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1

A Two-Parameter Extension of Urbanik’s Product Convolution Semigroup

Berg Christian

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 2

441--458

EN

We prove that sn(a, b) = Γ(an + b)/Γ(b), n = 0, 1,…, is an infinitely divisible Stieltjes moment sequence for arbitrary a, b > 0. Its powers sn (a, b)c, c > 0, are Stieltjes determinate if and only if ac ≤ 2. The latter was conjectured in a paper by Lin (2019) in the case b = 1. We describe a product convolution semigroup τc(a, b), c > 0, of probability measures on the positive half-line with densities ec (a, b) and having the moments sn (a, b)c. We determine the asymptotic behavior of ec (a, b)(t) for t → 0 and for t → ∞, and the latter implies the Stieltjes indeterminacy when ac > 2. The results extend the previous work of the author and López (2015) and lead to a convolution semigroup of probability densities (gc (a, b)(x))c>0 on the real line. The special case (gc (a, 1)(x))c>0 are the convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities gc (a, b)(x) lead to determinate Hamburger moment problems.

2

Application of the maxima distribution to the evaluation of the variability of flood risks in the Odra River basin on the basis of quarterly maxima of daily water levels

Szałata Łukasz, Kuźmiński Łukasz, Zwoździak Jerzy

Environment Protection Engineering

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2019

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Vol. 45, nr 2

127--140

EN

The method of assessing the flood risk variability has been presented based on maximum value distributions. Hydrological data were used in the form of daily water levels for the period 1981-2013. The collected data originate from the Malczyce Hydrological Station on the 300th km of the Odra River. To assess the risk of flooding based on the collected data quarterly highs were selected. As a measure of flood risk, the authors suggested the probability of exceeding the emergency level in the studied section of the river. This risk was calculated using the theoretical distribution function of the quarterly heights of the water level. The study used the Gumbel distribution. A special attention was paid to the possibility of using the presented solutions for an integrated flood risk management process in accordance with applicable national and European legislation.

3

Odra River in Lower Silesia: probabilistic analysis of flood risk dynamics as part of sustainable development of water management

Kuźmiński Łukasz, Halama Arkadiusz

Managerial Economics

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2018

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Vol. 19, no. 2

205--225

EN

One of most common natural catastrophes in Poland are undoubtedly floods. Climatic change contributes to more and more often and violent occurrences of the maximum flow in rivers, which increases flood damage. Inadequate land management and the unjustified belief in the effectiveness of technical flood control measures can also contribute to flood damage. The development of water management (including flood protection) should be carried out in a sustainable way by integrating social, environmental, and economic objectives. In flood protection, those measures that are least invasive to the natural environment should be used first; in particular, non-technical flood protection methods (e.g., flood risk assessment and management, and the proper definition and management of flood plains). One of the bases for the sustainable development of water management is the preparation of models that can help us calculate the likelihood of maximum flow and to identify areas that are at risk of flooding. On this basis, the proper spatial policy and prevention of flood effects will be possible. This article presents the probabilistic analysis carried out on the flood risk dynamics for a selected area of the Odra River basin. The authors based their risk dynamic assessment on the results from the distributions of the maximum values for a selected hydrological characteristic – the flow rate. Based on the daily flow data from the years of 1994–2013 collected at a hydrological station on the Odra River in Malczyce, the 30-day flow maxima were set individually for four 5-year periods. Then, a probabilistic model of the maximum flow was developed based on these peaks for each 5-year period. The resulting models were used to estimate flood risks and for analyzing the dynamics for the studied area.

4

Comparison of Design Wave Heights Determined on the Basis of Long- and Short-term Measurement Data

Szmytkiewicz M., Szmytkiewicz P., Marcinkowski T.

Archives of Hydro-Engineering and Environmental Mechanics

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2018

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Vol. 65, nr 2

123--142

EN

The objective of this study is to determine differences between design wave heights determined on the basis of short- and long-term wave trains. Wave parameters measured over a period of 7.47 years in the vicinity of Coastal Research Station Lubiatowo were used as short-term wave data, while wave parameters determined through the so-called wave reconstruction for a period of 44 years for the same region and depth were used as long-term wave data. The results of the calculations lead to the conclusions the significant wave height distributions obtained for short and long-term wave data are similar.

5

Zastosowanie wybranych rozkładów prawdopodobieństwa wartości ekstremalnych do szacowania ryzyka wystąpienia zagrożenia powodziowego w dorzeczu Odry na Dolnym Śląsku

Kuźmiński Ł., Szałata Ł., Zwoździak J.

Ochrona Środowiska

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2016

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Vol. 38, nr 3

35--39

PL

Analizie poddano maksymalne dobowe przepływy w Odrze zmierzone w latach 1971–2013 w stacji hydrologicznej w Malczycach. Z całego zbioru danych wyselekcjonowano maksymalne przepływy roczne i do ich rozkładu empirycznego dopasowano trzy rozkłady teoretyczne – Gumbela, Frecheta i logarytmiczno-normalny. Przedstawiony przykład analizy probabilistycznej zagrożenia powodziowego w dorzeczu Odry na Dolnym Śląsku może być skutecznym narzędziem w ocenie wystąpienia ryzyka powodziowego. Otrzymane w ten sposób wyniki mogą zostać wykorzystane do szacowania prawdopodobieństwa wystąpienia określonych zdarzeń związanych z zagrożeniem powodziowym, a także do porównywania wybranych obszarów dorzecza pod względem zagrożenia powodziowego.

EN

The maximum daily flows in the Oder river from the period of 1971 to 2013 were analyzed in Malczyce hydrological station. From the entire data set the maximum annual flows were selected and their empirical distributions were fitted against the three theoretical distribution models: Gumbel, Frechet and the lognormal. The presented case of probabilistic flood risk analysis for the Oder catchment in Lower Silesia may become an effective flood risk assessment tool. The results may be employed to probability assessment of certain events related to flood risk and also to compare the flood risk in selected catchment regions.