On scaling exponents of spatial peak flows from rainfall and river network geometry

The hypothesis of statistical self-similarity, or scaling invariance, in the spatial variability of rainfall, channel network structures and floods has been supported by recent advances in data analyses. This hypothesis is used here to calculate the statistical scaling exponents of peak river flows using a random cascade model of spatial rainfall intensities and the Peano basin as an idealized model of a river basin. The 'maximum contributing set' approximately determines the magnitudes of peak flows in a self-similar manner in different subbasins of the Peano basin. For an instantaneously applied random, spatially uniform rainfall, the Hausdorff dimension of the maximum contributing set appears as the statistical simple scaling exponent of peak flows. This result is generalized to an instantaneously applied cascade rainfall, and it is shown to give rise to statistical multiscaling in peak flows. The multiscaling exponent of peak flows is computed and interpreted as a Hausdorff dimension of a fractal set supporting rainfall intensity on the maximum contributing set of the Peano basin. Potential implications of this interpretation are illustrated using the regional food frequency analysis of the Appalachian flood data in the United States and a rainfall data set from the tropical Atlantic ocean. It is argued that the hypothesis of self-similarity identifies a powerful theoretical framework which can unify a statistical theory of regional flood frequency with important empirical features of topographic, rainfall and flood data sets and distributed rainfall-landform-runoff relationships.