Christian Lantuéjoul

TBSIM: A computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method

The simulation of spatially correlated Gaussian random fields is widespread in geologic, hydrologic and environmental applications for characterizing the uncertainty about the unsampled values of regionalized attributes. In this respect, the turning bands method has received attention among practitioners, for it allows multidimensional simulations to be generated at the CPU cost of one-dimensional simulations. This work provides and documents a set of computer programs for (i) constructing three-dimensional realizations of stationary and intrinsic Gaussian random fields, (ii) conditioning these realizations to a set of data and (iii) back-transforming the Gaussian values to the original attribute units. Such programs can deal with simulations over large domains and handle anisotropic and nested covariance models. The quality of the proposed programs is examined through an example consisting of a non-conditional simulation of a spherical covariance model. The artifact banding in the simulated maps is shown to be negligible when thousands of lines are used. The main parameters of the univariate and bivariate distributions, as well as their expected ergodic fluctuations, also prove to be accurately reproduced.

Non Conditional Simulation of Stationary Isotropic Multigaussian Random Functions

In this paper, the problem of simulating a 3-dimensional stationary, isotropic multigaussian random function with covariance C is considered. On the basis of the Central Limit Theorem, a possible procedure consists of simulating a large number of independent stationary random functions (not necessarily multigaussian) with covariance C. This procedure raises two questions: i) how to simulate a random function of covariance C? Several algorithms are possible (spectral, dilution, turning bands, tessellation, migration …). These algorithms are briefly presented and compared from various standpoints such as their range of validity and their efficiency. ii) what is the number of random functions that have to be simulated? Even if no magic number can be recommended, a helpful tool to answer this question is the Berry-Esseen theorem.

Can a Training Image Be a Substitute for a Random Field Model

In most multiple-point simulation algorithms, all statistical features are provided by one or several training images (TI) that serve as a substitute for a random field model. However, because in practice the TI is always of finite size, the stochastic nature of multiple-point simulation is questionable. This issue is addressed by considering the case of a sequential simulation algorithm applied to a binary TI that is a genuine realization of an underlying random field. At each step, the algorithm uses templates containing the current target point as well as all previously simulated points. The simulation is validated by checking that all statistical features of the random field (supported by the simulation domain) are retrieved as an average over a large number of outcomes. The results are as follows. It is demonstrated that multiple-point simulation performs well whenever the TI is a complete (infinitely large) realization of a stationary, ergodic random field. As soon as the TI is restricted to a limited domain, the statistical features cannot be obtained exactly, but integral range techniques make it possible to predict how much the TI should be extended to approximate them up to a prespecified precision. Moreover, one can take advantage of extending the TI to reduce the number of disruptions in the execution of the algorithm, which arise when no conditioning template can be found in the TI.

Substitution random functions

The construction of the new family of random functions presented here was inspired by cartographic techniques. In exactly the same way that a map is a combination of spatial information and an appropriate representation, a substitution random function is a combination of a spatial random function and a coding process. Explicit calculations can be derived on substitution random functions. Moreover, non conditional and conditional simulations can be carried out to produce outcomes that respect some prescribed morphology.

Setting up the General Methodology for Discrete Isofactorial Models

A general methodology for building change of support models for discrete variables has been proposed by Matheron (1984a). This methodology is based on discrete diffusion processes (i.e. birth and death processes). As the marginal distribution as well as the diffusion coefficients are arbitrary, the method is applicable to a wide range of phenomena, but the inference of the parameters is a challenging problem. This article describes some methods for overcoming this problem.

Local contributions to the Euler–Poincaré characteristic of a set

The Euler–Poincaré characteristic (EPC) of a polyconvex subset X of Rd can be evaluated by covering the subset with an auxiliary tessellation, measuring its contribution within each cell of the tessellation and adding all contributions. Two different ways are proposed to define the contribution of a cell to the EPC of X. These contributions turn out to be related by duality formulae. Finally, three applications are given: the measurement of the EPC on adjacent fields, the measurement of the EPC on discretized images and the detection of defects in atomic structures.

Gaussian random function

An extremely useful consequence of the central limit theorem is the existence of a class of random functions whose spatial distribution depends only on their first two moments. These are Gaussian random functions. Their main statistical properties are reviewed, the texture of their realizations is examined, and algorithms are proposed to simulate them, conditionnally or not.

Estimation of the connectivity of a synthetic porous medium

– the genus G(S) of a surface S is the maximum number of non-homologous closed curves that can be drawn on S without dividing it into separate parts. In the case where S delimits one internal domain, an equivalent but more concrete definition can be given. It is the maximum number of fundamental cuts that can be made through the structure without splitting it into separate parts (DeHoff & Rhines, 1968). – the Euler–Poincaré characteristic (EPC) N3(X) of a set X is defined as the total number of distinct surfaces of X minus the sum of their genuses (Kronsbein & Steele, 1967; Barrett & Yust, 1970). In other words

Sampling of Orebodies with a Highly Dispersed Mineralization

This paper presents several results on how to sample highly dispersed type orebodies, e.g. alluvial gold or diamond deposits. Two major difficulties are encountered when sampling such orebodies. The first is of a geometric nature. Mineralization is not homogeneous, but is distributed in patches. The second difficulty is that the mineralization within the patches is highly variable. As a consequence, two different factors at two different scales account for the grade variability of the orebody. The first difficulty results from the spatial distribution of the patches and the second from the high dispersion of the mineralization within the patches. Both factors can be very important. In the case where the second factor is dominant, it is possible to sample so as to reduce its impact. This can be achieved by choosing a sample support appropriate to the size of the patches.

Basic morphological concepts

The purpose of this chapter is to introduce the two basic concepts of mathematical morphology, namely dilations and erosions. Three possible uses are given. Firstly, dilation and erosion can be combined to produce two other morphological concepts (openings and closings) which have rich structural content. Secondly, the Hausdorff distance between objects has a simple morphological interpretation. Finally, the chance of detecting an object by regular sampling can be simply expressed in terms of dilations.