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1.
Large cokriging systems arise in many situations and are difficult to handle in practice. Simplifications such as simple kriging,
strictly collocated and multicollocated cokriging are often used and models under which such simplifications are, in fact,
equivalent to cokriging have recently received attention. In this paper, a two-dimensional second-order stationary random
process with known mean is considered and the redundancy of certain components of the data at certain locations vis-à-vis
the solution to the simple cokriging system is examined. Conditions for the simple cokriging weights of these components at
these locations are set to zero. The conditions generalise the notion of the autokrigeability coefficient and can, in principle,
be applied to any data configuration. In specific sampling situations such as the isotopic and certain heterotropic configurations,
models under which simple kriging, strictly collocated, multicollocated and dislocated cokriging are equivalent to simple
cokriging are readily identified and results already available in the literature are obtained. These are readily identified
and the results are already available in the literature. The advantage of the approach presented here is that it can be applied
to any data configuration for analysis of permissible simplifications in simple cokriging. 相似文献
2.
Multivariable spatial prediction 总被引：1，自引：0，他引：1
For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given. 相似文献
3.
Multivariable variogram and its application to the linear model of coregionalization 总被引：1，自引：0，他引：1
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
4.
5.
Joint Consistent Mapping of High-Dimensional Geochemical Surveys 总被引：1，自引：0，他引：1
Geochemical surveys often contain several tens of components, obtained from different horizons and with different analytical techniques. These are used either to obtain elemental concentration maps or to explore links between the variables. The first task involves interpolation, the second task principal component analysis (PCA) or a related technique. Interpolation of all geochemical variables (in wt% or ppm) should guarantee consistent results: At any location, all variables must be positive and sum up to 100 %. This is not ensured by any conventional geostatistical technique. Moreover, the maps should ideally preserve any link present in the data. PCA also presents some problems, derived from the spatial dependence between the observations, and the compositional nature of the data. Log-ratio geostatistical techniques offer a consistent solution to all these problems. Variation-variograms are introduced to capture the spatial dependence structure: These are direct variograms of all possible log ratios of two components. They can be modeled with a function analogous to the linear model of coregionalization (LMC), where for each spatial structure there is an associated variation matrix describing the links between the components. Eigenvalue decompositions of these matrices provide a PCA of that particular spatial scale. The whole data set can then be interpolated by cokriging. Factorial cokriging can also be used to map a certain spatial structure, eventually projected onto those principal components (PCs) of that structure with relevant contribution to the spatial variability. If only one PC is used for a certain structure, the maps obtained represent the spatial variability of a geochemical link between the variables. These procedures and their advantages are illustrated with the horizon C Kola data set, with 25 components and 605 samples covering most of the Kola peninsula (Finland, Norway, Russia). 相似文献
6.
On Some Simplifications of Cokriging Neighborhood 总被引：2，自引：0，他引：2
Jacques Rivoirard 《Mathematical Geology》2004,36(8):899-915
Choosing the cokriging neighborhood is often difficult. A poor choice, ignoring influent data, can result in a loss of information as well as in artifacts in simulations based on cokriging. Then it is convenient to use if possible, or to refer to models that lead to simplified cokriging neighborhood. We essentially consider the case of two stationary variables, a target variable and an auxiliary one. By examining possible simplifications, we set up a list of models (essentially models with residuals) that, in general or under specific configurations, lead to simplifications of cokriging neighborhood. Collocated, dislocated, and other types of neighborhood are identified, that are optimal in some models and configurations. Possible extensions to cokriging with unknown means, and to more variables, are included. 相似文献
7.
Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities. 相似文献
8.
Ordinary Cokriging Revisited 总被引：12，自引：0，他引：12
P. Goovaerts 《Mathematical Geology》1998,30(1):21-42
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
9.
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
10.
《Mathematical Geology》1997,29(6):779-799
Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations.
They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors
without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms
are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the
usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary
sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear
coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization
matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models. 相似文献
11.
On the Equivalence of the Cokriging and Kriging Systems 总被引：2，自引：0，他引：2
Simple cokriging of components of a p-dimensional second-order stationary random process is considered. Necessary and sufficient conditions under which simple cokriging is equivalent to simple kriging are given. Essentially this condition requires that it should be possible to express the cross-covariance at any lag series h using the cross-covariance at |h|=0 and the auto-covariance at lag series h. The mosaic model, multicolocated kriging and the linear model of coregionalization are examined in this context. A data analytic method to examine whether simple kriging of components of a multivariate random process is equivalent to its cokriging is given 相似文献
12.
Many applications are multivariate in character and call for stochastic images of the joint spatial variability of multiple variables conditioned by a prior model of covariances and cross- covariances. This paper presents an algorithm to perform cosimulation of such spatially intercorrelated variables. This new algorithm builds on a Markov-type hypothesis whereby collocated information screens further away data of the same type, allowing cosimulation without the burden of a full cokriging. The proposed algorithm is checked against a synthetic multi-Gaussian reference dataset, then against a multi-Gaussian cosimulation approach using full cokriging. The results indicate that the proposed algorithm perform as well as the full cokriging approach in reproducing the univariate and bivariate statistics of the reference set, yet at less cpu cost. 相似文献
13.
Novel basin modelling concept for simulating deformation from mechanical compaction using level sets
Sean McGovern Stefan Kollet Claudius M. Bürger Ronnie L. Schwede Olaf G. Podlaha 《Computational Geosciences》2017,21(5-6):835-848
As sedimentation progresses in the formation and evolution of a depositional geologic basin, the rock strata are subject to various stresses. With increasing lithostatic pressure, compressional forces act to compact the porous rock matrix, leading to overpressure buildup, changes in the fluid pore pressure and fluid flow. In the context of petroleum systems modelling, the present study concerns the geometry changes that a compacting basin experiences subject to deposition. The purpose is to track the positions of the rock layer interfaces as compaction occurs. To handle the challenge of potentially large geometry deformations, a new modelling concept is proposed that couples the pore pressure equation with a level set method to determine the movement of lithostratigraphic interfaces. The level set method propagates an interface according to a prescribed speed. The coupling term for the pore pressure and level-set equations consists of this speed function, which is dependent on the compaction law. The two primary features of this approach are the simplicity of the grid and the flexibility of the speed function. A first evaluation of the model concept is presented based on an implementation for one spatial dimension accounting for vertical effective stress. Isothermal conditions with a constant fluid density and viscosity were assumed. The accuracy of the implemented numerical solution for the case of a single stratigraphic unit with a linear compaction law was compared to the available analytical solution [38]. The multi-layer setup and the nonlinear case were tested for plausibility. 相似文献
14.
大型对称变带宽方程组的Cholesky分解法 总被引：6，自引：3，他引：3
作者针对地球物理数值模拟中常碰到的大型稀疏变带宽方程组的求解问题,介绍了 大型稀疏变带宽矩阵的存储方法及Ｃｈｏｌｅｓｋｙ分解法,该方法的特点是用二个一维数组,其中一个输入时存储对称稀疏矩阵变带宽内的元素,输出时存储Ｃｈｌｏｅｓｋｙ下三角短阵带宽内的元素;另一个存储对称变带宽矩阵对角线元素在前一维数组中的位置,大大节约了所占的计算内存空间。 相似文献
15.
以重力位在场源内部满足泊松方程为依据,以重力矢量满足第三类边界条件为切入点,推导了与三度体重力矢量满足的边值问题相对应的变分问题,进而利用有限单元法实现了对变分问题的求解.立方体模型试验结果表明:文中提出的新的系数矩阵存储方式较之传统方式能够更有效地节约存储空间,且为利用预条件共轭梯度技术更加快速地求解线性方程组提供了保障;重力矢量的计算精度与边界长度及单元网格的边长息息相关,其计算效率则主要取决于所要计算的节点总数和大型稀疏线性方程组求解算法的优劣;一般情况下,当单元的边长小于场源体边长的1/10、边界长度大于场源体长度的7.5倍时,能够获得理想的结果. 相似文献
16.
轴对称多层可压缩渗透各向异性岩基固结分析 总被引：1，自引：0，他引：1
从轴对称固结基本方程出发,通过对时间t、坐标r的Laplace-Hankel变换,再对坐标z的Laplace变换,得到三元一次方程组,解此方程组,并进行Laplace逆变换,得到了单层可压缩渗透各向异性岩基轴对称固结问题的传递矩阵,然后利用传递矩阵法,结合层间连续性条件和边界条件,得到了多层可压缩渗透各向异性岩基轴对称固结问题在积分变换域内的解。最后应用Laplace-Hankel逆变换技术得到轴对称固结问题在物理域内的理论解。编制了相应的计算程序,并进行了数值计算与分析,讨论了可压缩性和渗透各向异性对岩基固结的影响,结果表明：可压缩性越大,岩基瞬时沉降越大;渗透各向异性对固结过程影响明显,但对初始和最终沉降的影响很小。 相似文献
17.
Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization 总被引：1，自引：0，他引：1
Spatial and temporal behavior of hydrochemical parameters in groundwater can be studied using tools provided by geostatistics.
The cross-variogram can be used to measure the spatial increments between observations at two given times as a function of
distance (spatial structure). Taking into account the existence of such a spatial structure, two different data sets (sampled
at two different times), representing concentrations of the same hydrochemical parameter, can be analyzed by cokriging in
order to reduce the uncertainty of the estimation. In particular, if one of the two data sets is a subset of the other (that
is, an undersampled set), cokriging allows us to study the spatial distribution of the hydrochemical parameter at that time,
while also considering the statistical characteristics of the full data set established at a different time.
This paper presents an application of cokriging by using temporal subsets to study the spatial distribution of nitrate concentration
in the aquifer of the Lucca Plain, central Italy. Three data sets of nitrate concentration in groundwater were collected during
three different periods in 1991. The first set was from 47 wells, but the second and the third are undersampled and represent
28 and 27 wells, respectively. Comparing the result of cokriging with ordinary kriging showed an improvement of the uncertainty
in terms of reducing the estimation variance. The application of cokriging to the undersampled data sets reduced the uncertainty
in estimating nitrate concentration and at the same time decreased the cost of the field sampling and laboratory analysis.
Received: 23 July 1997 · Accepted: 31 March 1998 相似文献
18.
A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure 总被引：1，自引：0，他引：1
Peter Bastian 《Computational Geosciences》2014,18(5):779-796
In this paper, we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential/capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes, the resulting systems of nonlinear algebraic equations are solved with Newton’s method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust, and efficient. In particular, no postprocessing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1,000 processors. 相似文献
19.
Kriging without negative weights 总被引：1，自引：0，他引：1
Under a constant drift, the linear kriging estimator is considered as a weighted average ofn available sample values. Kriging weights are determined such that the estimator is unbiased and optimal. To meet these requirements, negative kriging weights are sometimes found. Use of negative weights can produce negative block grades, which makes no practical sense. In some applications, all kriging weights may be required to be nonnegative. In this paper, a derivation of a set of nonlinear equations with the nonnegative constraint is presented. A numerical algorithm also is developed for the solution of the new set of kriging equations. 相似文献
20.
Jacques Rivoirard 《Mathematical Geology》2002,34(7):797-808
Kriging with external drift allows one to estimate a target variable, accounting for a densely sampled auxiliary variable. Contrary to cokriging, kriging with external drift does not make explicit the structural link between target variable and auxiliary variable, for the latter is considered to be deterministic. In this paper, we show that kriging with external drift assumes implicitly an absence of spatial dependence between the auxiliary variable and the residual of the linear regression of target variable on auxiliary variable at same point. This is the simple model with orthogonal residual, where cokriging is collocated and coincides with kriging of the residual. In this model, the cross-structure is proportional to the structure of the auxiliary variable, and the linear regression of target variable on auxiliary variable does not depend on the support. 相似文献