Complexities in power-law modeling of river discharge time series

Distributional analysis of river discharge time series is an important task in many areas of hydrological engineering, including optimal design of water storage and drainage networks, management of extreme events, risk assessment for water supply, and environmental flow management, among many others. Having diverging moments, heavy-tailed power-law distributions have attracted widespread attention, especially for the modeling of the likelihood of extreme events such as floods and droughts. River discharge data may be sampled with different resolutions, such as 15 minutes, 1 day, 1 month, or even 1 year. It has not been examined in hydrology how the estimated power-law exponent in river discharge series may depend on the resolution of the data. Using river flow data from continental U.S.A. and Europe, we show that the dependence of power-law exponent on resolution is substantial. This is further corroborated by aggregating independent Pareto random variables. River discharge series are more complicated than independent Pareto random variables, however, since flow data have strong seasonal cycle and long-range correlations. These factors result in an increase in the estimate of power-law scaling exponent for coarser temporal resolutions. Such an increased estimate of the scaling exponent could cause under-representation of flooding risk.

Straightforward distributional analysis does not connect well with the complicated dynamics of river flows, including fractal and multifractal behavior, chaos-like dynamics, and seasonality. To better reflect river flows' dynamics, we propose to carry out distributional analysis of river flow time series according to three “flow seasons”: dry, wet, and transitional. We present a concrete statistical procedure to partition river flow data into such three seasons, and fit data in these seasons using two types of distributions, power-law and lognormal. We show that while both power-law and lognormal distributions are relevant to dry seasons, river flow data in wet seasons are typically better fitted by lognormal distributions than by power-law distributions.>