基于分形几何理论的DFN模型构建方法研究

以重庆梁-忠（梁平县-忠县）高速公路礼让隧道为工程研究背景,通过测线法调查现场节理,获得了节理产状分布概率密度函数,并从分形几何学的角度分析了节理间距及迹长的分形分布规律,推导出能反映节理间距及迹长分布状态的分形维数D及分形分布概率密度函数。在该基础上采用Matlab软件以及Monte-Carlo随机分析方法,产生节理参数随机数,结合3DEC中最新模块离散裂隙网络(DFN)技术,建立了能反映节理裂隙分布特征的离散裂隙网络模型并验证了模型的有效性,结果表明,分形分布比负指数分布包含更多的间距、迹长分布信息,更接近于实际分布;分形维数D反映了节理间距、迹长在其变化范围内的分布特征,分形维数的大小取决于小间距、小迹长部分数量在总节理数量中的比例,为节理裂隙岩体网络模型构建提供了一种新方法。     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

Uncertainty in fractal dimension estimated from power spectra and variograms

The reliability of using fractal dimension (D) as a quantitative parameter to describe geological variables is dependent mainly on the accuracy of estimated D values from observed data. Two widely used methods for the estimation of fractal dimensions are based on fitting a fractal model to experimental variograms or power-spectra on a log-log plot. The purpose of this paper is to study the uncertainty in the fractal dimension estimated by these two methods. The results indicate that both spectrum and variogram methods result in biased estimates of the D value. Fractal dimension calculated by these two methods for the same data will be different unless the bias is properly corrected. The spectral method results in overestimated D values. The variogram method has a critical fractal dimension, below which overestimation occurs and above which underestimation occurs. On the bases of 36,000 simulated realizations we propose empirical formulae to correct for biases in the spectral and variogram estimated fractal dimension. Pitfalls in estimating fractal dimension from data contaminated by white noise or data having several fractal components have been identified and illustrated by simulated examples.     

Oscillatory Zoning in an Agate from Kazakhstan: Autocorrelation Functions and Fractal Statistics of Trace Element Distributions

The results of numerical treatment of proton-induced X-ray emission (PIXE) data for trace element (Fe, Sr, Zn) contents in agate from the environment surrounding the city of Pavlodar (Kazakhstan) are presented in this paper. In order to mathematically characterize the zoning pattern, fractal geometry, correlation, and Fourier analyses were used for investigation of trace element distributions. Autocorrelation calculations show that oscillations in Fe, Sr, and Zn along the profile have periodic components. It is shown that trace element distributions can be described in terms of fractal geometry. The measured Hurst exponents by methods of the width and the power spectrum are mostly in the range 0.14–0.28, indicating fractal scaling and antipersistent behavior of trace elements along the profile.     

On the practice of estimating fractal dimension

Coastlines epitomize deterministic fractals and fractal (Hausdorff-Besicovitch) dimensions; a divider [compass] method can be used to calculate fractal dimensions for these features. Noise models are used to develop another notion of fractals, a stochastic one. Spectral and variogram methods are used to estimate fractal dimensions for stochastic fractals. When estimating fractal dimension, the objective of the analysis must be consistent with the method chosen for fractal dimension calculation. Spectal and variogram methods yield fractal dimensions which indicate the similarity of the feature under study to noise (e.g., Brownian noise). A divider measurement method yields a fractal dimension which is a measure of complexity of shape.     

分形分维理论在地质灾害发育及空间分布规律中的应用——以长安区滑坡、崩塌地质灾害为例

长安区区内地层岩性、地质构造多样,加之人类经济工程活动、降雨、地震等诱发影响,秦岭北坡及黄土台塬边缘滑坡崩塌地质灾害发育,是陕西省地质灾害较严重的县(区)之一。本文以长安区为研究区域,在对长安区自然地理及区域地质环境条件等资料充分收集的基础上,结合长安区滑坡崩塌灾害点的实际调查资料,详细分析了长安区的滑坡崩塌灾害发育特征及空间分布规律,并采用网络覆盖法,对长安区滑坡崩塌地质灾害的空间分布进行了分形分维计算。求得地质灾害的空间分布分维值为0.876 6,反应出长安区滑坡崩塌地质灾害空间分布特性的复杂性。研究结果表明,分形分维理论能合理评价地质灾害空间分布特征,对类似地区的地质灾害空间分布特征分析具有借鉴意义。     

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.     

Characterization and Quantification of Uncertainty in?Coregionalization Analysis

Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.     

重磁场的变差函数特征及应用

用变差函数研究重磁场的区域变化特征.变差函数的变程反映重磁场的相干范围, 块金效应反映随机干扰, 基台值反映变异程度.重磁场的理论模拟说明: 重力场的相干范围大于磁场, 重磁场变程主要取决于场源深度, 浅源重磁场变差函数近似为球状模型或指数模型, 深源重磁场近似为连续性更好的高斯模型.磁场场源深度近似等于变程的一半, 重力场场源深度近似等于变程的四分之一.湖北大冶铁矿垂直分量磁异常具有几何各向异性, 北西-南东走向, 变差函数推测磁铁矿平均深度为250m.磁异常小波多尺度分解细节和逼近部分磁场具有协调几何各向异性, 变差函数的各阶场源深度估计结果与功率谱估计结果吻合.      

泥石流堆积物粒度分布特征影响因素分析及分形维数预测

泥石流堆积物作为泥石流发育最终的产物,含有大量与泥石流发生过程和发育特征相关的信息,能够反映泥石流灾害程度和活动强度。研究表明,泥石流堆积物颗粒具有明显的自相似性和无标度区间,运用分形理论,计算泥石流堆积物颗粒分布的分维数。分析分维数与主沟长度、泥砂补给段长度比、主沟平均比降、流域最大相对高差和松散物源量的关系,结果表明分维数与各因素之间存在较强的非线性响应关系。以乌东德库区泥石流实测数据为例,以上述的5个因素作为输入单元,建立了泥石流堆积物分维数支持向量机预测模型,并对分维数进行了预测,其预测结果的最大误差为1.25%,说明预测值与实测值吻合度较高。综合表明支持向量机预测模型能够较好地模拟和泛化数据,是一种行之有效的泥石流堆积物分形维数预测方法,可用于不具备筛析条件的泥石流堆积物粒度分布特征的预测与研究,进而可为研究泥石流的形成机理、类型、危险度和堆积物的形成演化特征及物理力学性质提供一个新思路。     

分形在储层参数井间预测中的应用

现有地震资料难以实现储层薄层的预测，分形理论的引入，可弥补这一不足．以具有连续性和等间距采样点的测井资料为基础，以分形分析理论为手段，结合生产实际进行了研究．根据分形地质统计学中对储层参数的研究，得出其分形理论模型，并通过VisualBasic语言编程，在计算机上实现了某地区4口井储层参数（包括孔隙度、渗透率和泥质含量）的井间预测，其结果达到了预期的效果，说明将分形理论用于储层参数井间预测具有可行性．     

一种新的岩石节理面三维粗糙度分形描述方法

研究并提出一种新的岩石节理面三维粗糙度分形描述方法。首先,基于激光扫描数据将节理表面离散成三角网,并建立与剪切方向相关的三维均方根抵抗角的计算方法。其次,运用分形数学理论,提出一种新的基于三维均方根抵抗角的节理面粗糙度分形描述方法。最后,采用新方法对天然玄武岩节理和花岗岩张拉型节理的粗糙特性进行分析。研究结果表明,提出的新方法能够较全面地反映节理面的三维几何形貌信息,并能描述节理粗糙度的各向异性特性。研究成果为进一步建立岩石节理面的三维剪切强度公式和剪切本构理论奠定了基础。     

Characterization and Quantification of Uncertainty in Coregionalization Analysis

Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.     

不同类型地貌的各向异性分形与多重分形特征研究

利用标准偏差法和固定质量法，研究了新疆天山地区3种不同构造地貌单元地形剖线的自仿射分形和多重分形特征。结果表明，在所研究的标度范围内，不同构造地貌单元剖线分维值总体上呈现出高山区大于中低山区大于盆地区特点，同一个地貌研究区不同方向的剖线呈现不同程度的各向异性。多重分形谱Dq的形态和值域范围也呈现出不同特征。研究认为，地貌表面形态并不是完全随机的，而是一种各向异性的分形布朗运动，分维值表征了内外营力作用的方向和强度，渡越长度是自仿射分形研究中的另一个重要参量。利用上述分形特征可定量研究地貌的发展阶段和地貌动力学。     

基于膨润土凝胶分形结构的屈服强度理论

在核废料处置库安全使用的设计年限（数万年至数十万年）内,膨润土遇水侵蚀,导致缓冲/回填层致密性降低、渗透性增加,危及核废料处置库安全。膨润土侵蚀是以膨润土凝胶形式迁移,膨润土凝胶的屈服强度就是侵蚀的临界剪切应力。膨润土凝胶颗粒之间的联结靠颗粒间的长程作用,即van der Waals力,颗粒间的联结作用取决于凝胶的结构。本文基于膨润土凝胶结构的分形模型,假设凝胶的屈服强度等于冲刷面上单位面积的van der Waals力的总和,导出了膨润土凝胶的屈服强度（σ_y）的表达式,表示为凝胶的固体体积率（φ_s）的幂函数,即σ_y=σ_y0φ_sm,幂函数的指数是凝胶结构分维的函数。膨润土凝胶的屈服强度理论得到了试验数据的验证。     

BLP approach for the estimation of variogram parameters

Variograms of hydrologic characteristics are usually obtained by estimating the experimental variogram for distinct lag classes by commonly used estimators and fitting a suitable function to these estimates. However, these estimators may fail the conditionally positive-definite property and the better results for the statistics of cross-validation, which are two essential conditions for choosing a valid variogram model. To satisfy these two conditions, a multi-objective bilevel programming estimator (MOBLP) which is based on the process of cross-validation has been developed for better estimate of variogram parameters. This model is illustrated with some rainfall data from Luan River Basin in China. The case study demonstrated that MOBLP is an effective way to achieve a valid variogram model.     

SD法在西藏甲玛铜多金属矿床资源量估算中的应用

SD(最佳结构曲线断面积分储量计算和储量审定计算)法诞生于20世纪80年代,是在对传统法和样条函数法改进并与距离幂次反比法等方法结合的基础上发展起来的新方法,在储量估算领域内具有一定的先进性。多金属矿多种元素共生的情况使资源量的计算变得复杂。对西藏甲玛铜多金属矿使用SD法和平行断面法进行资源量估算,结果对比证明SD方法能较好地适用于多金属矿的资源量计算。     

Determination of Joint Roughness Coefficients Using Roughness Parameters

This study used precisely digitized standard roughness profiles to determine roughness parameters such as statistical and 2D discontinuity roughness, and fractal dimensions. Our methods were based on the relationship between the joint roughness coefficient (JRC) values and roughness parameters calculated using power law equations. Statistical and 2D roughness parameters, and fractal dimensions correlated well with JRC values, and had correlation coefficients of over 0.96. However, all of these relationships have a 4th profile (JRC 6–8) that deviates by more than ±5 % from the JRC values given in the standard roughness profiles. This indicates that this profile is statistically different than the others. We suggest that fractal dimensions should be measured within the entire range of the divider, instead of merely measuring values within a suitable range. Normalized intercept values also correlated with the JRC values, similarly to the fractal dimension values discussed above. The root mean square first derivative values, roughness profile indexes, 2D roughness parameter, and fractal dimension values decreased as the sampling interval increased. However, the structure function values increased very rapidly with increasing sampling intervals. This indicates that the roughness parameters are not independent of the sampling interval, and that the different relationships between the JRC values and these roughness parameters are dependent on the sampling interval.