Conditioning of the stationary kriging matrices for some well-known covariance models

In this paper, the condition number of the stationary kriging matrix is studied for some well-known covariance models. Indeed, the robustness of the kriging weights is strongly affected by this measure. Such an analysis can justify the choice of a covariance function among other admissible models which could fit a given experimental covariance equally well.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

Conditional LAS stochastic simulation of regionalized variables in random fields

Ore reserves forecasts are required to aid in investment decisions, mine design and valuation, short and long term production plans and proper and efficient mill design. In random multivariable fields with limited data and high levels of uncertainty, the kriged block estimates produce a smoothing effect resulting in underestimating high values and overestimating low values. The modified conditional simulation (MCS) methodology solves these problems by simulating the random field to preserve its mean and the variance structure. The simulation model is conditioned to reproduce the data at known sample points to minimize the variability between the simulated data and the true field data. In this study, the authors develop the MCS methodology to simulate ore reserve grades using the best linear unbiased estimation (BLUE) and the local average subdivision (LAS) techniques. The MCS methodology is applied to simulate block grades in a section of the Sabi Gold Project in Zimbabwe. The results are compared with the kriged estimates for this section. Analysis of the results shows that the MCS methodology reproduces the known sample grades with minimum estimation error of 0.001 while the estimation error associated with the kriged estimates is 1.104, a 100% efficiency of the MCS method over the kriging technique.     

Estimation of covariance parameters in kriging via restricted maximum likelihood

In kriging, parametric approaches to covariance (or variogram) estimation require that unknown parameters be inferred from a single realization of the underlying random field. An approach to such an estimation problem is to assume the field to be Gaussian and iteratively minimize a (restricted) negative loglikelihood over the parameter space. In doing so, the associated computational burden can be considerable. Also, it is usually not easy to check whether or not the minimum achieved is global. In this note, we show that in many practical cases, the structure of the covariance (or variogram) function can be exploited so that iterative minimizing algorithms may be advantageously replaced by a procedure that requires the computation of the roots of a simple rational function and the search for the minimum of a function depending on one variable only. As a consequence, our approach allows one to observe in a straightforward fashion the presence of local minima. Furthermore, it is shown that insensitivity of the likelihood function to changes in parameter value can be easily detected. The note concludes with numerical simulations that illustrate some key features of our estimation procedure.     

On the estimation of the generalized covariance function

The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables.     

Interval-valued random functions and the kriging of intervals

Estimation procedures using data that include some values known to lie within certain intervals are usually regarded as problems of constrained optimization. A different approach is used here. Intervals are treated as elements of a positive cone, obeying the arithmetic of interval analysis, and positive interval-valued random functions are discussed. A kriging formalism for interval-valued data is developed. It provides estimates that are themselves intervals. In this context, the condition that kriging weights be positive is seen to arise in a natural way. A numerical example is given, and the extension to universal kriging is sketched.     

The random coin method: Solution of the problem of the simulation of a random function in the plane

This paper presents a new geostatistical method to obtain realizations of stationary random functions in the plane.     

Computationally efficient restricted maximum likelihood estimation of generalized covariance functions

Computational aspects of the estimation of generalized covariance functions by the method of restricted maximum likelihood (REML) are considered in detail. In general, REML estimation is computationally intensive, but significant computational savings are available in important special cases. The approach taken here restricts attention to data whose spatial configuration is a regular lattice, but makes no restrictions on the number of parameters involved in the generalized covariance nor (with the exception of one result) on the nature of the generalized covariance function's dependence on those parameters. Thus, this approach complements the recent work of L. G. Barendregt (1987), who considered computational aspects of REML estimation in the context of arbitrary spatial data configurations, but restricted attention to generalized covariances which are linear functions of only two parameters.     

The combination of sampling and kriging in the regional estimation of coal resources

Two-dimensional systematic sampling of small plots followed by the kriging of those plots may be employed to obtain regional estimates of coal resources and measures of the accuracy of the estimates. The use of sampling makes large savings in computation possible. Two case studies involving the estimation of coal tonnage are discussed.     

基于GPU的并行条件模拟算法及其在储量估算中的应用

条件模拟是一种计算非常耗时的高精度三维插值算法。针对串行条件模拟算法计算时间过长的问题,提出基于GPU的并行条件模拟算法,并进行储量估算。对条件模拟算法进行并行分析,利用GPU的高度并行性,构建CUDA通用计算开发环境,实现串行条件模拟算法到并行条件模拟算法的转换,使条件模拟算法的时间复杂度从O(n)降至O(logn)。并对西藏甲玛铜矿进行了储量估算。实验结果表明,在安装普通NVIDIA显卡的计算机以及估算精度不下降的情况下,GPU并行条件模拟的计算效率比CPU串行条件模拟的计算效率提高了60倍以上。     

On the estimation of scale of fluctuation in geostatistics

Describing how soil properties vary spatially is of particular importance in stochastic analyses of geotechnical problems, because spatial variability has a significant influence on local material and global geotechnical response. In particular, the scale of fluctuation θ is a key parameter in the correlation model used to represent the spatial variability of a site through a random field. It is, therefore, of fundamental importance to accurately estimate θ in order to best model the actual soil heterogeneity. In this paper, two methodologies are investigated to assess their abilities to estimate the vertical and horizontal scales of fluctuation of a particular site using in situ cone penetration test (CPT) data. The first method belongs to the family of more traditional approaches, which are based on best fitting a theoretical correlation model to available CPT data. The second method involves a new strategy which combines information from conditional random fields with the traditional approach. Both methods are applied to a case study involving the estimation of θ at three two-dimensional sections across a site and the results obtained show general agreement between the two methods, suggesting a similar level of accuracy between the new and traditional approaches. However, in order to further assess the relative accuracy of estimates provided by each method, a second numerical analysis is proposed. The results confirm the general consistency observed in the case study calculations, particularly in the vertical direction where a large amount of data are available. Interestingly, for the horizontal direction, where data are typically scarce, some additional improvement in terms of relative error is obtained with the new approach.     

条件模拟原理和技术进展及在水资源系统中的应用

在简要介绍条件模拟基本原理基础上,重点评述了国外20世纪90年代新的数学方法和计算技术进展及在国内外水资源系统应用的主要成果,并对今后学科和技术发展作了展望.     

Using robust kriging and sequential Gaussian simulation to delineate the copper- and lead-contaminated areas of a rapidly industrialized city in Yangtze River Delta,China

A total of 540 topsoil samples (0–15 cm), 188 subsoil samples (20–40 cm), and four individual soil profiles were collected in this study for mapping the Cu- and Pb-contaminated areas in soils of Zhangjiagang city, an industrialized city in the Yangtze River Delta region of China. Robust geostatistical methods were applied for identifying possible spatial outliers of Cu and Pb data, and then a sequential Gaussian simulation was employed for delineating the potential areas where Cu or Pb concentration was affected by diffuse pollution. The results showed that the spatial outliers of Cu and Pb were strongly associated with various types of factories. The anthropogenic input of Cu to soils at local hotspots was closely related to emissions of printing and dyeing, metallurgical, and chemical factories, whereas a lead oxide factory and a chemical factory resulted in a considerable increase of Pb in the topsoil of the study area. Approximately 30% of the total land area of the study was at potential risk from the Cu or Pb diffuse pollution resulting from rapid industrialization of the area over the past 20 years.     

Empirical Orthogonal Functions Analysis Applied to the Inverse Problem in Hydrogeology: Evaluation of Uncertainty and Simulation of New Solutions

To fulfil the need to generate more realistic solutions, stochastic inverse simulations in hydrogeology are now constrained on both piezometric head and hydraulic conductivity data. These inverse techniques, often based on geostatistics, allow modifications of an initial solution conditioned only on hydraulic conductivity data to arrive at a final solution that also matches observed heads. By repeating the process as many times as necessary with different initial solutions, one generates an ensemble of final solutions thereby addressing the uncertainty of the inverse problem. This requires a method able to handle the whole ensemble and to work on its relevant characteristics. From this standpoint, the analysis by Empirical Orthogonal Functions (EOF) appears promising. The method builds an orthogonal decomposition of the covariance matrix, calculated over the whole set of solutions, and the areas in space where the first functions have a greater influence corresponding to locations of maximum uncertainty in the solutions. These locations depend both on the hydraulic characteristics of the flow problem and on the spatial distribution of available data. The EOF analysis is used on a synthetic problem that mimics a possible behavior of the Culebra aquifer of the Waste Isolation Pilot Plant (WIPP, New Mexico). The method also allows new solutions to be generated at lower computational cost by a random composition of the functions obtained by the EOF analysis. These new solutions keep the main characteristics of the initial ensemble and because they can be conditioned, they return very good results when they are used to solve the direct problem.     

Hydraulic Effects of Shales in Fluvial-Deltaic Deposits: Ground-Penetrating Radar, Outcrop Observations, Geostatistics, and Three-Dimensional Flow Modeling for the Ferron Sandstone, Utah

Ground-penetrating radar (GPR) surveys, outcrop measurements, and cores provide a high-resolution 3D geologic model to investigate the hydraulic effects of shales in marine-influenced lower delta-plain distributary channel deposits within the Cretaceous-age Ferron Sandstone at Corbula Gulch in central Utah, USA. Shale statistics are computed from outcrop observations. Although slight anisotropy was observed in mean length and variogram ranges parallel and perpendicular to pale of low , the anisotropy is not statistically significant and the estimated mean length is 5.4 m. Truncated Gaussian simulation was used to create maps of shales that are placed on variably dipping stratigraphic surfaces interpreted from high-resolution 3D GPR surveys, outcrop interpretations, and boreholes. Sandstone permeability is estimated from radar responses calibrated to permeability measurements from core samples. Experimentally designed flow simulations examine the effects of variogram range, shale coverage fraction, and trends in shale coverage on predicted upscaled permeability, breakthrough time, and sweep efficiency. Approximately 1500 flow simulations examine three different geologic models, flow in the 3 coordinate directions, 16 geostatistical parameter combinations, and 10 realizations for each model. ANOVA and response models computed from the flow simulations demonstrate that shales decrease sweep, recovery, and permeability, especially in the vertical direction. The effect on horizontal flow is smaller. Flow predictions for ideal tracer displacements at Corbula Gulch are sensitive to shale-coverage fraction, but are relatively insensitive to twofold variations in variogram range or to vertical trends in shale coverage. Although the hydraulic effects of shale are statistically significant, the changes in flow responses rarely exceed 20%. As a result, it may be reasonable to use simple models when incorporating analogous shales into models of reservoirs or aquifers.     

冲断构造与正反转构造物理模拟实验的研究进展

冲断构造和正反转构造是我国西部叠合盆地中典型的构造样式,其构造物理模拟实验是研究和模拟自然界冲断构造（正反转构造）变形特征和动力学过程的一种有效的实验方法。在查阅大量相关资料的基础上,本文概述了物理模拟实验的发展历史,以及国内外在实验理论、实验技术和从二维到三维转变等方面的研究现状。目前,构造物理模拟实验已被广泛应用于构造地质学、石油构造地质学等众多研究领域,是油气勘探研究由定性描述跨入半定量-定量分析的有效途径之一。本文分别阐述了冲断构造和正反转构造在近年来取得的进展：1）在冲断构造物理模拟实验方面,介绍了双指向冲断构造和推覆体、地表作用对冲断构造的影响以及冲断构造中地层缩短量和应变等3方面的进展;2）在正反转构造物理模拟实验方面主要讲述了基底对反转构造演化型式的制约以及反转临界条件的三维构造模拟取得的进展。同时,认为物理模拟在未来的发展过程中应紧密结合数值模拟、高精度成像技术和数据采集技术等。作为研究构造变形机制的重要媒介,冲断构造物理模拟在塑性变形对冲断构造的影响、如何反映现代构造地质学成果以及模拟过程中如何加入化学物质迁移等问题方面仍存在不足。此外,高端实验室的建设和模拟技术也是构造物理模拟实验面临的一个问题。     

伊通河冬季冰冻浓缩状态下无机污染物分配规律的室内模拟

为研究冬季冰冻浓缩状态下无机污染物分配规律,采集长春市伊通河冬季水样,在分析伊通河不同季节水质基础上,选取F －、TFe、TP、COD 四项指标进行冰冻浓缩状态下室内模拟实验。实验结果表明: 不同冰冻程度情况下,初次冰冻上层冰中的COD、TP、TFe、F － 值均低于河水和冰下层水; 用冰冻3 h 后的冰下层水进行二次冰冻,上层冰中的F －、TFe 值仍低于冰下层水,而COD 值高于冰下层水; 随着pH 值及离子初始浓度的不同,TFe、TP － 在冰上层离子含量有明显变化,说明在冰冻浓缩状态下,可以有效控制结冰层水质。     

卸荷条件下三峡地下厂房大型联合块体稳定性的三维数值模拟分析

块体稳定性分析一般采用只考虑重力条件下的刚体极限平衡方法来进行计算,而没有考虑开挖卸荷后块体的二次应力场对块体的稳定性的影响。文中按照实际施工工序,对三峡地下电站主厂房的两大型顶拱边墙联合块体进行三维数值模拟分析。分析结果表明:块体的顶拱部分在开挖过程中垂直厂房轴线的水平方向始终为受压状态且压应力随开挖过程逐渐增大(其很难沿结构面“爬坡”失稳,因此对结构面的起伏度进行了研究,并统计确定了其在结构面强度参数中的贡献),而垂直应力在顶拱开挖结束时卸荷基本结束,变形相对较小,沿滑面向开挖区滑移;块体的边墙部分在两个方向均表现为卸荷,特别是垂直厂房轴线的水平方向卸荷明显,在其表部甚至出现拉应力区,变形以向开挖区水平变形为主,水平方向变形量为竖向变形量的7～9倍,且总变形量较顶拱部分大得多,因此块体边墙部分在失稳前并非沿块体滑面滑移,而是向开挖区水平卸荷回弹为主。基于数值模拟结果,对块体的可能失稳的变形规律和应力分布特征进行了分析,并且提出了块体稳定度的概念,从而对其稳定性作出相应的评价。     

Autochthonous block fields in southern Norway: Implications for the geometry,thickness, and isostatic loading of the Late Weichselian Scandinavian ice sheet

The consistent geographical and altitudinal distribution of autochthonous block fields (mantle of bedrock weathered in situ) and trimlines in southern Norway suggests a multi-domed and asymmetric Late Weichselian ice sheet. Low-gradient ice-sheet profiles in the southern Baltic region, in the North Sea, and along the outer fjord areas of southern Norway, are best explained by movement of ice on a bed of deforming sediment, although water lubricated sliding or a combination of the two, may not be excluded. The ice-thickness distribution of the Late Weichselian Scandinavian ice sheet is not in correspondence with the modern uplift pattern of Fennoscandia. Early Holocene crustal rebound was apparently determined by an exponential, glacio-isostatic rise. Later, however, crustal movements appear to have been dominated by large-scale tectonic uplift of the Fennoscandian Shield, centred on the Gulf of Bothnia, the region of maximum lithosphere thickness.     

神狐海域天然气水合物成藏系统数值模拟与试采矿体优选

为了进一步了解南海北部陆坡神狐海域勘探区天然气水合物成藏系统特征,优选天然气水合物优势矿体,基于经过矿体的地震剖面资料,结合区域成藏地质条件,分别构建了勘探区W17和W18两个矿体的二维地质模型,从天然气水合物成藏的稳定域、气源形成、运移输导及储集成藏进行了系统数值模拟。结果表明：①神狐勘探区成矿气源丰富,来自浅层的生物成因气和深层的热成因气都可作为天然气水合物成藏的气源;②神狐勘探区流体输导条件良好,深部大断裂可以作为油源断裂沟通深部源岩,是连接深部热成因气的主要通道,浅部调节性断裂和渗透性砂岩一起作为横向+垂向复合输导;③在神狐勘探区稳定域内,区域构造部位相对高、断块封闭性相对好、渗透率相对大的区域为天然气水合物成藏的有利储集层;④综合分析认为,W17矿体比W18矿体在气源、运移及储层特性上更具优势,应作为优先考虑的试采矿体。