Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

稳健协同克立格因子分析及其在化探中的应用

稳健协同克立格因子分析集稳健统计、协同克立格和因子分析的优点于一体,可同时研究多个变量不同方向的结构差异及变化性,用该方法研究次生晕指示的结构特征可在一定程度反映原生晕的特征,揭示其深层次信息．对团结沟金矿区次生晕分析表明：在矿产预测中矿床原生晕组合标志并不适用于次生晕及分散流的研究,本方法对大（中）比例尺次生晕、分散流与原生晕的信息转换和关联有独到之处．     

Comparison between ordinary kriging (OK) and inverse distance weighted (IDW) based on estimation error. Case study: Dardevey iron ore deposit,NE Iran

Estimation of mineral resources and reserves with low values of error is essential in mineral exploration. The aim of this study is to compare inverse distance weighted (IDW) and ordinary kriging (OK) methods based on error estimation in the Dardevey iron ore deposit, NE Iran. Anisotropic ellipsoid and variograms were calculated and generated for estimation of Fe distribution by both methods. Density, continuity of ore and waste, the number of points involved, and the discretization factor in the estimation of ore and waste boundaries were determined and the resource estimated by IDW and OK methods. Estimation errors were classified based on JORC standard, and both methods were compared due to distribution of error estimation. Results obtained by the study indicate that error estimation of OK method is less than IDW method and that the results of OK method are reliable.     

Extending Multivariate Space-Time Geostatistics for Environmental Data Analysis

Environmental studies require multivariate data such as chemical concentrations with space-time coordinates. There are two general conditions related to such data: the existence of correlations among the coregionalized variables and the differences in numbers of data which occur because of insufficient data caused by measurement error or bad weather conditions. This study proposes geostatistical techniques for space-time multivariate modeling that take into consideration these correlations and data absences. These techniques consist of suitable modeling of semivariograms and cross-semivariograms for quantifying correlation structures among multivariables and of extending standardized ordinary cokriging. The tensor product cubic smoothing surface method is used for space-time semivariogram modeling. These methods are applied to the chemical component data of the Ariake Sea, a typical closed sea in southwest Japan. In order to clarify environmental changes in the Ariake Sea, the concentration data of four nutritive salts (NO₂–N, NO₃–N, NH₄–N, and PO₄–P) at 38 stations over 25 years are used as environmental indicators. For each of the kinds of data, there are spaces and times for which there is no data available. The effectiveness of the modeling of space-time semivariograms and the high estimation capability of the extended cokriging are demonstrated by cross-validation. Compared with ordinary kriging for a single variable, multivariate space-time standardized ordinary cokriging can provide a more detailed concentration map of nutritive salts and while elucidating their temporal changes over sparsely spaced data areas. In the space-time models by ordinary kriging, on the other hand, smooth trends are obvious.     

Multivariable spatial prediction

For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.     

Ordinary Cokriging Revisited

This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation.     

Ordinary Cokriging Revisited

This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation.     

河南偃龙煤田深部铝土等矿种的相变关系及资源量估算方法

偃龙煤田深部铝土矿矿床是豫西典型的隐伏铝土矿床,通过该沉积型矿产铝土矿、黏土矿、铁矾土、菱铁矿等诸矿产的含矿岩系特征认识,对该矿床的垂向上和纵向上的岩相相变特征和元素含量变化特征进行深入的分析和研究,元素含量变化与岩相相变关系密切,矿种之间存在相变过渡关系的特征.根据该认识,对沉积型铝土矿的诸矿种资源量估算方法提出了科学合理的认识.     

Successive Nonparametric Estimation of Conditional Distributions

Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities.     

新疆哈密野马泉西金矿区资源量数据集

康古尔塔格韧性剪切带是金矿重要控矿带,产有康古尔、马头滩、石英滩等重要矿床。野马泉西金矿即位于该韧性剪切带的东延苦水-雅满苏韧性剪切带上,属韧性剪切带控矿型金矿。本文在野外地质调查基础上采集了野马泉西金矿探矿工程坐标、工程内样品的分析测试成果等数据,采用地质块段法对野马泉西金矿资源量进行了估算,形成了数据集。数据集测试样品均委托有国家甲级资质的测试中心进行,内外检合格率90%以上,数据质量可靠,对周边同类型矿床的资源量估算具参考意义。     

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.     

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.     

贵州修文小山坝铝土矿中镓等伴生元素分布规律研究

通过ICP-MS等分析方法对贵州小山坝铝土矿中伴生元素的研究,发现该矿床TiO2、Ga、Ge、Sc、Nb、Li和REE等有用伴生元素在铝土矿矿体中均有较高富集,其含量大多接近或超过了工业综合利用标准,具有较大的经济价值,特别是矿区富铁铝土矿石中Ga和Sc等含量异常高.贵州铝土矿资源丰富,其伴生元素在铝土矿冶炼过程中所产生的赤泥中更为富集,因此,贵州铝土矿综合利用前景十分巨大,这些元素潜在的经济价值,在某种程度上,甚至会超过主导产品氧化铝的价值.开展铝土矿中镓等伴生有用元素的分布及赋存状态的研究,对于镓等稀散元素在特殊地质地球化学环境中超常富集机制、矿产资源综合价值的重新认识评估和矿产资源的综合利用将十分重要,不仅可为铝工业注入新的活力,增加新的经济增长点,而且还可以解决大量赤泥堆放和利用问题,同时获得较大的社会效益与环境效益.     

Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization

 Spatial and temporal behavior of hydrochemical parameters in groundwater can be studied using tools provided by geostatistics. The cross-variogram can be used to measure the spatial increments between observations at two given times as a function of distance (spatial structure). Taking into account the existence of such a spatial structure, two different data sets (sampled at two different times), representing concentrations of the same hydrochemical parameter, can be analyzed by cokriging in order to reduce the uncertainty of the estimation. In particular, if one of the two data sets is a subset of the other (that is, an undersampled set), cokriging allows us to study the spatial distribution of the hydrochemical parameter at that time, while also considering the statistical characteristics of the full data set established at a different time. This paper presents an application of cokriging by using temporal subsets to study the spatial distribution of nitrate concentration in the aquifer of the Lucca Plain, central Italy. Three data sets of nitrate concentration in groundwater were collected during three different periods in 1991. The first set was from 47 wells, but the second and the third are undersampled and represent 28 and 27 wells, respectively. Comparing the result of cokriging with ordinary kriging showed an improvement of the uncertainty in terms of reducing the estimation variance. The application of cokriging to the undersampled data sets reduced the uncertainty in estimating nitrate concentration and at the same time decreased the cost of the field sampling and laboratory analysis. Received: 23 July 1997 · Accepted: 31 March 1998     

Multivariate Intrinsic Random Functions for Cokriging

In multivariate geostatistics, suppose that we relax the usual second-order-stationarity assumptions and assume that the component processes are intrinsic random functions of general orders. In this article, we introduce a generalized cross-covariance function to describe the spatial cross-dependencies in multivariate intrinsic random functions. A nonparametric method is then proposed for its estimation. Based on this class of generalized cross-covariance functions, we give cokriging equations for multivariate intrinsic random functions in the presence of measurement error. A simulation is presented that demonstrates the accuracy of the proposed nonparametric estimation method. Finally, an application is given to a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site used for atomic-bomb testing.     

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.     

Comparing sequential Gaussian and turning bands algorithms for cosimulating grades in multi-element deposits

Stochastic simulation is increasingly used to map the spatial variability in the grades of elements of interest and to assess the uncertainty in the mineral resources and ore reserves. The practical implementation requires specifying a stochastic model, which describes the spatial distribution of the grades, and an algorithm to construct realizations of these grades, viewed as different possible outcomes or scenarios. In the case of the Gaussian random field model, a variety of algorithms have been proposed in the past decades, but their ability to reproduce the model statistics is often unequal. In this paper, we compare two such algorithms, namely the turning bands and the sequential algorithms. The comparison is hold through a synthetic case study and a real case study in a porphyry copper deposit located in southeastern Iran, in which it is of interest to jointly simulate the copper, molybdenum, silver, lead and zinc grades. Statistical testing and graphical validations are realized to check whether or not the realizations reproduce the features of the true grades, in particular their direct and cross variograms. Sequential simulation based on collocated cokriging turns out to poorly reproduce the cross variograms, while turning bands proves to be accurate in all the analyzed cases.     

丽江石鼓马工村铜矿资源评价

矿床为典型的层控砂岩铜矿,重新估算铜矿床资源量,提出难选氧化矿石的“氨浸搅拌浸出→萃取→电积→浸渣浮选”加工处理流程,综合回收率提高至80％以上,并对矿床资源经济进行有效评价,为资源开发及回收利用提供可靠依据。     

Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity

This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.