共查询到20条相似文献，搜索用时 62 毫秒
1.
Olivier Dubrule 《Mathematical Geosciences》2017,49(4):441-465
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. 相似文献
2.
Jinchi Chu 《Mathematical Geology》1996,28(7):923-936
Sequential Indicator Simulation (SIS), although widely used, is relatively slow, and requires tedious inference of a large number of indicator variogram models. SIS is designed only to estimate class proportions and to reproduce indicator variogram models; the statistics of the continuous attribute being simulated,z-histogram and variogram, may be poorly reproduced. Several implementations of the SIS algorithm are proposed resulting in better reproduction of statistics yet with better CPU performance. 相似文献
3.
James R. Carr 《Mathematical Geology》1994,26(5):605-621
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. 相似文献
4.
Comparative performance of indicator algorithms for modeling conditional probability distribution functions 总被引：1，自引：0，他引：1
P. Goovaerts 《Mathematical Geology》1994,26(3):389-411
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. 相似文献
5.
On Visualization for Assessing Kriging Outcomes 总被引：7，自引：0，他引：7
James R. Carr 《Mathematical Geology》2002,34(4):421-433
Extant opinion about kriging is that all weights should be positive. Visualizations rendered by converting kriged grids to digital images are presented to show that negative weights may be beneficial to some spatial problems. In particular, variogram models with zero-valued nuggets, already well known to minimize smoothing through kriging, result in a visual resolution substantially superior to that from kriging with a variogram model having a nonzero nugget value in application to satellite acquired data. Negative weights are more likely when using variogram models with zero-valued nuggets, but resultant visualizations often show a smoother transition between extreme data values. This is true even when a variogram model having a nugget value of zero is not optimum with respect to mean square error, as is demonstrated using a nitrate data set. An analogy to digital image processing is used to suggest that the influence of negative weights in kriging is similar to a high-boost kernel. 相似文献
6.
Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented. 相似文献
7.
Heuristic models for indicator variograms and their parameters (practical nugget effect and range) are proposed for a bivariate normal distribution with spherical correlogram. These models can be used conveniently as a preliminary check for bivariate normality. In the general non-Gaussian case, indicator variogram models for multiple threshold values must verify a certain number of order relations (inequalities) established directly from the properties of a general bivariate cumulative distribution function. An interesting, little-known maximum hole effect for indicator correlation is pointed out. 相似文献
8.
Nonstationarity of the Mean and Unbiased Variogram Estimation: Extension of the Weighted Least-Squares Method 总被引：1，自引：0，他引：1
When concerned with spatial data, it is not unusual to observe a nonstationarity of the mean. This nonstationarity may be modeled through linear models and the fitting of variograms or covariance functions performed on residuals. Although it usually is accepted by authors that a bias is present if residuals are used, its importance is rarely assessed. In this paper, an expression of the variogram and the covariance function is developed to determine the expected bias. It is shown that the magnitude of the bias depends on the sampling configuration, the importance of the dependence between observations, the number of parameters used to model the mean, and the number of data. The applications of the expression are twofold. The first one is to evaluate a priori the importance of the bias which is expected when a residuals-based variogram model is used for a given configuration and a hypothetical data dependence. The second one is to extend the weighted least-squares method to fit the variogram and to obtain an unbiased estimate of the variogram. Two case studies show that the bias can be negligible or larger than 20%. The residual-based sample variogram underestimates the total variance of the process but the nugget variance may be overestimated. 相似文献
9.
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. 相似文献
10.
Teacher''s Aide Variogram Interpretation and Modeling 总被引：13，自引：0，他引：13
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. 相似文献
11.
Four variogram models for regional groundwater geochemical data are presented. These models were developed from an empirical study of the sample variograms for more than 10 elements in groundwaters from two geologic regions in the Plainview quandrangle, Texas. A procedure is given for the estimation of the variogram in the isotropic and anisotropic case. The variograms were found useful for quantifying the differences in spatial variability for elements within a geologic unit and for elements in different geologic units. Additionally, the variogram analysis enables assessment of the assumption of statistical independence of regional samples which is commonly used in many statistical procedures. The estimated variograms are used in computation of kriged estimates for the Plainview quadrangle data. The results indicate that an inverse distance weighting model was superior for prediction than simple kriging with the particular variograms used. 相似文献
12.
I. C. Lemmer 《Mathematical Geosciences》1986,18(7):605-623
The theory of mononodal variography developed in the preceeding paper is checked against a simulated deposit consisting of 60,500 grade values, called Stanford II. In the case of this deposit at least, assumptions underlying the concept of mononodal variography are borne out accurately. In particular, a linear relationship does exist indeed between indicator and grade variogram values of Stanford II at corresponding lags. Furthermore, such grade-indicator plots, and the information deduced from them, are robust under reduction of data at the mononodal cutoff. The method thus has predictive potential for grade variograms of highly variant deposits. Forecasting a grade variogram from the associated mononodal indicator variogram and grade-indicator plot is illustrated. Agreement with the experimental variogram is shown to be excellent. 相似文献
13.
Is the ocean floor a fractal? 总被引：1，自引：0，他引：1
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. 相似文献
14.
以多重分形理论为基础,对中天山乌拉斯台地区铜多金属元素的岩屑测量数据,采用C-A法获得铜多金属的异常下限值,将其作为阈值进行指示克里格插值,绘制研究区的铜多金属地球化学异常图。研究显示,基于该方法获得的Cu矿化异常高值区主要集中在华力西早期第三侵入次的石英闪长岩和花岗闪长岩岩体中,受北西向和次级北东向断裂构造控制明显,该异常区可以作为寻找热液型铜多金属矿产的重要远景区。该方法对于地球化学数据空间变异性强烈的地区,较之普通克里格插值法具有更好的地球化学异常识别能力和高值信息重建能力,所得结果的最高累计频率值范围与已知矿化点的空间位置吻合度更高,在地球化学异常信息提取工作中具有推广意义。 相似文献
15.
Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains 总被引：10，自引：0，他引：10
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. 相似文献
16.
I. C. Lemmer 《Mathematical Geology》1986,18(7):605-623
The theory of mononodal variography developed in the preceeding paper is checked against a simulated deposit consisting of 60,500 grade values, called Stanford II. In the case of this deposit at least, assumptions underlying the concept of mononodal variography are borne out accurately. In particular, a linear relationship does exist indeed between indicator and grade variogram values of Stanford II at corresponding lags. Furthermore, such grade-indicator plots, and the information deduced from them, are robust under reduction of data at the mononodal cutoff. The method thus has predictive potential for grade variograms of highly variant deposits. Forecasting a grade variogram from the associated mononodal indicator variogram and grade-indicator plot is illustrated. Agreement with the experimental variogram is shown to be excellent.This paper is based in part on a PhD thesis submitted to the Department of Applied Earth Sciences, Stanford University, Stanford, California 94305, in 1984 (unpublished). 相似文献
17.
Estimating Variogram Uncertainty 总被引：10，自引：0，他引：10
The variogram is central to any geostatistical survey, but the precision of a variogram estimated from sample data by the method of moments is unknown. It is important to be able to quantify variogram uncertainty to ensure that the variogram estimate is sufficiently accurate for kriging. In previous studies theoretical expressions have been derived to approximate uncertainty in both estimates of the experimental variogram and fitted variogram models. These expressions rely upon various statistical assumptions about the data and are largely untested. They express variogram uncertainty as functions of the sampling positions and the underlying variogram. Thus the expressions can be used to design efficient sampling schemes for estimating a particular variogram. Extensive simulation tests show that for a Gaussian variable with a known variogram, the expression for the uncertainty of the experimental variogram estimate is accurate. In practice however, the variogram of the variable is unknown and the fitted variogram model must be used instead. For sampling schemes of 100 points or more this has only a small effect on the accuracy of the uncertainty estimate. The theoretical expressions for the uncertainty of fitted variogram models generally overestimate the precision of fitted parameters. The uncertainty of the fitted parameters can be determined more accurately by simulating multiple experimental variograms and fitting variogram models to these. The tests emphasize the importance of distinguishing between the variogram of the field being surveyed and the variogram of the random process which generated the field. These variograms are not necessarily identical. Most studies of variogram uncertainty describe the uncertainty associated with the variogram of the random process. Generally however, it is the variogram of the field being surveyed which is of interest. For intensive sampling schemes, estimates of the field variogram are significantly more precise than estimates of the random process variogram. It is important, when designing efficient sampling schemes or fitting variogram models, that the appropriate expression for variogram uncertainty is applied. 相似文献
18.
I. C. Lemmer 《Mathematical Geosciences》1986,18(7):589-604
For any distribution of grades, a particular cutoff grade is shown here to exist at which the indicator covariance is proportional to the grade covariance to a very high degree of accuracy. The name “mononodal cutoff” is chosen to denote this grade. Its importance for robust grade variography in the presence of a large coefficient of variation—typical of precious metals—derives from the fact that the mononodal indicator variogram is then linearly related to the grade variogram yet is immune to outlier data and is found to be particularly robust under data information reduction. Thus, it is an excellent substitute to model in lieu of a difficult grade variogram. A theoretical expression for the indicator covariance is given as a double series of orthogonal polynomials that have the grade density function as weight function. Leading terms of this series suggest that indicator and grade covariances are first-order proportional, with cutoff grade dependence being carried by the proportionality factor. Kriging equations associated with this indicator covariance lead to cutoff-free kriging weights that are identical to grade kriging weights. This circumstance simplifies indicator kriging used to estimate local point-grade histograms, while at the same time obviating order relations problems. 相似文献
19.
G. Bourgault 《Mathematical Geology》1996,28(6):723-734
A combination of factorial kriging and probability field simulation is proposed to correct realizations resulting from any simulation algorithm for either too high nugget effect (noise) or poor histogram reproduction. First, a factorial kriging is done to filter out the noise from the noisy realization. Second, the uniform scores of the filtered realization are used as probability field to sample the local probability distributions conditional to the same dataset used to generate the original realization. This second step allows to restore the data variance. The result is a corrected realization which reproduces better target variogram and histogram models, yet honoring the conditioning data. 相似文献
20.
R. M. Lark 《Mathematical Geosciences》2008,40(7):729-751
The effect of outliers on estimates of the variogram depends on how they are distributed in space. The ‘spatial breakdown
point’ is the largest proportion of observations which can be drawn from some arbitrary contaminating process without destroying
a robust variogram estimator, when they are arranged in the most damaging spatial pattern. A numerical method is presented
to find the spatial breakdown point for any sample array in two dimensions or more. It is shown by means of some examples
that such a numerical approach is needed to determine the spatial breakdown point for two or more dimensions, even on a regular
square sample grid, since previous conjectures about the spatial breakdown point in two dimensions do not hold. The ‘average
spatial breakdown point’ has been used as a basis for practical guidelines on the intensity of contaminating processes that
can be tolerated by robust variogram estimators. It is the largest proportion of contaminating observations in a data set
such that the breakdown point of the variance estimator used to obtain point estimates of the variogram is not exceeded by
the expected proportion of contaminated pairs of observations over any lag. In this paper the behaviour of the average spatial
breakdown point is investigated for cases where the contaminating process is spatially dependent. It is shown that in two
dimensions the average spatial breakdown point is 0.25. Finally, the ‘empirical spatial breakdown point’, a tool for the exploratory
analysis of spatial data thought to contain outliers, is introduced and demonstrated using data on metal content in the soils
of Sheffield, England. The empirical spatial breakdown point of a particular data set can be used to indicate whether the
distribution of possible contaminants is likely to undermine a robust variogram estimator. 相似文献