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.     

The variations of stable carbon isotope ratio of land plant-derived n-alkanes in deep-sea sediments from the Bering Sea and the North Pacific Ocean during the last 250,000 years

Two piston cores, one located far from the continents (The North Pacific Ocean: ES core), and another located comparatively closer to the continents (The Bering Sea: BOW-8a core) were investigated to reconstruct environmental changes on source land areas. The results show significant contribution of terrestrial organic matter to sediments in both cores. The δ¹³C values of n-C₂₇, n-C₂₉, and n-C₃₁ alkanes in sediments from the North Pacific ES core show significant glacial to interglacial variation whereas those from the Bering Sea core do not. Variations of δ¹³C values of land plant n-alkanes are related to the environmental or vegetational changes in the source land areas. Environmental changes, especially, aridity, rainfall, and pCO₂ during glacial/interglacial transitional periods can affect vegetation, and therefore C₃ / C₄ plant ratios, resulting in δ¹³C changes in the preserved land plant biomarkers. Maximum values of δ¹³C as well as maximum average chain length values of long chain n-alkanes in the ES core occur mostly at the interglacial to glacial transition zones reflecting a time lag related to incorporation of living organic matter into soil and transportation into ocean basins via wind and/or ability of C₄ plants to adapt for a longer period before being replaced by C₃ plants when subjected to gradual climatic changes. Irregular variations with no clear glacial to interglacial trends in the BOW-8a core may result from complex mixture of aerosols from westerly winds and riverine organic matter from the Bering Sea catchments. In addition, terrestrial organic matter entering the Bering Sea could originate from multiple pathways including eolian, riverine, and ice rafted debris, and possibly be disturbed by turbidity and other local currents which can induce re-suspension and re-sedimentation causing an obliterated time relation in the Bering Sea biomarker records.     

Stochastic characteristics of sedimentation in evaporite basins: Potassium strata of Solikamsk deposit

The mode of alternation of halopelite, rock salt, and sylvinite beds in the Solikamsk deposit may be expressed in terms of Markov chains where each state is the lithologic composition of the bed. Both ² and likelihood tests confirm the similarity of the Markov model with the actual geologic sections and suggest that evaporite sedimentation can be modeled by a renewal process.     

Bayesian Analysis of the Discovery Process Model Using Markov Chain Monte Carlo

Based upon the Bayesian framework for analyzing the discovery sequence in a play, a Markov chain Monte Carlo sampler—the Metropolis–Hastings algorithm, is employed to sample model parameters and pool sizes from their joint posterior distribution. The proposed sampling scheme ensures that the parameter space of changing dimension can be traversed in spite of the unknown number of pools. The equal sample weights make it easy to obtain the confidence intervals and assess the statistical error in the estimates, so that the statistical behaviors of the discovery process modeling can be well understood. Two application examples of the Halten play in Norwegian Sea and the Bashaw reef play in the Western Canada Basin show that, the computational advantage of this method to the simple Monte Carlo integration is considerable. In order to increase the convergence speed of the sample chains to the posterior distributions, several parallel simulations with different starting values are recommended.     

用遗传算法确定马尔柯夫随机场的最佳参数

对现有马尔柯夫随机场 (MRF)参数的解求方法作了详细分析。针对现有方法存在的缺点 ,提出基于遗传算法的MRF参数解求方法 ;分析讨论了求解的稳定性 ;并利用解的稳定性作为纹理分类的判据 ,作了航空影像纹理的分类实验。实验结果表明 ,求解的方法是正确的。用解的稳定性作为判据的纹理分类效果 ,通过对比实验证明是有效的。     

On outlier-hiding effects in specific Gauss–Markov models: geodetic examples

 A specific subclass of Gauss–Markov models has been defined as containing the models for which the disturbance/response matrix, determined under the assumption of uncorrelated observations, consists of independent diagonal blocks. A proposed modification of reliability assessment procedure for such models is presented By the appropriate reduction of a given full covariance matrix for the observations, the proposal allows the assessment to be made in the resulting model which, in contrast to the initial model, is free from outlier-hiding effects of the type not occurring in ordinary models. The theoretical findings are demonstrated using simple numerical examples. All the proofs supporting the proposal are gathered in Appendixes. The proposal, which is not without its own weak points, is an attempt to associate the reliability assessment in specific Gauss–Markov models with effective outlier detection. Received: 23 June 1998 / Accepted: 5 July 2000     

A New Approach to Gravity Anomaly Separation: Differential Markov Random Field (DMRF)

In this paper the authors introduce a novel approach to stochastic image processing, denoted as Differential Markov Random Field (DMRF), which has been applied to gravity anomaly separation problems. The advantages of the method are that it introduces only little distortion into the shape of the original image and that it is not affected significantly by factors such as the overlap power spectra of regional and residual fields. Testing of the proposed meth using synthetic examples gave excellent results.     

土性高斯平稳场-马尔可夫场在渗流随机分析中的应用研究

不同类别土之间的性质差异导致了土性空间变异性的增大,需要建立一种模型以更真实地反映变异产生机制。引入基于马尔可夫场的多元指示地质统计模型,推导了含多类土的土层特性参数统计特征的理论表达式,分析了多类土的存在对K-L(Karhunen-Loeve)展开的影响,应用谱模型对土层进行了渗流力学随机分析。模拟结果证实了高斯平稳场-马尔可夫场的正确性和谱模型的有效性。     

天津蓟县雾迷山旋回层基本模式及其马尔柯夫链分析

津蓟县的中元古界雾迷山组,是一套碳酸盐岩地层,具近似对称相序组构的环潮坪型碳酸盐米级旋回层序特别以育。其近似对称的相序组构及普遍的1：4叠加形态,表明它们与短偏心率旋回具有成因关联,被命名为雾迷山旋回层来代表真正的碳酸盐沉积旋回。雾迷山旋回层是与高频率产面变化相关的环境加深及环境变浅过程的产物。马尔柯夫链分析的结果表明了雾迷山旋回层的基本相序模式是客观存在的,由于它们以瞬时暴露间断面为界,因而与Vail等（1977）的层序地层概念体系中所定义的“准层序”存在明显的差异。尽管难以断定前寒武纪地球轨道效应旋回的周期安全与显生宙一致,但是,七级韵律层、六级旋回层及五级准层序组之间的垂直叠置形态,以及由它们所指示的周期时限,与显生宙温室效应时期的轨道效应旋回周期是大体一致的。     

黑河流域水资源动态变化及其趋势的灰色Markov链预测

根据有关水文气象台站的观测数据,对黑河流域水资源时空分布特征及变化规律进行了分析,并基于GM(1,1)模型和Markov链原理,提出了一个用于黑河流域的水资源变化趋势预测的灰色离散随机过程模型。结果表明,从1944年有观测记录以来的近60a时间里,以黑河干流为代表的黑河流域天然径流经历了几个长度不等的丰水段与枯水段,而目前正处于1996年开始第5个丰水段的下降段的最低点或自2000年开始的第6个枯水段的起始点。未来几年里,黑河流域天然水资源的变化总体上将呈现一种偏枯或平水偏枯的状态,但径流偏枯的幅度将不会很大。