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 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.     

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.     

Global sensitivity analysis for subsoil parameter estimation in mechanized tunneling

The present paper validates two alternative global sensitivity analysis methods, namely variance-based and elementary effect, for the purpose of detecting key subsoil parameters that influence the output of mechanized tunnel finite element simulation. In the elementary effect method, a strategy for considering the dependencies, that result from a set of constraints between different parameters, is proposed. Moreover, because the numerical implementation of variance-based sensitivity estimates, in particular, has been proven to require intensive evaluations of the system under investigation, a practical surrogate modeling technique is utilized. This technique is based on quadratic polynomial regression and represents a reliable approximation of the computationally expensive mechanized tunnel simulation. Furthermore, a convergence analysis based on Central Limit Theorem for the numerical implementation of the methods is introduced. The adopted analysis highlights model evaluations needed for the sensitivity measures to converge, as well as the uncertainty involved in these measures.     

The estimation of the experimental and basic components of the observed variance of rock strength

SummaryThe Estimation of the Experimental and Basic Components of the Observed Variance of Rock Strength The intercept of the strength correlelogram over distance is used to estimate the proportions of the observed strength variance which can be attributed to the testing (experimental) procedure and to the basic differences between the strengths of the specimens. The method is applied to sandstone uniaxial strength data (Wiid, 1968) where approximately half of the observed variance is experimental and half basic. A spatial trend accounts for most of the basic variance. The uniaxial strength is also spatially correlated with the triaxial strength. It is shown that the results of testing procedures of equal reliability will be poorly correlated if the experimental components of variance are large.

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ZusammenfassungDie Abschätzung der Anteile an den Streuungen der Festigkeiten von Fels, die den Versuchsverfahren bzw die den tatsächlichen Streuungen zuzuordnen sind Der Achsabschnitt im Druckfestigkeitskorrelelogramm wird verwendet, um die Anteile der beobachteten Druckfestigkeitsstreuungen, welche dem Versuchsverfahren bzw. den tatsächlichen Unterschieden zwischen den Druckfestigkeiten der Proben zuzuordnen sind, zu schätzen.Die Methode wird bei einachsialen Druckfestigkeiten eines Sandsteines (Wiid, 1968) angewandt, wobei ungefähr je die Hälfte der beobachteten Streuungen auf die Versuchstechnik bzw. auf tatsächliche Unterschiede zurückzuführen sind. Die einachsiale Druckfestigkeit wird auch mit der dreiachsialen Druckfestigkeit räumlich korreliert.Es wird gezeigt, daß die Ergebnisse der Testverfahren von gleicher Zuverlässigkeit schlechter korreliert werden können, wenn die experimentalen Komponenten an den Streuungen größer sind.

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RésuméL'estimation des parties constituantes expérimentales et fondamentales de la variance observée de la résistance de roche La valeur pour distance zéro de résistance du correlélogramme (en fonction de distance) est utilisée pour faire une estimation des proportions de la variance observée qui peuvent être attribuées au procédé expérimental ainsi qu'aux différences fondamentales entre les résistances des specimens. La méthode est appliquée à des données de résistance uniaxe de grès, ou à peu près la moitié de la variance observée est dûe à l'expérimentateur et le reste est fondamental. Une tendance spatiale explique la plûpart de la variance fondamentale. La résistance uniaxe est aussi corrélée spatialement avec la résistance triaxe. Il est démontré que les résultats des procédés expérimentaux qui sont de même fidélité seront mal corrélés si les partiés constituantes de variance expérimentales sont élevées.

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With 8 Figures     

A variance of geostatisticians

Different individuals will take different approaches to the analysis and interpretation of data. This study attempted to quantify the effect of such individual differences on the quality of geostatistical spatial estimates. Identical spatial data sets were sent to 12 investigators, who independently analyzed the data and produced spatial interpolations. The results varied considerably. Differences in the interpolations could be attributed to differences in choice of methodology, differences in data interpretation, and, in a few cases, errors in procedure. The potential differences in economic and societal costs between decisions based on good vs. bad interpolations warrant a systematic approach to the identification and testing of interpolation methods.Although the research described in this article has been supported by the U.S. Environmental Protection Agency, it has not been subjected to Agency review and no official endorsement should be inferred.     

Magnesium-iron compositional variance in hornblende pyroxene granulites

Hornblende-orthopyroxene-calcic pyroxene-plagioclase (andesine-labradorite) — magnetite-ilmenite±quartz assemblages are evidently the most common granulite facies representatives of basic rocks throughout the world, and they must represent a variance of more than one. Ramberg (1948) invoked an additional degree of freedom arising out of unequal fractionation of Mg and Fe in the ferromagnesian silicates in such rocks. This is examined in detail on the basis of chemical data on twentytwo hornblende-orthopyroxenecalcic pyroxene triads, half of them from the type charnockite area near Madras.Theoretical consideration of a magnesium-iron distribution model shows that in quartz-bearing hornblende pyroxene granulites, the Mg/Mg+Fe ratios of hornblende, orthopyroxene and calcic pyroxene are uniquely determined by temperature at constant pressure. But these assemblages contain quartz rarely and Mg-Fe distribution among the three can be best described by three exchange equilibria where, at constant temperature, there is a fixed relation between the Mg/Mg+Fe ratios of the minerals. Among these, the exchange equilibria between hornblende-orthopyroxene and hornblende-calcic pyroxene appear to be interdependent; however, they are known to be significantly modified by varying tetrahedral aluminium contents of hornblendes. Comparison of molecular Mg/(Mg+Fe) ratios of hornblendes and parent rocks strongly points to an absence of Mg-Fe compositional variance in the total system, a petrologically important corollary being that the hornblendes in these rocks are highly unlikely to be secondary after pyroxenes. The general variance of the assemblages is also discussed.     

The variogram sill and the sample variance

The relationship between the sill of the variogram and the sample variance is explored. The common practice of using the sample variance as an estimate of the variogram sill is questioned, and a conceptual framework for determining the appropriateness of this heuristic is constructed.     

Locating maximum variance segments in sequential data

An automated method is presented for the identification of peaks in sets of sequential data. The method is based upon the location of those segments with maximum variance and has the advantage of guarding against the masking of small-scale effects by large-scale effects. The procedure is illustrated with data taken as part of the National Uranium Resource Evaluation project.This work was performed under the auspices of the U.S. Energy Research and Development Administration, Grand Junction Office.     

Second-order approximation to the variance of a proportion

The second-degree Taylor polynomial for a ratio is used to derive approximations to the means, variances, and covariances of proportions.     

方差体技术在地震勘探中的应用

三维地震数据体反映了地下一个规则网格的反射情况。当遇到地下有断层或某个局部区域地层不连续变化时，一些地震道的反射特征就会与附近地震道出现差异而导致地震首局部的不连续性。三维方差体技术就是求取三维数据体所有样点的方差值来反映这种不连续性信息。通过该技术在济宁2号井某采区的实际应用，可看出，方差体技术在三维地震信息的自动拾取以及提高地震资料解释成果精度等方面有明显优势，对断层、陷落柱有良好的自动识别能力。     

测流不确定度的插值方差估计法

目前,测流不确定度通过误差试验或通过经验数值来确定,但这些方式存在着工作量大或不确定估计不足等局限性。为解决此问题,对基于实测数据和统计理论的插值方差估计法在不同测流条件下进行了验证,选取白河、襄阳和沙洋3个流量站进行了实测数据的不确定度分析,同时对白河站进行了Monte Carlo试验,比较插值方差估计法得到的不确定度与真实误差的差异。结果表明,插值方差估计法能较好地反映水位变化的影响,插值方差估计法所得到的不确定度与真实测流误差的相关系数达0.64,与断面水位变化的Spearman相关系数达0.79,高、中水位情况下插值方差估计法的不确定度估计结果较为合理,低水位情况下偏高。     

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.     

方差技术在生物礁识别中的应用

在海相碳酸盐岩勘探中,生物礁滩的识别对确定储层有利相带具有重要意义.方差技术是检测地下断层、地层不连续变化现象及异常地质体卓有成效的一种技术,这里首先介绍了它的基本原理,然后重点展示和分析该技术在YB生物礁识别,沉积相带刻画中的应用成果,证实其技术优势和良好的应用前景.     

Global climatic changes

Book reviewed in this article:
Berger, A. (ed.): Climatic Variations and Variability: Fits and Thfonrs