Chantal de Fouquet

Conditioning a Gaussian model with inequalities

Functions with mixed discrete/continuous distribution, random sets or coregionalizations between a function and a random set can be modelled using gaussian random functions. In these cases, the conditions at experimental points are expressed in terms of inequalities: 1 ≤ i ≤ n, a i ≤ X i ≤ b i . The simulation with composite constraints (equalities and inequalities) is performed in two steps: firstly we ensure that the inequality constraints are validated, secondly we perform a classic conditional simulation. To validate the inequalities, we propose two methods: the first one is based on the acceptance-rejection technique but can be used only to validate very few inequalities; in the second, we sample the stationary distribution of a Markov chain. This latter case is illustrated with a few examples.

Combining geostatistics and flow simulators to identify transmissivity

The reconstruction of the transmissivity field from the more numerous experimental hydraulic head data, an inverse problem, remains the focus of continuing stochastic-based research. The difficulty of this problem arises not only from the complexity of the diffusion equation that links the two variables, but also from taking into account the physical aspects of the site under study; e.g. the boundary conditions, the effective recharge, and the geology. In practical applications, the validity of purely analytical techniques proposed to date is limited by certain simplifying assumptions, like the linearization of the flow equation, made in order to obtain a solution. For this reason, a hybrid methodology combining geostatistical techniques with deterministic numerical flow simulators is proposed. This combination allows the numerical calculation of the direct and cross covariances needed to cokrige the transmissivity from both the transmissivity and hydraulic head data. The flexibility of numerical flow simulators takes away the need for the simplifying assumptions of analytical techniques to apply the proposed methodology.

Applications of kriging to image sequence coding

Abstract Kriging is a linear interpolation method used by geostatisticians. In this paper we show that it can be successfully applied to image sequence coding. First we apply it to texture coding and we improve these preliminary results with an optimization method that we have named ‘inverse kriging’. Then we apply kriging and inverse kriging to motion vector fields, which are by essence smooth within an object. We show an application of these tools in the framework of the active mesh coding scheme developed within the RACE/Morpheco project.

REMINDERS ON THE CONDITIONING KRIGING

Conditioning simulations by using kriging was proposed by G. Matheron at the start of the 70s. The following presentation only recalls results that should be already well-known.

Mapping the Centimeter-Scale Spatial Variability of PAHs and Microbial Populations in the Rhizosphere of Two Plants.

Rhizoremediation uses root development and exudation to favor microbial activity. Thus it can enhance polycyclic aromatic hydrocarbon (PAH) biodegradation in contaminated soils. Spatial heterogeneity of rhizosphere processes, mainly linked to the root development stage and to the plant species, could explain the contrasted rhizoremediation efficiency levels reported in the literature. Aim of the present study was to test if spatial variability in the whole plant rhizosphere, explored at the centimetre-scale, would influence the abundance of microorganisms (bacteria and fungi), and the abundance and activity of PAH-degrading bacteria, leading to spatial variability in PAH concentrations. Two contrasted rhizospheres were compared after 37 days of alfalfa or ryegrass growth in independent rhizotron devices. Almost all spiked PAHs were degraded, and the density of the PAH-degrading bacterial populations increased in both rhizospheres during the incubation period. Mapping of multiparametric data through geostatistical estimation (kriging) revealed that although root biomass was spatially structured, PAH distribution was not. However a greater variability of the PAH content was observed in the rhizosphere of alfalfa. Yet, in the ryegrass-planted rhizotron, the Gram-positive PAH-degraders followed a reverse depth gradient to root biomass, but were positively correlated to the soil pH and carbohydrate concentrations. The two rhizospheres structured the microbial community differently: a fungus-to-bacterium depth gradient similar to the root biomass gradient only formed in the alfalfa rhizotron.

Adapting geostatistical transmissivity simulations to finite difference flow simulators

The validity of hydrogeological studies is often dependant on the correct estimation of the flux (or, equivalently, the equivalent transmissivity) of the field. However, depending on the level of fine-scale heterogeneity of the porous media considered, flow simulators based on the finite differences algorithm can be shown to underestimate the equivalent transmissivity. This result is shown to depend on the value on the intermesh transmissivities and not the mesh transmissivities. When working with geostatistically simulated point support transmissivities, intermesh transmissivities can be directly calculated so that the finite differences result obtained is consistent with the spatial transmissivity distribution.

Environmental Statistics Revisited: Is the Mean Reliable?

The sample mean of data collected during critical pollution periods is a biased estimator of the annual mean. The bias can be corrected by weighting techniques, which take into account the measurement dates. The differences with standard calculations become important to characterize the temporal evolution when sampling changes with time. Sound statistical methods are therefore needed.

Can Measurement Errors Be Characterized from Replicates

Sample measurements (of grade, depth, etc.) are almost inevitably affected by errors. Several error models were studied in the literature. But the interest of replicates for selecting the error model received limited attention. If measurement errors are supposed to be additive, homoscedastic, without correlation between them, and spatially not correlated with the exact values, the variances of the measurement errors are computable from the sample, simple, and cross-variograms of replicate data sets, even if the variogram of the exact value is pepitic (Aldworth W, Spatial prediction, spatial sampling, and measurement error. Retrospective Theses and Dissertations. Paper 11842. Iowa State University Digital Repository @ Iowa State University, 1998; Faucheux et al. Characterisation of a hydrocarbon polluted soil by an intensive multi-scale sampling. Geostats 2008, proceedings of the 8th international geostatistics congress, 1–5 Dec. 2008, Santiago, Chile. Ortiz J-M, Emery X (eds) for an example, 2008). But what about the other cases? When the error is additive, its correlation with the exact value can remain undetectable. The variance of the measurement errors is thus not always computable. It’s the same for an error of multiplicative type. Except in some special cases, keeping the different measurement values rather than their average improves the precision of the estimation.

Large-Scale CO2 Storage in a Deep Saline Aquifer: Uncertainties in Predictions Due to Spatial Variability of Flow Parameters and Their Modeling

Scarce data and uncertainties in the spatial variation of geological properties lead to different possible models of these heterogeneities. The aim of this study is to compare the pressure results and CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}