The Analytic Method And Application Of The Extended Fractal Dimension For Linear Rivers

Fractal Geometry is a new non-linear mathematical method applied to researching the phenomenon which is complex and irregular as well as self-similar. Numerous geographical phenomenons in nature appear with complexity and self-similarities on their spatial form and structure, so it has become an important part in fractal research.River is the natural linear channel which has regular or intermittent flow on the land surface, and it has the close relationship with much other phenomenon of nature, culture and society including terrain developing, human migration, the expansion of urban and traffic distribution, biodiversity and so on. Thereby, it becomes one of the common research objects of Geography, cartography and fractal geometry. According to its figure and the change of the fractal dimension, the complexity and diversity of the river form performance in the two sides of the fractal character as that, on the one hand, namely the fractal character on the different local spatial sections of the whole river channel changes, which means the dependence of the fractal character on the space; On the other hand, the fractal character under the different observation scales changes too, which means the dependence of the fractal character on the scale. The paper analyzed the deficiencies of the traditional fractal analysis method in the study of cartography which is on the ideal fractal objects, and set forth the need and the main idea for extending the fractal dimension analysis method, based on the feature of the inverse "S" shape.The paper extended the analysis scope to the whole observation scale interval for the Richardson curve, and established the fractal simulation model adopting the function which appears the inverse "S" such as Cubic Polynomial Model with Derivative and the inverse Logistic model. In this way, the non-linear relationship between the curve length and the observation scale has been established. Secondly, the paper applied the meta fractal curve into the research for the river's morphology based on the phenomenon of its local fractal feature. As for the arbitrariness of determining the size of the sliding window in preliminary studies, the paper put forward an automatic method to determine the size of the sliding window on the basis of scale-analysis on the spectrum of fractal dimension. Analysis of the test showed that the method enhanced the automatic degree of the application for the Meta fractal curve. Finally, the paper took the seven major rivers in China especially Yangtze River as example, achieved the extended fractal measurement model for the rivers' length, and analyzed the local morphological feature of the rivers on the different scales map. Furthermore, the paper applied Meta fractal dimension curve into the segmentation for the whole Yangtze River, the result turned out that it is consistent with the scope of the geographical structure unit. The research in the paper showed that the EFDA method can play an important role in the analysis for the form of linear river on the map. At the same, it will promote the fractal geometry in the application of the geography and cartography.