协同泛克立格法—数据处理的新方法

协同克立格法是多元地质统计学的一个重要方法，它以协同区域化变量为研究对象，充分考虑了变量的空间相关性和变量间的统计相关性，因而有着单变量克立格法无可比拟的优点。非平稳条件下的协同泛克立格法不仅能对变量进行线性、无偏、最优估计，而且能对漂移和剩余进行估计。协同泛克立格法在数据处理中有着十分乐观的发展前景。     

协同克立格法在广西林旺金矿化探数据处理中的应用

协同克立格法,同时兼具化探数据多元性及克立格法表征空间属性的特点。考虑到地质变量两个以上的空间属性,在化探数据处理中运用协同克立格法,可以进一步提高估计精度。运用协同克立格法对广西林旺矿区中金元素成矿进行预测,经相关性分析显示,在矿区中元素的基本组合是Au、Ag、As、Hg,协同克立格插值以Au为主要变量,伴生元素Ag、As、Hg为次要变量。把插值计算后的结果与传统多元统计方法和普通克立格法的计算结果进行比较,结果协同克立格法得到的估值误差较小,预测精度较高,在成矿元素的预测中具有一定程度的优越性。     

指示克立格法的理论及方法

地质、物化探数据的分布常出现一个长尾巴,这种分布特征影响了矿产储量计算及数据处理的精度.引起这种非正态分布的因素有:数据中有特异值,它的出现表明矿床具有高品位的矿化作用,这对储量计算及数据处理是很重要的;数据由若干个总体组成,即所谓的混合总体.对这种数据,普通克立格法是不稳健的.为此,本文介绍一种新的地质统计学方法——指示克立格法(非参数方法).文中讨论了(1)指示函数及其二阶矩;(2)指示半变异函数;(3)指示克立格方程组;(4)待估块段品位的指示克立格估计.最后,还介绍1个应用实例,并给出了计算方法和步骤.     

克立格法在采场品位估计中的应用

地下矿山的采场形态变化较大，且不规则，施行克立格法估值有一定的难度。本文从矿化模型的构造入手，了如何直接对地下矿山的采场实施克立格法估值，以便是要场储量，品位和估计精度。     

限制性克立格法在阿舍勒铜矿储量计算中的研究与应用

为提高矿产储量计算的精确性，本文运用了限制性克立格法。给出了限制性克立格法的计算理论，并设计出限制性克立格法在阿舍勒铜矿中的储量计算流程，实现了对阿舍勒铜矿体的限制性克立格储量计算。结果表明，限制性克立格法的储量计算结果（783654．39t）要比普通克立格法的储量计算结果（706541．53t）更接近于实际勘探值（919454t），限制性克立格储量计算方法能够取得较为理想的预测结果。     

克立格法在地下水观测网优化设计中的应用

回顾了克立格法在地下水观测网优化设计中的应用和进展,介绍了克立格法的基本原理,论述了克立格法在地下水观测网中的应用思路与实现方法,举例说明了利用克立格法优化调整河北平原地下水观测网。     

次生晕数据的对数正态泛克立格法研究及异常评价

首先介绍了对数正态泛克立格法的基本理论和方法,包括对数正态分布,三参数对数正态分布,求解估值(Z_v)~*所需的权系数λ_α.及求解漂移值(m_v)~*所需的权系数ρ_α的对数正态泛克立格方程组.其次,笔者应用对数正态泛克立格法探讨了华北某测区的次生晕数据、元素统计分布特征以及变异函数及结构分析.最后,给出测区估计结果,元素综合异常图,划分出5个异常区,并结合地质条件对异常进行了综合评价.     

克立格法在地下水数值模型中的应用

回顾了克立格法在水文地质学中的应用现状及进展，介绍了克立格法的基本原理，论述了克立格法在地下水模拟数值模型中应用的思路与实现方法，举例说明了利用克里格法插值处理含水层结构与初始水头分布。     

克立格法在几个矿床应用的体会

一、引言与回顾 克立格法总称为地质统计学。在四年的过程中,通过学习和交流,及七个矿床的实例运算,我们初步掌握了克立格法的方法原理,搞通了普通克立格法的基本问题。我们以归纳体会为主要内容,整理出此文。 二、克立格法的优点 克立格法在西方世界目前已广泛地用于储量计算和数据处理,表明克立格法有突出的优点。我们体会,这些优点至少可以概括为5点。     

归来庄金矿床化探信息金异常的协同克立格法研究

根据收集到的归来庄金矿床的化探资料,选取与Au相关系数较大的W、As、Sb等区域化探元素作为变量,对该区金异常空间分布进行协同克立格法研究.在对4个区域化变量进行空间结构分析的基础上,建立空间结构模型并进行估计.综合化探金异常的协同克立格法研究结果表明,该区有多处金异常存在,具有较好的找矿前景.     

Kriging without negative weights

Under a constant drift, the linear kriging estimator is considered as a weighted average ofn available sample values. Kriging weights are determined such that the estimator is unbiased and optimal. To meet these requirements, negative kriging weights are sometimes found. Use of negative weights can produce negative block grades, which makes no practical sense. In some applications, all kriging weights may be required to be nonnegative. In this paper, a derivation of a set of nonlinear equations with the nonnegative constraint is presented. A numerical algorithm also is developed for the solution of the new set of kriging equations.     

拓扑克里格法与普通克里格法在区域洪水频率分析中的比较研究

依据北江(珠江流域支流)流域6个水文测站年最大洪峰流量资料,分别用Top-kriging(拓扑克里格法)和普通克里格法进行区域洪水频率估计。采用均方根误差作为频率分布线型拟合优度指标。运用线性矩法进行单站洪水频率分析,确定10、50、100、1000年一遇设计洪水值。在此基础上,从Topkriging和普通克里格法设计洪水估计不确定性和相对线性矩法单站洪水频率的估计误差两个方面比较Top-kriging和普通克里格法。结果表明:(1)Top-kriging法是更好的线性无偏估计,相比普通克里格法更适合于区域洪水频率估计;(2)Top-kriging法设计洪水估计不确定性明显小于普通克里格法;(3)Top-kriging法设计洪水估计结果更接近线性矩法单站洪水频率分析结果。     

中天山乌拉斯台地区矿化异常提取及评价

以多重分形理论为基础,对中天山乌拉斯台地区铜多金属元素的岩屑测量数据,采用C-A法获得铜多金属的异常下限值,将其作为阈值进行指示克里格插值,绘制研究区的铜多金属地球化学异常图。研究显示,基于该方法获得的Cu矿化异常高值区主要集中在华力西早期第三侵入次的石英闪长岩和花岗闪长岩岩体中,受北西向和次级北东向断裂构造控制明显,该异常区可以作为寻找热液型铜多金属矿产的重要远景区。该方法对于地球化学数据空间变异性强烈的地区,较之普通克里格插值法具有更好的地球化学异常识别能力和高值信息重建能力,所得结果的最高累计频率值范围与已知矿化点的空间位置吻合度更高,在地球化学异常信息提取工作中具有推广意义。     

Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.     

Kriging in terms of projections

In the last few years, an increasing number of practical studies using so-called kriging estimation procedures have been published. Various terms, such as universal kriging, lognormal kriging, ordinary kriging, etc., are used to define different estimation procedures, leaving a certain confusion about what kriging really is. The object of this paper is to show what is the common backbone of all these estimation procedures, thus justifying the common name of kriging procedures. The word kriging (in French krigeage) is a concise and convenient term to designate the classical procedure of selecting, within agiven class of possible estimators, the estimator with a minimum estimation variance (i.e., the estimator which leads to a minimum variance of the resulting estimation error). This estimation variance can be seen as a squared distance between the unknown value and its estimator; the process of minimization of this distance can then be seen as the projection of the unknown value onto the space within which the search for an estimator is carried out.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

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.     

Simplicial Indicator Kriging

Indicator kriging (IK) is a spatial interpolation technique devised for estimating a conditional cumulative distribution function at an unsampled location. The result is a discrete approximation, and its corresponding estimated probability density function can be viewed as a composition in the simplex. This fact suggested a compositional approach to IK which, by construction, avoids all its standard drawbacks (negative predictions, not-ordered or larger than one). Here, a simple algorithm to develop the procedure is presented.     

Kriging in a global neighborhood

The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.     

Correcting the Smoothing Effect of Ordinary Kriging Estimates

The smoothing effect of ordinary kriging is a well-known dangerous effect associated with this estimation technique. Consequently kriging estimates do not reproduce both histogram and semivariogram model of sample data. A four-step procedure for correcting the smoothing effect of ordinary kriging estimates is shown to be efficient for the reproduction of histogram and semivariogram without loss of local accuracy. Furthermore, this procedure provides a unique map sharing both local and global accuracies. Ordinary kriging with a proper correction for smoothing effect can be revitalized as a reliable estimation method that allows a better use of the available information.