Cokriging nonstationary data

Universal cokriging is used to obtain predictions when dealing with multivariate random functions. An important type of nonstationarity is defined in terms of multivariate random functions with increments which are stationary of orderk. The covariance between increments of different variables is modeled by means of the pseudo-cross-covariance function. Criteria are formulated to which the parameters of pseudo-cross-covariance functions must comply so as to ensure positive-definiteness. Cokriging equations and the induced cokriging equations are given. The study is illustrated by an example from soil science.     

Ordinary Cokriging Revisited

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.     

Ordinary Cokriging Revisited

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.     

Cokriging of regionalized compositions

To avoid spurious spatial correlation when analyzing the spatial covariance structure of regionalized compositions, additive-log-ratio transformation can be used. Here, the additive-log-ratio cokriging estimator, derived in a natural way from this transformation, is shown to be invariant under permutation of components of the untransformed regionalized composition. It leads, as expected, to an exact interpolation. As original data, predicted values of the regionalized composition at unknown points add up to the same constant c and lie between 0 and c.     

On Some Simplifications of Cokriging Neighborhood

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.     

Which Models for Collocated Cokriging?

When a target variable is sparsely sampled, compared to a densely sampled auxiliary variable, cokriging requires simplifications. In its strict sense, collocated cokriging makes use of the auxiliary variable only at the current point where the target variable is to be estimated; in the multicollocated form, it also makes use of the auxiliary variable at all points where the target variable is available. This paper looks for the models that support these collocated cokrigings, i.e., the models in which the simplification resulting from the collocated forms does not result in any loss of information. In these models, the cross-structure between the two variables is shown to be proportional to the structure of the auxiliary variable, not to the structure of the target variable as is often assumed (except, of course, when all structures are proportional). The target variable depends on the auxiliary variable and on a spatially uncorrelated residual. Collocated cokriging simplifies to the simple method, which consists in kriging this residual. The strictly collocated cokriging corresponds to the particular case where the residual has a pure nugget structure, but it is then reduced to the single regression at the target point. Except for this trivial case, there are no models in which strictly collocated cokriging is exactly a cokriging.     

Multivariate Intrinsic Random Functions for Cokriging

In multivariate geostatistics, suppose that we relax the usual second-order-stationarity assumptions and assume that the component processes are intrinsic random functions of general orders. In this article, we introduce a generalized cross-covariance function to describe the spatial cross-dependencies in multivariate intrinsic random functions. A nonparametric method is then proposed for its estimation. Based on this class of generalized cross-covariance functions, we give cokriging equations for multivariate intrinsic random functions in the presence of measurement error. A simulation is presented that demonstrates the accuracy of the proposed nonparametric estimation method. Finally, an application is given to a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site used for atomic-bomb testing.     

3D Property Modeling of Void Ratio by Cokriging

Void ratio measures compactness of ground soil in geotechnical engineering. When samples are collected in certain area for mapping void ratios, other relevant types of properties such as water content may be also analyzed. To map the spatial distribution of void ratio in the area based on these types of point, observation data interpolation is often needed. Owing to the variance of sampling density along the horizontal and vertical directions, special consideration is required to handle anisotropy of estimator. 3D property modeling aims at predicting the overall distribution of property values from limited samples, and geostatistical method can he employed naturally here because they help to minimize the mean square error of estimation. To construct 3D property model of void ratio, cokriging was used considering its mutual correlation with water content, which is another important soil parameter. Moreover, K-D tree was adopted to organize the samples to accelerate neighbor query in 3D space during the above modeling process. At last, spatial configuration of void ratio distribution in an engineering body was modeled through 3D visualization, which provides important information for civil engineering purpose.     

月尺度的协克立格模型在水文数据插补延长中的应用

月尺度的协同克立格模型能够对水文变量进行线性、无偏和最佳估计[2]。针对水文数据在时间上富足而空间上缺乏以及单变量具有相依性、多变量之间又有相关性的特点,以滦河流域内蒙古段的大河口、多伦和花塘沟三个雨量站的实测月降水量资料为基础,用月尺度的协克立格模型对缺测数据进行了插补延长。经验证明估值精度较高,结果可信。同时还对影响估值精度的数据结构进行了讨论。结果表明,数据结构对估值精度的影响主要取决于观测数据间存在的相关性。     

Mapping Soil Particle Size Fractions Using Compositional Kriging,Cokriging and Additive Log-ratio Cokriging in Two Case Studies

Information on the spatial distribution of soil particle-size fractions (psf) is required for a wide range of applications. Geostatistics is often used to map spatial distribution from point observations; however, for compositional data such as soil psf, conventional multivariate geostatistics are not optimal. Several solutions have been proposed, including compositional kriging and transformation to a composition followed by cokriging. These have been shown to perform differently in different situations, so that there is no procedure to choose an optimal method. To address this, two case studies of soil psf mapping were carried out using compositional kriging, log-ratio cokriging, cokriging, and additive log-ratio cokriging; and the performance of Mahalanobis distance as a criterion for choosing an optimal mapping method was tested. All methods generated very similar results. However, the compositional kriging and cokriging results were slightly more similar to each other than to the other pair, as were log-ratio cokriging and additive log-ratio cokriging. The similar results of the two methods within each pair were due to similarities of the methods themselves, for example, the same variogram models and prediction techniques, and the similar results between the two pairs were due to the mathematical relationship between original and log-ratio transformed data. Mahalanobis distance did not prove to be a good indicator for selecting an optimal method to map soil psf.     

Collocated Cokriging Based on Merged Secondary Attributes

There exist many secondary data that must be considered in in reservoir characterization for resource assessment and performance forecasting. These include multiple seismic attributes, geological trends and structural controls. It is essential that all secondary data be accounted for with the precision warranted by that data type. Cokriging is the standard technique in geostatistics to account for multiple data types. The most common variant of cokriging in petroleum geostatistics is collocated cokriging. Implementations of collocated cokriging are often limited to a single secondary variable. Practitioners often choose the most correlated or most relevant secondary variable. Improved models would be constructed if multiple variables were accounted for simultaneously. This paper presents a novel approach to (1) merge all secondary data into a single super secondary variable, then (2) implement collocated cokriging with the single variable. The preprocessing step is straightforward and no major changes are required in the standard implementation of collocated cokriging. The theoretical validity of this approach is proven, that is, the results are proven to be identical to a “full” approach using all multiple secondary variables simultaneously.     

On the Equivalence of the Cokriging and Kriging Systems

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     

Data Configurations and the Cokriging System: Simplification by Screen Effects

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.     

Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity

This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.     

归来庄金矿床化探信息金异常的协同克立格法研究

根据收集到的归来庄金矿床的化探资料,选取与Au相关系数较大的W、As、Sb等区域化探元素作为变量,对该区金异常空间分布进行协同克立格法研究.在对4个区域化变量进行空间结构分析的基础上,建立空间结构模型并进行估计.综合化探金异常的协同克立格法研究结果表明,该区有多处金异常存在,具有较好的找矿前景.     

A Hybrid Estimation Technique Using Elliptical Radial Basis Neural Networks and Cokriging

Mathematical Geosciences - Mineral resource estimation is an integral part of making informed decisions while evaluating a mining operation’s feasibility. Geostatistical tools estimate...     

Multidimensional spectrum estimation for nonstationary processes

A spectrum estimation method for multidimensional nonstationary processes has been developed. The estimation method has been applied to a nonstationary two-dimensional random field. The numerical findings show that the results of the estimations are about what can be expected from a multidimensional estimation for a nonstationary process.     

A nonstationary parameter model for the sandstone creep tests

The rock masses of hydro-fluctuation belt experience seepage pressure following impoundment in the Three Gorges Reservoir; its creep behaviors are significant for reservoir bank slopes. To study the creep behaviors under seepage pressure (0, 1.45, and 1.75 MPa), we performed creep tests using representative landslide sandstone in the Three Gorges Reservoir and investigated the sandstone creep behaviors under the coupling effects of seepage pressure and stress. Previous researches on rocks have usually regarded the creep constitutive parameter as a constant; however, in this study, a nonlinear, nonstationary, plastic-viscous (NNPV) creep model which can describe the curve of sandstone creep tests is proposed. The rock-creep parameters under three levels of seepage pressure were identified, and theoretical curves using the NNPV model agreed well with the experimental data, indicating that the new model cannot only describe the primary creep and secondary creep stages under varying seepage pressures but also, in particular, perfectly describes the tertiary creep stage. Finally, the sensitivity of the NNPV model parameters is analyzed, and the result shows that the nonstationary coefficient α and the nonlinear coefficient m are main parameters affecting the tertiary creep stage.