地学现象的分维与标度关系——以中国大陆山系为例

分形理论产生以后,已经在许多领域里取得了较为广泛的应用,但是在分形维数的计算方面仍存在一定的问题,最根本的问题之一是大量相关研究往往忽略了标度问题的影响,而将计算出的分维结果随意比较。本文则是以中国大陆山系为例,对地学现象的分维与标度的关系进行了研究,得出:(1)中国大陆山系的分形性质是客观存在的,其分形性质是不以测算标度而出现变化的,是始终存在的;(2)虽然在不同的测算标度下,中国大陆山系的分形性质始终是客观存在的,但是随着测算标度范围的变化,中国大陆山系的分维也会相应地出现一定的变化,即中国大陆山系客观存在的分形性质是绝对的,而分维则是一个相对的值,是随着测算标度而出现变化的;(3)分维不仅随着测算标度的变化而出现一定的变化,而且,分维是测算平均标度的函数,即D=f(r)(式中:r为测算平均标度,D为分维)关系成立;(4)因为分维值会随着不同标度范围的变化而出现相应的变化,所以,如果要进行分维之间的比较,则应选择相同的标度范围和标度值点。

On Relationship Between the Fractal and Scales --A Case Study of China's Continental Mountain Ridges

Mandelbrot enunciated the uncertainty of the length of a coastline in his paper"How long is the coastline of Britain?"published on Science in 1967. The fractal concept was presented for the first time in that paper. The fractal theory has been applied to a wide variety of fields after it was produced. The use of fractal analysis to geographical research has increased (Mandelbrot,1967;Xu et al.,1993; Turcotte,1996;Nina et al.,1993;Andrle,1996;Gao et al.,1996;Zhu et al.,2000;Wilson,2001;Christensen et al.,2002),but large numbers of geographical phenomena still need to be studied,and many problems still remain unsolved. Although fractal dimensions of lots of phenomena were calculated by the box-counting method,the possible influence of series of square grids on them is ignored all the while. The issue is systematically discussed as a case study of China's continental mountains.Finally,some significant conclusions are drawn as follows:(1)Although the fractal character objectively exists in China's continental mountains and doesn't vary with the changes of a series of square grids,the fractal dimensions of these mountains vary with these changes.(2)The fractal dimensions of China's continental mountains vary with the average length of sides of a series of square grids. The fractal dimensions of these mountains are the function of the average length of sides of square grids. They conform to the formula D=f(r) (where D is the fractal dimension,and r is the average length of side of square grid).(3)Different dots of data collection can affect the fractal dimensions of China's continental mountains.(4)The same range of length of side of square grid and dots of data collection can ensure the comparison of fractal dimensions of China's continental mountains,which is just the same as to the other fractal studies and can undoubtedly help to deepen the understanding of fractal and fractal dimensions.