Flood frequency analysis supported by the largest historical flood

Abstract. The use of non-systematic flood data for statistical purposes depends on the reliability of the assessment of both flood magnitudes and their return period. The earliest known extreme flood year is usually the beginning of the historical record. Even if one properly assesses the magnitudes of historic floods, the problem of their return periods remains unsolved. The matter at hand is that only the largest flood (XM) is known during whole historical period and its occurrence marks the beginning of the historical period and defines its length (L). It is common practice to use the earliest known flood year as the beginning of the record. It means that the L value selected is an empirical estimate of the lower bound on the effective historical length M. The estimation of the return period of XM based on its occurrence (L), i.e. ^M = L, gives a severe upward bias. The problem arises that to estimate the time period (M) representative of the largest observed flood XM.

From the discrete uniform distribution with support 1, 2, ... , M of the probability of the L position of XM, one gets ^L = M/2. Therefore ^M = 2L has been taken as the return period of XM and as the effective historical record length as well this time. As in the systematic period (N) all its elements are smaller than XM, one can get ^M = 2t( L+N).

The efficiency of using the largest historical flood (XM) for large quantile estimation (i.e. one with return period T = 100 years) has been assessed using the maximum likelihood (ML) method with various length of systematic record (N) and various estimates of the historical period length ^M comparing accuracy with the case when systematic records alone (N) are used only. The simulation procedure used for the purpose incorporates N systematic record and the largest historic flood (XM_i) in the period M, which appeared in the L_i year of the historical period. The simulation results for selected two-parameter distributions, values of their parameters, different N and M values are presented in terms of bias and root mean square error RMSEs of the quantile of interest are more widely discussed.