An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

Dual Kriging with Local Neighborhoods: Application to the Representation of Surfaces

Ordinary kriging, in its common formulation, is a discrete estimator in that it requires the solution of a kriging system for each point in space in which an estimate is sought. The dual formulation of ordinary kriging provides a continuous estimator since, for a given set of data, only a kriging system has to be estimated and the resulting estimate is a function continuously defined in space. The main problem with dual kriging up to now has been that its benefits can only be capitalized if a global neighborhood is used. A formulation is proposed to solve the problem of patching together dual kriging estimates obtained with data from different neighborhoods by means of a blending belt around each neighborhood. This formulation ensures continuity of the variable and, if needed, of its first derivative along neighbor borders. The final result is an analytical formulation of the interpolating surface that can be used to compute gradients, cross-sections, or volumes; or for the quick evaluation of the interpolating surface in numerous locations.     

A Non-Homogeneous Model for Kriging Dosimetric Data

Mathematical Geosciences - This paper deals with kriging-based interpolation of dosimetric data. Such data typically show some inhomogeneities that are difficult to take into account by means of...     

DNAPLs污染含水层多相流数值模拟模型的替代模型

针对表面活性剂强化的重非水相流体(DNAPLs)污染的含水层修复问题,在建立多相流数值模拟模型的基础上,应用拉丁超立方采样(LHS)方法,在多相流模拟模型可控输入变量的可行域内采样,有效提高了采样效率和覆盖程度。根据采集的样品数据集,运用多元回归分析方法建立多相流模拟模型的替代模型--双响应面模型,为DNAPLs污染含水层修复过程的优化设计的耦合技术探索新的理论和方法。经检验,替代模型计算结果的相对误差均小于10%,精度较高,说明其在功能上充分逼近模拟模型。运用替代模型实现模拟模型与优化模型的连接,可以大幅度减少优化模型计算过程中直接多次反复调用模拟模型所引起的庞大计算负荷。     

An Empirical Comparison of Kriging Methods for Nonlinear Spatial Point Prediction

Spatial prediction is a problem common to many disciplines. A simple application is the mapping of an attribute recorded at a set of points. Frequently a nonlinear functional of the observed variable is of interest, and this calls for nonlinear approaches to prediction. Nonlinear kriging methods, developed in recent years, endeavour to do so and additionally provide estimates of the distribution of the target quantity conditional on the observations. There are few empirical studies that validate the various forms of nonlinear kriging. This study compares linear and nonlinear kriging methods with respect to precision and their success in modelling prediction uncertainty. The methods were applied to a data set giving measurements of the topsoil concentrations of cobalt and copper at more than 3000 locations in the Border Region of Scotland. The data stem from a survey undertaken to identify places where these trace elements are deficient for livestock. The comparison was carried out by dividing the data set into calibration and validation sets. No clear differences between the precision of ordinary, lognormal, disjunctive, indicator, and model-based kriging were found, neither for linear nor for nonlinear target quantities. Linear kriging, supplemented with the assumption of normally distributed prediction errors, failed to model the conditional distribution of the marginally skewed data, whereas the nonlinear methods modelled the conditional distributions almost equally well. In our study the plug-in methods did not fare any worse than model-based kriging, which takes parameter uncertainty into account.     

基于克里金模型的边坡稳定可靠度分析方法

边坡工程的复杂性不仅表现为各岩土参数的变异性和非确定性,而且还在于其极限状态功能函数的非解析性及隐式性。以Janbu 法为例,研究了隐式功能函数下易于执行的边坡工程稳定可靠度计算方法。首先,调用边坡极限平衡模式获得岩土基本参数,并利用拉丁超立方试验设计抽取影响边坡稳定性基本参数的适量初始样本。其次,采用地质统计学中的克里金（Kriging）各向异性关联映射方法,将边坡功能函数值表述为随机过程。然后,结合主动学习方法,基于搜索规则调整训练样本,通过反复迭代循环确定满足实际工程精度的随机过程所表示的边坡功能函数。最后,调用随机过程函数通过验算点法（JC法）获得边坡的失效概率。工程算例分析表明,文中方法的求解精度与蒙特卡洛模拟方法相当,但计算过程简明,效率高,更具工程实用性。     

水库调洪演算的随机数学模型

以介于微分方程和概率论之间的边缘数学分支随机微分方程的数学模型,对水库调洪过程中的随机现象和规律进行数学描述和分析,试图全面正确地综合各种不确定性因素对库水位随机过程的影响.根据水库蓄洪量具有Wiener过程特性的分析,推导了带有随机输入项和随机初始条件的调洪演算Ito方程.在此基础上,运用Fokker-Planck向前方程,求介了调洪过程库水位的概率密度分布.计算成果表明,运用随机微分方程进行水库的调洪演算,有利于正确分析水库调洪的随机过程和进一步开展水库泄洪风险分析.     

A Stochastic Object Model Conditioned to High-Quality Seismic Data

We present an approach for modeling facies bodies in which a highly constrained stochastic object model is used to integrate detailed seismic interpretation of the reservoir’s sedimentological architecture directly in a three-dimensional reservoir model. The approach fills the gap between the use of seismic data in a true deterministic sense, in which the facies body top and base are resolved and mapped directly, and stochastic methods in which the relationship between seismic attributes and facies is defined by conditional probabilities. The lateral geometry of the facies bodies is controlled by seismic interpretations on horizon slices or by direct body extraction, whereas facies body thickness and cross-sectional shape are defined by a mixture of seismic data, well data, and user defined object shapes. The stochastic terms in the model are used to incorporate local geometric variability, which is used to increase the geological realism of the facies bodies and allow for correct, flexible well conditioning. The result is a set of three-dimensional facies bodies that are constrained to the seismic interpretations and well data. Each body is defined as a parametric object that includes information such as location of the body axis, depositional direction, axis-to-margin normals, and external body geometry. The parametric information is useful for defining geologically realistic intrabody petrophysical trends and for controlling connectivity between stacked facies bodies.     

Hybrid Model for Urban Air Pollution Forecasting: A Stochastic Spatio-Temporal Approach

Air pollution is usually driven by a complex combination of factors in which meteorology, physical obstacles, and interactions between pollutants play significant roles. Considering the characteristics of urban atmospheric pollution and its consequent impacts on human health and quality of life, forecasting models have emerged as an effective tool to identify and forecast air pollution episodes. The overall objective of the present work is to produce forecasts of pollutant concentrations with high spatio-temporal resolution and to quantify the uncertainty in those forecasts. Therefore, a new approach was developed based on a two-step methodology. Firstly, neural network models were used to generate short-term temporal forecasts based on air pollution and meteorology data. The accuracy of those forecasts was then evaluated against an independent set of historical data. Secondly, local conditional distributions of the observed values with respect to the predicted values were used to perform spatial stochastic simulations for the entire geographic area of interest. With this approach the spatio-temporal dispersion of a pollutant can be predicted, while accounting for both the temporal uncertainty in the forecast (reflecting the neural networks efficiency at each monitoring station) and the spatial uncertainty as revealed by the spatial variograms. Based on an analysis of the results, our proposed method offers a highly promising alternative for the characterization of urban air quality.     

Combining Knowledge from Diverse Sources: An Alternative to Traditional Data Independence Hypotheses

Consider the assessment of any unknown event A through its conditional probability P(A | B,C) given two data events B, C of different sources. Each event could involve many locations jointly, but the two data events are assumed such that the probabilities P(A | B) and P(A | C) can be evaluated. The challenge is to recombine these two partially conditioned probabilities into a model for P(A | B,C) without having to assume independence of the two data events B and C. The probability P(A | B,C) is then used for estimation or simulation of the event A. In presence of actual data dependence, the combination algorithm provided by the traditional conditional independence hypothesis is shown to be nonrobust leading to various inconsistencies. An alternative based on a permanence of updating ratios is proposed, which guarantees all limit conditions even in presence of complex data interdependence. The resulting recombination formula is extended to any number n of data events and a paradigm is offered to introduce formal data interdependence.     

Lower Crustal Magma Genesis and Preservation: a Stochastic Framework for the Evaluation of Basalt-Crust Interaction

We present a quantitative assessment of the thermal and dynamicresponse of an amphibolitic lower crust to the intrusion ofbasaltic dike swarms in an arc setting. We consider the effectof variable intrusion geometry, depth of intrusion, and basaltflux on the production, persistence, and interaction of basalticand crustal melt in a stochastic computational framework. Distinctmelting and mixing environments are predicted as a result ofthe crustal thickness and age of the arc system. Shallow crustal(30 km) environments and arc settings with low fluxes of mantle-derivedbasalt are likely repositories of isolated pods of mantle andcrustal melts in the lower crust, both converging on daciticto rhyodacitic composition. These may be preferentially rejuvenatedin subsequent intrusive episodes. Mature arc systems with thickercrust (50 km) produce higher crustal and residual basaltic meltfractions, reaching 0·4 for geologically reasonable basaltfluxes. The basaltic to basaltic andesite composition of bothcrustal and mantle melts will facilitate mixing as the networkof dikes collapses, and Reynolds numbers reach 10^–4–1·0in the interiors of dikes that have been breached by ascendingcrustal melts. This may provide one mechanism for melting, assimilation,storage and homogenization (MASH)-like processes. Residual mineralassemblages of crust thickened by repeated intrusion are predictedto be garnet pyroxenitic, which are denser than mantle peridotiteand also generate convective instabilities where some of thecrustal material is lost to the mantle. This reconciles thethinner than predicted crust in regions that have undergonea large flux of mantle basalt for a prolonged period of time,and helps explain the enrichment of incompatible elements suchas K₂O, typical of mature arc settings, without the associatedmass balance problem. KEY WORDS: crustal anatexis; delamination; lower crust; magma mixing; thermal model     

克里金参数估值法及其在参数估计分析中的应用

为考虑岩土介质参数的空间分布的结构性和随机性等不确定因素,引入了克里金参数估值法。采用变异函数描述参数在空间结构上的变化,建立其空间变异规律的数学模型,从而实现对岩土参数的估值。通过实例分析揭示了克里金估值法具有反映“过滤效应”和“集团效应”的优点,对不同位置的数据赋予不同的权重系数,能够有效地反映参数空间变异结构,有利于对参数的合理化分析     

Stochastic modeling based on deterministic formulation: An example

In the context of seismology, an example of stochastic modeling on the basis of an established deterministic formulation is presented. The advantages of this approach to modeling over those based solely on statistical fit are discussed. It is demonstrated that the result of applying this procedure is a model whose main parameter has a physical interpretation, and therefore a validation based on criteria other than statistical goodness of fit is also possible. Statistical inference together with some demonstrating examples are also included.     

Compositional Kriging: A Spatial Interpolation Method for Compositional Data

Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable.     

基于嵌入离散裂缝的页岩气藏视渗透率模型

页岩储层具有不同类型的储集空间,但综合考虑不同储集空间,对页岩储层渗透率进行评价的模型未见报道.基于嵌入离散裂缝模型,建立的页岩气藏视渗透率模型包括4个步骤:（1）构建天然裂缝、有机质和无机质的空间分布模型;（2）筛选不同类型储集空间的渗透率计算方法;（3）基于嵌入离散裂缝模型,结合空间分布模型和渗透率计算方法,建立数值模拟模型;（4）在模型的入口和出口端施加压差,求得一定压差下通过该岩心的气体流量,采用达西定律得到该页岩气藏的视渗透率.其计算结果与文献报道的渗透率实验值吻合较好.通过对不同因素的探讨,结果表明,天然裂缝对页岩气藏视渗透率的贡献大于无机质和有机质孔隙.因此,计算页岩视渗透率时有必要对天然裂缝、有机质和无机质孔隙进行综合考虑.      

四川呷村火山成因块状硫化物矿床的综合找矿模式

呷村火山成因块状硫化物矿床是“三江”地区的超大型含金富银多金属矿床,其综合找矿模式的建立对“三江”有色、贵金属成矿带的成矿预测具有重要意义。文章在大量实测物探资料的基础上,结合区域地质、遥感和矿床地质资料,从6个方面总结了该矿床的综合找矿模式：即大地构造背景：局部（矿奁）成矿地质环境;地质找矿标志;区域物探、化探、遥感示矿要素;岩石、矿石物性特征;综合物（化）探异常特征。并给出了针对呷村型矿床的勘探程序和最佳勘探方法组合,对指导区域打矿及勘查工作部署具有重要意义。     

An Initial Guess for the Levenberg–Marquardt Algorithm for Conditioning a Stochastic Channel to Pressure Data

A standard procedure for conditioning a stochastic channel to well-test pressure data requires the minimization of an objective function. The Levenberg–Marquardt algorithm is a natural choice for minimization, but may suffer from slow convergence or converge to a local minimum which gives an unacceptable match of observed pressure data if a poor initial guess is used. In this work, we present a procedure to generate a good initial guess when the Levenberg–Marquardt algorithm is used to condition a stochastic channel to pressure data and well observations of channel facies, channel thickness, and channel top depth. This technique yields improved computational efficiency when the Levenberg–Marquardt method is used as the optimization procedure for generating realizations of the model by the randomized maximum likelihood method.     

An Interactive Sampling Strategy Based on Information Analysis and Ordinary Kriging for Locating Hot Spot Regions

This study proposes an interactive sampling strategy for locating the hot spot or maximum regions of a concerned attribute in a given area of survey. In the proposed strategy, information analysis is performed based on the ordinary kriging from the existing sample data to suggest a new batch of samples under the criterion of the highest information free energy. The information free energy (F) is a function of information energy (U) and information entropy (S) through F = U - TS, where T is information temperature and is used to coordinate the contribution of U and S to F. Information energy is the value of the concerned attribute, and information entropy is the transformed error variance of kriging and therefore measures the evenness and density of coverage of samples over the area under survey. At early sampling batches, information temperature is high and information entropy dominates the information free energy, and samples are suggested to give an even and dense enough coverage of the whole area under investigation. As samples accumulate, information temperature decreases to enlarge the contribution of information energy, and future samples are taken toward the locations with high attribute values. Two examples demonstrate the efficiency and effectiveness of the proposed sampling strategy in locating the hot spot regions of various fields: (1) a heavy metal contaminated site reproduced by modeling on 55 real field data; (2) a simulated two-dimensional field by the random phase volume (RPV) model. The results show that the proposed strategy, a robust interactive sampling procedure, is able to locate hot spot regions without compromising with the overall profile of an under-survey area.