Quantifying the Generalized Self-Similarity of Spatial Patterns for Mineral Resource Assessment
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摘要: 尺度不变性(scaleinvariance) 包括自相似性(各向同性)、自仿射性(成层结构)、广义自相似性(各向异性标度不变性), 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.尺度不变性可用分形和多重分形模型来表征.这些尺度特征的定量化可为刻画地质空间模式和模式识别提供有力的工具.例如, 热液矿床的群聚现象可以用局部分形特征(局部奇异性) 来刻画.通过在特征空间中(如频率空间) 识别空间模式的广义自相似性, 可以将空间混合模式进行分解或异常的识别.介绍了几种相关的分形模型和方法, 包括度量空间模式广义尺度独立性(GSI) 的线性模型; 基于广义尺度独立性的异常分解S-A方法; 度量空间模式的局部奇异性方法; 以及如何利用分形特征预测未发现矿床的2种方法.有些方法已应用于许多矿产资源评价实例中.给出了对加拿大Nova Scotia省西南部湖泊沉积物样品中的4种元素As、Pb、Zn和Cu的地球化学数据处理分析结果, 证明了局部奇异性分析和S-A异常分解方法对地球化学异常的增强和分离的有效性.研究表明: 由S-A方法分解的异常往往具有多重分形的特点, 而且普遍具有局部奇异性.研究区内具有明显奇异性的地区(元素含量富集区) 是金矿异常区域, 它们与金矿成矿作用和已知矿床的赋存密切相关.Abstract: Scale invariance, including self-similarity (isotropic), self-affinity (stratification), and generalized self-similarity (anisotropy), is a common property of spatial patterns generated from various geological processes and events. Scale invariance can be described by means of fractal and multifractal models. Quantifying the scale invariance properties of spatial patterns may provide a powerful tool for characterizing geological processes and events. For example, the clustering distribution of hydrothermal mineral deposits can be characterized by means of local singularity analysis. The identification of distinct generalized self-similarity in the Fourier domain can be used to decompose spatial patterns into separate components such as anomalies from background patterns. The current paper introduces a number of relevant multifractal models and methods, including a linear model for generalized scale invariance (GSI); a spectrum-area method (S-A) for anomaly separation; a local singularity analysis method; and methods for predicting undiscovered mineral deposits on the basis of fractal and multifractal properties. Some of these methods have been applied in various case studies. The case study introduced in the current paper demonstrates the application of S-A anomaly separation and local anomaly enhancement in analyzing lake sediment geochemical data (As, Pb, Zn and Cu) for gold mineral resource prediction. It has been shown that the areas delineated by a strong singularity in As, Pb, Zn and Cu are spatially associated with the location of known gold mineral deposits.
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图 1 加拿大Nova Scotia省西南部地区地质简图和已知矿床分布(Chatterjee, 1983)
Fig. 1. Simplified geology and distribution of known mineral deposits in southwestern Nova Scotia, Canada
图 2 加拿大Nova Scotia省西南部地区湖泊沉积物地球化学取样点分布(N=1 948)
(数据来自加拿大Nova Scotia省能源部矿产资源处)
Fig. 2. Location of the lake sediment geochemical samples (N=1 948), southwestern Nova Scotia, Canada
图 3 砷的奇异性指数α值分布
圆点表示金矿床, 多边形表示SMB和早期和晚期的花岗岩相
Fig. 3. Estimated singularity values (α) for As
图 4 在GeoDAS GSI (Cheng, 2000b) 软件中使用快速傅立叶变换(FFT) 计算所得的能谱密度分布(图的中心处波数为0)
Fig. 4. Power energy spectrum calculated using fast Fourier transform (FFT) implemented in GeoDAS GSI (Cheng, 2000b). The center of the map is where wave number equals zero. It is a two-dimensional function of wave numbers in horizontal and vertical axes
图 5 能谱密度-面积(S-A) 双对数图
本结果由GeoDAS软件(Cheng, 2000b) 计算所得.在能谱密度的3个区间(-125 284), (125 284-178 129)和(178 129-) 上使用最小二乘法拟合直线段.这3条直线段的斜率分别为-1.966, -2.85, -1.17, 截距分别为26.6, 36.9和16.6.以125 284为阀值定义2个滤波器: 异常滤波器, S < 125 284;背景滤波器, S > 125 284
Fig. 5. S-A plot showing relationship between power energy spectrum S and A (> S) on log-log paper
图 6 用图 5中所定义的背景滤波器滤波并经傅立叶逆变换得到的砷背景分布
图中圆点表示金矿床(点), 暗色图案表示As高背景区
Fig. 6. Background component decomposed from the As power spectrum density map using inverse Fourier transform with background filter as defined in
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