Minimum-variance unbiased quadratic estimation of covariances of regionalized variables

The parameters of covariance functions (or variograms) of regionalized variables must be determined before linear unbiased estimation can be applied. This work examines the problem of minimum-variance unbiased quadratic estimation of the parameters of ordinary or generalized covariance functions of regionalized variables. Attention is limited to covariance functions that are linear in the parameters and the normality assumption is invoked when fourth moments of the data need to be calculated. The main contributions of this work are (1) it shows when and in what sense minimum-variance unbiased quadratic estimation can be achieved, and (2) it yields a well-founded, practicable, and easy-to-automate methodology for the estimation of parameters of covariance functions. Results of simulation studies are very encouraging.     

A nonparametric view of generalized covariances for kriging

Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data.     

Conditioning of the stationary kriging matrices for some well-known covariance models

In this paper, the condition number of the stationary kriging matrix is studied for some well-known covariance models. Indeed, the robustness of the kriging weights is strongly affected by this measure. Such an analysis can justify the choice of a covariance function among other admissible models which could fit a given experimental covariance equally well.     

Nonparametric estimation of generalized covariances by modeling on-line data

Structural analysis of data displaying trends may be performed with the help of generalized increments, the variance of these increments being a function of a generalized covariance. Generalized covariances are estimated primarily by parametric methods (i. e., methods searching for the best coefficients of a predetermined function), but also may be computed by one known nonparametric alternative. In this paper, a new nonparametric method is proposed. It is founded on the following principles: (1) least-squares residues are generalized increments; and (2) the generalized covariance is not a unique function, but a family of functions (the system is indeterminate). The method is presented in a general context of a k order trend in R^d, although the full solution is given only fork = I in Rⁱ. In Rⁱ, higher order trends may be developed easily with the equations included in this paper. For higher dimensions in space, the problem is more complex, but a research approach is proposed. The method is tested on soil pH data and compared to a parametric and nonparametric method.     

Consolidation analysis of non-homogeneous soil by the weak form quadrature element method

A weak form quadrature element formulation for consolidation analysis of non-homogeneous saturated soil is established based on Biot’s theory. Numerical examples are given, and the results are compared with the analytical solutions available or those from the commercial finite element software ABAQUS, demonstrating accuracy and rapid convergence of quadrature element solutions. The disparity between various treatments of non-homogenous soil is discussed, and the effectiveness and advantages of the quadrature element formulation in consolidation analysis of non-homogeneous soil are demonstrated.     

阿拉善地块性质和归属的再认识

阿拉善地块被认为是一个前寒武纪微陆块,其归属和演化历史长期存在争议。一部分学者认为其为华北克拉通的组成部分,是华北克拉通的西延;一部分学者认为与扬子或塔里木克拉通具有亲缘性;还有一些学者认为具有独立的地质演化历史。本文综合近年来阿拉善地块的研究进展,结合我们新的研究资料,梳理了阿拉善地块自新太古代到显生宙的岩石组成、构造热事件及年代格架,得出以下主要认识:1)阿拉善地块的早前寒武纪变质基底记录了2.7~3.0Ga陆壳生长、~2.5Ga TTG岩浆-变质事件,2.0~2.3Ga岩浆事件,以及1.9~1.95Ga和1.80~1.85Ga两期重要的变质事件,具有与华北克拉通相似的岩石组成和地壳演化历史;2)阿拉善地块发育与华北克拉通相似的中-新元古代沉积岩系,其碎屑锆石具有来源华北克拉通本身的物源区特征,反映出阿拉善地块直到中元古代-新元古代早期仍可能是华北克拉通的一部分;3)阿拉善地块显生宙的多期岩浆-构造-变质事件可能主要是中亚造山带早古生代以来造山作用的响应,而不是阿拉善地块与华北克拉通碰撞的产物。从奥陶纪到早二叠世,阿拉善地块可能处在古亚洲洋向南俯冲的活动大陆边缘环境;4)河西走廊-贺兰山南部早古生代沉积岩系(大黄山群和香山群)不属于阿拉善地块,而是祁连造山带的组成部分,其碎屑锆石组成和代表的物源特征不能作为阿拉善地块与华北克拉通在早古生代彼此分离的证据。     

连续正交地面磁测用于评价含油性

分析了石油勘探中高精度磁测未能获得高品质资料的原因,认为高精度资料与高品质资料并非等同,进而提出连续正交地面磁测观测技术。指出连续的磁力观测能够增大信息密度,避免因抽样引起品质降低;正交观测可以较好地控制垂直地质构造走向或称顺测线方向有意义的局部异常,消除由于测量原因导致的顺测线干扰,同时通过交点平差可提高观测精度。利用剖面与平面数据的抽样与统计证实了方法的有效性,并引用一个失败的例子说明所提出方法的必要性。认为现阶段开展连续正交地面磁测已具备了仪器与技术基础,具有可行性。     

斜轴高斯投影在跨带物探长剖面测量中的应用

在高精度物探重力测量中,需要提供高精度的平面位置和高程。在物探长剖面测量中,对东西跨度较大的线路进行控制测量时,高斯-克吕格投影分带和高程引起的投影变形较大,不能满足长距离重力测量对精度的要求。基于高斯-克吕格投影的基本理论,采用斜轴变形椭球高斯投影方法,结合最小二乘法、坐标转换理论及椭球变换,将原始椭球构建斜轴变形椭球,可以减小高斯投影横坐标和高程投影变形的影响,避免高斯投影分带过多对应用的影响。以漠河—呼和浩特物探长剖面测地数据为例,利用GPS快速静态测量获得平面和高程位置,测点距离约1km,通过斜轴高斯投影进行投影,最大平面精度为67.87mm/km,最大高程精度为53.039mm/km,最大投影综合变形的中误差为88.51 mm/km,大大减小了投影变形,提高了地图投影精度。因此,该投影在跨度物探长剖面测量中的应用具有一定优势。     

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.     

以高斯理论为基础计算弯曲钻孔节点空间坐标

为了拟合出更加接近钻孔实际的曲线,结合钻孔测斜实际,以严密的高斯理论为基础,给出了以钻孔天顶角和倾角为参数的弯曲钻孔的参数方程。并在此基础上,推出了钻孔曲线各点的计算公式,更准确地描述了钻孔的实际情况,为后续工作剖面图的绘制和储量计算提供了基础。这里使用高斯和折线法作为一个实例,对比计算了赤土店铅锌矿区ZK13002钻孔。计算结果表明:在钻孔曲线形状、累计弧长、插值点特征等多个方面的高斯法都优于折线法,而因算法的改变引起的工作量又是很小的。     

Spatial analyses of groundwater levels using universal kriging

For water levels, generally a non-stationary variable, the technique of universal kriging is applied in preference to ordinary kriging as the interpolation method. Each set of data in every sector can fit different empirical semivariogram models since they have different spatial structures. These models can be classified as circular, spherical, tetraspherical, pentaspherical, exponential, gaussian, rational quadratic, hole effect, K-bessel, J-bessel and stable. This study aims to determine which of these empirical semivariogram models will be best matched with the experimental models obtained from groundwater-table values collected from Mustafakemalpasa left bank irrigation scheme in 2002. The model having the least error was selected by comparing the observed water-table values with the values predicted by empirical semivariogram models. It was determined that the rational quadratic empirical semivariogram model is the best fitted model for the studied irrigation area.     

Estimation of proportions of lithotypes using auxiliary information

The problem of estimating the proportion of lithological (ore) types in volumes of mineral deposits is considered when an auxiliary variable Z(z) is observed along with the lithology L(x) at x. The process {Z(x)} is assumed to be stationary and an assumption on the statistical relationship between {Z(x)} and {L(x)} processes is made. Consequently, the second-order stationarity of {L(x)} is not required. A method using the kriged distribution of the auxiliary variable is proposed. A case study illustrating the procedure is given.     

不同采样点数量下土壤有机质含量空间预测方法对比

运用普通克里格、泛克里格、协同克里格和回归克里格4种方法,结合由DEM获取的高程因子以及土壤全氮和阳离子交换量（CEC）,预测了黑龙江省海伦市耕地有机质含量的空间分布。不同样点数量下海伦市土壤有机质含量的空间变异结构分析表明,样点数量多并不一定能够识别土壤有机质含量的结构性连续组分,最优化的布置采样点位置可能比单纯增加...     

Solving the kriging problem by using the Gram-Schmidt orthogonalization

This paper presents an approach to solve the kriging problem, defined in terms of projections, by using Gram-Schmidt orthogonalization. The Gram-Schmidt orthogonalization allows us to find an optimal approximationY* in then-dimensional subspaceH _n of any vectorY element of a Hilbert spaceH. This approach requiresO(n ³) multiplication operations to obtain an orthogonal basis, andO(n ²) multiplications needed to calculate kriging solution for a given point.     

Definition of geochemical patterns on the soil of Paul de Arzila using correspondence analysis

The aim of this study is to define geochemical and geological patterns, not only inside of Paul de Arzila natural reserve but in the surrounding area. The Correspondence Analysis was used as a discriminate technique to separate previously identified groups, in this case three types of soils (cambisols + luvisols + fluvisols). The purpose was to identify the main chemical association representative for the local types of soils. The results allow us to separate the chemical elements that are related to natural sources from the anthropogenic sources.     

Geostatistical analysis of arsenic concentration in groundwater in Bangladesh using disjunctive kriging

The National Hydrochemical Survey of Bangladesh sampled the water from 3,534 tube wells for arsenic throughout most of Bangladesh. It showed that 27% of the shallow tube wells (less than 150 m deep) and 1% of the deep tube wells (more than 150 m deep) exceeded the Bangladesh standard for arsenic in drinking water (50 µg L^–1). Statistical analyses revealed the main characteristics of the arsenic distribution. Concentrations ranged from less than the detection limit (0.5 µg L^–1), to as much as 1,600 µg L^–1, though with a very skewed distribution, and with spatial dependence extending to some 180 km. Disjunctive kriging was used to estimate concentrations of arsenic in the shallow ground water and to map the probability that the national limit for arsenic in drinking water was exceeded for most of the country (the Chittagong Hill Tracts and the southern coastal region were excluded). A clear regional pattern was identified, with large probabilities in the south of the country and small probabilities in much of the north including the Pleistocene Tracts. Using these probabilities, it was estimated that approximately 35 million people are exposed to arsenic concentrations in groundwater exceeding 50 µg L^–1 and 57 million people are exposed to concentrations exceeding 10 µg L^–1 (the WHO guideline value).     

Geostatistical case studies of the advantages of lognormal-de Wijsian kriging with mean for a base metal mine and a gold mine

Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.