Investigation on the properties of the relationship between rare and extreme rainfall and flood volumes, under some distributional restrictions

Abstract

The fact that rainfall data are usually more abundant and more readily regionalized than streamflow data has motivated hydrologists to conceive methods that incorporate the hydrometeorologial information into flood frequency analyses. Some of them, particularly those derived from the French GRADEX method, involve assumptions concerning the relationship between extreme rainfall and flood volumes, under some distributional restrictions. In particular, for rainfall probability distributions exhibiting exponential-like upper tails, it is possible to derive the shape and scale of the probability distribution of flood volumes by hypothesizing the basic properties of such a relationship, under rare and/or extreme conditions. This paper focuses on a parametric mathematical model for the relationship between rare and extreme rainfall and flood volumes under exponentially-tailed distributions. The model is analyzed and fitted to rare and extreme events derived from hydrological simulation of long stochastically-generated synthetic series of rainfall and evaporation for the Indaiá River basin, located in south-central Brazil. The paper also provides a sensitivity analysis of the model parameters in order to better understand flood events under rare and extreme conditions. By working with hydrologically plausible hypothetical events, the modeling approach proved to be a useful way to explore extraordinary rainfall and flood events. The results from this exploratory analysis provide grounds to derive some conclusions regarding the relative positions of the upper tails of the probability distributions of rainfall and flood volumes.

Understanding Heavy Tails of Flood Peak Distributions

“…He suggested that for very large return period floods, for example, beyond 500 years, the distribution of the maximum mean rainfall intensity over a duration in the order of the time of concentration of the catchment should be considered as the possible flood peak asymptotic distribution, as schematically illustrated in Figure 5. The suggestion that the flood tail heaviness follows the tail heaviness of precipitation for large return periods is also the basic assumption of the GRADEX method which is widely used in practice, particularly in France (Naghettini et al., 2012). The return period beyond which the flood distribution follows the rainfall distribution is typically set to much lower values, for instance, to 10–20 years for relatively impermeable watersheds and up to 50 years for watersheds with high infiltration capacity (Naghettini et al., 2012).…”

“…The suggestion that the flood tail heaviness follows the tail heaviness of precipitation for large return periods is also the basic assumption of the GRADEX method which is widely used in practice, particularly in France (Naghettini et al., 2012). The return period beyond which the flood distribution follows the rainfall distribution is typically set to much lower values, for instance, to 10–20 years for relatively impermeable watersheds and up to 50 years for watersheds with high infiltration capacity (Naghettini et al., 2012). Such low values are in contrast to McCuen and Smith (2008) and Gaume (2006) who suggested that the tail behavior of the flood distribution was inherited from the rainfall distribution in the extreme case only, that is, for very large return periods.…”

What controls the tail behaviour of flood series: Rainfall or runoff generation?

“…While we cannot directly transfer the GEV shape parameter from rainfall to flood peak distributions, rainfall still has an important role: in his analytical analysis, Gaume (2006) states that "the shape of the flood peak distribution is asymptotically controlled by the rainfall statistical properties, given limited and reasonable assumptions concerning the rainfall-runoff process". A similar assumption is the basis of the GRADEX method which is used in practice, for example, in France (Naghettini et al, 2012). The method assumes that beyond a certain return period, the upper tail of a flood peak distribution is the same as the upper tail of the rainfall distribution (Naghettini et al, 2012).…”

“…A similar assumption is the basis of the GRADEX method which is used in practice, for example, in France (Naghettini et al, 2012). The method assumes that beyond a certain return period, the upper tail of a flood peak distribution is the same as the upper tail of the rainfall distribution (Naghettini et al, 2012). While in the GRADEX method, this return period is usually assumed to be between 10 and 50 years (Naghettini et al, 2012), Gaume (2006) estimates it to be beyond 500 years.…”

“…The method assumes that beyond a certain return period, the upper tail of a flood peak distribution is the same as the upper tail of the rainfall distribution (Naghettini et al, 2012). While in the GRADEX method, this return period is usually assumed to be between 10 and 50 years (Naghettini et al, 2012), Gaume (2006) estimates it to be beyond 500 years. Merz et al (2022) state in their review study on heavy tails that runoff generation processes strongly modulate tail behaviour of streamflow -but only up to a certain return period -and that for very high return periods the flood tail tends to be dominated by the rainfall tail.…”