The U-WEDGE Transformation Method for Multivariate Geostatistical Simulation

To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. Spatial decorrelation methods are usually based on the diagonalisation of the variance/covariance and semivariogram matrices of the set of attributes for a chosen family of lag spacings. These matrices are symmetric and there are several efficient methods for the approximate joint diagonalisation of a family of symmetric matrices. One of these is the uniformly weighted exhaustive diagonalisation with Gauss iterations (U-WEDGE) method. In contrast to the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed, U-WEDGE can be applied directly to the set of experimental semivariogram matrices without having to place restrictions on the number of structures in the linear model of coregionalisation, thus removing one of the restrictions placed on the subsequent modelling of the spatial structure of the factors. We use an iron-ore data set to illustrate the method and present a comparison between the simulated attributes obtained from U-WEDGE and MAF with the full co-simulation of the attributes.     

Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions

Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since, in general, convolution does not provide closed-form integration, we discuss efficient computation. We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze a trivariate pollution dataset from California.     

Direct Pattern-Based Simulation of Non-stationary Geostatistical Models

Non-stationary models often capture better spatial variation of real world spatial phenomena than stationary ones. However, the construction of such models can be tedious as it requires modeling both statistical trend and stationary stochastic component. Non-stationary models are an important issue in the recent development of multiple-point geostatistical models. This new modeling paradigm, with its reliance on the training image as the source for spatial statistics or patterns, has had considerable practical appeal. However, the role and construction of the training image in the non-stationary case remains a problematic issue from both a modeling and practical point of view. In this paper, we provide an easy to use, computationally efficient methodology for creating non-stationary multiple-point geostatistical models, for both discrete and continuous variables, based on a distance-based modeling and simulation of patterns. In that regard, the paper builds on pattern-based modeling previously published by the authors, whereby a geostatistical realization is created by laying down patterns as puzzle pieces on the simulation grid, such that the simulated patterns are consistent (in terms of a similarity definition) with any previously simulated ones. In this paper we add the spatial coordinate to the pattern similarity calculation, thereby only borrowing patterns locally from the training image instead of globally. The latter would entail a stationary assumption. Two ways of adding the geographical coordinate are presented, (1) based on a functional that decreases gradually away from the location where the pattern is simulated and (2) based on an automatic segmentation of the training image into stationary regions. Using ample two-dimensional and three-dimensional case studies we study the behavior in terms of spatial and ensemble uncertainty of the generated realizations.     

A Geostatistical Simulation Approach to a Pollution Case in Northeastern Italy

The study zone is located in the city of Padova (the Veneto region, NE Italy). The industrial activities present in this area since 1950 have produced very high concentrations of Pb, Zn, Cu, Pcb and oils until a depth of 7 m. The unsaturated and the saturated zones are both polluted. Because of the abundance of Pb values it was decided to analyze the lead distribution in the studied zone. In many studies of the polluted sites, the geometry of the investigated volume is highly anisotropic. Generally we have an extension of some hectares in the horizontal plane and of a few meters in depth. It is likely that different horizontal spatial continuity structures in pollution distribution are found at different depths both for the layered nature of the medium and for the transition between unsaturated and saturated conditions. In such conditions the decision to divide a 3D problem into 1D and 2D problems can be useful. Initially the studied volume was divided into seven layers up to 5 m in depth; the study was then approached in two phases. First, the Pb values in the vertical direction were analyzed, considering a derive along z, and estimating the values using the Kriging with Trend (KT) method. Thus it was possible to increase the data in the z direction, especially in the deeper layers. Second, 500 realizations of the Pb distribution for each of the seven layers were simulated using the simulated annealing procedure. Finally, results were presented and discussed for each layer in terms of median and probability maps.     

Geostatistical Simulation of Regionalized Pore-Size Distributions Using Min/Max Autocorrelation Factors

In many fields of the Earth Sciences, one is interested in the distribution of particle or void sizes within samples. Like many other geological attributes, size distributions exhibit spatial variability, and it is convenient to view them within a geostatistical framework, as regionalized functions or curves. Since they rarely conform to simple parametric models, size distributions are best characterized using their raw spectrum as determined experimentally in the form of a series of abundance measures corresponding to a series of discrete size classes. However, the number of classes may be large and the class abundances may be highly cross-correlated. In order to model the spatial variations of discretized size distributions using current geostatistical simulation methods, it is necessary to reduce the number of variables considered and to render them uncorrelated among one another. This is achieved using a principal components-based approach known as Min/Max Autocorrelation Factors (MAF). For a two-structure linear model of coregionalization, the approach has the attractive feature of producing orthogonal factors ranked in order of increasing spatial correlation. Factors consisting largely of noise and exhibiting pure nugget–effect correlation structures are isolated in the lower rankings, and these need not be simulated. The factors to be simulated are those capturing most of the spatial correlation in the data, and they are isolated in the highest rankings. Following a review of MAF theory, the approach is applied to the modeling of pore-size distributions in partially welded tuff. Results of the case study confirm the usefulness of the MAF approach for the simulation of large numbers of coregionalized variables.     

Direct Multiple-Point Geostatistical Simulation of Edge Properties for Modeling Thin Irregularly Shaped Surfaces

Thin, irregularly shaped surfaces such as clay drapes often have a major control on flow and transport in heterogeneous porous media. Clay drapes are often complex, curvilinear three-dimensional surfaces and display a very complex spatial distribution. Variogram-based stochastic approaches are also often not able to describe the spatial distribution of clay drapes since complex, curvilinear, continuous, and interconnected structures cannot be characterized using only two-point statistics. Multiple-point geostatistics aims to overcome the limitations of the variogram. The premise of multiple-point geostatistics is to move beyond two-point correlations between variables and to obtain (cross) correlation moments at three or more locations at a time using training images to characterize the patterns of geological heterogeneity. Multiple-point geostatistics can reproduce thin irregularly shaped surfaces such as clay drapes, but this is often computationally very intensive. This paper describes and applies a methodology to simulate thin, irregularly shaped surfaces with a smaller CPU and RAM demand than the conventional multiple-point statistical methods. The proposed method uses edge properties for indicating the presence of thin irregularly shaped surfaces. Instead of pixel values, edge properties indicating the presence of irregularly shaped surfaces are simulated using snesim. This method allows direct simulation of edge properties instead of pixel properties to make it possible to perform multiple-point geostatistical simulations with a larger cell size and thus a smaller computation time and memory demand. This method is particularly valuable for three-dimensional applications of multiple-point geostatistics.     

Multivariate Categorical Modeling with Hierarchical Truncated Pluri-Gaussian Simulation

Multiple categorical variables such as mineralization zones, alteration zones, and lithology are often available for geostatistical modeling. Each categorical variable has a number of possible categorical outcomes. The current approach for numerical modeling of categorical variables is to either combine the categorical variables or to model them independently. The collapse of multiple categorical variables into a single variable with all combinations is impractical due to the large number of combinations. In some cases, lumping categorical variables is justified in terms of stationary domains; however, this decision is often due to the limitations of existing techniques. The independent modeling of each categorical variable will fail to reproduce the collocated joint categorical relationships. A methodology for the multivariate modeling of categorical variables utilizing the hierarchical truncated pluri-Gaussian approach is developed and illustrated with the Swiss Jura data set. The multivariate approach allows for improved reproduction of multivariate relationships between categorical variables.

     

Measures of Parameter Uncertainty in Geostatistical Estimation and Geostatistical Optimal Design

Studies of site exploration, data assimilation, or geostatistical inversion measure parameter uncertainty in order to assess the optimality of a suggested scheme. This study reviews and discusses measures for parameter uncertainty in spatial estimation. Most measures originate from alphabetic criteria in optimal design and were transferred to geostatistical estimation. Further rather intuitive measures can be found in the geostatistical literature, and some new measures will be suggested in this study. It is shown how these measures relate to the optimality alphabet and to relative entropy. Issues of physical and statistical significance are addressed whenever they arise. Computational feasibility and efficient ways to evaluate the above measures are discussed in this paper, and an illustrative synthetic case study is provided. A major conclusion is that the mean estimation variance and the averaged conditional integral scale are a powerful duo for characterizing conditional parameter uncertainty, with direct correspondence to the well-understood optimality alphabet. This study is based on cokriging generalized to uncertain mean and trends because it is the most general representative of linear spatial estimation within the Bayesian framework. Generalization to kriging and quasi-linear schemes is straightforward. Options for application to non-Gaussian and non-linear problems are discussed.     

Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions

Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation.     

An Application of Multivariate Simulation in the Cement Industry

The profitability of a cement plant depends largely on its capacity to produce homogeneous cement with chemical composition close to specified targets for the cement type produced. One crucial step is the mixing of limestone with other raw materials in proportions calculated to meet these targets. Major design and operation decisions depend on the efficiency of this homogenizing step. The adequate modeling of the mixing process requires simulation of representative cross-correlated time series of chemical compositions of the raw materials involved. The chemical composition signals are obtained by multivariate geostatistical simulation using an LU (Cholesky) decomposition of the covariance matrix. Modifications to the usual LU method are presented. First, the effect on the raw covariance matrix of the closure property of chemical analysis is imposed. Second, the problem of memory space limitations in the LU method is tackled by using overlapping sliding neighbourhoods. The simulation algorithm is applied to the Joppa cement plant owned by Lafarge North America. The simulated raw material input streams are fed into the quality mix control (QMC), a proprietary software that models and controls the mixing operation to produce an output stream with cement characteristics as close as possible to desired targets. Two signal series are studied, one autocorrelated with a moderate temporal range and one with no autocorrelation. The QMC produces C₃S output signals having comparable short scale periodic variograms except that the variance of the uncorrelated signal is four times greater than those of the autocorrelated signal and the real Joppa data. The raw material feeder variograms have the same sill for both the white noise and the autocorrelated signals. However, the autocorrelated signal feeder variogram presents lower short term dispersion variance, a characteristic feature of Joppa operations. Our results show the importance of simulating the right temporal structure of the raw materials to realistically forecast the behavior of the output signals. We also discuss some practical implications of these findings for the design and operation of a cement plant.     

Geostatistical estimation of multi-phase structures

A method is proposed for the characterization of the disjoint shapes of a multi-phase set. The method uses a global structural function and provides estimates of the complete mosaic of phases, honoring the individual volume proportions inferred from the experimental samples. The estimates of shapes can be improved by local conditioning to the covariance of each phase and to geometrical characteristics such as spatial orientation of the different strata. The mapping of uncertainty zones for individual phases is one advantage of using a geostatistical approach to characterize the morphology of qualitative (non-numerical) variables.     

Using Multivariate Statistical and Geostatistical Methods to Identify Spatial Variability of Trace Elements in Agricultural Soils in Dongguan City,Guangdong,China

Dongguan (东莞) City, located in the Pearl River Delta, South China, is famous for its rapid industrialization in the past 30 years. A total of 90 topsoil samples have been collected from agricultural fields, including vegetable and orchard soils in the city, and eight heavy metals (As, Cu, Cd,Cr, Hg, Ni, Pb, and Zn) and other items (pH values and organic matter) have been analyzed, to evaluate the influence of anthropie activities on the environmental quality of agricultural soils and to identify the spatial distribution of trace elements and possible sources of trace elements. The elements Hg, Pb, and Cd have accumulated remarkably here, incomparison with the soil background content of elements in Guangdong (广东) Province. Pollution is more serious in the western plain and the central region, which are heavily distributed with industries and rivers. Multivariate and geostatistical methods have been applied to differentiate the influences of natural processes and human activities on the pollution of heavy metals in topsoils in the study area. The results of cluster analysis (CA) and factor analysis (FA) show that Ni, Cr, Cu, Zn, and As are grouped in factor F1,Pb in F2, and Cd and Hg in F3, respectively. The spatial pattern of the three factors may be well demonstrated by geostatistical analysis. It is shown that the first factor could be considered as a natural source controlled by parent rocks. The second factor could be referred to as "industrial and traffic pollution sources". The source of the third factor is mainly controlled by long-term anthropic activities ,ad a consequence of agricultural fossil fuel consumption and atmospheric deposition.     

Uncertainties in Facies Proportion Estimation II: Application to Geostatistical Simulation of Facies and Assessment of Volumetric Uncertainties

Geostatistical simulations of lithotypes or facies are commonly used to create a geological model and to describe the heterogeneities of petroleum reservoirs. However, it is difficult to handle such models in the framework of multiple realizations to assess the uncertainty of hydrocarbon in place. Indeed, the hydrocarbon in place is correlated with the facies proportions, which are themselves uncertain. The uncertainty model of facies proportions is not easy to describe because of closure relationships. A previous attempt was made with a nonparametric approach using the resampling technique. It has been successful in a stationary case but it is difficult to extend it to nonstationary cases. In this paper, we have applied the vectorial beta parametric model or Dirichlet model. It has provided much more realistic uncertainties on volumetrics in very different geological and geostatistical contexts.     

Time-Lapse Analysis of Methane Quantity in the Mary Lee Group of Coal Seams Using Filter-Based Multiple-Point Geostatistical Simulation

Coal seam degasification and its success are important for controlling methane, and thus for the health and safety of coal miners. During the course of degasification, properties of coal seams change. Thus, the changes in coal reservoir conditions and in-place gas content as well as methane emission potential into mines should be evaluated by examining time-dependent changes and the presence of major heterogeneities and geological discontinuities in the field. In this work, time-lapsed reservoir and fluid storage properties of the New Castle coal seam, Mary Lee/Blue Creek seam, and Jagger seam of Black Warrior Basin, Alabama, were determined from gas and water production history matching and production forecasting of vertical degasification wellbores. These properties were combined with isotherm and other important data to compute gas-in-place (GIP) and its change with time at borehole locations. Time-lapsed training images (TIs) of GIP and GIP difference corresponding to each coal and date were generated by using these point-wise data and Voronoi decomposition on the TI grid, which included faults as discontinuities for expansion of Voronoi regions. Filter-based multiple-point geostatistical simulations, which were preferred in this study due to anisotropies and discontinuities in the area, were used to predict time-lapsed GIP distributions within the study area. Performed simulations were used for mapping spatial time-lapsed methane quantities as well as their uncertainties within the study area. The systematic approach presented in this paper is the first time in literature that history matching, TIs of GIPs and filter simulations are used for degasification performance evaluation and for assessing GIP for mining safety. Results from this study showed that using production history matching of coalbed methane wells to determine time-lapsed reservoir data could be used to compute spatial GIP and representative GIP TIs generated through Voronoi decomposition. Furthermore, performing filter simulations using point-wise data and TIs could be used to predict methane quantity in coal seams subjected to degasification. During the course of the study, it was shown that the material balance of gas produced by wellbores and the GIP reductions in coal seams predicted using filter simulations compared very well, showing the success of filter simulations for continuous variables in this case study. Quantitative results from filter simulations of GIP within the studied area briefly showed that GIP was reduced from an initial ～73 Bcf (median) to ～46 Bcf (2011), representing a 37 % decrease and varying spatially through degasification. It is forecasted that there will be an additional ～2 Bcf reduction in methane quantity between 2011 and 2015. This study and presented results showed that the applied methodology and utilized techniques can be used to map GIP and its change within coal seams after degasification, which can further be used for ventilation design for methane control in coal mines.     

Multivariate Block-Support Simulation of the Yandi Iron Ore Deposit,Western Australia

Mineral deposits frequently contain several elements of interest that are spatially correlated and require the use of joint geostatistical simulation techniques in order to generate models preserving their spatial relationships. Although joint-simulation methods have long been available, they are impractical when it comes to more than three variables and mid to large size deposits. This paper presents the application of block-support simulation of a multi-element mineral deposit using minimum/maximum autocorrelation factors to facilitate the computationally efficient joint simulation of large, multivariable deposits. The algorithm utilized, termed dbmafsim, transforms point-scale spatial attributes of a mineral deposit into uncorrelated service variables leading to the generation of simulated realizations of block-scale models of the attributes of interest of a deposit. The dbmafsim algorithm is utilized at the Yandi iron ore deposit in Western Australia to simulate five cross-correlated elements, namely Fe, SiO₂, Al₂O₃, P and LOI, that are all critical in defining the quality of iron ore being produced. The block-scale simulations reproduce the direct- and cross-variograms of the elements even though only the direct variograms of the service variables have to be modeled. The application shows the efficiency, excellent performance and practical contribution of the dbmafsim algorithm in simulating large multi-element deposits.     

利用地质统计学方法模拟岩石裂隙网络的三维空间分布——以云南个旧高松矿田为例

裂隙在地学的诸多领域中均具有重要意义,其空间分布可以使用地质统计学方法进行模拟,同时考虑裂隙的方向(走向和倾角)。利用序贯高斯模拟方法可以估计裂隙密度的空间分布,并根据裂隙密度数值随机产生裂隙位置的空间分布。裂隙方向被划分成若干(非)均等的方向组,将裂隙方向归属到其所属方向组,表示为由若干二值变量组成的指示形式,0和1分别代表该裂隙方向不属于和属于该组。为了便于计算,减少方向指示变量的成分数目,使用主成分分析法求出方向指示变量的主成分,用普通克里格法估计各主成分的空间分布。把估计结果反演为原始的指示形式,并找出其中数值最大的方向组且将其赋值为1。按照对应方向组内裂隙方向的累积密度函数,随机产生裂隙的方向。根据估计结果,将符合一定距离和角度标准的裂隙元连接为一个裂隙面,从而形成裂隙网络。根据在云南个旧锡矿高松矿田白云岩中进行裂隙网络模拟的应用,可见该方法由于组合了序贯高斯模拟法、普通克里格法和主成分分析法,可以较好地对岩石裂隙位置和方向进行合理的模拟。     

地质统计学方法在地下水水位估值中应用

对于许多区域水资源或水环境问题,地下水水流模拟往往要采用数值方法,需给出每个节点上初始水位值,以反映流场的初始状态。另外,地下水水位动态长期监测分析,需由观测点水位估计任一点的水位。文中阐述了地下水水位估值的地质统计学方法－泛克立格法原理,以河南省焦作市修武段地下水数值模拟分析区为例,分析了用一次、二次漂移的泛克立格方法模拟地下水初始流场的估值情况和对真实流场特征的反映情况。指出在进行区域地下水位