L'analyse discriminante, un puissant moyen de validation des hypothèses hydrogéologiques

L'étude des tableaux de données hydrochimiques acquises au cours de campagnes synchrones (« instantanés ») ou de suivis diachroniques à pas régulier (hebdomadaire, quotidien, horaire) s'opère généralement en résumant l'information par des méthodes statistiques. Ces méthodes descriptives, qui négligent nécessairement une partie de l'information initiale, permettent l'interprétation de la structure du tableau de données en termes de fonctionnement hydrocinématique (BAKALOWICZ, 1979, 1982, MUDRY et BLAVOUX, 1988, ROLET et SEGUIN, 1986 8 et b).Ces méthodes, fondées sur l'analyse d'une seule population statistique (bi ou multidimensionnelle) impliquent un mode de raisonnement déductif.Leur application, ainsi que l'examen du tableau des données brutes (ou de ses représentations graphiques), peut mettre en évidence des sous-groupes fondés sur des critères hydrogéologiques. La réalité de ces sous-groupes peut être testée à l'aide de méthodes statistiques basées sur l'analyse de la variance. Certaines méthodes utilisent le rapport des variances qu'elles comparent à la distribution de Snedecor (analyse de la variance à une ou deux voies), d'autres comparent des variances multidimensionnelles intraclasses à la variance interclasses, c'est le cas de l'analyse discriminante. Les sous-groupes constituent une variable qualitative dont la pertinence peut être démontrée par la calcul. L'analyse discriminante apparaît donc comme un outil décisionnel. Le présent article présente brièvement la méthode du point de vue statistique et montre deux exemples d'application à des sources karstiques.Le premier exemple traite de l'appartenance chimique d'une phase de basses eaux à la petite crue qui la précède et non à un tarissement au sens hydrocinématique. L'analyse discriminante permet d'affirmer qu'une recharge peu perceptible sur l'hydrogramme de la source amène une évolution chimique irréversible de l'eau de la réserve, responsable des phénomènes d'hystérésis observés sur les courbes concentration-débit. Ce cas est celui de l'aquifère de la Fontaine de Vaucluse (Sud-Est de la France) pendant un suivi quotidien d'étiage.Le second exemple permet de rattacher, par son comportement physico-chimique hebdomadaire, une émergence karstique à une autre et non à une troisième. Ce cas est celui du karst de la Rochefoucauld (Charente), avec les sources du Bouillant, de la Font de Lussac et de la Lèche. Les sources du Bouillant et de la Font de Lussac ont un comportement physico-chimique semblable, alors que la Lèche réagit de manière totalement indépendante. Elle constitue un système globalement distinct du point de vue hydrocinématique, ce qui permet de minimiser les relations mises en évidence par traçage artificiel entre les deux systèmes.The study of the hydrochemical data tables obtained during synchronous sampling (weekly, daily or hourly) is generally carried out by resuming the information by statistical methods. These methods, that disregard part of the initial information, allow to explain the structure of the data table in terms of hydrokinematics (BAKALOWICZ, 1979, 1982; MUDRY et BLAVOUX, 1986; ROLET et SEGUIN, 1986 a et b).These descriptive methods, based on the analysis of a single (bi or multivariate) statistical population, imply deductive reasoning. Their application, as well as the study of the untreated data table (or of its scattergrams), can show the presence of sub-groups based on hydrogeological criteria. The existence of such sub-groups can be tested by statistical methods based on variance analysis. Several methods use the variance ratio and compare it to Snedecor's distribution (single or double path variance analysis), others compare multidimensional intragroup variances to intergroup variance. This is the case with discriminant analysis.This paper describes the method from a point of view of statistics and presents two examples of application to karst springs.The first study deals with the chemical relationship of a low-water period with the preceding period of small floods and not to a hydrokinematical water drying up. The discriminant analysis allows to say Chat a recharge less visible on the hydrograph of the spring induces a non reversible chemical evolution of the reserve water. This is the case of the Fontaine de Vaucluse karst spring (Southeastern France) during a daily low water sampling.The second study allows to relate a karst spring to another one, thanks to its weekly physico-chemical behaviour, and not to a third one : this is the case of the La Rochefoucauld karst system (Charente, Western France) with the Bouillant, Font de Lussac and Lèche springs. The Bouillant and Font de Lussac springs behave in the same way hydrochemically, whereas the Lèche works independently. It is, hydrokinematically, a separate karst system, even if there is a certain relationship between all of them

Boundary criticality at the Anderson transition between a metal and a quantum spin Hall insulator in two dimensions

Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\nu$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a $Z_2$ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents which we show (numerically) to depend on whether the insulating state is topologically trivial or not.Comment: 12 pages, 13 figures, selected for an Editors' Suggestion in PR

Quarks, Gluons and Frustrated Antiferromagnets

The Contractor Renormalization Group method (CORE) is used to establish the equivalence of various Hamiltonian free fermion theories and a class of generalized frustrated antiferromagnets. In particular, after a detailed discussion of a simple example, it is argued that a generalized frustrated SU(3) antiferromagnet whose single-site states have the quantum numbers of mesons and baryons is equivalent to a theory of free massless quarks. Furthermore, it is argued that for slight modification of the couplings which define the frustrated antiferromagnet Hamiltonian, the theory becomes a theory of quarks interacting with color gauge-fields.Comment: 21 pages, Late

Topological Hubbard model and its high-temperature quantum Hall effect

The quintessential two-dimensional lattice model that describes the competition between the kinetic energy of electrons and their short-range repulsive interactions is the repulsive Hubbard model. We study a time-reversal symmetric variant of the repulsive Hubbard model defined on a planar lattice: Whereas the interaction is unchanged, any fully occupied band supports a quantized spin Hall effect. We show that at 1/2 filling of this band, the ground state develops spontaneously and simultaneously Ising ferromagnetic long-range order and a quantized charge Hall effect when the interaction is sufficiently strong. We ponder on the possible practical applications, beyond metrology, that the quantized charge Hall effect might have if it could be realized at high temperatures and without external magnetic fields in strongly correlated materials.Comment: 11 pages, 5 figure

Instanton correlators and phase transitions in two- and three-dimensional logarithmic plasmas

The existence of a discontinuity in the inverse dielectric constant of the two-dimensional Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a three-dimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.Comment: 15 pages, 16 figure

Influence of higher d-wave gap harmonics on the dynamical magnetic susceptibility of high-temperature superconductors

Using a fermiology approach to the computation of the magnetic susceptibility measured by neutron scattering in hole-doped high-Tc superconductors, we estimate the effects on the incommensurate peaks caused by higher d-wave harmonics of the superconducting order parameter induced by underdoping. The input parameters for the Fermi surface and d-wave gap are taken directly from angle resolved photoemission (ARPES) experiments on Bi{2}Sr{2}CaCu{2}O{8+x} (Bi2212). We find that higher d-wave harmonics lower the momentum dependent spin gap at the incommensurate peaks as measured by the lowest spectral edge of the imaginary part in the frequency dependence of the magnetic susceptibility of Bi2212. This effect is robust whenever the fermiology approach captures the physics of high-Tc superconductors. At energies above the resonance we observe diagonal incommensurate peaks. We show that the crossover from parallel incommensuration below the resonance energy to diagonal incommensuration above it is connected to the values and the degeneracies of the minima of the 2-particle energy continuum.Comment: 13 pages, 7 figure

Validation des écoulements par l'analyse discriminante

L'étude des écoulements souterrains et du drainage est généralement réalisée par l'établissement des cartes et des profils piézométriques (en effectuant des campagnes de relevés piézométriques dans les forages et les piézomètres captant la nappe). On essaye ici de confirmer les résultats obtenus à partir de l'interprétation de la piézométrie par le traitement statistique multidimensionnel des données hydrogéochimiques des sources, et plus précisément l'analyse factorielle discriminante, et de caractériser l'importance et la dominance de l'un des paramètres responsables de la minéralisation des eaux karstiques : les phénomènes climatiques (saisons), et les caractéristiques du système (lithologie, structure du drainage, gradients). L'application à la nappe des calcaires du Bathonien-Bajocien du plateau de Besançon (Jura externe, France) a permis de démontrer que la variabilité chimique des eaux aux exutoires karstiques est conditionnée plus par les caractéristiques hydrogéologiques de l'aquifère (unités hydrogéologiques) que par les évènements hydroclimatiques (effet "campagne" ou période d'étude), et de resituer les forages par rapport aux sources testées ; l'association d'un groupe de forages avec certaines sources est à relier avec des circulations rapides en zone libre, ce cas est celui du regroupement de la source du Trébignon et les puits de Chailluz (situés dans la zone libre de l'aquifère) qui présentent le même comportement hydrochimiques. Alors que d'autres forages sont groupés avec le fonctionnement de la nappe en zone captive, ce cas correspond à celui de l'affectation des puits de Thise (implantés dans la partie captive de l'aquifère) à la source de Roche- lez-Beaupré. Ces résultats corroborent ceux obtenus par l'étude piézométrique.The plateau of Besançon (Eastern France) spreads between the large valley of the Ognon to the north and the enclosed valley of the Doubs to the south. Stratigraphically, the principal formations belong to the middle Jurassic era and the region is marked by a brittle tectonic characterized by NE-SW faults. Bajocian and Bathonian limestones constitute a karstic aquifer (200 m in thickness), which is free for much of its extent but becomes confined under Oxfordian marls towards the Doubs valley. The hydrogeological and structural boundaries of the drainage basin are complex, as are its hydrodynamic functioning and the hydrochemical responses observed at the outlets. The reservoir is drained principally by the Mouillère, Trébignon and Roche-lez-Beaupré springs, these outlets being situated mainly in the Doubs valley. The region is largely covered by dense vegetation, essentially forests, which constitute a groundwater recharge zone, and by the conglomeration of Besançon. One of the most karstified regions in France, the site includes two groups of wells that supply the town of Besançon with drinking water: the first group is situated on the unconfined zone of the aquifer (Chailluz), and the second is located on the confined part of the karstic aquifer (Thise). The wells are equipped with piezometers to monitor water levels continuously.The rainfall distribution in this karstic region is homogeneous. During the 1991- 1992 hydrological year, the observed precipitation was 1144 mm at Besançon, with an evapotranspiration constituting about 50 % of the measured precipitation. On the Besançon plateau, the water mineralization of the different karstic springs is related to the nature of the underlying geological media (lithology, importance of karstification, etc.), to the water residence time and to seasonal variables (characteristics of the rainfall episode, hydroclimatic events, etc.). The water chemistry is relatively stable during dry intervals, but exhibits high fluctuations during flood periods. Outlets were sampled every 7 to 10 days from January 1991 to August 1992, in order to follow variations in water physico-chemistry. The sampling frequencies covered diverse water-level situations in the aquifer (low-water period, high-water period, drying-up period, etc.), and allowed the application of some multidimensional statistical tools.The piezometric profile and lithologic correlations between the boreholes of Chailluz and Thise show that the highest hydraulic gradient is calculated between the Chailluz boreholes and the Trébignon spring. The hydraulic gradient is low in the confined part of the aquifer (between the Thise boreholes and the spring of Roche-lez- Beaupré). Two zones can be identified: 1. a free zone distinguished by flow and drainage relationships between the Chailluz boreholes and the Trébignon spring; 2. a confined zone linking the Thise boreholes and the Roche-lez-Beaupré spring, which corresponds to a common water circulation. To confirm the results obtained from the piezometric study, we have used multidimensional statistical techniques to process the hydrochemical data from the karstic springs.Factorial discriminant analysis is a multivariate method that allows the testing of hypotheses. It belongs to the field of decisional statistics, and is based on the comparison of multidimensional intragroup variances to intergroup variance. These methods can show the presence of certain relationships between a qualitative explained criterion and a group of quantitative explanatory characters, and they allow one to describe these latter relationships. Cross discriminant analysis can be used to evaluate the pertinence of a given hypothesis and to identify the dominant factor(s). The introduction of a qualitative variable within a population allows the division of this population into different groups, with each individual assigned to one group. Discrimination of the groups consists of maximizing the variance between their centres of gravity (a principal components analysis is performed on the matrix of the centres of gravity of the defined groups a priori); one can then clarify the properties that distinguish the different groups. If the individual is close to the centre of gravity of its group, it is "well classified". In the case where the distance to the centre of gravity of its group is superior to that to the centre of gravity of another group, the individual is "poorly classified" and it will be reassigned to this other group.Another advantage of these methods is the use of supplementary individuals, which are not involved in the definition of the groups. Supplementary individuals are reassigned to the groups to which they are statistically linked. In karstic hydrogeology, cross factorial discriminant analysis of a population enables one to validate the classification criteria based on the analysis of flows and concentrations. These units of classification deal with the hydrokinematic phases (seasonal variable) or with the intrinsic systems (geographic variable). Therefore, in our case we have crossed two qualitative factors, "system" (Trébignon, Roche-lez-Beaupré) and "season" (spring, summer, autumn, winter) by undertaking two different analyses on the same series of data.The application of these techniques to the limestone aquifer of the Bathonian- Bajocian shows that intrinsic factors predominate in determining the mineralization of the outlet water. On the other hand, this method enables one to associate borehole chemistry to tested springs: the association of a group of boreholes with particular springs is explained by the rapid circulation and drainage in the unconfined zone of the karstic media. This is the case of the association of the Chailluz boreholes (situated in the unconfined zone of the aquifer) with the Trébignon spring, which behave hydrochemically in the same manner. Other boreholes are grouped with the confined aquifer; this case corresponds to the assignment of the Thise boreholes (located in the confined zone of the aquifer) to the Roche-lez-Beaupré spring. These results corroborate those obtained from the piezometric study. Factorial discriminant analysis allowed the verification of the appropriation of the predefined groups to the discriminant factors: hydrogeologic character (lithology, structure of the drainage, gradients, etc.) and hydroclimatic phenomena (season, hydrometeorological episodes, etc.). The choice of the property groups based on the geographic and geologic criteria proved pertinent

Incidences de pluies exceptionnelles sur un aquifère libre côtier en zone semi-aride (Chaouia, Maroc)

Les eaux souterraines qui représentent les seules ressources en eau de la Chaouia côtière, sont très minéralisées : intrusion marine, évaporation et recyclage des eaux d'irrigation chargées en sels. L'évolution spatio-temporelle piézométrique et hydrochimique de cet aquifère a été observée sur 179 puits, de 1991 à 1998 qui inclue une période exceptionnellement pluvieuse. Une comparaison par rapport à l'état de la nappe en 1971, date du début de l'exploitation des eaux souterraines, a été faite. Les suivis ont montré qu'à la suite des pluies importantes de 1996 (943 mm), un rehaussement important du niveau de la nappe et des dilutions plus ou moins retardées de tous les sels en solution dans l'eau, sont mesurés dans tous les puits. En effet, la nappe qui a accusé des baisses de 10 à 20 m entre 1971 et 1995, remonte en 1996 de 4,5 à 12,5 m en moyenne surtout au centre et à l'amont de la plaine. Quant à la qualité des eaux qui s'était dégradée, elle a connu des améliorations importantes avec des variations de 0,5 à plus de 4 ms/cm au niveau de la conductivité électrique. Ces faits montrent une grande sensibilité de la nappe aux apports par les pluies qui constituent sa principale alimentation.A study of regional pollution in the coastal aquifer system between Oum-er-Rbia River and Bir Jdid (Coastal Moroccan Meseta, Morocco) was based on data gathered from 1995 to 1998. The study improved the hydrogeological and hydrochemical understanding of the aquifers. The study examined the effects of significant rains on the quantity and quality of coastal groundwater. These waters exhibited high sensitivity to the rain input.Measurements undertaken since 1991 on 179 wells in the study area reveal that the groundwaters are highly mineralized - conductivity reaches more than 10 mS/cm, depending on season and well location. These results also show high chloride concentrations (more than 3500 mg/l), sodium concentrations frequently in the 500 - 1000 mg/l range and nitrates between 150 and 250 mg/l.In coastal Chaouia, these high concentrations of dissolved mineral salts aggravate the problem of supplying quality water to the rural population for drinking or even market-gardening irrigation purposes. The area is characterized by increased irrigated surfaces (more than 16000 hectares with 4125 m3 /s) and a demographic rise that has triggered the chaotic boring of more than 2000 wells into a heterogeneous aquifer sensitive to salinity. Possible sources of this high salinity include :Seawater intrusion into coastal aquifer sectors (mainly into a two-kilometre strip of coastline). The degree and the length of the marine intrusion were exacerbated by intensive pumping for irrigation needs, particularly during the dry season. Other factors such as coastal aquifer permeability, saturation zone thickness and basement depth also affect the degree and length of the marine intrusion. Seawater pollution is more marked in the southwestern coastal area.Reuse of irrigation saline waters, especially as groundwater circulates in the coastal and eastern sectors deep in the soil (to depths of 10 m).Several important factors have been highlighted concerning the origin of the chemical elements in solution and the mechanism of hydrochemical distribution: evaporation, lithology of tapped aquifers, water table depth, distance of wells from the coast, type of soil, use of fertilizers, frequency of pumping operations, and rainfall, among others.A regular network of 179 wells was monitored and surveyed in this study. Spatial and temporal changes in the water table and in the hydrochemistry of the aquifer were monitored between 1991 and 1998, a span which included an exceptionally rainy period (1996). Data are compared with those from 1971, the year groundwater exploitation began.The studied region is characterized by a semi-arid climate, mean annual rainfall of 391 mm (between 1977 and 1998), mean temperature of 17.8 °C, and mean rainfall input of 142 mm. In the dry season, high temperature combined with low rainfall and intensive pumping operations give rise to salinization of shallow groundwater. Groundwater in the area of study circulates in two principal hydrogeological matrices :1. Sandy-calcareous Plio-Quaternary and paleozoic strata in costal sectors and the eastern part located between Tnine Chtouka and Bir Jdid. These hydrogeological strata are characterized by significant porosity and permeability and are exploited at a shallow depth, generally less than 14 m, under sandy-clayed soil.2. A Cretaceous aquifer in marly limestone located between Oum-er-Rbia River and Tnine Chtouka. This aquifer is characterized by a low permeability and a water table generally exceeding 24 m in depth.Monitoring showed that after the heavy rains of 1996 (943 mm, with infiltration of 142 mm), the water table rose markedly in all study wells and a lagged dilution (3 to 6 months) was noted for all mineral salts dissolved in water. In fact, the water table, which had dropped between 1971 and 1995 (10 m in the costal sectors and 20 m in the others parts of the studied region), rose by an average 4.5 to 12.5 m in the central and up-gradient parts of the plain. Several factors were at work between 1971 and 1995: decreased pumping times (about 15 minutes) combined with sharp decreases in the thickness of the saturated zone; appearance of a closed piezometric level in the western region between Oum-er-Rbia River and Tnine Chtouka; and desiccation of 59 wells. With the aid of a polynomial model, annual level evolution of the five coastal test piezometers showed correlation factors of 0.76 to 0.93 respectively for years 1996 and 1997. Other quantitative effects of exceptional rains on groundwater identified include: increased pumping time (up from less than 15 minutes prior to the heavy rains of 1996); increased saturated zone thickness; rises in the water-table surpassing former sea-level heights in certain wells ; general advancement of the piezometric levels towards the coastline; and increased aquifer thickness stemming from longer pumping times.In terms of quality, groundwater showed gradually increasing salinity (by a factor of 2) between 1971 and 1995 and significant improvements after the rains of 1996. Between 1971 and 1995, increased mineralization was observed mainly in coastal and eastern parts of the aquifer between Tnine Chtouka and Bir Jdid. Conductivity of the waters in some coastal wells increased from 5 mS/cm to more than 10 mS/cm. After the exceptional rains of 1996, differences ranging from 0.5 to more than 4 mS/cm were observed in 71 % of the 179 wells analysed. Over the same period, chloride concentrations decreased by average values ranging from 150 to 500 mg/l (higher decreases surpassed 1000 mg/l). By contrast, spatial distributions of conductivity and of the main mineral salts remained unchanged and coastal and eastern groundwaters were still the most vulnerable to high salinity.The quantitative and qualitative effects of heavy rain point to the high sensitivity of these groundwaters to rain input, which is the principal recharge source of the studied aquifer