Discrete multifractals

The concept of multifractal modeling has been used intensively in various fields of science for characterizing measures with self- similarity. It has been shown that multifractal modeling provides powerful tools for characterizing patterns in the spatial distribution of geological quantities and objects. Existing multifractal models were proposed for the purpose of handling spatially intertwined fractals with continuous fractal spectrum f(α) (or continuous codimension function C(γ)). In this paper, these conventional multifractals are termed “continuous multifractals” whereas multifractals with discrete fractal dimensions are termed “discrete multifractals.” The properties of discrete multifractals are investigated. It is shown by various artificial examples and a case study of stratigraphy of Ocean Drilling Program (ODP) site 645 that spatially intertwined fractals/multifractals indeed can have discrete fractal dimensions. Histogram-and moment-based techniques are proposed for discrete multifractal modeling and illustrated using the artificial examples. The new concept of discrete multifractals and associated multifractal modeling yields not only techniques for characterizing multifractals with discrete fractal dimensions but it also provides insight into the relationships between fractals, bifractals, and multifractals.     

Markov Processes and Discrete Multifractals

Fractals and multifractals are a natural consequence of self-similarity resulting from scale-independent processes. Multifractals are spatially intertwined fractals which can be further grouped into two classes according to the characteristics of their fractal dimension spectra: continuous and discrete multifractals. The concept of multifractals emphasizes spatial associations between fractals and fractal spectra. Distinguishing discrete multifractals from continuous multifractals makes it possible to describe discrete physical processes from a multifractal point of view. It is shown that multiplicative cascade processes can generate continuous multifractals and that Markov processes result in discrete multifractals. The latter result provides not only theoretical evidence for existence of discrete multifractals but also a fundamental model illustrating the general properties of discrete multifractals. Classical prefractal examples are used to show how asymmetrical Markov process can be applied to generate prefractal sets and discrete multifractals. The discrete multifractal model based on Markov processes was applied to a dataset of gold deposits in the Great Basin, Nevada, USA. The gold deposits were regarded as discrete multifractals consisting of three spatially interrelated point sets (small, medium, and large deposits) yielding fractal dimensions of 0.541 for the small deposits (<25 tons Au), 0.296 for the medium deposits (25--500 tons Au), and 0.09 for the large deposits (>500 tons Au), respectively.     

Multifractal Modeling and Lacunarity Analysis

The so-called gliding box method of lacunarity analysis has been investigated for implementing multifractal modeling in comparison with the ordinary box-counting method. Newly derived results show that the lacunarity index is associated with the dimension (codimension) of fractal, multifractal and some types of nonfractals in power-law relations involving box size; the exponent of the lacunarity function corresponds to the fractal codimension (E – D) for fractals and nonfractals, and to the correlation codimension (E – lpar;2)) for multifractals. These results are illustrated with two case studies: De Wijs's zinc concentration values from the Pulacayo sphalerite-quartz vein in Bolivia and Cochran's tree seedlings example. Both yield low lacunarities and slightly depart from translational invariance.     

n维自仿射分形及其在地球化学中的应用

分形概念应用在地球科学中来刻画地质量和物体的自相似特征。研究表明分形模型常常提供有力工具来刻画地质量和物体的基本空间分布结构。本文提出了n维自仿射分形的检验与定量评定方法。通过实例，说明n维自仿射分形的方法在实际问题中的方法和步骤，并解释了分维数的实际意义。分维数足反映区域化变量在某方向变化程度的定量指标。该方法不仅适用于地球化学金元素和银元素数据，而且还适用于其他元素和地质数据，具有普遍的意义。     

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.     

Non-Linear Theory and Power-Law Models for Information Integration and Mineral Resources Quantitative Assessments

Singular physical or chemical processes may result in anomalous amounts of energy release or mass accumulation that, generally, are confined to narrow intervals in space or time. Singularity is a property of different types of non-linear natural processes including cloud formation, rainfall, hurricanes, flooding, landslides, earthquakes, wildfires, and mineralization. The end products of these non-linear processes can be modeled as fractals or multifractals. Hydrothermal processes in the Earth’s crust can result in ore deposits characterized by high concentrations of metals with fractal or multifractal properties. Here we show that the non-linear properties of the end products of singular mineralization processes can be applied for prediction of undiscovered mineral deposits and for quantitative mineral resource assessment, whether for mineral exploration or for regional, national and global planning for mineral resource utilization. In addition to the general theory and framework for the non-linear mineral resources assessment, this paper focuses on several power-law models proposed for characterizing non-linear properties of mineralization and for geoinformation extraction and integration. The theories, methods, and computer system discussed in this paper were validated using a case study dealing with hydrothermal Au mineral potential in southern Nova Scotia, Canada.     

A two-dimensional fuzzy random model of soil pore structure

A new conceptual model for soil poresolid structure is formalized. Soil poresolid structure is proposed to comprise spatially abutting elements each with a value which is its membership to the fuzzy set “pore, ” termed porosity. These values have a range between zero (all solid) and unity (all pore). Images are used to represent structures in which the elements are pixels and the value of each is a porosity. Twodimensional random fields are generated by allocating each pixel a porosity by independently sampling a statistical distribution. These random fields are reorganized into other poresolid structural types by selecting parent points which have a specified local region of influence. Pixels of larger or smaller porosity are aggregated about the parent points and within the region of interest by controlled swapping of pixels in the image. This creates local regions of homogeneity within the random field. This is similar to the process known as simulated annealing. The resulting structures are characterized using one and twodimensional variograms and functions describing their connectivity. A variety of examples of structures created by the model is presented and compared. Extension to three dimensions presents no theoretical difficulties and is currently under development.     

土壤水力特性分形特征的研究进展

从土壤结构特征的描述、土壤水力特性的研究等方面,综合评述了80年代以来国内外应用分形理论描述土壤水力特性参数及分形特征的研究进展,并探讨了其发展方向和前景。     

Error propagation in equations for geochemical modeling of radiogenic isotopes in two-component mixing

This paper presents error propagation equations for modeling of radiogenic isotopes during mixing of two components or end-members. These equations can be used to estimate errors on an isotopic ratio in the mixture of two components, as a function of the analytical errors or the total errors of geological field sampling and analytical errors. Two typical cases (“Small errors” and “Large errors”) are illustrated for mixing of Sr isotopes. Similar examples can be formulated for the other radiogenic isotopic ratios. Actual isotopic data for sediment and basalt samples from the Cocos plate are also included to further illustrate the use of these equations. The isotopic compositions of the predicted mixtures can be used to constrain the origin of magmas in the central part of the Mexican Volcanic Belt. These examples show the need of high quality experimental data for them to be useful in geochemical modeling of magmatic processes.     

滑坡位移多重分形特征与滑坡演化预测

在系统分析滑坡位移监测资料和位移演化特征的基础上,根据多重分形理论基本原理,对滑坡位移演化所具有的复杂性、突变性和非线性特征进行了分析和研究。单一分形维数对滑坡位移的演化趋势预测存在不足,文中分别以新滩滑坡、丹巴滑坡和黄蜡石滑坡为例,计算了滑坡位移时序演化的多重分维数演化特征。分析和评价位移演化规律与多重分维数演化特征的关系发现,多重分维数D1 > D2 > … > D∞时,滑坡趋于稳定;D1 < D2 < … < D∞时,滑坡向失稳破坏演化。当滑坡位移时序多重分维数演化特征出现拐点时,即分维数由D1 > D2 > … > D∞,经D1 > … > Dn < Dn+1 < … < D∞到 D1 < D2 < … < D∞的演化过程时,滑坡向不稳定的状态演化;当分维数由D1 < D2 < … < D∞,经D1 > … > Dn D2 > … > D∞的演化过程时,滑坡向趋于稳定的状态演化。研究表明,可以运用多重分维数演化特征对滑坡位移演化趋势与规律进行评价与预测。     

Stationarity Scores on Training Images for Multipoint Geostatistics

This research introduces a novel method to assess the validity of training images used as an input for Multipoint Geostatistics, alternatively called Multiple Point Simulation (MPS). MPS are a family of spatial statistical interpolation algorithms that are used to generate conditional simulations of property fields such as geological facies. They are able to honor absolute “hard” constraints (e.g., borehole data) as well as “soft” constraints (e.g., probability fields derived from seismic data, and rotation and scale). These algorithms require 2D or 3D training images or analogs whose textures represent a spatial arrangement of geological properties that is presumed to be similar to that of a target volume to be modeled. To use the current generation of MPS algorithms, statistically valid training image are required as input. In this context, “statistical validity” includes a requirement of stationarity, so that one can derive from the training image an average template pattern. This research focuses on a practical method to assess stationarity requirements for MPS algorithms, i.e., that statistical density or probability distribution of the quantity shown on the image does not change spatially, and that the image shows repetitive shapes whose orientation and scale are spatially constant. This method employs image-processing techniques based on measures of stationarity of the category distribution, the directional (or orientation) property field and the scale property field of those images. It was successfully tested on a set of two-dimensional images representing geological features and its predictions were compared to actual realizations of MPS algorithms. An extension of the algorithms to 3D images is also proposed. As MPS algorithms are being used increasingly in hydrocarbon reservoir modeling, the methods described should facilitate screening and selection of the input training images.     

Some basic notions of theoretical hydrochemistry

This report gives the conceptual approaches to the development of the theory of the formation of natural water composition in view of the primary properties of aquatic ecosystems: the “principle of uncertainty” in the estimation of the conditions, the “continuity” of the changes proceeding, and the “plurality” of water’s natural structure and components (biotic and abiotic), as well as their interaction processes. The importance and features of each of these basical criteria for characterizing the conditions of an aquatic ecosystems are substantiated in theory.     

Some features of twisting of the Earth in geological history: the tectonophysical aspect

In this work some examples of “twisting” of the Earth in the geologic past are given: displacement of the northern parts of the global submeridional structures to the west relative to their southern parts; the rotation of the “geodynamic pair” of Siberia-Laurentia in the Proterozoic; sinistral displacement of the Northern Hemisphere relative to the Southern Hemisphere after the collapse of the last Pangaea; the equatorial rotation of the continental plates; oblique orientation of the global network of planetary fracturing; an inclination of the axis of submeridional compression; sinistral “beveling”; the dextral “twisting” of Venus. All these examples confirm the idea of possible sinistral “twisting” of the Earth that has been proposed by many authors. The cause of such “twisting” is unclear, although it is likely connected with the Earth’s rotation around its axis. Some of these examples show that many paleomagnetic reconstructions can be usefully discussed in a tectonophysical aspect. Moreover, in connection with this data, the development of a new scientific field, called “paleomagnetic tectonophysics”, is possible.     

新疆东准噶尔地区断裂的多重分形研究

结合东准噶尔地区强应变构造带的空间展布情况和断裂的空间分布特征 ,用多重分形模型测算了各带的多重分维值。研究结果表明 :强应变构造带在不同尺度上具有自相似性 ;断裂的空间分布为多重分形结构 ,断裂的演化有从复杂几何结构的次级断裂组合向单一连续的大断裂过渡而分维值逐渐减小的趋势 ;不同的大地构造单元有不同的分维值 ,而且有靠近板块缝合带 (或海沟 )分维值减小、远离板块缝合带 (或海沟 )分维值增大的趋势     

Predicting Lahar-Inundation Zones: Case Study in West Mount Pinatubo, Philippines

This paper demonstrates techniques for pre-eruption prediction of lahar-inundation zones in areas where a volcano has not erupted within living memory and/or where baseline geological information about past lahars could be scarce or investigations to delimit past lahars might be incomplete. A lahar source (or proximal lahar-inundation) zone is predicted based on ratio of vertical descent to horizontal run-out of eruptive deposits that spawn lahars. Immediate post-eruption distal lahar-inundation zones are predicted based on “pre-eruption” distal lahar-inundation zones and on spatial factors derived from a digital elevation model. Susceptibility to distal lahar-inundation is estimated by weights-of-evidence, by logistic regression and by evidential belief functions. Predictive techniques are applied using a geographic information system and are tested in western part of Pinatubo volcano (Philippines). Predictive maps are compared with a forecast volcanic-hazard map through validation against a field-based volcanic-hazard map. The predictive model of proximal lahar-inundation zone has “true positive” prediction accuracy, “true negative” prediction accuracy, “false positive” prediction error and “false negative” prediction error that are similar to those of the forecast volcanic-hazard map. The predictive models of distal lahar inundation zones have higher “true positive” prediction accuracy and lower “false negative” prediction error than the forecast volcanic-hazard map, although the latter has higher “true negative” prediction accuracy and lower “false positive” prediction error than the former. The results illustrate utility of proposed predictive techniques in providing geo-information could be used, howbeit with caution, for planning to mitigate potential lahar hazards well ahead of an eruption that could generate substantial source materials for lahar formation.     

Comparison of Mathematical Methods of Potential Modeling

Various attempts are known to turn the “catalogue” of mineral deposit models compiled by Cox and Singer (1986) operational, and have initiated activities called “potential mapping”, “potential modeling”, or “targeting”. The common ultimate objective is to estimate the probability for a given location that a mineralization of a given type occurred. The mathematics range from “weights of evidence” and others featuring a Bayesian approach to logistic regression by maximum likelihood, and include other realizations by means of fuzzy methods, genetic programming, and artificial neural nets. Once developed and coded, applications are not restricted to mineral prospection and exploration but include any kind of occurrences and their estimated probabilities, e.g., risk assessment of land slides and many others.     

Carlin-type gold deposits in Qinling and some related problems

The sediment-hosted disseminated gold deposits in the Oinling region are of sedimentation-slight-metamorphic origin superimposed by hydrothermal reworking at moderate-low temperatures and are well comparable with the typical Carlin gold deposits in the United States.In view of the confusing concept concerning the “sediment-hosted”and “Carlin-type” gold deposits,the authors propose that the term“sediment-hosted gold deposit”should be used in a broad sense which encompasses at least the four subtypes,i.e.,the Carlin type,the metamorphic fine clastic type,the hydrothermal sedimentary type and the vein type.In oter words,the “Carlin-type”should not be used as a synonym for “sediment-hosted”but is recommended as a subtype under the general category of “sediment-hosted gold deposits”     

Multifractal modeling and spatial statistics

In general, the multifractal model provides more information about measurements on spatial objects than a fractal model. It also results in mathematical equations for the covariance function and semivariogram in spatial statistics which are determined primarily by the second-order mass exponent. However, these equations can be approximated by power-law relations which are comparable directly to equations based on fractal modeling. The multifractal approach is used to describe the underlying spatial structure of De Wijs 's example of zinc values from a sphalerite-bearing quartz vein near Pulacayo, Bolivia. It is shown that these data are multifractal instead of fractal, and that the second-order mass exponent (=0.979±0.011 for the example) can be used in spatial statistical analysis.     

Characterization of Geochemical Distributions Using Multifractal Models

The use of multifractals in the applied sciences has proven useful in the characterization and modeling of complex phenomena. Multifractal theory has also been recently applied to the study and characterization of geochemical distributions, and its relation to spatial statistics clearly stated. The present paper proposes a two-dimensional multifractal model based on a trinomial multiplicative cascade as a proxy to some geochemical distribution. The equations for the generalized dimensions, mass exponent, coarse Lipschitz–Hölder exponent, and multifractal spectrum are derived. This model was tested with an example data set used for geochemical exploration of gold deposits in Northwest Portugal. The element used was arsenic because a large number of sample assays were below detection limit for gold. Arsenic, however, has a positive correlation with gold, and the two generations of arsenopyrite identified in the gold quartz veins are consistent with different mineralizing events, which gave rise to different gold grades. Performing the multifractal analysis has shown problems arising in the subdivision of the area with boxes of constant side length and in the uncertainty the edge effects produce in the experimental estimation of the mass exponent. However, it was possible to closely fit a multifractal spectrum to the data with enrichment factors in the range 2.4–2.6 and constant K₁ = 1.3. Such parameters may give some information on the magnitude of the concentration efficiency and heterogeneity of the distribution of arsenic in the mineralized structures. In a simple test with estimated points using ordinary lognormal kriging, the fitted multifractal model showed the magnitude of smoothing in estimated data. Therefore, it is concluded that multifractal models may be useful in the stochastic simulation of geochemical distributions.     

非线性成矿预测理论:多重分形奇异性-广义自相似性-分形谱系模型与方法

介绍了“奇异性-广义自相似性-分形谱系”(“3S”: Singularity-generalized self-Similarity-fractal Spectrum)为核心的多重分形现代成矿预测理论与模型(Multifractal Mineralization Prediction Theory and Models)的基本内容和前沿研究方向.讨论了作为非线性、复杂性理论的重要领域之一, 多重分形理论所提供的“奇异性-广义自相似性-分形谱系”等概念和相关的模型.这些新概念和模型不仅能够合理地描述成矿系统、成矿过程、成矿富集规律、矿产资源时空分布, 还提供了定量模拟和识别成矿异常(地质、地球物理、地球化学、遥感异常)的有效模型和实用方法.将多重分形原理与成矿过程、矿产资源分布规律、矿产资源信息获取研究相结合, 可形成具有良好应用前景的现代成矿预测理论与模型.采用该多重分形矿产资源预测理论和在此基础上所开发的专用地学非线性空间信息GeoDASGIS技术, 对国内外多个金属成矿区带进行了矿产资源勘查与评价, 均取得了较理想的预测效果, 表明对开展矿产资源勘查和评价是有效和可行的.