P. A. Dowd

The Variogram and Kriging: Robust and Resistant Estimators

Previously published techniques for robust and resistant variogram estimation are reviewed and several alternative methods are introduced. The performance of various estimators is assessed on two typical practical examples.

A new computer code for discrete fracture network modelling

The authors describe a comprehensive software package for two- and three-dimensional stochastic rock fracture simulation using marked point processes. Fracture locations can be modelled by a Poisson, a non-homogeneous, a cluster or a Cox point process; fracture geometries and properties are modelled by their respective probability distributions. Virtual sampling tools such as plane, window and scanline sampling are included in the software together with a comprehensive set of statistical tools including histogram analysis, probability plots, rose diagrams and hemispherical projections. The paper describes in detail the theoretical basis of the implementation and provides a case study in rock fracture modelling to demonstrate the application of the software.

Lognormal kriging—the general case

A theoretical study of the general case of the estimation of regionalized variables with a lognormal distribution is presented. The results of this study are compared to those obtained assuming conservation of lognormality. The numerical significance of the different solutions is illustrated by several simple examples.

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.

CONNEC3D: a computer program for connectivity analysis of 3D random set models

Geostatistical simulation is used in risk analysis studies to incorporate the spatial uncertainty of experimental variables that are significantly under-sampled. For example, the values of hydraulic conductivity or porosity are critical in petroleum reservoir production modelling and prediction, in assessing underground sites as waste repositories, and in modelling the transport of contaminants in aquifers. In all these examples connectivity of the permeable phase or permeable lithofacies is a critical issue. Given an indicator map on a regular two- or three-dimensional grid, which can be obtained from continuous-valued or from categorical variables, CONNEC3D performs a connectivity analysis of the phase of interest (coded 0 or 1 by an indicator function). 3D maps of multiple indicators, categories or continuous variables can also be analysed for connectivity by suitable coding of the input map. Connectivity analysis involves the estimation of the connectivity function τ(h) for different spatial directions and a number of connectivity statistics. Included in the latter are the number of connected components (ncc), average size of a connected component (cc), mean length of a cc in the X, Y and Z directions, size of the largest cc, maximum length of a cc along X, Y and Z and the numbers of percolating components along X, Y and Z. In addition, the program provides as output a file in which each cc is identified by an integer number ranging from 1 to ncc. The implementation of the program is demonstrated on a random set model generated by the sequential indicator algorithm. This provides a means of estimating the computational time required for different grid sizes and is also used to demonstrate computationally that when the semi-variogram of the indicator function is anisotropic the connectivity function is also anisotropic. There are options within the program for 6-connectivity analysis, 18-connectivity analysis and 26-connectivity analysis. The software is provided in two formats, as a stand-alone program that can perform connectivity analysis of an indicator map and as a subroutine that can be repeatedly called in order to calculate averages of connectivity analyses of a large number of realizations of indicator maps, or to identify critical realizations generated by conditional simulations of continuous variables or categorical variables.

A simulated annealing approach to mine production scheduling

Increasing global competition, quality standards, environmental awareness and decreasing ore prices impose new challenges to mineral industries. Therefore, the extraction of mineral resources requires careful design and scheduling. In this research, simulated annealing (SA) is recommended to solve a mine production scheduling problem. First of all, in situ mineral characteristics of a deposit are simulated by sequential Gaussian simulation, and averaging the simulated characteristics within specified block volumes creates a three-dimensional block model. This model is used to determine optimal pit limits. A linear programming (LP) scheme is used to identify all blocks that can be included in the blend without violating the content requirements. The Lerchs–Grosmann algorithm using the blocks identified by the LP program determines optimal pit limits. All blocks that lie outside of the optimal pit limit are removed from the system and the blocks within the optimal pit are submitted to the production scheduling algorithm. Production scheduling optimization is carried out in two stages: Lagrangean parameterization, resulting in an initial sub-optimal solution, and multi-objective SA, improving the sub-optimal schedule further. The approach is demonstrated on a Western Australian iron ore body.

A review of recent developments in geostatistics

Abstract This paper reviews developments in geostatistics in the period 1987 to mid-1991. The developments which are regarded as significant by the author fall broadly under six headings: simulation, indicator kriging, interval estimation, applications to hydrocarbon reservoirs and hydrology, incorporation of prior information in spatial estimation, and fuzzy kriging. A summary of significant contributions under each of these headings is given together with an assessment of their importance and application.

Plurigau: a computer program for simulating spatial facies using the truncated Plurigaussian method

Truncated plurigaussian simulation is a useful method for simulating spatial categorical variables, such as facies, in a geological context. The method is an extension of the truncated Gaussian method that retains the main advantages of the latter (mainly that it produces permissible sets of indicator semi-variograms and cross-semi-variograms) but overcomes its limitations (the truncated Gaussian method only reproduces sequentially ranked categories). The method is based on the truncation of two Gaussian random functions that may, or may not, be correlated. PLURIGAU is an ANSI Fortran-77 computer program for performing conditional or unconditional truncated plurigaussian simulations of spatial categories. The number of facies, spatial relations between the facies, proportions of each facies, indicator semi-variograms and indicator cross-semi-variograms must be known or estimated from experimental data. The program calculates the four thresholds for each of the facies (two for each of the Gaussian random functions) and the covariance models for the two Gaussian random functions.The simulation of the Gaussian random functions may be done using any of the methods available. Conditioning has been implemented by a simple acceptance-rejection technique embedded within sequential Gaussian simulation algorithm. A case study is provided so that the implementation of the programs can be checked and the results are discussed.

The Second-Order Stationary Universal Kriging Model Revisited

Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of ‘good’ estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.

FACOTR2D: a computer program for factorial cokriging

Spatial variables in the geosciences often display different patterns of variability at different spatial scales. When these variables are cross-correlated, the magnitude of the cross-correlation may differ at the different scales of spatial variability. This scale-dependent spatial variability and cross-correlation can be investigated by geostatistical factor analysis, which consists of four steps: • Semi-variogram analysis. Semi-variograms and cross-semi-variograms are estimated from the experimental data and are used to identify the scales of variability. The different scales of variability can be interpreted and can often be linked to physical causes. • Fitting a model to the set of semi-variograms and cross-semi-variograms. A consistent model, such as the linear model of coregionalization, is fitted to the set of experimental semi-variograms and cross-semi-variograms. • Factorial cokriging is used to estimate an individual factor, an individual component or a combination of components of any of the coregionalized variables. • Interpretation of the estimated factor or component.The purpose of this paper is to describe programs for performing the second and third steps. Programs are also provided for automatic fitting of a linear coregionalization model and for estimating, by factorial cokriging, a single factor, a single component or a combination of components. Although programs for factorial kriging and programs for cokriging can be found elsewhere the software described here (FACTOR2D) provides a complete solution by combining factorial cokriging with a program for fitting a linear coregionalization model (LCMFIT2). The use and performance of the programs is demonstrated on simulated and real case studies.