Fractal relation of medial axis length to the water body area

Abstract

As it was demonstrated by Mandelbrot (1982), the power value is not 2 for non-standard shapes, fractals, the relation between the length of medial axis (Serra, 1982), L, and area, A, of surface water body is taken in the form ofL Ab, where, b d/2 is an exponent fitted to the data. The fractal relation between water body area, A, to the length ofmedial axis of water body is proposed. Through this fractal relation it is indicated that L Ad/2. More than 160 surface water bodies have been traced from satellite remote sensing data, covering the area between the geographical co-ordinates of 1800 to 18 30’north latitudes and 83 15’to 83 45’east longitudes. These traced water bodies are digitised and converted the data intowater body and no-water body regions. The lengths of medial axis and areal extents of 160 water bodies, in pixel units, have been computed. A graph is plotted between logarithm of area and logarithm of length of medial axis of 160 water bodies. The fractal dimension of medial axis length, d, is computed as 1.113+ 0.01 for 160 water bodies. The data are well fitted as the power law LA 0556, which equals d/2. b-d/2 value almost tallied with derived exponent power value, ie, 0.556. The value 2b is almostequal to 1.113: t: 0.01 which is computed as fractal dimension of medial axis length. Strikingly this is close to the value of 1.136, the fractal dimension of thelength of the main water course, which is hypothesisedby Mandelbrot.