Fractal Singular-Value (Egin-Value) Decomposition Method for Geophysical and Geochemical Anomaly Reconstruction
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摘要: 地球物理和地球化学异常是找矿的重要依据.地球物理和地球化学异常取决于地层、构造在成矿时间上的多样性与空间上排列、叠置的复杂性.地层、构造因素是构造、岩浆、沉积与成矿地球化学等多种动力学过程的综合反映.这些岩石和构造的因素以及动力过程相互渗透和影响决定了最终地质、地球物理与地球化学场.本文提出的在GIS环境下实现的分形奇异值分解(MSVD) 异常重建方法, 不仅可以提取地球物理和地球化学等异常, 而且能够进一步刻画其中的线性和环状构造、细微的局部纹理结构特征.该方法首先对地球物理和地球化学等网格数据进行二维矩阵的奇异值分解, 之后用左特征向量矩阵与右特征向量矩阵的直积构造一个正交完备基.地球物理和地球化学二维数据可以投影到该正交基上, 其投影系数是矩阵的奇异值.在该正交完备空间的某些子空间上对地球物理和地球化学等数据进行滤波.为了选择子空间, 本文定义了上述正交完备基中的能谱密度、能谱半径(或尺度) 与能量测度.在此基础上与空间域及频率域类比, 探讨了能量测度与能谱密度呈现分形(fractal和bifractal) 规律.利用分形关系的间断点, 设计分形奇异值重建算子, 实现对地球物理和地球化学异常的分解.以加拿大NovaScotia南部布格重力异常与As地球化学异常为例, 采用MSVD方法分解Au、Wu -Sn -U等已知矿有关的地球化学异常.发现重建异常能很好地用于解释已知矿点的分布规律.重建的地球化学异常显现了地球化学中的线状和环状异常; 重建的布格重力异常有效勾勒出原图中不易发现的纹理结构, 这些纹理结构可以合理地解释已知矿点在侵入岩体内及其周围的分布规律.应用实例表明, 该方法不仅可以从起因复杂的异常中区分出背景、异常场, 还可以识别代表了成矿源岩、流体、运移通道、赋存空间等异常因素引起的纹理、结构与构造特征.同时实现了GIS环境下交互可视化的MSVD处理与解释系统, 增强了地质异常定量分析的实用性与可操作性.Abstract: Geochemical and geophysical anomalies are originated from geological processes. These processes involve a great deal of complexity temporally and spatially. It is critical to improve the current anomaly extraction methods from the standpoint of the association of geophysical and geochemical anomalies for mineral exploration. The fractal singular-value-decomposition (MSVD) in GIS environment developed in this study is demonstrated superb in extracting linear and circular geophysical and geochemical anomalies as well as the detailed structural and textural information from 2D geochemical and geophysical maps. The MSVD method constructs a self-contained orthogonal basis using the outer product of left and right eigenvector matrixes decomposed from 2D geochemical or geophysical maps. A power-law relationship based on fractal theory has been suggested to associate the spectrum density and spectrum radius (or spectrum scale) defined in the paper. Multiple power-law relationships observed between the spectrum density and spectrum radius can help to group singular values and their corresponding eigenvectors. Each of these groups can be used to reconstruct the geophysical and geochemical maps to reflect decomposed components. The component reconstructed with relative large singular values may correspond to background and those obtained with relatively small singular values may represent anomalies. This method has been demonstrated using datasets from Nova Scotia, Canada. The results obtained for As and other elements from lake sediment samples, gravity anomalies and airborne magnetic anomalies have shown that the power-law relationship might exist between spectrum density and spectrum radius. Several different exponents are observed from the datasets which can be based to separate the anomalies from background.
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Key words:
- anomaly reconstruction /
- fractal SVD (MSVD) /
- MSVD plot /
- GIS
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图 1 奇异值分解能量测度定义(a) 以及As的分形奇异值分解图解(b)
Fig. 1. Definition of accumulated energy (b) and MSVD plot (a)
图 3 研究区位置、岩性单元和矿点分布
Fig. 3. Location of study area, simplified lithology units and locations of the known mineral depositsore and mineral occurrences (Xu, 2001)
图 5 地球化学As元素奇异值分解积分能量贡献(a) 与能量密度(b) 随奇异值的变化
Fig. 5. Accumulated energy (a) and energy spectrum density (b) of As versus the number of singular-value number
图 6 As地球化学图(对数变换) (a)、As异常图(b) 以及解释(c)
Fig. 6. log-transformed As concentration map (a), MSVD anomaly (b) and interpretation (c)
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