Jean-Paul Chilès

Geostatistical modeling of sound propagation: Principles and a field application experiment

The assessment of noise sources for environmental purposes requires reliable methods for mapping. Numerical models are well adapted for sophisticated simulations and sensitivity analyses; however, real-time mapping of large frequency bands must be based on fast and acceptable computations and honor in situ measurements. In this paper, a real-time mapping procedure of noise exposure is proposed. The procedure is based on geostatistical modeling of spatial variations and applied to a case study taken from an experimental campaign, where a point source was placed on a flat meadow. An analytical approximation of the acoustic field was first computed with the Embleton model. The difference between this approximation and the actual measurements (L(eq15 min) 1/3-octave bands samples from 19 microphones spread over the meadow) showed spatial structure, which has been modeled with a variogram. Finally, the geostatistical technique of kriging with external drift provided an optimal interpolation of the acoustic field data while encapsulating the first approximation from the Embleton model. Systematic geostatistical inference and real-time mapping with the proposed procedure can be envisaged in simple cases.

Some Applications of Geostatistics to Civil Engineering

The geostatistical approach is more and more often used in civil engineering project, where many constraints must be satisfied in terms of risk and security. The study of the tunnel project under the English Channel, shows how geostatistics can be applied to combine data from wells, sonic and seismic surveys, taking account of various measurement errors, to produce contour maps and section maps as accurate as possible. The calculation and modelling of local variograms produce reliable results (estimation and standard deviation), that enable the geostatistician to optimize the tunnel alignment and to define an optinun design of new surveys in sensitive areas. Some simpler case studies are also presented.

Channel Tunnel: Geostatistical Prediction Facing the Ordeal of Reality

Geometrical modelling of geological layers in the Channel Tunnel area used geostatistical techniques for the computing of predictions and confidence intervals. The results, especially those concerning the top of the Gault Clay, were used to design the route of the tunnel. During the tunnelling, subvertical boreholes were drilled downwards from inside the tunnels at various intervals, to check the real level and dip of the top of the Gault clay. This provided an outstanding opportunity to confront the geostatistical prediction with reality. A critical analysis of the discrepancies assessed the quality and reliability of the model, and validated the geostatistical methods.

History Matching of a Stochastic Multifractal Subesismic Fault Model

Many geostatistical methods have been developed to generate realistic models of subseismic fault networks. The problem is that these models are difficult to history match. In this work, we present an original methodology in which history matching is performed through a modification of fault positions. We first propose a multifractal methodology to generate the faults. The method has been specifically developed to allow history matching. The model parameters are derived from the analysis of the seismic faults. A stochastic algorithm is used to generate 3D subseismic fault realizations that are constrained to the seismic faults. The fracture network is then discretized on a dual porosity simulation grid. Equivalent flow parameters are computed using an analytical method. Last, full field simulations are performed using a multiphase flow simulator. Then, we introduce a method to gradually change the locations of faults while preserving multifractal properties. Changes in locations are driven from a reduced number of parameters. These parameters are gradually modified to optimize the geometry of the realization and compel the initial fault model to reproduce the hydrodynamic behaviour observed on the field. The potential of the methodology is demonstrated on a 3D case study.

Fifty Years of Kriging

Random function models and kriging constitute the core of the geostatistical methods created by Georges Matheron in the 1960s and further developed at the research center he created in 1968 at Ecole des Mines de Paris, Fontainebleau. Initially developed to avoid bias in the estimation of the average grade of mining panels delimited for their exploitation, kriging received progressively applications in all domains of natural resources evaluation and earth sciences, and more recently in completely new domains, for example, the design and analysis of computer experiments (DACE). While the basic theory of kriging is rather straightforward, its application to a large diversity of situations requires extensions of the random function models considered and sound solutions to practical problems. This chapter presents the origins of kriging as well as the development of its theory and its applications along the last fifty years. More details are given for methods presently in development to efficiently handle kriging in situations with a large number of data and a nonstationary behavior, notably the Gaussian Markov random field (GMRF) approximation and the stochastic partial differential (SPDE) approach, with a synthetic case study concerning the latter.