A theoretically based Wakeby distribution for annual flood series

Abstract

The exact distribution of the ratio of any magnitude to the sum of all magnitudes in an annual flood series satisfying the usual distribution-free assumptions of independence and identical distribution, and the additional parametric assumption of exponential tail behaviour with truncation, is shown to be a beta distribution of the first kind. A two-parameter linear transformation of the beta distribution completes the derivation and yields a Wakeby distribution which has the number of members in a series as a given parameter. The Wakeby distribution is developed to illustrate how, in principle, some perceived deficiencies in current flood frequency analysis may be met: more complex parametric assumptions should lead to distributions of wider application. In particular, the distribution has a secure theoretical basis and is hydrologically more realistic because it bounds the variate and requires the definition of a temporally finite annual series. Analytical expressions are obtained for estimating the two distribution parameters; the quantité standard error and a plotting rule. An example is given of the application of the distribution to the design flood problem and an annual flood series is modelled. It is further suggested that a suitable design value for the largest flood to be withstood by a protection work is a statistic of the largest flood occurring during its lifetime. For the derived Wakeby distribution this criterion specifies risk and probability of non-exceedance of the design flood once a lifetime is selected.Une distribution théorique de Wakeby pour une série annuelle de crues Résumé On montre que la distribution exacte du rapport de n'importe quelle grandeur à la somme de toutes les grandeurs dans une série annuelle de crues répondant aux hypothèses habituelles d'indépendance et de distribution identique ainsi qu'à l'hypothèse paramétrique du comportement en courbe exponentielle tronquée, est une distribution beta du premier ordre. Une transformation linéaire à deux paramètres complète la dérivée et conduit à une distribution de Wakeby dont le nombre de membres dans une sérié correspond à un paramètre donné. On développe cette distribution afin d'illustrer comment, enOpen for discussion until I December 1989 231George A. Griffiths 232 principe, certaines insuffisances dans l'analyse de la fréquence des débits de crue peuvent être rencontrées: d'autres hypotheses plus complexes devraient conduire à des distributions plus largement utilisables. En particulier, la distribution possède une base théorique sûre et est hydrologiquement parlant plus réaliste, car elle donne une limite à la variable et nécessite la définition d'une série annuelle définie dans le temps. Des expressions analytiques sont obtenues pour estimer les deux paramètres de distribution, l'écart-type et une règle pour la mise en place des points représentatifs. On donne un exemple de l'application de la distribution au problème de l'étude de la crue du projet et l'on modélise une série annuelle des crues. On suggère encore qu...

Variability case study based on in-situ rebound hardness testing of concrete. Part 1. Statistical analysis of inherent variability parameters

“…Wakeby distribution is typically used for flood frequency analysis and for other extreme events in hydrology [22]. Wakeby distribution has five parameters, therefore, it can mimic the shapes of many commonly used skewed distributions.…”

Probabilistic Distributions for <i>Acacia Mearnsii</i> De Wild Total Height and the Influence of Environmental Factors

“…The Wakeby distribution has been mainly used for modeling extreme events and in the most cases for hydrology studies, for example researches developed by [40][41][42][43][44][45]. The distribution has five parameters, more than the most used ones.…”

Estimation of the Parameters of Wakeby Distribution by a Numerical Least Squares Method and Applying it to the Annual Peak Flows of Turkish Rivers

“…Houghton (1978) is the first researcher to practice this distribution for flood frequency analysis. Landwehr et al (1979a, b), Kumar and Chander (1987), Rao and Arora (1987), Griffiths (1989), Haktanir (1992), Vogel et al (1993), Haktanir and Horlacher (1993), Zalina et al (2002), and Öztekin (2007) are some of the others who employed the W5 for flood frequency and rainfall analyses.…”

“…More theoretical information about Wakeby distribution with an application to a design flood problem is given in Griffiths (1989). However, in Griffiths (1989), three-parameter case of Wakeby distribution (γ = 0 and δ = 0) (equivalent to the three parameter generalized Pareto distribution) was employed.…”