Fractal Theory and the Estimation of Extreme Floods

Floods and draughts constitute extreme values of great consequence to society. A wide variety of statistical techniques have been applied to the evaluation of the flood hazard. A primary difficulty is the relatively short time span over which historical data are available, and quantitative estimates for palcofloods are generally suspect. It was in the context of floods that Hurst introduced the concept of the rescaled range. This was subsequently extended by Mandelbrot and his colleagues to concepts of fractional Gaussian noises and fractional Brownian walks. These studies introduced the controversial possibility that the extremes of floods and droughts could be fractal. An extensive study of flood gauge records at 1200 stations in the United States indicates a good correlation with fractal statistics. It is convenient to introduce the parameter F which is the ratio of the 10 year flood to the 1-year flood; for fractal statistics F is also the ratio of the 100 year flood to the 10 year flood and the ratio of the 1000 year flood to the 100 year flood. It is found that the parameter F has strong regional variations associated with climale. The acceptance of power-law statistics rather than exponentially based statistics would lead to a far more conservative estimate of future flood hazards.