地球化学场的分形与多重分形特征
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摘要
地球化学场(geochemical fields)时-空结构研究是区域成矿作用动力学的重要内容，对区域和矿区矿产预测和新一轮国土资源调查发挥着十分重要的作用。分形(fractal)与多重分形(multifractal)理论作为非线性科学的一个分支，是研究复杂系统时-空结构特征的基本理论和有效方法，已广泛应用于地球科学的各个领域 ，也是地球化学家关注的焦点，许多地球化学过程，如成矿作用、火山活动等都被认为具有自相似（self-similarity）或自仿射(self-affinity)的性质，即具有分形或多重分形的特征。因此，用分形与多重分形的方法研究地球化学场的时空结构具有重要的意义。
本文主要运用两种典型的分形与多重分形方法，即矩分析方法(the method of moments)和经过改进的浓度面积法（ACAF model），辅助以频率分析法QQ 图（QQ plot）和直方图(histogram)，分析了地球化学场可能的分形与多重分形特征。论文主要包括三大部分，即不同的De Wijs模型模拟、实例分析和Monte Carlo模拟。实例分析主要选择了粤北4292km2内1448个基岩样品的25项元素、安徽省江南江北各约22000 km2 和 18100 km2内5489个和4524个水系沉积物样品的14项元素、塔里木盆地４个含油气区的6418个土壤样品的酸解烃(C1～C5)、紫外、荧光、蚀变碳酸盐和汞等共计330,000余个地球化学数据。所涉及地球化学指标之多、地域之广、样品数量之大，在国内外众多研究中都是空前的。
第一部分De Wijs模型及其改进模型的模拟结果均加深了我们对地球化学场分形与多重分形特征的认识。
通过二维De Wijs模型模拟具有不同浓集系数(enrichment factor)、有局部削减(local suppression)和局部叠加(local superimposition)的地球化学场，本文首先分析了其多重分形谱(multifractal spectrum)的形态特征、ACAF分布模式及频率检验（probability test）结果。结果显示：①基本的二维De Wijs模型能产生完美的连续多重分形(continuous multifractal)， De Wijs模拟结果在空间上具有标度不变性(scale-invariant)。②传统的De Wijs模拟结果其多重分形谱完全对称(symmetric)，参数???（多重分形谱最高点的奇异指数（singularity exponent）值）和 ???（奇异指数的宽度）随着浓集系数的增加单调增加。任意高值或低值部分的削减都会打破这种对称性，从而造成多重分形谱函数f(?)曲线右偏或左偏，即右偏多重分形(right-deviated multifractal，RM)和左偏多重分形(left-deviated multifractal，LM)。以??为界，??可以分成左右两部分Δ?L和Δ?R，其比率R（非对称系数，asymmetry index）对于刻画潜在的局部富集与贫化十分有效。如果将较高浓集系数d2的De Wijs模拟结果局部叠加于另一具有较小浓集系数d1的De Wijs模拟结果上，最后结果其多重分形谱函数左偏。当d1不变时，随着d2的增大， f(?) 曲线左偏越来越明显，相应地Δ?,Δ?L 以及R 的值随之增加，但多重分形谱函数右侧改变甚少。③在双对数坐标图（log-log diagram）上，上述多重分形的ACAF分析结果具有两种分布模式，即简单的两直线段拟合模式(bi-segment pattern，BS pattern)和多直线段拟合模式(multi-segment pattern，MS pattern)。任何有局部叠加的De Wijs模拟结果均服从MS分布模式，而基本的De Wijs模拟结果却服从BS分布模式。④基本的De Wijs模拟结果主要服从对数正态(lognormal distribution)分布。
金属地球化学场（metallic geochemical fields）与油气地球化学场(oil/gas geochemical
fields)实例分析结果表明：①地球化学场具有连续多重分形特征，这一规律具有普适性。同时，金属地球化学场与油气地球化学场在场的空间结构上具有很大的差异。②对主要的成油指标（major oil/gas forming indexes）,其质量指数（mass exponent）与矩级次(order, moment)之间呈一定程度的线性相关，其多重分形谱函数较窄，部分指标呈单钩状(single spike)。所有这些证实，主要的成油气指标具有较弱的多重分形特征，部分显示为明显的单一分形。而主要的成矿元素（major ore-forming elements）的质量指数与矩级次之间呈一曲线，多重分形谱宽而连续，因此主要的成矿指标在金属地球化学场中具有连续多重分形特征。主要成矿指标非对称的多重分形谱函数正预示着在长期的形成过程中，地球化学场曾经历过不同的局部叠加或贫化。非对称指数R可望用于成矿潜力(potential)评价中。③网格化数据分析结果表明，过密或者过稀的网格密度均不能较好地刻划地球化学场的本质特征，这是因为，较小的网格密度可能会产生一系列“假”数据，而网格密度太大又可能导致部分原始数据被“平滑”甚至忽略，这样得出的结果极有可能偏离实际。本论文通过不同网格密度的网格化与数据实际数据结果的对比研究，发现，平均每个网格内取1.5个数据比较合适，这样使得采样较密区域的网格内有2个或2个以上样品，采样较稀区域的网格内有1个样品，部分网格内没有样品，即为采样空白区。④在双对数坐标图上，主要的成矿元素和主要油气化探指标均不同程度地显示服从MS分布模式，而其他多表现为BS 分布模式。不同的分布模式可望用于区域含矿含油性和区域环境评价，
It's of great significance to study geochemical element distribution patterns by using fractal and multifractal methods. This dissertation will apply two fractal and multifractal methods called the method of moments and ACAF model to investigate whether it is a universal feature that a geochemical field is fractal or multifractal and whether lognormal or normal distribution data will yield a multifractal or single fractal with the additional use of QQ plot and histogram. Three important and interesting parts will be contained in this dissertation, including De Wijs modeling, Case studies and Monte Carlo Simulations.
Based on simulations of two-dimensional De Wijs models with different parameters, high- and low-value suppression and local superimposition of other De Wijs models, this dissertation first simulated different geochemical element distribution patterns and then investigated their multifractal spectrum function shapes, ACAF distribution patterns and probability testing results. It can be drawn out that: ①Basic two-dimensional De Wijs modeling can produce a perfect continuous multifractal and such De Wijs construction is scale-invariant in space. If the spatial structure keeps the same, the multifractal spectrum shape will not change as neither relatively less or more iteration times nor the global division do have any effect on the multifractal shape.②Conventional De Wijs models exhibit symmetric multifractal spectra and parameters ???and ???increase monotonously with "enrichment factor" d. Any suppression of high values or low values in a simulating geochemical field will break this symmetry and make the spectrum curves f(?) deviate to right or to left, resulting in right-deviated multifractal(RM) and left-deviated multifractal(LM), respectively. The parameterΔ??can then be split intoΔ?L and Δ?R and their ratio R is called asymmetry index by the author and proves to be a pivotal parameter characterizing the underlying enrichment or pauperization mechanism. Local superimposition of De Wijs model of d1 by another De Wijs model of d2 will make the multifractal spectrum curves deviate left. With the increase of enrichment factor d2 of superimposed component, f(?) curves deviate more violently, and hence the parametersΔ?,Δ?L and asymmetric index R increases systematically with the enrichment factor of superimposed component, leaving its right side almost unchanged.③When the concentration bins and the corresponding frequency are drawn on log-log diagrams, the continuous multifractals of simulated geochemical fields following De Wijs construction can be categorized by two patterns: bi-segment pattern (BS pattern) and multi-segment pattern(MS pattern). Any basic De Wijs model follows BS pattern, whereas the De Wijs model with local superimposition obeys MS pattern, which is in agreement with the results of the method of moments.④The principle part of such kind of basic De Wijs modeling data is authentically of lognormal distribution. Whether the tails of it follow Pareto distribution is still in issue as there are so many repeated concentrations in such construction data that it's nearly impossible to detect the distribution patterns of any small part, such as the tails.
For case studies, two distinct data sets will be employed, including elements in metallic
geochemical fields, as well as petroleum data from oil/gas regions. The former data set contains concentrations of 25 elements in 1448 whole rock samples from a region of 4290km2 in the north Guangdong Province, South China; 14 elements of 5489 and 4524 stream sediments in South and North Anhui Province, respectively, South China; and the latter with more than 15 hydrocarbons and other indices in 6418 soil samples from 4 oil/gas fields with total survey area 7000 km2 in Tarim Basin, North China. The results indicated that: ①Fractal and multifractal properties are quite universal in geochemical fields. However, metallic geochemical fields and oil/gas geochemical fields are intrinsically different from each other in spatial structu
本文主要运用两种典型的分形与多重分形方法，即矩分析方法(the method of moments)和经过改进的浓度面积法（ACAF model），辅助以频率分析法QQ 图（QQ plot）和直方图(histogram)，分析了地球化学场可能的分形与多重分形特征。论文主要包括三大部分，即不同的De Wijs模型模拟、实例分析和Monte Carlo模拟。实例分析主要选择了粤北4292km2内1448个基岩样品的25项元素、安徽省江南江北各约22000 km2 和 18100 km2内5489个和4524个水系沉积物样品的14项元素、塔里木盆地４个含油气区的6418个土壤样品的酸解烃(C1～C5)、紫外、荧光、蚀变碳酸盐和汞等共计330,000余个地球化学数据。所涉及地球化学指标之多、地域之广、样品数量之大，在国内外众多研究中都是空前的。
第一部分De Wijs模型及其改进模型的模拟结果均加深了我们对地球化学场分形与多重分形特征的认识。
通过二维De Wijs模型模拟具有不同浓集系数(enrichment factor)、有局部削减(local suppression)和局部叠加(local superimposition)的地球化学场，本文首先分析了其多重分形谱(multifractal spectrum)的形态特征、ACAF分布模式及频率检验（probability test）结果。结果显示：①基本的二维De Wijs模型能产生完美的连续多重分形(continuous multifractal)， De Wijs模拟结果在空间上具有标度不变性(scale-invariant)。②传统的De Wijs模拟结果其多重分形谱完全对称(symmetric)，参数???（多重分形谱最高点的奇异指数（singularity exponent）值）和 ???（奇异指数的宽度）随着浓集系数的增加单调增加。任意高值或低值部分的削减都会打破这种对称性，从而造成多重分形谱函数f(?)曲线右偏或左偏，即右偏多重分形(right-deviated multifractal，RM)和左偏多重分形(left-deviated multifractal，LM)。以??为界，??可以分成左右两部分Δ?L和Δ?R，其比率R（非对称系数，asymmetry index）对于刻画潜在的局部富集与贫化十分有效。如果将较高浓集系数d2的De Wijs模拟结果局部叠加于另一具有较小浓集系数d1的De Wijs模拟结果上，最后结果其多重分形谱函数左偏。当d1不变时，随着d2的增大， f(?) 曲线左偏越来越明显，相应地Δ?,Δ?L 以及R 的值随之增加，但多重分形谱函数右侧改变甚少。③在双对数坐标图（log-log diagram）上，上述多重分形的ACAF分析结果具有两种分布模式，即简单的两直线段拟合模式(bi-segment pattern，BS pattern)和多直线段拟合模式(multi-segment pattern，MS pattern)。任何有局部叠加的De Wijs模拟结果均服从MS分布模式，而基本的De Wijs模拟结果却服从BS分布模式。④基本的De Wijs模拟结果主要服从对数正态(lognormal distribution)分布。
金属地球化学场（metallic geochemical fields）与油气地球化学场(oil/gas geochemical
fields)实例分析结果表明：①地球化学场具有连续多重分形特征，这一规律具有普适性。同时，金属地球化学场与油气地球化学场在场的空间结构上具有很大的差异。②对主要的成油指标（major oil/gas forming indexes）,其质量指数（mass exponent）与矩级次(order, moment)之间呈一定程度的线性相关，其多重分形谱函数较窄，部分指标呈单钩状(single spike)。所有这些证实，主要的成油气指标具有较弱的多重分形特征，部分显示为明显的单一分形。而主要的成矿元素（major ore-forming elements）的质量指数与矩级次之间呈一曲线，多重分形谱宽而连续，因此主要的成矿指标在金属地球化学场中具有连续多重分形特征。主要成矿指标非对称的多重分形谱函数正预示着在长期的形成过程中，地球化学场曾经历过不同的局部叠加或贫化。非对称指数R可望用于成矿潜力(potential)评价中。③网格化数据分析结果表明，过密或者过稀的网格密度均不能较好地刻划地球化学场的本质特征，这是因为，较小的网格密度可能会产生一系列“假”数据，而网格密度太大又可能导致部分原始数据被“平滑”甚至忽略，这样得出的结果极有可能偏离实际。本论文通过不同网格密度的网格化与数据实际数据结果的对比研究，发现，平均每个网格内取1.5个数据比较合适，这样使得采样较密区域的网格内有2个或2个以上样品，采样较稀区域的网格内有1个样品，部分网格内没有样品，即为采样空白区。④在双对数坐标图上，主要的成矿元素和主要油气化探指标均不同程度地显示服从MS分布模式，而其他多表现为BS 分布模式。不同的分布模式可望用于区域含矿含油性和区域环境评价，
It's of great significance to study geochemical element distribution patterns by using fractal and multifractal methods. This dissertation will apply two fractal and multifractal methods called the method of moments and ACAF model to investigate whether it is a universal feature that a geochemical field is fractal or multifractal and whether lognormal or normal distribution data will yield a multifractal or single fractal with the additional use of QQ plot and histogram. Three important and interesting parts will be contained in this dissertation, including De Wijs modeling, Case studies and Monte Carlo Simulations.
Based on simulations of two-dimensional De Wijs models with different parameters, high- and low-value suppression and local superimposition of other De Wijs models, this dissertation first simulated different geochemical element distribution patterns and then investigated their multifractal spectrum function shapes, ACAF distribution patterns and probability testing results. It can be drawn out that: ①Basic two-dimensional De Wijs modeling can produce a perfect continuous multifractal and such De Wijs construction is scale-invariant in space. If the spatial structure keeps the same, the multifractal spectrum shape will not change as neither relatively less or more iteration times nor the global division do have any effect on the multifractal shape.②Conventional De Wijs models exhibit symmetric multifractal spectra and parameters ???and ???increase monotonously with "enrichment factor" d. Any suppression of high values or low values in a simulating geochemical field will break this symmetry and make the spectrum curves f(?) deviate to right or to left, resulting in right-deviated multifractal(RM) and left-deviated multifractal(LM), respectively. The parameterΔ??can then be split intoΔ?L and Δ?R and their ratio R is called asymmetry index by the author and proves to be a pivotal parameter characterizing the underlying enrichment or pauperization mechanism. Local superimposition of De Wijs model of d1 by another De Wijs model of d2 will make the multifractal spectrum curves deviate left. With the increase of enrichment factor d2 of superimposed component, f(?) curves deviate more violently, and hence the parametersΔ?,Δ?L and asymmetric index R increases systematically with the enrichment factor of superimposed component, leaving its right side almost unchanged.③When the concentration bins and the corresponding frequency are drawn on log-log diagrams, the continuous multifractals of simulated geochemical fields following De Wijs construction can be categorized by two patterns: bi-segment pattern (BS pattern) and multi-segment pattern(MS pattern). Any basic De Wijs model follows BS pattern, whereas the De Wijs model with local superimposition obeys MS pattern, which is in agreement with the results of the method of moments.④The principle part of such kind of basic De Wijs modeling data is authentically of lognormal distribution. Whether the tails of it follow Pareto distribution is still in issue as there are so many repeated concentrations in such construction data that it's nearly impossible to detect the distribution patterns of any small part, such as the tails.
For case studies, two distinct data sets will be employed, including elements in metallic
geochemical fields, as well as petroleum data from oil/gas regions. The former data set contains concentrations of 25 elements in 1448 whole rock samples from a region of 4290km2 in the north Guangdong Province, South China; 14 elements of 5489 and 4524 stream sediments in South and North Anhui Province, respectively, South China; and the latter with more than 15 hydrocarbons and other indices in 6418 soil samples from 4 oil/gas fields with total survey area 7000 km2 in Tarim Basin, North China. The results indicated that: ①Fractal and multifractal properties are quite universal in geochemical fields. However, metallic geochemical fields and oil/gas geochemical fields are intrinsically different from each other in spatial structu
引文
1 Agterberg F P, Cheng Q, Wright D F. Fractal modeling of mineral deposits[A]. International Symposium on the Application of Computers and Operations Research in the Mineral Industries[C]. Canada, Montreal: Canadian Institute of Mining Metallurgy and Petroleum, 1993. 43-53.
2 Agterberg F P, Cheng Q, Brown.A, Good,D. Multifractal modeling of fractures in the Lac Du Boonet Batholith, Manitoba. Computer & Geosciences, 1996,22(5):497-507.
3 Agterberg F P. How nature works: the science of self-organized criticality, book review[J]. Computer & Geosciences, 1998,24: 205-207
4 Agterberg F P. Book Reviews. Computers and Geosciences,1999,25(1):93-99.
5 Agterberg F P.Multifractal Simulation of Geochemical Map Patterns. Proceedings of the international Symposium on Diversity of Mineralization and its Prediction and Assessment, Beijing and Wuhan, 2001,6-14.
6 Ahrens L H. Element distributions in igneous rocks-VII-Geoch.et Cosmoch. Acta, 28,p.271 -290,1964. (Cited in Yu C.,1987).
7 Ahrens L H. Distribution of the elements in our planet,1965. (Cited in Yu C.,1987).
8 Allegre C J, Lewin E. Scaling Laws and Geochemical Distribution[J]. Earth Planet Sci Letters,1995,132:1-13.
9 Andre Robert. Statistical Properties of Sediment Bed Profiles in Alluvial Channels. Math Geol[J].1988,20(3):205~225.
10 An Zhen-wen, Yang Cui-hua, Wang Lin-ying et al. Studies on the multifractals of time and space clustering of earthquakes[J](In Chinese with English Abstract). Letters of Geophysics, 2000,43(1):74~80.
11 Atmanspacher,H., Scheigraber,H., Wiedenmann, G.,1989. Determination of f(?) for a limited random point set. Physical Review, 1989,40(7):3954-3963.
12 Bak P. How nature works [M]. New York: Springer Verlag,1996, 1-212
13 Bao Zheng-yu, Li Jin-yao, Su Jiang-yu et al. Geochemical field analysis and oil/gas geochemical exploration[J](In Chinese with English Abstract). Petroleum exploration and development, 1997,18(1)：39-41.
14 Bao Zheng-yu, Li Fang-lin, Jia Xian-qiao. Methodology of Temporal-spatial structure of geochemical fields[J](In Chinese with English Abstract). Earth Sci- Journal of China University of Geosciences, 1999,24(3): 282-286.
15 Binder,K.. Monte Carlo Methods in Statistical Physics[M]. Germany: Berlin Helderberg,1986:1-411.
16 Blenkinsop T. Spatial distribution of gold deposit: two examples from the Zimbabwe Archaen Craton[A]. Barton C C. Fractal in Petroleum Geology and Earth Processes[M]. New York and London:Plonum Press,1994. 247-258.
17 Block A, Von-Bloh W, Klenke T,et al. Multifractal analysis of the microdistribution of elements in sedimentary structure using images from scanning electron microscopy and energy dispersive X ray spectrometry[J]. Journal of Geophysical Research, B,Solid Earth and Planets,1991,96(10):16223-16230.
Brian Klinkenberg, A review of methods used to determine the fractal dimension of linear
18 features, Math. Geology. 26 （1）23-46
19 B?lviken, B. , Stokke, P. R. , Feder, J. and J?ssang, T. , 1992. The fractal nature of geochemical landscapes. J. Geochem. Explor. , 43:91-109.
20 Carlson C A. Spatial distribution of ore deposit[J]. Geology,1991,19(2):107-117.
21 Chang Yinfo, Liu Xiangpei, Wu Yenchang. The Copper-Iron Belt of the Lower and Middle Reaches of the Changjiang River[M]. Beijing:Press of Geology, 1991,379p.
22 Chen Chun-zi, Lan Chao-li. Fractal Theory and its Application in Earth Science[J](In Chinese with English Abstract). Earth Sci Explor, 1998, 6:123-128
23 Chen Jian-guo, Wang Ren-duo, Chen Yong-qing. The extraction of concealed fractal information and the delineation of geochemical anomalies by fractal methods[J] (In Chinese with English Abstract). Earth Sci- Journal of China University of Geosciences, 1998,23(2):175-178.
24 Chen Jian-ping, Wang Qing, Xiao Shu-fang. Computer modeling of the fractal dimension of the rock mass fracture network[J] (In Chinese with English Abstract). Journal of Engineering Geology, 1995,3(3)79-859.
25 Cheng Q,Agterberg F P, Ballatyne S B.The separation of geochemical anomalied from background by fractal methods [J]. Explor Geochem,1994,51: 109-130
26 Cheng Q. The perimeter-area fractal model and its application to geology. Math. Geol. 1995a, 27:69-82
27 Cheng Q, Agterberg F P.Multifractall modeling and spatial point processes[J]. Math Geol,1995b,27:831-845.
28 Cheng Q, Agterberg F P.Multifractall modeling and spatial statistics[J]. Math Geol,1996a,28:1-16.
29 Cheng Q, Agterberg F P. Comparision between types of multifractal modeling. Math.Geol.1996b,28:1001-1015
30 Cheng Q , Agterberg F P, Bonham-Carter G F. A spatial analysis method for geochemical anomaly seperation [J]. Explor Geochem, 1996c,56:183-195
31 Cheng Q, Bonham-Carter G F, Hall G E M. Statistical study of trace elements in the soluble organic and amorphous Fe-Mn phases of surficial analysis [J]. Explor Geochem,1997a,59: 27-46
32 Cheng Q. Discrete multifractals[J]. Math Geol,1997b,29(2):245-266.
33 Cheng Q.Multifractal modelling and lacunarity analysis[J]. Math Geol,1997c,29:919-932.
34 Cheng Q, Bonham G F, Hall G E M, Bajc A. Statistical study of trace elements in the soluble organic and amorphous Fe-Ma phases of surficial sediments,Sudbury Basin;1,Multivariate and spatial analysis. J. Geochem. Explor,1997d,59(1):27-46
35 Cheng Q. Multifractality and spatial statistics [J]. Computer & Geosciences, 1999a,25(9): 949-962
36 Cheng Q.Gliding box method and multifractal modelling [J]. Computer & Geosciences, 1999b,25(9): 1073-1079.
37 Cheng Q. Spatial and scaling modeling for geochemical anomaly separation. J. Geochem. Explor. ,1999c, 65:175-194.
38 Cheng Q. Markov processes and discrete multifractals[J]. Math Geol,1999d,31(4):455-469.
Cheng Q,Xu Y,Eric Grunsky. Intergrated spatial and spectrum method for geochemical
39 anomaly separation. Natural Resources[J].2000a,9(1):43-51.
40 Cheng Q. Multifractal theory and Geochemical Element Distribution Patterns[J]. Earth Science-Journal of China University of Geosciences, 2000b,25(3): 311-318
41 Cheng Q. Selection of Multifractal Scaling Breaks and Separation of Geochemical and Geophysical Anomaly.Proceedings of the international Symposium on Diversity of Mineralization and its Prediction and Assessment,Beijing and Wuhan, 2001a,23-28.
42 Cheng Q. Multifractal and Geostatistic Methods for Characterizing Local Structure and Singularity Properties of Exploration Geochemical Anomalies[J](In Chinese). Earth Science-Journal of China University of Geosciences,2001b,26(2):161-166.
43 Cheng Q. Spatial Self-similarity and Geophysical and Geochemical Anomaly Decomposition[J](In Chinese). Advance of Geophysics,2001c,16(2): 8-17.
44 Cheng Xiao-jiu, Lu Jian-hang, Song Liang-ming. The meaning and program of fractal dimension of Pb and An[J](In Chinese with English Abstract). Geol Prospect, 1994, 30(5):32-35.
45 Chiles,J.P.. Fractal and Geostatistical Methods for Modeling of Fracture Network. Math Geol[J].1988,20(6):631~665.
46 De Wijs H J. Statistics of ore distribution: (1) frequency distribution of assay values[J]. Geol Mijnbouw,1951,13:365-375.
47 De Wijs H J. Statistics of ore distribution: (2) theory of binomial distribution applied to sampling and engineering problems[J]. Geol Mijnbouw,1953,15:12-24.
48 E.A.Yfantis,G.T.Flatman, E.J.Englund. Simulation of Geological Surfaces Using Fractals. Math Geol[J].1988,20(6):667~672.
49 Evertsz C J G,Mandelbrot B B. Multifratal measures (Appendix B) [A]. In:Peitgen H-O,Jurgens H, Saupe D,eds.Chaos and fractals[C]. New York:Springer Verlag, 1992, 922~953.
50 Feder J. Fractals[M]. New York:Plenum,1988, 1~283.
51 Fowler A. D., and Roach D. E., Dimensionality analysis of tine-series data: nonlinear methods, Computers & Geosciences, 1993, 19 （1）: 41-52
52 Halsey T C,Jensen M H,Kadanoff L P, et al. Fractal measures and their singularities, the characterization of strange sets[J]. Phys Rev,1986,33:1141-1151.
53 Hardy,H.H. The Fractal Character of Photoes of Slabbed Cores. Math Geol[J].1992,24(1):76~97.
54 Hu Yuan-lai, Jiang Ruo-wei, Hu Chun-tao et al. Algorithm of fuzzy fractional dimension and its application in reservoir[J](In Chinese with English Abstract). Geophysical prospecting for Petroleum, 1995,30(5):645-653.
55 Ioannis K.Koukouvelas, Marios Asimakopoulos, Theodoros T Doutsos. Fractal characteristics of active normal faults: an example of the eastern Gulf of Corinth, Greece. Tectonophysics[J],1999,308, 263-274.
56 Jaggi,S., Dale A. Quattrochi, and Nina Siu-Ngan Lam, Implementation and operation of three fractal measurement algorithms for analysis of remote-sensing data, Computers & Geosciences, 1993, 19（6）: 745-767
57 James R.Carr, William B.Benzer. On the Practice of Estimation Fractal Dimension. Math Geol[J].1988,20(7):945~958.
58 Jiang Jia-yu, Duan Yu-shun. Application of fractal theory to hydrocarbon detection[J](In Chinese with English Abstract).Geophysical prospecting for Petroleum, 1995，34(1)47-51.
59 Jin You-yu, Meng Xian-guo. Quantitative analysis of Time and Space Series in Geology. Wuhan: Press of China University of Geosciences, 1992,1-169.
60 Joost H.J.Van Opheusden. The origin of an increasing or decreasing multifractal spectrum. Physica[J]. 1998,22:10-22.
61 Journel A G, Huijbregts C. Mining Geostatistics, Academic Press,1978,1-690.
62 Juan M.Garcia-Ruiz, Fermin Otalora. Fractal Trees and Horton's Laws. Math Geol[J].1992,24(1):61~71.
63 Kitaev,N.A.. Multidimensional Analysis of Geochemical Fields. Math Geol[J].1991,23(1): 16-32.
64 Laherrere, J. Distributions de type "fractal parabolique" dans la nature. Comptes Rendus de l'Academie des Sciences, Serie II. Sciences de la Terre et des Planetes(In French, Abstract in English), 1996, 322(7): 535-541.
65 Li C, Ma T, Xu Y. A new view: chaotic dynamical nature in mineralization. In:30th international geological congress abstracts,1996a,3 of 3.
66 Li C,Wang G. Isotopic geochemistry of Chinese fluorite depostis. International Geol. Rev.,1996b,38: 1054-1067
67 Li Chang-jiang, Jiang Xu-liang, Xu You-lang,et al. Fractal Study on Mesozoic hydrothermal deposits in Zhejiang Province[J](In Chinese with English Abstract). Sci Geol Sinica, 1996c,22(1): 264-273.
68 Li Chang-jiang, Ma Shi-hua, Zhu Xing-sheng et al. Fractals, Chaos and ANN in Mining Exploration[M](In Chinese). Beijing: Geological Press, 1999,1-140.
69 M.Ann Piech, Kenneth R.Piech. Fingerprints and Fractal Terrain. 1990,22(4):458-485.
70 Ma Tu-hua, Zhu Xing-sheng, Li Chang-jiang. Spatial distribution of fluorite deposits in Zhejiang Province[J](In Chinese with English Abstract). Mineral deposits, 2000,19(3):281-288.
71 Mao Ling-bo, Gui Zhi-xian, Zhu Guang-sheng. Application of fractal geometry to prediction of reservior parameters between two wells[J](In Chinese with English Abstract). Journal of Jianghan Petroleum Institute,1995,17(2): 43-47.
72 Mandelbrot B B. Stochastic models for the Earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. Proceedings of the National Academy of Sciences(USA):72,1975,3825-3828.
73 Mandelbrot B B. The Fractal Geometry of Nature. San Francisco: Freeman, 1983
74 Mandelbrot B B., New "anomalous" multiplicative multifractals: left-sided f(?) and the modeling of DLA, Physica A168,1990:95-111.
75 Mandelbrot B B. Fractals: Form, Chance and Dimension. San Francisco: Freeman, 1997
76 Mandelbrot B B. Multifractals and 1/f Noise. New York: Springer-Verlag, 1999
77 Meng Xian-guo, Zhao Peng-da. Fractal Structures of Geological data[J](In Chinese with English Abstract). Earth Sci-Journal of China University of Geosciences, 1991a, 16(2):207-211.
Meng Xian-guo. R/S Analysis and Fractal Processing of Geochemical Data[J](In Chinese with English Abstract). Earth Sci-Journal of China University of Geosciences, 1991b,
78 16(3):281-286.
79 Nicholas M.S.Rock. Summary Statistics in Geochemistry: A study of the Performance of Robust Estimates. Math Geol[J].1988,20(3):243~275.
80 Paladin, G., Vulpiani,A.. Anomalous scaling laws in multifractal objects[J]. Physical Reports, 1987,156(4):147-225.
81 Paredes, C., Elorza, F.J.. Fractal and Multifractal analysis of fractured geological media:surface-subsurface correlation. Computer & Geosciences, 1999,25(9): 1081-1096.
82 Park.H,Lee.S. Effect of multifractalities of catalyst surface: a Monte Carlo study of catalytic CO oxidation. Surface Science[J].1998,411:1-9.
83 Pei Tao. Spatial Structure Analysis of Geochemical Field of Oil and Gas and its application to Exploration in North Tarim Basin. Ph.D Dissertation of China University of Geosciences, 1998,1-89.
84 Peitgen H O, Jurgens H, Saupe D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.
85 Pillet R. Mise en evidence du comportement multifractal de la distribution frequence-magnitude d'un echantillon sismique( loi de Gutenberg-Richter). Comptes Rendus de l'Academie des Sciences, Serie II. Sciences de la Terre et des Planetes, 1997, 324(7): 805-810.
86 Prokoph, A.. Fractal, multifractal and scaling window correlation dimension analysis of sedimentary time series. Computer & Geosciences, 1999,25(9): 1009-1021.
87 Qin Chang-xing, Zhai Yu-sheng. Some self-similar phenomena and their meaning in economic geology[J](In Chinese with English Abstract). Mineral deposits,1992,11(3): 259-266.
88 Razumovsky,N.K. On the r?le of the logarithmically normal law of frequency distribution in petrology & geochemistry-Comp.Rend.(Doklady) de l'Acad, de Sciences de l'URSS,33(1),p.48-49,1941 (Cited in Yu C.,1987)
89 Richardson, W.A. The frequency distribution of igneous rocks, part II: the law of distribution in relation to petrogenic theoried---Mineral.Mag.,1923,20,p.1-4.(Cited in Yu C.,1987)
90 Sanderson D J, Roberts S A. Fractal relationship between vein thickness and gold grade in drill core from La Codosera, Spain[J]. Economic Geology,1994,89:168-173.
91 Shen Bu-ming, Shen Yuan-chao. Fractal dimension of some gold deposit in Xinjiang and its geological application[J](In Chinese with English Abstract). Sci China(B), 1993,23(3)297-302.
92 Shen Wei. Application of Fractal and Chaos in Geology[J](In Chinese with English Abstract). Geological Science and Technology Information, 1996,15(2):103-109.
93 Shen Wei. A new method of estimating fractal dimension in fractal model of minerogenetic prediction[J](In Chinese with English Abstract). Journal of Changchun University of Earth Sci, 1997,27(1):86-91.
94 Shi Jun-fa, Wang Chun-ning. Fractal Analysis of Gold Deposits in China: implication for giant deposit exploration[J](In Chinese with English Abstract). Geological Science and Technology Information, 1998,23(6):616-619.
95 Shi Yu-tian, Pan Bao-zhi. The application of fractal: The quantitative description of fractures[J](In Chinese with English Abstract). Geophysical and Geochemical Exploration, 2000,24(6):426-430.
96 Shi Jun-fa. Element Geochemical Distribution from Zhuji Area, Zhejiang Province and Scaling laws[J](In Chinese with English Abstract). Earth Sci- Journal of China University of Geosciences,2001,26(2): 167-171
97 Sim B L, Agterberg F P, Beaudry C. Determining the cutoff between background and relative base metal smeller contamination levels using multifractal. Comput. Geosci. 1999,25(9):1023-1041
98 Standley H E, Meakin P. Multifractal phenomena in physics and chemistry[J]. Nature,1998,335(6189): 405~409.
99 T.Ju.Spivak, V.I.Grabenko. Multidimensional Statistical Methods Used for Prediction of Hydrocarbon Accumulations in Turkmenistan. Math Geol[J].1989,21(6):625~629.
100 Turcotte D L. Fractals and Chaos in Geology and Geophysics[M]. Cambridge: Cambridge University Press, 1997, second edition.
101 Vistelius,A.B. The skew frequency distribution and the fundamental law of the geochemical processes--- Jour.Geol.,68,No.1,1960. (Cited in Yu C.,1987)
102 Xie Xue-jin, Yin B. Geochemical patterns from local to global[J]. Explor Geochem,1993,47: 109-129
103 Xu Shun-shan, Wang Gan-guo, Jiang Wan, Zhang Da, Cehn Bei-lin, Xie Yan-xia. Studying on Fractals of Zijinshan Copper-Gold Deposit[J](In Chinese). Geology and Exploration,1999,35(5):50-52.
104 Y.Gueguen, J.Dienes. Transport Properties of Rocks from Statistics and Percolation. Math Geol[J].1989,21(1): 1-13.
105 Yi Chui-xiang. Unlinear Science and Its Application in Earth Science[M](In Chinese). Beijing: Weather Press, 1995,4:226-258.
106 Yu Chong-wen, Luo Ting-chuan, Bao Zhengyu,et al. Regional geochemistry of the Nanling District. Beijing: Geology Press, 1987,1-543(in Chinese).
107 Yu Chong-wen, Chen Kuang, Bao Zheng-yu, et al.. Ore-forming dynamics[M](In Chinese). Beijing: Geology Press,1998.
108 Yu Chong-wen. Large ore deposits and metallogenic zones at the edge of chaos[J](In Chinese with English Abstract). Earth Sci Frontier, 1999, 3:85-102
109 Yue You-xi, Wang Cai-jing. The applications of multifractals in Fracture Research[J](In Chinese with English Abstract). Geophysical prospecting for Petroleum, 1998,37(1):109-114.
110 Zhai Yu-sheng et al. Ore-forming laws of iron-copper(gold) deposits in middle-lower Yangtze River[M].Beijing: Press of Geology, 1992:p235.
111 Zhang Jian-long. Study on Fractal Dimension Characteristics of Au Grade of Gala Gold deposit and its periphery[J](In Chinese with English Abstract).Computing Techniques for Geophysical and Geochemical Exploration,1196,18(1):78-81.
112 Zhang Jun, Zhou Qiaowei. Application of Fractal Method to Analysis of Temporal and Spatial Structure of gold mineralization[J](IN Chinese). Geoscience- Journal of Graduate school, China University of Geosciences, 2000,14(1):56-60.
113 Zhang Zhe-ru, Mao Hua-hai. Fractal theory and ore-forming[J](In Chinese). Earth Sci Frontiers, 2000, 7(1):195~204.
114 Zhang Zheru, Mao Huahai, Qiuming Cheng. Fractal Geometry of Element Distribution on Mineral Surfaces, Mathematical geology[J],2001,33(2):217-228.
115 Zheng Ming-xin, Wang Gong-xian, Wang Lan-sheng. Study on Application of Theory in Landslide's Prediction[J](In Chinese). Journal of Geological Hazards and Environment Preservation, 1998,9(2):18-26.
116 Zhong Bao-rong, Wang Ze-zhong, Li Long-yan et al. Fractal Analysis of Reservior pore Texture Accomplished By Computer[J](In Chinese with English Abstract). Oil and Gas Geology,1995,16(4):379-383.
2 Agterberg F P, Cheng Q, Brown.A, Good,D. Multifractal modeling of fractures in the Lac Du Boonet Batholith, Manitoba. Computer & Geosciences, 1996,22(5):497-507.
3 Agterberg F P. How nature works: the science of self-organized criticality, book review[J]. Computer & Geosciences, 1998,24: 205-207
4 Agterberg F P. Book Reviews. Computers and Geosciences,1999,25(1):93-99.
5 Agterberg F P.Multifractal Simulation of Geochemical Map Patterns. Proceedings of the international Symposium on Diversity of Mineralization and its Prediction and Assessment, Beijing and Wuhan, 2001,6-14.
6 Ahrens L H. Element distributions in igneous rocks-VII-Geoch.et Cosmoch. Acta, 28,p.271 -290,1964. (Cited in Yu C.,1987).
7 Ahrens L H. Distribution of the elements in our planet,1965. (Cited in Yu C.,1987).
8 Allegre C J, Lewin E. Scaling Laws and Geochemical Distribution[J]. Earth Planet Sci Letters,1995,132:1-13.
9 Andre Robert. Statistical Properties of Sediment Bed Profiles in Alluvial Channels. Math Geol[J].1988,20(3):205~225.
10 An Zhen-wen, Yang Cui-hua, Wang Lin-ying et al. Studies on the multifractals of time and space clustering of earthquakes[J](In Chinese with English Abstract). Letters of Geophysics, 2000,43(1):74~80.
11 Atmanspacher,H., Scheigraber,H., Wiedenmann, G.,1989. Determination of f(?) for a limited random point set. Physical Review, 1989,40(7):3954-3963.
12 Bak P. How nature works [M]. New York: Springer Verlag,1996, 1-212
13 Bao Zheng-yu, Li Jin-yao, Su Jiang-yu et al. Geochemical field analysis and oil/gas geochemical exploration[J](In Chinese with English Abstract). Petroleum exploration and development, 1997,18(1)：39-41.
14 Bao Zheng-yu, Li Fang-lin, Jia Xian-qiao. Methodology of Temporal-spatial structure of geochemical fields[J](In Chinese with English Abstract). Earth Sci- Journal of China University of Geosciences, 1999,24(3): 282-286.
15 Binder,K.. Monte Carlo Methods in Statistical Physics[M]. Germany: Berlin Helderberg,1986:1-411.
16 Blenkinsop T. Spatial distribution of gold deposit: two examples from the Zimbabwe Archaen Craton[A]. Barton C C. Fractal in Petroleum Geology and Earth Processes[M]. New York and London:Plonum Press,1994. 247-258.
17 Block A, Von-Bloh W, Klenke T,et al. Multifractal analysis of the microdistribution of elements in sedimentary structure using images from scanning electron microscopy and energy dispersive X ray spectrometry[J]. Journal of Geophysical Research, B,Solid Earth and Planets,1991,96(10):16223-16230.
Brian Klinkenberg, A review of methods used to determine the fractal dimension of linear
18 features, Math. Geology. 26 （1）23-46
19 B?lviken, B. , Stokke, P. R. , Feder, J. and J?ssang, T. , 1992. The fractal nature of geochemical landscapes. J. Geochem. Explor. , 43:91-109.
20 Carlson C A. Spatial distribution of ore deposit[J]. Geology,1991,19(2):107-117.
21 Chang Yinfo, Liu Xiangpei, Wu Yenchang. The Copper-Iron Belt of the Lower and Middle Reaches of the Changjiang River[M]. Beijing:Press of Geology, 1991,379p.
22 Chen Chun-zi, Lan Chao-li. Fractal Theory and its Application in Earth Science[J](In Chinese with English Abstract). Earth Sci Explor, 1998, 6:123-128
23 Chen Jian-guo, Wang Ren-duo, Chen Yong-qing. The extraction of concealed fractal information and the delineation of geochemical anomalies by fractal methods[J] (In Chinese with English Abstract). Earth Sci- Journal of China University of Geosciences, 1998,23(2):175-178.
24 Chen Jian-ping, Wang Qing, Xiao Shu-fang. Computer modeling of the fractal dimension of the rock mass fracture network[J] (In Chinese with English Abstract). Journal of Engineering Geology, 1995,3(3)79-859.
25 Cheng Q,Agterberg F P, Ballatyne S B.The separation of geochemical anomalied from background by fractal methods [J]. Explor Geochem,1994,51: 109-130
26 Cheng Q. The perimeter-area fractal model and its application to geology. Math. Geol. 1995a, 27:69-82
27 Cheng Q, Agterberg F P.Multifractall modeling and spatial point processes[J]. Math Geol,1995b,27:831-845.
28 Cheng Q, Agterberg F P.Multifractall modeling and spatial statistics[J]. Math Geol,1996a,28:1-16.
29 Cheng Q, Agterberg F P. Comparision between types of multifractal modeling. Math.Geol.1996b,28:1001-1015
30 Cheng Q , Agterberg F P, Bonham-Carter G F. A spatial analysis method for geochemical anomaly seperation [J]. Explor Geochem, 1996c,56:183-195
31 Cheng Q, Bonham-Carter G F, Hall G E M. Statistical study of trace elements in the soluble organic and amorphous Fe-Mn phases of surficial analysis [J]. Explor Geochem,1997a,59: 27-46
32 Cheng Q. Discrete multifractals[J]. Math Geol,1997b,29(2):245-266.
33 Cheng Q.Multifractal modelling and lacunarity analysis[J]. Math Geol,1997c,29:919-932.
34 Cheng Q, Bonham G F, Hall G E M, Bajc A. Statistical study of trace elements in the soluble organic and amorphous Fe-Ma phases of surficial sediments,Sudbury Basin;1,Multivariate and spatial analysis. J. Geochem. Explor,1997d,59(1):27-46
35 Cheng Q. Multifractality and spatial statistics [J]. Computer & Geosciences, 1999a,25(9): 949-962
36 Cheng Q.Gliding box method and multifractal modelling [J]. Computer & Geosciences, 1999b,25(9): 1073-1079.
37 Cheng Q. Spatial and scaling modeling for geochemical anomaly separation. J. Geochem. Explor. ,1999c, 65:175-194.
38 Cheng Q. Markov processes and discrete multifractals[J]. Math Geol,1999d,31(4):455-469.
Cheng Q,Xu Y,Eric Grunsky. Intergrated spatial and spectrum method for geochemical
39 anomaly separation. Natural Resources[J].2000a,9(1):43-51.
40 Cheng Q. Multifractal theory and Geochemical Element Distribution Patterns[J]. Earth Science-Journal of China University of Geosciences, 2000b,25(3): 311-318
41 Cheng Q. Selection of Multifractal Scaling Breaks and Separation of Geochemical and Geophysical Anomaly.Proceedings of the international Symposium on Diversity of Mineralization and its Prediction and Assessment,Beijing and Wuhan, 2001a,23-28.
42 Cheng Q. Multifractal and Geostatistic Methods for Characterizing Local Structure and Singularity Properties of Exploration Geochemical Anomalies[J](In Chinese). Earth Science-Journal of China University of Geosciences,2001b,26(2):161-166.
43 Cheng Q. Spatial Self-similarity and Geophysical and Geochemical Anomaly Decomposition[J](In Chinese). Advance of Geophysics,2001c,16(2): 8-17.
44 Cheng Xiao-jiu, Lu Jian-hang, Song Liang-ming. The meaning and program of fractal dimension of Pb and An[J](In Chinese with English Abstract). Geol Prospect, 1994, 30(5):32-35.
45 Chiles,J.P.. Fractal and Geostatistical Methods for Modeling of Fracture Network. Math Geol[J].1988,20(6):631~665.
46 De Wijs H J. Statistics of ore distribution: (1) frequency distribution of assay values[J]. Geol Mijnbouw,1951,13:365-375.
47 De Wijs H J. Statistics of ore distribution: (2) theory of binomial distribution applied to sampling and engineering problems[J]. Geol Mijnbouw,1953,15:12-24.
48 E.A.Yfantis,G.T.Flatman, E.J.Englund. Simulation of Geological Surfaces Using Fractals. Math Geol[J].1988,20(6):667~672.
49 Evertsz C J G,Mandelbrot B B. Multifratal measures (Appendix B) [A]. In:Peitgen H-O,Jurgens H, Saupe D,eds.Chaos and fractals[C]. New York:Springer Verlag, 1992, 922~953.
50 Feder J. Fractals[M]. New York:Plenum,1988, 1~283.
51 Fowler A. D., and Roach D. E., Dimensionality analysis of tine-series data: nonlinear methods, Computers & Geosciences, 1993, 19 （1）: 41-52
52 Halsey T C,Jensen M H,Kadanoff L P, et al. Fractal measures and their singularities, the characterization of strange sets[J]. Phys Rev,1986,33:1141-1151.
53 Hardy,H.H. The Fractal Character of Photoes of Slabbed Cores. Math Geol[J].1992,24(1):76~97.
54 Hu Yuan-lai, Jiang Ruo-wei, Hu Chun-tao et al. Algorithm of fuzzy fractional dimension and its application in reservoir[J](In Chinese with English Abstract). Geophysical prospecting for Petroleum, 1995,30(5):645-653.
55 Ioannis K.Koukouvelas, Marios Asimakopoulos, Theodoros T Doutsos. Fractal characteristics of active normal faults: an example of the eastern Gulf of Corinth, Greece. Tectonophysics[J],1999,308, 263-274.
56 Jaggi,S., Dale A. Quattrochi, and Nina Siu-Ngan Lam, Implementation and operation of three fractal measurement algorithms for analysis of remote-sensing data, Computers & Geosciences, 1993, 19（6）: 745-767
57 James R.Carr, William B.Benzer. On the Practice of Estimation Fractal Dimension. Math Geol[J].1988,20(7):945~958.
58 Jiang Jia-yu, Duan Yu-shun. Application of fractal theory to hydrocarbon detection[J](In Chinese with English Abstract).Geophysical prospecting for Petroleum, 1995，34(1)47-51.
59 Jin You-yu, Meng Xian-guo. Quantitative analysis of Time and Space Series in Geology. Wuhan: Press of China University of Geosciences, 1992,1-169.
60 Joost H.J.Van Opheusden. The origin of an increasing or decreasing multifractal spectrum. Physica[J]. 1998,22:10-22.
61 Journel A G, Huijbregts C. Mining Geostatistics, Academic Press,1978,1-690.
62 Juan M.Garcia-Ruiz, Fermin Otalora. Fractal Trees and Horton's Laws. Math Geol[J].1992,24(1):61~71.
63 Kitaev,N.A.. Multidimensional Analysis of Geochemical Fields. Math Geol[J].1991,23(1): 16-32.
64 Laherrere, J. Distributions de type "fractal parabolique" dans la nature. Comptes Rendus de l'Academie des Sciences, Serie II. Sciences de la Terre et des Planetes(In French, Abstract in English), 1996, 322(7): 535-541.
65 Li C, Ma T, Xu Y. A new view: chaotic dynamical nature in mineralization. In:30th international geological congress abstracts,1996a,3 of 3.
66 Li C,Wang G. Isotopic geochemistry of Chinese fluorite depostis. International Geol. Rev.,1996b,38: 1054-1067
67 Li Chang-jiang, Jiang Xu-liang, Xu You-lang,et al. Fractal Study on Mesozoic hydrothermal deposits in Zhejiang Province[J](In Chinese with English Abstract). Sci Geol Sinica, 1996c,22(1): 264-273.
68 Li Chang-jiang, Ma Shi-hua, Zhu Xing-sheng et al. Fractals, Chaos and ANN in Mining Exploration[M](In Chinese). Beijing: Geological Press, 1999,1-140.
69 M.Ann Piech, Kenneth R.Piech. Fingerprints and Fractal Terrain. 1990,22(4):458-485.
70 Ma Tu-hua, Zhu Xing-sheng, Li Chang-jiang. Spatial distribution of fluorite deposits in Zhejiang Province[J](In Chinese with English Abstract). Mineral deposits, 2000,19(3):281-288.
71 Mao Ling-bo, Gui Zhi-xian, Zhu Guang-sheng. Application of fractal geometry to prediction of reservior parameters between two wells[J](In Chinese with English Abstract). Journal of Jianghan Petroleum Institute,1995,17(2): 43-47.
72 Mandelbrot B B. Stochastic models for the Earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. Proceedings of the National Academy of Sciences(USA):72,1975,3825-3828.
73 Mandelbrot B B. The Fractal Geometry of Nature. San Francisco: Freeman, 1983
74 Mandelbrot B B., New "anomalous" multiplicative multifractals: left-sided f(?) and the modeling of DLA, Physica A168,1990:95-111.
75 Mandelbrot B B. Fractals: Form, Chance and Dimension. San Francisco: Freeman, 1997
76 Mandelbrot B B. Multifractals and 1/f Noise. New York: Springer-Verlag, 1999
77 Meng Xian-guo, Zhao Peng-da. Fractal Structures of Geological data[J](In Chinese with English Abstract). Earth Sci-Journal of China University of Geosciences, 1991a, 16(2):207-211.
Meng Xian-guo. R/S Analysis and Fractal Processing of Geochemical Data[J](In Chinese with English Abstract). Earth Sci-Journal of China University of Geosciences, 1991b,
78 16(3):281-286.
79 Nicholas M.S.Rock. Summary Statistics in Geochemistry: A study of the Performance of Robust Estimates. Math Geol[J].1988,20(3):243~275.
80 Paladin, G., Vulpiani,A.. Anomalous scaling laws in multifractal objects[J]. Physical Reports, 1987,156(4):147-225.
81 Paredes, C., Elorza, F.J.. Fractal and Multifractal analysis of fractured geological media:surface-subsurface correlation. Computer & Geosciences, 1999,25(9): 1081-1096.
82 Park.H,Lee.S. Effect of multifractalities of catalyst surface: a Monte Carlo study of catalytic CO oxidation. Surface Science[J].1998,411:1-9.
83 Pei Tao. Spatial Structure Analysis of Geochemical Field of Oil and Gas and its application to Exploration in North Tarim Basin. Ph.D Dissertation of China University of Geosciences, 1998,1-89.
84 Peitgen H O, Jurgens H, Saupe D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.
85 Pillet R. Mise en evidence du comportement multifractal de la distribution frequence-magnitude d'un echantillon sismique( loi de Gutenberg-Richter). Comptes Rendus de l'Academie des Sciences, Serie II. Sciences de la Terre et des Planetes, 1997, 324(7): 805-810.
86 Prokoph, A.. Fractal, multifractal and scaling window correlation dimension analysis of sedimentary time series. Computer & Geosciences, 1999,25(9): 1009-1021.
87 Qin Chang-xing, Zhai Yu-sheng. Some self-similar phenomena and their meaning in economic geology[J](In Chinese with English Abstract). Mineral deposits,1992,11(3): 259-266.
88 Razumovsky,N.K. On the r?le of the logarithmically normal law of frequency distribution in petrology & geochemistry-Comp.Rend.(Doklady) de l'Acad, de Sciences de l'URSS,33(1),p.48-49,1941 (Cited in Yu C.,1987)
89 Richardson, W.A. The frequency distribution of igneous rocks, part II: the law of distribution in relation to petrogenic theoried---Mineral.Mag.,1923,20,p.1-4.(Cited in Yu C.,1987)
90 Sanderson D J, Roberts S A. Fractal relationship between vein thickness and gold grade in drill core from La Codosera, Spain[J]. Economic Geology,1994,89:168-173.
91 Shen Bu-ming, Shen Yuan-chao. Fractal dimension of some gold deposit in Xinjiang and its geological application[J](In Chinese with English Abstract). Sci China(B), 1993,23(3)297-302.
92 Shen Wei. Application of Fractal and Chaos in Geology[J](In Chinese with English Abstract). Geological Science and Technology Information, 1996,15(2):103-109.
93 Shen Wei. A new method of estimating fractal dimension in fractal model of minerogenetic prediction[J](In Chinese with English Abstract). Journal of Changchun University of Earth Sci, 1997,27(1):86-91.
94 Shi Jun-fa, Wang Chun-ning. Fractal Analysis of Gold Deposits in China: implication for giant deposit exploration[J](In Chinese with English Abstract). Geological Science and Technology Information, 1998,23(6):616-619.
95 Shi Yu-tian, Pan Bao-zhi. The application of fractal: The quantitative description of fractures[J](In Chinese with English Abstract). Geophysical and Geochemical Exploration, 2000,24(6):426-430.
96 Shi Jun-fa. Element Geochemical Distribution from Zhuji Area, Zhejiang Province and Scaling laws[J](In Chinese with English Abstract). Earth Sci- Journal of China University of Geosciences,2001,26(2): 167-171
97 Sim B L, Agterberg F P, Beaudry C. Determining the cutoff between background and relative base metal smeller contamination levels using multifractal. Comput. Geosci. 1999,25(9):1023-1041
98 Standley H E, Meakin P. Multifractal phenomena in physics and chemistry[J]. Nature,1998,335(6189): 405~409.
99 T.Ju.Spivak, V.I.Grabenko. Multidimensional Statistical Methods Used for Prediction of Hydrocarbon Accumulations in Turkmenistan. Math Geol[J].1989,21(6):625~629.
100 Turcotte D L. Fractals and Chaos in Geology and Geophysics[M]. Cambridge: Cambridge University Press, 1997, second edition.
101 Vistelius,A.B. The skew frequency distribution and the fundamental law of the geochemical processes--- Jour.Geol.,68,No.1,1960. (Cited in Yu C.,1987)
102 Xie Xue-jin, Yin B. Geochemical patterns from local to global[J]. Explor Geochem,1993,47: 109-129
103 Xu Shun-shan, Wang Gan-guo, Jiang Wan, Zhang Da, Cehn Bei-lin, Xie Yan-xia. Studying on Fractals of Zijinshan Copper-Gold Deposit[J](In Chinese). Geology and Exploration,1999,35(5):50-52.
104 Y.Gueguen, J.Dienes. Transport Properties of Rocks from Statistics and Percolation. Math Geol[J].1989,21(1): 1-13.
105 Yi Chui-xiang. Unlinear Science and Its Application in Earth Science[M](In Chinese). Beijing: Weather Press, 1995,4:226-258.
106 Yu Chong-wen, Luo Ting-chuan, Bao Zhengyu,et al. Regional geochemistry of the Nanling District. Beijing: Geology Press, 1987,1-543(in Chinese).
107 Yu Chong-wen, Chen Kuang, Bao Zheng-yu, et al.. Ore-forming dynamics[M](In Chinese). Beijing: Geology Press,1998.
108 Yu Chong-wen. Large ore deposits and metallogenic zones at the edge of chaos[J](In Chinese with English Abstract). Earth Sci Frontier, 1999, 3:85-102
109 Yue You-xi, Wang Cai-jing. The applications of multifractals in Fracture Research[J](In Chinese with English Abstract). Geophysical prospecting for Petroleum, 1998,37(1):109-114.
110 Zhai Yu-sheng et al. Ore-forming laws of iron-copper(gold) deposits in middle-lower Yangtze River[M].Beijing: Press of Geology, 1992:p235.
111 Zhang Jian-long. Study on Fractal Dimension Characteristics of Au Grade of Gala Gold deposit and its periphery[J](In Chinese with English Abstract).Computing Techniques for Geophysical and Geochemical Exploration,1196,18(1):78-81.
112 Zhang Jun, Zhou Qiaowei. Application of Fractal Method to Analysis of Temporal and Spatial Structure of gold mineralization[J](IN Chinese). Geoscience- Journal of Graduate school, China University of Geosciences, 2000,14(1):56-60.
113 Zhang Zhe-ru, Mao Hua-hai. Fractal theory and ore-forming[J](In Chinese). Earth Sci Frontiers, 2000, 7(1):195~204.
114 Zhang Zheru, Mao Huahai, Qiuming Cheng. Fractal Geometry of Element Distribution on Mineral Surfaces, Mathematical geology[J],2001,33(2):217-228.
115 Zheng Ming-xin, Wang Gong-xian, Wang Lan-sheng. Study on Application of Theory in Landslide's Prediction[J](In Chinese). Journal of Geological Hazards and Environment Preservation, 1998,9(2):18-26.
116 Zhong Bao-rong, Wang Ze-zhong, Li Long-yan et al. Fractal Analysis of Reservior pore Texture Accomplished By Computer[J](In Chinese with English Abstract). Oil and Gas Geology,1995,16(4):379-383.