首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献，搜索用时 31 毫秒
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.
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   

3.
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.  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.
On a controversial method for modeling a coregionalization   总被引：2，自引：0，他引：2  
This paper reviews two alternative approaches for modeling the (cross) variograms in a coregionalization: (1) fitting the traditional linear model of coregionalization. or (2) deducing the crossvariogram model as a linear combination of prior direct (auto) variogram models while checking the Cauchy-Schwarz inequalities. We show that the second approach has no practical advantage over the traditional one and may not be valid if more than two variables are involved. In such case. Cauchy-Schwarz inequalities are necessary but not sufficient conditions for validity of a coregionalization model.  相似文献   

7.
The aim of this short article is to stress the importance of using only positive-definite functions as models for covariance functions and variograms.The two examples presented show that a negative variance can easily be obtained when a nonadmissible function is chosen for the variogram model.  相似文献   

8.
We present the results of spectroscopy of 71 objects with steep and ultra-steep spectra (α < −0.9, S ∝ ν α) from the “Big Trio” (RATAN-600-VLA-BTA) project, performed with the “Scorpio” spectrograph on the 6-m telescope of the Special Astrophysical Observatory (Russian Academy of Sciences). Redshifts were determined for these objects. We also present several other parameters of the sources, such as their R magnitudes, maximum radio sizes in seconds of arc, flux densities at 500, 1425, and 3940 MHz, radio luminosities at 500 and 3940 MHz, and morphology. Of the total number of radio galaxies studied, four have redshifts 1 ≤ z < 2, three have 2 ≤ z < 3, one has 3 ≤ z < 4, and one has z = 4.51. Thirteen sources have redshifts 0.7 < z < 1 and 15 have 0.2 < z < 0.7.Of all the quasars studied, five have redshifts 0.7 < z < 1, seven have 1 ≤ z < 2, four have 2 ≤ z < 3, and one has z = 3.57. We did not detect any spectral lines for 17 objects.  相似文献   

9.
 The structural behavior of stuffed derivatives of quartz within the Li1− x Al1− x Si1+ x O4 system (0 ≤ x ≤ 1) has been studied in the temperature range 20 to 873 K using high-resolution powder synchrotron X-ray diffraction (XRD). Rietveld analysis reveals three distinct regimes whose boundaries are defined by an Al/Si order-disorder transition at x=∼0.3 and a β–α displacive transformation at x=∼0.65. Compounds that are topologically identical to β-quartz (0 ≤ x < ∼0.65) expand within the (0 0 1) plane and contract along c with increasing temperature; however, this thermal anisotropy is significantly higher for structures within the regime 0 ≤ x < ∼0.3 than for those with compositions ∼0.3 ≤ x < ∼0.65. We attribute this disparity to a tetrahedral tilting mechanism that occurs only in the ordered structures (0 ≤ x < ∼0.3). The phases with ∼0.65 ≤ x ≤ 1 adopt the α-quartz structure at room temperature, and they display positive thermal expansion along both a and c from 20 K to their α–β transition temperatures. This behavior arises mainly from a rotation of rigid Si(Al)-tetrahedra about the <100> axes. Landau analysis provides quantitative evidence that the charge-coupled substitution of Li+Al for Si in quartz dampens the α–β transition. With increasing Li+Al content, the low-temperature modifications exhibit a marked decrease in spontaneous strain; this behavior reflects a weakening of the first-order character of the transition. In addition, we observe a linear decrease in the α–β critical temperature from 846 K to near 0 K as the Li+Al content increases from x=0 to x=∼0.5. Received: 26 June 2000 / Accepted: 1 December 2000  相似文献   

10.
A hierarchical geostatistical analysis is conducted on a high-resolution, multiscale hydraulic conductivity (ln K) map, created by scaling up an experimental stratigraphy. Unlike a previous study which evaluates ln K variograms within individual depositional environments, this study analyzes deposits (or samples) that incorporate multiple depositional environments. Based on conductivity cutoffs selected from a global ln K histogram, an indicator map is created to divide the deposits into 4 categories: sand, silty sand, clayey silt, and clay (Hierarchy-I). Based on facies and facies assemblage types selected using geological criteria, two more indicator maps are created at a higher hierarchy (Hierarchy-II) to divide the deposits into 14 units and 2 units, respectively. For each sample, its experimental ln K variogram is decomposed into 4 auto- and cross-transition component variograms. The decomposition characteristics are then evaluated against the underlying heterogeneity and specific division rule. The analysis reveals that: (1) ln K cutoffs (sand contents of the physical stratigraphy) can be used to distinguish the shifts in dominant deposition mode; (2) sample univariate modes depend on the choice of hierarchical division; (3) sample variograms exhibit smooth-varying correlation structures (exponential-like variograms are observed in samples with a large variance in mean facies length); (4) the decomposition characteristics are sensitive to the division based on conductivity cutoffs, but not sensitive to the division based on depositional environment (For all samples, with appropriate division, the sample variogram is closely approximated by the sum of the cross-transition component variograms.); and (5) at the Hierarchy-II level, the 2-unit division gives similar decomposition characteristics as the 14-unit division. For the select samples, parsimony in hierarchical division is achieved at the facies assemblage scale.  相似文献   

11.
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   

12.
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   

13.
Positive definiteness is not enough   总被引：2，自引：0，他引：2  
Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed.  相似文献   

14.
In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.  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.

A spectral algorithm is proposed to simulate an isotropic Gaussian random field on a sphere equipped with a geodesic metric. This algorithm supposes that the angular power spectrum of the covariance function is explicitly known. Direct analytic calculations are performed for exponential and linear covariance functions. In addition, three families of covariance functions are presented where the calculation of the angular power spectrum is simplified (shot-noise random fields, Yadrenko covariance functions and solutions of certain stochastic partial differential equations). Numerous illustrative examples are given.

  相似文献   

17.
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   

18.
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.  相似文献   

19.
Geostatistical analyses require an estimation of the covariance structure of a random field and its parameters jointly from noisy data. Whereas in some cases (as in that of a Matérn variogram) a range of structural models can be captured with one or a few parameters, in many other cases it is necessary to consider a discrete set of structural model alternatives, such as drifts and variograms. Ranking these alternatives and identifying the best among them has traditionally been done with the aid of information theoretic or Bayesian model selection criteria. There is an ongoing debate in the literature about the relative merits of these various criteria. We contribute to this discussion by using synthetic data to compare the abilities of two common Bayesian criteria, BIC and KIC, to discriminate between alternative models of drift as a function of sample size when drift and variogram parameters are unknown. Adopting the results of Markov Chain Monte Carlo simulations as reference we confirm that KIC reduces asymptotically to BIC and provides consistently more reliable indications of model quality than does BIC for samples of all sizes. Practical considerations often cause analysts to replace the observed Fisher information matrix entering into KIC with its expected value. Our results show that this causes the performance of KIC to deteriorate with diminishing sample size. These results are equally valid for one and multiple realizations of uncertain data entering into our analysis. Bayesian theory indicates that, in the case of statistically independent and identically distributed data, posterior model probabilities become asymptotically insensitive to prior probabilities as sample size increases. We do not find this to be the case when working with samples taken from an autocorrelated random field.  相似文献   

20.
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   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号