Disjunctive kriging revisited: Part II

In order to find distributions other than infinitely divisible distributions which are suitable for disjunctive kriging, infinitesimal generators are used. In addition to distributions developed in Part I, this leads to development of suitable models for the beta (), hypergeometric, and binomial distributions.     

An application of disjunctive kriging for earthquake ground motion estimation

For earthquake ground motion studies, the actual ground motion distribution should be reproduced as accurately as possible. For optimal estimation of ground motion, kriging has been shown to provide accurate estimates. Although kriging is accurate for this application, some estimates it provides are underestimates. This has dire consequences for subsequent design for earthquake resistance. Kriging does not provide enough information to allow an analysis of each estimate for underestimation. For such an application, disjunctive kriging is better applied. This advanced technique quantifies the probability that an estimate equals or exceeds particular levels of ground motion. Furthermore, disjunctive kriging can provide improved estimation accuracy when applied for local estimation of ground motion.     

Disjunctive Kriging with Hard and Imprecise Data

This paper presents a methodology for assessing local probability distributions by disjunctive kriging when the available data set contains some imprecise measurements, like noisy or soft information or interval constraints. The basic idea consists in replacing the set of imprecise data by a set of pseudohard data simulated from their posterior distribution; an iterative algorithm based on the Gibbs sampler is proposed to achieve such a simulation step. The whole procedure is repeated many times and the final result is the average of the disjunctive kriging estimates computed from each simulated data set. Being data-independent, the kriging weights need to be calculated only once, which enables fast computing. The simulation procedure requires encoding each datum as a pre-posterior distribution and assuming a Markov property to allow the updating of pre-posterior distributions into posterior ones. Although it suffers some imperfections, disjunctive kriging turns out to be a much more flexible approach than conditional expectation, because of the vast class of models that allows its computation, namely isofactorial models.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

Disjunctive kriging, universal kriging, or no kriging: Small sample results with simulated fields

This paper provides a comparison between linear (universal) and nonlinear (disjunctive) kriging estimators when they are computed from small samples chosen randomly on simulated stationary and nonstationary fields. Point estimation results are reported. In all cases considered, kriging estimators were found better than a local mean estimator, with universal kriging either better than or as good as disjunctive kriging. The latter, which is suited to handle stationary fields, did not provide more accurate estimates because the use of small samples led to inconsistencies in the assumed bivariate model. Universal kriging was particularly better with nonstationary fields.     

An Empirical Bayes Analysis of Volcanic Eruptions

This paper describes a simple empirical Bayes approach to inference about the eruption rate of a volcano under a Poisson process model. Under this approach, the prior distribution of the eruption rate is estimated from the eruption records of a group of similar volcanoes. The approach is illustrated using the eruption records of six Central American volcanoes.     

Comparing some approximate methods for building local confidence intervals for predicting regionalized variables

Approximate local confidence intervals can be produced by nonlinear methods designed to estimate indicator variables. The most precise of these methods, the conditional expectation, can only be used in practice in the multi-Gaussian context. Theoretically, less efficient methods have to be used in more general cases. The methods considered here are indicator kriging, probability kriging (indicator-rank co-kriging), and disjunctive kriging (indicator co-kriging). The properties of these estimators are studied in this paper in the multi-Gaussian context, for this allows a more detailed study than under more general models. Conditional distribution approximation is first studied. Exact results are given for mean squared errors and conditional bias. Then conditional quantile estimators are compared empirically. Finally, confidence intervals are compared from the points of view of bias and precision.     

Comparison of two techniques for applying disjunctive kriging: The Gaussian anamorphosis model versus the direct statistical inference of the bivariate distributions

Indicator cokriging is an alternative to disjunctive kriging for estimation of spatial distributions. One way to determine which of these techniques is more accurate for estimation of spatial distributions is to apply each to a particular type of data. A procedure is developed for evaluation of disjunctive kriging and indicator cokriging for such an application. Application of this procedure to earthquake ground motion data found disjunctive kriging to be at least as accurate as indicator cokriging for estimation of spatial distributions of peak horizontal acceleration. Indicator cokriging was superior for all other types of earthquake ground motion data.     

Distribution of kriging error and stationarity of the variogram in a coal property

If a particular distribution for kriging error may be assumed, confidence intervals can be estimated and contract risk can be assessed. Contract risk is defined as the probability that a block grade will exceed some specified limit. In coal mining, this specified limit will be set in a coal sales agreement. A key assumption necessary to implement the geostatistical model is that of local stationarity in the variogram. In a typical project, data limitations prevent a detailed examination of the stationarity assumption. In this paper, the distribution of kriging error and scale of variogram stationarity are examined for a coal property in northern West Virginia.     

A parametric study of robustness of kriging variance as a function of range and relative nugget effect for a spherical semivariogram

In geostatistics, an estimation of blocks of a deposit is reported along with the variance of error made in their estimation. This calculation is based on the model chosen for the semivariogram of the deposit so that mistakes in its estimation can manifest themselves in the perception of accuracy with which blocks are known. Changes in kriging variance resulting from various amounts of error in modeling the relative nugget effect and range of the semivariogram are investigated for an extensive set of spherical semivariograms.     

Indicator kriging for unit vectors: Rock joint orientations

Indicator kriging (IK) is extended to analyze three-dimensional random unit vectors and evaluate the local probability distribution of rock joint orientations in geological formations. The pole vector representing joint orientations is regionalized and projected on a plane normal to the mean attitude of the joint family and centered at the mean. A two-dimensional cutoff system is developed to define the indicator variable, and corresponding indicator variograms and indicator kriging. The cutoff system defines probability regions similar to those of a bivariate distribution, concentric rings sliced into radial sectors. A case study made on an open pit mine proved positively the efficiency of IK and encourages its applications to localized probabilistic structural modeling for geotechnical or geohydrological analysis and oil and gas reservoir analysis.     

Restricted kriging: A link between sample value and sample configuration

Restricted kriging provides a simple and quick remedy for the problem known as the weight independence of data in ordinary kriging. A major consequence of this problem is the effect of over-smearing in estimates, which, in turn, adds one uncertain factor to subsequent mine decisions. A detailed count is reported here on a restricted kriging system that incorporates two restrictions—one for high-grade samples and the other for low-grade samples. The restriction of high grade samples is because of their low priori likelihoods, whereas the main reason to restrict low grade samples is their nature as being waste and low analysis precisions. The two constraints reinforce each other in terms of enhancing the variables of estimates. A detailed case study on an epithermal gold deposit is carried out in terms of both cross validation and block modeling, showing that restricted kriging is superior over OK in mimicking the variables of original data.     

Variography of submarine morphology: Problems of deregularization, and cartographical implications

During the German Antarctic Expedition VI (leg 3, December 1987 to March 1988), bathymetric surveys were made in the Weddell Sea by the SEABEAM sonar system. For the first time geostatistical methods were applied in the SEABEAM-postprocessing. The investigations of variography that were necessary prior to the cartographical-geomorphological evaluation shed new light on classical geostatistical concerns. SEABEAM data provide a good example of a mean square, differentiable regionalized variable, where data are sampled over a two-dimensional support due to the technique of the sonar device. By deregularizations of the sample variograms, spatial continuity can be shown to be a property of seafloor depth as well as a point variable. The results are discussed in a sedimentological context. As an application of the regional variogram analyses, large-scale kriged bathymetric maps are presented.     

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。     

基于ANN与GIS技术的区域岩溶塌陷稳定性预测———以桂林西城区为例

岩溶地面塌陷是岩溶区常见的一种地质灾害, 塌陷区域预测是进行国土规划、资源开发与灾害防治的必要工作.由于岩溶塌陷的影响因素众多且相互作用, 发展过程复杂, 加之各评价因子的数值获取困难, 致使长期以来塌陷区域定量预测成为一个难以解决的课题.现行的区域预测模型不能描述塌陷形成模式的非线性特征, 也难以克服评价因子权重确定过程中人为经验因素的影响.神经网络技术的自学习、自适应与高度非线性映射特点显示了其在塌陷区域预测领域中应用的前景.根据研究区内地面塌陷空间聚集分布的特征, 提出了不同因子组合条件下塌陷发生可能性的定量化方法, 结合选定的评价因子类别确定了神经网络预测模型的结构, 利用312个塌陷点样本中的292个进行网络训练, 余下的20个样本的校验结果表明该模型具有较高的可信度.运用GIS技术将研究区进行评价单元划分, 并获取各评价因子的取值, 输入到训练好的网络中进行预测.将各单元的输出值进行归并处理后得到研究区岩溶塌陷的稳定级分区图.      

An anisotropic Model for Structured Soils: Part I: Theory

The paper presents an incremental plasticity constitutive Model for Structured Soils (MSS–2) to describe the effects of structure (stress history and bonding) on the mechanical behaviour of cohesive soils. Such effects are high initial stiffness, dilatancy and peak strength, their appreciable reduction upon strain-induced de-structuring and the evolution of material anisotropy. The proposed model advances present practice in incremental elasto-plasticity for structured soils by incorporating: (a) distorted ellipsoids, rotated with respect to the isotropic axis, for the Structure Strength Envelope (bounding surface) and the Plastic Yield Envelope (yield surface) to describe the evolution of structure and anisotropy, (b) an Intrinsic Compressibility Framework and a corresponding Intrinsic Strength Envelope which represents a lower bound of the Structure Strength Envelope and is used as reference locus for the structureless material, (c) an improved damage mechanism to model structure degradation by plastic strains and (d) a non-associated flow rule controlled by structure. The proposed model is modular, its features can be activated simultaneously or selectively, and the 3-D tensorial formulation facilitates direct implementation in finite elements codes.     

Three-dimensional quantitative textural analysis of metamorphic rocks using high-resolution computed X-ray tomography: Part I. Methods and techniques

Macroscopic textures resulting from different atomic-scale mechanisms for metamorphic crystallization display different degrees of order, clustering, intergrowth and relative isolation of porphyroblasts. Data on the sizes and locations of thousands of crystals in a three-dimensional volume are required to identify reliably the mechanisms governing nucleation and growth of porphyroblasts from these textural features. These data can now be acquired by means of high-resolution computed X-ray tomography. Numerical models that simulate porphyroblast formation governed by either interface-controlled or diffusion-controlled reaction mechanisms indicate that quantitative textural analysis can discriminate between these possibilities. These numerical models also allow a comparison between textures predicted for different crystallization mechanisms and textures measured in natural samples, from which inferences can be drawn concerning the relative importance of these mechanisms in nature. An independent test of the validity of such inferences is possible for porphyroblasts such as garnet that may preserve prograde growth zoning and allow the examination of normalized radius–rate relations.     

Towards a Higher Comparability of Geothermometric Data obtained by Raman Spectroscopy of Carbonaceous Material. Part I: Evaluation of Biasing Factors

Raman spectroscopy of carbonaceous material (RSCM) is frequently used to determine metamorphic peak temperatures from the structural order of carbonaceous material enclosed in metasediments. This method provides a quick, robust and relatively cheap geothermometer. However, the comparability of the RSCM parameter is low as there are at least three major sources of biasing factors. These sources are the spectral curve‐fitting procedure, the sample characteristics itself and the experimental design including the used Raman system. To assess the impacts of the biasing factors on RSCM, a series of experiments was performed. The experiments showed that curve‐fitting is strongly influenced by individual operator‐bias and the degrees of freedom in the model, implying the need for a standardised curve‐fitting procedure. Due to the diversity of components (optics, light detection device, gratings, etc.) and their combinations within the Raman systems, different Raman instruments generally give differing results. Consequently, to estimate comparable metamorphic temperatures from RSCM data, every Raman instrument needs its own calibration. This demands a reference material series that covers the entire temperature calibration range. Although sample heterogeneity will still induce some variation, a reference material series combined with standardised curve‐fitting procedures will significantly increase the overall comparability of RSCM data from different laboratories.