The updated kriging variance and optimal sample design

A number of criteria based on kriging variance calculations may be used for infill sampling design in geologic site characterization. Searching for the best new sample locations from a set of candidate locations can result in excessive computation time if these criteria and the naive rekriging are used. The relative updated kriging estimate and variance for universal kriging estimation are demonstrated as a simple kriging estimate and variance, respectively. The updated kriging variance is demonstrated as the multiplication of two kriging variances. Using these updated kriging variance equations can increase the computational speed for selecting the best new sample locations. The application results for oil rock thickness in an oilfield indicate that minimizing the average relative updated kriging variance is a useful alternative to the other criteria based on kriging variance in optimal infill sampling design for geologic site characterization.     

General polygonal variogram functions: Evaluation of estimation variance integrals

The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a general polygonal function the requisite integrals may be easily computed by a closed form representation of simple integrals. This paper provides the integration formulas for two-dimensional variogram functions whose domain is represented as a finite collection of rectangles. The integration formulas essential for a fully developed polygonal approach to an extensive statistical evaluation of geostatistical quantities are presented.     

The Droimchogaidh sill,Connacht, Ireland

The Droimchogaidh sill, south-central Mayo, is sheeted at high-level in the Ordovician sedimentary pile of the South Mayo Trough. This association now lies raised on the eastern fringe of the Connemara-Murrisk Tertiary uplift. The sill is basin-shaped with a maximum thickness of about 100 m. It is composed of isotropic or weakly layered olivine gabbro, and subordinate, locally-developed picritic cumulates and syenitic differentiates. The chemistry of the gabbro reveals the effects of post-emplacement hydrothermal circulation, but the magma that supplied the sill was probably similar in composition to that of the analcime-olivine dolerite dikes further west. The ⁴⁰Ar/³⁰Ar dating gives an age of 55 ± 1 Ma for the sill, consistent with a younger set of ages from the Irish/British Tertiary province.     

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.     

Exact formulas for approximating variogram function integrals

The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a polygonal function the integral may be easily approximated by closed form representations of polygonal integrals. This approach provides a basis for more extensive statistical evaluation not evident in existing approximation methods. This paper provides the closed form representations for two-dimensional variogram functions whose domain is represented by a finite collection of rectangles.     

Some exact sampling distributions for variogram estimators

For equally spaced observations from a one-dimensional, stationary, Gaussian random function, the characteristic function of the usual variogram estimator for a fixed lag k is derived. Because the characteristic function and the probability density function form a Fourier integral pair, it is possible to tabulate the sampling distribution of a function of a using either analytic or numerical methods. An example of one such tabulation is given for an underlying model that is simple transitive.     

变异函数在异常空间插值中的应用

针对数据空间架构中异常点对地质数据解译过程的干扰，以协方差数学理论为基础，提出界定空间数据分布方法-变异函数。着重阐述变异函数的求取过程，对新疆贝克滩水系沉积物化探数据处理，预测误差均值为0．1032、平均预测标准差为21．12、标准均方根预测误差为0．9841。预测值与样品实测值误差较小，界定的空间数据分布与实际矿床（点）分布吻合较好。     

Petrology,structure and age of the Rough Point sill,Donegal

The Rough Point sill, which is one of the thickest of a large number of sills intruded into the Appin Group metasediments of the Dalradian sequence in north Donegal, contains a number of internal features including mafic:felsic banding, pegmatite bands and significant variations in grain size. There are also appreciable variations in mineralogy, particularly the proportions of plagioclase and amphibole, and in the major and trace element chemistry. Together these features indicate that the sill is, like the metasediments which it intrudes, inverted, and is therefore older than the Caledonian F₂ structures which caused the inversion. Disruption of the sediments at the sill contact is also earlier than the F₂ event. The timing of the basic magmatic episode together with the geochemistry of the sill invite comparison with the metadolerites of the southwest Highlands of Scotland.     

The variogram method for a fractal model of a rock joint surface

The variogram method can be used to make a fractal model of a rock joint surface. However, it has been found that the range of lag satisfying the power law is very small, that is, less than about 10% of the profile length. The cause of this has been investigated mathematically. The main cause of this problem is due to the fact that the profile length is assumed infinite for the theory, but it is finite for the actual calculation. This discrepancy between the actual calculation and the theory yields a significant error and causes the problem when the lag is large. To confirm the validity of this conclusion, it has been demonstrated that the range of lag satisfying the power law increases with an increase in the profile length by applying the variogram method to profiles cut from a long profile. In addition, the range of lag was investigated mathematically and it has been clarified that the range of lag increases with an increase in the fractal dimension. These results suggest that the profile length, the sampling interval and the removal of the linear trend are items to which we must pay attention when we use the variogram method.     

The age of the Lugar sill and a discussion of the Late-Carboniferous/Early-Permian sill complex of S.W. Scotland

The incremental heating ⁴⁰Ar/³⁹Ar method has been used to date a kaersutite separate from the main theralite-picrite unit of the Lugar sill. Ar-release spectra show discordance mainly due to the presence of excess Ar in alteration phases intimately associated with the primary kaersutite. Plateaus in the spectra allow an emplacement age of 288 ± 6 Ma (2γ) to be assigned to the sill. The results are used to discuss the age relations of the alkaline Permo-Carboniferous sill complex of southwest Scotland. It is concluded that these sills could have been emplaced over a shorter period of time than implied by the available conventional K/Ar data. The classification of the alkaline sills into kylite and teschenite types has little petrogenetic significance.     

Studies of the effects of outliers and data transformation on variogram estimates for a base metal and a gold ore body

Variograms for gold and lead values from the Loraine and Prieska mines, respectively, indicate that data outliers can seriously distort and/or mask the real variogram patterns. Studies show that this problem is best overcome for these mines by logarithmic transformation of the data, and/or a suitable screening out of such outliers, and/or more robust variogram estimation procedures; the benefits are particularly significant when the basic data is limited.     

变差函数在赣杭构造带火山岩型铀矿田(床)评价中的应用

在简要介绍变差函数的原理、算法和研究区地质情况的基础上,将变差函数引入火山岩型铀成矿作用与地形相关性研究中。详细描述了赣杭构造带中不同成矿规模矿田(床)区地形高程变差统计的过程,初步分析了变差统计结果的地质意义。研究结果表明,地形高程值变差函数的长轴方向可以反映研究区的主要构造方向,变差函数的长短轴半径与铀矿床的成矿规模在一定程度上具有对应关系。     

Variance as a function of sample support size

The effect of sample support size on variance is examined and evaluated. Results based on variograms and geostatistics are compared to the classical relationship developed by H. F. Smith in 1938; that is, that the variance is reduced fromV ₁ toV ₁ /n ^b as the support area increases from I ton plots for uniformity trials. The exponentb is between zero and one. Theoretical results are based on use of auxiliary functions and account for the size and shape of the sample support and the overall field geometry. Results are given in terms of approximations by rational functions for ease of calculation. Experimental results for uniformity trials, infiltration measurements, and spectral data from satellites are compared to theoretical and empirical results. Applications include not only uniformity trials, but also measurement theory.     

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.     

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.     

An application of the deterministic variogram to design-based variance estimation

When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.     

Multivariable variogram and its application to the linear model of coregionalization

In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.     

The multivariate (co)variogram as a spatial weighting function in classification methods

The multivariate variogram and the multivariate covariogram are used as spatial weighting functions for forming spatially homogeneous groups automatically. The groups are created after either deflating similarities between distant samples with the multivariate covariogram or by inflating dissimilarities between distant samples with the multivariate variogram. These approaches can be seen as generalization of the Oliver and Webster proposal. Two data sets show the efficiency of the two weighting functions when compared to the classical approach which does not take spatial information into account. In one case study, the weighting of similarities by the multivariate covariogram showed more interpretable results than the weighting of dissimilarities by the multivariate variogram.