地球化学图纹理的多重分形模拟

Frederik P.Agterberg

地球化学图纹理的多重分形模拟

Frederik P.Agterberg

加拿大地质调查所, 渥太华 K1A 0E8, 加拿大

基金项目: 加拿大自然科学基金(NSERCProject)

详细信息

作者简介:
Frederik P.Agterberg, 男, 64岁, 博士, 加拿大国家地调所高级研究员, 渥太华大学兼职教授, 博士生导师, 国际数学地质学会副主席, 国际定量地层学会主席, 荷兰皇家科学院外籍院士, 主要从事数学地质、矿产资源评价和定量地层学研究

中图分类号: P628⁺.1
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出版历程
- 收稿日期: 2001-02-18
- 刊出日期: 2001-04-25

MULTIFRACTAL SIMULATION OF GEOCHEMICAL MAP PATTERNS

Frederik P. Agterberg

Geological Survey of Canada, 601 Booth Street, Ottawa, K1A 0E8, Canada

摘要

摘要: 利用一个简单的基于De Wijs模型的多重分形模型, 可以模拟元素富集值的各种地球化学纹理.每种纹理在平均值上是自相似的, 因为将乘积阶次模型(multiplicative cascade model) 应用到任何子区均能得出类似的纹理样式.在其他的试验中, 通过叠加一个二维趋势纹理(2-dimensional trend pattern) 以及把它与一个常值富集模型混合, 原始的自相似纹理就产生畸变.本文将要研究这些畸变是如何改变用三步矩(3-step method of moments) 所估测的多重分形谱(multifractal spectrum).推导出了满足De Wijs模型纹理的离散和连续频率分布模型.这些模拟纹理满足离散频率分布模型, 当乘积阶次模型(multipicative cascade model) 无限细分时, 假设离散频率分布模型的上界是一连续频率分布, 这个离散分布就在形式上逼近该连续频率分布的上边界.这一极限分布在中心是对数正态的, 但有两个巴利多(Pareto) 分布的尾.这种方法在矿产和油气评价中有重要的潜在意义.
- 分形 /
- 多重分形 /
- De Wijs模型 /
- 地球化学 /
- 图形纹理 /
- 计算机模拟
Abstract: Using a simple multifractal model based on the model De Wijs, various geochemical map patterns for element concentration values are being simulated. Each pattern is self-similar on the average in that a similar pattern can be derived by application of the multiplicative cascade model used to any small subarea on the pattern. In other experiments, the original, self-similar pattern is distorted by superimposing a 2-dimensional trend pattern and by mixing it with a constant concentration value model. It is investigated how such distortions change the multifractal spectrum estimated by means of the 3-step method of moments. Discrete and continuous frequency distribution models are derived for patterns that satisfy the model of De Wijs. These simulated patterns satisfy a discrete frequency distribution model that as upper bound has a continuous frequency distribution to which it approaches in form when the subdivisions of the multiplicative cascade model are repeated indefinitely. This limiting distribution is lognormal in the center and has Pareto tails. Potentially, this approach has important implications in mineral and oil resource evaluation.
- Fractal /
- multifractal /
- model of De Wijs /
- geochemistry /
- map pattern /
- computer simulation.

HTML全文

图 1 元素富集品位(整体平均值等于1) 两种模拟图纹的三维图

a.原始模型的128×128个数据矩阵; b.根据幂次律函数向原始模型数据矩阵叠加了趋势的矩阵, 幂次律函数是通过一因子10使得矩阵最大值出现在原始数据最小值的相反方向处; c.b中矩阵的一部分. (这些是用Mathematica 4软件获得的彩色图的黑白版本)

Fig. 1. Three-dimensional plots of two simulated map patterns for element concentration values (overall mean value is equal to 1) obtained by means of a stochastic version of the model of De Wijs; largest values truncated at upper end

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图 2 图 1a中128×128个品位值的图样作的集群分割函数与分割单元边长度的双对数(底为2) 映射图

最小单元的边log₂ε=1.仅仅当q为整数时, 计算结果显示出来了.直线的斜率在图 3a中给出

Fig. 2. Log-log plot (base 2) of mass-partition function versus length of cell side for pattern of 128×128 concentration values of Fig.1a

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图 3 对图 2继续运用计算矩的方法

a.主体指数τ (q) 和q之间的关系; b.奇异指数α (q) 和q之间的关系; c.多重分形谱值f (α) 与奇异指数α之间的关系

Fig. 3. Method of moments continued from

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图 4 d取不同值时直方图法在图 1a纹理中的应用

a.d=0.4, n=14;b.d=0.4, n=30

Fig. 4. Fig.4 Histogram method applied to pattern of Fig.1a with different values d, n

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图 5 与3个实验半方差图相对比的理论多重分形半方差图

它来自纹理的128行数据, 这些纹理与图 1a相似.实验半方差图与连续曲线之间的偏差相对较大, 但可能没有很大的差别

Fig. 5. Theoretical form of multifractal semivariogram in comparison with three experimental semivariograms

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图 6 把所有的品位加上(图 1a) 一个很小的值(0.01) 得到的结果

与图 3c相比, 仅在图右区有差异

Fig. 6. A small value (0.01) was added to all concentration values (cf. Fig. 1a)

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图 7 根据图 1b中纹理的128×128个品位, 得到的集群分割函数与分割单元边长度在双对数坐标图(底为2) 的映射关系

前3个点的连线线段仅仅用于矩法.符号的说明在图 2中

Fig. 7. Log-log plot (base 2) of mass-partition function versus length of cell side for pattern of 128×128 concentration values of Fig.1b

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图 8 从图 1b纹理开始用矩法计算得到的多元谱系

与图 3c相比, 在左区上有所区别

Fig. 8. Multivariate spectrum obtained by means of the method of moments starting from pattern shown in Fig.1b

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图 9 a.直方图法在d取0.6、n取20时在品位数据中的应用; b.与a中两个多元谱相关的频率分布曲线; 极限形式的频率差不多接近对数二项频率, 但不同的是0在中心和端点处; c.在Q-Q坐标图上, 上界频率分布的对数正态分布

Fig. 9. a: Histogram method illustrated in Fig.4 applied to concentration values with d=0.6 and n=20; b: Frequency distribution curves corresponding to the two multivariate spectra shown in Fig.9a; frequences of limiting form slightly exceed logbinomial frequencies but difference is zero in the center and at the endpoints; c: Lognormal Q-Q plot of upper bound frequency distribution shown in Fig.9b

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