Modelling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency,Walther’s Law and Entropy

This paper is an extension of the two-dimensional coupled Markov chain model developed by Elfeki and Dekking (2001) supplemented with extensive simulations. We focus on the development of various coupled Markov chains models: the so-called fully forward Markov chain, fully backward Markov chain and forward–backward Markov chain models. We addressed many issues such as: sensitivity analysis of optimal sampling intervals in horizontal and lateral directions, directional dependency, use of Walther’s law to describe lateral variability, effect of conditioning on number of boreholes on the model performance, stability of the Monte Carlo realizations, various implementation strategies, use of cross validation techniques to evaluate model performance and image division for statistically non-homogeneous deposits are addressed. The applications are made on three sites; two sites are located in the Netherlands, and the third is in the USA. The purpose of these applications is to show under which conditions the Markov models can be used, and to provide some guidelines for the practice. Entropy maps are good tools to indicate places where high uncertainty is present, so can be used for designing sampling networks to reduce uncertainty at these locations. Symmetric and diagonally dominant horizontal transition probabilities with proper sampling interval show plausible results (fits with geologists prediction) in terms of delineation of subsurface heterogeneous structures. Walther’s law can be utilised with a proper sampling interval to account for the lateral variability.     

Markov Chain Random Fields for Estimation of?Categorical Variables

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.     

Markov Chain Random Fields for Estimation of Categorical Variables

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.     

A Fixed-Path Markov Chain Algorithm for Conditional Simulation of Discrete Spatial Variables

The Markov chain random field (MCRF) theory provided the theoretical foundation for a nonlinear Markov chain geostatistics. In a MCRF, the single Markov chain is also called a “spatial Markov chain” (SMC). This paper introduces an efficient fixed-path SMC algorithm for conditional simulation of discrete spatial variables (i.e., multinomial classes) on point samples with incorporation of interclass dependencies. The algorithm considers four nearest known neighbors in orthogonal directions. Transiograms are estimated from samples and are model-fitted to provide parameter input to the simulation algorithm. Results from a simulation example show that this efficient method can effectively capture the spatial patterns of the target variable and fairly generate all classes. Because of the incorporation of interclass dependencies in the simulation algorithm, simulated realizations are relatively imitative of each other in patterns. Large-scale patterns are well produced in realizations. Spatial uncertainty is visualized as occurrence probability maps, and transition zones between classes are demonstrated by maximum occurrence probability maps. Transiogram analysis shows that the algorithm can reproduce the spatial structure of multinomial classes described by transiograms with some ergodic fluctuations. A special characteristic of the method is that when simulation is conditioned on a number of sample points, simulated transiograms have the tendency to follow the experimental ones, which implies that conditioning sample data play a crucial role in determining spatial patterns of multinomial classes. The efficient algorithm may provide a powerful tool for large-scale structure simulation and spatial uncertainty analysis of discrete spatial variables.     

基于不同邻域系统的马尔可夫链模型的储层岩相随机模拟

针对在油气储层随机模拟中马尔可夫链模型的不同方向的转移概率矩阵求取困难的问题,提出一种二维剖面中不同方向的转移概率矩阵求取方法,这种方法的提出使得不同阶次的各向同性和各向异性的邻域系统的转移概率矩阵的求取变得容易可行。随后,对不同邻域系统的马尔可夫链模型采用蒙特卡罗抽样方法进行了储层岩相随机模拟试验。最后比较了不同邻域系统岩相模拟的结果并探讨了在储层研究中的适用性。     

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。     

Transition probability-based indicator geostatistics

Traditionally, spatial continuity models for indicator variables are developed by empirical curvefitting to the sample indicator (cross-) variogram. However, geologic data may be too sparse to permit a purely empirical approach, particularly in application to the subsurface. Techniques for model synthesis that integrate hard data and conceptual models therefore are needed. Interpretability is crucial. Compared with the indicator (cross-) variogram or indicator (cross-) covariance, the transition probability is more interpretable. Information on proportion, mean length, and juxtapositioning directly relates to the transition probability: asymmetry can be considered. Furthermore, the transition probability elucidates order relation conditions and readily formulates the indicator (co)kriging equations.     

Reducing geological uncertainty by conditioning on boreholes: the coupled Markov chain approach

The CMC (coupled Markov chain) model, which is based on the extension of Markov chains in two-dimensions, is used in the reduction of uncertainty in geological structures when conditioned (i.e., honours the data and their location) on a number of boreholes. The model has been applied to an unconsolidated aquifer deposit located in the central Rhine-Meuse delta (the Gorkum study area) in the Netherlands. A comparison is also made between the CMC and the SIS (sequential indicator simulation) model, which is based on Kriging and co-Kriging theories on the same deposit. The results show the potential applicability of the CMC model in reducing the uncertainty in geological configurations when a sufficient number of boreholes is available. Reproduction of the global geological features requires relatively few boreholes (in this case study, nine boreholes with 30-m spacing over a distance of 240 m). However, reproduction of the proportion of each state requires a relatively large number of boreholes (in this case study 31 boreholes with 8-m spacing over a distance of 240 m). It has been shown that variograms can be deceptive in modeling the spatial pattern and that they reflect only part of the complete spatial structure in the field. The use of transition probabilities via the CMC model provides a better alternative approach, because it uses multiple point information. Amro M. M. Elfeki on leave from Department of Irrigation and Hydraulics, Faculty of Engineering, Mansoura University, Mansoura, Egypt     

Three-dimensional slope reliability and risk assessment using auxiliary random finite element method

This paper aims to propose an auxiliary random finite element method (ARFEM) for efficient three-dimensional (3-D) slope reliability analysis and risk assessment considering spatial variability of soil properties. The ARFEM mainly consists of two steps: (1) preliminary analysis using a relatively coarse finite-element model and Subset Simulation, and (2) target analysis using a detailed finite-element model and response conditioning method. The 3-D spatial variability of soil properties is explicitly modeled using the expansion optimal linear estimation approach. A 3-D soil slope example is presented to demonstrate the validity of ARFEM. Finally, a sensitivity study is carried out to explore the effect of horizontal spatial variability. The results indicate that the proposed ARFEM not only provides reasonably accurate estimates of slope failure probability and risk, but also significantly reduces the computational effort at small probability levels. 3-D slope probabilistic analysis (including both 3-D slope stability analysis and 3-D spatial variability modeling) can reflect slope failure mechanism more realistically in terms of the shape, location and length of slip surface. Horizontal spatial variability can significantly influence the failure mode, reliability and risk of 3-D slopes, especially for long slopes with relatively strong horizontal spatial variability. These effects can be properly incorporated into 3-D slope reliability analysis and risk assessment using ARFEM.     

沉积模拟的基本数学模型及其在第四纪研究中的应用

内容提要本文扼要介绍了沉积模拟的基本数学模型,其中包括十种随机模拟数学模型和十种确定模拟数学模型,阐明其简要原理和应用范围,并讨论它们在第四纪研究中的应用现状和前景。本文涉及的应用范围主要为:第四纪沉积环境、第四纪沉积物特征和成因类型。最后讨论了地质过程数学模拟的特点和意义、第四纪沉积过程数学模拟对提高第四纪地质学研究定量化水平的作用,以及进一步开展工作的方向。     

An Improved Gradual Deformation Method for Reconciling Random and Gradient Searches in Stochastic Optimizations

Constraining stochastic models of reservoir properties such as porosity and permeability can be formulated as an optimization problem. While an optimization based on random search methods preserves the spatial variability of the stochastic model, it is prohibitively computer intensive. In contrast, gradient search methods may be very efficient but it does not preserve the spatial variability of the stochastic model. The gradual deformation method allows for modifying a reservoir model (i.e., realization of the stochastic model) from a small number of parameters while preserving its spatial variability. It can be considered as a first step towards the merger of random and gradient search methods. The gradual deformation method yields chains of reservoir models that can be investigated successively to identify an optimal reservoir model. The investigation of each chain is based on gradient computations, but the building of chains of reservoir models is random. In this paper, we propose an algorithm that further improves the efficiency of the gradual deformation method. Contrary to the previous gradual deformation method, we also use gradient information to build chains of reservoir models. The idea is to combine the initial reservoir model or the previously optimized reservoir model with a compound reservoir model. This compound model is a linear combination of a set of independent reservoir models. The combination coefficients are calculated so that the search direction from the initial model is as close as possible to the gradient search direction. This new gradual deformation scheme allows us for reducing the number of optimization parameters while selecting an optimal search direction. The numerical example compares the performance of the new gradual deformation scheme with that of the traditional one.     

Assessment of alluvial aquifer heterogeneity and development of stochastic hydrofacies models for the Hat Yai Basin in Southern Thailand

In previous studies, the groundwater flow models formulated for the Hat Yai Basin were conventional and deterministic because the geologic heterogeneity of the alluvial aquifer system in the basin had not yet been assessed. This paper describes an effort to develop hydrofacies models, such that the spatial variability of the aquifer system can be represented in a systematic way. Variogram parameters that characterize the alluvial aquifer heterogeneity were determined. Based on these variogram parameters, an indicator-based geostatistical approach was used to develop hydrofacies models using sequential indicator simulation. The hydrofacies models indicate three distinct aquifer units, namely Hat Yai, Khu Tao, and Kho Hong aquifers, which is in good agreement with a conceptual model, and incorporates spatial variability as observed in field data from borehole logs. The hydrofacies models can be used in groundwater modeling and simulations.     

南盘江盆地中三叠统复理石韵律的马尔科夫链特征及其地质意义

本文利用南盘江盆地中三叠统复理石韵律的野外测量数据,从马尔柯夫链原理出发,对其进行了频数转移矩阵、概率转移矩阵、极限概率矩阵和环流矩阵等沉积旋回最优分解综合分析.分析过程中发现并命名了循环链和二级循环链,并对循环链进行了特殊处理.通过分析模拟获得的状态循环模式图,建立了不同地点实测段的代表性韵律结构模式并绘制了韵律结构...     

Use of random-genetic models in the study of sedimentary processes

To study sedimentary phenomena, we introduce random-genetic models in which genetic hypotheses and structural random elements occur for the main part. Starting from geologic hypotheses we choose principal factors which may be random functions or random variables. These factors are: depth, nature of the facies, sedimentation rate, and subsidence. Equations of evolution link the factors. Depth is a Markov process, but generally the resultant sequence does not make a Markov chain or Markov process. Three examples of such models are given with the results of simulations.     

Indicator Variogram Models: Do We Have Much Choice?

Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.     

Monitoring and predicting land use change in Tripoli Metropolitan City using an integrated Markov chain and cellular automata models in GIS

The rapid development of cities in developing countries results in deteriorating of agricultural lands. The majority of these agricultural lands are converted to urban areas, which affects the ecosystems. In this research, an integrated model of Markov chain and cellular automata models was applied to simulate urban land use changes and to predict their spatial patterns in Tripoli metropolitan area, Libya. It is worth mentioning that there is not much research has been done about land use/cover change in Libyan cities. In this study, the performance of integrated CA–Markov model was assessed. Firstly, the Markov chain model was used to simulate and predict the land use change quantitatively; then, the CA model was applied to simulate the dynamic spatial patterns of changes explicitly. The urban land use change from 1984 to 2010 was modelled using the CA–Markov model for calibration to compute optimal transition rules and to predict future land use change. In validation process, the model was validated using Kappa index statistics which resulted in overall accuracy more than 85 %. Finally, based on transition rules and transition area matrix produced from calibration process, the future land use changes of 2020 and 2025 were predicted and mapped. The findings of this research showed reasonably good performance of employed model. The model results demonstrate that the study area is growing very rapidly especially in the recent decade. Furthermore, this rapid urban expansion results in remarkable continuous decrease of agriculture lands.     

Facies Modeling Using a Markov Mesh Model Specification

The spatial continuity of facies is one of the key factors controlling flow in reservoir models. Traditional pixel-based methods such as truncated Gaussian random fields and indicator simulation are based on only two-point statistics, which is insufficient to capture complex facies structures. Current methods for multi-point statistics either lack a consistent statistical model specification or are too computer intensive to be applicable. We propose a Markov mesh model based on generalized linear models for geological facies modeling. The approach defines a consistent statistical model that is facilitated by efficient estimation of model parameters and generation of realizations. Our presentation includes a formulation of the general framework, model specifications in two and three dimensions, and details on how the parameters can be estimated from a training image. We illustrate the method using multiple training images, including binary and trinary images and simulations in two and three dimensions. We also do a thorough comparison to the snesim approach. We find that the current model formulation is applicable for multiple training images and compares favorably to the snesim approach in our test examples. The method is highly memory efficient.     

大尹格庄金矿床矿体三维建模与矿化分布规律

文章根据勘探线地质剖面图钻孔样品化验数据和中段地质平面图坑道样品化验数据,利用Vulcan软件对大尹格庄金矿床矿体建立了两种三维实体模型,并将这两种模型结合起来,利用Datamine软件得到综合后的块体模型;运用地质统计学方法及Surpac软件,分析了①、②号两个矿（脉）体群的Au品位空间变化结构,并求得了搜索椭球体,实现了矿体和Au品位变化形态的空间分布展示;结合三维建模成果与矿体空间变化结构,分析得出了矿化分布规律.②号矿（脉）体群沿NE向继续展布的区域,是寻找深部隐伏矿体的有利部位.     

Back analysis of slope failure with Markov chain Monte Carlo simulation

Field observed performance of slopes can be used to back calculate input parameters of soil properties and evaluate uncertainty of a slope stability analysis model. In this paper, a new probabilistic method is proposed for back analysis of slope failure. The proposed back analysis method is formulated based on Bayes’ theorem and solved using the Markov chain Monte Carlo simulation method with a Metropolis–Hasting algorithm. The method is very flexible as any type of prior distribution can be used. The method is also computationally efficient when a response surface method is employed to approximate the slope stability model. An illustrative example of back analysis of a hypothetical slope failure is presented. Effects of jumping distribution functions and number of samples on the efficiency of Markov chains are studied. It is found that the covariance matrix of the jumping function can be set to be one half of the covariance of the prior distribution to achieve a reasonable acceptance rate and that 80,000 samples seem to be sufficient to obtain robust posterior statistics for the example. It is also found that the correlation of cohesion and friction angle of soil does not affect the posterior statistics and the remediation design of the slope significantly, while the type of the prior distribution seems to have much influence on the remediation design.