Fitting the Linear Model of Coregionalization by Generalized Least Squares

In geostatistical studies, the fitting of the linear model of coregionalization (LMC) to direct and cross experimental semivariograms is usually performed with a weighted least-squares (WLS) procedure based on the number of pairs of observations at each lag. So far, no study has investigated the efficiency of other least-squares procedures, such as ordinary least squares (OLS), generalized least squares (GLS), and WLS with other weighing functions, in the context of the LMC. In this article, we compare the statistical properties of the sill estimators obtained with eight least-squares procedures for fitting the LMC: OLS, four WLS, and three GLS. The WLS procedures are based on approximations of the variance of semivariogram estimates at each distance lag. The GLS procedures use a variance–covariance matrix of semivariogram estimates that is (i) estimated using the fourth-order moments with sill estimates (GLS₁), (ii) calculated using the fourth-order moments with the theoretical sills (GLS₂), and (iii) based on an approximation using the correlation between semivariogram estimates in the case of spatial independence of the observations (GLS₃). The current algorithm for fitting the LMC by WLS while ensuring the positive semidefiniteness of sill matrix estimates is modified to include any least-squares procedure. A Monte Carlo study is performed for 16 scenarios corresponding to different combinations of the number of variables, number of spatial structures, values of ranges, and scale dependence of the correlations among variables. Simulation results show that the mean square error is accounted for mostly by the variance of the sill estimators instead of their squared bias. Overall, the estimated GLS₁ and theoretical GLS₂ are the most efficient, followed by the WLS procedure that is based on the number of pairs of observations and the average distance at each lag. On that basis, GLS₁ can be recommended for future studies using the LMC.     

Combination of Dependent Realizations Within the Gradual Deformation Method

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.     

Teacher''s Aide: Modeling Hole-Effect Variograms of Lithology-Indicator Variables

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.     

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.     

Fitting matrix-valued variogram models by simultaneous diagonalization (Part II: Application)

As an application, we demonstrate a proposed variogram modeling scheme using a spatial data set. Because the scheme relies on a procedure for simultaneously diagonalizing several matrices, we briefly describe the FG and least-squares algorithms. The model obtained by our scheme is used to cokrige the data. In addition, the proposed scheme is compared to more traditional methods.     

The equivalence of predictions from universal kriging and intrinsic random-function kriging

A proof is provided that the predictions obtained from kriging based on intrinsic random functions of orderk are identical to those obtained from anappropriate universal kriging model. This is a theoretical result based on known variability measures. It does not imply that people performing traditional universal kriging will get the same predictions as those using intrinsic random functions, because traditionally these methods differ in how variability is modeled. For intrinsic random functions, the same proof shows that predictions do not depend on the specific choice of the generalized covariance function. It is argued that the choice between these methods is really one of modeling and estimating the variability in the data.     

序贯模拟方法在储层建模中的应用研究

条件模拟方法是地质统计学发展的一个主要方向和研究热点。作者通过序贯模拟方法在储层建模中的应用，对模拟结果进行了参数统计和变异函数分析；并对克立格和序贯模拟方法进行了比较。应用研究的结果证明了该方法的各种特点，弥补了条件模拟方法在理论研究上的不足，为进一步的方法改进和拓宽其应用范围提供了理论依据和实例参考。     

Conditional simulation of intrinsic random functions of order k

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.     

Bayesian Inference of Spatial Covariance Parameters

This paper shows the application of the Bayesian inference approach in estimating spatial covariance parameters. This methodology is particularly valuable where the number of experimental data is small, as occurs frequently in modeling reservoirs in petroleum engineering or when dealing with hydrodynamic variables in groundwater hydrology. There are two main advantages of Bayesian estimation: firstly that the complete distribution of the parameters is estimated and, from this distribution, it is a straightforward procedure to obtain point estimates, confidence regions, and interval estimates; secondly, all the prior information about the parameters (information available before the data are collected) is included in the inference procedure through their prior distribution. The results obtained from simulation studies are discussed.     

Random vectors and spatial analysis by geostatistics for geotechnical applications

Geostatistics is extended to the spatial analysis of vector variables by defining the estimation variance and vector variogram in terms of the magnitude of difference vectors. Many random variables in geotechnology are in vectorial terms rather than scalars, and its structural analysis requires those sample variable interpolations to construct and characterize structural models. A better local estimator will result in greater quality of input models; geostatistics can provide such estimators: kriging estimators. The efficiency of geostatistics for vector variables is demonstrated in a case study of rock joint orientations in geological formations. The positive cross-validation encourages application of geostatistics to spatial analysis of random vectors in geoscience as well as various geotechnical fields including optimum site characterization, rock mechanics for mining and civil structures, cavability analysis of block cavings, petroleum engineering, and hydrologic and hydraulic modelings.     

The Necessity of a Multiple-Point Prior Model

Any interpolation, any hand contouring or digital drawing of a map or a numerical model necessarily calls for a prior model of the multiple-point statistics that link together the data to the unsampled nodes, then these unsampled nodes together. That prior model can be implicit, poorly defined as in hand contouring; it can be explicit through an algorithm as in digital mapping. The multiple-point statistics involved go well beyond single-point histogram and two-point covariance models; the challenge is to define algorithms that can control more of such statistics, particularly those that impact most the utilization of the resulting maps beyond their visual appearance. The newly introduced multiple-point simulation (mps) algorithms borrow the high order statistics from a visually and statistically explicit model, a training image. It is shown that mps can simulate realizations with high entropy character as well as traditional Gaussian-based algorithms, while offering the flexibility of considering alternative training images with various levels of low entropy (organized) structures. The impact on flow performance (spatial connectivity) of choosing a wrong training image among many sharing the same histogram and variogram is demonstrated.     

Canonical variate analysis with unequal covariance matrices: Generalizations of the usual solution

Canonical variate analysis is extended for use when the covariance matrices are not equal. Linear combinations of variates are derived by generalizing either a weighted between-groups approach or the likelihood-ratio test and the associated noncentrality matrix. The usual solution and the two generalizations are compared via generated data for a few typical configurations of means in a situation in which the covariance matrices are in fact equal. The MSE of the canonical variate coefficients and group means for the generalizations are approximately three times those for the usual solution, due to corresponding changes in the variances. Two examples are discussed.     

Estimation of a multidimensional covariance function in case of anisotropy

A model of a multivariate covariance function with an ellipsoidal directional correlation scale has been developed. The axes of the ellipsoidal scale are related to the eigenvalues and eigenvectors of a matrix B which characterizes the ellipsoid of the range of influence. The matrix B is found to be related to a matrix T which can be estimated directly from sparse sampling data and can be used to determine estimates of the matrix B. The method has been applied to both two-dimensional and three-dimensional cases. The numerical results show that the satisfactory accuracy is obtained with sparse sampling data from an anisotropic random function.     

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.     

基于多元自适应回归样条的青藏高原温泉区域的冻土分布制图

以探地雷达、电磁测深、钻探等技术方法获得野外数据及数字高程（DEM）遥感数据为基础,通过聚类分析和相关性分析对高程、坡度、坡向等因素对多年冻土分布的影响进行了定量化研究.利用非线性的多元自适应回归样条（MARS）方法建立了基于高程、太阳辐射的多年冻土分布模型,通过自身的交叉验证及对比年平均地温模型和逻辑回归模型的总体分...     

Minimum norm quadratic estimation of components of spatial covariance

Quadratic estimators of components of a nested spatial covariance function are presented. Estimators are unbiased and possess a minimum norm property. Inversion of a covariance matrix is required but, by assuming that spatial correlation is absent, a priori, matrix inversion can be avoided. The loss of efficiency that results from this assumption is discussed. Methods can be generalized to include estimation of components of a generalized polynomial covariance assuming the underlying process to be an intrinsic random function. Particular attention is given to the special case where just two components of spatial covariance exist, one of which represents a nugget effect.     

组合切削具产生的预破碎区对钻进效果的影响

从组合切削具将在岩石中产生预破碎区的论点出发,通过实验定量研究了预破碎区深度与掏槽刃切入深度的关系,得出了预破碎区有利于降低岩石强度及岩石破碎能耗的结论,并用生产试验结果进行了验证。提出了孕镶金刚石钻头钻进Ⅶ～Ⅷ级硬岩时仍存在预破碎区和切削、微切削破岩方式,以及预破碎区并非越大越好的学术观点。     

The origins of kriging

In this article, kriging is equated with spatial optimal linear prediction, where the unknown random-process mean is estimated with the best linear unbiased estimator. This allows early appearances of (spatial) prediction techniques to be assessed in terms of how close they came to kriging.     

海洋环境因素极值组合及设计标准

由于海洋环境条件的复杂性、多变性及随机性,设计标准的选取是决定工程结构安全度、造价、效益及合理型式的主要因素。传统的设计标准,无法考虑海洋环境条件的随机组合,往往过高估计环境条件设计标准,造成不必要的浪费,甚至使具有开发前景的油田失去开采价值。以实测和后报资料为基础,使用多维联合概率的随机模拟技术,结合不同结构型式的极值响应及不同资料样本的选择方法,提出了海洋工程结构物上的风、浪、流、潮联合荷载及相应的联合概率水平问题,用以作为海洋工程环境荷载设计标准。     

福建马坑外围铁矿深部勘查钻探工艺组合应用研究

针对福建马坑外围铁矿深部复杂地层钻探技术难题,开展钻探工艺的组合应用探讨,通过采用绳索取心液动锤、多功能孔底反循环单动双管取心技术等以及复杂地层护壁堵漏的技术组合（套管、泥浆、水泥）,初步解决矿区复杂地层取心难、孔壁故障频发、易斜等问题,总结了一套对后续施工具有参考价值的钻探工艺和措施。