Mise en oeuvre du krigeage sur arbre.

International audienceUn modèle de fonctions aléatoires définies sur une topologie d'arbre (de Fouquet & Bernard-Michel, 2006) a été développé pour l'estimation de concentrations le long d'un réseau hydrographique. Le principe consiste à décomposer le réseau en filets élémentaires joignant chaque " source " à " l'exutoire ". La concentration Z au point x s'exprime comme une combinaison linéaire Z(x)=\Sum wiYi(x) de variables aléatoires élémentaires Yi définies sur ces filets, dont les coefficients wi dépendent de la position sur l'arbre. Le krigeage des concentrations au point x revient donc à l'estimation d'une combinaison linéaire des Yi(x) à partir d'autres combinaisons linéaires, les concentrations mesurées aux points expérimentaux $x_\alpha$; : Z(x_\alpha)= \Sum wi(x_\alpha)Yi(x_\alpha). Le krigeage dépend des hypothèses sur les concentrations, qui se ramènent à des hypothèses sur les variables élémentaires Yi et sur les coefficients wi. Ces hypothèses déterminent les conditions requises pour l'estimation : le nombre minimum de mesures et leur répartition par arête. Ce modèle est appliqué aux concentrations en nitrates sur un réseau constitué d'une petite portion de la Seine (de l'amont de Paris à l'estuaire) et de la Marne. Tout d'abord, la simulation déterministe ProSe, disponible en tout point (Even et al., 1998), permet de tester les hypothèses. Ensuite, le krigeage est effectué à partir des mesures aux stations effectivement disponibles

Optimal spatial design for air quality measurement surveys: what criteria?

International audienceIn this work, we present a spatial statistical methodology to design benzene air concentration measurement surveys at the urban scale. In a first step, we define an a priori modeling based on an analysis of data coming from previous campaigns on two different agglomerations. More precisely, we retain a modeling with an external drift which consists of a drift plus a spatially correlated residual. The statistical analysis performed leads us to choose the most relevant auxiliary variables and to determine an a priori variogram model for the residual. An a priori distribution is also defined for the variogram parameters, whose values appear to vary from a campaign to another. In a second step, we optimize the positioning of the measuring devices on a third agglomeration according to a Bayesian criterion. Practically, we aim at finding the design that minimizes the mean over the urban domain of the universal kriging variance, whose parameters are based on the a priori modeling, while accounting for the prior distribution over the variogram parameters. Two optimization algorithms are then compared: simulated annealing and a particle filter based algorithm

Influence of geological parameters on CO2 storage prediction in deep saline aquifer at industrial scale

International audienceConsequences of uncertainties on geological parameters are examined using 2D models at large extension. Reduction of the uncertainties on predictions is also investigated, either because parameter's influence is negligible for the project design, or by showing for which parameters additional data will significantly increase the quality of prediction. TOUGH2/ECO2N is used to simulate the injection of millions tonnes of CO2 for the specific case of the Dogger aquifer (carbonates aquifer in the Paris Basin) with high lateral and vertical heterogeneities and for which few data are available. Studied parameters are spatial variability and correlation length of permeability, value of absolute permeability, pore compressibility, caprock permeability and relative permeability curves. Several numerical models of permeability are constructed: two uniform cases (two values of permeability) and 200 geostatistical initial realizations, which are modified according to the studied parameters. Results are compared in terms of propagation of pressure perturbations, injectivity (pressure in the vicinity of the well) and in terms of gas migration and dissolution. The results indicate, for the specific scale and values, that: (1) Pore compressibility, absolute value and spatial variability of permeability have the main influence on pressure propagation and injectivity. Relative permeability curves and correlation lengths have a weaker influence for the peak of pressure but tend to increase the variations for maximum/minimum cases. (2) Relative permeability curves and heterogeneities have a significant impact on prediction of gas dissolution and migration. At last, we also investigate the possibility to reduce the number of runs

Equivalent block transmissivity in an irregular 2D polygonal grid for one-phase flow: a sensitivity analysis

International audienceUpscaling is needed to transform the representation of non additive space-dependent variables, such as permeability, from the fine grid of geostatistical simulations (to simulate small scale spatial variability) to the coarser, generally irregular grids for hydrodynamic transport codes. A new renormalisation method is proposed, based on the geometric properties of a Voronoï grid. It is compared to other classic methods by a sensitivity analysis (grid, range and sill of the variogram, random realisation of a simulation); the criterion is the flux of a tracer at the outlet. The effect of the upscaling technique on the results appears to be of second order compared to the spatial discretisation, the choice of variogram, and the realisation

The influence of spatial variability on 2D reactive transport simulations

International audienceIn reactive transport simulations, the effects of the spatial variability of geological media are generally neglected. The impact of this variability is systematically examined here in 2D simulations, with a simple geometry and chemistry with a positive feedback: increase of porosity and of permeability during calcite dissolution. The results highlight the leading role in these conditions of: (i) the correlation length of porosity and of permeability; and (ii) the kinematic dispersivity, whose effects are dominant compared to those of variance and reaction kinetics. The impact of stochastic variability (between several randomdraws) is also significant, as it is of the same order of magnitude as the impact of the range and dispersivity

Apport croisé de la modélisation déterministe et géostatistique. Exemple des concentrations en nitrates de la Seine.

5pages+résuméLe long des réseaux hydrographiques, l'espacement des stations de mesure des concentrations et l' hétérogénéité des mesures compliquent le calage des modèles géostatistiques nécessaires à l'estimation des concentrations (Bernard-Michel et de Fouquet, 2006). Le recours à un modèle déterministe peut-il permettre de remédier à la rareté des mesures, en fournissant débit, concentrations sur tout le domaine simulé ? Nous examinons par bief les résultats du modèle PROSE qui décrit la Seine et la Marne peu avant leur confluence jusqu'à Poses. Temporellement les mesures et la simulation PROSE sont ajustables en modèle linéaire de corégionalisation. Spatialement, le cokrigeage permet de recaler les conditions aux limites, et donc la simulation, aux mesures de contrôle. reproduisant la variabilité spatiale, mais faiblement corrélé aux mesures, le modèle PROSE s'interprète alors en terme de simulation non conditionnelle. Enfin, un modèle de variogramme tri-variable débit-flux-concentrations est ajusté spatialement

Caractérisation spatiale et temporelle des "Masses d'Eau Cours d'Eau". Spatial and Temporal characterization of "River Water Bodies".

International audienceThis article aims to understand how to extrapolate in space and time discrete measurements in order to calculate physico-chemical indicators in rivers, which are required by the Water Framework Directive. Linked to this issue, few questions are addressed. Does the French National Basin Network provide enough information in order to make consistent water quality maps? How does the temporal indicator - the 90 percentile - vary in space? The outputs of the ProSe model applied to the Seine River are used to compare two different methods for calculating the 90 percentile: the classical method based on the empirical percentile function and a method that aims to reduce the estimation bias of the 90 percentile. This second method includes temporal weighting and linearization o the empirical percentile function, and therefore its application is a little more complex. But with this method the bias induced by irregular and/or few measurements is reduced. Three methods for spatializing the 90 percentiles have been tested in order to obtain occurrence percentages of the percentiles for each quality class. The first one is based on the "failure principle" and consists in keeping only the worst site for the considered "River Water Body". The second one respects the proportion of percentiles located in each quality class, while the third one allocates an influence segment to each measurement site. Spatializing temporal percentiles in "River Water Bodies" by influence segments leads to a marked improvement of occurrence percentage estimations and reveals the necessity to take into account the spatial configuration of measurement sites when calculating a quality indicator

Spatial representativeness of an air quality monitoring station. Application to NO2 in urban areas

International audienceThe present study aims at setting up a geostatistical methodology that could be implemented in an operational context to assess the spatial representativeness of a measurement station. In the proposed definition, a point is considered as belonging to the area of representativeness of a station if its concentration differs from the station measurement by less than a given threshold. Additional criteria related to distance or environmental characteristics may also be introduced. Concentrations are first estimated at each point of the domain applying kriging techniques to passive sampling data obtained from measurement surveys. The standard deviation of the estimation error is then used, making a hypothesis on the error distribution, to select the points, at a fixed risk, where the difference of concentration with respect to the station is below the threshold. The methodology is then applied to NO2 experimental datasets for different French cities

Caractérisation spatiale et temporelle de la qualité des « Masses d’Eau Cours d’Eau »

Focalisé sur les indicateurs physico-chimiques soutenant la biologie des cours d’eau, l’article examine l’interpolation de ce type de mesures, dans le temps et l’espace, pour le calcul des indices légaux requis par la Directive Cadre Européenne sur l’Eau. En effet, le calcul d’indicateurs statistiques, à partir d’une information très lacunaire, pose problème. Différentes méthodes de calcul du quantile 90 par station sont-elles équivalentes? Comment cet indicateur varie-t-il spatialement? Le Réseau National de Bassin français fournit-il suffisamment d’information pour une caractérisation pertinente de la qualité des eaux?Les sorties du modèle déterministe ProSe appliqué à la Seine, à pas de temps journalier, sont utilisées pour comparer différentes méthodes de calcul des indicateurs. Les résultats déduits du modèle exhaustif sont comparés à ceux calculés après un échantillonnage simulant celui du réseau de surveillance.Deux calculs du quantile 90 temporel par station sont examinés : le calcul classique fondé sur la fonction de quantile empirique, et une méthode légèrement plus complexe, avec une pondération temporelle et une linéarisation de la fonction de quantile, qui atténue effectivement les biais induits par l’échantillonnage irrégulier durant l’année, ou découlant du nombre restreint de mesures.Trois méthodes de « spatialisation » sont ensuite testées afin d’obtenir des pourcentages d’occurrence des quantiles par classe de qualité dans chaque « Masses d’Eau Cours d’Eau » : le « principe de défaillance » retient la station la plus défavorable; la deuxième méthode calcule la proportion des stations par classe de qualité; la dernière pondère chaque station par son « segment d’influence ». La spatialisation par segments d’influence des quantiles temporels au sein des « Masses d’Eau Cours d’Eau » améliore nettement les estimations des pourcentages d’occurrence, montrant la nécessité de la prise en compte de la localisation des stations lors du calcul d’un indice de qualité.This research aimed to understand how to interpolate discrete measurements, in space and time, in order to calculate physico-chemical indicators in rivers, which are required by the European Water Framework Directive. Linked to this issue, several questions were addressed. Are the different methods used to calculate temporal 90th-percentiles at a given site equivalent? How does this legal indicator vary in space? Does the French National Basin Network provide enough information to make consistent water quality characterization?The daily outputs of the ProSe model applied to the Seine River were used as proxies to compare different calculation methods of the 90th-percentile. The results deduced from the exhaustive model were compared to those calculated, after sampling the outputs according to the monitoring network sampling scheme. Two calculations of the temporal 90th-percentile at a given site were examined: the classical method based on the empirical percentile function and a slightly more complex method that includes temporal weighting and linearization of the empirical percentile function. This second method reduced the estimation bias of the 90th-percentile induced by irregular and/or few measurements.Three methods for spatializing the 90th-percentiles were tested to obtain occurrence percentages of the percentiles for each quality class in each “Stream Water Body”: the “failure principle” consists in keeping only the worst site; the second approach calculates the proportion of sites located in each quality class; the third method allocates an influence segment to each measurement site. Spatializing temporal percentiles in “Stream Water Bodies” by influence segments led to a marked improvement in occurrence percentage estimations and revealed the need to take into account the spatial configuration of measurement sites when calculating a quality indicator