Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions

This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions.     

Uncertainties in Facies Proportion Estimation I. Theoretical Framework: The Dirichlet Distribution

To simulate geological models comprising several litho-types—or facies—we need first to estimate their proportions, which are often poorly known. The corresponding uncertainties can be modelled using a Bayesian approach for inverting the multinomial distribution. The result obtained is known as the Dirichlet distribution. It can be simulated by decomposition into independent conditional distributions. Application of the model is extended to the case of nonstationary proportions and, with some approximation, to the case of correlated spatial data. The mathematical developments presented in the appendices provide a more precise and general definition of the distribution, several decomposition formulae into independent variables, the determination of remarkable stability properties, and the resulting consequences for the conditional and marginal distributions.     

Using non-Gaussian distributions in geostatistical simulations

Parametric geostatistical simulations such as LU decomposition and sequential algorithms do not need Gaussian distributions. It is shown that variogram model reproduction is obtained when Uniform or Dipole distributions are used instead of Gaussian distributions for drawing i. i.d. random values in LU simulation, or for modeling the local conditional probability distributions in sequential simulation. Both algorithms yield simulated values with a marginal normal distribution no matter if Gaussian, Uniform, or Dipole distributions are used. The range of simulated values decreases as the entropy of the probability distribution decreases. Using Gaussian distributions provides a larger range of simulated normal score values than using Uniform or Dipole distributions. This feature has a negligible effect for reproduction of the normal scores variogram model but have a larger impact on the reproduction of the original values variogram. The Uniform or Dipole distributions also produce lesser fluctuations among the variograms of the simulated realizations.     

Successive Nonparametric Estimation of Conditional Distributions

Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities.     

Approximate local confidence intervals under change of support

Approximate local confidence intervals are constructed from uncertainty models in the form of the conditional distribution of the random variable Z given values of variables [Z_i, i=1,...,n]. When the support of the variable Z is any support other than that of the data, the conditional distributions require a change of support correction. This paper investigates the effect of change of support on the approximate local confidence intervals constructed by cumulative indicator kriging, class indicator kriging, and probability kriging under a variety of conditions. The conditions are generated by three simulated deposits with grade distributions of successively higher degree of skewness; a point support and two different block supports are considered. The paper also compares the confidence intervals obtained from these methods using the most used measures of confidence interval effectiveness.     

Conditioning Fractal (fBm/fGn) Porosity and Permeability Fields to Multiwell Pressure Data

Stochastic fractal (fGn and fBm) porosity and permeability fields are conditioned to given variogram, static (or hard), and multiwell pressure data within a Bayesian estimation framework. Because fGn distributions are normal/second-order stationary, it is shown that the Bayesian estimation methods based on the assumption of normal/second-order stationary distributions can be directly used to generate fGn porosity/permeability fields conditional to pressure data. However, because fBm is not second-order stationary, it is shown that such Bayesian estimation methods can be used with implementation of a pseudocovariance approach to generate fBm porosity/permeability fields conditional to multiwell pressure data. In addition, we provide methods to generate unconditional realizations of fBm/fGn fields honoring all variogram parameters. These unconditional realizations can then be conditioned to hard and pressure data observed at wells by using the randomized maximum likelihood method. Synthetic examples generated from one-, two-, and three-dimensional single-phase flow simulators are used to show the applicability of our methodology for generating realizations of fBm/fGn porosity and permeability fields conditioned to well-test pressure data and evaluating the uncertainty in reservoir performance predictions appropriately using these history-matched realizations.     

Experimental Assessment of Gradual Deformation Method

Uncertainty in future reservoir performance is usually evaluated from the simulated performance of a small number of reservoir realizations. Unfortunately, most of the practical methods for generating realizations conditional to production data are only approximately correct. It is not known whether or not the recently developed method of Gradual Deformation is an approximate method or if it actually generates realizations that are distributed correctly. In this paper, we evaluate the ability of the Gradual Deformation method to correctly assess the uncertainty in reservoir predictions by comparing the distribution of conditional realizations for a small test problem with the standard distribution from a Markov Chain Monte Carlo (MCMC) method, which is known to be correct, and with distributions from several approximate methods. Although the Gradual Deformation algorithm samples inefficiently for this test problem and is clearly not an exact method, it gives similar uncertainty estimates to those obtained by MCMC method based on a relatively small number of realizations.     

Significance of conditioning to piezometric head data for predictions of mass transport in groundwater modeling

Transmissivity and head data are sampled from an exhaustive synthetic reference field and used to predict the arrival positions and arrival times of a number of particles transported across the field, together with an uncertainty estimate. Different combinations of number of transmissivity data and number of head data used are considered in each one of a series of 64 Monte-Carlo analyses. In each analysis, 250 realizations of transmissivity fields conditioned to both transmissivity and head data are generated using a novel geostatistically based inverse method. Pooling the solutions of the flow and transport equations in all 250 realizations allows building conditional frequency distributions for particle arrival positions and arrival times. By comparing these fresquency distributions, we can assess the incremental gain that additional head data provide. The main conclusion is that the first few head data dramatically improve the quality of transport predictions.     

Assessing the accuracy of sequential Gaussian simulation and cosimulation

Sequential Gaussian simulation is widespread in Earth Science applications to quantify the uncertainty about regionalized properties. Its practical implementation relies on the screen effect approximation in order to determine the successive conditional distributions by considering only the information available in the neighborhood of the target location. A methodology is presented to assess the accuracy of sequential Gaussian simulation, by calculating the theoretical moments (expectation and variance–covariance matrix) of the simulated random vector and comparing them with the moments of the underlying model. The methodology can be applied in both the conditional and non-conditional contexts, as well as for univariate or multivariate simulation. It is helpful to determine appropriate implementation parameters, in particular about the visiting sequence and the design of the moving neighborhood for selecting relevant conditioning information, prior to performing simulation.     

Quantifying Uncertainty in Mineral Resources by Use of Classification Schemes and Conditional Simulations

Mineral inventory determination consists of estimating the amount of mineral resources on a block-by-block basis and classifying individual blocks into categories with increasing level of geologic confidence. Such classification is a crucial issue for mining companies, investors, financial institutions, and authorities, but it remains subject to some confusion because of the wide variations in methodologies and the lack of standardized procedures. The first part of this paper considers some of the criteria used to classify resources in practice and their impact through a sensitivity study using data from a Chilean porphyry copper deposit. Five classification criteria are compared and evaluated, namely: Search neighborhoods, absolute and relative kriging variances, absolute and relative conditional simulation variances. It is shown that some classification criteria either favor or penalize the high-grade areas if the grade distribution presents a proportional effect. In the second part of the paper, conditional simulations are used to quantify the uncertainty on the overall mineral resources. This approach is promising for risk analysis and decision-making. Unlike linear kriging, simulations allow inclusion of a cutoff grade in the calculation of the resources and also provide measures of their joint uncertainty over production volumes.     

Generation of multivariate cross-correlated geotechnical random fields

A geotechnical problem that involves several spatially correlated parameters can be best described using multivariate cross-correlated random fields. The joint distribution of these random variables cannot be uniquely defined using their marginal distributions and correlation coefficients alone. This paper presents a generic methodology for generating multivariate cross-correlated random fields. The joint distribution is rigorously established using a copula function that describes the dependence structure among the individual variables. The cross-correlated random fields are generated through Cholesky decomposition and conditional sampling based on the joint distribution. The random fields are verified regarding the anisotropic scales of fluctuation and copula parameters.     

Dependence of earthquake recurrence times and independence of magnitudes on seismicity history

The fulfillment of a scaling law for earthquake recurrence–time distributions is a clear indication of the importance of correlations in the structure of seismicity. In order to characterize these correlations we measure conditional recurrence–time and magnitude distributions for worldwide seismicity as well as for Southern California during stationary periods. Disregarding the spatial structure, we conclude that the relevant correlations in seismicity are those of the recurrence time with previous recurrence times and magnitudes; in the latter case, the conditional distribution verifies a scaling relation depending on the difference between the magnitudes of the two events defining the recurrence time. In contrast, with our present resolution, magnitude seems to be independent on the history contained in the seismic catalogs (except perhaps for Southern California for very short time scales, less than about 30 min for the magnitude ranges analyzed).     

Revisiting Multi-Gaussian Kriging with the Nataf Transformation or the Bayes’ Rule for the Estimation of Spatial Distributions

A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space.     

An Empirical Comparison of Kriging Methods for Nonlinear Spatial Point Prediction

Spatial prediction is a problem common to many disciplines. A simple application is the mapping of an attribute recorded at a set of points. Frequently a nonlinear functional of the observed variable is of interest, and this calls for nonlinear approaches to prediction. Nonlinear kriging methods, developed in recent years, endeavour to do so and additionally provide estimates of the distribution of the target quantity conditional on the observations. There are few empirical studies that validate the various forms of nonlinear kriging. This study compares linear and nonlinear kriging methods with respect to precision and their success in modelling prediction uncertainty. The methods were applied to a data set giving measurements of the topsoil concentrations of cobalt and copper at more than 3000 locations in the Border Region of Scotland. The data stem from a survey undertaken to identify places where these trace elements are deficient for livestock. The comparison was carried out by dividing the data set into calibration and validation sets. No clear differences between the precision of ordinary, lognormal, disjunctive, indicator, and model-based kriging were found, neither for linear nor for nonlinear target quantities. Linear kriging, supplemented with the assumption of normally distributed prediction errors, failed to model the conditional distribution of the marginally skewed data, whereas the nonlinear methods modelled the conditional distributions almost equally well. In our study the plug-in methods did not fare any worse than model-based kriging, which takes parameter uncertainty into account.     

Nonparametric estimation of multivariate joint and conditional spatial distributions

To estimate the recoverable reserves of deposits containing more than one metal, linear combinations of sample grades based on the economic importance of each metal have been employed. These linear combinations, called equivalent grades, have inherent problems at least one of which is that the contribution of an individual metal is confounded with all others. As an alternative to equivalent grades, multivariable joint and conditional spatial distributions may be estimated using indicator variable methods. These spatial distributions may then be used to determine the joint or conditional amounts of the metals in the estimated recoverable reserve.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

航空电磁法在水文地质调查中的应用

为了对水文地质调查提供多方法、多深度和多参数解释,提高航空电磁法在水文地质调查中的应用效果,以河北曹碑店地区、定陵—北京十三陵水库地区、黑龙江宝清地区、澳大利亚桥维拉地区、山东黄河口地区的航空电磁资料为基础,建立了利用航空电磁法研究海侵程度、水质填图、寻找浅层淡水的定性和定量解释方法。结果表明: 航空电磁法可以快速高效地研究不同深度的地下水及海侵程度空间分布特征,快速计算地下水矿化度,尤其可为大范围水文地质调查提供多层次的海侵底界面和水平界线、含水层和隔水层分布及水质分布信息,在多个研究区取得了较好的水文地质勘查效果。     

Geochemical-focusing of manganese in lake sediments — An indicator of deep-water oxygen conditions

The lateral distributions of Mn concentrations in the sediments of two Swiss lakes under varying oxygen conditions have been determined. The comparison of Mn distribution patterns with oxygen in the deep-water provides strong evidence for a geochemical-focusing effect, which is driven by the redox cycle of manganese. Conditions essential for this process to occur are anoxic sediments in contact with oxic deep-water. Average sedimentary manganese concentrations determined for different water-depth ranges are directly proportional to the area of shallower sediments. This result indicates that geochemical-focusing of manganese in lake sediments is a promising proxy indicator for the reconstruction of oxygen conditions during deposition.     

Geostatistics for Power Models of Gaussian Fields

This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions.     

Probability field for the post-processing of stochastic simulations

A combination of factorial kriging and probability field simulation is proposed to correct realizations resulting from any simulation algorithm for either too high nugget effect (noise) or poor histogram reproduction. First, a factorial kriging is done to filter out the noise from the noisy realization. Second, the uniform scores of the filtered realization are used as probability field to sample the local probability distributions conditional to the same dataset used to generate the original realization. This second step allows to restore the data variance. The result is a corrected realization which reproduces better target variogram and histogram models, yet honoring the conditioning data.     

Two Ordinary Kriging Approaches to Predicting Block Grade Distributions

Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables (transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates of the recoverable reserves that do not utilize the mean value of the normal score data. The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary indicator kriging, which show strong deviations from conditional expectation.