007 LabROSA cover song detection system[END_REF] to make the matching robust to tempo change by assuming knowledge of the tempo (provided by a tempo and beat estimation algorithm). Each song is then represented as a sequence of chroma, each representing the tonal content of one beat. The matching score is then given by the maximal cross-correlation between the possibly transposed chromas sequences. Performance may be enhanced by tuning the pre-processing step, e.g. the beat tracker. Different biases for the average BPM were tested and it was found that 120 BPM was the best performing. As this bias may also influence the beat positions, several matching can be made if considering several BPM assumptions (120 and 240) for each song. Keeping the best one between the four possible correlations further improved the results, see Table 1. [START_REF] Serrà | Cross recurrence quantification for cover song identification[END_REF] suggest to evaluate similarity between two songs by means of their longest common subsequence, computed using dynamic programming. They start by transposing the query into the tonality of the candidate song by estimating the "Optimal Transposition Index", as described in [START_REF] Serra | Chroma binary similarity and local alignment applied to cover song identification[END_REF].

Matching using dynamic programming

The similarity estimation then consists in finding a path in a binary similarity matrix S k , called "Cross Recurrence Plot" and defined as:

S k i,j = (q i ∈ NN k (c j , Q)) ∧ (c j ∈ NN k (q i , C)), (1)
where q i (resp. c i ) is the i-th element of the query (resp. candidate) sequence and NN k (c j , Q) denotes the set of k-nearest neighbours of vector c j among the set of vectors in the query sequence Q.

In order to find flexible paths in this matrix, allowing for local edition operations such as insertions, deletions or replacements, the similarity is defined using a specifically tailored dynamic programming algorithm.

Modeling temporality

An alternative approach is to build a model of the temporal evolution of the features for each song. The prediction capability of this model given another song indicate whether the two songs could match. Then, the cross-prediction error can be regarded as a measure of similarity of the two songs to be compared. The main difficulty resides in the choice of the temporal model. In [START_REF] Serrà | Model-based cover song detection via threshold autoregressive forecast[END_REF], Thresholded Auto Regressive (TAR) models [START_REF] Tong | Threshold autoregression, limit cycles and cyclical data[END_REF] are considered which consists of a collection of AR models where each single one is valid only in a certain domain of the reconstructed state space identified using a clustering technique.

Indexing approaches

In [START_REF] Buccio | A Scalable Cover Identification Engine[END_REF], a Bag Of Features (BOF) approach is considered to describe songs. First, features are quantized in order to encode the indices of their (ordered) 3 most energetic dimensions. Then, each song is arbitrarily segmented into chunks of equal length and each of these chunks is represented by the multiset of its quantized features. Quantization indices are the keys of an inverted file used to index the database. Then, at query time, cosine similarity between the multiset M Qi of the i-th chunk of the query and the multiset M Cj of the j-th multiset of the database candidate is computed by querying the inverted file for each element in M Qi .

Finally, the similarity between sequences Q and C is computed as the product of cosine scores between segments Q i and their best match in C.

Proposed Approach

Our approach is based on an approximate (temporally) local k-nearest neighbour search and an adequate voting strategy. Temporal information is deliberately ignored at the first step so as to allow fast comparison, which efficiently produces the list of nearest neighbors for each descriptor of the query track. The voting strategy builds upon the results of this comparison to filter out temporally incoherent matches and ponderate scores.

Note that this process is repeated for each of the d possible transposition indices for the query. In the following, we describe our algorithm for a given transposition index, the last step of our similarity evaluation being to retain the highest score obtain among all indices, corresponding to the best matching transposition.

Local nearest-neighbour approximation

To improve the efficiency of our scheme, we use an approximate nearest neighbor (ANN) search algorithm trading accuracy against efficiency. The search is performed in an indexing structure containing the set of vectors associated with all the database songs.

The most popular methods for this kind of search are Locality Sensitive Hashing [START_REF] Datar | Locality-sensitive hashing scheme based on p-stable distributions[END_REF] and the FLANN algorithm [START_REF] Muja | Fast approximate nearest neighbors with automatic algorithm configuration[END_REF]. However these methods require to access the full vectors for post-verification, which limits the number of descriptors and therefore the bumber of songs that can be considered without accessing disk storage. We, instead, use a recent method which was shown successful to index billions of vectors [START_REF] Jégou | Searching in one billion vectors: re-rank with source coding[END_REF]. It is an extension of the product quantization method [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], which is based on inverted lists and quantization codes associated with the indexed vectors. The key parameters of this product quantization method, referred to as IVFADC in [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF], are:

• The total number of inverted lists. It is set to 1024 in our case;

• The number M of inverted lists visited per query descriptor. This parameter has the most impact on efficiency. We use M =4.

• The number of bytes per descriptor in the inverted file system, which is fixed to 6 in our case (+4 bytes for the descriptor identifier).

Although the approach of [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF] returns mostly correct neighbors, the distance values obtained from the codes are not very accurate. We, therefore, use the vector re-ranking stage used in [START_REF] Jégou | Searching in one billion vectors: re-rank with source coding[END_REF] to improve the distance estimation and improve the quality of the nearest neighbors. In essence, this re-ranking stage avoids using the raw chroma descriptors by encoding the error vector using quantization codes. This introduces an additional memory cost of 6 bytes per descriptor depending on the desired precision. This parameter has an important impact on memory, but little impact on efficiency, as the re-ranking is performed on a short-list of 2k neighbors only, where k is the number of nearest neighbours we want to retrieve.

Overall, each descriptor is represented by b = 16 bytes. This is a crucial point that motivates the use of this approach in our system. The method used in this paper has successfully been used to index a billion-sized dataset without resorting to costly disk accesses. When transposed into a cover song detection framework, where sequences are in the order of 1,000 features long, this means that a million songs could be efficiently indexed this way, using around 20GB memory. In contrast, Locality Sensitive Hashing [START_REF] Datar | Locality-sensitive hashing scheme based on p-stable distributions[END_REF] requires a lot of hash functions to achieve good performance, and needs the raw descriptors to establish the final ranking.

On output of this stage, we have a list of nearest neighbors, which are then sorted by database song and query timestamp. For a given database song, we will denote in the following the matches by L m = {m 1 , . . . , m n } where m i = (t i q , t i c , s i ) is made of the timestamp t i q of the vector in the query that matches a vector at timestamp t i c in the candidate sequence with a score s i . The matches in the list L m are therefore ordered in ascending order of t i q and, if necessary, t i c .

Voting strategy

Once a set of approximate k-nearest neighbours is computed for each vector of the query sequence, we compute a score for each match based on considerations of burstiness and then perform temporal robustification so as to filter out incoherent matches.

Handling Multiple matches. For a given vector of the query track, it often occurs that several vectors from the same database song are returned. These vectors correspond to different timestamps in the database song. These vectors have therefore a higher contribution than what would be desirable in the cover similarity measure based on nearest neigbhors. In order to limit their contribution, if multiple matches of the same database are associated with the same query vector, then their vote score is downweighted. This is done by choosing a voting score of 1 divided by the square root of the number of matches of the same song that have voted for this particular query vector. For instance, if two vectors of the database vector are nearest neighbors of the query vector, then their voting contribution is 1/ √ 2 ≈ 0.707 instead of 1 for an isolated match. Temporal robustification. Let us consider a candidate sequence C that shares some matches with the query sequence. The aim is then to select the largest set of consistent matches in L m . A pair (m i , m j ) of matches is consistent if and only if it satisfies the following two inequalities:

(t i q -t j q ) × (t i c -t j c ) > 0 (2) t i q -t j q t i c -t j c ∈ 1 r * ; r * (3)
These inequations1 ensure that timestamps of i and j are coherent in the sense that if the query timestamp in i is lower (resp. greater) than the one in j, the candidate timestamp has to be lower (resp. greater) in i than in j too. Moreover, the parameter r * controls the expected maximum beat ratio between two songs. A set of matches is said consistent if any pair of matches in this set satisfies this consistency check.

In order to assess similarity between sequences, we compute the maximum total score of all sets of consistent matches in L m . Algorithm 1 presents a way to compute such a score while only maintaining a table of scores of length n, the number of matches.

Figure 1 illustrates the set of matches found by this algorithm for two particular query/database tuples. Here, the song is a cover of the query, therefore the total number of matches is relatively high. The line corresponds to the path connecting the matches that have been selected as belonging to the best set of consistent matches. Note the small proportion of matches that are temporally consistent, which justifies the use of such a filtering step. Score normalization. The score obtained by the voting scheme is not regularized. Songs with many descriptors are, therefore, favored compared to those with fewer descriptors. Different normalization schemes have been evaluated [START_REF] Philbin | Lost in quantization: Improving particular object retrieval in large scale image databases[END_REF] in the context of image retrieval, some of which are applicable in a BOF framework only, such as the L2 normalization of the BOF vector. In our voting framework, we divide the score by the square root of the number of descriptors, Algorithm 1 Scoring algorithm that sums consistent matches together. Require: L m = {m 1 , . . . , m n }, r * 1: bestsofar ← zeros (n) 2: for i = 1:n do 3:

for j = (i + 1):n do 4: if (m i , m j ) is consistent w.r.t. r * then 5: score ← bestsofar[i] + s i 6: bestsofar[j] ← max(bestsofar[j], score) 7:
end if

8:
end for 9:

bestsofar[i] ← bestsofar[i] + s i 10: end for 11: return max(bestsofar)

t b t q (a) Cover song t b t q (b) Non-cover song
Figure 1: Temporal consistency. On input, our algorithm takes a set of matches retrieved by approximate search (displayed by points). On output, it produces the best subset of matches that are consistent, here represented by a line connecting the selected matches. The same query is considered for both figures. On the left, the database song is a cover of the query song, while it is not the case for the right-hand side figure. Note than even though the candidate song in (b) has much more matches, it is ranked after the one of (a) due to both our anti-burstiness strategy and our temporal consistency checking.

i.e. the normalized similarity s(a, b) is obtained as:

s(a, b) = s(a, b) √ n a √ n b , (4)
where n a and n b correspond to the number of descriptors in song a and b, respectively. This square root normalization has the desirable property that s(a, b) = 1 if each descriptor of a is matched with itself only, in which case s(a, a) = n a . Note that this property is also satisfied in a bag-of-word framework if the cosine similarity is used. The complexity of this last stage may appear costly, as it includes dynamic programming and is therefore quadratic in the number of matches obtained for a particular query/base tuple. However, in practice this number of matches is very small for the vast majority of database elements. This is because the total number of nearest neighbors retrieved by the ANN algorithm is k × n q , where n q is the number of descriptor in the query. Only a limited fraction of these neighbors are associated with a particular database song. In the worst case, all the neighbors would be associated with the same song2 , leading to a total complexity of k 2 × n 2 q to compare the query with all the database elements.

Experiments

Datasets

We evaluate our algorithm on the well-known Covers80 dataset [START_REF] Ellis | The "covers80" cover song data set[END_REF]. This dataset is made of 80 pairs of pop songs, each pair being made of two different versions of a same song. All covers are performed by different artists from the original versions. We used the software of [START_REF] Ellis | Identifying "cover songs" with chroma features and dynamic programming beat tracking[END_REF] to extract 12-dimensional chroma features, using a hop size of 125ms and an analysis window size of 250ms.

Evaluation protocol

The search quality is measured as the recall@k, i.e., the proportion of queries whose nearest neighbor is ranked in the first k positions. Also, the recall@k is the fraction of queries for which the nearest neighbor would be retrieved correctly if a short-list of k songs was verified using a perfect re-ranking strategy. The efficiency is measured by actual timings.

Evaluation of the proposed approach

In the following, we use, for our algorithm, a coarse quantizer dividing the feature space into 1024 cells and explore, for each query vector, the 4 closest cells. Doing so, we compute approximate distances for 1/256 th of the database. As stated above, each feature from the database is encoded using 16 bytes.

Figure 2: Influence of the maximum tempo ratio on the quality of results. We study in the following the influence of the number k of nearest neighbors retrieved and the tempo ratio r * . When not explicited, the former is set to 20 and the latter to 1.4. We compare our approach to the results obtained by the 2007 LabROSA cover detection system [START_REF] Ellis | The 2007 LabROSA cover song detection system[END_REF] on the covers80 dataset, so as to assess its quality. We also report timings showing the efficiency of our algorithm on this dataset.

Method

Results obtained for are presented in Table 1 and show that using only the first few nearest neighbours for each query vector performs better, which allows cheap (in terms of time consumption) temporal robustification, as the latter is a function of the number of matches (see Section 3.2 for more details). Note however that tuning parameters on such a small dataset is difficult as the results can be noisy due to the small number of songs to retrieve. Moreover, when considering a larger database, as votes are distributed among a higher number of songs, it is expected that scores would be lower, implying the need for larger values of k so as to perform a more appropriate consistency check.

We obtain comparable results to the baseline of LabROSA, which is positive, as our algorithm enables scaling up without comparing a query song with every candidate song in the database. Also, our algorithm only considers little temporal information: the only prior knowledge is that the ratio of tempo between two versions of a song r * should be in the range [1/r * ; r * ].

Figure 2 shows that setting reasonable values for r * has little impact on the quality of retrieval, while an optimum is reached for r * = 1.5.

Table 2 also reports timings for the same values of k, divided into the time spent for the approximate nearest neighbour search and the one for the voting scheme. Notice that the latter is always much lower than the former, which confirms the statement made in Section 3.2 that its complexity is not a limiting factor Table 2: Timings to compare a query with all the 80 covers of Covers-80, averaged over the 80 queries and measured on 1 core. The timings of the first column depend on the database size (almost independent from k). The time consistency check (second column) mainly depends on k. Note that using our multi-threaded implementation (12 cores, 1 per transposition), we obtain query times of about 1 second for k = 20.

Discussion

In this paper we introduced a new approach for cover song identification. It is based on a local estimation of nearest neighbors and a temporal consistency check tailored to filter out matches that do not correspond to a realistic alignment of song segments. The results obtained on a small dataset are encouraging as the achieved search quality is comparable to already published results on the same dataset, at a very low computing cost. We aim at testing our algorithm on the recently published Second Hand Dataset, on which the large-scale capabilities of our algorithm could be enlightened.

Table 1 :

 1 Results obtained on the covers80 dataset. The varying parameter is the number of nearest neighbours considered for the vote.

		Variant	recall@10
		k = 5	0.463
	Our approach	k = 20	0.513
		k = 50	0.500
		Baseline	0.513
	LabROSA	Tempo bias = 120 BPM	0.600
		Dual tempo levels	0.675

In practice, checking if these criteria are satisfied is done based on a single test, by removing the absolute value in the second inequality.

In practice, this is obviously not the case

Note that these timings are reported when running our experiments on a single core, while our code is parallelized and can achieve much lower timings when run on several cores.