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.     

Simple and Ordinary Multigaussian Kriging for Estimating Recoverable Reserves

Multigaussian kriging is used in geostatistical applications to assess the recoverable reserves in ore deposits, or the probability for a contaminant to exceed a critical threshold. However, in general, the estimates have to be calculated by a numerical integration (Monte Carlo approach). In this paper, we propose analytical expressions to compute the multigaussian kriging estimator and its estimation variance, thanks to polynomial expansions. Three extensions are then considered, which are essential for mining and environmental applications: accounting for an unknown and locally varying mean (local stationarity), accounting for a block-support correction, and estimating spatial averages. All these extensions can be combined; they generalize several known techniques like ordinary lognormal kriging and uniform conditioning by a Gaussian value. An application of the concepts to a porphyry copper deposit shows that the proposed “ordinary multigaussian kriging” approach leads to more realistic estimates of the recoverable reserves than the conventional methods (disjunctive and simple multigaussian krigings), in particular in the nonmineralized undersampled areas.     

基于条件期望的全概率公式及其应用

基于条件期望和随机事件A的示性函数IA，推导了离散型和连续型随机变量的全概率公式，要解决诸如离散型和连续型等一类随机问题时，若采用传统的全概率公式来解，势必带来很大的困难。但是，运用本文所研究的全概率公式。结果令人满意，具有较强的实用性，从而拓展了全概率公式的应用范围和解题思路，值得推广应用，同时该公式在可靠性理论研究中也有重要应用。     

Influence of the improbable in earth sciences

The special significance of unanticipated extreme data (in space and time), especially for both genetic and economic predictions, is emphasised with reference to numerous dissimilar geological examples. However, commonly, data for measured variables falling near, or beyond, the tails of unimodal distribution curves have not been incorporated in inverse or forward models used previously in the earth sciences.     

Correcting the Smoothing Effect of Estimators: A Spectral Postprocessor

The postprocessing algorithm introduced by Yao for imposing the spectral amplitudes of a target covariance model is shown to be efficient in correcting the smoothing effect of estimation maps, whether obtained by kriging or any other interpolation technique. As opposed to stochastic simulation, Yao's algorithm yields a unique map starting from an original, typically smooth, estimation map. Most importantly it is shown that reproduction of a covariance/semivariogram model (global accuracy) is necessarily obtained at the cost of local accuracy reduction and increase in conditional bias. When working on one location at a time, kriging remains the most accurate (in the least squared error sense) estimator. However, kriging estimates should only be listed, not mapped, since they do not reflect the correct (target) spatial autocorrelation. This mismatch in spatial autocorrelation can be corrected via stochastic simulation, or can be imposed a posteriori via Yao's algorithm.     

长江上游干支流洪水遭遇分析

目前洪水遭遇研究仅限于简单的实测资料统计分析,尚缺系统有效的理论方法.基于多维Copula函数分别建立金沙江屏山站、岷江高场站、嘉陵江北碚站以及长江宜昌站的洪水发生时间和量级的联合分布.以此方法估计干支流洪水发生时间和量级的遭遇可能性及条件概率,并与实测资料的统计结果进行比较.结果表明,模型计算结果与实测相吻合,说明该...     

基于CPT资料的土性参数随机场特性研究

通过CPT资料研究土性参数随机场特性是一种比较理想的方法。土性参数随机场的平稳性是后续研究的先决条件之一,可以通过Bartlett统计量进行检验,但采用不同的模型拒绝域相差比较大,这样给实际应用造成困难,新模型LGS由于适应范围广可以解决这一问题。从随机场的基本理论出发,介绍了新相关模型LGS的数值模拟方法。基于LGS相关模型,采用修正的Bartlett统计量对随机场平稳性进行检验,统一了拒绝标准。用两组数据进行了实例分析,通过编制的程序进行计算,结果表明,这种相关模型对不同形式的相关函数有较好的适应性,可以解决拒绝域不统一的问题。     

Models for Support and Information Effects: A Comparative Study

The recoverable reserves in an ore deposit depend on several factors, in particular the size of the selective mining units (support effect) and the misclassifications when sending these units to mill or dump according to their estimated grade (information effect). Both effects imply a loss of selectivity and have to be correctly forecasted. In this work, several models are reviewed and applied to a synthetic ore deposit characterized by a highly skewed grade histogram and a spatial connectivity of high grades. The affine correction, mosaic correction, and discrete Gaussian model are compared when assessing the global recoverable reserves, whereas local estimations are performed by indicator kriging with affine correction, bigaussian disjunctive kriging, and multigaussian conditional expectation. Despite their convenience and simplicity, distribution-free methods like affine correction or indicator kriging have a poorer accuracy than the other methods. In the global framework, the discrete Gaussian model is a better alternative and is based on mild assumptions. Local estimations are not accurate and may be improved by resorting to a more suitable parametric model or to conditional simulations.     

基于copula函数的三峡水库预泄对鄱阳湖防洪影响分析

三峡工程的运行对鄱阳湖防洪形势存在潜在影响。以三峡-鄱阳湖系统为典型,采用基于copula理论的多维联合分布函数,建立三峡工程运行前长江-鄱阳湖-"五河"(赣江、抚河、信江、饶河、修河)系统中水文要素之间的联合概率分布及条件概率分布,并假设该条件分布关系在三峡工程运行前后保持不变;估计三峡工程运行后长江水文要素的概率分布,结合前面的条件概率分布,可以得到三峡工程运行后研究变量的概率分布;对比分析前后概率分布的变化,即可从统计角度评价三峡水库运行对鄱阳湖水文情势的影响。研究表明:三峡工程运行对鄱阳湖水位有一定影响;5、6月份三峡预泄,将增高鄱阳湖水位,其中,平均水位的增幅大于最高水位增幅,低水增幅大于高水增幅;三峡预泄影响下,湖区圩堤堤前水位没有超过原有堤防设计水位,没有降低湖区圩堤的防洪标准。     

Geographical Information Systems and Geostatistics for Modelling Radioactively Contaminated Land Areas

The Chernobyl plume contaminated vast lands of Europe with radiocaesium (¹37Cs) in 1986 because of the deposition of radionuclides on the ground by wet and dry deposition processes. Nevertheless, in a nuclear emergency, contamination data may be very sparse and there is need to make rapid and scientifically supported decisions. Here we analyze the rainfall field, an important precursor of the wet deposition, during the passage of the plume. Thus, estimating rainfall spatial variability can help to identify possible contaminated areas and associated risks when rainfall exceeded a given threshold. In this paper, we show that the conditional probabilities of exceeding threshold rainfall values could be spatially assessed using the mutual benefits of linking geostatistical and geographical information system (GIS) to quantify the evaluation of the risk involved in decision making. In particular, the non-parametric geostatistic technique, termed Indicator Kriging (IK), enables one to efficiently estimate the probability that the true value exceeds the threshold values by means of the indicator coding transform. Afterward, GIS has been used to find the areas probably affected by wash-out (probability >0.5 that rainfall is above a certain threshold). The experimental study has been focused on a test site in Beneventan agroecosystem (Southern Italy) to model the spatial uncertainty over a continuous area from sparse rainfall data. This enabled to generate probability maps delineating area potentially affected by to contamination to be monitored after wet deposition of Chernobyl releases.     

Hydraulic Effects of Shales in Fluvial-Deltaic Deposits: Ground-Penetrating Radar, Outcrop Observations, Geostatistics, and Three-Dimensional Flow Modeling for the Ferron Sandstone, Utah

Ground-penetrating radar (GPR) surveys, outcrop measurements, and cores provide a high-resolution 3D geologic model to investigate the hydraulic effects of shales in marine-influenced lower delta-plain distributary channel deposits within the Cretaceous-age Ferron Sandstone at Corbula Gulch in central Utah, USA. Shale statistics are computed from outcrop observations. Although slight anisotropy was observed in mean length and variogram ranges parallel and perpendicular to pale of low , the anisotropy is not statistically significant and the estimated mean length is 5.4 m. Truncated Gaussian simulation was used to create maps of shales that are placed on variably dipping stratigraphic surfaces interpreted from high-resolution 3D GPR surveys, outcrop interpretations, and boreholes. Sandstone permeability is estimated from radar responses calibrated to permeability measurements from core samples. Experimentally designed flow simulations examine the effects of variogram range, shale coverage fraction, and trends in shale coverage on predicted upscaled permeability, breakthrough time, and sweep efficiency. Approximately 1500 flow simulations examine three different geologic models, flow in the 3 coordinate directions, 16 geostatistical parameter combinations, and 10 realizations for each model. ANOVA and response models computed from the flow simulations demonstrate that shales decrease sweep, recovery, and permeability, especially in the vertical direction. The effect on horizontal flow is smaller. Flow predictions for ideal tracer displacements at Corbula Gulch are sensitive to shale-coverage fraction, but are relatively insensitive to twofold variations in variogram range or to vertical trends in shale coverage. Although the hydraulic effects of shale are statistically significant, the changes in flow responses rarely exceed 20%. As a result, it may be reasonable to use simple models when incorporating analogous shales into models of reservoirs or aquifers.     

基于系统评价的块体可靠度指标分析及应用

利用条件概率评价及系统工程理论,将关键块体作为块体系统一个重要组成部分,分析了关键块体可靠度指标评价办法,探讨了关键块体与次生关键块可靠度内在关系,由此类推,提出块体系统评价方法,推导出块体系统可靠度计算公式,实现对块体系统的失稳概率的评价计算。并选择一工程实例,对评价方法予以验证。结果表明：传统关键块体可靠度评价办法忽略块体间内在影响,结果偏向风险;而文中方法综合考虑块体之间影响,并建立了系统评价模型,对块体系统评价更接近实际情况,对块体系统理论的研究有一定借鉴意义。     

Toward a general procedure for analysis of extreme random events in the earth sciences

Engineering and scientific approaches to design magnitude estimation are briefly revisited. Some defense is offered for use of annual maxima in design as if they were variables from a common distribution. However, to assume any particular form of distribution tail beyond the largest data value is not justifiable, regardless of the degree of data support over the main body of the distribution. An alternative approach to the design problem is suggested through use of parameter-free nonparametric estimation using the kernel method. Some simulation results are presented which suggest that the parameter-free approach is worthy of further development. A particular advantage of nonparametric methods is that competing estimators can be checked against parametric distributions, leading to a progressive improvement in estimator accuracy.This paper was presented (by title) at Engineering Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

不同组合模型区域滑坡易发性及精度分析

为探索区域滑坡易发性评价模型的适用性和评价结果的合理性,以滑坡灾害高发的白龙江流域为研究区,首先选取坡度、地形起伏度、距断层距离、地层岩性、流域沟壑密度、植被指数等6项影响滑坡发生的孕灾因子作为易发性的评价指标,以研究区2 093处滑坡灾害点为样本数据,依据各指标条件下的信息量值、确定性系数值和证据权重值曲线突变规律,并结合滑坡面积及分级面积频率比曲线作为等级划分的临界值来确定因子分级状态;其次,基于指标因子状态分级和相关性分析结果,采用信息量法、确定性系数法、证据权法分别与逻辑回归组合的3种模型开展区域滑坡灾害易发性评价,并从模型结果、适用性和精度等方面采用多手段对3种组合模型进行比较和讨论。研究结果表明：在区域滑坡易发性评价方面,3组模型均表现较为理想,信息量和逻辑回归组合模型的预测精度为94.6%,其预测精度和准确性优于其他2种组合模型。笔者以白龙江流域中游及其岷江支流段为例,开展滑坡灾害易发性评价模型适用性、评价结果分析以及预测精度评价对比和研究等,成果可为该区地质灾害防灾减灾和国土空间用途管制规划决策提供参考。     

Some thoughts on the automated generation and selection of hypotheses in the earth sciences

Many major scientific problems in the earth sciences can be expressed in terms of finding a sequence of past events which can explain an observed configuration. The configuration might be anything from distribution of continents to a particular landform pattern. Because the past is unobservable, explanations take the form of one or more subjective hypotheses for a specific configuration origin. The subjective approach allows free rein of the human scientific spirit, but the way also is open for personal bias and construction of needlessly elaborate hypotheses. As an alternative, an objective process of hypothesis generation should be achievable by way of a generalized expert system incorporating all geological environments. The generality is required in order to allow for the possibility of producing surprising hypotheses, which might not have been anticipated in the context of any specific geological environment under study. In selecting from many available hypotheses, the criterion of simplicity is a useful means by which a hypothesis subset can be created and listed. However, no guarantee exists that subset members will better approximate the truth. The rationale is rather in terms of a working rule for avoiding unnecessary complexity in explanations. Creation of a generalized expert system would be a major project involving a team effort. Such a project would have the advantage of raising the scientific profile of mathematical geology, a subdiscipline which at present has something of an image of solving little problems elegantly.     

随机介质模型的模拟与混合型随机介质

讨论了随机介质模型的基本概念及指数型和高斯型椭圆自相关函数所描述的随机介质模型的特点, 并提出了混合型随机介质模型的概念, 该随机介质模型能更加灵活、准确地描述实际介质.通过选择在水平方向和垂直方向上的自相关长度a、b以及粗糙度r, 可以产生出各种不同形式的混合型随机介质模型.模拟结果显示, 混合型随机介质模型能更加灵活地描述实际介质, 具有适应性强, 使用方便、灵活, 能有效地模拟油气藏细节的优点.      

Reliability based risk index for the design of reinforced earth structures

The design methods currently used for earth reinforcement are mostly based on deterministic properties of both the soil and the construction materials used. Nowadays, however, the general trend is designing at a specific degree of reliability. This is even more true where the raw data such as soil properties exhibit significant variation. Deterministic solutions, in this case, may not suffice. Therefore, this paper will attempt to use probabilistic formulations thereby modifying the existing design procedure of reinforced earth retaining walls to account for uncertainties and variabilities. Through a first order Taylor's series expansion about the mean, the mean and variance of the strip reinforcing components, namely width and length, are derived in terms of the variations in the soil properties. Design charts that enable estimation of both mean and variance are developed to avoid extensive partial differentiation involved in the computations. Using appropriate probability distributions along with the mean and variance, the final design outputs are determined for a selected failure probability by introducing what is refered to as 'risk index'. The results indicate that the risk index increases with an increase in the coefficient of variations and a decrease in failure probability. Furthermore, it is shown that in some cases, depending on the variabilities of the soil properties, the classical design technique produced a relatively high failure probability. This revised version was published online in July 2006 with corrections to the Cover Date.