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.     

Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances

A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features that could not be generated with traditional kriging.     

Geostatistical Mapping with Continuous Moving Neighborhood

An issue that often arises in such GIS applications as digital elevation modeling (DEM) is how to create a continuous surface using a limited number of point observations. In hydrological applications, such as estimating drainage areas, direction of water flow is easier to detect from a smooth DEM than from a grid created using standard interpolation programs. Another reason for continuous mapping is esthetic; like a picture, a map should be visually appealing, and for some GIS users this is more important than map accuracy. There are many methods for local smoothing. Spline algorithms are usually used to create a continuous map, because they minimize curvature of the surface. Geostatistical models are commonly used approaches to spatial prediction and mapping in many scientific disciplines, but classical kriging models produce noncontinuous surfaces when local neighborhood is used. This motivated us to develop a continuous version of kriging. We propose a modification of kriging that produces continuous prediction and prediction standard error surfaces. The idea is to modify kriging systems so that data outside a specified distance from the prediction location have zero weights. We discuss simple kriging and conditional geostatistical simulation, models that essentially use information about mean value or trend surface. We also discuss how to modify ordinary and universal kriging models to produce continuous predictions, and limitations using the proposed models.     

Geological Controls of Variograms in a Complex Carbonate Reservoir, Eastern Province, Saudi Arabia

To derive meaningful results from a geostatistical study, it is extremely important to establish the relationship between geology and variograms. However, it is not always easy to establish such a relationship, mainly because of the inadequacy of quantitative or qualitative information. Large amounts of reliable well-log porosity data and the detailed geological information from a complex carbonate reservoir located in the Eastern Province of Saudi Arabia provided an ideal case study of the relationships between geology and variograms. The reservoir under consideration is in the form of a gently dipping, elongated anticline and consists of three productive zones which have been subdivided into several lithologically distinct layers. Results, which indicate an excellent match between geology and porosity variograms, are summarized as follows: (1) use of stratigraphic distances resulted in considerable improvement in the behavior of variograms, (2) structurally controlled geometrical anisotropy is the most obvious feature of variograms, and (3) layers consisting of more complex lithologic framework provided variograms with relatively greater nugget variances and shorter ranges. These results provide further insight into geology and form a meaningful basis for estimation and simulation of the reservoir. They also could be used as a general guide in the geostatistical study of similar carbonate reservoirs.     

黄河济南以下河段某些异常地质作用及几种典型沉积构造研究

黄河自 1972年发生首次断流以来,以后 2 6年间,发生断流年数就有 19年,近年则愈演愈裂。由此引发了一系列特殊的地质作用,我们称之为异常地质作用,如汛期高含沙河水骤涨骤落,致使下游河床淤积严重,且不断抬高、展宽,并向二级悬河发育;床底不稳、河道迁移频繁,心滩、边滩上水道发育;而断流期则风的作用明显,风蚀地貌发育。水风地质作用混合,相克相伴,构成了黄河三角洲极为复杂的地质景观。又因黄河以沉积细粒物质为主,在多种因素的综合控制下,其沉积物层面或层内发育的沉积构造具有独特性,如变形层理、泥沙钟乳、液化 (泄水 )构造、气胀构造、植物划痕构造、特殊波痕及特殊流痕等,是典型的沉积构造。正确识别和分析黄河断流后的异常地质作用及由其形成的沉积构造对古河流相分析有重要参考意义.     

Robustness of Kriging weights to non-bias conditions

In this paper, the effect on Kriging weights of non-bias conditions, when the same residual covariance model is used, has been studied by the l₂ norm of the weights difference between Ordinary Kriging and Kriging with a trend model. Four covariance models, in 1-D and 2-D, and in interpolation and extrapolation conditions are examined. Situations in which both algorithms yield the same results are pointed out.     

Correcting the Smoothing Effect of Ordinary Kriging Estimates

The smoothing effect of ordinary kriging is a well-known dangerous effect associated with this estimation technique. Consequently kriging estimates do not reproduce both histogram and semivariogram model of sample data. A four-step procedure for correcting the smoothing effect of ordinary kriging estimates is shown to be efficient for the reproduction of histogram and semivariogram without loss of local accuracy. Furthermore, this procedure provides a unique map sharing both local and global accuracies. Ordinary kriging with a proper correction for smoothing effect can be revitalized as a reliable estimation method that allows a better use of the available information.     

Evaluation of Interpolation Accuracy of Neural Kriging with Application to Temperature-Distribution Analysis

An interpolation method based on a multilayer neural network (MNN), has been examined and tested for the data of irregular sample locations. The main advantage of MNN is in that it can deal with geoscience data with nonlinear behavior and extract characteristics from complex and noisy images. The training of MNN is used to modify connection weights between nodes located in different layers by a simulated annealing algorithm (one of the optimization algorithms of the network). In this process, three types of errors are considered: differences in values, semivariograms, and gradients between sample data and outputs from the trained network. The training is continued until the summation of these errors converges to an acceptably small value. Because the MNN trained by this learning criterion can estimate a value at an arbitrary location, this method is a form of kriging and termed Neural Kriging (NK). In order to evaluate the effectiveness of NK, a problem on restoration ability of a defined reference surface from randomly chosen discrete data was prepared. Two types of surfaces, whose semivariograms are expressed by isotropic spherical and geometric anisotropic gaussian models, were examined in this problem. Though the interpolation accuracy depended on the arrangement pattern of the sample locations for the same number of data, the interpolation errors of NK were shown to be smaller than both those of ordinary MNN and ordinal kriging. NK can also produce a contour map in consideration of gradient constraints. Furthermore, NK was applied to distribution analysis of subsurface temperatures using geothermal investigation loggings of the Hohi area in southwest Japan. In spite of the restricted quantity of sample data, the interpolation results revealed high temperature zones and convection patterns of hydrothermal fluids. NK is regarded as an interpolation method with high accuracy that can be used for regionalized variables with any structure of spatial correlation.     

Empirical Maximum Likelihood Kriging: The General Case

Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution.     

Kriging法在区域滑坡危险性评价中的应用

区域滑坡危险性评价是进行区域滑坡风险性研究的基础.由于滑坡演变机制的复杂性,使得目前基于独立分析各因素对滑坡影响的“白箱”型评价模式具有一定风险性,同时这类评价方法要求对滑坡演变和研究区地质地理背景进行非常细致的监测和调查.为了克服这些问题,文章提出了一种基于Kriging插值理论的“黑箱”型评价方法.在利用该方法对历史滑坡点的规模进行评价的基础上,利用Kriging插值法获取研究区的滑坡危险性区划,并以四川省苍溪县为例,验证了运用该方法进行区域滑坡危险性评价的可行性.     

F-K反偏移中的插值映射

均匀介质中的F-K反偏移是实现一般介质下F-K反偏移的基础,而均匀介质中的F-K反偏移的实质是一个插值映射过程。根据插值对象为复函数的特点,采用了实部虚部分别插值、去振荡因子法、模和相位分别插值、直接频率域插值4种插值方法对点脉冲模型进行了试算。结果表明:截断的直接频率域插值在适应性和精度综合衡量下为最佳;去振荡因子法精度虽高,但可应用的模型太具理想,不适合一般模型中的应用;实部虚部分别插值与模和相位分别插值两者精度较低。     

离散数据地球化学编图中克里格法的应用-以津巴布韦超低密度地球化学图为例

自由网离散数据的地球化学编图以任意方式进行网格化成图存在明显不确定性,会导致地球化学特征的改变或歪曲。克里格法提供了一种更科学合理的空间估计插值方法。笔者运用该方法对津巴布韦超低密度地球化学调查数据进行了空间分析并编制了地球化学图件,研究和讨论了自由网离散数据的地球化学编图方法,评价了采用克里格法编制地球化学图的效果。     

An Experimental Comparison of Ordinary and Universal Kriging and Inverse Distance Weighting

A factorial, computational experiment was conducted to compare the spatial interpolation accuracy of ordinary and universal kriging and two types of inverse squared-distance weighting. The experiment considered, in addition to these four interpolation methods, the effects of four data and sampling characteristics: surface type, sampling pattern, noise level, and strength of small-scale spatial correlation. Interpolation accuracy was measured by the natural logarithm of the mean squared interpolation error. Main effects of all five factors, all two-factor interactions, and several three-factor interactions were highly statistically significant. Among numerous findings, the most striking was that the two kriging methods were substantially superior to the inverse distance weighting methods over all levels of surface type, sampling pattern, noise, and correlation.     

Kriging插值方法在地层模型生成中的应用

为了建立三维数字地层,采用了一种适合城市工程地质和岩土工程特点的地层数据模型-基于钻孔信息的3棱柱模型。由于钻孔之间的距离稀疏程度、方向、数据值存在差异,钻孔以外未知的地质特性需要插值和推断,传统的数理统计方法无法很好地解决空间样本点的选取、空间估值和2组以上空间数据的关系等问题。借鉴地质统计学的Kriging方法给出一种距离加权插值算法,即先根据空间数据得到统计特征,再根据统计特征进行插值。通过对地层模型插值结果的观察,得出该算法可以获得良好的插值效果。     

多重分形与Kriging插值在地层模型生成中的应用

为了建立三维数字地层,采用钻孔信息构建三棱柱,以适合工程地质和岩土工程特点。钻孔数据具有局部的、随机的、异常的特征,另外,钻孔数据具有一定的结构特点。传统的数理统计方法无法很好地解决空间样本点的选取、空间估值和2组以上空间数据的关系等问题,借鉴地质统计学的Kriging方法给出全局区域内一种距离加权插值算法,先根据空间数据得到统计特征,再根据统计特征进行插值,并引入多重分形理论弥补Kriging方法的滑动加权平均缺陷,度量区域化变量的局部奇异性。通过对地层模型插值结果的观察,得出该算法可以获得良好的全局和局部特征插值效果     

Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping

Fractional Brownian surfaces have been widely discussed as an appropriate model for the statistical behavior of topographic surfaces. The fractals model proposes that topographic surfaces are statistically self-similar, and that a single parameter, the fractal dimension, applies at all scales. This paper presents the results of empirical examinations of 17 topographic samples. Only one of these samples shows the statistical behavior predicted by the fractals model; however, in 15 of the 17 samples, the surfaces' variograms could be adequately described by ranges of scales having constant fractal dimension, separated by distinct scale breaks. For scale ranges between adjacent breaks, surface behavior should be that predicted by the fractals model; the breaks represent characteristic horizontal scales, at which surface behavior changes substantially. These scale breaks are especially important for cartographic representations and digital elevation models, since they represent scales at which there is a distinct change in the relation between sampling interval and the associated error.     

近南北向构造在塔里木盆地的踪迹

为厘清近南北向构造在塔里木盆地中的性质、地位和作用,笔者通过地质—地球物理分析,确认了以反转断层及相关褶皱为主导的挤压型构造是塔里木近南北向构造的主要表现形式。它发生发展于大陆内部,具有分布的广泛性和等距性、延伸的线性和断续性、纵深的陡倾性和穿透性、主断面的逆冲等特征。它的发育通常是基于先存的NS(—NNE)向张裂带和NNW(—NW)或NNE(—NE)向剪裂带,前者多演化为大、中型南北向构造,而后者多演化为中、小型南北向构造。近南北向构造的反转断层呈现为伸展—走滑—挤缩的演化顺序;最终定型于新生代;其驱动力主要源于太平洋—菲律宾板块向西的推挤动力。近南北向反转构造是种复合型的封闭性构造,对于石油和天然气地质研究,具有实际意义。     

东北三省月降水量的时空克里金插值研究

为了实现对时空场任意点进行时空插值,以东北三省1999~2008年的月降水数据为研究对象,选用了一类比较实用的积和式时空变异函数模型进行时空克里金插值。每个站点的降水量都是时间序列,插值之前对各站点的降水进行了时序分解和去季节项处理。在分别得到纯空间和纯时间的变异函数基础上建立时空变异函数。将二维的普通克里金插值扩展为三维,并对1999年10月各站点进行降水量估计,同时与单纯空间克里金插值效果进行比较。结果表明时空插值效果理想,因为同时考虑了空间、时间的相关性,插值精度较空间克里金更高。     

高精度人工地震波层析成像技术勘探地下构造

应用人工爆破地震波层析成像方法勘探地下速度结构。在海域、山地多种地质条件下进行了地震波层析成像方法定量勘探研究。依勘探目标体而异,通过设计震源频率、震源和观测仪器布局,提高观测分辨率,实现定量高精度速度结构勘探研究。勘探最大深度可达Moho界面;山地最高分辨率可达3 m。结果显示了该方法在近海区域和山区不同地质条件的适用性及不同勘探深度与勘探精度的实用性。在日本九州近海地域勘测中,同时使用了反射与折射2种走时资料,可以同时改善水平及垂向的分辨率。成像图显示了地下结构详细特征,如任意斜向的凹凸形状,或高角度逆冲错断构造,分辨率达到10 m。同时显示出这些复杂几何体的埋藏深度。结果表明,诸如地下断层形状,资源矿藏储舱结构异常体的复杂几何形状,都可期望在层析成像技术的勘探分辨能力之内。它对于地下结构研究,矿产资源勘探,工程环境地质研究都有重要意义。结果也显示了成像方法的运算收敛速度和计算稳定性在结构勘探中是适用的。