Multifractal Distribution of Eigenvalues and Eigenvectors from 2D Multiplicative Cascade Multifractal Fields

Two-dimensional fields (maps) generated by isotropic and anisotropic multiplicative cascade multifractal processes are common in many fields including oceans, atmosphere, the climate and solid earth geophysics. Modeling the anisotropic scaling property and heterogeneity of these types of fields are essential for understanding the underlying processes. The paper explicitly derives the eigenvalues and eigenvectors from these types of fields and proves that the eigenvalues and eigenvectors are described by non-conservative multifractal distributions. This results in a new multifractal model implemented in eigen domain to characterize 2D fields not only with respect to overall heterogeneity and singularity as characterized by the ordinary multifractal model applied to the field itself, but also with respect to orientational heterogeneity of the field. It may also result in a new way to characterize the distribution of extreme large eigenvalues involved in other studies such as principal component analysis. A newly defined operator and its properties as derived in this paper may be useful for studying other types of multifractal cascade processes.     

基于栅格数据的局部奇异性分析迭代方法及其软件实现

近些年,弱缓化探异常识别已成为成矿预测和勘查评价中十分关键的问题。多重分形理论的局部奇异性分析因其崭新的思路、简便的方法、优良的效果而在弱缓异常识别中受到广泛关注。在深入剖析局部奇异性分析基本思想及计算方法的基础上,引入正规化尺度参数L,提出了迭代方法来改进局部奇异性指数的估值。给出了奇异性迭代算法并用C＋＋编程实现,软件功能强劲,操作灵活,不仅适用于各向同性情形,还适用于各向异性尺度的奇异性计算和方向性奇异性计算。软件的动态链接库版本已在Geo DAS矿产资源定量预测专业软件中应用。     

Anisotropic Scaling Models of Rock Density and the Earth’s Surface Gravity Field

In this paper we consider an anisotropic scaling approach to understanding rock density and surface gravity which naturally accounts for wide range variability and anomalies at all scales. This approach is empirically justified by the growing body of evidence that geophysical fields including topography and density are scaling over wide range ranges. Theoretically it is justified, since scale invariance is a (geo)dynamical symmetry principle which is expected to hold in the absence of symmetry breaking mechanisms. Unfortunately, to date most scaling approaches have been self-similar, i.e., they have assumed not only scale invariant but also isotropic dynamics. In contrast, most nonscaling approaches recognize the anisotropy (e.g., the strata), but implicitly assume that the latter is independent of scale. In this paper, we argue that the dynamics are scaling but highly anisotropic, i.e., with scale dependent differential anisotropy. By using empirical density statistics in the crust and a statistical theory of high Prandtl number convection in the mantle, we argue that is a reasonable model for the 3-D spectrum (K is the horizontal wavevector and K is its modulus, k _z is a vertical wavenumber), (s,H _z ) are fundamental exponents which we estimate as (5.3,3), (3,3) in the crust and mantle, respectively. We theoretically derive expressions for the corresponding surface gravity spectrum. For scales smaller than ≈100 km, the anisotropic crust model of the density (with flat top and bottom) using empirically determined vertical and horizontal density spectra is sufficient to explain the (Bouguer) g _z spectra. However, the crust thickness is highly variable and the crust-mantle density contrast is very large. By considering isostatic equilibrium, and using global gravity and topography data, we show that this thickness variability is the dominant contribution to the surface g _z spectrum over the range ≈100–1000 km. Using estimates of mantle properties (viscosity, thermal conductivity, thermal expansion coefficient, etc.), we show that the mantle contribution to the mean spectrum is strongest at ≈1000 km and is comparable to the variable crust thickness contribution. Overall, we produce a model which is compatible with both the observed (horizontal and vertical) density heterogeneity and surface gravity anomaly statistics over a range of meters to several thousand kilometers.     

Facies Modeling Using a Markov Mesh Model Specification

The spatial continuity of facies is one of the key factors controlling flow in reservoir models. Traditional pixel-based methods such as truncated Gaussian random fields and indicator simulation are based on only two-point statistics, which is insufficient to capture complex facies structures. Current methods for multi-point statistics either lack a consistent statistical model specification or are too computer intensive to be applicable. We propose a Markov mesh model based on generalized linear models for geological facies modeling. The approach defines a consistent statistical model that is facilitated by efficient estimation of model parameters and generation of realizations. Our presentation includes a formulation of the general framework, model specifications in two and three dimensions, and details on how the parameters can be estimated from a training image. We illustrate the method using multiple training images, including binary and trinary images and simulations in two and three dimensions. We also do a thorough comparison to the snesim approach. We find that the current model formulation is applicable for multiple training images and compares favorably to the snesim approach in our test examples. The method is highly memory efficient.     

A New Model for Quantifying Anisotropic Scale Invariance and for Decomposition of Mixing Patterns

A new power–law function has been derived to represent the relationship between area of the set consisting of wave numbers with spectral energy density above S (A(>S)) on the two-dimensional frequency plane and S. The power–law relation holds if the field concerned possessing isotropic scale invariance or generalized scaling invariance involves rotational and ratio-scale changing transforms. The equation is valid for dealing with common exploration geophysical and geochemical fields encountered in mineral exploration and environmental assessment. This power–law function not only provides a new model for characterizing anisotropic scaling invariance for generalized scaling field, for example, estimating the power exponent of power spectrum of generalized scale invariance measure in frequency domain, but also forms a theoretical base for the S–A filtering technique developed for decomposing a mixing field into components on the basis of distinct scaling properties in the frequency domain. It is demonstrated that the method has potential to become a general technique for image processing and pattern recognition.     

Mortar coupling and upscaling of pore-scale models

Pore-scale models are becoming increasingly useful as predictive tools for modeling flow and transport in porous media. These models can accurately represent the 3D pore-structure of real media. Currently first-principles modeling methods are being employed for obtaining qualitative and quantitative behavior. Generally, artificial, simple boundary conditions are imposed on a model that is used as a stand-alone tool for extracting macroscopic parameters. However, realistic boundary conditions, reflecting flow and transport in surrounding media, may be necessary for behavior that occurs over larger length scales or including pore-scale models in a multiscale setting. Here, pore-scale network models are coupled to adjacent media (additional pore-scale or continuum-scale models) using mortars. Mortars are 2D finite-element spaces employed to couple independent subdomains by enforcing continuity of pressure and flux at shared boundary interfaces. While mortars have been used in the past to couple subdomains of different models, physics, and meshes, they are extended here for the first time to pore-scale models. The approach is demonstrated by modeling single-phase flow in coupled pore-scale models, but the methodology can be utilized to model dynamic processes and perform multiscale modeling in 3D continuum simulators for flow and transport.     

Numerical Analysis of Fluvial Landscapes

The Smith and Bretherton model for fluvial landsurfaces consists of a pair of partial differential equations: one governing water flow and one governing sediment flow. Numerical solutions of these equations have been shown to provide realistic models of the evolution of fluvial landscapes. Further analysis of these equations shows that they possess scaling laws (Hack’s Law) that are known to exist in nature. The preservation of these scaling laws in simulations is highly dependent on the numerical method used. Two numerical methods, both optimized for overland flow, have been used to simulate these surfaces. The implicit method exhibits the correct scaling laws, but the explicit method fails to do so. These equations, and the resulting models, help to bridge the gap between the deterministic and the stochastic theories of landscape evolution. Slight modifications have been made to this model to make the resulting surfaces more realistic. The most successful of these was the addition of an abrasion term to assist in the channelization of rivers.     

奇异性理论在个旧锡铜矿产资源预测中的应用: 成矿弱信息提取和复合信息分解

本研究的目的是应用非线性理论和高新信息处理技术获取矿产资源预测综合信息, 开展以有色金属和贵金属矿产资源潜力评价和预测靶区圈定, 提交个旧及周边地区矿产资源潜力分布图.围绕该研究任务, 重点开展了如何应用奇异性理论和方法, 对比个旧东西矿区的异同.由于区域构造和岩体分布等空间变化性, 导致东西区成矿背景存在较大差异, 受出露地表或近地表矿体分布和矿山开采的影响, 东西区的成矿异常强度和大小都存在较大差异, 东区总体呈高背景而西区为低背景, 因而, 对东西区的成矿信息对比研究和异常圈定相对困难.采用局部奇异性分析方法从地球化学分形密度的角度圈定了局部异常, 在东西区均较好地反映了致矿地球化学异常的分布, 同时采用广义自相似分析方法分解了综合地球化学异常和背景.结果表明, 东西区地球化学背景差异悬殊, 而局部异常具有显著的自相似性.据此在东西区同时圈定的局部异常具有内在的相似性和表现形式上的多样性, 以此为依据所圈定的靶区均具有找矿意义.      

地理空间数据的尺度转换

尺度一般是指空间范围的大小,地理空间数据的尺度转换是尺度研究的重要问题之一。针对地理信息系统(GIS)技术支持下的地理空间数据尺度转换问题,首先回顾了尺度转换的理论基础,即等级理论、分形理论、区域化随机变量理论、地理学第一定律等理论的基本内涵;然后总结了地理学不同研究领域内主要的尺度转换方法,重点分析了重采样法、变异函数法、分形分维法及小波分析法的基本原理、模型方法与典型应用案例;最后介绍了地理空间数据尺度转换效应研究的进展。基于上述总结和分析认为:构建一套无级变换的、系统的尺度转换方法,整合不同学科领域的数据与过程模型、形成数据模型同化的技术体系,这是地理空间数据尺度转换研究的重要课题。     

Multiscale modeling of ice deformation behavior

Understanding the flow of ice in glaciers and polar ice sheets is of increasing relevance in a time of potentially significant climate change. The flow of ice has hitherto received relatively little attention from the structural geological community. This paper aims to provide an overview of methods and results of ice deformation modeling from the single crystal to the polycrystal scale, and beyond to the scale of polar ice sheets. All through these scales, various models have been developed to understand, describe and predict the processes that operate during deformation of ice, with the aim to correctly represent ice rheology and self-induced anisotropy. Most of the modeling tools presented in this paper originate from the material science community, and are currently used and further developed for other materials and environments. We will show that this community has deeply integrated ice as a very useful “model” material to develop and validate approaches in conditions of a highly anisotropic behavior. This review, by no means exhaustive, aims at providing an overview of methods at different scales and levels of complexity.     

二维混合场的各向异性尺度不变性量化方法及其应用

由不同尺度过程或作用叠加而形成的混合场在地学领域很常见,研究如何量化这些场的尺度不变性以及如何刻画其各项异性特征具有重要意义.介绍了近期研发的图像各向异性尺度不变性模拟和分解方法,并将其应用于遥感图像处理中.该方法是将各向异性尺度不变性模拟(SIG)和分形滤波(S-A)方法融合的,对于任意二维场,先用S-A模型判断其是否为混合场.如果是混合场,用S-A模型对其进行模式分解,然后再运用SIG模型量化分解后的各组分的各向异性尺度不变性,并描述其具体变换特征.基于一幅混合遥感影像的应用实例表明,该方法能够有效地量化二维混合场的各向异性尺度不变性.此外,混合场只有在正确分解成不同尺度的组分之后才能得到合理的利用.      

Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method

The space domain version of the turning bands method can simulate multidimensional stochastic processes (random fields) having particular forms of covariance functions. To alleviate this limitation a spectral representation of the turning bands method in the two-dimensional case has shown that the spectral approach allows simulation of isotropic two-dimensional processes having any covariance or spectral density function. The present paper extends the spectral turning bands method (STBM) even further for simulation of much more general classes of multidimensional stochastic processes. Particular extensions include: (i) simulation of three-dimensional processes using STBM, (ii) simulation of anisotropic two- or three-dimensional stochastic processes, (iii) simulation of multivariate stochastic processes, and (iv) simulation of spatial averaged (integrated) processes. The turning bands method transforms the multidimensional simulation problem into a sum of a series of one-dimensional simulations. Explicit and simple expressions relating the cross-spectral density functions of the one-dimensional processes to the cross-spectral density function of the multidimensional process are derived. Using such expressions the one-dimensional processes can be simulated using a simple one-dimensional spectral method. Examples illustrating that the spectral turning bands method preserves the theoretical statistics are presented. The spectral turning bands method is inexpensive in terms of computer time compared to other multidimensional simulation methods. In fact, the cost of the turning bands method grows as the square root or the cubic root of the number of points simulated in the discretized random field, in the two- or three-dimensional case, respectively, whereas the cost of other multidimensional methods grows linearly with the number of simulated points. The spectral turning bands method currently is being used in hydrologic applications. This method is also applicable to other fields where multidimensional simulations are needed, e.g., mining, oil reservoir modeling, geophysics, remote sensing, etc.     

A finite volume scheme with improved well modeling in subsurface flow simulation

We present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion, and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest modification of the nonlinear method takes into account the nonlinear (e.g., logarithmic) singularity of the pressure in the near-well region and introduces a correction to improve accuracy of the pressure and the flux calculation. In this paper, we consider a linear version of the nonlinear method waiving its monotonicity for sake of better accuracy. The new method is generalized for anisotropic media, polyhedral grids and nontrivial cases such as slanted, partially perforated wells or wells shifted from the cell center. Numerical experiments show noticeable reduction of numerical errors compared to the original monotone nonlinear FV scheme with the conventional Peaceman well model or with the given analytical well rate.     

Application of Stochastic Fracture Network with Numerical Fluid Flow Simulations to Groundwater Flow Modeling in Fractured Rocks

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A comparative study of ANN and Neuro-fuzzy for the prediction of dynamic constant of rockmass

Physico-mechanical properties of rocks have great significance in all operational parts in mining activities, from exploration to final dispatch of material. Compressional wave velocity (p-wave velocity) and anisotropic behaviour of rocks are two such properties which help to understand the rock response under varying stress conditions. They also influence the breakage mechanism of rock. There are different methods to determine thep-wave velocity and anisotropyin situ and in the laboratory. These methods are cumbersome and time consuming. Fuzzy set theory, Fuzzy logic and Neural Networks techniques seem very well suited for typical geotechnical problems. In conjunction with statistics and conventional mathematical methods, hybrid methods can be developed that may prove to be a step forward in modeling geotechnical problems. Here, we have developed and compared two different models, Neuro-fuzzy systems (combination of fuzzy and artificial neural network systems) and Artificial neural network systems, for the prediction of compressional wave velocity.     

Stochastic simulation of patterns using Bayesian pattern modeling

In this paper, a Bayesian framework is introduced for pattern modeling and multiple point statistics simulation. The method presented here is a generalized clustering-based method where the patterns can live on a hyper-plane of very low dimensionality in each cluster. The provided generalizationallows a remarkable increase in variability of the model and a significant reduction in the number of necessary clusters for pattern modeling which leads to more computational efficiency compared with clustering-based methods. The Bayesian model employed here is a nonlinear model which is composed of a mixture of linear models. Therefore, the model is stronger than linear models for data modeling and computationally more effective than nonlinear models. Furthermore, the model allows us to extract features from incomplete patterns and to compare patterns in feature space instead of spatial domain. Due to the lower dimensionality of feature space, comparison in feature space results in more computational efficiency as well. Despite most of the previously employed methods, the feature extraction filters employed here are customized for each training image (TI). This causes the features to be more informative and useful. Using a fully Bayesian model, the method does not require extensive parameter setting and tunes its parameters itself in a principled manner. Extensive experiments on different TIs (either continuous or categorical) show that the proposed method is capable of better reproduction of complex geostatistical patterns compared with other clustering-based methods using a very limited number of clusters.     

The effect of particle shape and grain‐scale properties of shale: A micromechanics approach

Traditional approaches for modeling the anisotropic elasticity response of the highly heterogeneous clay fabric in shale have mainly resorted to geometric factors such as definitions of particles shapes and orientations. However, predictive models based on these approaches have been mostly validated using macroscopic elasticity data. The recent implementation of instrumented indentation aimed at probing nano‐scale mechanical behaviors has provided a new context for characterizing and modeling the anisotropy of the porous clay in shale. Nanoindentation experimental data revealed the significant contribution of the intrinsic anisotropy of the solid clay to the measured elastic response. In this investigation, we evaluate both the effects of geometric factors and of the intrinsic anisotropic elasticity of the solid clay phase on the observed anisotropy of shale at multiple length scales through the development of a comprehensive theoretical micromechanics approach. It was found that among various combinations of these sources of anisotropy, the elastic response of the clay fabric represented as a granular ensemble of aligned effective clay particles with spherical morphology and anisotropic elasticity compares satisfactorily to nanoindentation and ultrasonic pulse velocity measurements at nano‐ and macroscopic length scales, respectively. Other combinations of sources of anisotropy could yield comparable predictions, particularly at macroscopic scales, at the expense of requiring additional experimental data to characterize the morphology and orientations of particles. Copyright © 2009 John Wiley & Sons, Ltd.     

空间模式的广义自相似性分析与矿产资源评价

尺度不变性(scale invariance)包括自相似性(各向同性)、自仿射性(成层结构)、广义自相似性(各向异性标度不变性),是由各种地质过程和地质事件所产生的地质特征和模式的本质属性．尺度不变性可用分形和多重分形模型来表征．这些尺度特征的定量化可为刻画地质空问模式和模式识别提供有力的工具．例如。热液矿床的群聚现象可以用局部分形特征(局部奇异性)来刻画．通过在特征空问中(如频率空问)识别空问模式的广义自相似性．可以将空间混合模式进行分解或异常的识别．介绍了几种相关的分形模型和方法。包括度量空问模式广义尺度独立性(GSI)的线性模型;基于广义尺度独立性的异常分解S—A方法;度量空问模式的局部奇异性方法;以及如何利用分形特征预测未发现矿床的2种方法．有些方法已应用于许多矿产资源评价实例中．给出了对加拿大Nova Scotia省西南部湖泊沉积物样品中的4种元素As、Pb、Zn和Cu的地球化学数据处理分析结果。证明了局部奇异性分析和S—A异常分解方法对地球化学异常的增强和分离的有效性．研究表明：由S—A方法分解的异常往往具有多重分形的特点,而且普遍具有局部奇异性．研究区内具有明显奇异性的地区(元素含量富集区)是金矿异常区域。它们与金矿成矿作用和已知矿床的赋存密切相关．     

2.5维各向异性介质中地震波场数值模拟

这里实现了在2.5维各向异性介质中地震波传播的数值模拟。首先从2.5维一阶速度-应力弹性各向异性波动方程出发,得出了该方程的拟谱法数值解法;然后通过对Marmousi速度模型进行数值模拟,表明了拟谱法的有效性;最后计算并分析了均匀和混合各向异性介质中波的传播快照和理论模拟的地震记录,进一步认识了波在各向异性介质中的传播规律。     

Numerical and intelligent modeling of triaxial strength of anisotropic jointed rock specimens

The strength of anisotropic rock masses can be evaluated through either theoretical or experimental methods. The latter is more precise but also more expensive and time-consuming especially due to difficulties of preparing high-quality samples. Numerical methods, such as finite element method (FEM), finite difference method (FDM), distinct element method (DEM), etc. have been regarded as precise and low-cost theoretical approaches in different fields of rock engineering. On the other hand, applicability of intelligent approaches such as fuzzy systems, neural networks and decision trees in rock mechanics problems has been recognized through numerous published papers. In current study, it is aimed to theoretically evaluate the strength of anisotropic rocks with through-going discontinuity using numerical and intelligent methods. In order to do this, first, strength data of such rocks are collected from the literature. Then FlAC, a commercially well-known software for FDM analysis, is applied to simulate the situation of triaxial test on anisotropic jointed specimens. Reliability of this simulation in predicting the strength of jointed specimens has been verified by previous researches. Therefore, the few gaps of the experimental data are filled by numerical simulation to prevent unexpected learning errors. Furthermore, a sensitivity analysis is carried out based on the numerical process applied herein. Finally, two intelligent methods namely feed forward neural network and a newly developed fuzzy modeling approach are utilized to predict the strength of above-mentioned specimens. Comparison of the results with experimental data demonstrates that the intelligent models result in desirable prediction accuracy.