On the character of distribution of chemical elements in nature

Distribution of chemical elements in nature is controlled by the general physical-chemical state of natural processes, a result of the effort to reach thermodynamic equilibrium. Natural processes, forming geologic objects (e.g., rocks, ores) usually do not reach total chemical equilibrium. This is reflected in distributions of chemical elements in nature. We recognize symmetric as well as asymmetric distributions. The question arises, what distribution in nature is most commonly followed? Evidently, this should be a distribution of chemical elements in thermodynamic equilibrium throughout the system, thus with maximum entropy. This paper notes that Gauss's normal distribution may be derived from statistic explanation of entropy as a function of contnt of chemical elements in a system with application of Einstein's formula.     

Use of the truncated shifted Pareto distribution in assessing size distribution of oil and gas fields

The truncated shifted Pareto (TSP) distribution, a variant of the two-parameter Pareto distribution, in which one parameter is added to shift the distribution right and left and the right-hand side is truncated, is used to model size distributions of oil and gas fields for resource assessment. Assumptions about limits to the left-hand and right-hand side reduce the number of parameters to two. The TSP distribution has advantages over the more customary lognormal distribution because it has a simple analytic expression, allowing exact computation of several statistics of interest, has a J-shape, and has more flexibility in the thickness of the right-hand tail. Oil field sizes from the Minnelusa play in the Powder River Basin, Wyoming and Montana, are used as a case study. Probability plotting procedures allow easy visualization of the fit and help the assessment.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

A review of lognormal estimators forin situ reserves

The term lognormal kriging does not correspond to a single well defined estimator. In fact, several types of lognormal estimators forin situ reserves are available, and this may cause confusion. These estimators are based on different assumptions—that is, different models. This paper presents a review of these estimators.     

基于FLAC3D模拟的大规模采空区破坏概率研究

采空区稳定性的问题由来已久,但目前对采空区的整体稳定性评价结论几乎仍停留在定性层面上,特别是对于条件复杂的大规模地下矿山采空区,若能将其量化则在很大程度上将会对矿山安全提供一定的保障。针对大规模采空区稳定性的难题,在数值模拟的基础上引入采空区破坏概率的概念试图解决这一问题。在FLAC3D后处理的过程中,结合统计学中概率的基本原理,利用Excel的风险分析插件@Risk对FLAC3D网格单元的计算结果进行统计分析,并以95%的置信区间为准,从主应力、剪切破坏判据Fs以及弹性应变能等方面深入研究分析采空区的破坏概率,进而将采空区稳定性问题得以量化,同时得出了大量岩石网格单元主应力和弹性应变能分布服从韦布尔分布(Weibull)和对数正态分布(lognorm)的规律。量化不仅解决了空区稳定性的问题,也将为数值模拟开拓更为广阔的应用空间。     

基于围岩力学参数概率分布模型的变形敏感性灰关联分析

为了给布伦口-公格尔水电站地下洞室某标段围岩稳定性分析的参数选取提供可靠的理论依据,综合考虑岩体参数的空间变异性,针对常规敏感性分析方法所存在的不足,运用三维离散元计算程序,提出了基于围岩力学参数概率分布模型的变形敏感性灰关联分析方法。该方法以岩体密度?、弹性模量E、泊松比?、黏聚力c、内摩擦角φ及节理内摩擦角φj等6个围岩力学参数作为因素序列,拱顶下沉量作为目标序列,分析影响因素在整个定义区间内的变化对围岩拱顶下沉的敏感程度。结果表明：密度是最敏感因素,其次是弹性模量、黏聚力和泊松比,而内摩擦角以及节理内摩擦角敏感性最小。最后,将常规灰关联敏感性分析与该方法计算结果进行对比,结果表明：除密度、内摩擦角和节理内摩擦角一致外,其余参数敏感性均与结论不一致。因此,文中方法在考虑实际参数概率分布的基础上能够更加准确、合理地对参数进行综合评价。     

Turbidite bed thickness distributions: methods and pitfalls of analysis and modelling

Turbidite bed thickness distributions are often interpreted in terms of power laws, even when there are significant departures from a single straight line on a log–log exceedence probability plot. Alternatively, these distributions have been described by a lognormal mixture model. Statistical methods used to analyse and distinguish the two models (power law and lognormal mixture) are presented here. In addition, the shortcomings of some frequently applied techniques are discussed, using a new data set from the Tarcău Sandstone of the East Carpathians, Romania, and published data from the Marnoso‐Arenacea Formation of Italy. Log–log exceedence plots and least squares fitting by themselves are inappropriate tools for the analysis of bed thickness distributions; they must be accompanied by the assessment of other types of diagrams (cumulative probability, histogram of log‐transformed values, q–q plots) and the use of a measure of goodness‐of‐fit other than R², such as the chi‐square or the Kolmogorov–Smirnov statistics. When interpreting data that do not follow a single straight line on a log–log exceedence plot, it is important to take into account that ‘segmented’ power laws are not simple mixtures of power law populations with arbitrary parameters. Although a simple model of flow confinement does result in segmented plots at the centre of a basin, the segmented shape of the exceedence curve breaks down as the sampling location moves away from the basin centre. The lognormal mixture model is a sedimentologically intuitive alternative to the power law distribution. The expectation–maximization algorithm can be used to estimate the parameters and thus to model lognormal bed thickness mixtures. Taking into account these observations, the bed thickness data from the Tarcău Sandstone are best described by a lognormal mixture model with two components. Compared with the Marnoso‐Arenacea Formation, in which bed thicknesses of thin beds have a larger variability than thicknesses of the thicker beds, the thinner‐bedded population of the Tarcău Sandstone has a lower variability than the thicker‐bedded population. Such differences might reflect contrasting depositional settings, such as the difference between channel levées and basin plains.     

层状地质体与地下工程开挖三维可视化

地质体与人造工程几何形态极其复杂,其三维可视化是地学的一个难题。通过多层DEM形成层状地质体界面,在此基础上,借助中轴变换技术概念,用地层厚度和地层中层面将地层的形状特征及拓扑关系表现出来,建立了一个判断地层任意位置属性值的方法,藉此运用空间插值技术、等值线生成技术、扫描表示技术等可视化技术将地层以及地下工程开挖绘制出来,避免了复杂的开挖体元属性及形体判断和开挖面与体元求交过程。采用此方法实现了某水电站地层与厂房基坑开挖三维可视化。     

Models of breakage and selection for particle size distributions

It is generally agreed that particle size distributions of sediments tend ideally to approximate the form of the lognormal probability law, but there is no single widely accepted explanation of how sedimentary processes generate the form of this law. Conceptually, and in its simplest form, sediment genesis involves the transformation of a parent rock mass into a particulate end product by processes that include size reduction and selection during weathering, transportation, and deposition. The many variables that operate simultaneously during this transformation can be shown to produce a distribution of particle sizes that approaches asymptotically the lognormal form when the effect of the variables is multiplicative. This was first shown by Kolmogorov (1941). Currently available models combine breakage and selection in differing degrees, but are similar in treating the processes as having multiplicative effects on particle sizes. The present paper, based on careful specification of the initial state, the nth breakage rule and the nth selection rule, leads to two stochastic models for particle breakage, and for both models the probability distributions of particle sizes are obtained. No attempt is made to apply these models to real world sedimentary processes, although this topic is touched upon in the closing remarks.     

北塘水库库底地层渗透系数的随机特性分析

为研究渗透系数的随机分布特征,以滨海平原水库也是南水北调调节及事故备用水库的北塘水库库底地层为研究对象,通过野外取样、室内达西渗透实验和颗粒分析实验等手段,获取了库底地层渗透系数的两组数据及其空间分布,借助于SPSS软件的P—P图以及单样本K-S检验,对渗透系数的空间随机分布进行了分析。结果表明,与正态分布相比较,北塘水库库底地层渗透系数更接近对数正态分布,这为库区渗漏量计算的可靠性分析、库底沉积物渗透系数的空间变异特征分析等奠定了基础。     

基于偏正态概率分布的粒度分布次总体分离及其沉积环境指示意义

沉积物颗粒是某种沉积环境和水动力条件下多个沉积过程的最终产物。粒度分布是原始沉积信息的载体,是来自不同沉积过程的多个次总体的叠加,频率曲线可能表现为双峰或者多峰特征。传统的沉积学粒度分析方法并未深入研究次总体;常见的概率分布模型在分离次总体后无法全面计算统计参数。本文以214份鄱阳湖现代沉积物的粒度分布数据为例,利用偏正态概率分布模型共分离提取977个次总体,计算各个次总体的统计参数,并对比分析了不同沉积环境中次总体参数的异同。结果表明：① 次总体均值、方差、偏度、峰度、所占百分比和最大频率等参数规律明显;② 从曲流河河道到河流末端、在河流末端顺流方向上和河道左右两侧远离河道方向上,粒度分布中主要次总体粒度均值逐渐减小,河道间洼地和湖区沉积物粒度分布的各个次总体占比接近;③ 江心洲的河道砂和河漫滩细粒粒度分布分别由3种和5种不同类型次总体组成。该方法可为沉积环境的定量判断和沉积过程的定量研究提供参考。     

Estimation of statistical parameters for censored lognormal hydraulic conductivity measurements

Field tests of hydraulic conductivity (e.g., injection test, pumping test, etc.) in low permeability formations are subject to censoring due to the detection limit of the instruments used. An iterative method of estimating the mean and variance of hydraulic conductivity data with a presumed log-normal distribution function is presented. This method accounts for the data that are actually below the lower detection limit (called truncated data) and thus gives distribution parameters that are more representative for the underlying distribution. The proposed method is then tested on two simulated normally distributed random datasets having different variances. The results show that the means and variances estimated by the proposed method are very accurate. Finally, the method is used to estimate the mean and variance of hydraulic conductivity data from single hole water injection tests in a fractured geological formation.     

基于偏正态概率分布的粒度分布次总体分离及其沉积环境指示意义

沉积物颗粒是某种沉积环境和水动力条件下多个沉积过程的最终产物。粒度分布是原始沉积信息的载体,是来自不同沉积过程的多个次总体的叠加,频率曲线可能表现为双峰或者多峰特征。传统的沉积学粒度分析方法并未深入研究次总体;常见的概率分布模型在分离次总体后无法全面计算统计参数。本文以214份鄱阳湖现代沉积物的粒度分布数据为例,利用偏正态概率分布模型共分离提取977个次总体,计算各个次总体的统计参数,并对比分析了不同沉积环境中次总体参数的异同。结果表明：① 次总体均值、方差、偏度、峰度、所占百分比和最大频率等参数规律明显;② 从曲流河河道到河流末端、在河流末端顺流方向上和河道左右两侧远离河道方向上,粒度分布中主要次总体粒度均值逐渐减小,河道间洼地和湖区沉积物粒度分布的各个次总体占比接近;③ 江心洲的河道砂和河漫滩细粒粒度分布分别由3种和5种不同类型次总体组成。该方法可为沉积环境的定量判断和沉积过程的定量研究提供参考。     

若尔盖县大骨节病分布特征及其与地质环境的关系

大骨节病是一种地方性、变形性骨关节病。国内外学者历经160年的研究,病因假说上百种,但至今仍无定论。若尔盖县是中国大骨节病发病率较高的病区之一,通过在病区开展野外调查、岩土水样测试等工作,本文总结了该区大骨节病分布特征和自然与地质环境特征,探讨了大骨节病分布与地层岩性、土壤、地形地貌、饮水水源等地质环境的关系。研究表明:基岩岩性及其土壤母质成分是影响病情分布的基本因素;病区土壤中Se、Mo、Ba等元素含量较非病区低,N、P、Pb、S元素含量高于非病区;饮用地表水中的矿化度、硬度、腐殖酸含量病区与非病区差异不显著,病区硒含量低于轻、非病区;大骨节病的致病影响因素复杂,除与岩、土、水地球化学环境因素有关外,亦与饮食结构等因素有关;大骨节病与地质环境关系密切,其病因需进一步探索。     

大样本岩土参数概率分布的正态信息扩散推断

在重要的岩土工程中,确定大样本岩土参数的概率分布对岩土工程稳定性和可靠性分析有着极其重要的意义。为此,基于积分均方误差最小计算得到的最优窗宽,提出了推断大样本岩土参数概率密度函数的正态信息扩散法。该方法基于信息扩散原理,从试验样本和信息论的角度出发,充分利用样本提供的数据信息,而不是先假定成经典的概率分布曲线拟合检验,数学意义和物理意义更加充分和严密。以推断压缩指数Cc的概率密度函数为例,分析了窗宽对信息扩散估计结果的影响规律,说明了该方法在大样本岩土参数概率密度函数推断方面的合理性。用该方法推断了大样本岩土抗剪强度参数的概率密度函数,通过K-S检验,验证了该方法的正确性和实用性。     

拟合三参数对数正态分布时第三参数c的优选方法探讨

在地质数据中，许多数据常呈正偏或负偏分布，但并不服从对数或反对娄正态态分布。这时可考虑它们是否服从三参数对数或反对数正态分布。本给出一种估计第三参数ｃ的方法。它可快速、客观、较优地确定此参数，弥补以往凭经验多次选择、反复试算等缺陷，为检验地质数据的统计分布型式，掌握分布规律，从而用于统计预测等提供了方便。     

Target intersection probabilities for parallel-line and continuous-grid types of search

The expressions for calculating the probability of intersection of hidden targets of different sizes and shapes for parallel-line and continuous-grid types of search can be formulated by vsing the concept of conditional probability. When the prior probability of the orientation of a widden target is represented by a uniform distribution, the calculated posterior probabilities are identical with the results obtained by the classic methods of probability. For hidden targets of different sizes and shapes, the following generalizations about the probability of intersection can be made: (1) to a first approximation, the probability of intersection of a hidden target is proportional to the ratio of the greatest dimension of the target (viewed in plane projection) to the minimum line spacing of the search pattern; (2) the shape of the hidden target does not greatly affect the probability of the intersection when the largest dimension of the target is small relative to the minimum spacing of the search pattern, (3) the probability of intersecting a target twice for a particular type of search can be used as a lower bound if there is an element of uncertainty of detection for a particular type of tool; (4) the geometry of the search pattern becomes more critical when the largest dimension of the target equals or exceeds the minimum spacing of the search pattern; (5) for elongate targets, the probability of intersection is greater for parallel-line search than for an equivalent continuous square-grid search when the largest dimension of the target is less than the minimum spacing of the search pattern, whereas the opposite is true when the largest dimension exceeds the minimum spacing; (6) the probability of intersection for nonorthogonal continuous-grid search patterns is not greatly different from the probability of intersection for the equivalent orthogonal continuous-grid pattern when the orientation of the target is unknown. The probability of intersection for an elliptically shaped target can be approximated by treating the ellipse as intermediate between a circle and a line. A search conducted along a continuous rectangular grid can be represented as intermediate between a search along parallel lines and along a continuous square grid. On this basis, an upper and lower bound for the probability of intersection of an elliptically shaped target for a continuous rectangular grid can be calculated. Charts have been constructed that permit the values for these probabilities to be obtained graphically. The use of conditional probability allows the explorationist greater flexibility in considering alternate search strategies for locating hidden targets.This paper was presented at Symposium 116.3, Quantitative Strategy for Exploration, held as part of the 25th International Geological Congress, Sydney, Australia, August 1976.     

岩体不连续面迹长与直径间的概率关系模型分析

由于“随机”概念的模糊性,在假定不连续面为薄圆盘及弦长作为迹长的前提下,基于迹线端点在圆周上均匀分布、迹线中点在直径上均匀分布、迹线中点在圆面域内均匀分布这3种明确而又不同的“随机”含义,可以得到3种合理的迹长 与直径 间的概率关系模型。通过对这3种模型进行分析认为：其在各自的“随机”含义下都是正确的,只是适用条件各异;野外实际可行的条件是介于模型2 和模型3之间的,合理地分析应该同时考虑这两种模型。在实际应用中,应该首先对所有取样面产状之间的关系进行分析,根据其结果来分配给模型2和模型3以某种权重,这样得到的迹长与不连续面直径之间概率关系更具合理性,其结果弥补了目前国内外进行的相关研究仅仅考虑到第2类模型的不足。     

花岗岩锆石U-Pb年龄与全岩Rb-Sr等时线年龄对比研究及其地球化学意义

对国内外32个花岗岩体的锆石U-Pb年龄与全岩Rb-Sr等时线年龄之间差值（Δt）进行的频数统计分析表明：Δt呈对称正态分布（偏度系数CSK=0.36;峰度系数CKU=2.99）;年龄差（Δt）既呈正值又有负值,其均值为2.08Ma;相对年龄差（Rt）小于5%。采用最小二乘法计算,花岗岩体锆石U-Pb年龄（tZr）对全岩Rb-Sr等时线年龄（tRb）拟合出相关系数很高（r=0.998）,回归系数接近l（α=1.003）的线性回归方程（tRb =1.003tZr +1.258）。这些统计特征表明,从总体来看,花岗岩体的Rb-Sr等时线定年测定结果与锆石U-Pb定年测定结果是一致的,花岗岩全岩Rb-Sr等时线定年方法是成熟、可信的,同时也为花岗岩锆石U-Pb年龄代表结晶年龄而不代表花岗岩侵位年龄提供了依据。     

矿床储量规模分布的分形模型及其分维的数理分析

矿床储量规模预测在矿产勘查和评价中占有十分重要的位置,而其预测的最主要手段是建立矿床储量规模分布的数学模型.根据矿床储量的统计自相似性,应用概率统计方法建立了矿床储量规模的分形模型.同时,通过模型及其分维值的数理分析,得出不同分维值的地学解释,讨论了分维的线性与非线性最小二乘回归估计法.