Une méthodologie générale de comparaison de modèles d'estimation régionale de crue

L'estimation du débit QT de période de retour T en un site est généralement effectuée par ajustement d'une distribution statistique aux données de débit maximum annuel de ce site. Cependant, l'estimation en un site où l'on dispose de peu ou d'aucune données hydrologiques doit être effectuée par des méthodes régionales qui consistent à utiliser l'information existante en des sites hydrologiquement semblables au site cible. Cette procédure est effectuée en deux étapes: (a) détermination des sites hydrologiquemcnt semblables(b) estimation régionalePour un découpage donné (étape a), nous proposons trois approches méthodologiques pour comparer les différentes méthodes d'estimation régionale. Ces approches sont décrites en détail dans ce travail. Plus particulièrement il s'agit de- simulation par la méthode du bootstrap - analyse de régression ou Bayes empirique - méthode bayésienne hiérarchiqueEstimation of design flows with a given return period is a common problem in hydrologic practice. At sites where data have been recorded during a number of years, such an estimation can be accomplished by fitting a statistical distribution to the series of annual maximum floods and then computing the (1-1/T) -quantile in the estimated distribution. However, frequently there are no, or only few, data available at the site of interest, and flood estimation must then be based on regional information. In general, regional flood frequency analysis involves two major steps:- determination of a set of gauging stations that are assumed to contain information pertinent to the site of interest. This is referred to as delineation of homogeneous regions.- estimation of the design flood at the target site based on information from the sites ofthe homogeneous region.The merits of regional flood frequency analysis, at ungauged sites as well as at sites where some local information is available, are increasingly being acknowledged, and many research papers have addressed the issue. New methods for delitneating regions and for estimating floods based on regional information have been proposed in the last decade, but scientists tend to focus on the development of new techniques rather than on testing existing ones. The aim ofthis paper is to suggest methodologies for comparing different regional estimation alternatives.The concept of homogeneous regions has been employed for a long time in hydrology, but a rigorous detinition of it has never been given. Usually, the homogeneity concerns dimensionless statistical characteristics of hydrological variables such as the coefficient of variation (Cv) and the coefficient of skewness (Cs) of annual flood series. A homogeneous region can then be thought of as a collection of stations with flood series whose statistical properties, except forscale, are not significantly different from the regional mean values. Tests based on L-moments are at present much applied for validating the homogeneity of a given region. Early approaches to regional flood frequency analysis were based on geographical regions, but recent tendencies are to deline homogeneous regions from the similarity of basins in the space of catchment characteristics which are related to hydrologic characteristics. Cluster analysis can be used to group similar sites, but has the disadvantage that a site in the vicinity ofthe cluster border may be closer to sites in other clusters than to those ofits ovm group. Burn (1990a, b) has recently suggested a method where each site has its owm homogeneous region (or region of influence) in which it is located at the centre of gravity.Once a homogeneous region has been delineated, a regional estimation method must be selected. The index flood method, proposed by Dalrymple (1960), and the direct regression method are among the most commonly used procedures. Cunnane (1988) provides an overview of several other methods. The general performance of a regional estimation method depends on the amount of regional information (hydrological as well as physiographical and climatic), and the size and homogeneity of the region considered relevant to the target site. Being strongly data-dependent, comparisons of regional models will be valid on a local scale only. Hence, one cannot expect to reach a general conclusion regarding the relative performance of different models, although some insight may be gained from case studies.Here, we present methodologies for comparing regional flood frequency procedures (combination of homogeneous regions and estimation methods) for ungauged sites. Hydrological, physiographical and climatic data are assumed to be available at a large number of sites, because a comparison of regional models must be based on real data. The premises of these methodologies are that at each gauged site in the collection of stations considered, one can obtain an unbiased atsite estimate of a given flood quantile, and that the variance of this estimate is known. Regional estimators, obtained by ignoring the hydrological data at the target site, are then compared to the at-site estimate. Three difrerent methodologies are considered in this study:A) Bootstrap simulation of hydrologic dataIn order to preserve spatial correlation of hydrologic data (which may have an important impact on regional flood frequency procedures), we suggest performing bootstrap simulation of vectors rather than scalar values. Each vector corresponds to a year for which data are available at one or more sites in the considered selection of stations; the elements ofthe vectors are the different sites. For a given generated data scenario, an at-site estimate and a regional estimate at each site considered can be calculated. As a performance index for a given regional model, one can use, for example, the average (over sites and bootstrap scenarios) relative deviation ofthe regional estimator from the at-site estimator.B) Regression analysisThe key idea in this methodology is to perform a regression analysis with a regional estimator as an explanatory variable and the unknown quantile, estimated by the at-site method, as the dependent variable. It is reasonable to assume a linear relation between the true quantiles and the regional estimators. The estimated regression coeflicients express the systematic error, or bias, of a given regional procedure, and the model error, estimated for instance by the method of moments, is a measure of its variance. It is preferable that the bias and the variance be as small as possible, suggesting that these quantities be used to order different regional procedures.C) Hierarchical Bayes analysisThe regression method employed in (B) can also be regarded as the resultfrom an empirical Bayes analysis in which point estimates of regression coeflicients and model error are obtained. For several reasons, it may be advantageous to proceed with a complete Bayesian analysis in which bias and model error are considered as uncertain quantities, described by a non-informative prior distribution. Combination of the prior distribution and the likelihood function yields through Bayes, theorem the posterior distribution of bias and model error. In order to compare different regional models, one can then calculate for example the mean or the mode of this distribution and use these values as perfonnance indices, or one can compute the posterior loss

Quantum order by disorder in frustrated diamond lattice antiferromagnets

We present a quantum theory of frustrated diamond lattice antiferromagnets. Considering quantum fluctuations as the predominant mechanism relieving spin frustration, we find a rich phase diagram comprising of six phases with coplanar spiral ordering in addition to the N\'eel phase. By computing the specific heat of these ordered phases, we obtain a remarkable agreement between $(k,k,0)$-spiral ordering and the experimental specific heat data for the diamond lattice spinel compounds MnSc$_2$S$_4$, Co$_3$O$_4$ and CoRh$_2$O$_4$, i.e. specific heat data is a strong evidence for $(k,k,0)$-spiral ordering in all of these materials. This prediction can be tested in future neutron scattering experiments on Co$_3$O$_4$ and CoRh$_2$O$_4$, and is consistent with existing neutron scattering data on MnSc$_2$S$_4$. Based on this agreement we infer a monotonically increasing relationship between frustration and the strength of quantum fluctuations.Comment: 4 pages, 2 figures, version published in PR

Modélisation de l'évolution du couvert nival, sur le sous-bassin LG4 de la rivière La Grande dans le nord du Québec, à l'aide du modèle français CROCUS

Au Québec, l'accumulation de la neige au sol prend une grande importance dans bon nombre d'activités humaines comme la production d'hydroélectricité. Face à cette importance de la neige, il devient fort intéressant de faire appel à des outils de modélisation dans le but de faire le suivi complet du manteau neigeux par la simulation des différents processus physiques agissant sur le couvert nival. En ce sens, la présente recherche vise à évaluer la capacité d'adaptation aux conditions environnementales du Québec, du modèle français multicouches CROCUS de suivi et de fonte du couvert nival, en vue de son utilisation ultérieure. Le choix du site d'étude s'accorde avec la présence de stations d'acquisition de données sur le couvert nival et la disponibilité d'un large éventail de données météorologiques essentielles aux modélisations.Suite au prétraitement des données d'entrée, le modèle CROCUS a pu être étalonné dans le but d'évaluer ses aptitudes à simuler l'évolution de l'épaisseur et de l'équivalent en eau du couvert nival, de même que les profils de densité et de température de ce couvert. Compte tenu des données disponibles, les résultats des périodes d'étalonnage et de vérification peuvent être considérés comme très bons et très encourageants pour des applications ultérieures de CROCUS au Québec.In Québec, snow accumulation on the ground is important for a large number of activities, including hydroelectric production. Snow being an important environmental aspect to look at for operations in cold regions, making use of adequate models able to simulate the various physical processes affecting the characteristics of the snowpack throughout the entire snow season becomes also quite important. So, the present project aims to evaluate the ability of the CROCUS French multilayer snow model to be adapted to Northern Quebec environmental conditions. The choice of the study area has been done on the basis of data availability on both the temporal evolution the snowpack at three automatic stations and the meteorological variables necessary as input to the model. Following preprocessing of input data, the CROCUS model has been calibrated in order to evaluate his capability to simulate the temporal evolution not only of the thickness and water equivalent of the snowpack, but also of the density and temperature profiles within the pack. Given the available data, the results for both the calibration and verification periods can be considered as good and very encouraging for future applications of the CROCUS model in Québec

Estimation de l'équivalent en eau du couvert nival au moyen d'images radar satellitaires

L'objectif de cette étude est de vérifier le potentiel des images radar à synthèse d'ouverture (RSO) pour estimer l'équivalent en eau du couvert nival sur le bassin de la rivière La Grande (Baie de James, Québec). Il s'agit d'un milieu dominé par une forêt ouverte d'épinettes noires, des brûlis et des tourbières. Cette information intéresse grandement Hydro-Québec qui gère plusieurs installations hydro-électriques dans cette région subarctique. Durant deux ans, six campagnes de terrain ont été réalisées sur le bassin de la rivière La Grande et une dizaine d'images RSO du satellite européen ERS-1 ont été acquises, étalonnées et géoréférencées, afin de déterminer la relation entre les coefficients de rétrodiffusion des images radar (hiver et automne) et la résistance thermique du couvert nival. Cette relation constitue la première partie d'un algorithme d'estimation de l'équivalent en eau. Elle utilise plus spécifiquement le rapport de rétrodiffusion, qui est la différence entre une image avec neige et une image sans neige. La deuxième partie de cette algorithme déduit l'équivalent en eau du couvert de neige à partir de sa résistance thermique et de sa densité, en se basant sur la relation physique établie par les mesures de terrain. L'équivalent en eau du couvert nival a donc été estimé pour quatre images de février et mars 1994 et 1995. L'erreur moyenne sur l'estimation de l'équivalent en eau de la neige au sol est de 2% à 3% (-5 à 7mm) sur l'ensemble des sites d'échantillonnage avec un écart-type de 14 à 19% (-35 à 45mm). Ces résultats ont encouragé Hydro-Québec à poursuivre la recherche avec les données du satellite canadien RADARSAT (opérationnel depuis le 1er avril 1996) et à développer un prototype pour la cartographie de l'équivalent en eau du couvert nival à partir d'images radar.The goal of this study was to evaluate the potential of synthetic aperture radar (SAR) images for estimating the snow water equivalent (SWE) on the La Grande river watershed (James Bay area, Québec). This information is of major interest for Hydro-Québec, which exploits many hydroelectric complexes throughout this subarctic region. The La Grande watershed is composed of moderately dense to opened black spruce forests, opened areas, burned areas and peat bogs. Over two years (1994-1995), six field campaigns were carried out on a study site located between the LG4 and Laforge1 reservoir, in the center of the La Grande river watershed. The field measurements were of two types: 20 snow lines (depth, snow water equivalent (SWE), density) and 8 snow profiles (depth, density, grain size, temperature, dielectric constant). With these data, the thermal resistance of the snowpack was calculated for every test-site, using the depth, density and thermal conductivity of each layer. Concurrently, more than 10 SAR images (European Satellite ERS-1) of the study site were acquired, calibrated and georeferenced. The backscattering coefficients of all winter images were extracted. Using a reference image (snow-free), backscattering ratios were calculated. They are the difference between a winter image and a snow-free image. This process is used to reduce the impact of vegetation and topography. Then, the relationship between the backscattering ratios and the snowpack thermal resistance of february and march 1994 are established, as the first part of an algorithm developed to estimate the snow water equivalent. The second part of the algorithm infers the snowpack water equivalent from its thermal resistance and density, based on the physical relationship established with field data. This approach is based on studies conducted by INRS-Eau in a southern Quebec agricultural area (BERNIER and FORTIN (1998)). The hypothesis are based on the following: - The snowpack characteristics influence the underlying soil temperature;- The dielectric constant of the soil varies with the soil temperature under 0°C;- The radar signal is influenced by the soil dielectric constant;- Thus, the snowpack characteristics (thermal resistance) influence the radar signal. However, due to variations of soil humidity on the date of the reference image (september 1994), two slightly different relationships were obtained. One for open areas and open forests and one for burned areas and peat bogs. This shows the importance of using a good reference image, with homogeneous soil conditions. It could be better to obtain an image later in the fall, when the soil is frozen. The relationships established here are preliminary, as they use a small dataset. It is estimated that a better regression should be obtained with the acquisition of more images and with a greater range of snow characteristics. However, the algorithm is applied to test the applicability of the method. First, the algorithm was applied on the test-sites, using the images from February and March of 1994 and 1995. The mean error on the estimation of the snow water equivalent is 2% to 3% ( 5 to 7mm), with a deviation of 14% to 19% ( 35 to 45mm). The results are comparable for both years, even if the algorithm is based on 1994 data only. Secondly, the algorithm is applied on the whole images. A classification of a Landsat-TM image is used to identify the land cover of every pixel, which determines the regression and the snow density to be used in the algorithm. Four maps of the SWE are produced and resampled to a resolution of 500m. These are compared with the field measurements from the four nearest Hydro-Quebec snow survey sites. The SWE measured by Hydro-Quebec are all within the most dominant SWE class of each map. Further validation of the results will be possible when the algorithm can be applied on a sub-watershed, which is the actual scale used by Hydro-Quebec. However, the results of this study were sufficiently promising to Hydro-Quebec to support a follow up research with data from the canadian satellite RADARSAT (operational since april 1996). Meanwhile, to improve the algorithm, it is important to obtain a good and homogeneous reference image, to better assess the impact of the land cover and to acquire a dataset with a greater range of snow characteristics

Biaxial spin-nematic phase of two dimensional disordered rotor models and spin-one bosons in optical lattices

We show that the ground state of disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense. We present a mean-field analysis of the model and demonstrate that the biaxial phase is stable against small quantum fluctuations. We point out the possibility of experimental realization of such rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice in the limit of a large number of atoms per site and also suggest an imaging experiment to detect the biaxial nematicity in such systems.Comment: revtex file 7 pages, 2 figures, version published in PR

Utilisation de l'information historique en analyse hydrologique fréquentielle

L'utilisation de l'information historique dans une analyse fréquentielle permet de mieux mobiliser l'information réellement disponible et devrait donc permettre d'améliorer l'estimation des quantiles de grande période de retour. Par information historique, on entend ici de l'information relative à des grandes crues qui se sont produites avant le début de la période de mesure (dite période de jaugeage systématique) des niveaux et débits des lacs et rivières. On observe de manière générale que l'utilisation de l'information historique conduit à une diminution de l'impact des valeurs singulières dans les séries d'enregistrements systématiques et à une diminution de l'écart-type des estimations. Dans le présent article on présente les méthodes statistiques qui permettent la modélisation de l'information historique.Use of information about historical floods, i.e. extreme floods that occurred prior to systematic gauging, can often substantially improve the precision of flood quantile estimates. Such information can be retrieved from archives, newspapers, interviews with local residents, or by use of paleohydrologic and dendohydrologic traces. Various statistical techniques for incorporating historical information into frequency analyses are discussed in this review paper. The basic hypothesis in the statistical modeling of historical information is that a certain perception water level exists and that during a given historical period preceding the period of gauging, all exceedances of this level have been recorded, be it in newpapers, in people's memory, or trough traces in the catchment such as sediment deposits or traces on trees. No information is available on floods that did not exceed the perception threshold. It is further assumed that a period of systematic gauging is available. Figure 1 illustrates this situation. The U.S. Water Resources Council (1982) recommended the use of the method of adjusted moments for fitting the log Pearson type III distribution. A weighting factor is applied to the data below the threshold observed during the gauged period to account for the missing data below the threshold in the historical period. Several studies have pointed out that the method of adjusted moments is inefficient. Maximum likelihood estimators based on partially censored data have been shown to be much more efficient and to provide a practical framework for incorporating imprecise and categorical data. Unfortunately, for some of the most common 3-parameter distributions used in hydrology, the maximum likelihood method poses numerical problems. Recently, some authors have proposed use of the method of expected moments, a variant of the method of adjusted moments which gives less weight to observations below the threshold. According to preliminary studies, estimators based on expected moments are almost as efficient as maximum likelihood estimators, but have the advantage of avoiding the numerical problems related to the maximization of likelihood functions. Several studies have emphasized the potential gain in estimation accuracy with the use of historical information. Because historical floods by definition are large, their introduction in a flood frequency analysis can have a major impact on estimates of rare floods. This is particularly true when 3-parameter distributions are considered. Moreover, use of historical information is a means to increase the representativity of a outlier in the systematic data. For example, an extreme outlier will not get the same weight in the analysis if one can state with certainty that it is the largest flood in, say, 200 years, and not only the largest flood in, say, 20 years of systematic gauging.Historical data are generally imprecise, and their inaccuracy should be properly accounted for in the analysis. However, even with substantial uncertainty in the data, the use of historical information is a viable means to improve estimates of rare floods

Long-time Behavior of a Two-layer Model of Baroclinic Quasi-geostrophic Turbulence

We study a viscous two-layer quasi-geostrophic beta-plane model that is forced by imposition of a spatially uniform vertical shear in the eastward (zonal) component of the layer flows, or equivalently a spatially uniform north-south temperature gradient. We prove that the model is linearly unstable, but that non-linear solutions are bounded in time by a bound which is independent of the initial data and is determined only by the physical parameters of the model. We further prove, using arguments first presented in the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing ball in appropriate function spaces, and in fact the existence of a compact finite-dimensional attractor, and provide upper bounds for the fractal and Hausdorff dimensions of the attractor. Finally, we show the existence of an inertial manifold for the dynamical system generated by the model's solution operator. Our results provide rigorous justification for observations made by Panetta based on long-time numerical integrations of the model equations

Synthèse de modèles régionaux d'estimation de crue utilisée en France et au Québec

De nombreuses méthodes régionales ont été développées pour améliorer l'estimation de la distribution des débits de crues en des sites où l'on dispose de peu d'information ou même d'aucune information. Cet article présente une synthèse de modèles hydrologiques utilisés en France et au Québec (Canada), à l'occasion d'un séminaire relatif aux " méthodes d'estimation régionale en hydrologie " tenu à Lyon en mai 1997. Les modèles français sont fortement liés à une technique d'extrapolation de la distribution des crues, la méthode du Gradex, qui repose sur l'exploitation probabiliste conjointe des séries hydrométriques et pluviométriques. Ceci explique les deux principaux volets d'études régionales pratiquées en France : les travaux liés à la régionalisation des pluies et ceux liés à la régionalisation des débits. Les modèles québecois comprennent généralement deux étapes : la définition et la détermination de régions hydrologiquement homogènes, puis l'estimation régionale, par le transfert à l'intérieur d'une même région de l'information des sites jaugés à un site non-jaugé ou partiellement jaugé pour lequel on ne dispose pas d'information suffisante. Après avoir donné un aperçu des méthodes pratiquées dans les deux pays, une discussion dégage les caractéristiques principales et les complémentarités des différentes approches et met en évidence l'intérêt de développer une collaboration plus étroite pour mieux tenir compte des particularités et des complémentarités des méthodes développées de part et d'autre. Une des pistes évoquées consiste à combiner l'information régionale pluviométrique (approche française) et hydrométrique (approche québécoise).Design flood estimates at ungauged sites or at gauged sites with short records can be obtained through regionalization techniques. Various methods have been employed in different parts of the world for the regional analysis of extreme hydrological events. These regionalization approaches make different assumptions and hypotheses concerning the hydrological phenomena being modeled, rely on various types of continuous and non-continuous data, and often fall under completely different theories. A research seminar dealing with " regional estimation methods in hydrology " took place in Lyon during the month of May 1997, and brought together various researchers and practitioners mainly from France and the Province of Quebec (Canada). The present paper is based on the conferences and discussions that took place during this seminar and aims to review, classify, comparatively evaluate, and potentially propose improvements to the most prominent regionalization techniques utilized in France and Quebec. The specific objectives of this paper are :· to review the main regional hydrologic models that have been proposed and commonly used during the last three decades ;· to classify the literature into different groups according to the origin of the method, its specific objective, and the technique it adopts ; · to present a comprehensive evaluation of the characteristics of the methods, and to point out the hypotheses, data requirements, strengths and weaknesses of each particular one ; and · to investigate and identify potential improvements to the reviewed methods, by combining and extending the various approaches and integrating their particular strengths.Regionalization approaches adopted in France include the Gradex method which represents a simplified rainfall-runoff model which provides estimates of flood magnitudes of given probabilities and is based on rainfall data which often cover longer periods and are more reliable than flow data (Guillot and Duband, 1967 ; CFGB, 1994). It is based on the hypotheses that beyond a given rainfall threshold (known as the pivot point), all water is transformed into runoff, and that a rainfall event of a given duration generates runoff for the same length of time. These hypotheses are equivalent to assuming that, beyond the pivot point, the rainfall-runoff relationship is linear and that the precipitation and runoff probability curves are parallel on a Gumbel plot.In Quebec (and generally in North America), regional flood frequency analysis involves usually two steps : delineation of homogeneous regions, and regional estimation. In the first step, the focus is on identifying and regrouping sites which seem sufficiently homogeneous or sufficiently similar to the target ungauged site to provide a basis for information transfer. The second step of the analysis consists in inferring flood information (such as quantiles) at the target site using data from the stations identified in the first step of the analysis. Two types of " homogeneous " regions can be proposed : fixed set regions (geographically contiguous or non-contiguous) and neighborhood type of regions. The second type includes the methods of canonical correlation analysis and of the regions of influence. Regional estimation can be accomplished using one of two main approaches : index flood or quantile regression methods.The results of this work indicate that the philosophies of regionalization and the methods utilized in France and Quebec are complementary to each other and are based on different needs and outlooks. While the approaches followed in France are characterized by strong conceptual and geographic aspects with an emphasis on the utilization of information related to other phenomena (such as precipitations), the approaches adopted in Quebec rely on the strength of their statistical and stochastic components and usually condense the spatial and temporal information to a realistic functional form. This dissimilarity in the approaches being followed on either side may be originated by the distinct topographic and climatic characteristics of each region (France and Quebec) and by the differences in basin sizes and hydrometeorologic network densities. The conclusions of the seminar point to the large potential of improvements in regional estimation methods, which may result from an enhanced exchange between scientists from both sides : indeed, there is much to gain from learning about the dissimilarities between the various approaches, comparing their performances, and devising new methods that combine their individual strengths. Hence, the Gradex method for example could benefit from an increased utilization of regional flood information, while flood regionalization methods utilized in Quebec could gain much from the formalization of the use of rainfall information and from the integration of an improved modeling of physical hydrologic phenomena. This should result in the enhancement of the efficiency of regional estimation methods and their ability to handle various practical conditions.It is hoped that this research will contribute towards closing the gap between French and Quebec literature, and more generally between the European and the North American hydrological schools of thought, by narrowing the large literature that is available, by providing the necessary cross-evaluation of regional flood analysis models, and by providing comprehensive propositions for improved approaches for regional hydrologic modeling