GIS-Based Estimation Of Box Dimension And Horton Fractal Dimension Of Water System

Water systems have fractal features and their fractal dimension value reflects the complexity of the river.There are two main methods to estimate the fractal dimension of water system at present.One is Box dimension based on the fractal theory definition and the other is the Horton fractal dimension based on the Horton law.Based on DEMs with two precisions of 30 meter and 90 meter,the ArcGis platform and its Model Builder were used to extract the water systems in the Siheshui basin and the Hanjiang River basin and its nine sub-basins.The computing method for Box dimension and Horton fractal dimension of the water systems under a series of thresholds and the influence factors of fractal dimension are analyzed.Main research works and achievements were introduced as follows.(1)The Box dimension method is to place the river network in different size cell,and to count the number of grids which the river network contained in.The absolute value of the slope of the double logarithmic graph of the grid size and the grid number is fractal dimension value.The results showed that:(1)The correlation index of drainage network fractal dimension in Siheshui basin were greater than 0.98 and that of Hanjiang River basin were greater than 0.97,showing that the study basins were homogeneous in scale-free range and had good fractal features respectively.(2)Resolution of DEMs,such as 30 meter and 90 meter,had little effect on fractal value based on box-counting method without considering the accuracy of river network.(3)Fractal dimension values varied from the confluence thresholds of the drainage networks,and decreased sharply along with the increases of thresholds at the beginning,and then decreased gently to be value 1,but greater than 1.(4)The actual fractal dimension value of the river was unique and should be calculated based on real drainage networks.In another way,selecting the appropriate confluence threshold to make digital river network and the actual river network matches,the fractal dimension value of the digital river network could represent the actual dimension value of the real river to some extent.(5)When the drainage system was homogeneous,it had strong statistical selfsimilarity,so that the fractal dimension of the whole basin was similar to that of each sub basin.Therefore,drainage fractal dimension values were not related with the basin area.(6)The elevation distribution reflected the difference of terrain,affected the structure network and the density of the river network,and thus affected the fractal dimension of the water system.The elevation distribution was uniform,and the water system of the river basin had a larger fractal dimension value.While the height distribution was steep and the low ground was flat,the fractal dimension value of the river basin was small.(2)Horton fractal dimension is the ratio of the logarithm of the branch ratio to the length ratio of the river.The results showed that:(1)The river branch structure was not obvious in Siheshui basin for its small scale,long strip river network structure and small number of river channels;Under a series of thresholds,the branching ratios of the Hanjiang River basin were stable;its length ratios fluctuated;and its fractal dimensions varied in a large range,which averaged 2.21.(2)The increase of the confluence threshold caused the discard of small rivers,and the river orders decreased.At this time,the ratio of branch ratio and length of the river had changed greatly.(3)When the data of the highest level were not counted,the average branching ratio was the same as that of the whole river network in Hanjiang River basin,while the average length ratio was larger and the average fractal dimension was 1.53.(4)According to the structures of the river networks and the features of the channels in the nine sub-basins of the Hanjiang River,the data of the whole networks were statistically analyzed for Guanyinqiao,Xikou,Shanghang,Yangjiafang and Chaoan,while the data of the highestlevel channels were not counted for Baokeng,Hekou,Shuikou and Hengshan.The results showed an average branching ratio of 4.51,an average length ratio of 2.66 and an average fractal dimension of 1.54,which were relatively consistent with the channel features and fractal dimension of the whole Hanjiang River basin when its channel of the highest level was not counted.(3)Comparison between Box dimension method and Horton fractal dimension method showed that:(1)The definition of the fractal dimension of Box dimension and Horton fractal dimension were equivalent.(2)The calculation process of the Box dimension method only considered the distribution of graphs and was not strongly related to the branch structure of the actual river network.However,Horton fractal dimension method took the river network structure into account,such as branching ratio,length ratio,and had strong correlation with actual fractal structure or branch structure of river network.(3)The fractal dimension value calculated by Horton fractal dimension method was larger than that calculated by Box dimension method.