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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   

2.
When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.  相似文献   

3.
Calculation of Uncertainty in the Variogram   总被引：6，自引：0，他引：6  
There are often limited data available in early stages of geostatistical modeling. This leads to considerable uncertainty in statistical parameters including the variogram. This article presents an approach to calculate the uncertainty in the variogram. A methodology to transfer this uncertainty through geostatistical simulation and decision making is also presented.The experimental variogram value for a separation lag vector h is a mean of squared differences. The variance of a mean can be calculated with a model of the correlation between the pairs of data used in the calculation. The data here are squared differences; therefore, we need a measure of a 4-point correlation. A theoretical multi-Gaussian approach is presented for this uncertainty assessment together with a number of examples. The theoretical results are validated by numerical simulation. The simulation approach permits generalization to non-Gaussian situations.Multiple plausible variograms may be fit knowing the uncertainty at each variogram point, . Multiple geostatistical realizations may then be constructed and subjected to process assessment to measure the impact of this uncertainty.  相似文献   

4.
The Second-Order Stationary Universal Kriging Model Revisited   总被引：3，自引：0，他引：3  
Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.  相似文献   

5.
On the Use of Non-Euclidean Distance Measures in Geostatistics   总被引：4，自引：0，他引：4  
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.  相似文献   

6.
In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on measurements taken at various spatial locations. The classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the regular unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in IR2 , as well as some simulations on irregular grids. In order to study the global effects of perturbations on variogram estimators, further simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% on average. Theoretical results confirm the previous statement on data in IRd , d 1.  相似文献   

7.
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   

8.
In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters.  相似文献   

9.
    
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.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   

10.
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   

11.
The relative variogram has been employed as a tool for correcting a simple kind of nonstationarity, namely that in which local variance is proportional to local mean squared. In the past, this has been linked in a vague way to the lognormal distribution, although if {Zt; t D}is strongly stationary and normal over a domain D,then clearly {exp (Zt); t D}will stillbe stationary, but lognormal. The appropriate link is made in this article through a universal transformation principle. More general situations are considered, leading to the use of a scaled variogram.  相似文献   

12.
Summary The reflectance of oriented crystal faces parallel (100) and (001) of ilvaite was measured in air and in oil at different wavelengths with linearly polarized light. Refractive indices and absorption constants were calculated from the reflectance values. In contrast ton andn ,n has a strong dispersion. For the calculation ofn the absorption constant can be neglected. According to the unit cell ofBelov andMokeeva (1954) with the lattice constantsa 0=8.82,b 0=13.07,c 0=5.86 Å,n vibrates parallel to [001]n parallel to [100] andn parallel to [010]. Ilvaite is optically negative.
Auflichtuntersuchungen zur Optik des Ilvaits
Zusammenfassung Auf orientiert geschliffenen (100) und (001) Kristallplatten von Ilvait wurde das Reflexionsvermögen in Luft und in Öl mit linear polarisiertem Licht bei verschiedenen Wellenlängen gemessen. Aus den Reflexionswerten wurden die Brechungsindices und Absorptionskonstanten berechnet.n zeigt im Gegensatz zun undn eine auffallend starke Dispersion. Für die Berechnung vonn kann die Absorptionskonstante vernachlässigt werden. Nach der Aufstellung der Elementarzelle vonBelov undMokeeva (1954) mit den Gitterkonstantena 0=8,82,b 0=13,07,c 0=5,86 Å schwingtn parallel [001],n parallel [100] undn parallel [010]. Der Ilvait ist optisch negativ.

With 2 Figures  相似文献   

13.
In the context of spatial statistics, the classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. If data are Gaussian with constant mean, then the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix and the variance matrix. When data are independent with unidimensional and regular support, an explicit formula for this correlation is available. The same is true for a multidimensional and regular support as can be shown by using Kronecker products of matrices. As variogram fitting is a crucial stage for correct spatial prediction, it is proposed to use a generalized least squares method with an explicit formula for the covariance structure (GLSE). A good approximation of the covariance structure is achieved by taking account of the explicit formula for the correlation in the independent situation. Simulations are carried out with several types of underlying variograms, as well as with outliers in the data. Results show that this technique (GLSE), combined with a robust estimator of the variogram, improves the fit significantly.  相似文献   

14.
Principal component analysis (PCA) is commonly applied without looking at the spatial support (size and shape, of the samples and the field), and the cross-covariance structure of the explored attributes. This paper shows that PCA can depend on such spatial features. If the spatial random functions for attributes correspond to largely dissimilar variograms and cross-variograms, the scale effect will increase as well. On the other hand, under conditions of proportional shape of the variograms and cross-variograms (i.e., intrinsic coregionalization), no scale effect may occur. The theoretical analysis leads to eigenvalue and eigenvector functions of the size of the domain and sample supports. We termed this analysis growing scale PCA, where spatial (or time) scale refers to the size and shape of the domain and samples. An example of silt, sand, and clay attributes for a second-order stationary vector random function shows the correlation matrix asymptotically approaches constants at two or three times the largest range of the spherical variogram used in the nested model. This is contrary to the common belief that the correlation structure between attributes become constant at the range value. Results of growing scale PCA illustrate the rotation of the orthogonal space of the eigenvectors as the size of the domain grows. PCA results are strongly controlled by the multivariate matrix variogram model. This approach is useful for exploratory data analysis of spatially autocorrelated vector random functions.  相似文献   

15.
Summary As suggested bySmith (1968) and supported by most structural data published since, in alkali feldspar the ** plot can be used for estimating (**), i. e. the difference in Al confent between 0 andm subsites. The present study investigates the topologically identical plot on the basis of the configuration of the alkali feldspar tetrahedral framework. Changes in Al content ofT-sites are functionally related to changes in cosines of and . While the total Al causing changes in cos is directly equal to the difference in Al content between 0 andm subsites, the total Al causing changes in cos is expressed by a complicated function which is equal with a very good approximation to three fourths of the difference in Al content betweenm and 0 subsites. This relation of quasiproportionality. like the feasible substitution in alkali feldspar of the diagram cos vs. cos by the plot, represents two simplifying assumptions which permit the difference in Al content to be calculated, as previously predicted.
Strukturelle Deutung für (**) aus dem */* der Alkalifeldspäte
Zusammenfassung Nach einem Vorschlag vonSmith (1968) und in Übereinstimmung mit den meisten seither publizierten Strukturdaten kann man in Alkalifeldspäten das */* zur Abschätzung von (**), also des Unterschiedes im Al-Gehalt auf der 0- undm-Position benützen. Die vorliegende Arbeit untersucht das topologisch idente /-Diagramm auf der Basis der Gestalt des Tetraederverbandes der Alkalifeldspäte. änderungen im Al-Gehalt derT-Position sind mit Änderungen im Kosinus von und korreliert. Während der die cos -änderungen verursachende Al-Gesamtgehalt unmittelbar dem Unterschied im Al-Gehalt derO-undm-Position entspricht, ist der die cos -Änderungen verursachende Al-Gesamtgehalt durch eine komplizierte Funktion ausgedrückt, die aber mit sehr guter Näherung drei Vierteln der Differenz im Al-Gehalt auf denm-und 0-Poisitionen entspricht. Diese quasi-Proportionalität und die Ersetzbarkeit des cos /cos -Diagrammes durch das /-Diagramm bei den Alkalifeldspäten stellen zwei Vereinfachungen bei der Berechnung des Al-Gehaltes dar.
  相似文献   

16.
Seafloor classification is aimed at quantitatively characterizing seafloor properties such as roughness and anisotropy, and at using such spatial characteristics to distinguish geological provinces automatically. From geostatistical principals, a variogram method is developed for seafloor classification and it is demonstrated for data from the western flank of the Mid-Atlantic Ridge at 25°45N to 26°40N. This study uses HYDROSWEEP bathymetric data which have been ping-edited to flag erroneous data records, and navigation corrected. The classification method can handle the resultant data gaps inside the survey swaths as well as interpret data from several swaths. For a suite of test areas representative of different geological provinces, directional variograms are calculated, and characteristic parameters are extracted for the classification. Examples include a sediment pond, abyssal hill terrain in several segments and of variable spacing, inside and outside corners of ridge discontinuities, and mixed morphological forms. The dependency of the results on random or regular subsampling and on the size of the test area is investigated.  相似文献   

17.
Summary Integrated X-ray powder diffraction, scanning electron microscopy, electron probe, and transmission electron microscopy studies have identified the rare contact assemblage calaverite–sylvanite–hessite in a sample of gold ore from the Golden Mile deposit, Kalgoorlie, Australia. The presence of coexisting calaverite–hessite at Kalgoorlie is a non-equilibrium assemblage whereby the stable hessite-bearing assemblage is hessite–sylvanite, which formed from the breakdown of the -phase or -phase below 120°C, stützite+-phase, or sylvanite+stützite+-phase, as predicted by Cabri (1965).  相似文献   

18.
Under the intrinsic coregionalization model if both primary and secondary measurements are available at all sample locations, the conventional geostatistical wisdom is that cokriging provides exactly the same solution as univariate kriging on the primary process alone. However, recent eamples have been given where nonzero secondary cokriging weights have accurred under this spatial dependence structure. This note identifies the conditions under which secondary information is useful under the assumption of intrinsic coregionalization. An illustration is given using a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site.  相似文献   

19.
Starting with a stochastic differential equation with random coefficients describing steady-state flow, the effective hydraulic conductivity of 1-, 2-, and 3-dimensional aquifers is derived. The natural logarithm of hydraulic conductivity (lnK) is assumed to be heterogeneous, with a spatial trend, and isotropic. The effective conductivity relates the mean specific discharge in an aquifer to the mean hydraulic gradient, thus its importance in predicting Darcian discharge when field data represent mean or average values of conductivity or hydraulic head. Effective conductivity results are presented in exact form in terms of elementary functions after the introduction of special sets of coordinate transformations in two and three dimensions. It was determined that in one, two, and three dimensions, for the type of aquifer heterogeneity considered, the effective hydraulic conductivity depends on: (i) the angle between the gradient of the trend of lnK and the mean hydraulic gradient (which is zero in the one-dimensional situation); (2) (inversely) on the product of the magnitude of the trend gradient of lnK, b, and the correlation scale of lnK, and (3) (proportionally) on the variance of lnK, f 2 . The productb plays a central role in the stability of the results for effective hydraulic conductivity.  相似文献   

20.
Activity-composition relationships for pyrope-grossular garnet   总被引：1，自引：0，他引：1  
Activity coefficients () for grossular in pyrope-grossular garnet have been determined experimentally using the divariant assemblage garnet-anorthite-sillimanite (kyanite)-quartz. Values of for garnets with 10–12 mole % grossular have been obtained at 1000 °, 1100 °, 1200 ° and 1300 ° C at pressures between 15 and 21 Kb. The data are consistent with a symmetrical regular solid model for grossular-pyrope solid solutions. The interaction parameter (W) increases linearly with decreasing temperature and is given by W = 7460-4.3 T cals (T in °K). A solvus in the pyrope-grossular solid solution is predicted with a temperature of critical mixing of 629°C±90 ° C.  相似文献   

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