Andre G. Journel

Nonparametric estimation of spatial distributions

The indicator approach, whereby the data are used through their rank order, allows a nonparametric approach to the data bivariate distribution. Such rich structural information allows a nonparametric risk-qualified, estimation of local and global spatial distributions.

Geostatistical Software Library and User's Guide

This book will be an important text to most of geostatisticians, including graduate students and experts in the field of practical geostatistics. The guts of this volume are the two highdensity IBM disks which come with it and contain 37 programs which can be run in both UNIX and DOS environments but are not machine specific. The programs are aimed at three major areas of geostatistics: quantifying spatial variability (variograms), generalized linear regression techniques (kriging), and stochastic simulation. In all there are some 80 source files included with the distribution diskettes. The programs are not execuable but require to be compiled before running them. A machine with a fortran compiler is required. The intent of the authors is to make this suite of programs accessible to anyone who wants to use them. The source code of these programs has been assembled, developed, tested, and tried at Stanford University over a period of some 12 years. Though this library of programs is not intended as a commercial product it represents a gold mine to those who need a jump start into the field of geostatistics.

Geostatistical Space–Time Models: A Review

Geostatistical space–time models are used increasingly for addressing environmental problems, such as monitoring acid deposition or global warming, and forecasting precipitation or stream flow. Each discipline approaches the problem of joint space–time modeling from its own perspective, a fact leading to a significant amount of overlapping models and, possibly, confusion. This paper attempts an annotated survey of models proposed in the literature, stating contributions and pinpointing shortcomings. Stochastic models that extend spatial statistics (geostatistics) to include the additional time dimension are presented with a common notation to facilitate comparison. Two conceptual viewpoints are distinguished: (1) approaches involving a single spatiotemporal random function model, and (2) approaches involving vectors of space random functions or vectors of time series. Links between these two viewpoints are then revealed; advantages and shortcomings are highlighted. Inference from space–time data is revisited, and assessment of joint space–time uncertainty via stochastic imaging is suggested.

Combining knowledge from diverse sources: An alternative to traditional data independence hypotheses

Consider the assessment of any unknown event A through its conditional probability P(A | B,C) given two data events B, C of different sources. Each event could involve many locations jointly, but the two data events are assumed such that the probabilities P(A | B) and P(A | C) can be evaluated. The challenge is to recombine these two partially conditioned probabilities into a model for P(A | B,C) without having to assume independence of the two data events B and C. The probability P(A | B,C) is then used for estimation or simulation of the event A. In presence of actual data dependence, the combination algorithm provided by the traditional conditional independence hypothesis is shown to be nonrobust leading to various inconsistencies. An alternative based on a permanence of updating ratios is proposed, which guarantees all limit conditions even in presence of complex data interdependence. The resulting recombination formula is extended to any number n of data events and a paradigm is offered to introduce formal data interdependence.

When do we need a trend model in kriging

Under usual estimation practice with local search windows for data and for interpolation situations, universal kriging and ordinary kriging yield the same estimates, using a data set with apparent trend, for both the unknown attribute and its trend component. Modeling the trend matters only in extrapolation situations. Because conditions of the case study presented arise most frequently in practice, the simpler ordinary kriging is the preferred option.

Joint Sequential Simulation of MultiGaussian Fields

The sequential simulation algorithm can be used for the generation of conditional realizations from either a multiGaussian random function or any non-Gaussian random function as long as its conditional distributions can be derived. The multivariate probability density function (pdf) that fully describes a random function can be written as the product of a set of univariate conditional pdfs. Drawing realizations from the multivariate pdf amounts to drawing sequentially from that series of univariate conditional pdfs. Similarly, the joint multivariate pdf of several random functions can be written as the product of a series of univariate conditional pdfs. The key step consists of the derivation of the conditional pdfs. In the case of a multiGaussian fields, these univariate conditional pdfs are known to be Gaussian with mean and variance given by the solution of a set of normal equations also known as simple cokriging equations. Sequential simulation is preferred to other techniques, such as turning bands, because of its ease of use and extreme flexibility.

Geostatistics: models and tools for the Earth Sciences

The probability construct underlying geostatistical methodology is recalled, stressing that stationary is a property of the model rather than of the phenomenon being represented. Geostatistics is more than interpolation and kriging(s) is more than linear interpolation through ordinary kriging. A few common misconceptions are addressed.

Integrating Seismic Data in Reservoir Modeling: The Collocated Cokriging Alternative

The two sources of information commonly available for modeling the top of a structure, depth data from wells and geophysical measurements from seismic surveys, are often Miicult to integrate. W bile, the well data provide t he most accurate measurements of depths there are rarely enough wells to permit an accurate appraisal from well data alone. On the other hand, the seismic data are generally less precise but more abundant. Two geoatatistical methods, “external drift~ and “collocated cokrigingfi, are proposed to integrate the two sources of information. A case study is used to document the strengths and weaknesses of both approaches for constructing contour maps cf the top structure and assessing the uncertainty on such maps through stochastic simulations.

Joint simulation of multiple variables with a Markov-type coregionalization model

Many applications are multivariate in character and call for stochastic images of the joint spatial variability of multiple variables conditioned by a prior model of covariances and cross- covariances. This paper presents an algorithm to perform cosimulation of such spatially intercorrelated variables. This new algorithm builds on a Markov-type hypothesis whereby collocated information screens further away data of the same type, allowing cosimulation without the burden of a full cokriging. The proposed algorithm is checked against a synthetic multi-Gaussian reference dataset, then against a multi-Gaussian cosimulation approach using full cokriging. The results indicate that the proposed algorithm perform as well as the full cokriging approach in reproducing the univariate and bivariate statistics of the reference set, yet at less cpu cost.

New Method for Reservoir Mapping

The sequential indicator simulation (SIS) algorithm allows building alternative, equiprobable, numerical models of reservoir heterogeneities that reflect spatial-connectivity patterns of extreme values (e.g., permeability) and honor data values at their locations. This paper presents a case study of a sampled slab of Berea sandstone.