Multifractal Theory Based Local Singularity Analysis Method And Its Application In Spatial Information Extraction For Mineral Exploration

Interdisciplinary research involving non-linear theory, complex theory 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 can be described by the power-law model.The exponent a is termed the local singularity exponent in the local singularity analysis. Within a 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 capable of the spatial (temporal) localization for the anomaly. This method provides a simple and direct strategy for detecting and characterizing 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 using optimal 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 construction of a set of the anisotropic windows to calculate the local singularity. These techniques ensure the evolution from LSA to GLSA by 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, and it also takes more potential needs into account.The extended algorithm based on raster model has been implemented by MATLAB and the 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 charting techniques 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 local singularity 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^*, r₀,β₀,θ₀) 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. The local optimum U^*-value can be regarded as another contrast anomaly index; the scale r₀-value shows the local connectivity of the field; the compressed ratioβ₀-value characterizes the local intensity of the 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.