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.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

重磁场的变差函数特征及应用

用变差函数研究重磁场的区域变化特征.变差函数的变程反映重磁场的相干范围, 块金效应反映随机干扰, 基台值反映变异程度.重磁场的理论模拟说明: 重力场的相干范围大于磁场, 重磁场变程主要取决于场源深度, 浅源重磁场变差函数近似为球状模型或指数模型, 深源重磁场近似为连续性更好的高斯模型.磁场场源深度近似等于变程的一半, 重力场场源深度近似等于变程的四分之一.湖北大冶铁矿垂直分量磁异常具有几何各向异性, 北西-南东走向, 变差函数推测磁铁矿平均深度为250m.磁异常小波多尺度分解细节和逼近部分磁场具有协调几何各向异性, 变差函数的各阶场源深度估计结果与功率谱估计结果吻合.      

Limiting stochastic operations for stationary spatial processes

A natural extrapolation of stochastic operations (continuity and differentiation) already described in time domain (one-dimensional case) is established for spatial processes (two- or three-dimensional case). If stationarity decision is assumed, the continuity and differentiability (in the mean square sense) of a spatial process depends on the continuity and differentiability of the correlation function at the origin. Spatial processes described by stationary random functions are not continuous (in the mean square sense) when the covariance function presents a nugget effect, and they are not differentiable when the same covariance function is described by a spherical or an exponential covariance (models which are often used in geostatistics).     

Stochastic Simulation of Nonstationary Rainfall Fields, Accounting for Seasonality and Atmospheric Circulation Pattern Evolution

A model for generating daily spatial correlated rainfall fields suitable for evaluating the impacts of climate change on water resources is presented. The model, termed Stochastic Rainfall Generating Process, is designed to incorporate two major nonstationarities: changes in the frequencies of different precipitation generating mechanisms (frontal and convective), and spatial nonstationarities caused by interactions of mesoscale atmospheric patterns with topography (orographic effects). These nonstationarities are approximated as discrete sets of the time-stationary Stochastic Rainfall Generating Process, each of which represents the different spatial patterns of rainfall (including its variation with topography) associated with different atmospheric circulation patterns and times of the year (seasons). Each discrete Stochastic Rainfall Generating Process generates daily correlated rainfall fields as the product of two random fields. First, the amount of rainfall is generated by a transformed Gaussian process applying sequential Gaussian simulation. Second, the delimitation of rain and no-rain areas (intermittence process) is defined by a binary random function simulated by sequential indicator simulations. To explore its applicability, the model is tested in the Upper Guadiana Basin in Spain. The result suggests that the model provides accurate reproduction of the major spatiotemporal features of rainfall needed for hydrological modeling and water resource evaluations. The results were significantly improved by incorporating spatial drift related to orographic precipitation into the model.     

Correlations between global features of terrestrial fields

Studies of correlation coefficients between different sets of global geophysical data may lead to useful inferences concerning their relationship or independence. If one data set is allowed to rotate with respect to another, the statistical theory is complicated and extra care is required before one can conclude that there is any statistical significance to a maximized correlation coefficient. If, for some relative rotation, two spherical harmonic fields are significantly correlated, then their individual degree component harmonics of dominant power must also be significantly correlated. Rotations can be found that result in high correlations between the dominant low-degree spherical harmonics of the geomagnetic and tertestrial gravity field potentials, but rotations can also be found that result in equally high, yet meaningless, correlations if the lunar gravity field is substituted for the geomagnetic field. To explain such high correlations, the theoretical correlation distribution function between normally distributed component harmonics is derived and then verified for lowdegree harmonics by using a Monte Carlo technique which takes into account the three-dimensional rotation group. Some curious properties surface: (1)the correlation distribution function for all possible relative orientations is almost the same between identical and uncorrelated fields; and (2)a system for determining the correlation distribution function from randomly selected fields or from randomly rotated fields is almost ergodic.     

Spatial orthogonality of the principal components computed from coregionalized variables

Within the frame of the linear model of coregionalization, this paper sets up equations relating the variogram matrix of the principal components extracted from the variance-covariance matrix to the diagonal variogram matrices of the regionalized factors. The spatial orthogonality of the principal components is investigated in three situations: the intrinsic correlation, two basic structures with independent nugget components, three basic structures with independent nugget components and uncorrelated subsets of variables. Two examples point out that the correlation between the principal components may be nonnegligible at short distances, especially if the correlation structure changes according to the spatial scale considered. For one of the two case studies, an orthogonal varimax rotation of the first principal components is found to greatly reduce the spatial correlation between some of them.     

柞水银硐子特大型银多金属矿床矿物地球化学研究

本文主要讨论银硐子特大型银多金属矿床矿物地球化学特征,显微构造的矿物地球化学研究表明银,铜主要富集于不同成分的沉积纹层过渡部位、斜交层面的密集劈理区、Ｓ形,Ｘ形裂隙构造中,本矿床的成矿成晕过程可划分为三个成矿期,五个成矿阶段,泥盆纪海底热水同生沉积成矿期中,第一阶段为菱铁矿－铁白云岩热水沉积－交代成矿阶段;第二阶段为银－多金属海底热水混合同生沉各成矿阶段;第三阶段为铜、银－铜、钡海底热水同生沉积成     

Probabilistic evaluation of tunnel face stability in spatially random soils using sparse polynomial chaos expansion with global sensitivity analysis

The sparse polynomial chaos expansion is employed to perform a probabilistic analysis of the tunnel face stability in the spatially random soils. A shield tunnel under compressed air is considered which implies that the applied pressure is uniformly distributed on the tunnel face. Two sets of failure mechanisms in the context of the limit analysis theory with respect to the frictional and the purely cohesive soils are used to calculate the required face pressure. In the case of the frictional soils, the cohesion and the friction angle are modeled as two anisotropic cross-correlated lognormal random fields; for the purely cohesive soils, the cohesion and the unit weight are modeled as two anisotropic independent lognormal random fields. The influences of the spatial variability and of the cross-correlation between the cohesion and the friction angle on the probability density function of the required face pressure, on the sensitivity index and on the failure probability are discussed. The obtained results show that the spatial variability has an important influence on the probability density function as well as the failure probability, but it has a negligible impact on the Sobol’s index.     

Characteristic behavior and order relations for indicator variograms

Heuristic models for indicator variograms and their parameters (practical nugget effect and range) are proposed for a bivariate normal distribution with spherical correlogram. These models can be used conveniently as a preliminary check for bivariate normality. In the general non-Gaussian case, indicator variogram models for multiple threshold values must verify a certain number of order relations (inequalities) established directly from the properties of a general bivariate cumulative distribution function. An interesting, little-known maximum hole effect for indicator correlation is pointed out.     

中国狗头金分布特征与成因讨论

中国大部分狗头金分布在湖南、四川、黑龙江、青海与山东五省。狗头金分布与水系有密切关系。狗头金可划分为两大类成因类型:表生成矿型(物理风化、化学溶解、再生-齐汞化)与原生成矿型(原生金块与原生改造)。含金地貌与新构造运动在成矿作用中占有重要地位。     

随机介质热弹性力学模型球坐标问题的解析解

针对随机介质热弹性力学模型,在随机韦泊分布和指数分布下,提出了一种解决随机概率分布用连续函数表示的方法,并推导出该模型球坐标问题的解析解。通过算例,对随机模型与均质模型热应力的区别做了详细研究,结果表明随机概率分布参数m对随机模型的热应力有重要影响。     

直角域中凸起和孔洞对SH波的散射与地震动

按照波函数展开法和镜像方法,对直角域中半圆形凸起和圆形孔洞对SH波的散射进行了分析,得到其稳态解。对含孔洞和凸起的直角域做分区,等效为一个含孔洞与凹陷的直角域和一个圆域的契合,其在分界面上满足位移和应力的连续性条件,即契合条件,分别构造两个区域内的位移波函数,按照孔洞边界柱面上的应力自由和契合条件定解波函数展开式的系数。按Fourier级数展开法,得到定解条件的线性代数方程组,截断求解,进而得到问题的解析解。数值算例给出圆形孔洞边沿动应力和地表位移幅值的分布情况,得到直角域自由边界、凸起、孔洞对散射和地震动的影响。     

德阳须家河组四段沉积相特征和砂体分布规律

孝泉—新场—合兴场地区上三叠统须家河组四段以发育扇三角洲沉积体系为主,其砂体成因类型为扇三角洲前缘水下辫状分流河道(含砾)中—粗粒砂岩夹少量碎屑流沉积砾岩。须家河组四段可划分为1个长期、3个中期和18个短期基准面旋回层序,主要砂体合并为6套砂组。各砂组分布与由基准面变化引起的可容纳空间和沉积物供给量比值密切相关:低位体系域沉积期,基准面上升缓慢,沉积物供给(远)大于可容纳空间,沉积作用以主动进积为主,砂体不断向湖盆方向推进;湖侵体系域沉积期,基准面快速上升,沉积物供给量逐渐减少而(远)小于可容纳空间,沉积作用由进积逐渐转入加积和退积;高位体系域沉积期,基准面由缓慢上升逐渐进入到快速下降,可容纳空间由缓慢增加突变为迅速减小,而沉积物供给由小于或略等于可容纳空间逐渐变为(远)大于可容纳空间,沉积作用由弱进积、加积迅速变为强迫进积。     

Observation of Pore Spaces and Microcracks Using a Fluorescent Technique in Some Reservoir Rocks of Oil, Gas and Geothermal Fields in the Green Tuff Region, Japan

Abstract: Pore spaces and microcracks in representative oil, gas and geothermal reservoir rocks from the Green Tuff region, Japan, were examined using a fluorescent technique. This technique was developed to visualize microscopically pore spaces and microcracks filled with synthetic resin mixed with fluorescent paint under ultraviolet light. Various morphology of pore spaces and microcracks was clearly identified. Spaces in studied reservoir rocks are classified into following three types: pore spaces in matrix, pore spaces in particles, and microcracks. It is observed that valuable oil and gas reservoir rocks relatively include many pore spaces, while microcracks are important for geothermal rocks. Correlation between textural characteristics and porosity or permeability was found in the oil reservoir rocks. Effective permeability depends upon pore spaces in matrix more than upon other components such as pore spaces in particles and microcracks. Looseness in matrix caused by larger grain size of particles is strongly correlated with permeability. Pore spaces play an important role as a reservoir in oil and gas fields, but are less important in geothermal field. Instead, microcracks are important for geothermal reservoir system.     

The Fourier Integral Method: An efficient spectral method for simulation of random fields

The Fourier Integral Method (FIM) of spectral simulation, adapted to generate realizations of a random function in one, two, or three dimensions, is shown to be an efficient technique of non-conditional geostatistical simulation. The main contribution is the use of the fast Fourier transform for both numerical calculus of the density spectral function and as generator of random finite multidimensional sequences with imposed covariance. Results obtained with the FIM are compared with those obtained by other classic methods: Shinozuka and Jan Method in 1D and Turning Bands Method in 2D and 3D, the points for and against different methodologies are discussed. Moreover, with the FIM the simulation of nested structures, one of which can be a nugget effect and the simulation of both zonal and geometric anisotropy is straightforward. All steps taken to implement the FIM methodology are discussed.     

西藏墨竹工卡地区甲玛矽卡岩型铜多金属矿体的变化性

以甲玛矽卡岩型矿体为研究对象,通过对矿体主要区域化变量矿石品位半变异函数的分析,建立了相应的半变异函数的球状模型,从而求得矿体主要变化方向上的滞后距、块金值、变程等参数。根据这些参数,求得矽卡岩型矿体的铜钼矿体主元素Cu、Mo品位在各方向上的变化性质系数为0.45,表明铜钼矿体中主元素品位在走向、倾向和垂向上都具有明显的坐标性变化;而铅锌矿体中Pb在各方向上的变化性质系数为0.58,Zn变化性质系数为0.5,显示铅锌矿体中主元素品位在走向、倾向和垂向上均具有一定的随机性变化。Cu、Mo、Pb、Zn品位变化程度系数在走向、倾向方向均大于0.9,表明矿体内的矿石品位变化程度是极复杂的。Cu、Mo在垂向上的变化程度系数值均在0.6~0.8之间,变化程度中等;Pb、Zn垂向上的变化程度系数值介于0.3~0.6之间,变化程度较简单。据此认为,控制矿体变化的因素主要是由斑岩-矽卡岩型矿床的成因所决定的,矿体形成时存在多阶段的矿化叠加,使有用矿物、成矿元素具有明显的分带性。     

Complex-Valued Random Fields for Vectorial Data: Estimating and Modeling Aspects

Complex-valued random fields represent a natural extension of real-valued random fields and can be useful for modeling vectorial data in two dimensions (i.e., a wind field). In such a case, some theoretical issues arise concerning generating and fitting complex covariance functions to be used for prediction purposes. In this paper, some general aspects and properties of complex-valued random fields are summarized and a procedure to fit complex stationary covariance functions is proposed. A case study for analyzing wind speed data is presented.     

Spectral Simulation of Isotropic Gaussian Random Fields on a Sphere

A spectral algorithm is proposed to simulate an isotropic Gaussian random field on a sphere equipped with a geodesic metric. This algorithm supposes that the angular power spectrum of the covariance function is explicitly known. Direct analytic calculations are performed for exponential and linear covariance functions. In addition, three families of covariance functions are presented where the calculation of the angular power spectrum is simplified (shot-noise random fields, Yadrenko covariance functions and solutions of certain stochastic partial differential equations). Numerous illustrative examples are given.

     

New spaces,ordinary practices: Circulating and sharing within diverse economies of provisioning

This article draws upon two distinct UK case studies to explore how alternative modes of provisioning employ ordinary practices of sharing and circularity. Speaking to debates about alterity, diverse economies (Gibson-Graham, 2008) and emerging literature on the circular and shared economy, these two small and informal based models, one food based, the other clothing, are put forward as examples of the vast array of contemporary ‘alternative’ forms of consumption and provisioning taking place across the UK. The article illuminates how diverse economies are ‘made material’ through their materials and practices. In doing so I make three key arguments: firstly, and overall, that studying materiality is one way to illuminate these new and emerging spaces of provisioning, highlighting their practices, intimacies and ambiguities. Secondly, this material focus illustrates how the practices of provisioning – in particular, sharing and circulating - are not new, but are instead organised in original and novel ways; and this has wider implications for contemporary debates on circular and shared economy. Thirdly, that the materials of provisioning can be both beneficial and troublesome to provisioning organisations’ practices of circulating and sharing and the extent to which they tackle issues of social exclusion, financial hardship and sustainable resource use.