Multifractal analysis of daily river flows including extremes for basins of five to two million square kilometres, one day to 75 years

Abstract

Multifractal analysis of the daily river flow data from 19 river basins of watershed areas ranging from 5 to 1.8 × 10 6 km 2 from the continental USA was performed. This showed that the daily river flow series were multifractal over a range of scales spanning at least 2 3 to 2 16 days. Although no outer limit to the scaling was found (and for one series this was as long as 74 years duration) for most of the rivers, there is a break in the scaling regime at a period of about one week which is comparable to the atmosphere's synoptic maximum, the typical lifetime of planetary-scale atmospheric structures. For scales longer than 8 days, the universal multifractal parameters characterizing the infinite hierarchy of scaling exponents were estimated. The parameter values were found to be close to those of (small basin) French rivers studied by Tessier et al. (1996). The multifractal parameters showed no systematic basin-to-basin variability; our results are compatible with random variations. The three basic universal multifractal parameters are not only robust over wide ranges of time scales, but also over wide ranges in basin size, presumably reflecting the space—time multiscaling of both the rainfall and runoff processes. Multifractal processes are generically characterized by first-order multifractal phase transitions: qualitatively different behavior is shown for the extreme events in which the probability distributions display algebraic fall-offs associated with (nonclassical) self-organized critical (SOC) behavior. Using the observed flow series, the corresponding critical exponents were estimated. These were used to determine maximum flow volume exponents and hence to theoretically predict maximum flow volumes over aggregation periods ranging from 2 3 to 2 16 days. These theoretical predictions are based on four empirical parameters which are valid over the entire range of aggregation periods and compare favourably with the standard (GEV) method for predicting the extremes, even though the latter implicitly involve many more parameters: three different exponents for each aggregation period. While the standard approach is essentially ad hoc and assumes independent random events and exponential probability tails (which, we show, systematically underestimate the extremes), the multifractal approach is based on the clear physical principle of scale invariance which (implicitly) involves long-range dependencies, and which (typically) involves nonclassical algebraic probabilities.