Multiscaling theory of flood peaks: Regional quantile analysis

Abstract

We study the spatial random field of peak flows indexed by a channel network. Invariance of the probability distributions of peak flows under translation on this indexing set defines statistical homogeneity in a network. It implies that floods can be indexed by the network magnitude, or equivalently the drainage area, which serves as a scale parameter. This definition generalizes to homogeneity of flows in a geographical region containing several networks which are not necessarily the subnetworks of a single network. The widely used quantile regression method of the United States Geological Survey (USGS) provides one simple criterion to approximately designate homogeneous geographic regions. It is argued that the redefinition of homogeneity via the constancy of the coefficient of variation (CV) of floods implied by the index flood assumption is ad hoc. Invariance of the probability distributions of peak flows under scale change is used to develop the simple scaling and the multiscaling theories of regional floods in terms of their quantiles. The simple scaling theory predicts a constant CV and log‐log linearity between flood quantiles and drainage areas such that the slopes in these equations do not change with the probability of exceedance. Multiscaling theory of floods is developed to exhibit differences in floods between small and large basins. This theory shows that the CV for small basins increases, and for large basins it decreases, as area increases. Moreover, the quantiles do not obey log‐log linearity with respect to drainage areas. However, for large basins an approximate log‐log linearity between quantiles and drainage areas is shown to hold. The slopes in these equations decrease as p decreases; i.e., larger floods have smaller slopes than smaller floods. This approximation provides a theoretical interpretation of the results of the empirical quantile regression method in homogeneous regions where simple scaling or the index flood assumption does not hold. Recent results on physical interpretations of the scaling theories are also summarized here. A simple nonlinear method is developed to estimate the parameters in the multiscaling theory. This method and other features of the theory are illustrated using flood data from central Appalachia in the United States.