Iwona Markiewicz

Generalized Exponential Distribution in Flood Frequency Analysis for Polish Rivers

Many distributions have been used in flood frequency analysis (FFA) for fitting the flood extremes data. However, as shown in the paper, the scatter of Polish data plotted on the moment ratio diagram shows that there is still room for a new model. In the paper, we study the usefulness of the generalized exponential (GE) distribution in flood frequency analysis for Polish Rivers. We investigate the fit of GE distribution to the Polish data of the maximum flows in comparison with the inverse Gaussian (IG) distribution, which in our previous studies showed the best fitting among several models commonly used in FFA. Since the use of a discrimination procedure without the knowledge of its performance for the considered probability density functions may lead to erroneous conclusions, we compare the probability of correct selection for the GE and IG distributions along with the analysis of the asymptotic model error in respect to the upper quantile values. As an application, both GE and IG distributions are alternatively assumed for describing the annual peak flows for several gauging stations of Polish Rivers. To find the best fitting model, four discrimination procedures are used. In turn, they are based on the maximized logarithm of the likelihood function (K procedure), on the density function of the scale transformation maximal invariant (QK procedure), on the Kolmogorov-Smirnov statistics (KS procedure) and the fourth procedure based on the differences between the ML estimate of 1% quantile and its value assessed by the method of moments and linear moments, in sequence (R procedure). Due to the uncertainty of choosing the best model, the method of aggregation is applied to estimate of the maximum flow quantiles.

Comparison of Two Nonstationary Flood Frequency Analysis Methods within the Context of the Variable Regime in the Representative Polish Rivers

Changes in river flow regime resulted in a surge in the number of methods of non-stationary flood frequency analysis. Common assumption is the time-invariant distribution function with time-dependent location and scale parameters while the shape parameters are time-invariant. Here, instead of location and scale parameters of the distribution, the mean and standard deviation are used. We analyse the accuracy of the two methods in respect to estimation of time-dependent first two moments, time-invariant skewness and time-dependent upper quantiles. The method of maximum likelihood (ML) with time covariate is confronted with the Two Stage (TS) one (combining Weighted Least Squares and L-moments techniques). Comparison is made by Monte Carlo simulations. Assuming parent distribution which ensures the asymptotic superiority of ML method, the Generalized Extreme Value distribution with various values of linearly changing in time first two moments, constant skewness, and various time-series lengths are considered. Analysis of results indicates the superiority of TS methods in all analyzed aspects. Moreover, the estimates from TS method are more resistant to probability distribution choice, as demonstrated by Polish rivers’ case studies.

Flood Quantile Estimates Related to Model and Optimization Criteria

Flood frequency analysis (FFA) provides information about the probable size of flood flows. Empirical methods are more commonly employed in engineering design and planning, and among empirical methods the at-site frequency analysis is by far the most commonly used method. When applying the methods of at-site flood frequency analysis, it is clear that the role of hydrology seems minor at best and the role of statistics seems to be the lead one, whereas it should be the other way round. FFA entails the estimation of the upper quantiles of an assumed form of a probability density function of the annual or partial duration maximum flows, as the true function is not known. In the paper, the five two-parameter models and their three-parameter counterparts have been assumed successively for describing the annual peak flows for Nowy Targ gauging station on the Dunajec River. The 1 % quantile has been estimated by four optimization criteria. To find the best fitting model, three discrimination procedures have been applied. The best fitting model and, thus, hydrological design value depends on the optimization criterion and the procedure of discrimination. It is characteristic for hydrological size of samples. At the same time, the designers of the hydraulic structures want to have a unique value, not accepting the uncertainty. It seems essential that we should go back and start examining the way in which we have been doing the hydrological frequency analysis.