Selection of the probabilistic model of extreme floods: The case of the River Tiber in Rome

Summary

The probability distribution of the annual maximum peak flows on the River Tiber at the town of Rome (Ripetta) can be evaluated using the information available since the XV century. In this paper the probability distribution of two series of annual peak flows maxima observed at the Ripetta gauge is analysed: the systematic series, including all the data observed since the beginning of the systematic stage record in 1782, and the censored series of the exceptional floods that inundated the town of Rome, starting from the XV century. In order to find a criterion for choosing the “optimum” distribution law for the high return period quantiles, an index that takes into account both the accuracy and the uncertainty of the quantiles estimation is proposed.

Introduction

The definition of flood protection plans requires the evaluation of design floods with return periods up to 200–500 years. The following steps are involved in the frequency analysis of peak flows records: (1) selection of peak flows data; (2) choice of an appropriate probability distribution model, fitted using appropriate estimation techniques; (3) testing the goodness-of-fit; and (4) estimation of flood quantiles and construction of confidence intervals.

Very few stage gauges were installed in Italy before the XIX century, and since most were installed after the beginnings of the XX century, only seldom the record lengths reach 100 years, and are usually much shorter. Thus the estimation of high flood quantiles requires the extrapolation of the probability distributions far beyond the usual sample length, involving high estimation uncertainties.

Some tools are available to reduce this uncertainty: (1) using physically based rainfall–runoff models in continuous simulation; (2) combining data from several gauges in a regionalization procedure; and (3) incorporating the “non-systematic” information at a gauge into the instrumental records (Kidson and Richards, 2005). Non-systematic information, related to events occurred before the beginning of systematic recording, includes historical floods and paleofloods. The magnitude of large “historical floods” can be determined when these extraordinary events are described in extant historical accounts or recorded by manufactured physical marks. “Paleoflood” data can be obtained by analyzing physical evidence of the occurrence of large floods, as erosions, sediment deposits and botanical specimens. Because only large values are usually recorded by the non-systematic information, historical flood and paleoflood data may at best constitute censored samples (Stedinger and Cohn, 1986, Hirsch and Stedinger, 1987, Francés, 1998, Cohn et al., 2001, Martins and Stedinger, 2001, England et al., 2003, Kidson and Richards, 2005, Stedinger and Griffis, 2008), but they provide all the same an unique information about the upper tail of the probability distributions, even if the data are usually less reliable than those of the systematic data.

Several probability distribution functions have been used in flood frequency analysis (e.g. Kidson and Richards, 2005). The proliferation of the employed distributions is itself symptomatic of the weak theoretical basis of the probabilistic models of hydrologic phenomena.¹ As a consequence, theoretical considerations are rarely employed to justify the selection of the probabilistic model (Singh and Strupczewski, 2002, Kidson and Richards, 2005).

In several countries the approach to flood frequency analysis follows a national standardized procedure, enforced for planning and insurance purposes. The use of a fixed probability model for all catchments is based on the rationale that the differences between the quantile estimators resulting from reasonable alternative distributions are substantially lower than the uncertainty of the quantile estimator themselves, and that in any case the true underlying probability distribution cannot be identified. Thus the main issues are the use of a reasonable distribution, consistent with the available data, and fitting the distribution by the best available procedure (Stedinger and Griffis, 2008). A remarkable example is the official adoption as a standard procedure in the USA of the Log-Pearson 3 distribution, that is considered a reasonable and flexible model within the range of the parameter values typical of US flood series (IACWD, 1982).

The standardized use of a single model can be interesting from the administrative point of view; however, it still remains that no theoretical argument justifies the use of a simple distribution, even for a single sample. That involves several questions: is a certain degree of climatic uniformity a sufficient reason to assume some kind of uniformity in the flood distribution of different basins? How relevant are the observations of ordinary floods for the estimation of high return period quantiles? Is there some common limiting behaviour of extreme floods?

Recent studies have shown that the flood samples are the outcome of different combinations of separate identifiable processes. Mixed populations, and hence mixed distributions, are the outcome of several possible factors, such as different types of flood producing storms, rainfall and snowmelt floods, inundations and floodplain flow, antecedent basin soil moisture, conditions of vegetal cover (Singh et al., 2005). More complex multi-parametric probability distribution functions are often required to describe flood series including subset generated by different processes. Nevertheless there are many arguments against the use of multi-parametric models in flood frequency analysis, such as the difficulty in parameter estimation, the increasing estimates uncertainties as the sample size decreases and the number of the parameters increases, and the limited number of extreme events. In some cases, simulation experiments showed higher accuracy in upper quantiles estimates using a simple model than by using the true multi-parameter distribution (Singh and Strupczewski, 2002). In some cases, parameters parsimony may suggest fitting simple distributions on the subsets generated by distinct processes, notwithstanding the smaller sample sizes (Kidson and Richards, 2005).

The identification of the “optimum” distribution law for high return period quantiles is often based on the use of a goodness-of-fit tests, that verify whether or not a probability model fits the data to a specific level of confidence. But the results obtained depend on the assumed significance level and are not always conclusive, because several models can pass the tests.

A promising approach, rarely applied in flood frequency analysis, is the use of model selection techniques, that lead objectively to the selection of a single probability distribution (Zucchini, 2000). The application of model selection techniques, combined with the knowledge of their performances, appears to increase the efficiency of design floods estimation (Mitosek et al., 2006, Di Baldassarre et al., 2008).

In this paper the annual maxima of peak flows series on the River Tiber in the town of Rome (Ripetta) are analysed using the information available since the XV century. In fact, the systematic stage measurements began at the end of the XVIII century, while the historical record of the peak stages of the floods that inundated the town goes back to the Renaissance. This yields a unique occasion to study the long-term behaviour of floods, provided that a method can be devised to estimate the stage-discharge relationships at the ancient Ripetta landing, where the stages of the exceptional floods were recorded.

The paper is organized as follows. The available systematic and historical data samples are described in detail in “Available data”. Two data samples are available: a systematic sample of 185 events from 1782 to 1989 and a censored sample of 20 historical floods exceeding a perception threshold from 1422 to 1989. The flood frequency analysis is developed in “Flood frequency analysis”. Owing to the complex behaviour shown by the systematic sample, we propose fitting both a heterogeneous multi-parameter probability law, estimated using the whole sample, and a simple two-parameters probability law, estimated using only the upper quantiles. A simple criterion for the selection of the “optimum” model for the estimation of upper quantiles is proposed in “Selection of the probabilistic model”. Some conclusions are drawn in “Conclusion”.