Comparing the robustness of ordinary kriging and lognormal kriging: Outlier resistance

Ordinary kriging is well-known to be optimal when the data have a multivariate normal distribution (and if the variogram is known), whereas lognormal kriging presupposes the multivariate lognormality of the data. But in practice, real data never entirely satisfy these assumptions. In this article, the sensitivity of these two kriging estimators to departures from these assumptions and in particular, their resistance to outliers is considered. An outlier effect index designed to assess the effect of a single outlier on both estimators is proposed, which can be extended to other types of estimators. Although lognormal kriging is sensitive to slight variations in the sill of the variogram of the logs (i.e., their variance), it is not influenced by the estimate of the mean of the logs.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Robustness of variograms and conditioning of kriging matrices

Current ideas of robustness in geostatistics concentrate upon estimation of the experimental variogram. However, predictive algorithms can be very sensitive to small perturbations in data or in the variogram model as well. To quantify this notion of robustness, nearness of variogram models is defined. Closeness of two variogram models is reflected in the sensitivity of their corresponding kriging estimators. The condition number of kriging matrices is shown to play a central role. Various examples are given. The ideas are used to analyze more complex universal kriging systems.Research performed while on leave at Centre de Geóstatistique et de Morphologie Mathématique, Fontainebleau.     

Spatial prediction and ordinary kriging

Suppose data {Z(s _i ):i=1, ..., n} are observed at spatial locations {s _i :i=1, ..., n}. From these data, an unknownZ(s ₀) is to be predicted at a known locations ₀c, or, ifZ(s₀) has a component of measurement error, then a smooth versionS(s ₀) should be predicted. This article considers the assumptions needed to carry out the spatial prediction using ordinary kriging, and looks at how nugget effect, range, and sill of the variogram affect the predictor. It is concluded that certain commonly held interpretations of these variogram parameters should be modified.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

含水层渗透性空间分布的指示克立格估值

详细介绍了指示克立格估值计算的理论和方法。以指示变异函数为基本工具分析了华北某地区第四系含水层渗透性空间分布的结构特征,结果表明该地区含水层渗透性存在明显的各向异性特征。水平方向上,X轴方向的相关性较Y轴方向的好,Z轴的相关性最差。用指示克立格法对未采样点处进行估值,估值结果显示含水层渗透性由山前向滨海逐渐变低,在垂直方向上,渗透性变化不明显,浅部比深部略好;同时给出了估计精度,并认为对估计精度不高的区域可通过增加适当的工程加以控制。最后用交叉验证法对估值结果进行了检验,证明建立的指示变异函数模型合理且估值效果较好。这一实际应用表明指示克立格法可以很好地描述第四系含水层渗透性的空间分布规律。     

Robust estimation of the variogram: I

It is a matter of common experience that ore values often do not follow the normal (or lognormal) distributions assumed for them, but, instead, follow some other heavier-tailed distribution. In this paper we discuss the robust estimation of the variogram when the distribution is normal-like in the central region but heavier than normal in the tails. It is shown that the use of a fourth-root transformation with or without the use of M-estimation yields stable robust estimates of the variogram.Visiting Scientist, NRIMS, during the period in which this work was carried out.     

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.     

Comparison between different kriging estimators

Six different geostatistical estimators (linear kriging, lognormal kriging, and disjunctive kriging, each with and without a nonbias, i.e., universality condition) were compared using data from a polymetallic deposit in Algeria. The differences between estimators with and without the nonbias condition were far more pronounced than between the different kriging methods. This highlights the importance of choosing an appropriate stationarity model for the data. The criterion concerning kriging weight of the mean in simple kriging, proposed by Remacre (1984, 1987) and Rivoirard (1984) was found to be helpful for determining blocks where the choice of the stationarity hypothesis was critical.     

On unbiased backtransform of lognormal kriging estimates

Lognormal kriging is an estimation technique that was devised for handling highly skewed data distributions. This technique takes advantage of a logarithmic transformation that reduces the data variance. However, backtransformed lognormal kriging estimates are biased because the nonbias term is totally dependent on a semivariogram model. This paper proposes a new approach for backtransforming lognormal kriging estimates that not only presents none of the problems reported in the literature but also reproduces the sample histogram and, consequently, the sample mean.     

Application of spatial filter theory to kriging

Data-processing requirements for remotely sensed, digital images include spatial filtering to suppress image noise, enhance edges/contacts, and improve image clarity. Spatial filter theory demonstrates that the addition of a high-pass filtered image to a low-pass filtered image yields the original digital image. Application of this principle in kriging can be accomplished by using the same covariance matrix to solve for two weighting vectors to yield a result analogous to low- and high-pass filtering. The addition of kriged estimates calculated using both weighting vectors is analogous to summing high-, and low-pass filtered digital images. This modified method of kriging yields estimates associated with less smoothing compared to ordinary kriging. Statistical moments of original sample data are better preserved through estimation by this method.     

The combination of sampling and kriging in the regional estimation of coal resources

Two-dimensional systematic sampling of small plots followed by the kriging of those plots may be employed to obtain regional estimates of coal resources and measures of the accuracy of the estimates. The use of sampling makes large savings in computation possible. Two case studies involving the estimation of coal tonnage are discussed.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

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.     

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.     

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.     

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.     

Restricted kriging for mixture of grade models

A modified type of kriging, referred to as restricted kriging (RK), is proposed in this study. The method incorporates constraints on different grade classes to restrict the influence of the samples having different likelihoods in estimation. RK is motivated by the estimation of mineral reserves when grades have highly skewed distributions. Ordinary kriging tends to produce an overly smoothed interpolated surface by underestimating high grades and overestimating low grades. The fact that ordinary kriging gives a uniform prior treatment to all samples independent of their values is a major factor associated with this smoothing effect. The new approach differentiates each grade portion by preselected cutoffs. RK is developed for a single cutoff and then extended into a general form for any finite number of cutoffs. Restricted cokriging (RCK) is also formulated to simultaneously estimate a set of random functions with restriction conditions. Methods are suggested for determination of the probabilities of occurrence of different grade portions. Finally, the new approach is demonstrated on a case study of an epithermal gold deposit.     

Compensating for estimation smoothing in kriging

Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft² slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging.     

A comparison between cokriging and ordinary kriging: Case study with a polymetallic deposit

Cokriging is applied to the estimation of mineral resources in a polymetallic deposit. Several major steps, which should be taken in using cokriging, are highlighted as necessary practical considerations. The case study is related to an ultramafic copper-nickel deposit. Six elements, Cu, Ni, Au, Ag, Pt, and Pd, occurring in the deposit, are partitioned into three subgroups and the elements within each group are simultaneously estimated on the basis of over 4000 drill assays. A comparison was made between ordinary kriging and cokriging methods through cross-validation. The results show that cokriging has significantly improved the estimates of resources by reducing the overall estimation error by over 15% and the variance of error by over 20%.     

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.     

地质统计学方法在地下水水位估值中应用

对于许多区域水资源或水环境问题,地下水水流模拟往往要采用数值方法,需给出每个节点上初始水位值,以反映流场的初始状态。另外,地下水水位动态长期监测分析,需由观测点水位估计任一点的水位。文中阐述了地下水水位估值的地质统计学方法－泛克立格法原理,以河南省焦作市修武段地下水数值模拟分析区为例,分析了用一次、二次漂移的泛克立格方法模拟地下水初始流场的估值情况和对真实流场特征的反映情况。指出在进行区域地下水位