Pseudo-cross variograms, positive-definiteness, and cokriging

Cokriging allows the use of data on correlated variables to be used to enhance the estimation of a primary variable or more generally to enhance the estimation of all variables. In the first case, known as the undersampled case, it allows data on an auxiliary variable to be used to make up for an insufficient amount of data. Original formulations required that there be sufficiently many locations where data is available for both variables. The pseudo-cross-variogram, introduced by Clark et al. (1989), allows computing a related empirical spatial function in order to model the function, which can then be used in the cokriging equations in lieu of the cross-variogram. A number of questions left unanswered by Clark et al. are resolved, such as the availability of valid models, an appropriate definition of positive-definiteness, and the relationship of the pseudo-cross-variogram to the usual cross-variogram. The latter is important for modeling this function.     

Positive definiteness is not enough

Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed.     

Use of indicator variograms for an enhanced spatial analysis

Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general.     

Variogram models must be positive-definite

The aim of this short article is to stress the importance of using only positive-definite functions as models for covariance functions and variograms.The two examples presented show that a negative variance can easily be obtained when a nonadmissible function is chosen for the variogram model.     

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.     

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.     

Robustness of variograms and conditioning of kriging matrices

Current ideas of robustness in geostatistics concentrate upon estimation of the experimental variogram. However, predictive algorithms can be very sensitive to small perturbations in data or in the variogram model as well. To quantify this notion of robustness, nearness of variogram models is defined. Closeness of two variogram models is reflected in the sensitivity of their corresponding kriging estimators. The condition number of kriging matrices is shown to play a central role. Various examples are given. The ideas are used to analyze more complex universal kriging systems.Research performed while on leave at Centre de Geóstatistique et de Morphologie Mathématique, Fontainebleau.     

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.     

Fast sequential indicator simulation: Beyond reproduction of indicator variograms

Sequential Indicator Simulation (SIS), although widely used, is relatively slow, and requires tedious inference of a large number of indicator variogram models. SIS is designed only to estimate class proportions and to reproduce indicator variogram models; the statistics of the continuous attribute being simulated,z-histogram and variogram, may be poorly reproduced. Several implementations of the SIS algorithm are proposed resulting in better reproduction of statistics yet with better CPU performance.     

Automatic Modeling of (Cross) Covariance Tables Using Fast Fourier Transform

Covariance models provide the basic measure of spatial continuity in geostatistics. Traditionally, a closed-form analytical model is fitted to allow for interpolation of sample Covariance values while ensuring the positive definiteness condition. For cokriging, the modeling task is made even more difficult because of the restriction imposed by the linear coregionalization model. Bochner's theorem maps the positive definite constraints into much simpler constraints on the Fourier transform of the covariance, that is the density spectrum. Accordingly, we propose to transform the experimental (cross) covariance tables into quasidensity spectrum tables using Fast Fourier Transform (FFT). These quasidensity spectrum tables are then smoothed under constraints of positivity and unit sum. A backtransform (FFT) yields permissible (jointly) positive definite (cross) covariance tables. At no point is any analytical modeling called for and the algorithm is not restricted by the linear coregionalization model. A case study shows the proposed covariance modeling to be easier and much faster than the traditional analytical covariance modeling, yet yields comparable kriging or simulation results.     

Uncertainty in fractal dimension estimated from power spectra and variograms

The reliability of using fractal dimension (D) as a quantitative parameter to describe geological variables is dependent mainly on the accuracy of estimated D values from observed data. Two widely used methods for the estimation of fractal dimensions are based on fitting a fractal model to experimental variograms or power-spectra on a log-log plot. The purpose of this paper is to study the uncertainty in the fractal dimension estimated by these two methods. The results indicate that both spectrum and variogram methods result in biased estimates of the D value. Fractal dimension calculated by these two methods for the same data will be different unless the bias is properly corrected. The spectral method results in overestimated D values. The variogram method has a critical fractal dimension, below which overestimation occurs and above which underestimation occurs. On the bases of 36,000 simulated realizations we propose empirical formulae to correct for biases in the spectral and variogram estimated fractal dimension. Pitfalls in estimating fractal dimension from data contaminated by white noise or data having several fractal components have been identified and illustrated by simulated examples.     

Kriging with imprecise (fuzzy) variograms. II: Application

The geostatistical analysis of soil liner permeability is based on 20 measurements and imprecise prior information on nugget effect, sill, and range of the unknown variogram. Using this information, membership functions for variogram parameters are assessed and the fuzzy variogram is constructed. Both kriging estimates and estimation variances are calculated as fuzzy numbers from the fuzzy variogram and data points. Contour maps are presented, indicating values of the kriged permeability and the estimation variance corresponding to selected membership values called levels.     

Kriging with imprecise (fuzzy) variograms. I: Theory

Imprecise variogram parameters are modeled with fuzzy set theory. The fit of a variogram model to experimental variograms is often subjective. The accuracy of the fit is modeled with imprecise variogram parameters. Measurement data often are insufficient to create good experimental variograms. In this case, prior knowledge and experience can contribute to determination of the variogram model parameters. A methodology for kriging with imprecise variogram parameters is developed. Both kriged values and estimation variances are calculated as fuzzy numbers and characterized by their membership functions. Besides estimation variance, the membership functions are used to create another uncertainty measure. This measure depends on both homogeneity and configuration of the data.     

A method for seafloor classification using directional variograms, demonstrated for data from the western flank of the Mid-Atlantic Ridge

Seafloor classification is aimed at quantitatively characterizing seafloor properties such as roughness and anisotropy, and at using such spatial characteristics to distinguish geological provinces automatically. From geostatistical principals, a variogram method is developed for seafloor classification and it is demonstrated for data from the western flank of the Mid-Atlantic Ridge at 25°45N to 26°40N. This study uses HYDROSWEEP bathymetric data which have been ping-edited to flag erroneous data records, and navigation corrected. The classification method can handle the resultant data gaps inside the survey swaths as well as interpret data from several swaths. For a suite of test areas representative of different geological provinces, directional variograms are calculated, and characteristic parameters are extracted for the classification. Examples include a sediment pond, abyssal hill terrain in several segments and of variable spacing, inside and outside corners of ridge discontinuities, and mixed morphological forms. The dependency of the results on random or regular subsampling and on the size of the test area is investigated.     

A Study of Spatial Variability on Aquifer Hydraulic Conductivity K

A structural analysis of K of an aquifer system in the study area is presented, and the main direction and degree of the variability of K are found by using the unstationary regionalized variable theory of geostatistics. Optimal estimation of K has been made by universal kriging method (U K M ). Both spatial variability distribution map and division map of K are given.     

含水层渗透系数K的空间变异性研究

本文结合桥梁实例对公路双曲拱桥出现的一些主要病害进行了初步分析，并进而对桥梁状况提出评定意见和处理建议，仅供参考。     

岩土性质的空间信息统计分析

运用地质统计学的原理与方法 ,以变差函数为工具 ,在工程实测数据的基础上对静止水位进行了分析 ,得到不同方向上的实验变差函数及理论变差函数模型 ,并提出了一个变异性综合指标 ,实现了对岩土性质空间变异性的定量化分析。     

Fixed-Domain Asymptotics for Variograms Using Subsampling

The variogram is a measure of the local variation in space of a random field. For large geostatistical data sets, the traditional empirical variogram may be hard to compute. This article presents, for processes with a fixed domain, the effect of using a subsample of the available data on the performance of the empirical variogram. The motivation of this work, apart from the saving on computation, is to study how dense the observations need to be in the bounded sampling region to obtain most of the information we would get from continuous observations in the fixed domain.     

地质统计学中变差函数参数估计的新方法

遗传算法是一种模拟生物进化规律的全局优化算法。对传统的遗传算法进行改进,并应用于地质统计学变差函数参数估计中。实例分析表明,该方法简便、通用,具有较高拟合精度,是非线性、不连续可微模型参数估计的方法