A nonparametric view of generalized covariances for kriging

Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data.     

Computationally efficient restricted maximum likelihood estimation of generalized covariance functions

Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance function's dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.     

On the estimation of the generalized covariance function

The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables.     

Estimation of the generalized covariance function: II. A response surface approach

A procedure is proposed that employs first-moment estimation (kriging), cross-validation, and response surface analysis to estimate parameters of a generalized covariance function. Results from application of this procedure to two data sets are given.     

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.     

Minimum-variance unbiased quadratic estimation of covariances of regionalized variables

The parameters of covariance functions (or variograms) of regionalized variables must be determined before linear unbiased estimation can be applied. This work examines the problem of minimum-variance unbiased quadratic estimation of the parameters of ordinary or generalized covariance functions of regionalized variables. Attention is limited to covariance functions that are linear in the parameters and the normality assumption is invoked when fourth moments of the data need to be calculated. The main contributions of this work are (1) it shows when and in what sense minimum-variance unbiased quadratic estimation can be achieved, and (2) it yields a well-founded, practicable, and easy-to-automate methodology for the estimation of parameters of covariance functions. Results of simulation studies are very encouraging.     

基于广义S变换时频滤波的MT数据去噪

针对油气勘探中大地电磁(MT)数据易受各类干扰的污染,且信噪难以分离的问题,把基于广义S变换的时频滤波技术应用于MT数据处理中来,得到MT数据的S域时频分布,分析受噪MT数据在S域的时频分布特征,再在S变换时频域进行时频阈值去噪,并对滤波后的S域时频谱进行逆变换重构,分离得到去噪后的MT数据。给出了基于广义S域时频滤波的方法原理与应用步骤,对被污染的仿真和实测MT数据进行了时频阈值滤波,并与小波阈值去噪方法进行了比较研究。结果表明：基于广义S变换的时频滤波方法可有效抑制MT数据中的干扰,从噪声信号中分离出有效的大地电磁数据,且减少了人为参与,提高了MT勘测的数据质量。     

Trend surface analysis as a special case of IRF-k kriging

This short note establishes the equivalence between trend surface analysis with polynomials of orderk and IRF-k (intrinsic random function of orderk) kriging with a nugget effect covariance model.     

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.     

A modification of minimum norm quadratic estimation of a generalized covariance function for use with large data sets

Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency.     

An example of cluster analysis applied to a large geologic data set: Aerial radiometric data from Copper Mountain, Wyoming

One objective of the aerial radiometric surveys flown as part of the U.S. Department of Energy's National Uranium Resource Evaluation (NURE) program was to ascertain the spatial distribution of near-surface radioelement abundances on a regional scale. Some method for identifying groups of observations with similar -ray spectral signatures and radioelement concentration values was therefore required. It is shown in this paper that cluster analysis can identify such groups with or without a priori knowledge of the geology of an area. An approach that combines principal components analysis with convergentk-means cluster analysis is used to classify 6991 observations (each observation comprising three radiometric variables) from the Precambrian rocks of the Copper Mountain, Wyoming area. This method is compared with a convergentk-means analysis that utilizes available geologic knowledge. Both methods identify four clusters. Three of the clusters represent background values for the Precambrian rocks of the area, and the fourth represents outliers (anomalously high²¹⁴Bi). A segmentation of the data corresponding to geologic reality as interpreted by other methods has been achieved by perceptive quantitative analysis of aerial radiometric data. The techniques employed are composites of classical clustering methods designed to handle the special problems presented by large data sets.     

Use of the geometric average for effective permeability estimation

The geometric average is often used to estimate the effective (large-scale) permeability from smaller-scale samples. In doing so, one assumes that the geometric average is a good estimator of the geometric mean. Problems with this estimator arise, however, when one or more of the samples has a very low value. The estimate obtained becomes very sensitive to the small values in the sample set, while the true effective permeability may be only weakly dependent on these small values. Several alternative methods of estimating the geometric mean are suggested. In particular, a more robust estimator of the geometric mean, the j^th Winsorized mean, is proposed and several of its properties are compared with those of the geometric average.     

Intrinsic random function of order<Emphasis Type="Italic"> k</Emphasis> kriging of electrical resistivity data for estimating the extent of saltwater intrusion in a coastal aquifer system

A saltwater intrusion study using electrical resistivity distribution was conducted in a coastal aquifer system in the southeastern part of Busan, Korea. This aquifer system is divided into four layers according to the hydrogeologic characteristics, and the horizontal extent of intruded saltwater is determined at each layer through the geostatistical interpretation of electrical resistivity data. Intrinsic random function of order k (IRF-k) kriging is performed with covariance models to produce the plane of spatial mean resistivities. The kriged estimates are evaluated by cross validation, and they showed a good agreement with the true values. The statistics of cross validation represented low errors for the estimates. In the resistivity contour maps more than 5 m below the surface, a dominant direction of saltwater intrusion was observed beginning from the east side. These results led the authors to conclude that the application of geostatistical technique to electrical resistivity data is useful for assessing saltwater intrusion in a coastal aquifer system.     

煤炭资源/储量估算地质块段法误差及其改进方法

地质块段法是煤炭资源/储量估算中最常用和最重要的方法。块段的具体划分方法在相关规范和规定中并无明确的具体规定,而块段的划分方法不同,估算的资源/储量结果亦不同。结合生产实际,通过实例应用,定量分析了地质块段不同划分方法所产生的误差,并对地质块段不同划分方法进行了定量评价。经过分析和确定误差的影响因素,提出了最佳块段划分的具体原则与方法,对提高煤田勘查中资源/储量估算的准确度具有现实意义。     

Palaeoclimatic signals recognized by chemometric treatment of molecular stratigraphic data

The high-resolution stratigraphy of various marker compounds has been studied, using GC, HPLC and GC-MS, in a 13 m gravity core recovered from the Kane Gap region, eastern equatorial Atlantic, which provides a record of the glacial/interglacial episodes over the last million years. Downhole variations in many presumed source-specific components are observed (e.g. in n-alkanes from terrigenous land plants and dinosterol from dinoflagellates), which may be due to perturbations or cyclicities resulting from climatic change. Fluctuations in the unsaturation of alkenones attributable to variations in water temperatures show correlations with the glacial/interglacial cycles recorded in the δ¹⁸O values for planktonic foraminifera, thereby providing a potential organic geochemical measure of past climates. These molecular abundance data can be linked to the palaeotemperature record, following computer treatments using principal component and spectral analyses. Molecular stratigraphy shows promise as a new chemostratigraphical tool where other means of stratigraphy fail, for example, through calcium carbonate dissolution.     

利用三维地震勘探资料在GMS平台下实现地层实体建模的方法研究

三维地震勘探成果数据体包含丰富的地质信息，任意物性反射层均可作为钻孔地层划分的依据。物性反射层等深线数据文件与GMS中钻孔数据文件通过点的空间坐标相互关联，可高效地构建GMS中地层实体建模所需的虚拟钻孔。实践表明，该方法具有很高的地层建模精度，能成倍提高工作效率，同时也为三维地震勘探成果中其它物性特征的获取利用提供了一个可借鉴方式。     

Lithospheric structure of the Tornquist Zone resolved by nonlinear P and S teleseismic tomography along the TOR array

The main aim of the TOR project is to study the lithospheric–asthenospheric boundary structure under the Sorgenfrei–Tornquist Zone, across northern Germany, Denmark and southern Sweden. Relative arrival-time residuals of teleseismic P and S phases from 51 earthquakes, recorded by 150 seismic stations along the TOR array, were used to delineate the transition zone in the studied area. The effects of crustal structures were investigated by correcting the teleseismic residuals for travel-time variations in the crust based on a 3D crustal model derived from other data. The inversion was carried out for S phases. The results were then compared with the corresponding P-wave models. As expected, the derived models show that the relatively old and cold Baltic Shield has higher velocity at depth than the younger lithosphere farther South. The models show two sharp and distinct increases in depth to velocities which are low compared to our reference model, as we move from South to North. The location and sharpness of these boundaries suggests that the features resolved are, at least partially, compositional in origin, presumably related to mantle depletion. A sharp and steep subcrustal boundary is found roughly coincident with the southern edge of Sweden. This is below where the edge of the Baltic Shield is usually placed, based on surface geological evidence (the Sorgenfrei–Tornquist Zone). Another less significant transition is recognised more or less beneath the Elbe-lineament. Relatively high d(V_p / V_s) ratios under the central part of the profile (Denmark) indicate relatively low S-velocity in an area where a gravity high supports the hypothesis of extensive mafic intrusions.