Methodology for Assessing Multiple-Coincidence of Flood Wave Peaks in Complex River Systems

 Stevan Prohaska1, Aleksandra Ilić2, Vesna Tripković1

1 Jaroslav Černi Institute for the Development of Water Resources, 80 Jaroslav Černi Str., 11226 Pinosava - Belgrade, Serbia; e-mail: This e-mail address is being protected from spambots. You need JavaScript enabled to view it

2 Faculty of Civil Engineering and Architecture, University of Niš, Niš, Serbia



Assessment of the statistical significance of future floods in highly complex hydrological circumstances associated with the formation of flood waves, such as those which exist at the confluences of a main river and its major tributaries, as well as the selection of hydrologic design parameters for the sizing of flood defenses in these areas, constitute a priority task in contemporary hydrological science. The previously followed conventional approach, which involves one-dimensional exceedance probability analyses, does not provide satisfactory results for a flood risk assessment in a situation where floods on two or more rivers do not occur simultaneously. This approach needs to be developed further, to define the flood probability in a downstream area within the multi-dimensional scope of flood probabilities at the entry profiles of the main river and its tributaries. In other words, it is necessary to address the multivariate coincidence of floods in a complex river system. A practical solution to this problem is the objective of the research presented in this paper. A procedure has been defined for the generation of multiple probability functions for flood peaks on the main river and several tributaries, and in the specific case such a solution is proposed for the determination of the multiple-coincidence of flood waves on the Danube River and its main tributaries - the Tisa and the Sava. At this stage of development of a mathematical model for multiple-coincidence assessments, the flood wave peak was adopted as a representative parameter of the significance of the flood. The paper is illustrated with practical examples of established correlations between partial and multiple coincidences of flood waves on the Danube, the Tisa and the Sava, and contains an assessment of several historic floods in the considered region of Serbia.

Keywords: multiple-coincidence, flood wave, flood wave peak, two-dimensional distribution function, cumulative probability function, exceeding cumulative probability.




Flood protection, including both preventative and recovery measures, may at times require spending of a considerable portion of the national income. An effective flood protection system is comprised of a set of complex structural and non-structural measures (construction and maintenance of hydraulic structures, optimized system management, zoning, insurance, flood forecasting and warning, and the like), which complement each other. The prevailing requirement is to spend flood protection funds efficiently, for which a comprehensive assessment of flood causes and effects is needed.

Total flood protection is obviously not possible. Even if adverse anthropogenic impacts on the onset and progress of a flood event are disregarded, there is always a risk that the capacity of the flood defenses, which ensure a certain level of safety, will be exceeded. A higher level of protection may require a significant increase in spending.

As such, an optimal degree of flood protection must be selected, to make construction and maintenance spending proportionate to potential damage.

Knowledge of high water is extremely important for flood protection of a threatened area and is of overriding significance for flood defenses in terms of both construction and safety. The flood wave peak (or the highest river discharge) is the design criterion for the sizing of a number of structures and the selection of their conveyance capacity. To assess the effect of a river reservoir or river channel on the transformation of a flood wave, in addition to maximum discharge, one needs to address the volume of the flood wave and the shape of the flood hydrograph. Since hydraulic structures are designed with the goal of providing a certain level of safety downstream, the determination of the design flood involves the definition of the maximum discharge and/or other characteristics of the flood wave which correspond to a certain probability of occurrence, or flood risk.



The conventional approach to flood risk assessments involves the determination of the probability that a flood may exceed a pre-defined magnitude of the considered flood wave characteristic. In effect, this is equivalent to the establishment of the return period of the flood. The procedure generally implemented in this regard includes a statistical-probabilistic analysis of hydrological data collected by a nearby gauging station. From an engineering perspective, this approach yields satisfactory results for a large number of tasks, particularly in addressing flood protection issues in relatively simple cases such as where there are no tributaries in the protected area.

However, if the protected area includes the mouths of a number of major tributaries, the above approach will not produce a reliable assessment of the considered flood wave characteristic. Namely, as a rule, floods on two or more rivers in relative proximity occur differently, such that their maximum flood wave characteristics do not appear at the same time. Further, a flood wave on one river may have a considerable influence on the flow regime of another river. Additionally, hydrological data are usually collected by gauging stations located beyond the zone of interaction between the considered rivers and, as a result, analyses of available data cannot produce an assessment of the influence of one river on another. In such circumstances, it is particularly important to assess the coincident occurrence of flood waves on the main river and the tributary. If there are a number of major tributaries, the problem becomes rather complex and requires comprehensive insight into the hydrological and hydraulic conditions in the entire region, where random variables need to be treated as multi-dimensional phenomena in order to fully define the risk as the most common criterion for the sizing of flood defenses. The two-dimensional (and even multi-dimensional) random variable theory, as applied in this work, represents an approach which allows the outlined difficulties to be overcome or, more precisely, it enables the determination of the probability of concurrent flood events on two or more rivers.

Methodology for Assessing the Multi-Dimensional Coincidence of Flood Waves

The theoretical background for the development of a methodology for assessing the multi-dimensional coincidence of flood waves on the main river and its tributaries is presented in detail in (Prohaska et al., 2008). The present paper makes full use of and expands on that work. For convenience, the same nomenclature is used and the theoretical expansion is presented in the final section of this chapter.

The term "coincidence", as used in the present context, means the probability of simultaneous occurrence of floods on two or more rivers. The theoretical background is based on the practical application of multiple probability distribution functions, or their conditional probabilities. As relevant variables (random variables), the simultaneous quantitative characteristics of flood wave hydrographs of the main river, and one or more major tributaries, are considered. In general, these flood parameters are: maximum hydrograph ordinate (peak), flood wave volume, and flood wave duration or time lag between flood wave peaks of the main river and the tributary.

As a starting point, the proposed methodology for assessing the multi-dimensional coincidence of flood waves in complex river systems uses the methodology developed for defining the coincidence of flood waves on two neighboring rivers (Prohaska, 1999). "Coincidence" denotes events occurring at the same time and is equivalent to the probability of simultaneous occurrence of two random variables, X and Y, which stand for the considered random events (flood parameters) on neighboring rivers.