Efficient updating of kriging estimates and variances

This short note presents a method for efficiently updating ordinary kriging estimates and variances when one or more additional samples are incorporated into the kriging system. First, the foundation linear algebra result is presented. Then the update equations are derived. Finally, an illustrative application of updating is briefly discussed.     

Estimation of regionalized compositions: A comparison of three methods

A regionalized composition is a random vector function whose components are positive and sum to a constant at every point of the sampling region. Consequently, the components of a regionalized composition are necessarily spatially correlated. This spatial dependence—induced by the constant sum constraint—is a spurious spatial correlation and may lead to misinterpretations of statistical analyses. Furthermore, the cross-covariance matrices of the regionalized composition are singular, as is the coefficient matrix of the cokriging system of equations. Three methods of performing estimation or prediction of a regionalized composition at unsampled points are discussed: (1) the direct approach of estimating each variable separately; (2) the basis method, which is applicable only when a random function is available that can he regarded as the size of the regionalized composition under study; (3) the logratio approach, using the additive-log-ratio transformation proposed by J. Aitchison, which allows statistical analysis of compositional data. We present a brief theoretical review of these three methods and compare them using compositional data from the Lyons West Oil Field in Kansas (USA). It is shown that, although there are no important numerical differences, the direct approach leads to invalid results, whereas the basis method and the additive-log-ratio approach are comparable.     

地质统计学方法在井间储层参数预测中的应用

本文探讨了利用地质统计学中的三种克立格方法进行井间储层参数预测的原理及方法,并在某油田中应用,通过最优性检验,建立起了适合研究区地质特点的各种储层参数的最佳估计模型,揭示了储层参数在三维空间的变化规律,为建立三维储层预测模型开辟了新的方法。     

扇形沉积体生长的分形几何特征分析

扇形沉积体是我国内陆及沿海地区中、新生代含油气盆地中的重要油气勘探目标,对其平面分布形态和内部结构参数特征的定量表征是油气勘探部门极为关注的问题。随着分形几何理论在沉积学领域应用的日趋广泛,使人们可从新的角度来考虑这一问题的解决方法。扇形沉积体的生长是一个复杂的非线性动力学过程,且具有分形特征。分维数的大小反映了沉积体外部形态和内部结构的复杂程度,分维数的变化幅度预示着构造和沉积背景的某些信息。将分形和混沌理论相结合,也许是实施扇形沉积体非线性动力学建模和模拟的重要途经之一。     

Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.     

考虑权值非负约束的泛克立格算法

泛克立格法是地质统计学的一种重要估值方法,当区域化变量在空间变异几何域内非平稳时常用泛克立格法来估值。然而,用泛克立格方程组计算估计权值时由于未对权的符号作任何限制,从而使得计算出的权值经常出现负权现象,而负权的存在有许多弊端,所以在许多应用中有必要对权值作非负要求。本文基于线性规划方法提出了一种考虑权值非负约束的泛克立格算法,该算法既考虑到了权值的非负约束条件,又利用了线性规划方法求解简便快捷的优点。     

因子克里格分析支持的重金属元素空间多尺度变化特征——以北部湾东北部海域为例

将因子克里格分析方法应用于海底沉积物重金属元素空间多尺度变化的研究中,分析了北部湾东北部表层沉积物中7种重金属元素,Ni、Cu、Zn、Pb、Cr、As和Cd的空间多尺度结构特征,识别并分离了重金属元素不同尺度的空间变化,并探讨了各个尺度上重金属的物源及控制因素。结果显示,研究区重金属存在块金效应、变程为30km的球状结构(短变程尺度)和变程为140km的球状结构(长变程尺度)等三种尺度的空间结构,并拟合了研究区7种重金属元素的线性协同区域化模型。短变程尺度上,7种重金属元素的空间分布表现为"点状"或"条带状"的局部尺度的空间变化特征;长变程尺度上,7种重金属元素的空间分布反映出"片状"的区域尺度的空间变化特征。局部尺度上,Zn、Cr、Ni、Cu、Pb和Cd等元素主要源于海南岛、雷州半岛、广西大陆等陆地母岩物质,其空间分布特征受到了沉积物粒度的控制作用;而As主要受到海南岛、雷州半岛的人为污染成分的影响。区域尺度上,Zn、Cr、Ni和Cu主要来源于雷州半岛、海南岛的母岩物质,其次为广西大陆;As主要来源于海南岛的母岩物质,其次为雷州半岛、广西大陆。这些重金属元素在北部湾终年逆时针大环流的控制下,进行搬运、迁移并沉积在"汇聚中心",形成了研究区重金属元素区域尺度上NWW-NNW向的分布格局;而As不同于其他元素的亲陆的NNW向分布特征,主要源于其不同的地球化学行为。     

Bayesian area-to-point kriging using expert knowledge as informative priors

Area-to-point (ATP) kriging is a common geostatistical framework to address the problem of spatial disaggregation or downscaling from block support observations (BSO) to point support (PoS) predictions for continuous variables. This approach requires that the PoS variogram is known. Without PoS observations, the parameters of the PoS variogram cannot be deterministically estimated from BSO, and as a result, the PoS variogram parameters are uncertain. In this research, we used Bayesian ATP conditional simulation to estimate the PoS variogram parameters from expert knowledge and BSO, and quantify uncertainty of the PoS variogram parameters and disaggregation outcomes. We first clarified that the nugget parameter of the PoS variogram cannot be estimated from only BSO. Next, we used statistical expert elicitation techniques to elicit the PoS variogram parameters from expert knowledge. These were used as informative priors in a Bayesian inference of the PoS variogram from BSO and implemented using a Markov chain Monte Carlo algorithm. ATP conditional simulation was done to obtain stochastic simulations at point support. MODIS (Moderate Resolution Imaging Spectroradiometer) atmospheric temperature profile data were used in an illustrative example. The outcomes from the Bayesian ATP inference for the Matérn variogram model parameters confirmed that the posterior distribution of the nugget parameter was effectively the same as its prior distribution; for the other parameters, the uncertainty was substantially decreased when BSO were introduced to the Bayesian ATP estimator. This confirmed that expert knowledge brought new information to infer the nugget effect at PoS while BSO only brought new information to infer the other parameters. Bayesian ATP conditional simulations provided a satisfactory way to quantify parameters and model uncertainty propagation through spatial disaggregation.     

Statistical data fusion of multi-sensor AOD over the Continental United States

This article illustrates two techniques for merging daily aerosol optical depth (AOD) measurements from satellite and ground-based data sources to achieve optimal data quality and spatial coverage. The first technique is a traditional Universal Kriging (UK) approach employed to predict AOD from multi-sensor aerosol products that are aggregated on a reference grid with AERONET as ground truth. The second technique is spatial statistical data fusion (SSDF); a method designed for massive satellite data interpolation. Traditional kriging has computational complexity O(N³), making it impractical for large datasets. Our version of UK accommodates massive data inputs by performing kriging locally, while SSDF accommodates massive data inputs by modelling their covariance structure with a low-rank linear model. In this study, we use aerosol data products from two satellite instruments: the moderate resolution imaging spectrometer and the geostationary operational environmental satellite, covering the Continental United States.     

Effect of growing watershed imperviousness on hydrograph parameters and peak discharge

An increasing impervious area is quickly extending over the Wu‐Tu watershed due to the endless demands of the people. Generally, impervious paving is a major result of urbanization and more recently has had the potential to produce more enormous flood disasters than those of the past. In this study, 40 available rainfall–runoff events were chosen to calibrate the applicable parameters of the models and to determine the relationships between the impervious surfaces and the calibrated parameters. Model inputs came from the outcomes of the block kriging method and the non‐linear programming method. In the optimal process, the shuffled complex evolution method and three criteria were applied to compare the observed and simulated hydrographs. The tendencies of the variations of the parameters with their corresponding imperviousness were established through regression analysis. Ten cases were used to examine the established equations of the parameters and impervious covers. Finally, the design flood routines of various return periods were furnished through use of approaches containing a design storm, block kriging, the SCS model, and a rainfall‐runoff model with established functional relationships. These simulated flood hydrographs were used to compare and understand the past, present, and future hydrological conditions of the watershed studied. In the research results, the time to peak of flood hydrographs for various storms was diminished approximately from 11 h to 6 h in different decrements, whereas peak flow increased respectively from 127 m³ s^?1 to 629 m³ s^?1 for different storm intensities. In addition, this study provides a design diagram for the peak flow ratio to help engineers and designers to construct hydraulic structures efficiently and prevent possible damage to human life and property. Copyright © 2007 John Wiley & Sons, Ltd.