Generalized sequential kriging (GSK) for heterogeneous cell size: property modeling without matrices

Sequential kriging avoids the use of matrices and resolves the issue of unstable solutions. It allows for stepwise ways to get joint estimations and cosimulations that are equivalent to the simultaneous solution. The approach is proposed as the solution for geocellular modeling with variable cell size from heterogeneous structural properties (HSPs) as required for modeling with structural constraints. Rock properties are controlled by structural domains, regions, and structural geology parameters. In some cases, rock properties are cross-correlated to formation thickness, curvature of structures, and other structural attributes. Cell thickness may be proportional to formation thickness and may enter as a conditioning property in the estimation of rock property parameters for simulation. In addition, cell volume controls the upscaling of covariance structures (i.e., regularized variograms). Structural properties are priorly modeled. Perturbation response functions (PRFs) are computed for each cell vs all possible sample point locations to facilitate sequential kriging. Upscaled PRFs are modified following conditional updating after each new data value is included in the estimation of parameters. Generalized sequential kriging is expected to become the main tool for real-time spatial modeling of 3D cellular models with HSP. In addition, some new developments related to the sequential kriging algorithm are included. Sequential kriging can be used for the estimation of parameters for simulation in the so-called unstructured grids.     

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.     

Transmissivity estimation for highly heterogeneous aquifers: comparison of three methods applied to the Edwards Aquifer,Texas, USA

Obtaining reliable hydrological input parameters is a key challenge in groundwater modeling. Although many quantitative characterization techniques exist, experience applying these techniques to highly heterogeneous real-world aquifers is limited. Three geostatistical characterization techniques are applied to the Edwards Aquifer, a limestone aquifer in south-central Texas, USA, for the purposes of quantifying the performance in an 88,000-cell groundwater model. The first method is a simple kriging of existing hydraulic conductivity data developed primarily from single-well tests. The second method involves numerical upscaling to the grid-block scale, followed by cokriging the grid-block conductivity. In the third method, the results of the second method are used to establish the prior distribution for a Bayesian updating calculation. Results of kriging alone are biased towards low values and fail to reproduce hydraulic heads or spring flows. The upscaling/cokriging approach removes most of the systematic bias. The Bayesian update reduced the mean residual by more than a factor of 10, to 6 m, approximately 2.5% of the total head variation in the aquifer. This agreement demonstrates the utility of automatic calibration techniques based on formal statistical approaches and lends further support for using the Bayesian updating approach for highly heterogeneous aquifers.     

Revisiting Multi-Gaussian Kriging with the Nataf Transformation or the Bayes’ Rule for the Estimation of Spatial Distributions

A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space.     

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).     

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.     

Determining the best search neighbourhood in reserve estimation, using geostatistical method: A case study anomaly No 12A iron deposit in central Iran

Ordinary kriging and non-linear geostatistical estimators are now well accepted methods in mining grade control and mine reserve estimation. In kriging, the search volume or ‘kriging neighbourhood’ is defined by the user. The definition of the search space can have a significant impact on the outcome of the kriging estimate. In particular, too restrictive neighbourhood, can result in serious conditional bias. Kriging is commonly described as a ‘minimum variance estimator’ but this is only true when the neighbourhood is properly selected. Arbitrary decisions about search space are highly risky. The criteria to consider when evaluating a particular kriging neighbourhood are the slope of the regression of the ‘true’ and ‘estimated’ block grades, the number of kriging negative weights and the kriging variance. Search radius is one of the most important parameters of search volume which often is determined on the basis of influence of the variogram. In this paper the above-mentioned parameters are used to determine optimal search radius.     

The updated kriging variance and optimal sample design

A number of criteria based on kriging variance calculations may be used for infill sampling design in geologic site characterization. Searching for the best new sample locations from a set of candidate locations can result in excessive computation time if these criteria and the naive rekriging are used. The relative updated kriging estimate and variance for universal kriging estimation are demonstrated as a simple kriging estimate and variance, respectively. The updated kriging variance is demonstrated as the multiplication of two kriging variances. Using these updated kriging variance equations can increase the computational speed for selecting the best new sample locations. The application results for oil rock thickness in an oilfield indicate that minimizing the average relative updated kriging variance is a useful alternative to the other criteria based on kriging variance in optimal infill sampling design for geologic site characterization.     

Compensating for estimation smoothing in kriging

Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft² slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging.     

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.     

FuzzyKrig: a comprehensive matlab toolbox for geostatistical estimation of imprecise information

Using kriging has been accepted today as the most common method of estimating spatial data in such different fields as the geosciences. To be able to apply kriging methods, it is necessary that the data and variogram model parameters be precise. To utilize the imprecise (fuzzy) data and parameters, use is made of fuzzy kriging methods. Although it has been 30 years since different fuzzy kriging algorithms were proposed, its use has not become as common as other kriging methods (ordinary, simple, log, universal, etc.); lack of a comprehensive software that can perform, based on different fuzzy kriging algorithms, the related calculations in a 3D space can be the main reason. This paper describes an open-source software toolbox (developed in Matlab) for running different algorithms proposed for fuzzy kriging. It also presents, besides a short presentation of the fuzzy kriging method and introduction of the functions provided by the FuzzyKrig toolbox, 3 cases of the software application under the conditions where: 1) data are hard and variogram model parameters are fuzzy, 2) data are fuzzy and variogram model parameters are hard, and 3) both data and variogram model parameters are fuzzy.     

Notes on the robustness of the kriging system

The robustness of the kriging system with respect to uncertainty of the theoretical variogram is investigated. Inequalities for possible changes of the kriging estimator and the estimation variance are derived. Results of a numerical study show that changes of kriging weights can be predicted partly with the help of the maximal kriging weight.     

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.     

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.     

On the influence of kriging parameters on the cartographic output—A study in mapping subglacial topography

Geostatistics provides a suite of methods, summarized as kriging, to analyze a finite data set to describe a continuous property of the Earth. Kriging methods consist of moving window optimum estimation techniques, which are based on a least-squares principle and use a spatial structure function, usually the variogram. Applications of kriging techniques have become increasingly wide-spread, with ordinary kriging and universal kriging being the most popular ones. The dependence of the final map or model on the input, however, is not generally understood. Herein we demonstrate how changes in the kriging parameters and the neighborhood search affect the cartographic result. Principles are illustrated through a glaciological study. The objective is to map ice thickness and subglacial topography of Storglaciären, Kebnekaise Massif, northern Sweden, from several sets of radio-echo soundings and hot water drillings. New maps are presented.     

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.     

Mapping Curvilinear Structures with Local Anisotropy Kriging

Interpolating geo-data with curvilinear structures using geostatistics is often disappointing. Channels, for example, become disconnected sets of lakes when interpolated from point data. In order to improve the interpolation of geological structures (e.g., curvilinear structures), we present a new form of kriging, local anisotropy kriging (LAK). Local anisotropy kriging combines a gradient algorithm from image analysis with kriging in an iterative way. After an initial standard kriging interpolation, the gradient algorithm determines the local anisotropy for each cell in the grid using a search area around the cell. Subsequently, kriging is carried out with the spatially varying anisotropy. The anisotropy calculation and subsequent kriging steps will then succeed until the result is satisfactory in the way of reproducing the curvilinear structures. Depending on the size of the search area more or less detail in the geological structures can be reproduced with LAK. Using test examples we show that LAK interpolates data with curvilinear structures more realistically than standard kriging. In a real world case, using bathymetric data of the Oosterschelde estuary, LAK also proves to be quantitatively superior to standard kriging. Absolute interpolation errors are decreased by 23%. Local anisotropy kriging only uses information from point data, which makes the method very objective, it only presents “what the data can tell.”     

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.     

Kriging without negative weights

Under a constant drift, the linear kriging estimator is considered as a weighted average ofn available sample values. Kriging weights are determined such that the estimator is unbiased and optimal. To meet these requirements, negative kriging weights are sometimes found. Use of negative weights can produce negative block grades, which makes no practical sense. In some applications, all kriging weights may be required to be nonnegative. In this paper, a derivation of a set of nonlinear equations with the nonnegative constraint is presented. A numerical algorithm also is developed for the solution of the new set of kriging equations.