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.     

Application of universal kriging for estimation of earthquake ground motion: Statistical significance of results

Universal kriging is compared with ordinary kriging for estimation of earthquake ground motion. Ordinary kriging is based on a stationary random function model; universal kriging is based on a nonstationary random function model representing first-order drift. Accuracy of universal kriging is compared with that for ordinary kriging; cross-validation is used as the basis for comparison. Hypothesis testing on these results shows that accuracy obtained using universal kriging is not significantly different from accuracy obtained using ordinary kriging. Tests based on normal distribution assumptions are applied to errors measured in the cross-validation procedure;t andF tests reveal no evidence to suggest universal and ordinary kriging are different for estimation of earthquake ground motion. Nonparametric hypothesis tests applied to these errors and jackknife statistics yield the same conclusion: universal and ordinary kriging are not significantly different for this application as determined by a cross-validation procedure. These results are based on application to four independent data sets (four different seismic events).     

Mapping Soil Particle Size Fractions Using Compositional Kriging,Cokriging and Additive Log-ratio Cokriging in Two Case Studies

Information on the spatial distribution of soil particle-size fractions (psf) is required for a wide range of applications. Geostatistics is often used to map spatial distribution from point observations; however, for compositional data such as soil psf, conventional multivariate geostatistics are not optimal. Several solutions have been proposed, including compositional kriging and transformation to a composition followed by cokriging. These have been shown to perform differently in different situations, so that there is no procedure to choose an optimal method. To address this, two case studies of soil psf mapping were carried out using compositional kriging, log-ratio cokriging, cokriging, and additive log-ratio cokriging; and the performance of Mahalanobis distance as a criterion for choosing an optimal mapping method was tested. All methods generated very similar results. However, the compositional kriging and cokriging results were slightly more similar to each other than to the other pair, as were log-ratio cokriging and additive log-ratio cokriging. The similar results of the two methods within each pair were due to similarities of the methods themselves, for example, the same variogram models and prediction techniques, and the similar results between the two pairs were due to the mathematical relationship between original and log-ratio transformed data. Mahalanobis distance did not prove to be a good indicator for selecting an optimal method to map soil psf.     

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.     

Application of disjunctive kriging for estimating economic grade distribution in an iron ore deposit: A case study of the Choghart North Anomaly,Iran

Most significant iron ore deposits in Iran are located in Central Iran Zone. These deposits belong to the Bafq mining district. The Bafq mining district is located in the Early Cambrian Kashmar-Kerman volcanic arc of Central Iran. Linear estimation of regionalized variables (for example by inverse distance weighting or ordinary Kriging) results in relatively high estimation variances, i.e. the estimates have very low precision. Assessment of project economics (or other critical decision making) based on linear estimation is therefore risky. Non-linear estimation methods like disjunctive kriging perform better and the lower estimation variance allows less risky economic decision-making. Another advantage of disjunctive kriging is that it allows estimation of functions of the primary variable, which here is the grade (Fe %) of the ore. In particular it permits estimation of indicator functions defined using thresholds on the primary variable. This paper is devoted to application of disjunctive kriging method in Choghart North Anomaly iron ore deposit in Central Iran, Yazd province, Iran. In this study, the Fe concentration of Choghart North Anomaly iron ore deposit was modelled and estimated. The exploration data consists of borehole samples measuring the Fe concentration. A Gaussian isofactorial model is fitted to these data and disjunctive kriging was used to estimate the regionalized variable (Fe %) at unsampled locations and to assess the probabilities that the actual concentrations exceed a threshold value at a given location. Consequently a three dimensional model of probability of exceeding a threshold value and the estimated value are provided by disjunctive kriging to divide the ore into an economic and uneconomic part on the basis of estimation of indicator functions using thresholds grades defined on point support. The tools and concepts are complemented by a set of computer programs that are applied to the case study. The study showed that disjunctive kriging can be applied successfully for modeling the grade of an ore deposit. Results showed that the correlation between the estimated value and real value at locations close to each other is 81.9%.     

Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains

The continuous-lag Markov chain provides a conceptually simple, mathematically compact, and theoretically powerful model of spatial variability for categorical variables. Markov chains have a long-standing record of applicability to one-dimensional (1-D) geologic data, but 2- and 3-D applications are rare. Theoretically, a multidimensional Markov chain may assume that 1-D Markov chains characterize spatial variability in all directions. Given that a 1-D continuous Markov chain can be described concisely by a transition rate matrix, this paper develops 3-D continuous-lag Markov chain models by interpolating transition rate matrices established for three principal directions, say strike, dip, and vertical. The transition rate matrix for each principal direction can be developed directly from data or indirectly by conceptual approaches. Application of Sylvester's theorem facilitates establishment of the transition rate matrix, as well as calculation of transition probabilities. The resulting 3-D continuous-lag Markov chain models then can be applied to geo-statistical estimation and simulation techniques, such as indicator cokriging, disjunctive kriging, sequential indicator simulation, and simulated annealing.     

Disjunctive Kriging with Hard and Imprecise Data

This paper presents a methodology for assessing local probability distributions by disjunctive kriging when the available data set contains some imprecise measurements, like noisy or soft information or interval constraints. The basic idea consists in replacing the set of imprecise data by a set of pseudohard data simulated from their posterior distribution; an iterative algorithm based on the Gibbs sampler is proposed to achieve such a simulation step. The whole procedure is repeated many times and the final result is the average of the disjunctive kriging estimates computed from each simulated data set. Being data-independent, the kriging weights need to be calculated only once, which enables fast computing. The simulation procedure requires encoding each datum as a pre-posterior distribution and assuming a Markov property to allow the updating of pre-posterior distributions into posterior ones. Although it suffers some imperfections, disjunctive kriging turns out to be a much more flexible approach than conditional expectation, because of the vast class of models that allows its computation, namely isofactorial models.     

Comparative performance of indicator algorithms for modeling conditional probability distribution functions

This paper compares the performance of four algorithms (full indicator cokriging. adjacent cutoffs indicator cokriging, multiple indicator kriging, median indicator kriging) for modeling conditional cumulative distribution functions (ccdf).The latter three algorithms are approximations to the theoretically better full indicator cokriging in the sense that they disregard cross-covariances between some indicator variables or they consider that all covariances are proportional to the same function. Comparative performance is assessed using a reference soil data set that includes 2649 locations at which both topsoil copper and cobalt were measured. For all practical purposes, indicator cokriging does not perform better than the other simpler algorithms which involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is found to be higher for cokriging algorithms, especially when constraints on the kriging weights are applied.     

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。     

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

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.     

Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization

 Spatial and temporal behavior of hydrochemical parameters in groundwater can be studied using tools provided by geostatistics. The cross-variogram can be used to measure the spatial increments between observations at two given times as a function of distance (spatial structure). Taking into account the existence of such a spatial structure, two different data sets (sampled at two different times), representing concentrations of the same hydrochemical parameter, can be analyzed by cokriging in order to reduce the uncertainty of the estimation. In particular, if one of the two data sets is a subset of the other (that is, an undersampled set), cokriging allows us to study the spatial distribution of the hydrochemical parameter at that time, while also considering the statistical characteristics of the full data set established at a different time. This paper presents an application of cokriging by using temporal subsets to study the spatial distribution of nitrate concentration in the aquifer of the Lucca Plain, central Italy. Three data sets of nitrate concentration in groundwater were collected during three different periods in 1991. The first set was from 47 wells, but the second and the third are undersampled and represent 28 and 27 wells, respectively. Comparing the result of cokriging with ordinary kriging showed an improvement of the uncertainty in terms of reducing the estimation variance. The application of cokriging to the undersampled data sets reduced the uncertainty in estimating nitrate concentration and at the same time decreased the cost of the field sampling and laboratory analysis. Received: 23 July 1997 · Accepted: 31 March 1998     

Disjunctive kriging, universal kriging, or no kriging: Small sample results with simulated fields

This paper provides a comparison between linear (universal) and nonlinear (disjunctive) kriging estimators when they are computed from small samples chosen randomly on simulated stationary and nonstationary fields. Point estimation results are reported. In all cases considered, kriging estimators were found better than a local mean estimator, with universal kriging either better than or as good as disjunctive kriging. The latter, which is suited to handle stationary fields, did not provide more accurate estimates because the use of small samples led to inconsistencies in the assumed bivariate model. Universal kriging was particularly better with nonstationary fields.     

Order relation correction experiments for probability kriging

Consideration of order relations is key to indicator kriging, indicator cokriging, and probability kriging, especially for the latter two methods wherein the additional modeling of cross-covariance contributes to an increased chance of violating order relations. Herein, Gaussian-type curves are fit to estimates of the cumulative distribution function (cdf) at data quantiles to: (1) yield smoothed estimates of the cdf; and (2) to correct for violations of order relations (i.e., to correct for situations wherein the estimate of the cdf for a larger quantile is less than that for a smaller quantile). Smoothed estimates of the cdf are sought as a means to improve the approximation to the integral equation for the expected value of the regionalized variable in probability kriging. Experiments show that this smoothing yields slightly improved estimation of the expected value (in probability kriging). Another experiment, one that uses the same variogram for all indicator functions, does not yield improved estimates.Presented at the 25th Anniversary Meeting of the IAMG, Prague, Czech Republic, October 10–15, 1993.     

Comparison of kriging techniques in a space-time context

Space-time processes constitute a particular class, requiring suitable tools in order to predict values in time and space, such as a space-time variogram or covariance function. The space-time co-variance function is defined and linked to the Linear Model of Coregionalization under second-order space-time stationarity. Simple and ordinary space-time kriging systems are compared to simple and ordinary cokriging and their differences for unbiasedness conditions are underlined. The ordinary space-time kriging estimation then is applied to simulated data. Prediction variances and prediction errors are compared with those for ordinary kriging and cokriging under different unbiasedness conditions using a cross-validation. The results show that space-time kriging tend to produce lower prediction variances and prediction errors that kriging and cokriging.     

Multivariable spatial prediction

For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.     

Restricted kriging for mixture of grade models

A modified type of kriging, referred to as restricted kriging (RK), is proposed in this study. The method incorporates constraints on different grade classes to restrict the influence of the samples having different likelihoods in estimation. RK is motivated by the estimation of mineral reserves when grades have highly skewed distributions. Ordinary kriging tends to produce an overly smoothed interpolated surface by underestimating high grades and overestimating low grades. The fact that ordinary kriging gives a uniform prior treatment to all samples independent of their values is a major factor associated with this smoothing effect. The new approach differentiates each grade portion by preselected cutoffs. RK is developed for a single cutoff and then extended into a general form for any finite number of cutoffs. Restricted cokriging (RCK) is also formulated to simultaneously estimate a set of random functions with restriction conditions. Methods are suggested for determination of the probabilities of occurrence of different grade portions. Finally, the new approach is demonstrated on a case study of an epithermal gold deposit.     

Estimation of potential failure risks in a mine slope using indicator kriging

Indicator kriging has been applied to the study of failure mechanisms in a mine slope in Minas Gerais, Brazil, to estimate potential failure risks in limited areas along this slope. Timbopeba Mine, Vale Company, is an open pit iron mine situated in the Quadrilátero Ferrífero, a very important mining district in Minas Gerais. A slope excavated in quartzite with a maximum height of 200 m at the time of this study, has presented many failure problems involving the sliding of blocks formed by discontinuities. These blocks are of limited size in comparison to the dimensions of the overall slope. They appear along the entire slope, wherever discontinuity orientations have led to the kinematic feasibility of these blocks. Geostatistics permits the estimation of local failure probability distributions associated to these local failures, which would not be possible with traditional statistical models. The geostatistical method employed in this study, indicator kriging, is quite suitable because it is unnecessary to assume a particular global distribution of the phenomena being modeled. The model was used for locating areas with a great tendency for sliding failure, as it considers the local spatial variability of discontinuity orientations. Copyright © 2010 John Wiley & Sons, Ltd.     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

Indicator principal component kriging

An alternative to multiple indicator kriging is proposed which approximates the full coindicator kriging system by kriging the principal components of the original indicator variables. This transformation is studied in detail for the biGaussian model. It is shown that the cross-correlations between principal components are either insignificant or exactly zero. This result allows derivation of the conditional cumulative density function (cdf) by kriging principal components and then applying a linear back transform. A performance comparison based on a real data set (Walker Lake) is presented which suggests that the proposed method achieves approximation of the conditional cdf equivalent to indicator cokriging but with substantially less variogram modeling effort and at smaller computational cost.