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1.
Estimation of non-ergodic variograms and their sampling variance by design-based sampling strategies
Design-based sampling strategies based on classical sampling theory offer unprecedented potentials for estimation of non-ergodic variograms. Unbiased and uncorrelated estimates of the semivariance at the selected lags and of its sampling variance can be simply obtained. These estimates are robust against deviations from an assumed spatial autocorrelation model. The same holds for the variogram model parameters and their sampling (co)variances. Moreover, an objective measure for lack of fit of the fitted model can simply be derived. The estimators for two basic sampling designs, simple random sampling and stratified simple random sampling of pairs of points, are presented. The first has been tested in real world for estimating the non-ergodic variograms of three soil properties. The parameters of variogram models and their sampling (co)variances were estimated with 72 pairs of points distributed over six lags. 相似文献
2.
Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach. 相似文献
3.
Changik Han Jiyang Wang Mingguo Zheng Ende Wang Jianming Xia GwangSu Li Sunchol Choe 《Earth Science Informatics》2016,9(2):197-213
A critical step for kriging in geostatistics is estimation of the variogram. Traditional variogram modeling comprise of the experimental variogram calculation, appropriate variogram model selection and model parameter determination. Selecting of the variogram model and fitting of model parameters is the most controversial aspect of geostatistics. Shapes of valid variogram models are finite, and sometimes, the optimal shape of the model can not be fitted, leading to reduced estimation accuracy. In this paper, a new method is presented to automatically construct a model shape and fit model parameters to experimental variograms using Support Vector Regression (SVR) and Multi-Gene Genetic Programming (MGGP). The proposed method does not require the selection of a variogram model and can directly provide the model shape and parameters of the optimal variogram. The validity of the proposed method is demonstrated in a number of cases. 相似文献
4.
Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations. 相似文献
5.
Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations. 相似文献
6.
Sample schemes used in geostatistical surveys must be suitable for both variogram estimation and kriging. Previously schemes
have been optimized for one of these steps in isolation. Ordinary kriging generally requires the sampling locations to be
evenly dispersed over the region. Variogram estimation requires a more irregular pattern of sampling locations since comparisons
must be made between measurements separated by all lags up to and beyond the range of spatial correlation. Previous studies
have not considered how to combine these optimized schemes into a single survey and how to decide what proportion of sampling
effort should be devoted to variogram estimation and what proportion devoted to kriging
An expression for the total error in a geostatistical survey accounting for uncertainty due to both ordinary kriging and variogram
uncertainty is derived. In the same manner as the kriging variance, this expression is a function of the variogram but not
of the sampled response data. If a particular variogram is assumed the total error in a geostatistical survey may be estimated
prior to sampling. We can therefore design an optimal sample scheme for the combined processes of variogram estimation and
ordinary kriging by minimizing this expression. The minimization is achieved by spatial simulated annealing. The resulting
sample schemes ensure that the region is fairly evenly covered but include some close pairs to analyse the spatial correlation
over short distances. The form of these optimal sample schemes is sensitive to the assumed variogram. Therefore a Bayesian
approach is adopted where, rather than assuming a single variogram, we minimize the expected total error over a distribution
of plausible variograms. This is computationally expensive so a strategy is suggested to reduce the number of computations
required 相似文献
7.
The reliability of using fractal dimension (D) as a quantitative parameter to describe geological variables is dependent mainly
on the accuracy of estimated D values from observed data. Two widely used methods for the estimation of fractal dimensions
are based on fitting a fractal model to experimental variograms or power-spectra on a log-log plot. The purpose of this paper
is to study the uncertainty in the fractal dimension estimated by these two methods. The results indicate that both spectrum
and variogram methods result in biased estimates of the D value. Fractal dimension calculated by these two methods for the
same data will be different unless the bias is properly corrected. The spectral method results in overestimated D values.
The variogram method has a critical fractal dimension, below which overestimation occurs and above which underestimation occurs.
On the bases of 36,000 simulated realizations we propose empirical formulae to correct for biases in the spectral and variogram
estimated fractal dimension. Pitfalls in estimating fractal dimension from data contaminated by white noise or data having
several fractal components have been identified and illustrated by simulated examples. 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.
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
10.
In the present paper, we propose a new method for the estimation of the variogram, which combines robustness with efficiency under intrinsic stationary geostatistical processes. The method starts by using a robust estimator to obtain discrete estimates of the variogram and control atypical observations that may exist. When the number of points used in the fit of a model is the same as the number of parameters, ordinary least squares and generalized least squares are asymptotically equivalent. Therefore, the next step is to fit the variogram by ordinary least squares, using just a few discrete estimates. The procedure is then repeated several times with different subsets of points and this produces a sequence of variogram estimates. The final estimate is the median of the multiple estimates of the variogram parameters. The suggested estimator will be called multiple variograms estimator. This procedure assures a global robust estimator, which is more efficient than other robust proposals. Under the assumed dependence structure, we prove that the multiple variograms estimator is consistent and asymptotically normally distributed. A simulation study confirms that the new method has several advantages when compared with other current methods. 相似文献
11.
Non-stationary Geostatistical Modeling Based on Distance Weighted Statistics and Distributions 总被引：3，自引：0，他引：3
A common assumption in geostatistics is that the underlying joint distribution of possible values of a geological attribute at different locations is stationary within a homogeneous domain. This joint distribution is commonly modeled as multi-Gaussian, with correlations defined by a stationary covariance function. This results in attribute maps that fail to reproduce local changes in the mean, in the variance and, particularly, in the spatial continuity. The proposed alternative is to build local distributions, variograms, and correlograms. These are inferred by weighting the samples depending on their distance to selected locations. The local distributions are locally transformed into Gaussian distributions embedding information on the local histogram. The distance weighted experimental variograms and correlograms are able to adapt to local changes in the direction and range of spatial continuity. The automatically fitted local variogram models and the local Gaussian transformation parameters are used in spatial estimation algorithms assuming local stationarity. The resulting maps are rich in nonstationary spatial features. The proposed process implies a higher computational effort than traditional stationary techniques, but if data availability allows for a reliable inference of the local distributions and statistics, a higher accuracy of estimates can be achieved. 相似文献
12.
Guillaume Larocque Pierre Dutilleul Bernard Pelletier James W. Fyles 《Mathematical Geology》2007,39(3):263-288
Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an
increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters
in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized.
This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the
method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares
estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated
coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter,
the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and
determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that
the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the
presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty,
sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling
domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be
high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram
range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset,
the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty
very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty
levels and limits of the method. 相似文献
13.
Variograms of hydrologic characteristics are usually obtained by estimating the experimental variogram for distinct lag classes
by commonly used estimators and fitting a suitable function to these estimates. However, these estimators may fail the conditionally
positive-definite property and the better results for the statistics of cross-validation, which are two essential conditions
for choosing a valid variogram model. To satisfy these two conditions, a multi-objective bilevel programming estimator (MOBLP)
which is based on the process of cross-validation has been developed for better estimate of variogram parameters. This model
is illustrated with some rainfall data from Luan River Basin in China. The case study demonstrated that MOBLP is an effective
way to achieve a valid variogram model. 相似文献
14.
Guillaume Larocque Pierre Dutilleul Bernard Pelletier James W. Fyles 《Mathematical Geosciences》2007,39(3):263-288
Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method. 相似文献
15.
In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental
cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled
with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization
matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects
and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to
the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another
structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures
(one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated
with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic
and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed
tests are generally valid. 相似文献
16.
Understanding Anisotropy Computations 总被引：2，自引：0，他引：2
17.
Sandstones of different ages provide economically significant oil, gas, and water reservoirs. In sandstones where heterogeneities are not visually obvious, it is particularly difficult to predict the location of permeability barriers and the scale at which high and low permeability zones occur, yet this is critical in providing information on hydrocarbon reservoir performance. This study uses variogram analysis to investigate spatial variation in permeability in visually homogeneous reservoir sandstone successions. Air permeability measurements were taken using unsteady state probe permeametry following regular grid schemes with centimeter spacing. Spatial variation in permeability was characterized using omnidirectional and directional variograms. This study combines variography with geological interpretation to assess the degree of heterogeneity of permeability in visually homogeneous sandstone successions. Variography indicates spatial dependence and short-range variation at 1 cm grid spacings that is not apparent at a larger 5 cm grid spacing in the visually homogeneous sandstones studied. The range of the models fitted to the variograms provide a potentially important index of spatial variability in permeability for different depositional settings including aeolian, fluvial, shallow marine, and marine/mass- flow turbidite. 相似文献
18.
Kriging with imprecise (fuzzy) variograms. I: Theory 总被引：2，自引：0，他引：2
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. 相似文献
19.
黑河下游荒漠河岸林典型样带植被空间异质性 总被引：11，自引：1，他引：10
应用地统计学的理论与方法,分析了黑河流域下游荒漠河岸林主要种群,即老林胡杨(Popu luseuphratica)、幼林胡杨、柽柳(Tamarix chinensis)和苦豆子(Sophora alopecuroides)的空间异质性程度、异质性组成及尺度依赖问题.结果表明:黑河下游荒漠河岸林种群斑块格局明显,斑块内部异质性较小,斑块之间异质性增强,出现空间异质性变化较大的尺度为430m.而斑块内部因不同种群而变化,苦豆子和柽柳具有较大的空间异质性尺度,分别为43m和55m.老林和幼林胡杨种群的空间异质性尺度(8m和13m)较小,但异质性存在多尺度变化.采用30m分辨率的遥感数据能够较好地分析种群斑块格局,但是对种群内的空间异质性分析需要更高的分辨率. 相似文献
20.
Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general. 相似文献