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

Comité National Français de Géodésie et de Géophysique : rapport quadriennal 1991-1994

Au cours de quatre campagnes successives (1990-1993), l'expérience EPSAT-Niger a permis d'acquérir un ensemble de données pluviographiques et radar sans équivalent à ce jour sur l'Afrique de l'Ouest. Un panorama rapide des données acquises et des conditions climatiques qui ont prévalu durant ces 4 années d'expérience est donné ici, permettant de dégager quelques éléments dominants de la climatologie des précipitations au Sahel. Il est suivi d'une description des travaux menés en vue d'obtenir des estimations de lames d'eau au sol plus fiables sur des surfaces dont la taille peut varier selon les besoins des hydrologues et des modélisateurs du climat. Une analyse des champs pluviométriques associés aux systèmes convectifs de méso-échelle est ensuite présentée, montrant qu'ils sont constitués de structures emboîtées dont la trace se retrouve jusque dans les cartes de cumuls saisonniers. Néanmoins la possibilité de calculer des statistiques pertinentes pour ces structures est limitée par la nature intégrée des données fournies aussi bien par le radar que par les pluviographes. Un effort majeur reste donc à accomplir pour prendre en compte la distorsion introduite par ces effets d'intégration et développer des procédures d'estimation des paramètres de modèles de champs pluviométriques sahéliens. (Résumé d'auteur

Régionalisation des modules annuels et des régimes d'étiage du bassin hydrographique de la Moselle française : lien entre modèles régionaux

Les modélisations régionales proposées sur les modules annuels et les débits moyens d'étiage des sous-bassins de la Moselle française devraient contribuer à l'amélioration des connaissances sur le fonctionnement physique actuel des hydrosystèmes. Elles s'inscrivent dans le contexte des directives de la loi Pêche et de la loi sur l'Eau et plus récemment de la Directive Cadre Européenne. Quarante neuf sous-bassins répartis en trois jeux ont permis de caler et valider un modèle régional des modules annuels et un modèle régional d'étiage. Dans le cas du bassin hydrographique de la Moselle française une certaine dépendance existe entre modèles régionaux dont la loi statistique choisie est la loi de Weibull à 2 paramètres. Une pseudo-dépendance est observée entre la loi régionale des modules annuels et la loi régionale des débits moyens d'étiage pour les années moyennes à sèches. Cette propriété va permettre en particulier l'usage d'une procédure simplifiée commune, établie à partir de la connaissance de jaugeages épisodiques d'étiage, pour l'estimation des descripteurs de débit d'un sous-bassin non observé : le module médian /qa et le débit quotidien minimal médian /vcnd=1. Pour le modèle régional d'étiage un deuxième descripteur local est nécessaire. Il s'agit d'un temps caractéristique d'étiage du sous-bassin ∆e (j) permettant de généraliser le modèle à toute durée d. Le concept débit-durée-fréquence QdF appliqué aux étiages exploite la convergence observée des distributions de différentes durées d et est indépendant de la loi fréquentielle choisie. Le caractère opérationnel de ces modélisations régionales dépend essentiellement de la précision d'estimation des descripteurs de débit du sous-bassin étudié /qa, pour les modules annuels et /vcnd=1 pour les étiages. Ces descripteurs de débit ont été estimés selon deux approches : l'approche classique par régression multiple et selon une approche simple de recherche d'un coefficient de tendance k entre jaugeages épisodiques d'étiage concomitants au sous-bassin étudié (pas ou peu d'observations) et au sous-bassin de référence (chronique de débit continue). Pour cela, un choix de cinq jaugeages d'étiage par an sur les douze dernières années en moyenne a été fait. Le descripteur de débit du sous-bassin étudié est ensuite déduit du produit de k par le descripteur de débit du sous-bassin de référence. Pour /vcnd=1, sb. étudié nous observons dans la majorité des cas une nette amélioration de l'estimation obtenue par régression, notamment une forte réduction des écarts les plus importants. Une similitude des classes de superficie entre sous-bassin étudié et sous-bassin de référence n'est pas exigée. La proximité géographique des sous-bassins semble donner de meilleurs résultats. En ce qui concerne le module médian /qasb. étudié, son estimation par régression multiple est assez performante. Parallèlement à cela, le coefficient k de tendance permet, de même que pour, une estimation cohérente de /qasb. étudié. Ce résultat un peu inattendu laisse supposer que la pseudo-dépendance observée entre modèles régionaux a bien une réalité physique. Nous avons insisté sur cette démarche "de régionalisation" nécessitant un faible investissement en mesures de débit des sous-bassins non observés par rapport au réseau national de suivi hydrométrique. Elle se présente à notre avis comme une alternative (ou complémentarité) intéressante aux méthodes de régionalisation à bases géostatistiques : telles que l'identification du voisinage hydrologique homogène du sous-bassin étudié ou encore la prise en compte de l'effet structurant du réseau hydrographique dans la cartographie du descripteur de débit. L'ensemble des connaissances relative à cette recherche est repris dans un Système d'Information Géographique pour répondre éventuellement à la demande.Regional modelling of the annual mean discharge and average low-flow discharge of the French Moselle sub-basins should improve our knowledge of its hydrosystem functioning. These models are part of the directives of the Water Act and Fishery Act and more recently of the European Framework Directive. Forty-nine sub-basins divided into three groups were used to calibrate and validate a regional model of the annual mean discharge and a regional model of low-flow discharge.A two-parameter Weibull distribution was used to represent the regional models of the Moselle catchment. Results show a relative dependence between the regional models. A pseudo-dependence was observed between the regional annual mean discharge model and the regional average low-flow discharge model for the average years to the dry ones. This property allowed us to use a common simplified procedure, established from the low-flow episodic discharge measurements, to estimate the flow descriptors of an ungauged sub-basin, namely the median annual mean discharge /qa and the median minimal daily low-flow discharge /vcnd=1. For the low-flow regional model a second local descriptor was necessary, the characteristic time of low-flow discharges of the sub-basin ∆e (j), which allowed us to generalize the model for every duration (d). The QdF flow-duration-frequency concept, applied to the low-flow discharge, is independent of the selected frequency law and uses the observed convergence of the distributions of various durations (d). The operational character of these regional models depends strongly on the accuracy of the estimation of the flow descriptors of the studied sub-basin (/qa for the annual mean discharges and /vcnd=1 for the low-flow discharges). These flow descriptors were estimated according to two approaches: a traditional approach based on multiple regression and a simple approach based on a trend coefficient (k) calculated between low-flow episodic discharge measurements that are concomitant between the studied sub-basin (no or few observations) and the reference sub-basin (time series of continuous discharges). A choice of five low flow discharge measurements per year over the last twelve years (on average) was made. Knowing the flow descriptor of the reference sub-basin, the flow descriptor of the studied sub-basin was then deduced from the trend coefficient k. For /vcnd=1, sb. étudié we observe in the majority of cases a significant improvement of the estimation obtained by regression, especially a net reduction of the most important variations. A similarity of the surface classes between the studied sub-basin and the reference sub-basin is not required. Geographical proximity of the sub-basins seems to provide better results. The estimation of the median annual mean discharges /qasb. étudié using multiple regression was rather impressive. In addition, the trend coefficient k established for the low-flow discharges allowed a coherent estimation of /vcnd=1, sb. étudié and /qasb. étudié. This quite unexpected result suggests that the observed pseudo-dependence between regional models has a physical reality.The "regionalization" technique detailed herein does not require a significant investment in flow measurements of the ungauged sub-basin compared to the national hydrometric network. It is an interesting (or complementary) alternative to the regionalization approach based on geostatistical methods that, for instance, identify the homogeneous hydrological vicinity of each sub-basin or take into account the structuring effect of the hydrographic network in the cartography of the flow descriptor

Effets des erreurs dans les coefficients structuraux d’un modèle intersectoriel « rectangulaire ». Une approche de type Monte-Carlo

The most simple rectangular input-output models use two rectangular matrices: R a market coefficient matrix, A* a production coefficient matrix. A given exogenous demand Xo determines the sectorial activity levels X* = [I — RA*]-1Xo. We assume that A* is random with expectation A. We study the distribution of the "error" X* — X with X = [I — RA]-1Xo.(1) For the statistically independent elements of A*, we analytically prove that X < EX*.(2) In the more realistic case of statistically dependent elements of A*.(a) One submatrix of A* with T non zero elements is chosen. The probabilistic model which generates the T coefficients is as follows: a* = (1 — μ)a + μ(S/n) b* où a* is the vector of the T random elements, a is the expectation of a* whose components are observed values of a real input-output model, S is the sum of components of a, μ is a parameter between zero and one, b* is a multinomial random vector with T components and parameters n, number of drawings during an experiment, and a/S, the corresponding probabilities.We control the variability of a* through μ and n. For a given experiment, we get a realisation of A* and we compute X*. K independent experiments allow us to estimate the expectation and the variance-covariance matrix of X*, simultaneous confidence intervals for the expectation of the components of X*, and also a few global measures of errors on X*.The Canadian model for 1961 (16 productive sectors, 40 commodities), is tested with that model.The main result is: the relative errors, measured according to the variation coefficients, are greatly reduced when we pass from the "errors" on a* to the corresponding "errors" on X*.(b) The same random model is also simultaneously applied to 2 or 3 sub-matrices of A*

Adaptation de la mesure potentiométrique à l'estimation en continu de la teneur en cuivre de solutions aqueuses présentant de fortes variations physico-chimiques

La mesure en continu, sans prélèvement et sous conditions physico-chimiques variables, de la teneur en cuivre d'une solution aqueuse, constitue le cadre de ce travail. La méthode d'estimation proposée est basée sur la potentiométrie sélective à l'aide d'une électrode spécifique au cuivre. Le comportement de l'électrode est modélisé par une loi non linéaire s'inspirant de la relation de Nernst généralisée. Le modèle intègre les paramètres température, conductivité et pH de la solution. Il rend possible une estimation de la concentration totale de cuivre même en présence de complexations hydroxyles. La phase de modélisation est présentée en détail. Les performances de l'estimateur sont évaluées puis discutées.Ion Selective electrodes (ISEs) offer an attractive solution for continuously evaluating the content of certain ionic species in aqueous media. Manufacturers propose a wide range of electrodes specific to heavy metals (Cu2+, Pb2+ …). Because they eliminate the need for sampling, are of reasonable size and have few electronic parts, ISEs seem highly appropriate for continuous monitoring in urban purification systems. Measurements obtained by these sensors in controlled media in the laboratory are usually precise, reliable and reproducible. However, it is not so with complex and uncontrolled media. This work falls within the general scope of the continuous measurement of heavy metals in wastewater. More particularly, it is devoted to the description of the behaviour of a copper-selective electrode (ISECu) in a medium presenting wide physicochemical variations.Experimental set-upIn order to study ISE behaviour, we developed an experimental platform that allowed us to reproduce in a reactor the physicochemical variations observed in wastewater, particularly with regards to salinity and acidity. The reactor was fitted with a measuring set consisting of five electrodes that measured the following parameters: pH (ref. integrated Ag/Agcl), redox (red), ISECu (ECu), temperature (T) and conductivity (s). A computer system carried out the acquisition of the five signals with a 10-second sampling period. The species concentration in the reactor was determined by calculating the weight of the solutions extracted from or injected into the reactor. Controlling the temperature of the system was undertaken using a cryostat. Sequential tests allowed the pH, redox potential and conductivity of the medium to be varied and were carried out by successive injections of different chemical products. The response times of the conductivity probe and of the pH and redox electrodes are shown here; the short response time of the sensors (20 to 30 s) and the strong correlation between the measured pH and redox are noted. ISE modellingThe model used to explain the ISE response is based on a generalization of Nernst's Law that takes into account the temperature and the activity of the free ions (Cu2+). Taking into consideration chemical equilibria and mass equations allowed us to link the activity of the free copper ions to the total injected copper concentration |Cu2+|tot and to the pH. Redox, strongly correlated to pH, was ignored in the mathematical model. Since hydroxyl complexation is the major complexation reaction (compared to other copper-binding ligands), the potential measured with the ISE took the following form:ECu=b0+b1T.log[(ϒ2|Cu2+|tot) / (1+b2ϒ210pH+b3ϒ2102pH) + b4]The activity coefficient ϒ2 of the Cu2+ ions was calculated from the ionic strength (I) of the solution, using the Debye-Hückel approximation. Ionic strength was derived from conductivity corrected to 25 °C. In wastewater, the ranges of the physicochemical parameters were as follows: T from 5 to 35°C; pH from 4 to 9; Omega from 500 to 2000 mS/cm; redox from 400 to -400 mV/ENH; and copper concentrations 10-3 mol/dm3.In order to identify the bi coefficients of the model, we established an experimental plan comprising 108 measurement points that covered, with a minimal number of experiments, the ranges of variations of the parameters of influence. A dispersion diagram of measured and modelled values gave a linear adjustment coefficient close to 0.99 and a standard deviation of 8.8 mV, which corresponds to a 0.34 decadal standard error in the concentration estimate. With a temperature of 25 °C, the model has a sensitivity of -26.4 mV/decade, very close to the theoretical slope of an electrode sensitive to divalent ions.ISE measurement of the copper concentration with large pH variations pH is the parameter which exerts the greatest influence on ISE response, which is why tests simulating copper pollution with large variations of pH were carried out. These tests enabled us to evaluate the performances of the model in terms of the estimation of copper content. Four solutions of total copper concentration equal to 10-6, 10-5, 10-4, 10-3 mol/dm3 respectively, were used. Their temperature was 25 °C and their conductivity was fixed at approximately 500 mS/cm. We varied the pH of each solution between 4 and 10. For the four tests, we show the estimate of the copper concentration obtained with our model starting from the potential measured by the ISE.In the case of strong copper pollution (10-3 mol/dm3), the model yields an overestimated concentration below pH 7 with a decadal error of less than 0.5. Above pH 7, the concentration is underestimated while maintaining a decadal error of less than 0.5. At pH 7, a 0.04-decade minimal error is found. For pollution equal to or less than 10-4 mol/dm3, the model gives good results in an acid or neutral medium with a decadal error usually less than 0.3. In an alkaline medium, concentration is overestimated. In this case the error increases in a roughly linear manner with the pH and the co-logarithm of copper concentration. From the results of these tests, we defined a valid domain of ISE copper concentration measurement using our model. In conclusion, the suggested method, although not very accurate, could be used as an indicator of the copper concentration level in wastewater. The ISE-response correction model is currently being tested under operational conditions at a water treatment plant in Nancy-Maxéville (France)

L’estimation de l’emploi à partir de la distribution des établissements par taille

La rareté des données est un problème endémique dans les recherches économiques appliquées à l’échelle régionale, a fortiori à l’échelle inframétropolitaine. Cet article discute trois méthodes pragmatiques d’estimation de l’emploi à partir de la distribution des établissements par taille : la méthode du point milieu de l’intervalle (méthode PM), le lissage lognormal et le lissage log-logistique. On compare les méthodes au moyen de deux ensembles de données : celles du Recensement des établissements et de l’emploi de Montréal de 1996 (Banque de Données et d’Information urbaine, INRS-Urbanisation et Ville de Montréal) et celles de Statistique Canada sur les industries manufacturières à deux chiffres de la Classification Type des Industries (CTI) au Québec en 1995. Les résultats montrent que les modèles lognormal et logistique sont nettement supérieurs à la méthode PM, notamment à cause du biais à la hausse inhérent à cette méthode.The scarcity of data is a widespread problem in applied regional and metropolitan economic research. This note discusses three pragmatic methods to estimate employment from the size-distribution of establishments: the class-interval mid-point method (MP method), lognormal smoothing, and log-logistic smoothing. The methods are compared using data from two sources: the 1996 Recensement des établissements et de l’emploi de Montréal (Banque de Données et d’Information urbaine, INRS-Urbanisation and Ville de Montréal), and Statistics Canada two-digit Standard Industrial Classification (SIC) manufacturing industry data for Québec in 1995. Results show that the smoothing models are clearly superior to the simple MP method, partly because of that method's inherent upward bias

La fiabilité des pronostics conjoncturels en France

En France il n’existe pas d’évaluation post mortem systématique des prévisions macroéconomiques. Cet article montre que les opinions des chefs d’entreprises industrielles quant à l’évolution de la production, telles que l’INSEE et la Banque de France en opèrent séparément la synthèse, ne sont pas aussi pertinentes qu’on le croit parfois. S’agissant du Baromètre de L’Expansion et de l’indicateur hebdomadaire d’activité économique du C.O.E., malgré des erreurs graves et de sérieux biais, l’auteur conclut à leur utilité, surtout pour aider à estimer précocement les tendances récentes de l’économie.Systematic post mortem evaluation of the macroeconomic forecasts' accuracydoes not exist in France. This article shows that the performance of the qualitativesurvey forecasts amongst company managers, obtained separately by the INSEEand by Banque de France, are not very good predictors of industrial production.Turning next to the "Baromètre de L'EXPANSION" and to the composite indicator of the COE, the author proves that, despite severe biases and errors, they have some utility, especially as early and low-price estimators of the present stateof the economy

Estimation des modèles probit polytomiques : un survol des techniques

Parce qu’il admet des structures très générales d’interdépendance entre les modalités, le probit polytomique (MNP) fournit une des formes les plus intéressantes pour modéliser les choix discrets qui découlent d’une maximisation d’utilité aléatoire. L’obstacle majeur et bien connu dans l’estimation de ce type de modèle tient à la complexité que prennent les calculs lorsque le nombre de modalités considérées est élevé. Cette situation est due essentiellement à la présence d’intégrales normales multidimensionnelles qui définissent les probabilités de sélection. Au cours des deux dernières décennies, de nombreux efforts ont été effectués visant à produire des méthodes qui permettent de contourner les difficultés de calcul liées à l’estimation des modèles probit polytomiques. L’objectif de ce texte consiste à produire un survol critique des principales méthodes mises de l’avant jusqu’à maintenant pour rendre opérationnel le cadre MNP. Nous espérons qu’il éclairera les praticiens de ces modèles quant au choix de technique d’estimation à favoriser au cours des prochaines années.The Multinomial Probit (MNP) model provides the most general framework to allow for interdependent alternatives in discrete choice analysis. The primary impediment to this methodology is related to the dimensionality of the response probabilities which are multifold normal integrals of about the size of the choice set. During the last two decades, numerous researches have been devoted to develop practical methodologies to replace these hard to compute choice probabilities in the estimation process. The main objective of this paper is to survey the major and the most important of these techniques