城乡聚落动态变化的维数公式

城乡聚落动态变化的维数公式

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contributed at 2018-12-04

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城乡聚落动态变化的维数公式

 

式中,A(r)代表以r为量测尺度的某一样本的面积;P(r)为同一样本的周长;C为截距;2/D为斜率。维数D等于斜率除2,其值在12之间,D越大代表图形形状越复杂。当D等于1.5时,则代表图形处于布朗随机运动状态,越接近该值,稳定性越差。如果定义城市形态变化(一个随机过程)B(t)为布朗运动,并且B(t)满足Var[B(t2)-B(t1)]∝|t2-t1|2H由于B(t)的图形只是某一次的取样图形,对于一定的HB(t)的所有取样都具有相同的分数维。且为:D=2-H,参数H(0<H<1)在直观上表示了图形的粗糙度。

根据以上分析,可以把分形布朗运动分成3个完全不同的类:H=1/2时,即D=1.5时,B(t2)-B(t1)B(t3)-B(t2)(t1<t2)是独立的,两者相关为0,故图形处于最不稳定状态。H>1/2时,即D<1.5时,随机过程的增量间有正的相关值,故图形趋于简单。当H=1时,图形在t方向的扩展必须以图形的幅度方向上相同的因子的扩展来补偿才能保持看上去一致,此时的图形的分数维为2-H=1H<1/2时,随机过程的增量间存在一个负的相关,一般此时的图形更为奇特,有较大的分数维。当H=0时,便得到一个完全不同的随机行为,在t方向上任意的压缩或扩展B(t)图形,看上去还是一样的,也即B(t)的图形密集地充满了平面上的一个区域,即它的分数维2

 

参考文献

杨山:发达地区城乡聚落形态的信息提取与分形研究——以无锡市为例。自然资源学报,2000,55(6)

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《资源环境数学模型手册》 (2018). 城乡聚落动态变化的维数公式, Model Item, OpenGMS, https://geomodeling.njnu.edu.cn/modelItem/4d53181b-0ba6-484d-8362-705aa193cb08
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initial contribute: 2018-12-04

Authorship

Affiliation:  
岳天祥编著
Email:  
yue@lreis.ac.cn
Homepage:  
View
Is authorship not correct? Feedback

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Last modifier : 
Mingyuan Li
Last modify time : 
2020-11-22
Modify times : 
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