New distance measures: The route toward truly non-Gaussian geostatistics

The projection or minimum error norm algorithm does not require that the distance measure be a variogram. In non-Gaussian cases, the traditional variogram distance measure leading to minimization of an error variance offers no definite advantage. Other distance measures, more outlierresistant than the variogram, are proposed which fulfill the condition of the projection theorem. The resulting minimum error norms provide the same data configurations ranking as traditionally obtained from kriging variances. A case study based on actual digital terrain data is presented.This paper was presented by title at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Isotropic Variogram Matrix Functions on Spheres

This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ ², log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization.     

Robust Resampling Confidence Intervals for Empirical Variograms

The variogram function is an important measure of the spatial dependencies of a geostatistical or other spatial dataset. It plays a central role in kriging, designing spatial studies, and in understanding the spatial properties of geological and environmental phenomena. It is therefore important to understand the variability attached to estimates of the variogram. Existing methods for constructing confidence intervals around the empirical variogram either rely on strong assumptions, such as normality or known variogram function, or are based on resampling blocks and subject to edge effect biases. This paper proposes two new procedures for addressing these concerns: a quasi-block-bootstrap and a quasi-block-jackknife. The new methods are based on transforming the data to decorrelate it based on a fitted variogram model, resampling blocks from the decorrelated data, and then recorrelating. The coverage properties of the new confidence intervals are compared by simulation to a number of existing resampling-based intervals. The proposed quasi-block-jackknife confidence interval is found to have the best properties of all of the methods considered across a range of scenarios, including normally and lognormally distributed data and misspecification of the variogram function used to decorrelate the data.     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

Kriging with imprecise (fuzzy) variograms. I: Theory

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

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

Teacher's Aide Variogram Interpretation and Modeling

The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology.     

Geostatistically consistent history matching of 3D oil-and-gas reservoir models

An approach for geostatistically consistent matching of 3D flow simulation models and 3D geological models is proposed. This approach uses an optimization algorithm based on identification of the parameters of the geostatistical model (for example, the variogram parameters, such as range, sill, and nugget effect). Here, the inverse problem is considered in the greatest generality taking into account facies heterogeneity and the variogram anisotropy. The correlation dependence parameters (porosity-to-log permeability) are clarified for each single facies.     

Teacher''s Aide Variogram Interpretation and Modeling

The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology.     

Indicator Variogram Models: Do We Have Much Choice?

Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.     

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.     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

Flexible spectral methods for the generation of random fields with power-law semivariograms

Random field generators serve as a tool to model heterogeneous media for applications in hydrocarbon recovery and groundwater flow. Random fields with a power-law variogram structure, also termed fractional Brownian motion (fBm) fields, are of interest to study scale dependent heterogeneity effects on one-phase and two-phase flow. We show that such fields generated by the spectral method and the Inverse Fast Fourier Transform (IFFT) have an incorrect variogram structure and variance. To illustrate this we derive the prefactor of the fBm spectral density function, which is required to generate the fBm fields. We propose a new method to generate fBm fields that introduces weighting functions into the spectral method. It leads to a flexible and efficient algorithm. The flexibility permits an optimal choice of summation points (that is points in frequency space at which the weighting function is calculated) specific for the autocovariance structure of the field. As an illustration of the method, comparisons between estimated and expected statistics of fields with an exponential variogram and of fBm fields are presented. For power-law semivariograms, the proposed spectral method with a cylindrical distribution of the summation points gives optimal results.     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

The Necessity of a Multiple-Point Prior Model

Any interpolation, any hand contouring or digital drawing of a map or a numerical model necessarily calls for a prior model of the multiple-point statistics that link together the data to the unsampled nodes, then these unsampled nodes together. That prior model can be implicit, poorly defined as in hand contouring; it can be explicit through an algorithm as in digital mapping. The multiple-point statistics involved go well beyond single-point histogram and two-point covariance models; the challenge is to define algorithms that can control more of such statistics, particularly those that impact most the utilization of the resulting maps beyond their visual appearance. The newly introduced multiple-point simulation (mps) algorithms borrow the high order statistics from a visually and statistically explicit model, a training image. It is shown that mps can simulate realizations with high entropy character as well as traditional Gaussian-based algorithms, while offering the flexibility of considering alternative training images with various levels of low entropy (organized) structures. The impact on flow performance (spatial connectivity) of choosing a wrong training image among many sharing the same histogram and variogram is demonstrated.     

Highly Robust Variogram Estimation

The classical variogram estimator proposed by Matheron is not robust against outliers in the data, nor is it enough to make simple modifications such as the ones proposed by Cressie and Hawkins in order to achieve robustness. This paper proposes and studies a variogram estimator based on a highly robust estimator of scale. The robustness properties of these three estimators are analyzed and compared. Simulations with various amounts of outliers in the data are carried out. The results show that the highly robust variogram estimator improves the estimation significantly.     

多物源条件下的储层地质建模方法

濮城沙三中油藏具有两个主物源,分别为NE向与SE向。油藏数值模拟需要在一套地质网格中对其进行模拟。经典的地质统计学利用变差函数描述区域化变量的空间几何结构特性。变差函数的计算是基于两点进行统计的,对其描述主要涉及方位角、变程、块金值和基台值。为了在一套模拟网格中模拟出多个物源条件下储层的分布特征,必须在不同的位置设置不同的变差函数参数。文中给出了两种方法实现这一目的:一是采用人为分区,把不同物源影响的区域分成不同的区块,分别对不同的区块设置不同的变差函数参数;二是采用变方位角,即根据不同的位置设置不同的变差函数方位角。这两种方法都实现了在一套网格中模拟具有多个物源方向的储层分布,更真实地再现了储层的空间展布特征。     

On the Use of Non-Euclidean Distance Measures in Geostatistics

In many scientific disciplines, straight line, Euclidean distances may not accurately describe proximity relationships among spatial data. However, non-Euclidean distance measures must be used with caution in geostatistical applications. A simple example is provided to demonstrate there are no guarantees that existing covariance and variogram functions remain valid (i.e. positive definite or conditionally negative definite) when used with a non-Euclidean distance measure. There are certain distance measures that when used with existing covariance and variogram functions remain valid, an issue that is explored. The concept of isometric embedding is introduced and linked to the concepts of positive and conditionally negative definiteness to demonstrate classes of valid norm dependent isotropic covariance and variogram functions, results many of which have yet to appear in the mainstream geostatistical literature or application. These classes of functions extend the well known classes by adding a parameter to define the distance norm. In practice, this distance parameter can be set a priori to represent, for example, the Euclidean distance, or kept as a parameter to allow the data to choose the metric. A simulated application of the latter is provided for demonstration. Simulation results are also presented comparing kriged predictions based on Euclidean distance to those based on using a water metric.     

Variogram Matrix Functions for Vector Random Fields with Second-Order Increments

The variogram matrix function is an important measure for the dependence of a vector random field with second-order increments, and is a useful tool for linear predication or cokriging. This paper proposes an efficient approach to construct variogram matrix functions, based on three ingredients: a univariate variogram, a conditionally negative definite matrix, and a Bernstein function, and derives three classes of variogram matrix functions for vector elliptically contoured random fields. Moreover, various dependence structures among components can be derived through appropriate mixture procedures demonstrated in this paper. We also obtain covariance matrix functions for second-order vector random fields through the Schoenberg–Lévy kernels.     

Using Simultaneous Diagonalization to Identify a Space–Time Linear Coregionalization Model

Although there are multiple methods for modeling matrix covariance functions and matrix variograms in the geostatistical literature, the linear coregionalization model is still widely used. In particular it is easy to check to ensure whether the matrix covariance function is positive definite or that the matrix variogram is conditionally negative definite. One of the difficulties in using a linear coregionalization model is in determining the number of basic structures and the corresponding covariance functions or variograms. In this paper, a new procedure is given for identifying the basic structures of the space–time linear coregionalization model and modeling the matrix variogram. This procedure is based on the near simultaneous diagonalization of the sample matrix variograms computed for a set of spatiotemporal lags. A case study using a multivariate spatiotemporal data set provided by the Environmental Protection Agency of Lombardy, Italy, illustrates how nearly simultaneous diagonalization of the empirical matrix variograms simplifies modeling of the matrix variograms. The new methodology is compared with a previous one by analyzing various indices and statistics.