Research On The Application Of Inverse 'S' Mathematical Model In Fractal Analysis Of Map Objects

Map is the abstract model of spatial information and reflects the complicated geographic phenomena. Fractal geometry has a particular ability to describe the complex phenomena. Naturally fractal geometry is applied to the analysis of map objects. Through fractal measurement, we can obtain the Richardson curve (scale-measure sequence) of an object. When the range of observed scale is wide enough, the Richardson curve of a map object always has a shape like an inverse 'S'. This paper detailedly analyzes the cause of formation of this phenomenon, and considering this fractal character of map objects, we substitute an inverse 'S' mathematical model—Cubic Polynomial Model With Derivative for the original Richardson curve of a map object, which is called the Modeling of Richardson Curve. We apply the inverse 'S' mathematical model to the fractal analysis of map objects.The determination of fractal non-scale interval is a basic and key problem of traditional fractal analysis methods of a single fractal dimension. This paper gets the mathematical formula by mathematical deduction and proposes a new method to determine the fractal non-scale interval of a map object on the basis of this inverse 'S' mathematical model. The Richardson Curves of map objects are divided into 5 different sections. Each section shows the map object's morphological characteristics in a certain range of observed scale. According to both the characteristic of the inverse 's' mathematical model—The Cubic Polynomial Model With Derivative and the one of the Richardson Curve, we fit the middle three sections of the curve by the mathematical model. On the base of the fitting, the original Richardson Curve is simplified into a mathematical function. After the deduction, we get the calculating formulas of the non-scale interval of map objects. This new method simplifies the complex calculation of previous methods and avoids the artificial interference always existing in some sense. We tested the validity of this new method by some typical experiments and the results of the experiments showed that this method could work well and stably.On the other hand, the traditional fractal analysis methods of a single fractal dimension has limitation on describing characters of the objects. Extended fractal dimentsion model are the hotspot of current research. This paper will apply the inverse 'S' mathematical model in the establishing of extended fractal dimentsion models and optimize the two models.With all the efforts above, we can apply fractal geometry more widely and more effectively in the research on geographical information.