Quantitative analysis of randomness exhibited by river channels using chaos game technique: Mississippi, Amazon, Sava and Danube case studies

Abstract. This paper presents a numerical evaluation of the randomness which can be observed in the geometry of major river channels. The method used is based upon that of generating a Sierpinski triangle via the chaos game technique, played with the sequence representing the river topography. The property of the Sierpinski triangle is that it can be constructed only by playing a chaos game with random values. Periodic or chaotic sequences always produce an incomplete triangle. The quantitative data about the scale of the random behaviour of the river channel pathway was evaluated by determination of the completeness of the triangle, generated on the basis of sequences representing the river channel, and measured by its fractal dimension. The results show that the most random behaviour is observed for the Danube River when sampled every 715 m. By comparing the maximum dimension of the obtained Sierpinski triangle with the gradient of the river we can see a strong correlation between a higher gradient corresponding to lower random behaviour. Another connection can be seen when comparing the length of the segment where the river shows the most random flow with the total length of the river. The shorter the river, the denser the sampling rate of observations has to be in order to obtain a maximum degree of randomness. From the comparison of natural rivers with the computer-generated pathways the most similar results have been produced by a complex superposition of different sine waves. By adding a small amount of noise to this function, the fractal dimensions of the generated complex curves are the most similar to the natural ones, but the general shape of the natural curve is more similar to the generated complex one without the noise.