Multidimensional Approaches to Calculation of Design Floods at Confluences—PROIL Model and Copulas


In confluence areas, where river flow slowdowns can be substantial, the selection of design flows for the purpose of designing a flood control system directly depends on flood waves at the mainstem and its tributaries. The aim of this paper is twofold: to elaborate a procedure that will define the flood exceedance probability in the analyzed area within a multidimensional probability space and to develop a methodology for defining flood coincidence in the mainstem reach with its main tributaries and thus determine design flows in order to design the flood control system in the confluence area. Mathematical models are based on (1) two-dimensional probability distribution (PROIL model) and (2) the copula function (Archimedean class of copulas), and fitted to a practical application in a two-dimensional probability distribution space. In case of random variables, simultaneous quantitative characteristics of flood wave (water flow) hydrographs for the mainstem and a tributary are considered. The paper discusses the specific reach of the Danube, from the Hungarian-Serbian border to the city of Smederevo, which encompasses three significant tributaries—the Drava, the Tisa, and the Sava.

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The authors express their gratitude to the Ministry of Education, Science and Technological Development of the Republic of Serbia for its financial assistance and support.


The presented analyses and results are part of the outcomes of the research project “Assessment of Climate Change Impact on Serbia’s Water Resources” (TR-37005), 2011–2019, of the Ministry of Education, Science and Technological Development of the Republic of Serbia.

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Aleksandra Ilić designed the initial research, conducted research and wrote initial manuscript except Introduction chapter. Stevan Prohaska checked the statistical analysis and results. Dragan Radivojević presented the results graphically. Slavisa Trajković contributed in terms of improving the initial research, writing Introduction chapter, improving the written language of the manuscript and checking the overall logical flow of the manuscript. In addition, the authors pointed out necessary comments towards improving the quality of the final manuscript.

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Correspondence to Aleksandra Ilić.

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Appendix 1. Marginal Probabilities Q 1%

See Table 5, 6 , and 7.

Table 5 Marginal probabilities Q1% for node 1
Table 6 Marginal probabilities Q1% for node 2
Table 7 Marginal probabilities Q1% for node 3

Appendix 2. Copula Properties

See Table 8.

Table 8 Correlation coefficients, parameter of copulas, and goodness of fit p value

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Ilić, A., Prohaska, S., Radivojević, D. et al. Multidimensional Approaches to Calculation of Design Floods at Confluences—PROIL Model and Copulas. Environ Model Assess (2021).

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  • Confluence
  • PROIL model
  • Archimedean copulas
  • Design flood level
  • Flood control system