A theoretically based Wakeby distribution for annual flood series

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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 quantite 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.

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