Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

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.     

Comparing some approximate methods for building local confidence intervals for predicting regionalized variables

Approximate local confidence intervals can be produced by nonlinear methods designed to estimate indicator variables. The most precise of these methods, the conditional expectation, can only be used in practice in the multi-Gaussian context. Theoretically, less efficient methods have to be used in more general cases. The methods considered here are indicator kriging, probability kriging (indicator-rank co-kriging), and disjunctive kriging (indicator co-kriging). The properties of these estimators are studied in this paper in the multi-Gaussian context, for this allows a more detailed study than under more general models. Conditional distribution approximation is first studied. Exact results are given for mean squared errors and conditional bias. Then conditional quantile estimators are compared empirically. Finally, confidence intervals are compared from the points of view of bias and precision.     

Transition probability-based indicator geostatistics

Traditionally, spatial continuity models for indicator variables are developed by empirical curvefitting to the sample indicator (cross-) variogram. However, geologic data may be too sparse to permit a purely empirical approach, particularly in application to the subsurface. Techniques for model synthesis that integrate hard data and conceptual models therefore are needed. Interpretability is crucial. Compared with the indicator (cross-) variogram or indicator (cross-) covariance, the transition probability is more interpretable. Information on proportion, mean length, and juxtapositioning directly relates to the transition probability: asymmetry can be considered. Furthermore, the transition probability elucidates order relation conditions and readily formulates the indicator (co)kriging equations.     

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.     

Mapping by simple indicator kriging

The problem of predicting the type of an unsampled point in a two-type map is addressed using a procedure called simple indicator kriging. This procedure estimates the conditional probability that the point is of one type or the other given the types of sample points. Simple indicator kriging is applied to examples of certain map models. It is shown to perform well, and in some cases to be exact in a specified sense. Use of simple indicator kriging to estimate multivariate normal quandrant probabilities is explored.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

Interpolation and mapping of probabilities for geochemical variables exhibiting spatial intermittency

In monitoring a minor geochemical element in groundwater or soils, a background population of values below the instrumental detection limit is frequently present. When those values are found in the monitoring process, they are assigned to the detection limit which, in some cases, generates a probability mass in the probability density function of the variable at that value (the minimum value that can be detected). Such background values could distort both the estimation of the variable at nonsampled locations and the inference of the spatial structure of variability of the variable. Two important problems are the delineation of areas where the variable is above the detection limit and the estimation of the magnitude of the variables inside those areas. The importance of these issues in geochemical prospecting or in environmental sciences, in general related with contamination and environmental monitoring, is obvious. In this paper the authors describe the two-step procedure of indicator kriging and ordinary kriging and compare it with empirical maximum likelihood kriging. The first approach consists of using a binary indicator variable for estimating the probability of a location being above the detection limit, plus ordinary kriging conditional to the location being above the detection limit. An estimation variance, however, is not available for that estimator. Empirical maximum likelihood kriging, which was designed to deal with skew distributions, can also deal with an atom at the origin of the distribution. The method uses a Bayesian approach to kriging and gives intermittency in the form of a probability map, its estimates providing a realistic assessment of their estimation variance. The pros and cons of each method are discussed and illustrated using a large dataset of As concentration in groundwater. The results of the two methods are compared by cross-validation.     

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%.     

Factorial and indicator kriging methods using a geographic information system to delineate spatial variation and pollution sources of soil heavy metals

This study uses factorial and indicator kriging methods with a geographic information system (GIS), to overlay and delineate the variation and pollution sources of soil heavy metals in fields in the north of Changhua County, Taiwan. Accordingly, the spatial variation and pollution probability patterns of soil cadmium, chromium and copper are estimated to support further environmental monitoring, remediation and planning. The overlay maps of industrial plants, irrigation channels and kriging estimates, constitute a GIS. This reveals that the short- and long-range variations and soil pollution by Cd, Cr and Cu strongly corresponded to the locations of irrigation systems and industrial plants in the area of study. The maps also verify most locations of suspected pollution sources from the GIS database. Moreover, the probability spatial patterns estimated by indicator kriging, based on the pollution thresholds of Cd, Cr and Cu, were also highly correlated with irrigation channels and industrial plants. The GIS, with factorial and indicator kriging under different combinations of variations and pollution probabilities of soil Cd, Cr and Cu, provide alternative scenarios for delineating the areas that contain high variation and high pollution probability of these heavy metals, to enable decision makers to identify pollution sources, and to further investigate and remedy soil pollution.     

Monotone Regression and Correction for Order Relation Deviations in Indicator Kriging

The indicator kriging (IK) is one of the most efficient nonparametric methods in geo-statistics. The order relation problem in the conditional cumulative distribution values obtained by IK is the most severe drawback of it. The correction of order relation deviations is an essential and important part of IK approach. A monotone regression was proposed as a new correction method which could minimize the deviation from original quintiles value, although, ensuring all order relations.     

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.     

Importance of Orthogonalization Algorithm in Modeling Conditional Distributions by Orthogonal Transformed Indicator Methods

The orthogonal transformed indicator approach to conditional cumulative distribution functions is based on the decomposition of the indicator variogram matrix as a matrix product. This paper explores the manner in which the decomposition algorithm affects the conditional cumulative distribution function as estimated by orthogonal transformed indicator kriging. Five decomposition algorithms are considered: spectral, Cholesky, symmetric, Cholesky-spectral, and simultaneous decompositions. Impact of the algorithms on spatial orthogonality and order relations problems is examined and their performances together with indicator kriging are compared using a real dataset.     

Indicator Kriging without Order Relation Violations

Indicator kriging (IK) is a spatial interpolation technique aimed at estimating the conditional cumulative distribution function (ccdf) of a variable at an unsampled location. Obtained results form a discrete approximation to this ccdf, and its corresponding discrete probability density function (cpdf) should be a vector, where each component gives the probability of an occurrence of a class. Therefore, this vector must have positive components summing up to one, like in a composition in the simplex. This suggests a simplicial approach to IK, based on the algebraic-geometric structure of this sample space: simplicial IK actually works with log-odds. Interpolated log-odds can afterwards be easily re-expressed as the desired cpdf or ccdf. An alternative but equivalent approach may also be based on log-likelihoods. Both versions of the method avoid by construction all conventional IK standard drawbacks: estimates are always within the (0,1) interval and present no order-relation problems (either with kriging or co-kriging). Even the modeling of indicator structural functions is clarified.     

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.     

Confidence Intervals of Quantiles in Hydrology Computed by an Analytical Method

Fitting probability distributions to hydrologic data samples is widely used for quantile estimation purposes. The estimated quantile (X^_T) is related to a return period (T). The confidence interval associated with each of the estimates has been calculated empirically, up until now, supposing that the quantile estimator is normally distributed. In this study, it is shown that the confidence interval follows a normal distribution only in the central part of the distribution. The real confidence limits are computed analytically, by defining and integrating the probability density function of the confidence interval. The results with an important number of hydrologic samples show that the upper confidence limits are significantly underestimated towards the tail of the distribution, when determined using the normality approximation for the quantile estimator.     

Application of geostatistics to identify gold-rich areas in the Finisterre-Fervenza region,NW Spain

Three univariate geostatistical methods of estimation are applied to a geochemical data set. The studied methods are: ordinary kriging (cross-validation), factorial kriging, and indicator kriging. These techniques use the probabilistic and spatial behaviour of geochemical variables, giving a tool for identifying potential anomalous areas to locate mineralization. Ordinary kriging is easy to apply and to interpret the results. It has the advantage of using the same experimental grid points for its estimates, and no additional grid points are needed. Factorial kriging decomposes the raw variable into as many components as there are identified structures in the variogram. This, however, is a complex method and its application is more difficult than that of ordinary or indicator kriging. The main advantages of indicator kriging are that data are used by their rank order, being more robust about outlier values, and that the presentation of results is simple. Nevertheless, indicator kriging is incapable of separating anomalous values and the high values from the background, which have a behaviour different to the anomaly. In this work, the results of the application of these 3 kriging methods to a set of mineral exploration data obtained from a geochemical survey carried out in NW Spain are presented. This area is characterised by the presence of Au mineral occurrences. The kriging methods were applied to As, considered as a pathfinder of Au in this area. Numerical treatment of Au is not applicable, because it presents most values equal to the detection limit, and a series of extreme values. The results of the application of ordinary kriging, factorial kriging and indicator kriging to As make possible the location of a series of rich values, sited along a N–S shear zone, considered a structure related to the presence of Au.     

Two Ordinary Kriging Approaches to Predicting Block Grade Distributions

Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables (transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates of the recoverable reserves that do not utilize the mean value of the normal score data. The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary indicator kriging, which show strong deviations from conditional expectation.     

Application of turning bands technique to simulate values of copper ore deposit parameters in Rudna mine (Lubin-Sieroszowice region in SW part of Poland)

ABSTRACT

The turning bands simulation is a valuable and highly useful tool in solving various geological-mining, environmental and geological-engineering problems when it is essential to determine the uncertainty of the estimates of simulated values Z_s (realizations) and assess the risk. This paper presents an investigative methodology and the results of calculations connected with the use of conditional turning bands simulation and bundled indicator kriging, making it possible to analyse the risk at different levels of uncertainty in the solution of optimization of the exploitation problems encountered in the mining of the polymetallic copper ore deposits in the Lubin-Sieroszowice region (Foresudetic monocline, the SW part of Poland). Examples of the evaluation of simulated values Z_s and probability P average values Z* of the deposit parameters within the block located in the Rudna Mine (the block R-3) area are provided.     

Correcting the Smoothing Effect of Estimators: A Spectral Postprocessor

The postprocessing algorithm introduced by Yao for imposing the spectral amplitudes of a target covariance model is shown to be efficient in correcting the smoothing effect of estimation maps, whether obtained by kriging or any other interpolation technique. As opposed to stochastic simulation, Yao's algorithm yields a unique map starting from an original, typically smooth, estimation map. Most importantly it is shown that reproduction of a covariance/semivariogram model (global accuracy) is necessarily obtained at the cost of local accuracy reduction and increase in conditional bias. When working on one location at a time, kriging remains the most accurate (in the least squared error sense) estimator. However, kriging estimates should only be listed, not mapped, since they do not reflect the correct (target) spatial autocorrelation. This mismatch in spatial autocorrelation can be corrected via stochastic simulation, or can be imposed a posteriori via Yao's algorithm.