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1.
Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rn; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada.  相似文献   

2.
Structural analysis of data displaying trends may be performed with the help of generalized increments, the variance of these increments being a function of a generalized covariance. Generalized covariances are estimated primarily by parametric methods (i. e., methods searching for the best coefficients of a predetermined function), but also may be computed by one known nonparametric alternative. In this paper, a new nonparametric method is proposed. It is founded on the following principles: (1) least-squares residues are generalized increments; and (2) the generalized covariance is not a unique function, but a family of functions (the system is indeterminate). The method is presented in a general context of a k order trend in Rd, although the full solution is given only fork = I in Ri. In Ri, higher order trends may be developed easily with the equations included in this paper. For higher dimensions in space, the problem is more complex, but a research approach is proposed. The method is tested on soil pH data and compared to a parametric and nonparametric method.  相似文献   

3.
A coregionalization simulation consists of the generation of realizations of a group of spatially related random variables. The Fourier integral method is presented, modified to carry out such a multivariable simulation. This method allows the simulation of realizations with any specified symmetrical covariance matrix and it is not limited to the classic linear model of coregionalization. The results of gaussian nonconditinal simulations from a case study modeling the spatial characteristics of a layer of coal are given.  相似文献   

4.
Generalized cross-validation for covariance model selection   总被引：4，自引：0，他引：4  
A weighted cross-validation technique known in the spline literature as generalized cross-validation (GCV), is proposed for covariance model selection and parameter estimation. Weights for prediction errors are selected to give more importance to a cluster of points than isolated points. Clustered points are estimated better by their neighbors and are more sensitive to model parameters. This rational weighting scheme also provides a simplifying significantly the computation of the cross-validation mean square error of prediction. With small- to medium-size datasets, GCV is performed in a global neighborhood. Optimization of usual isotropic models requires only a small number of matrix inversions. A small dataset and a simulation are used to compare performances of GCV to ordinary cross-validation (OCV) and least-squares filling (LS).  相似文献   

5.
Journel (1974) developed the turning-bands method which allows a three-dimensional data set with specified covariance to be obtained by the simulation of several one-dimensional realizations which have an intermediate covariance. The relationship between the threedimensional and one-dimensional covariance is straightforward and allows the one-dimensional covariance to be obtained immediately. In theory a dense uniform distribution of lines in three-dimensional space is required along which the one-dimensional realizations are generated; in practice most workers have been content to use the fifteen axes of the regular icosahedron. Many mining problems may be treated in two dimensions, and in this paper a turning-bands approach is developed to generate two-dimensional data sets with a specified covariance. By working in two dimensions, the area on which the data is simulated may be divided as finely as desired by the lines on which the one-dimensional realizations are first generated. The relationship between the two-dimensional and one-dimensional covariance is derived as a nontrivial integral equation. This is solved analytically for the onedimensional covariance. The method is applied to the generation of a two-dimensional data set with spherical covariance.  相似文献   

6.
The turning bands method (TBM) generates realizations of isotropic Gaussian random fields by summing contributions from line processes. We consider two-dimensional simulations and study the correlation bias attributable to the use of only a finite number L of lines. Our analytical and numerical results confirm that the maximal bias is of order 1/L, and that L = 64 lines suffice for excellent covariance reproduction. The notorious banding observed in simulations with an insufficient number of lines is a related but different phenomenon and depends strongly on the choice of the line simulation technique. Clear-cut recommendations for the number of lines necessary to avoid the effect can only be based on practical experience with the specific code at hand.  相似文献   

7.
Natural processes encountered in mining, hydrogeologic, environmental, etc. applications usually are poorly known because of scarcity of data over the area of interest. Therefore, stochastic estimation techniques are the tool of choice for a careful accounting of the heterogeneity and uncertainty involved. Within such a framework, a better utilization of all available data concerning the process of interest and all other natural processes related to it, is of primary importance. Because many natural processes show complicated spatial trends, the hypothesis of spatial homogeneity cannot be invoked always, and the more general theory of intrinsic spatial random fields should be employed. Efficient use of secondary information in terms of the intrinsic model requires that suitable permissibility criteria for the generalized covariances and cross-covariances are satisfied. A set of permissibility criteria are presented for the situation of two intrinsic random fields. These criteria are more general and comprehensive than the ones currently available in the geostatistical literature. A constrained least-square technique is implemented for the inference of the generalized covariance and cross-covariance parameters, and a synthetic example is used to illustrate the methodology. The numerical results show that the use of secondary information can lead to significant reductions in the estimation errors.  相似文献   

8.
Conditional Spectral Simulation with Phase Identification   总被引：2，自引：0，他引：2  
Spectral simulation is used widely in electrical engineering to generate random fields with a given covariance spectrum. The algorithms used are fast particularly when based on Fast Fourier Transform (FFT). However, because of lack of phase identification, spectral simulation only generates unconditional realizations. Local data conditioning is obtained typically by adding a simulated kriging residual. This conditioning process requires an additional kriging at each simulated node thus forfeiting the speed advantage of FFT. A new algorithm for conditioning is proposed whereby the phase values are determined iteratively to ensure approximative data reproduction while reproducing the frequency spectrum, that is, the covariance model. A case study is presented to demonstrate the algorithm.  相似文献   

9.
Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance function's dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.  相似文献   

10.
    
A method of analysis of covariance structure proposed by A. P. Dempster complements, in some respects, a recently suggested procedure by the author. The method is based on the comparison of ratios of generalized statistical distances and distancelike quantities. An analysis of septivariate data on the foraminiferTextilina mexicana (Cushman) shows general differences in covariance structure in which the sample rest linear discriminators also differ. This difference in covariance structure is thought to be genetic in origin. Two species of Middle Devonian brachiopods,Martinia inflata (Schnur) andUncites gryphus von Schlotheim), also show differences in covariance structure—the former slightly, the latter strongly. This is further analyzed forUncites by the author's methods and good agreement between the two approaches obtained.  相似文献   

11.
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   

12.
Conditional Simulation of Random Fields by Successive Residuals   总被引：2，自引：0，他引：2  
This paper presents a new approach to the LU decomposition method for the simulation of stationary and ergodic random fields. The approach overcomes the size limitations of LU and is suitable for any size simulation. The proposed approach can facilitate fast updating of generated realizations with new data, when appropriate, without repeating the full simulation process. Based on a novel column partitioning of the L matrix, expressed in terms of successive conditional covariance matrices, the approach presented here demonstrates that LU simulation is equivalent to the successive solution of kriging residual estimates plus random terms. Consequently, it can be used for the LU decomposition of matrices of any size. The simulation approach is termed conditional simulation by successive residuals as at each step, a small set (group) of random variables is simulated with a LU decomposition of a matrix of updated conditional covariance of residuals. The simulated group is then used to estimate residuals without the need to solve large systems of equations.  相似文献   

13.
Gradual deformation is a parameterization method that reduces considerably the unknown parameter space of stochastic models. This method can be used in an iterative optimization procedure for constraining stochastic simulations to data that are complex, nonanalytical functions of the simulated variables. This method is based on the fact that linear combinations of multi-Gaussian random functions remain multi-Gaussian random functions. During the past few years, we developed the gradual deformation method by combining independent realizations. This paper investigates another alternative: the combination of dependent realizations. One of our motivations for combining dependent realizations was to improve the numerical stability of the gradual deformation method. Because of limitations both in the size of simulation grids and in the precision of simulation algorithms, numerical realizations of a stochastic model are never perfectly independent. It was shown that the accumulation of very small dependence between realizations might result in significant structural drift from the initial stochastic model. From the combination of random functions whose covariance and cross-covariance are proportional to each other, we derived a new formulation of the gradual deformation method that can explicitly take into account the numerical dependence between realizations. This new formulation allows us to reduce the structural deterioration during the iterative optimization. The problem of combining dependent realizations also arises when deforming conditional realizations of a stochastic model. As opposed to the combination of independent realizations, combining conditional realizations avoids the additional conditioning step during the optimization process. However, this procedure is limited to global deformations with fixed structural parameters.  相似文献   

14.
Two methods for generating representative realizations from Gaussian and lognormal random field models are studied in this paper, with term representative implying realizations efficiently spanning the range of possible attribute values corresponding to the multivariate (log)normal probability distribution. The first method, already established in the geostatistical literature, is multivariate Latin hypercube sampling, a form of stratified random sampling aiming at marginal stratification of simulated values for each variable involved under the constraint of reproducing a known covariance matrix. The second method, scarcely known in the geostatistical literature, is stratified likelihood sampling, in which representative realizations are generated by exploring in a systematic way the structure of the multivariate distribution function itself. The two sampling methods are employed for generating unconditional realizations of saturated hydraulic conductivity in a hydrogeological context via a synthetic case study involving physically-based simulation of flow and transport in a heterogeneous porous medium; their performance is evaluated for different sample sizes (number of realizations) in terms of the reproduction of ensemble statistics of hydraulic conductivity and solute concentration computed from a very large ensemble set generated via simple random sampling. The results show that both Latin hypercube and stratified likelihood sampling are more efficient than simple random sampling, in that overall they can reproduce to a similar extent statistics of the conductivity and concentration fields, yet with smaller sampling variability than the simple random sampling.  相似文献   

15.
This paper presents a new method of constructing random functions whose realizations can be evaluated efficiently. The basic idea is to blend, both stochastically and linearly, a limited set of independent initial realizations previously generated by any chosen simulation method. The blending stochastic coefficients are determined in such a way that the new random function so generated has the same mean and covariance functions as the random function used for generating the initial realizations.  相似文献   

16.
Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.  相似文献   

17.
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   

18.
The Markov chain random field (MCRF) theory provided the theoretical foundation for a nonlinear Markov chain geostatistics. In a MCRF, the single Markov chain is also called a “spatial Markov chain” (SMC). This paper introduces an efficient fixed-path SMC algorithm for conditional simulation of discrete spatial variables (i.e., multinomial classes) on point samples with incorporation of interclass dependencies. The algorithm considers four nearest known neighbors in orthogonal directions. Transiograms are estimated from samples and are model-fitted to provide parameter input to the simulation algorithm. Results from a simulation example show that this efficient method can effectively capture the spatial patterns of the target variable and fairly generate all classes. Because of the incorporation of interclass dependencies in the simulation algorithm, simulated realizations are relatively imitative of each other in patterns. Large-scale patterns are well produced in realizations. Spatial uncertainty is visualized as occurrence probability maps, and transition zones between classes are demonstrated by maximum occurrence probability maps. Transiogram analysis shows that the algorithm can reproduce the spatial structure of multinomial classes described by transiograms with some ergodic fluctuations. A special characteristic of the method is that when simulation is conditioned on a number of sample points, simulated transiograms have the tendency to follow the experimental ones, which implies that conditioning sample data play a crucial role in determining spatial patterns of multinomial classes. The efficient algorithm may provide a powerful tool for large-scale structure simulation and spatial uncertainty analysis of discrete spatial variables.  相似文献   

19.
Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data. Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity in the data used, in addition to high order.  相似文献   

20.
A method of analysis of covariance structure proposed by A. P. Dempster complements, in some respects, a recently suggested procedure by the author. The method is based on the comparison of ratios of generalized statistical distances and distancelike quantities. An analysis of septivariate data on the foraminiferTextilina mexicana (Cushman) shows general differences in covariance structure in which the sample rest linear discriminators also differ. This difference in covariance structure is thought to be genetic in origin. Two species of Middle Devonian brachiopods,Martinia inflata (Schnur) andUncites gryphus von Schlotheim), also show differences in covariance structure—the former slightly, the latter strongly. This is further analyzed forUncites by the author's methods and good agreement between the two approaches obtained.  相似文献   

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