Abstract
The assessment of rare maximum streamflow is one of the key tasks in engineering hydrology, since design discharges with exceedance probabilities of 1 %, 0,5 %, and lower are directly used in the design and operation of hydraulic structures, bridges, culverts, and flood protection systems. In engineering practice, such characteristics are commonly determined using parametric statistical distributions with extrapolation of empirical series into the domain of low exceedance probabilities. However, when observation records are of limited length and the distribution of maximum streamflow exhibits pronounced asymmetry, such extrapolation is associated with substantial statistical uncertainty and may lead to methodologically incorrect and physically difficult-to-interpret results. The methodological advantage of the truncated approach was assessed using RMSE criteria within the statistically reliable portion of the distribution, stability of tail extrapolation, and physical interpretability of calculated discharge values. Quantitative assessment of statistical uncertainty was performed using confidence intervals of exceedance probability under finite observation record length conditions. This study examines methodological approaches to the assessment of rare annual maximum streamflow of lowland rivers in Kazakhstan under conditions of statistical uncertainty. The analysis is based on long-term records of annual maximum discharges for the Zhaiyk (Ural), Esil, Tobol, and Ilek rivers. Along with the classical parametric approach based on the Pearson type III distribution, a truncated graphical–analytical method for constructing frequency curves and the generalized extreme value (GEV) distribution were applied. It is shown that the use of the Pearson type III distribution leads to a systematic underestimation of design discharges with 1% exceedance probability by 12.5...25% compared to the truncated frequency curve. It is also established that the application of the GEV distribution parameterized by the maximum likelihood estimation (MLE) method for lowland rivers may result in unstable behavior of the distribution tail and overestimation of rare design discharges, whereas GEV parameterization based on L-moments provides more stable estimates that are consistent with the results obtained using the truncated approach. The results demonstrate that for observation records with lengths of approximately 80...120 years, deep statistical extrapolation into the domain of exceedance probabilities below 1% is characterized by high uncertainty. This provides a methodological justification for limiting extrapolation when estimating rare maximum streamflow.