Block Kriging for Lognormal Spatial Processes

Lognormal spatial data are common in mining and soil-science applications. Modeling the underlying spatial process as normal on the log scale is sensible; point kriging allows the whole region of interest to be mapped. However, mining and precision agriculture is carried out selectively and is based on block averages of the process on the original scale. Finding spatial predictions of the blocks assuming a lognormal spatial process has a long history in geostatistics. In this article, we make the case that a particular method for block prediction, overlooked in past times of low computing power, deserves to be reconsidered. In fact, for known mean, it is optimal. We also consider the predictor based on the “law” of permanence of lognormality. Mean squared prediction errors of both are derived and compared both theoretically and via simulation; the predictor based on the permanence-of-lognormality assumption is seen to be less efficient. Our methodology is applied to block kriging of phosphorus to guide precision-agriculture treatment of soil on Broom's Barn Farm, UK.     

The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets

Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.     

A review of lognormal estimators forin situ reserves

The term “lognormal kriging” does not correspond to a single well defined estimator. In fact, several types of lognormal estimators forin situ reserves are available, and this may cause confusion. These estimators are based on different assumptions—that is, different models. This paper presents a review of these estimators.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

Comparing the robustness of ordinary kriging and lognormal kriging: Outlier resistance

Ordinary kriging is well-known to be optimal when the data have a multivariate normal distribution (and if the variogram is known), whereas lognormal kriging presupposes the multivariate lognormality of the data. But in practice, real data never entirely satisfy these assumptions. In this article, the sensitivity of these two kriging estimators to departures from these assumptions and in particular, their resistance to outliers is considered. An outlier effect index designed to assess the effect of a single outlier on both estimators is proposed, which can be extended to other types of estimators. Although lognormal kriging is sensitive to slight variations in the sill of the variogram of the logs (i.e., their variance), it is not influenced by the estimate of the mean of the logs.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Generalized Bootstrap Method for Assessment of?Uncertainty in Semivariogram Inference

The semivariogram and its related function, the covariance, play a central role in classical geostatistics for modeling the average continuity of spatially correlated attributes. Whereas all methods are formulated in terms of the true semivariogram, in practice what can be used are estimated semivariograms and models based on samples. A generalized form of the bootstrap method to properly model spatially correlated data is used to advance knowledge about the reliability of empirical semivariograms and semivariogram models based on a single sample. Among several methods available to generate spatially correlated resamples, we selected a method based on the LU decomposition and used several examples to illustrate the approach. The first one is a synthetic, isotropic, exhaustive sample following a normal distribution, the second example is also a synthetic but following a non-Gaussian random field, and a third empirical sample consists of actual raingauge measurements. Results show wider confidence intervals than those found previously by others with inadequate application of the bootstrap. Also, even for the Gaussian example, distributions for estimated semivariogram values and model parameters are positively skewed. In this sense, bootstrap percentile confidence intervals, which are not centered around the empirical semivariogram and do not require distributional assumptions for its construction, provide an achieved coverage similar to the nominal coverage. The latter cannot be achieved by symmetrical confidence intervals based on the standard error, regardless if the standard error is estimated from a parametric equation or from bootstrap.     

Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.     

Comparison of machine learning methods for copper ore grade estimation

In this study, machine learning methods such as neural networks, random forests, and Gaussian processes are applied to the estimation of copper grade in a mineral deposit. The performance of these methods is compared to geostatistical techniques, such as ordinary kriging and indicator kriging. To ensure that these comparisons are realistic and relevant, the predictive accuracy is estimated on test instances located in drill holes that are different from the training data. The results of an extensive empirical study in the Sarcheshmeh porphyry copper deposit in Southeastern Iran illustrate that specially designed Gaussian processes with a symmetric standardization of the spatial location inputs and an anisotropic kernel yield the most accurate predictions. Furthermore, significant improvements are obtained when, besides location, information on the rock type is included in the set of predictor variables. This observation highlights the importance of carrying out detailed studies of the geological composition of the deposit to obtain more accurate ore grade predictions.     

Predicting Threshold Exceedance by Local Block Means in Soil Pollution Surveys

Soil contamination by heavy metals and organic pollutants around industrial premises is a problem in many countries around the world. Delineating zones where pollutants exceed tolerable levels is a necessity for successfully mitigating related health risks. Predictions of pollutants are usually required for blocks because remediation or regulatory decisions are imposed for entire parcels. Parcel areas typically exceed the observation support, but are smaller than the survey domain. Mapping soil pollution therefore involves a local change of support. The goal of this work is to find a simple, robust, and precise method for predicting block means (linear predictions) and threshold exceedance by block means (nonlinear predictions) from data observed at points that show a spatial trend. By simulations, we compared the performance of universal block kriging (UK), Gaussian conditional simulations (CS), constrained (CK), and covariance-matching constrained kriging (CMCK), for linear and nonlinear local change of support prediction problems. We considered Gaussian and positively skewed spatial processes with a nonstationary mean function and various scenarios for the autocorrelated error. The linear predictions were assessed by bias and mean square prediction error and the nonlinear predictions by bias and Peirce skill scores.     

Geostatistics for Power Models of Gaussian Fields

This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions.     

Spatial interpolation of severely skewed data with several peak values by the approach integrating kriging and triangular irregular network interpolation

It was not unusual in soil and environmental studies that the distribution of data is severely skewed with several high peak values, which causes the difficulty for Kriging with data transformation to make a satisfied prediction. This paper tested an approach that integrates kriging and triangular irregular network interpolation to make predictions. A data set consisting of total Copper (Cu) concentrations of 147 soil samples, with a skewness of 4.64 and several high peak values, from a copper smelting contaminated site in Zhejiang Province, China. The original data were partitioned into two parts. One represented the holistic spatial variability, followed by lognormal distribution, and then was interpolated by lognormal ordinary kriging. The other assumed to show the local variability of the area that near to high peak values, and triangular irregular network interpolation was applied. These two predictions were integrated into one map. This map was assessed by comparing with rank-order ordinary kriging and normal score ordinary kriging using another data set consisting of 54 soil samples of Cu in the same region. According to the mean error and root mean square error, the approach integrating lognormal ordinary kriging and triangular irregular network interpolation could make improved predictions over rank-order ordinary kriging and normal score ordinary kriging for the severely skewed data with several high peak values.     

An Empirical Comparison of Kriging Methods for Nonlinear Spatial Point Prediction

Spatial prediction is a problem common to many disciplines. A simple application is the mapping of an attribute recorded at a set of points. Frequently a nonlinear functional of the observed variable is of interest, and this calls for nonlinear approaches to prediction. Nonlinear kriging methods, developed in recent years, endeavour to do so and additionally provide estimates of the distribution of the target quantity conditional on the observations. There are few empirical studies that validate the various forms of nonlinear kriging. This study compares linear and nonlinear kriging methods with respect to precision and their success in modelling prediction uncertainty. The methods were applied to a data set giving measurements of the topsoil concentrations of cobalt and copper at more than 3000 locations in the Border Region of Scotland. The data stem from a survey undertaken to identify places where these trace elements are deficient for livestock. The comparison was carried out by dividing the data set into calibration and validation sets. No clear differences between the precision of ordinary, lognormal, disjunctive, indicator, and model-based kriging were found, neither for linear nor for nonlinear target quantities. Linear kriging, supplemented with the assumption of normally distributed prediction errors, failed to model the conditional distribution of the marginally skewed data, whereas the nonlinear methods modelled the conditional distributions almost equally well. In our study the plug-in methods did not fare any worse than model-based kriging, which takes parameter uncertainty into account.     

Normal and lognormal estimation

A comprehensive theoretical study of the problem of estimation of regionalized variables with normal or lognormal distribution is presented. Unbiased linear estimators are derived, under both assumptions that the population mean is known and unknown, and their error variance is calculated. The minimum variance kriging estimators are studied in more detail and are compared with the conditional expectations. The emphasis is on the study of lognormally distributed variates. The derived mathematical formulas are applicable to the optimal contouring of sample values with the appropriate distribution, as well as the optimal estimation of blocks of ore in mineral deposits.     

Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.     

Simple and Ordinary Multigaussian Kriging for Estimating Recoverable Reserves

Multigaussian kriging is used in geostatistical applications to assess the recoverable reserves in ore deposits, or the probability for a contaminant to exceed a critical threshold. However, in general, the estimates have to be calculated by a numerical integration (Monte Carlo approach). In this paper, we propose analytical expressions to compute the multigaussian kriging estimator and its estimation variance, thanks to polynomial expansions. Three extensions are then considered, which are essential for mining and environmental applications: accounting for an unknown and locally varying mean (local stationarity), accounting for a block-support correction, and estimating spatial averages. All these extensions can be combined; they generalize several known techniques like ordinary lognormal kriging and uniform conditioning by a Gaussian value. An application of the concepts to a porphyry copper deposit shows that the proposed “ordinary multigaussian kriging” approach leads to more realistic estimates of the recoverable reserves than the conventional methods (disjunctive and simple multigaussian krigings), in particular in the nonmineralized undersampled areas.     

对数泛克立金法在处理化探数据中的应用

本文研究如何用地质统计学方法处理服从对数正态分布的化探数据的问题。经对湖北阳新岩体铜含量数据的大量电算、分析、对比,最后选用了对数泛克立金法。由计算机自动绘出的估值图和漂移、剩余图能很好地反映该区化探趋势和异常,既优于趋势面分析和滑动平均法,也优于泛克立金法,且能获得较稳健的变差图。故此法对化探数据自动成图很有用处。     

On unbiased backtransform of lognormal kriging estimates

Lognormal kriging is an estimation technique that was devised for handling highly skewed data distributions. This technique takes advantage of a logarithmic transformation that reduces the data variance. However, backtransformed lognormal kriging estimates are biased because the nonbias term is totally dependent on a semivariogram model. This paper proposes a new approach for backtransforming lognormal kriging estimates that not only presents none of the problems reported in the literature but also reproduces the sample histogram and, consequently, the sample mean.     

Inverse theory in the earth sciences—an introductory overview with emphasis on Gandin's method of optimum interpolation

In 1963, Gandin published a monograph on “optimum interpolation for the objective analysis of meteorological fields, ? a method that is similar mathematically to geodetical least-squares prediction and collocation, simple kriging, and spectral interpolation. The common problem is the interpolation or extrapolation or estimation of a continuous spatial property from finitely many observations. Gandin 's method is presented in an inverse-theoretical context with focus on a methodological comparison with related methods. Underlying mathematical assumptions as well as geological implications are discussed. An introductory overview of inverse methods in the earth sciences is given, with emphasis on methods with a structure analysis step.     

地质统计学及其在第四纪研究中的应用

地质统计学是数学地质领域最为活跃而实用的分支,它是以区域化变量理论为基础,以变异函数为基本工具,研究那些在空间分布上既具有随机性又具有结构性的自然现象的科学。在第四纪研究中的很多特征(变量)均可看成区域化变量进行地质统计学分析。作者在讨论了经典概率论及数理统计方法简单地应用于第四纪研究可能出现的问题后,着重介绍了用于第四纪研究中的若干地质统计学方法及基本理论,同时,对地质统计学方法应用于第四纪研究中的前景进行了分析。     

Empirical Maximum Likelihood Kriging: The General Case

Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution.