Nicolas Desassis

Automatic Variogram Modeling by Iterative Least Squares: Univariate and Multivariate Cases

In this paper, we propose a new methodology to automatically find a model that fits on an experimental variogram. Starting with a linear combination of some basic authorized structures (for instance, spherical and exponential), a numerical algorithm is used to compute the parameters, which minimize a distance between the model and the experimental variogram. The initial values are automatically chosen and the algorithm is iterative. After this first step, parameters with a negligible influence are discarded from the model and the more parsimonious model is estimated by using the numerical algorithm again. This process is iterated until no more parameters can be discarded. A procedure based on a profiled cost function is also developed in order to use the numerical algorithm for multivariate data sets (possibly with a lot of variables) modeled in the scope of a linear model of coregionalization. The efficiency of the method is illustrated on several examples (including variogram maps) and on two multivariate cases.

Microseismic Monitoring - Consequences of Velocity Model Uncertainties on Event Location Uncertainties

Among many factors that contribute to microseismic location errors, the largest contribution is due to the lack of knowledge of the wave-propagation medium. In spite of efforts to build the “best” velocity model derived from surface seismic and/or logging data, these models are very often not adapted to the microseismic context and are characterized by numerous uncertainties. These uncertainties are often enhanced due to the poor aperture of the microseismic monitoring networks. Precise location of hypocenters requires deriving a very accurate velocity model using calibration shots; the inversion to obtain this model is a difficult task but cannot be neglected. We propose a tomography algorithm using calibrations shots that does not produce only a unique “best” velocity model but all velocity models that explain the observed data within the traveltime picking uncertainties. This approach allows deriving location uncertainties associated to velocity model uncertainties. These maps show that the commonly used probability associated to the picking uncertainties must not be used to represent the probability associated to the velocity model uncertainties

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.