Xavier Emery

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.

Simulation of geological domains using the plurigaussian model: New developments and computer programs

The plurigaussian model is currently used for simulating geological domains (facies) in petroleum reservoirs and mineral deposits, with the aim of assessing the uncertainty in the domain boundaries and of improving the geological controls in the characterization of quantitative attributes. This paper discusses the main aspects of the model and provides a set of computer programs to perform its inference and conditional simulation. Two types of conditioning information are allowed: hard data for which one has an exact knowledge of the actual domain at sample locations, and soft data consisting of inequality constraints on the local domain proportions (probabilities of occurrence) at control points chosen by the mining or reservoir geologist. An application to a Chilean porphyry copper deposit is finally presented, in which three Gaussian fields are used to simulate the spatial distribution of five mineralogical domains: gravels, leached capping, oxides, primary and secondary sulfides. The model is constructed so as to honor the topological contacts between mineralogical domains, their spatial continuity, the information logged at exploration drill holes, as well as the vertical proportion curves that indicate the mineralization profile with depth.

Iterative algorithms for fitting a linear model of coregionalization

In geostatistical applications, automated or semi-automated procedures are often used for modeling the spatial correlation structure (simple and cross variograms) of multivariate data. This paper deals with the well-known linear model of coregionalization and presents three iterative algorithms to find out coregionalization matrices that minimize a weighted sum of the squared deviations between sample and modeled variograms. The first one is a variation of Goulard and Voltzs proposal for variogram fitting with no constraint other than mathematical consistency. The second one uses simulated annealing for fitting subject to constraints on the simple variogram sills. The third one is a nonlinear least squares algorithm for fitting a plurigaussian model. In all three algorithms, the sample variogram matrices need not be entirely known for every lag vector, a situation of interest with heterotopic samplings. To demonstrate the capabilities of the proposed algorithms, a set of computer programs is provided and applied to case studies in mineral resources evaluation.

A turning bands program for conditional co-simulation of cross-correlated Gaussian random fields

A Matlab program (TBCOSIM) is provided for co-simulating a set of stationary or intrinsic Gaussian random fields in R^3, whose simple and cross-covariance functions are fitted by a linear model of coregionalization. It relies on the turning bands method, which performs three-dimensional simulation via a series of one-dimensional simulations along lines that span R^3. There is no restriction on the number of random fields to simulate, on the number of basic structures used in the coregionalization model, and on the number and configuration of the locations where simulation has to be performed. Additionally, the realizations can be made conditional to data, back-transformed and averaged over a block support. TBCOSIM uses parallel simulation algorithms: at each location, the random fields are simulated simultaneously and a single co-kriging is needed for conditioning all the realizations. The capabilities of the program are illustrated with the analysis of a set of non-conditional realizations and with an application to a soil contamination dataset.

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.

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The “sequential isofactorial” method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.

Using the Gibbs sampler for conditional simulation of Gaussian-based random fields

This article presents models of random fields with continuous univariate distributions that are defined by simple operations on stationary or intrinsic Gaussian fields. Realizations of these models can be conditioned to a set of data by using iterative algorithms based on the Gibbs sampler, while parameter inference relies on the fitting of the sample univariate and bivariate distributions. The proposed models are suited to the description of regionalized variables with a spatial clustering of high or low values, patterns of connectivity and curvilinearity, or an asymmetry in the spatial correlation of indicator variables with respect to the median threshold. The simulation procedure is illustrated by a case study in environmental science dealing with nickel concentrations in the topsoil of a polluted site.

Simulating Large Gaussian Random Vectors Subject to Inequality Constraints by Gibbs Sampling

The Gibbs sampler is an iterative algorithm used to simulate Gaussian random vectors subject to inequality constraints. This algorithm relies on the fact that the distribution of a vector component conditioned by the other components is Gaussian, the mean and variance of which are obtained by solving a kriging system. If the number of components is large, kriging is usually applied with a moving search neighborhood, but this practice can make the simulated vector not reproduce the target correlation matrix. To avoid these problems, variations of the Gibbs sampler are presented. The conditioning to inequality constraints on the vector components can be achieved by simulated annealing or by restricting the transition matrix of the iterative algorithm. Numerical experiments indicate that both approaches provide realizations that reproduce the correlation matrix of the Gaussian random vector, but some conditioning constraints may not be satisfied when using simulated annealing. On the contrary, the restriction of the transition matrix manages to satisfy all the constraints, although at the cost of a large number of iterations.

Conditional co-simulation of continuous and categorical variables for geostatistical applications

The modeling of uncertainty in continuous and categorical regionalized variables is a common issue in the geosciences. We present a hybrid continuous/categorical model, in which the continuous variable is represented by the transform of a Gaussian random field, while the categorical variable is obtained by truncating one or more Gaussian random fields. The dependencies between the continuous and categorical variables are reproduced by assuming that all the Gaussian random fields are spatially cross-correlated. Algorithms and computer programs are proposed to infer the model parameters and to co-simulate the variables, and illustrated through a case study on a mining data set.

An improved spectral turning-bands algorithm for simulating stationary vector Gaussian random fields

We propose a spectral turning-bands approach for the simulation of second-order stationary vector Gaussian random fields. The approach improves existing spectral methods through coupling with importance sampling techniques. A notable insight is that one can simulate any vector random field whose direct and cross-covariance functions are continuous and absolutely integrable, provided that one knows the analytical expression of their spectral densities, without the need for these spectral densities to have a bounded support. The simulation algorithm is computationally faster than circulant-embedding techniques, lends itself to parallel computing and has a low memory storage requirement. Numerical examples with varied spatial correlation structures are presented to demonstrate the accuracy and versatility of the proposal.