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.     

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.     

An application of disjunctive kriging for earthquake ground motion estimation

For earthquake ground motion studies, the actual ground motion distribution should be reproduced as accurately as possible. For optimal estimation of ground motion, kriging has been shown to provide accurate estimates. Although kriging is accurate for this application, some estimates it provides are underestimates. This has dire consequences for subsequent design for earthquake resistance. Kriging does not provide enough information to allow an analysis of each estimate for underestimation. For such an application, disjunctive kriging is better applied. This advanced technique quantifies the probability that an estimate equals or exceeds particular levels of ground motion. Furthermore, disjunctive kriging can provide improved estimation accuracy when applied for local estimation of ground motion.     

Standard cumulative semivariograms of stationary stochastic processes and regional correlation

The standard cumulative semivariograms (SCS), obtained analytically from the currently employed stationary stochastic processes, provide a basis for the model identification and its parameter as well as regional correlation estimations. The analytical solutions for different stationary stochastic processes such as independent (IP), moving average (MA), autoregressive (AR), and autoregressive integrated moving average ARIMA (1,0,1) processes give rise to different types of SCSs which can be expressed in terms of the autocorrelation structure parameters only. The SCSs of independent and MA processes appear as linear trends whereas other type of processes have SCSs which are nonlinear for short distances but become linear at large distances. Irrespective of the stationary stochastic process type the linear portions of SCSs have unit slopes. The vertical distance between these linear portions and that of the IP cumulative semivariogram (CS), provide an indicator for measuring the regional correlation. In the case of stationary processes, the straight line portions of any CS are parallel to each other. Hence, it is possible to identify the model from the sample CS. Finally, necessary procedures are provided for the model parameters estimation. The methodology developed, herein, is applied to some hydrochemical ions in the groundwater of the Wasia aquifer in central part of Kingdom of Saudi Arabia.     

On counting the number of data pairs for semivariogram estimation

In planning spatial sampling studies for the purpose of estimating the semivariogram, the number of data pairs separated by a given distance is sometimes used as a comparative index of the precision which can be expected from a given sampling design. Because spatial data are correlated, this index can be unreliable. An alternative index which partially corrects for this correlation, themaximum equivalent uncorrelated pairs, is proposed for comparing spatial designs. The index is developed under the assumption that the underlying stochastic process is Gaussian and is appropriate when the (population) semivariogram is to be estimated by the sample semivariogram.     

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.     

Inner product matrices, kriging, and nonparametric estimation of variogram

Two important problems in the practical implementation of kriging are: (1) estimation of the variogram, and (2) estimation of the prediction error. In this paper, a nonparametric estimator of the variogram to circumvent the problem of the precise choice of a variogram model is proposed. Using orthogonal decomposition of the kriging predictor and the prediction error, a method for selecting, what may be considered, a statistical neighborhood is suggested. The prediction error estimates based on this scheme, in fact, reflects the true prediction error, thus leading to proper coverage for the corresponding prediction interval. By simulations and a reanalysis of published data, it is shown that the proposals made in this paper are useful in practice.     

A general form of probability kriging for estimation of the indicator and uniform transforms

Probability kriging is implemented in a general cokriging procedure (c.f. Myers, 1982) for estimatingboth the indicator and uniform transforms. Paired-sum semi-variograms are used to facilitate the modeling of the cross-covariance between the uniform transform and each indicator transform. Estimates of the uniform transform are averaged over all cutoffs, the average used to derive an estimate of the original data. This estimate can be biased with respect to the mean data value, but is unbiased with respect to the data median.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

Using robust kriging and sequential Gaussian simulation to delineate the copper- and lead-contaminated areas of a rapidly industrialized city in Yangtze River Delta,China

A total of 540 topsoil samples (0–15 cm), 188 subsoil samples (20–40 cm), and four individual soil profiles were collected in this study for mapping the Cu- and Pb-contaminated areas in soils of Zhangjiagang city, an industrialized city in the Yangtze River Delta region of China. Robust geostatistical methods were applied for identifying possible spatial outliers of Cu and Pb data, and then a sequential Gaussian simulation was employed for delineating the potential areas where Cu or Pb concentration was affected by diffuse pollution. The results showed that the spatial outliers of Cu and Pb were strongly associated with various types of factories. The anthropogenic input of Cu to soils at local hotspots was closely related to emissions of printing and dyeing, metallurgical, and chemical factories, whereas a lead oxide factory and a chemical factory resulted in a considerable increase of Pb in the topsoil of the study area. Approximately 30% of the total land area of the study was at potential risk from the Cu or Pb diffuse pollution resulting from rapid industrialization of the area over the past 20 years.     

南京地区表土镉汞铅含量的空间统计分析

在南京城区和周围郊区近2500km2的区域内,采集了670个表层土壤样品,分析了Cd、Hg、Pb等重金属含量。在ARCGIS等地理信息系统软件的支持下,用地统计学等空间分析方法,分析了数据的概率分布特征、空间自相关特征以及变异函数特征。结果表明,研究区内表层土壤中Cd、Hg、Pb等重金属元素平均含量显著高于其全国平均水平;并且不同的元素具有不同的空间自相关特征;变异函数分析显示,Cd和Hg的分布具有明显的空间异质性,而Pb的方向性不明显。元素空间分布特征的研究对研究污染源和污染扩散具有一定指示和引导作用。     

新疆三工河流域地下水矿化度的时空变异及其分形特征

对面积约670 km2的新疆三工河流域地下水矿化度进行了取样检测,应用地质统计学方法对取得的数据进行了半方差函数分析,计算了平均值、方差、标准差、变异系数等传统统计特征值,并指出了用该方法表示地下水矿化度所存在的不足.在此基础上,将分形理论与地质统计学原理相结合,计算了三工河流域在1998、2002年地下水矿化度的分维值,分别为1.75,1.59;其变异系数分别为0.51,0.54,即2002年地下水矿化度的均匀性要比1998年的稍差.研究结果表明,地下水矿化度并不是具有理想分形特征的介质,它只是在一定的空间尺度内才具有分形特征.通过对1998年及2002年地下水样品的分析,得出该地区地下水矿化度在时间与空间上皆存在明显的变异性.在空间尺度上,地下水矿化度在研究区的北部大于中部,中部大于南部;在时间尺度上,随着时间的推移研究区北部的地下水水质正向矿化度加重的方向发展.     

Appropriate Sampling for Optimised Measurement (ASOM), rather than the Theory of Sampling (TOS) Approach,to Ensure Suitable Measurement Quality: A Refutation of Esbensen and Wagner (2014)

The ‘Appropriate Sampling for Optimised Measurement’ (ASOM) approach considers measurement to be the focus of the sampling process, and sampling to be only the first part of the measurement process. To achieve ASOM, the uncertainty of measurements, including its contribution from sampling, needs to be estimated and optimised in order to achieve fitness‐for‐purpose. Such samples are then ‘sufficiently’ representative. The ‘Theory of Sampling’ (TOS) focuses on the processes of primary sampling and sample preparation and assumes that samples are ‘representative’ if they are correctly prepared by nominally ‘correct’ protocols. It defines around ten sampling ‘errors’, which are either modelled or minimised to improve sampling quality. It is argued that the ASOM approach is more effective in achieving appropriate measurement quality than in applying TOS to just the first part of the measurement process. The comparison is made less effective by the different objectives, scopes, terminology and assumptions of the two approaches. ASOM can be applied to in situ materials that are too variable to be modelled accurately, or where sources of uncertainty are unsuspected. The proposed integration of ASOM with TOS (Esbensen and Wagner 2014, Trends in Analytical Chemistry, 57, 93–106) is therefore effectively impossible. However, some TOS procedures can be useful within the ASOM approach.     

The geotherm of the lithosphere beneath Qilin, SE China: a re-appraisal and implications for P–T estimation of Fe-rich pyroxenites

The Qilin geotherm established by Xu et al. [Xu, X.S., O'Reilly, S.Y., Zhou, X.M. and Griffin, W.L., 1996. A xenolith-derived geotherm and the crust-mantle boundary at Qilin, southeastern China. Lithos, 38: 41–62.] using the Ellis and Green [Ellis D.J. and Green D.H., 1979. An experimental study of the effect of Ca upon garnet-clinopyroxene Fe-Mg exchange equilibria. Contrib. Mineral. Petrol., 71: 13–22]/Wood [Wood B.J., 1974. Solubility of alumina in orthopyroxene coexisting with garnet. Contrib. Mineral. Petrol. 46: 1-15] combination is in need of revision on the basis of re-evaluation of geothermobarometers, comparison of calculated pressures and temperatures with experimentally determined phase relationships and geological/geophysical data. The invalid assumption that all iron is present as Fe²⁺ in minerals, and the thermal destruction of equilibrium Fe–Mg exchange between clinopyroxene and garnet that may have resulted from heating of the Qilin xenoliths by the host magma resulted in unrealistically high temperatures estimated by the Ellis and Green's thermometer. An important implication arising from this study is that care must be taken when applying thermobarometers to Fe-rich pyroxenites for the purpose of geotherm construction and a comprehensive analysis is often required.     

Major and Trace Element Analysis of Silicate Rocks by XRF and Laser Ablation ICP-MS Using Lithium Borate Fused Glasses: Matrix Effects, Instrument Response and Results for International Reference Materials

Major and trace element compositions of fifteen silicate rock reference materials have been determined by a combined XRF and laser ablation ICP‐MS (LA‐ICP‐MS) technique on glasses prepared by fusing the sample with a lithium borate flux (sample:flux = 1:3). Advantages of this technique include the ability to measure major and trace element abundances on a single sample using a quick and simple preparation that attacks resistant phases such as zircon without the need for acid dissolution. The method is suitable for a wide variety of bulk compositions including mafic, intermediate and silicic rocks. Abundance‐normalized mass response patterns (the ratio of signal intensity to element concentration) of the LA‐ICP‐MS analyses vary systematically with major element composition, demonstrating the presence of a matrix effect that cannot be compensated by normalisation to a single internal standard element. Increasing the sampling distance between the ICP‐MS cone and the torch reduces the magnitude of this effect, suggesting that a mechanism related to residence time of ablated particles in the plasma may be at least partially responsible for the observed variations in mass response patterns. When using a matrix‐matched calibration, agreement of the LA‐ICP‐MS results with published reference values or those obtained by solution ICP‐MS is 10% relative. Analytical precision based on replicate analyses is typically 5% RSD. Procedural detection limits that include contributions from gas background and flux are 0.01‐0.1 μg g‐¹ for the heavy mass trace elements (Rb‐U). Major element analyses by XRF show excellent agreement with results obtained using a conventional heavy element absorbing flux. High quality major and trace element data for silicate rocks can be achieved by a combined XRF and LA‐ICP‐MS analysis of Li₂B₄O₇/LiBO₂ fused glasses provided an appropriate matrix‐matched calibration is adopted.     

湖相烃源岩原始有机质恢复与生排烃效率定量研究 ——以东营凹陷古近系沙河街组四段优质烃源岩为例

在烃源岩沉积有机相精细划分的基础上,优选东营凹陷古近系沙河街组四段的优质烃源岩样品,建立了完整的自然演化地球化学剖面,并应用物质平衡法,开展了原始有机质恢复和生排烃效率定量计算。计算结果表明,沙河街组四段主力供烃区间内的优质烃源岩具有高生排烃效率和原始有机质丰度恢复系数,其中深层高过成熟阶段排烃效率可达70%~80%,原始有机质丰度恢复系数可达1.5~2.0,较以往资源评价中采用的数值有较明显提高。据该结果推算,东营凹陷沙河街组四段的总排烃量总体上将有一定程度的提高,其中中浅层烃源岩的资源贡献有所降低,而深层烃源岩的资源贡献有较大幅度升高,主力烃源岩埋藏深度有下沉的趋势,这预示着深层具有丰富的资源基础和勘探前景。研究成果对于济阳坳陷乃至整个渤海湾盆地的油气资源潜力再认识和深层油气藏勘探具有重要的指导意义。