基于小波变换的二维地球化学景观多重分形建模
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摘要
国内外大量研究结果表明,尺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。多重分形理论所提供的奇异性、广义自相似性、多重分形谱等概念和相关的模型,不仅能够客观的描述成矿系统、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、地球化学和遥感异常)的有效模型和实用方法。
作为傅里叶分析的延续,小波分析既保留了数据空间位置或时间上的信息,同时也包括了频域中的信息,因此对于处理这些由各种地质过程和地质事件形成的非平稳、奇异的二维地学信息时,小波变换有它的内在优势。此外,小波分析与多重分形都是描述尺度特征的工具,多重分形是从整体上描述了关于测度的尺度所具有的统计规律,而小波变换是通过多分辨率的特点由粗到细的观察某一点上的尺度信息,相当于对数据进行频率分割,小波分析和多重分形在尺度分析方面就一点和全局达到了统一。
围绕近几年多重分形理论在固体地球科学中的发展和应用,结合另外一种尺度分析手段——小波分析理论,本文开展的主要研究工作和获得的主要结论包括以下几个方面:
(1)首先总结了近几年国内外基于小波系数的高阶矩的多重分形分析方法,主要包括小波系数,WTMM,WLs等方法,并通过经典矿物分割模型de Wijs模型与对应的盒子测度法相对比,结果表明,以小波系数为基础的分析在一些前提下是比较有效的,总体表现没有传统的盒子测度法表现稳定。
(2)其次引入了一种新的尺度分析方法——累积量分析,一种在湍流域中被证实的可以避免少数误差扰动,增强整体尺度性分析稳定的估计方法,通过基于累积量的低阶矩统计量与差分导数对比,参照高阶矩估计,证明对于一些由简单多重分形构成的数据,基于累积量估计还是比较有效的。
(3)作为C-A,S-A方法等基于幂律关系的滤波模型的延续,本文基于小波能谱与尺度之间的幂律关系,构建了W-A模型(小波能谱和面积)滤波器,并通过云南个旧地区水系沉积物Sn,Pb和Cu元素的异常滤波的分析验证,证明该方法是有效的。
Scale-invariance has been viewed as a fundamental property of the nature for describing the geological characteristics and patterns caused from various geological processes and events. It has been demonstrated that the concepts and models relevant to multifractal theory are useful not only for characterizing the fundamental properties of non-linearity of the mineralization processes, the singular distribution of mineral deposits and ore element concentrations in mineral districts, but also for singularity analysis and anomaly delineation.
As an extension to Fourier analysis, it has been demonstrated that singular or non-stationary signals such as geochemical or geological patterns under wavelet analysis shows more efficient than Fourier analysis based on the retaining both space or time position and frequency components. Wavelet analysis and multifractals are different descriptions concerned with scaling analysis, one focuses on the general similarity and the other focuses on the scale information of one point.
This dissertation mainly devoted to some contents as follows:
Firstly, various multifractal formalisms based on wavelet transforms are summarized including wavelet coefficients, WTMM, and WLs methods. It is demonstrated by the case study of de Wijs model and stream sediment elements concentrations that box-counting is superior to the WLs and wavelet coefficient.
Secondly, a new scaling analysis method called cumulant analysis is proposed, which has been demonstrated to be an efficient method for general scaling analysis in fully developed turbulence fields and efficiently overcome the influence of some noises. Meanwhile, the lower orders moment estimation derived from cumulant analysis, compared with direct polynomial expansion using difference derivatives, shows good behavior for simple multifractals.
Finally, a new filter is proposed based on the power laws existed between wavelet coefficients and scales extended from C-A, S-A models. It has been showed us that by the case of stream sediment elements concentrations that W-A model is an efficient model for anomalies identification and can effectively avoid the influence caused from pseudo-high frequency components.
作为傅里叶分析的延续,小波分析既保留了数据空间位置或时间上的信息,同时也包括了频域中的信息,因此对于处理这些由各种地质过程和地质事件形成的非平稳、奇异的二维地学信息时,小波变换有它的内在优势。此外,小波分析与多重分形都是描述尺度特征的工具,多重分形是从整体上描述了关于测度的尺度所具有的统计规律,而小波变换是通过多分辨率的特点由粗到细的观察某一点上的尺度信息,相当于对数据进行频率分割,小波分析和多重分形在尺度分析方面就一点和全局达到了统一。
围绕近几年多重分形理论在固体地球科学中的发展和应用,结合另外一种尺度分析手段——小波分析理论,本文开展的主要研究工作和获得的主要结论包括以下几个方面:
(1)首先总结了近几年国内外基于小波系数的高阶矩的多重分形分析方法,主要包括小波系数,WTMM,WLs等方法,并通过经典矿物分割模型de Wijs模型与对应的盒子测度法相对比,结果表明,以小波系数为基础的分析在一些前提下是比较有效的,总体表现没有传统的盒子测度法表现稳定。
(2)其次引入了一种新的尺度分析方法——累积量分析,一种在湍流域中被证实的可以避免少数误差扰动,增强整体尺度性分析稳定的估计方法,通过基于累积量的低阶矩统计量与差分导数对比,参照高阶矩估计,证明对于一些由简单多重分形构成的数据,基于累积量估计还是比较有效的。
(3)作为C-A,S-A方法等基于幂律关系的滤波模型的延续,本文基于小波能谱与尺度之间的幂律关系,构建了W-A模型(小波能谱和面积)滤波器,并通过云南个旧地区水系沉积物Sn,Pb和Cu元素的异常滤波的分析验证,证明该方法是有效的。
Scale-invariance has been viewed as a fundamental property of the nature for describing the geological characteristics and patterns caused from various geological processes and events. It has been demonstrated that the concepts and models relevant to multifractal theory are useful not only for characterizing the fundamental properties of non-linearity of the mineralization processes, the singular distribution of mineral deposits and ore element concentrations in mineral districts, but also for singularity analysis and anomaly delineation.
As an extension to Fourier analysis, it has been demonstrated that singular or non-stationary signals such as geochemical or geological patterns under wavelet analysis shows more efficient than Fourier analysis based on the retaining both space or time position and frequency components. Wavelet analysis and multifractals are different descriptions concerned with scaling analysis, one focuses on the general similarity and the other focuses on the scale information of one point.
This dissertation mainly devoted to some contents as follows:
Firstly, various multifractal formalisms based on wavelet transforms are summarized including wavelet coefficients, WTMM, and WLs methods. It is demonstrated by the case study of de Wijs model and stream sediment elements concentrations that box-counting is superior to the WLs and wavelet coefficient.
Secondly, a new scaling analysis method called cumulant analysis is proposed, which has been demonstrated to be an efficient method for general scaling analysis in fully developed turbulence fields and efficiently overcome the influence of some noises. Meanwhile, the lower orders moment estimation derived from cumulant analysis, compared with direct polynomial expansion using difference derivatives, shows good behavior for simple multifractals.
Finally, a new filter is proposed based on the power laws existed between wavelet coefficients and scales extended from C-A, S-A models. It has been showed us that by the case of stream sediment elements concentrations that W-A model is an efficient model for anomalies identification and can effectively avoid the influence caused from pseudo-high frequency components.
引文
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