分形统计模型的理论研究及其在地质学中的应用

本文提出了一般分形模型和一般分维数的概念,认为许多地质模型是一般分形模型的特例,指出幂函数分布和帕累托分布是分形统计模型的数学基础,论证了幂函数分布在高端截尾条件下具有尺度不变的分形性质,根据非线性回归模型参数估计的方法,提出了求分维数的新方法,该方法具有许多优点。通过在计算机上产生随机数对分形统计模型进行模拟研究,以及通过实例说明分形统计模型应用的方法及步骤,并解释了分维数的实际意义。

THE THEORY STUDY OF FRACTAL STATISTICAL MODEL AND ITS APPLICATION IN GEOLOGY

The fractal was founded by Mathematician B.B.Mandelbrot. A fractal is an object made of parts similar to the whole in some way, either exactly the same except for scale or statistically the same. The fractal geometry deals with irregular phenomena or objects in nature, such as topographic relief, fracture strength of rocks, earthquake magnitude etc.. It is difficult to describe them by classic mathematical methods. But there is a common characteristic among these phenomena or objects-self-similar. Fractal dimension measures the degree of irregularity based on self-similarity, and is also a numerical index that quantifies the self-similarity of complex phenomena. The fractal theory was applied to the minerogenetic prediction in the 1980’s. D.L.Turcotte proposed that there exists a fractal relation between average grade and cumulative ore reserves. B.B.Mandelbrot considers that high grade copper with non-uniform distribution may have the multifractal structure. Meng Xianguo and Zhao Pengda suggest that the fractal structures exist in the geological data. Fractal dimension and multifractal spectrum characterize the complex fractal structures quantitatively. In the current paper we advance the conceptions of the general fractal models and fractal dimension and consider that many geological models are the special cases of the general fractal models, pointing out that the Power-function distribution and the Pareto-function are the mathematical base of the statistical model and proofing that the Power-function distribution possesses the fractal property of scaling under upper truncation. A new method is developed on the basis of nonlinear regression to estimate the fractal parameters D. The new method of getting parameter C and D is more precise than traditional method and has many advantages.The fractal dimension D can indicate the structure of random number or sample by simulated study. We have established the surface ore body predictive model(7), the surface ore cluster predictive model(8)and ore reserves predictive model(9). Those predictive results fit the actual situation. The fractal dimension D describes quantitatively the change or trend of the density on the ore body distribution. The prefactor parameter C is the initial value of the ore-body distribution. They have important significance in mineral resources exploration, prediction and evaluation.