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.     

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.     

Production of conditional simulations via the LU triangular decomposition of the covariance matrix

This paper reviews the turning band method and fast Fourier transform method of producing a nonconditional simulation of a multinormal random function with a given covariance structure. A review of the two common methods of conditioning the simulation to honor the data shows that they are formally equivalent. Another method for directly pondering a conditional simulation based on the LU triangular decomposition of the covariance matrix is presented. Computational and implementation difficulties are discussed.     

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.     

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.     

On the efficiency of the indicator approach in geostatistics

A common problem in geostatistics is to determine whether or not the value of a random field at an unsampled location exceeds a specified threshold using observed values of the random field at sampled locations. Under the indicator approach, the only information used to classify the unobserved value is whether or not the observed values exceed the threshold. This note shows that the loss of information from applying the indicator approach may be modest in the case where the underlying random field is Gaussian.     

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.     

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.     

Uncertainty in Coal Property Valuation in West Virginia: A Case Study

Interpolated grids of coal bed thickness are being considered for use in a proposed method for taxation of coal in the state of West Virginia (United States). To assess the origin and magnitude of possible inaccuracies in calculated coal tonnage, we used conditional simulation to generate equiprobable realizations of net coal thickness for two coals on a 7 min topographic quadrangle, and a third coal in a second quadrangle. Coals differed in average thickness and proportion of original coal that had been removed by erosion; all three coals crop out in the study area. Coal tonnage was calculated for each realization and for each interpolated grid for actual and artificial property parcels, and differences were summarized as graphs of percent difference between tonnage calculated from the grid and average tonnage from simulations. Coal in individual parcels was considered minable for valuation purposes if average thickness in each parcel exceeded 30 inches. Results of this study show that over 75% of the parcels are classified correctly as minable or unminable based on interpolation grids of coal bed thickness. Although between 80 and 90% of the tonnages differ by less than 20% between interpolated values and simulated values, a nonlinear conditional bias might exist in estimation of coal tonnage from interpolated thickness, such that tonnage is underestimated where coal is thin, and overestimated where coal is thick. The largest percent differences occur for parcels that are small in area, although because of the small quantities of coal in question, bias is small on an absolute scale for these parcels. For a given parcel size, maximum apparent overestimation of coal tonnage occurs in parcels with an average coal bed thickness near the minable cutoff of 30 in. Conditional bias in tonnage for parcels having a coal thickness exceeding the cutoff by 10 in. or more is constant for two of the three coals studied, and increases slightly with average thickness for the third coal.     

Estimation of potential failure risks in a mine slope using indicator kriging

Indicator kriging has been applied to the study of failure mechanisms in a mine slope in Minas Gerais, Brazil, to estimate potential failure risks in limited areas along this slope. Timbopeba Mine, Vale Company, is an open pit iron mine situated in the Quadrilátero Ferrífero, a very important mining district in Minas Gerais. A slope excavated in quartzite with a maximum height of 200 m at the time of this study, has presented many failure problems involving the sliding of blocks formed by discontinuities. These blocks are of limited size in comparison to the dimensions of the overall slope. They appear along the entire slope, wherever discontinuity orientations have led to the kinematic feasibility of these blocks. Geostatistics permits the estimation of local failure probability distributions associated to these local failures, which would not be possible with traditional statistical models. The geostatistical method employed in this study, indicator kriging, is quite suitable because it is unnecessary to assume a particular global distribution of the phenomena being modeled. The model was used for locating areas with a great tendency for sliding failure, as it considers the local spatial variability of discontinuity orientations. Copyright © 2010 John Wiley & Sons, Ltd.     

Posterior identification of histograms conditional to local data

Stochastic simulation techniques which do not depend on a back transform step to reproduce a prior marginal cumulative distribution function (cdf)may lead to deviations from that distribution which are deemed unacceptable. This paper presents an algorithm to post process simulated realizations or any spatial distribution to reproduce the target cdfin the case of continuous variables or target proportions in the case of categorical variables, yet honoring the conditioning data. Validations conducted for both continuous and categorical cases show that. by adjusting the value of a correction level parameter , the target cdfor proportions can be well reproduced without significant modification of the spatial correlation patterns of the original simulated realizations.     

Sampling variation of principal components

The convenience of reducing the dimension of a data matrix by principal component analysis invites substantive interpretation of the coefficients of the components. To test the consistency of component coefficients, 10 samples of approximately 25, 50, 100, and 200 items each were randomly drawn, with replacement, from a source sample consisting of 2086 subalkaline asalt analyses. From each sample principal components were calculated using 9 major oxides as variables. Although the eigenvalues are remarkably consistent, both across and within sample size groups, the coefficients of the eigenvectors are subject to considerable sample variance. It is sometimes assumed that the coefficients of the components calculated from small samples are well enough known to be used in detailed petrological interpretation. Our results indicate that the validity of this assumption should be tested in each specific research even when rather large samples are used. The testing procedure used here is suitable of a sufficiently large reservoir of sample items is available; in the absence of such a reservoir complete simulation could be used.     

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.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

Geostatistical regionalization of glacial aquitard thickness in northwestern Germany, based on fuzzy kriging

In many instances hydrogeological parameters obtained by conventional methods for selected localities within an aquifer or an aquitard are not sufficient for adequate regionalization at the scale of the entire layer. Here, we demonstrate an application of the fuzzy kriging method in regionalization of hydrogeological data, in which the set of conventional, crisp values is supplemented by imprecise information subjectively estimated by an expert. It is believed that such an approach eventually may reflect the real-world conditions more closely than a traditional crisp-value approach, because the former does not impose exactness artificially on phenomena which are diffuse by their nature. Spatial interpolation was done for the thickness of one of the major aquitards (till and glaciolacustrine clay) in northwestern Germany. The dataset consists of 329 crisp values from boreholes supplemented by 172 imprecise values defined as fuzzy numbers. It is demonstrated that the reliability of regionalization was higher, compared to regionalization performed with the crisp dataset only. Fuzzy kriging was performed with FUZZEKS (Fuzzy Evaluation and Kriging System) developed at the Ecosystem Research Center at the University of Kiel.     

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。     

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...     

Comparison of two techniques for applying disjunctive kriging: The Gaussian anamorphosis model versus the direct statistical inference of the bivariate distributions

Indicator cokriging is an alternative to disjunctive kriging for estimation of spatial distributions. One way to determine which of these techniques is more accurate for estimation of spatial distributions is to apply each to a particular type of data. A procedure is developed for evaluation of disjunctive kriging and indicator cokriging for such an application. Application of this procedure to earthquake ground motion data found disjunctive kriging to be at least as accurate as indicator cokriging for estimation of spatial distributions of peak horizontal acceleration. Indicator cokriging was superior for all other types of earthquake ground motion data.     

Determination of support in soil sampling

Short-range variation of measurements of a soil characteristic is a function of the support (i.e., size, shape, and orientation of the physical sample taken at a sample point) of the soil samples. This short-range variance often shows up as a substantial component of the nugget effect in experimental semivariograms or estimated generalized covariance functions. Proper choice of support may substantially reduce short-range variance (or nugget) of measurements made at sample points and thereby reduce estimation variance in point or block kriging based on these measurements. Appropriate variance formulas and an example are given for determination of support when support is an array of vertical cores (or bucket auger samples) of equal depth and diameter.