MULTISCALING THEORY OF FLOOD PEAKS - REGIONAL QUANTILE ANALYSIS

We study the spatial random field of peak flows indexed by a channel n etwork. Invariance of the probability distributions of peak flows unde r translation on this indexing set defines statistical homogeneity in a network. It implies that hoods can be indexed by the network magnitu de, or equivalently the drainage area, which serves as a scale paramet er. This definition generalizes to homogeneity of flows in a geographi cal region containing several networks which are not necessarily the s ubnetworks of a single network. The widely used quantile regression me thod of the United States Geological Survey(USGS) provides one simple criterion to approximately designate homogeneous geographic regions. I t 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 an d the multiscaling theories of regional hoods in terms of their quanti les. The simple scaling theory predicts a constant CV and log-log line arity between flood quantiles and drainage areas such that the slopes in these equations do not change with the probability of exceedance. M ultiscaling theory of floods is developed to exhibit differences in fl oods between small and large basins. This theory shows that the CV for small basins increases, and for large basins it decreases, as area in creases; Moreover, the quantiles do not obey log-log linearity with-re spect to drainage areas. However, for large basins an approximate log- log linearity between quantiles and drainage areas is shown to hold. T he slopes in these equations decrease as p decreases; i.e., larger flo ods have smaller slopes than smaller floods. This approximation provid es a theoretical interpretation of the results of the empirical quanti le regression method in homogeneous regions where simple scaling or th e index flood assumption does not hold. Recent results on physical int erpretations of the scaling theories are also summarized here. A simpl e nonlinear method is developed to estimate the parameters in the mult iscaling theory. This method and other features of the theory are illu strated using flood data from central Appalachia in the United States.