多重分形局部奇异性分析方法及其在矿产资源信息提取中的应用.

多重分形局部奇异性分析方法及其在矿产资源信息提取中的应用非线性理论、复杂性理论、空间信息技术与矿床学、矿产资源勘查与评价研究的结合是国际新兴研究领域。

自从多重分形概念被引入到分形理论中以来各种多重分形模型被纷纷提出并广泛应用于自然科学和社会科学各个领域中。

在地学领域，许多地质过程具有尺度独立性特征，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)的有工“效模型和实用方法。

基于多重分形理论的局部奇异性分析是近年来迅速发展的前缘研究方向，不仅在矿产资源信息提取中具有重要的应用，而且在其它许多应用领域也有良好的应用前景。

奇异性问题的研究在科学技术的诸多领域都有所涉及，并且有着各自特定的含义。

为了采用奇异性的基本原理研究一般的奇异性事件或过程，本文中的奇异性定义为：将在很小的时间—空间范围具有巨大能量释放或巨量物质形成的现象称之为具有奇异性。

成矿作用可以认为是一种特殊的奇异事件，它引起成矿物质的巨量堆积和元素高度富集。

非线性理论和复杂性理论的最新研究结果表明，奇异性通常具有尺度不变性特征，奇异性现象往往是分形的或多重分形的。

在局部奇异性分析中，指数α被称为局部奇异性指数，在不同的位置上幂律关系可以具有不同的α值，α值表征了模式的密度分布随度量尺度的变化性。

在地球化学数据中，正奇异的地段(α＜2))对应于由于矿化作用或其他局部地质过程而引起的元素富集地段；负奇异的地段(α＞2)对应于元素相对亏损的地区；无奇异(α≈2)的地区对应于背景场，背景场在地球化学图中所占范围较大。

α值越小，表明正奇异性越强烈。

局部奇异性分析方法可以直接对局部异常进行空间(时间)定位，并在低缓异常识别中效果显著，在矿床空间丛聚分布度量、遥感信息处理中也已取得较好的效果，该分析方法的引进还产生了奇异地质统计学插值方法。

局部奇异性指数是局部奇异性分析计算中的关键指标，目前的计算方法还存在一些不足，这主要表现在：(1)如果局部奇异性指数α值完全地刻画了奇异性的强度，那么由局部系数所构成的c集就应成为一个非奇异性的成份，然而在常规的局部奇异性分析中，对于局部奇异性指数的计算没有考虑局部系数的作用，这影响了α值计算的精度水平。

(2)地学数据往往具有各向异性，对于各向异性局部奇异性指数的计算，虽有相关方法提出，但都是相对简单。

有的计算考虑了方位各向异性，但没考虑空间位置的不同而具有不同的各向异性；有的考虑了空间位置的差异但不同尺度的各向异性方位和压缩比都是固定的。

对于不同空间位置及不同尺度上的各向异性参数的获取方法尚未有深入研究。

针对上述问题，本文在追踪论文相关的研究现状后，对基于多重分形理论的局部奇异性原理进行了较全面的讨论，指出了奇异性的局部统计自相似性、各向异性、多样性三个基本特征。

作者在剖析局部奇异性分析基本方法(LSA)及其算法的基础上，提出了两个改善模型：局部奇异性分析迭代方法(I-LSA)和广义局部奇异性分析方法(GLSA)。

同时在基于窗口方法和基于等值线方法的局部奇异性算法基础上设计并实现了推广算法。

最后，将局部奇异性分析方法应用于个旧地区水系沉积物Cu地球化学异常信息的提取。

本文主要的研究工作和获得的主要结论有以下几个方面：(1)局部奇异性分析迭代方法研究基于滑动平移固定窗口局部奇异性的计算结果可通过迭代方式进行优化。

作者对迭代方法的推导和算法进行了详细讨论，从方法上说明了常规非迭代方法和迭代方法之间的联系和区别。

常规非迭代方法可认为是迭代方法的一种特殊情形。

de wijs数据、二维各向同性模拟数据WX数据集的处理结果表明，局部奇异性分析迭代方法可以提高局部奇异性指数的计算精度。

(2)广义局部奇异性分析研究广义奇异性分析应用空间U统计量法获取了局部各向异性的窗口系列并进行局部奇异性指数的计算。

本文探讨了三个方面的计算技术：(a)各向异性动态度量模型与椭圆覆盖结点矩阵模板库技术：(b)不同尺度局部最优U值与各向异性参数获取方法；(c)各向异性空间覆盖盒子系列构建方法与局部奇异性指数计算。

它们能够较好地满足将空间U统计量法引入到广义局部奇异性度量中的计算需要。

二维各向异性模拟数据WY数据集五种不同计算条件的广义计算结果表明，应用空间U统计量法进行广义奇异性分析是可行的、有效的。

(3)局部奇异性分析的推广算法研究推广算法综合了窗口方法和等值线法的优点，在混合计算、空间覆盖盒子系列构建、空间加权、边缘处理等多个方面进行了功能增强。

在多种方式的空间覆盖盒子构造方面，作者对空间覆盖盒子系列参数定义文件进行了设计，该方法不仅建立广义奇异性分析中空间U统计量结果与局部奇异性计算之间的联结关系，而且它还提供了一种用户可干预的一般化的窗口构建方式。

推广算法不仅考虑了局部奇异性迭代方法和广义局部奇异性分析的计算需要，而且还考虑了具有潜在应用价值的其它需求，具有普适性。

推广算法已基于栅格数据模型并由MATLAB编程实现，同时还提供了MATLAB和ArcGIS、MAPGIS等不同软件平台之间的数据交换功能。

(4)个旧地区水系沉积物铜地球化学异常信息提取研究X-Y-W散点渲染图叠加矿产作图和t(≤α)曲线图提供了实用的数据探查和异常信息提取制图技术。

在对局部奇异性分析三种计算方法LSA、I-LSA、GLSA的对比中，我们认为在采用固定的滑动窗口进行局部奇异性计算时，I-LSA方法略优于LSA方法；在采用空间U统计量方法来获取各向异性的空间覆盖盒子，那么GLSA方法则比前两者更优越。

对个旧地区Cu水系沉积物含量数据的局部奇异性分析研究结果表明，利用局部奇异性指数圈定异常范围是非常有效的一种多重分形方法，个旧地区具有很好的铜矿资源找矿前景。

(5)其它相关研究除局部奇异性指数外，我们还获取了一些相关的有益结果：(a)局部奇异性迭代方法所获取的局部系数满足非奇异性的优良性质。

(b)空间U统计量的局部最优U值U~*、局部最优椭圆等效半径r_0、局部最优椭圆压缩比β_0、局部最优椭圆主轴方位角θ_0参数提供了数据场的重要信息：U~*值具有“衬度”意义，可以用来分离异常区域和背景区域；r_0值反映了数据场分布的局部连通性；β_0值反映了数据场各向异性的局部强烈程度；θ_0值反映了数据场各向异性的局部最优方位。

这些结果不仅丰富了局部奇异性的信息内涵，而且可促进对空间奇异性插值技术、各向异性多重分形模型的性质的研究。

由此可见，局部奇异性分析方法具有良好的应用前景。

本文创新点主要体现在：(1)针对提高局部奇异性指数计算精度问题提出了迭代方法该方法不仅对于提高局部奇异性分析方法的应用水平有直接意义，而且对讨论多重分形模型的性质、奇异性插值问题等均有参考意义。

(2)利用空间U统计量法实现了广义局部奇异性分析该项工作首次实现了在空间域中度量各向异性局部奇异性指数的计算技术，对扩大局部奇异性分析方法的应用和丰富局部奇异性分析方法的内涵具有创新意义Interdisciplinary research involving non-linear theory, complextheory and spatial information technology, economic geology and mineral resource assessment and exploration has become a growing new field in the earth science. Since the concept of multifractal was introduced originally by Mandelbrot, various multifractal models have been developed and some of these have been widely used in various fields of science for characterizing measures with scaling properties. 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. The local singularity analysis based on multifractal theory has been a rapid developing research orientation of the non-linear theory recently.Singularity can be defined and characterized in different ways; for example, it can be explained in a purely mathematical context with mathematical notation, or from a physical point of view emphasizing the physical processes. From a geological application point of view, this paper defines singularity as a special phenomenon with anomalous energy release or material accumulation occurring within narrow spatial-temporal intervals. Taking hydrothermal mineralization as an example, this event usually occurs within a relatively short period of geological time and causes anomalous enrichment of elements in relatively small orebodies. From a modern non-linear theory point of view, within a multifractal context, the singularity can be associated with the distribution of self-similar fields. The singularity phenomenon canbe described by the power-law model.The exponent a is termed thelocal singularity exponent in the local singularity analysis. Withina given range, a given power-law relationship holds true. a-value can quantify the local scaling invariance property charactering the concave/convex properties of the neighborhood values. For example,the local geochemical anomalies caused by mineralization, can be separated from regional background. Areas with positive singularity (a＜2) may correspond to areas where the element concentration is elevated due to mineralization or other local geological processes, whereas areas with negative singularity (a＞2) may reflect areas with depleted element concentration. Areas with zero singularity (a≈2), which dominates the geochemical map, represent background concentration values. The smaller a-value, the more singular the measure in a small vicinity around the location and the "stronger"the positive singularity. The local singularity analysis is capableof the spatial (temporal) localization for the anomaly. This methodprovides a simple and direct strategy for detecting andcharacterizing singularities and has been successfully applied in many fields, such as anomaly enhancement and identification of geochemical data, and texture analysis of remote-sensing images. What\'s more, a new direction was opened for studying how to improve the interpolation results by combining spatial association with singularity.It is essential to note that the key to the application of singularity analysis is the estimation of the local singularity exponents. However, the current method has some shortcomings to be solved.(1) The local coefficient c, as well as the a-value, plays a central role in local singularity analysis. In theory, c-set should be a non-singular set. But the basic model does not take it into consideration, which lower the precision of the a-value.(2) Anisotropy is not only a common characteristic of geochemical and geophysical fields but also carries valuable information for image processing and pattern recognition. The calculations for the anisotropic local singularity exponent by the current methods are too simple. In the practical application, the anisotropic parameters should be different with the location and the scale.Considering the scientific problems above, the author gives a general discussion on the local singularity principle and points out three basic properties of the singularity, which are the local statistical similarity, anisotropy and diversity. Then, the author introduces the basic model and algorithm of the local singularity analysis (LSA) and provides two improved model: the iterative approach to local singularity analysis (I-LSA) and the generalized local singularity analysis (GLSA). The extended algorithm is designed and implemented as well which preserves the advantages of the windows-based algorithm and the contour-based algorithm for calculating local singularity. At last, the case study was used to demonstrate the application of these new approaches to the anomaly identification of Cu concentration values from the stream sediment samples in Gejiu area, Yunnan province, China.The main research contents and conclusions in the dissertation are follows:(1) The iterative approach to local singularity analysis (I-LSA)An improved model of local singularity analysis, using an iterative approach, is proposed, which directs us towards investigating the regularity of the local coefficients to estimate the optimum local singularity exponents. It is demonstrated by the case study of the de Wijs\'s zinc data and an isotropic simulation data (2D) that I-LSA is superior to LSA. The latter can be considered as a special case of the former.(2) The generalized local singularity analysis (GLSA) A spatial and scaling approach, called spatial U statistic method, is introduced to look into the local anisotropy association and to characterize the singularity properties usingoptimal shapes and sizes.The author discuss three key techniques: (a) the dynamic model measuring anisotropy of the field and the storage technique of the matrix template library for the nodes covered by the ellipse; (b) The acquisition method of the local optimum U value with different scale and the anisotropic parameters; (c) the constructionof a set of the anisotropic windows to calculate the localsingularity. These techniques ensure the evolution from LSA to GLSAby means of the spatial U-statistic method. It is demonstrated by the case study of an anisotropic simulation data (2D) that GLSA combing the local singularity analysis with the spatial U-statistic method is feasible and effective.(3) The extended algorithm of the local singularity analysisThe extended algorithm not only preserves the advantages of the windows-based algorithm and the contour-based algorithm, but also extends more functions, such as the mixed calculation, the weighted spatial locations, edge processing. The extended algorithm supports the calculations of I-LSA and GLSA, andit also takes more potential needs into account.The extendedalgorithm based on raster model has been implemented by MATLAB andthe program has the function of data exchange with different software, such ArcGIS, MAPGIS.(4) The case study of Cu concentration values in Gejiu areaThe mineral deposits-overlaied X-Y-W rendering scatter diagram and the t(≤a) curve provide the practical chartingtechniques of data exploration and anomaly information extraction. Among LSA, I-LSA and GLSA, we come to the conclusions that I-LSA is superior to LSA employing the regular moving windows to calculate the a-value, and GLSA is the best of all employing the anisotropic windows which are variable with the location and the scale.The case study was used to demonstrate the application of the localsingularity to the anomaly identification of Cu concentration values from the stream sediment samples in Gejiu area, Yunnan province, China. The geochemical anomalies delineated using the singularity analysis method has the significant spatial correlation with the mineral deposits by the t(≤a) curve. The results reveal that the Gejiu area has a good the prospecting potential for copper.(5) Association studiesSome valuable conclusions are drawn during the association studies.(a) the local coefficient c set has the an excellent property which is non singular.(b) Several important parameters (U~*, r_0,β_0,θ_0) could be estimated by taking into account the spatial properties, geometric properties by means of the spatial U-statistic method. These parameters have a clear physical meaning which reveal the important information of the field. Thelocal optimum U~*-value can be regarded as another contrast anomaly index; the scale r_0-value shows the local connectivity of the field; the compressed ratioβ_0-value characterizes the local intensity ofthe anisotropy of the field; and the azimuthθ_o-vlaue reflects the local optimum orientation of the anisotropy of the field.The main innovations in this paper are as follows: (1) Advancing I-LSA to improve the precision of theα-value for the first time;(2) Achieving GLSA by means of the spatial U-statistic method for the first time.In a word, the local singularity is of perfect application prospect. The author hopes it can be steadily advanced, generally recognized and widely used by many scientific communities in the near future.');。