泛克立格法和指示克立格法在地球化学探矿中的应用

泛克立格法是一种非平稳随机函数的最佳线性无偏估计方法，作者将之用于处理区域地球化学探矿数据，给出被测元素的估计值，漂移植和涨落值，后者为评估元素区域北景和异常特性提供了有用信息，作者用非参数地质统计学的指示克立格法对化探元素含量进行异值的分析及大于各级下限值的概率估计。     

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.     

Rock Kriging with the Microscope

An idea to consider rock textures from a geostatistical viewpoint is suggested. Mineral grains are coded by indicator functions. Four metrics are shown of interest for petrographic applications. The simplest one is used to calculate covariograms of indicators for platinum-bearing gabbronorite from the Pansky rock massif (Kola Peninsula, Russia) with maximal range of 2 units. This is generalized in the concept of a minimal cluster of mineral grains for the given rock. The theory allows us to combine grain-by-grain and cluster-by-cluster considerations of rock texture. It may be used to classify monotonous lithological series using nuances of rock textures.     

Compositional Kriging: A Spatial Interpolation Method for Compositional Data

Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable.     

Efficient updating of kriging estimates and variances

This short note presents a method for efficiently updating ordinary kriging estimates and variances when one or more additional samples are incorporated into the kriging system. First, the foundation linear algebra result is presented. Then the update equations are derived. Finally, an illustrative application of updating is briefly discussed.     

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.     

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.     

Adding bounds to kriging

The ordinary kriging interpolation algorithm is extended by the inclusion of explicit lower and upper bounds on the estimate. The associated estimation variance is written as the ordinary kriging variance plus a non-negative correction term.     

Robustness of Kriging weights to non-bias conditions

In this paper, the effect on Kriging weights of non-bias conditions, when the same residual covariance model is used, has been studied by the l₂ norm of the weights difference between Ordinary Kriging and Kriging with a trend model. Four covariance models, in 1-D and 2-D, and in interpolation and extrapolation conditions are examined. Situations in which both algorithms yield the same results are pointed out.     

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 a finite domain

Adopting a random function model {Z(u),u study areaA} and using the normal equations (kriging) for estimation amounts to assume that the study areaA is embedded within a infinite domain. At first glance, this assumption has no inherent limitations since all locations outsideA are of no interest and simply not considered. However, there is an interesting and practically important consequence that is reflected in the kriging weights assigned to data contiguously aligned along finite strings; the weights assigned to the end points of a string are large since the end points inform the infinite half-space beyond the string. These large weights are inappropriate when the finite string has been created by either stratigraphic/geological limits or a finite search neighborhood. This problem will be demonstrated with numerical examples and some partial solutions will be proposed.     

A geostatistical approach to predicting the washability characteristics of coal in situ

The washability characteristics of coal are dependent on two basic relations: the ash assay vs. relative density curve, and the distribution by weight of the relative densities of coal particles. Armstrong and Whitmore (1980) demonstrated that the ash content and the yield of coal floating at a particular density can be predicted with reasonable accuracy using a simple inverse proportionality relation for the ash assay vs. density curve and a lognormal model for the distribution. In this paper, geostatistical techniques are used in conjunction with the two models to predict the washability characteristics of coal in situ.     

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.     

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.     

Problems in space-time kriging of geohydrological data

Spatiotemporal variables constitute a large class of geohydrological phenomena. Estimation of these variables requires the extension of geostatistical tools into the space-time domain. Before applying these techniques to space-time data, a number of important problems must be addressed. These problems can be grouped into four general categories: (1) fundamental differences with respect to spatial problems, (2) data characteristics, (3) structural analysis including valid models, and (4) space-time kriging. Adequate consideration of these problems leads to more appropriate estimation techniques for spatiotemporal data.     

Some limitations in the geostatistical evaluation of ore deposits

Conclusions The foregoing discussion indicates that geostatistical estimation of ore deposits is not local; it is not objective; it is not sensitive to local data trends; and it is not unrestrained by the range of data values.Kriging, as an interpolation method, is a variant of IDW least squares linear fit. As such, it suffers from the limitations of all IDW linear interpolation methods that employ only data values.The estimation variance, currently used to calculate the confidence limits of values for individual mining blocks, is hypothetical and globally derived. It is more closely related to sampling density than to local variation in the data set.Geostatistical methods, of course, have a real place in ore deposit assessment, e.g. global, comparative evaluation to assist decisions on development and investment. What is questioned here is the validity of employing a global method to assess detail (mining blocks) within an ore deposit.     

Global estimation variance: Formulas and calculation

This paper discusses the combination of kriging variances, which have been considered heretofor unfeasible since linearity of the problem and considerable simplifications which follow were overlooked. A simplified expression for global estimation variance is presented and an algorithm discussed with respect to precision and computer cost. A case study is presented, and, finally, an optimum calculation method is recommended.