Variogramme | EPFL Graph Search

Le est une fonction mathématique utilisée en géostatistique, en particulier pour le krigeage. On parle également de , de par le facteur ½ de sa définition. L', , ou est l'estimation et l'étude d'un variogramme sur une variable aléatoire. Variogramme d'une fonction aléatoire Considérons une variable aléatoire, de la variable d'espace , et supposons-la stationnaire, c'est-à-dire que la moyenne et la variance de sont indépendantes de . On pose la grandeur: Comme est stationnaire, le membre de droite dépend uniquement de la distance entre les points et . Le variogramme à une distance est alors la demi moyenne des carrés des différences des réalisations de sur les points espacés de . Variogramme borné La réciproque est fausse : si est intrinsèque et de variogramme borné, alors est la somme d'une fonction aléatoire stationnaire de et d'une variable aléatoire réelle. Intérêt du variogramme Le variogramme est défini pour toute fonction aléatoi

Elaboration d'outils pour l'investigation et l'assainissement des stands de tirs

Bruno Chappuis

Switzerland has about 6,000 shooting ranges, of which 2,000 are still in use. These activities generate a significant lead pollution. FOEN estimated 200 tons of lead is fired each year. This Master Project has been realized with the consultancy office CSD Ingénieurs Conseils SA in Lausanne. The project is divided into three parts related to investigations and remediation of shooting ranges. Eleven shooting ranges of the army and two communal shooting ranges have formed the field study of this project. On these sites, 20 samples were taken for laboratory analysis. Nearly 350 field tests with a portable X-ray fluorescence (XRF) were also performed. The first part of the study focused on sample preparation methods. Based on several assumptions, they often give conflicting results. It has been shown that not taking into account the size fraction between 1 cm and 2 mm leads to underestimating the lead concentration by 10% on average. Taking into account the bullets, and how, is even more important. The original content of lead in standard military bullets is 74% of the weight. A common use is to assume this content for the bullets found in soils samples. The lead content measured in bullets is from 64% for those in good shape to 17% for fragments. Lead seems to spread more easily than the steel jacket of the bullets. These results are of major importance for the calculation of the lead content of samples. A strong correlation between lead and bullet amount in soils has not been found. The condition of the bullets seemed to be a significant factor, but the representativeness of the samples was not sufficient due too few mass samples. Taking into account these results, a preparation method was proposed. The second part of the study concern leachate analysis. In particular, OSites leachate and OTD1 (acid) leachate was studied. These leachates are used respectively for polluted sites risk assessment and for choosing treatment solution of excavated materials. Correlations were looked for, between concentrations in leachate and soils parameters. CEC, clay content, organic carbon content and aquous pH has been used as soils parameters. Leachate OTD1 (neutral) has been preferred to leachate Osites for economic reasons, these two tests provide similar results for lead. Good correlations were obtained with non-linear models taking into account all the soils parameters. Models based only on the lead content and aqueous pH (better correlation) give interesting results, but have greater error. The third part focused on the study of geostatistics shooting ranges with fixed targets. Variogram were made from field measurements. A metric sampling grid metric was used. The variogram showed a remedy made from a range of 3 meters. Although the site shows a clear anisotropy, an isotropic spherical variogram with nugget effect has been adjusted. The data were insufficient to fit an anisotropic model. Calculations showed that the kriging interpolation provides better results than those obtained with conventional interpolation. The study also showed that the uncertainty of the kriging interpolation decreases when the mesh sampling is lower. An optimum between accuracy and number of tests could not be found, because the variogram used has too much variability. The results are very promising and will continue to be worked out in order to produce the means of zoning and estimate for the remediation of shooting range.

2009

Spatial Breakdown Point of Variogram Estimators

In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on spatially located data sets. Therefore, the classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in R2. In order to study the effects of perturbations, several simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, then the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% in average. Theoretical results confirm the previous statement on data in Rd, d>1.

1996

Analysis of Rainfall Data by Robust Spatial Statistics using S+SpatialStats

Reinhard Furrer

This paper discusses the use of robust geostatistical methods on a data set of rainfall measurements in Switzerland. The variables are detrended via non-parametric estimation penalized with a smoothing parameter. The optimal trend is computed with a smoothing parameter based on cross-validation. Then, the variogram is estimated by a highly robust estimator of scale. The parametric variogram model is fitted by generalized least squares, thus taking account of the variance-covariance structure of the variogram estimates. Comparison of kriging with the initial measurements is completed and yields interesting results. All these computations are done with the software S+SpatialStats, extended with new functions in Splus that are made available.

1998

Elaboration d'outils pour l'investigation et l'assainissement des stands de tirs

Bruno Chappuis

2009