Witold G. Strupczewski

Non-stationary approach to at-site flood frequency modelling. I. Maximum likelihood estimation

For dealing with hydrological non-stationarity in flood frequency modelling (FFM) and hydrological design, it is necessary to account for trends. Taking the case of at-site FFM, statistical parametric techniques are discussed for investigation of the time-trend. The investigation entails (1) an identification of a probability distribution, and (2) development of a trend software. The Akaike Information Criterion (AIC) was used to identify the optimum distribution, i.e. the distribution and trend function, which enabled an identification of the optimum non-stationary FFM in a class of 56 competing models. The maximum likelihood (ML) method was used to estimate the parameters of the identified model using annual peak discharge series. A trend can be assumed in the first two moments of a probability distribution function and it can be of either linear or parabolic form. Both the annual maximum series (AMS) and partial duration series (PDS) approach were considered in the at-site frequency modeling.

Non-stationary approach to at-site flood frequency modelling. III. Flood analysis of Polish rivers

This paper, the last of a three-part paper, investigates into trends in annual maximum flows of Polish rivers using 39 flow series for a period of 70 years from 1921 to 1990. The gradients of trends in the mean and the standard deviation (SD) are estimated by the weighted least squares method and the best fitting linear model of trend is with the aid of the Akaike Information Criterion (AIC). It is shown that for every time series, a trend in the variance has a considerable effect on the trend estimators of the mean value. The analysis also includes seasonal peak flow series in order to obtain further insight into the detected non-stationarity of the peak flow series. Using the maximum likelihood method for parameter estimation and AIC for identification of an optimum model, the best fitting probability distribution and trend model are identified, following the methods discussed in part 1. Then, the 1% quantiles for the first and the last years of observation, denoted, respectively, as Q1%(1921) and Q1%(1990), are estimated and analysed. The influence of the distribution assumption on the values of estimators of time-dependent moments is demonstrated. In general, a decreasing tendency in both the mean and the SD of annual peak flow series is detected. This tendency is more pronounced on rivers with a high contribution of winter floods to the annual peak flow discharge series. Summer season peak flow series is found to be stationary.

Hydrodynamic derivation of storage parameters of the Muskingum model

Abstract The St. Venant equations for unsteady flow in open channels and the Muskingum method are written both in their conventional forms and in the state-space formulation. The hydrodynamic equation of motion is solved by the method of state trajectory variation and the result for the first-order variation in the state-space variables is used as a basis of linking the parameters of the Muskingum model with the hydraulic parameters of the open channel reach. The results are applicable to any shape of cross-section and to any type of friction law.

Non-stationary approach to at-site flood frequency modelling. II. Weighted least squares estimation

This is the second of the three-part paper and generalises the least squares method to the weighted least squares (WLS) method in order to deal with the trend in the first two moments. The generalised method applies when the assumption of constant variance does not hold and the functional form of a trend in the variance is given. In the generalised method, the parameters of trend in the mean and variance are estimated simultaneously. To keep the weights as power functions of variances only, the restrictions on distribution functions are formulated, which, in fact, are not difficult to fulfil in hydrological studies. It is shown that the WLS method coincides with the maximum likelihood method in the case of the normal distribution.

Asymptotic bias of estimation methods caused by the assumption of false probability distribution

Abstract Asymptotic bias in large quantiles and moments for four parameter estimation methods, including the maximum likelihood method (MLM), method of moments (MOM), method of L-moments (LMM), and least squares method (LSM), is derived when a probability distribution function (PDF) is falsely assumed. The first three estimation methods are illustrated using the lognormal and gamma distributions forming an alternative set of PDFs. It is shown that for every method when either the gamma or lognormal distribution serves as the true distribution, the relative asymptotic bias (RB) of moments and quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (cv), except that RB of moments for MOM is zero. The value of RB is the smallest for MOM and largest for MLM. The bias of LMM occupies an intermediate position. The value of RB from MLM is larger for the lognormal distribution as a hypothetical distribution with the gamma distribution being assumed to be the true distribution than it would be in the opposite case. For cv=1 and MLM, it equals 30, 600, 320% for mean, variance and 0.1% quantile, respectively, while for MOM, the moments are asymptotically unbiased and the bias for 0.1% quantile amounts to 35%. An analysis of 39 70-year long annual peak flow series of Polish rivers provides an empirical evidence for the necessity to include bias in evaluation of the efficiency of PDF estimation methods.

Muskingum method revisited

The assumptions of the Muskingum method and the opinions of hydrologists concerning the nature and the range of variability of model parameters are subject to revision. The particular topic of analysis is the interpretation of the parameter x of the Muskingum model. The scope of the present work has been attained by means of: (1) systems approach; (2) matching impulse responses of the Muskingum model and of the linear dynamic wave model; and (3) analysis of the nonlinear storage equation developed for uniform channels with rectangular cross-section under assumption of: (a) common looped rating curve for both the input and output from the reach and (b) linear changes of water table along the reach. It has been proved, that the range of variability of the parameter x reads (−∞,0.5]. The dependence of this parameter upon the physical characteristics of the systems and flow variables has been studied. The length of the characteristic reach for the Muskingum method and corresponding range of values of parameter x have been determined. The influence of the delay existing in the physical system upon the value of parameter x has been studied. Based on the nonlinear storage equation, the dependence of parameters of the Muskingum method upon the inflow and outflow rate has been analysed.

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf

Abstract Two probability density functions (pdf), popular in hydrological analyses, namely the log-Gumbel (LG) and log-logistic (LL), are discussed with respect to (a) their applicability to hydrological data and (b) the drawbacks resulting from their mathematical properties. This paper—the first in a two-part series—examines a classical problem in which the considered pdf is assumed to be the true distribution. The most significant drawback is the existence of the statistical moments of LG and LL for a very limited range of parameters. For these parameters, a very rapid increase of the skewness coefficient, as a function of the coefficient of variation, is observed (especially for the log-Gumbel distribution), which is seldom observed in the hydrological data. These probability distributions can be applied with confidence only to extreme situations. For other cases, there is an important disagreement between empirical data and theoretical distributions in their tails, which is very important for the characterization of the distribution asymmetry. The limited range of shape parameters in both distributions makes the analyses (such as the method of moments), that make use of the interpretation of moments, inconvenient. It is also shown that the often-used L-moments are not sufficient for the characterization of the location, scale and shape parameters of pdfs, particularly in the case where attention is paid to the tail part of probability distributions. The maximum likelihood method guarantees an asymptotic convergence of the estimators beyond the domain of the existence of the first two moments (or L-moments), but it is not sensitive enough to the upper tails shape.

The properties of the kernels of the Volterra series describing flow deviations from a steady state in an open channel

Abstract The deviation of the flow from a steady state in an open channel is described by a nonlinear state equation. This model is used to derive analytically the kernels of the Volterra series. The properties and the structure of the two first kernels are examined. The condition of convergence of the Volterra series depending on the magnitude of the inflow increase is also discussed.

Climate Change Impact on Hydrological Extremes: Preliminary Results from the Polish-Norwegian Project

This paper presents the background, objectives, and preliminary outcomes from the first year of activities of the Polish–Norwegian project CHIHE (Climate Change Impact on Hydrological Extremes). The project aims to estimate the influence of climate changes on extreme river flows (low and high) and to evaluate the impact on the frequency of occurrence of hydrological extremes. Eight “twinned” catchments in Poland and Norway serve as case studies. We present the procedures of the catchment selection applied in Norway and Poland and a database consisting of near-natural ten Polish and eight Norwegian catchments constructed for the purpose of climate impact assessment. Climate projections for selected catchments are described and compared with observations of temperature and precipitation available for the reference period. Future changes based on those projections are analysed and assessed for two periods, the near future (2021–2050) and the far-future (2071–2100). The results indicate increases in precipitation and temperature in the periods and regions studied both in Poland and Norway.

The distributed Muskingum model

Abstract This paper investigates the limiting form of the multiple Muskingum model when the number of reaches increases to infinity, while maintaining finite values for the first and second moments. Both the cumulants, and the amplitude and phase characteristics of this distributed Muskingum model (DMM) are derived. The model is compared to the solution of the linearised Saint-Venant equation for a semi-infinite uniform channel (LSV). The error of the DMM in predicting the third central moment of the LSV is shown to be independent of channel length in contrast to the classical Muskingum model in which the error increases rapidly with length of channel.