## Estimation of Coinciding Flood Discharges in the Extended Area of the Sava/Danube Confluence Applying Different Approaches – Proil Model and Copula Method

Stevan Prohaska^{1}, Aleksandra Ilić^{2}, Boris Pokorni^{1}

^{1} Institute for the Development of Water Resources "Jaroslav Černi", Belgrade, Serbia; E-mail:
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^{2} University of Niš, Faculty of Civil Engineering and Architecture, Niš, Serbia; E-mail:
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### Abstract

The junction of two major lowland rivers, such as the confluence of the Sava and the Danube, is highly problematic for determining design flood discharges for flood protection systems. Namely, in areas where a recipient and a tributary join, the flood process tends to be complex in view of the mutual influence of flood flows as a result of asynchronous maximum discharges of the two rivers. It is generally known that the creation and development of flood waves on neighboring rivers differs as a rule, such that the highest flood waves do not occur simultaneously. As a result, the mutual influence of the two rivers needs to be taken into account in order to properly select their design discharges, which is achieved by estimating the probability of simultaneous "coupled" events, or, in other words, the coincidence of maximum annual discharges of both rivers. The outcomes of a case study applying different procedures (methods) to estimate the coincidence of flood discharges in the extended area of the Sava/Danube confluence are presented in the paper. The estimation methods are only briefly described, given the fact that their theoretical backgrounds are explained in detail in cited references. In essence, two approaches are used: (i) estimation of exceedance probability in two-dimensional space, referred to as the PROIL model, and (ii) use of three families of so-called Archimedean copulas (COPULA): Gumbel's, Clayton's and Frank's, to determine common probability distributions of different random variables in multidimensional space of probabilities. The coincidence of flood discharges of the Sava and the Danube is estimated applying the above approaches to the extended area of their confluence, from the gauging stations at Sremska Mitrovica on the Sava and at Slankamen and Smederevo on the Danube. The results are presented numerically, via marginal probabilities of the considered combinations of variables for which coincidences are defined, as well as graphically by means of parallel lines of exceedance probability for all the cases addressed applying both methods.

**Keywords:** flood discharge coincidence, two-dimensional random variable, PROIL model, copulas, flood wave peak, exceedance probability.

Introduction

The selection of design flood discharges in the extended zone of the confluence of a recipient and a tributary, for designing river training works and planning structural flood protection measures, is complex from the viewpoint of both the origin of flood waves on the recipient and tributaries and the possible diverse mutual influences of the slowed-down flow on the water levels of both rivers.

As such, there is no realistic basis for the commonly followed procedure to select the highest discharge (flood wave peak) of a certain return period, estimated from historic data on maximum annual discharges, which is unfortunately common in both domestic and international hydraulic engineering practice. Given that the aim of hydraulic structures is to ensure the required downstream safety, the determination of design flood discharges needs to be reduced to the definition of maximum discharges and/or other flood wave characteristics of a certain probability of occurrence, which will provide a full range of possible mutual influences of the flood discharges of the recipient and the tributary.

New flood protection concepts discard old ideas of sustainability and instead of a static state opt for transience, flexibility and decentralization as the key features of the system to be developed.

Based on global, regional and local climate change research, a new water management approach is needed, which closely resembles how nature itself functions.

From the viewpoint of flood waves on river reaches that include the mouth of a tributary, the definition of multivariate dependency between parameters that describe the flood hydrograph (in this case maximum discharge) determines the real space in which the selected parameters in different combinations of probabilities of occurrence can be found. These dependencies are very convenient for assessing the probabilistic significance of both historic and future floods, and can also be used to produce real-time forecasts at the output cross-section of the recipient when hydrograph predictions at input cross-sections of the recipient and the tributaries are known.

Bender et al. (2016) cite several studies that deal with probability distribution in two-dimensional space, such as those of Moris and Calise 1987, Raynal and Salas 1987, but the list should definitely include chapters in monographs that address the problem in case studies of the Danube and its major tributaries (Prohaska et al., 1999; Prohaska and Ilić, 2010). All these studies point out that exceedance probabilities in two-dimensional space are derived from pre-defined probability distribution functions, which has generally been a constraint in practice.

Increasingly popular copulas have superseded the conventional approach of the pre-defined probability distribution function of a random variable and allows it to be selected from different groups of distributions.

A comparative analysis of the PROIL model and the COPULA method, as used to determine the design discharge for sizing flood protection systems in the immediate zone of influence of the recipient and the tributary, specifically the Danube and the Sava, is presented in the paper.

Calculation Methods

Past hydrological practice involved sizing of flood defenses based on univariate assessment of the exceedance probability. Solely hypothetically, a literature source (Prohaska, Marjanović, Čabrić, 1978) introduces the probability distribution of a complex multidimensional event calculated on the basis of individual (marginal) distributions of each component of the random process, for which a normal or log-normal distribution is assumed as a rule. The measure of correlation between two random variables is the correlation coefficient determined from measured/standardized values.

The problem of determining the simultaneous occurrence of random variables is solved graphically, using nomograms (Abramowitz and Stegun, 1972). The resulting probability is in fact the probability of exceedance of an event with pre-selected combinations of random variables. The outcome is a grid of points, each characterized by a probability of occurrence of the defined combination, based on which same-probability lines are calculated. This approach has encountered numerous difficulties related to defining of a common distribution based on ad hoc marginal probabilities. The entire concept of defining exceedance functions of pre-selected combinations of random variables is based on the grapho-analytical approach packed into a single software application called the PROIL model.

The difficulties and constraints of this approach began to be addressed by the use of copulas, an approach in the probability theory which determines common distributions of multidimensional random variables.

Marginal Probabilities

To determine the distribution density, f(x,y), the first step is to establish the marginal probabilities* f(x,•)* and *f(•,y)* as:

(1)

(2)

Then their cumulative probabilities are:

(3)

and

(4)

Several known probability distribution functions that will be tested as possible marginal probabilities need to be extracted.

The standard condition in the commonly used approach to determine the exceedance probability of a selected flood wave parameter in two-dimensional probability space is that the selected random variables adhere to the same probability distribution law. In this regard, Pearson Type 3 (P3), Log-Pearson Type 3 (LP3), Extreme Value Type 1 – Gumbel (EV1) and Log-normal (LOGN) were selected as potential marginal distributions. The distribution parameters were assessed by the conventional method of moments (MOM). The goodness-of-fit of the selected theoretical distribution functions to the empirical data was assessed by the χ2, Kolmogorov-Smirnov and nω2 tests.

When the copulas were constructed, the group of possible marginal distributions was expanded to include the General Extreme Value (GEV) probability density distribution. The distribution parameters were determined applying the L-moment (LMOM) method. L-moments are based on the idea of conventional moments, but modified to make use of weight functions as probabilities (Hosking, 1990). The reason for using them to determine the distribution function parameters is that they are less sensitive to sample outliers and in the case of small samples provide a more accurate assessment of the distribution parameters. The goodness-of-fit of the distribution function to the sample was assessed by the the Kolmogorov-Smirnov and Darling-Anderson tests, as well as the Root Mean Square Error (RMSE).

Exceedance Probability in Two-dimensional Probability Space

Bivariate normal distribution is a distribution whose probability density is defined as (Prohaska, Marjanović, Čabrić, 1978):

(5)

where:

*x* and *y* – simultaneous occurrence of random variables X and Y, respectively;

*μ _{x}* and

*μ*– mathematical expectations of X and Y;

_{y}*σ _{x}* and

*σ*– standard deviations of X and Y;

_{y}*ρ* – correlation coefficients of X and Y.

The cumulative probability distribution, *F(x,y)*, is defined as:

(6)

The next step is to determine the exceedance probability, *Φ(x,y)*, in two-dimensional space of probabilities (Prohaska, Marjanović, Čabrić, 1978):

(7)

The use of bivariate probability distributions in statistical analysis of different flood hydrograph parameters requires simplification for the described calculation procedure to be applicable.

The main simplification is related to the assumption that each of the considered hydrograph parameters adheres to the normal (log-normal) distribution law, which need not be the case. The detailed theoretical background of defining bivariate distribution functions applying the grapho-analytical method (Abramowitz and Stegun, 1972) can be found in the literature (Prohaska et al., 1999).

The strength of correlation is determined by the relation between the coefficient of linear correlation R and the standard correlation coefficient error σR (Yevjevich, 1972):

(8)

Copulas

The mathematical model is based on multivariate probability distribution functions, or their conditional probabilities. For relevant random variables, the simultaneous parameters of flood hydrographs of the recipient and the tributary are considered.

The complexity of the flood wave rise phenomenon and its assessment require relating of the marginal distributions of a number of variables to the goal of defining a common distribution that describes the flood.

Copulas are tools used to determine the correlations among multiple random variables. The origin of the word "copula" is Latin (copulare), meaning connection or linking. It was introduced by Sklar in 1959 and the main purpose was to describe the connection between several random variables (Schmith, 2006).

Improvements in applied mathematics have shown that copulas are useful tools for studying dependent variables. The use of copulas departed from the conventional approach, which involves a predefined probability distribution function of a random variable, and enabled selection from different groups of distributions.

Such an approach includes modeling of the probability of occurrence in multivariate space, where the correlation of the individual terms is modeled independently of the probability distribution law to which each of them adheres individually. The ultimate goal is to determine conditional probability distributions and the return period of a flood as a complex phenomenon (Nelsen, 2006).

Let* (X _{1}, X_{2}, ... , X_{N}) *∈

*R*be a random vector whose cumulative distribution is:

^{N}and whose marginal distributions are:

The copula C of the distribution F is defined as [0,1]^{N} and

The existence and uniqueness of copula C are ensured under certain conditions, which are met in the cases considered here. Several construction methods are available, but the most commonly followed approach involves known families of copulas that depend on a single parameter, determined on the basis of a sample.

As already stated above, the variables can have different marginal distributions and in view of the entire structure of the correlations among the variables in the copula, it is separate from the type of the marginal distribution of each dependent variable.

Three families of copulas were used in the present research: Gumbel's, Clayton's and Frank's, which belong to the group of "Archimedean copulas". All are single-parameter copulas and the parameter is *θ* (Table 1) (Nelsen, 2006).

In order to determine the exceedance probability of two uniformly-distributed unit random variables with copula C(u,v,θ), the copula C (so-called survival copula) is calculated from the relation:

(12)

Assessment of the Copulas' Goodness-of-fit to the Sample

The goodness-of-fit of the copula to the sample was assessed on the basis of the statistical test (Reddy, Ganguli, 2012) initially proposed by Genest, Remillard and Beaudoin in 2009.

The test is founded upon the parametric "bootstrap" method. For each copula *n* random independent samples are generated and a synthetic time series determined and then compared with synthetic time series generated by a Monte Carlo simulation using measured data. The test is based on actual data and does not depend on the selected marginal probability.

The size of the sample in the present research is *n=10000*.

The value* p* is used to assess the statistical significance. This value is calculated based on the Cramer von Mises statistic *Sn* (Reddy, Ganguli, 2012) and compared with the α level of significance.

If *p<α*, the result is statistically significant. If not, it is not statistically significant. The greater the value of *p*, the better the copula description of the correlation among the time series.

Study Area and Input Data

Study Area

The described approach for the calculation of coinciding flood discharges is most often used in the area of flood protection related to complex river systems, where the mutual influence of the recipient and the tributary is dominant. Namely, in such cases, like that of the Sava/Danube confluence in the present study, it is very important to take into account the nature of the simultaneous occurrence (coincidence) of flood peaks when sizing flood protection measures.

The primary criterion for flood defenses within the zone of the Sava/Danube confluence is economical sizing of all structural measures (Prohaska and Petković, 1989). In the present case study the main structural measures are levees (dykes). This sector of the Danube River, from the hydrologic gauging stations at Slankamen on the Danube and Sremska Mitrovica on the Sava to Smederevo on the Danube, is shown in Fig.1.

Theoretical river discharges of different return periods at the above gauging stations were used for sizing of the flood protection system and they were derived applying the conventional procedure for statistically significant coincidence using maximum annual discharge time-series from the period 1931-2014.

For a river reach whose water levels are a function of water level fluctuations of another river (recipient and tributary), the selection of design discharges for sizing flood protection measures directly depends on the coincidence (exceedance probability line) for a combination of variables, as set out below.

Figure 1: The Danube from Slankamen to Smederevo.

Definition of Variables

The theoretical (design) water levels in the extended area of the confluence are derived by hydraulic analysis of water level lines, based on adopted boundary conditions and adopted design flow rates.

The following data need to be available in order to define the design water levels:

a) time-series of maximum annual discharges at the entry cross-sections (of the recipient and the tributary) and the exit cross-section (of the recipient), and

b) results of flood coincidence calculations performed using the following combinations of variables:

- maximum annual discharge of the recipient – maximum annual discharge of the tributary;

- maximum annual discharge of the recipient – corresponding discharge of the tributary, and

- maximum annual discharge of the tributary – corresponding discharge of the recipient.

The dependencies of the combinations of variables shown in Table 2 (Prohaska et al., 1999) were used to analyze the coincidence of flood discharges of the recipient and the tributary. The pertinent data relates to the gauging stations at Slankamen (QIN) and Smederevo (QOUT) on the Danube and the gauging station at Sremska Mitrovica (QTR) on the Sava, from 1931 to 2014.

The results of the coincidence calculations are the same-probability line of the combinations of the selected flood wave parameter (differential distribution) and lines that define the exceedance probability of the same combinations of variables.

Assessment of Coinciding Flood Discharges of the Danube and the Sava

Configuration of Flood Defenses in the Extended Zone of the Sava/Danube Confluence

The main assumption is that the primary flood defenses in the extended zone of the Sava/Danube confluence are levees (dykes). As a rule, levees are longer than the zone of mutual influence of the Danube and the Sava at times of flood discharges, in this case from the entry gauging stations at Sremska Mitrovica on the Sava and Slankamen on the Danube to the exit cross-section at Smederevo on the Danube. The study area is schematically represented in Fig.2.

The coincidence of flood discharges along the considered sector of the Sava/Danube confluence was estimated applying two approaches – the PROIL model and the COPULA method – to define the design water levels.

Figure 2: Schematic representation of the flood protection sector in the extended zone of the Sava/Danube confluence.

Results of the PROIL model

Coinciding flood discharges of the Danube and the Sava

If the conventional approach is followed, without taking into account coinciding flood discharges, the basis for levee sizing would be theoretical river discharges of different return periods. These values at the studied gauging stations on the Danube and the Sava are shown in Table 3 according to the assessment methods described in Subsection 2.1.

However, immediately upstream from the confluence, within the zone of mutual influence of the two rivers, the design discharges for levee sizing are not those shown in Table 3, but derived quantities that depend and the strength of the coincidence of flood discharges of the Danube and the Sava, according to the criteria shown in Table 2. In principle, the optimal approach is to adopt the most likely constellation of coincidences of the discharges of the Danube and the Sava, from the coincidence exceedance curve for the selected level of safety (i.e. return period).

In the present case study, the coincidence of flood discharges of the Danube and the Sava was estimated for the following constellations of variables:

The results of estimation of coinciding flood discharges of the Danube and the Sava, applying the methods described in Section 2, are graphically represented in Figs. 3–11. They show exceedance probability lines (bivariate distribution functions) and the Gumbel copula, as well as the corresponding empirical points.

Figure 3: Coinciding maximum annual discharges of the Danube at Slankamen and Smederevo.

Figure 4: Coinciding maximum annual discharges of the Danube at Slankamen and corresponding discharges at Smederevo.

Figure 5: Coinciding maximum annual discharge of the Danube at Smederevo and corresponding discharge at Slankamen.

Figure 6: Coinciding maximum annual discharge of the Danube at Slankamen and maximum annual discharge of the Sava at Sremska Mitrovica.

Figure 7: Coinciding maximum annual discharge of the Danube at Slankamen and corresponding discharge of the Sava at Sremska Mitrovica.

Figure 8: Coinciding maximum annual discharge of the Sava at Sremska Mitrovica and corresponding discharge of the Danube at Slankamen.

Figure 9: Coinciding maximum annual discharge of the Danube at Smederevo and maximum annual discharge of the Sava at Sremska Mitrovica.

Figure 10: Coinciding maximum annual discharge of the Danube at Smederevo and corresponding discharge of the Sava at Sremska Mitrovica.

Figure 11: Coinciding maximum annual discharge of the Sava at Sremska Mitrovica and corresponding discharge of the Danube at Smederevo.

To assess the statistical significance, Table 4 shows the basic indicators of the strength of the established coincidence correlations, including the linear correlation coefficient and the standard correlation coefficient error according to Eq. (8). The plus sign indicates that the considered coincidence constellation is statistically significant.

The results of the copula's goodness-of-fit to the sample are shown in Table 5, based on the approach described in Subsection 3.4. The Gumbel copula best described the dependency of the variables defined in Subsection 3.2.

The results lead to the conclusion that there is a statistically significant coincidence between the maximum annual discharges of the Sava at Sremska Mitrovica and the corresponding discharges of the Danube at Smederevo. In general, the values in Table 4 are within limits of statistical significance which, according to the set criteria, fully corroborate the validity of applying the proposed approach to estimate the flood discharge coincidence in the studied sector of the Sava/Danube confluence.

Selection of design discharges for defining the water level line in the considered sector of the Danube and the Sava

According to the PROIL model

The design discharge combinations for the extended zone of the studied confluence were determined using the graphics of coinciding flood discharges of the Sava and the Danube of various probabilities (Subsection 4.2.1). The numerical values of design discharges along characteristic river reaches (upstream and downstream of the confluence) are shown in Table 5.

According to the COPULA method

To follow in parallel the coinciding flood discharges of the Sava and the Danube derived from the two models, the results are presented analogously. Figures 3 through 11 are graphical representations of all the considered combinations (constellations) of variables discussed in Subsection 4.2.1, for Gumbel's copula that best describes the real data. The corresponding numerical values of design discharges at the considered reaches (upstream and downstream from the confluence) are shown in Table 7.

Discussion

Practical application of the coincidence probabilities (Tables 6 and 7) in the case of sizing of levees in the extended area of the Sava/Danube confluence, for a safety level that corresponds to a 100-year return period, is comprised of the following:

The adopted design discharges for calculating the 100-year water surface level along the entire sector of the Danube from Slankamen to Smederevo are schematically represented in Fig 12.

Figure 12: Maximum discharges for calculating the 100-year water surface levels in the studied sector of the Danube.

For the reach of *the Sava upstream from its confluence with the Danube,* within the zone of their mutual influence, the design water level is the envelope of maximum water levels derived from water surface level calculations, based on discharges resulting from the following combinations of variables (values in parentheses refer to Gumbel's copula):

The adopted design discharges for calculating 100-year water surface levels in the sector of the Danube from Smederevo to its confluence with the Sava and up the Sava to Sremska Mitrovica is schematically represented in Fig. 13.

Figure 13: Design maximum discharges for calculating 100-year water surface levels along the Danube to its confluence with the Sava and up the Sava to Sremska Mitrovica.

Conclusion

The results of this research led to the conclusion that there is a statistically significant coincidence of the considered combinations of maximum discharges of the Danube at Smederevo and the maximum and corresponding discharges of the Sava at Sremska Mitrovica (Table 4).

Parallel representations of the exceedance lines of all the considered coincidences within the sector of the Danube from Slankamen to Smederevo in Serbia, including its confluence with the Sava, show that the isolines of bivariate probability distributions generally exceed those based on the copula theory for all return periods (Figs. 3 through 11). However, with regard to design discharges for sizing flood protection measures, Gumbel's copula showed higher values (Fig. 12), or somewhat lower values of corresponding discharges (Fig. 13).

The correlations established in Section 4 are an indicator of the inevitability of abandoning the conventional approach to design flood discharges for sizing flood defenses along river reaches joined by tributaries, based on univariate probabilities of occurrence of maximum annual discharges at gauging stations. The present study showed that the values derived from examining a flood within the zone of a recipient and a tributary are lower and this allows engineers to design structural measures more economically and thus leave enough room for non-structural flood protection measures which are becoming increasingly important. The authors of this paper recommend that floods be viewed as a complex phenomenon and that other flood wave parameters (e.g. volume) also be considered, in multidimensional space.

Acknowledgment

The presented analyses and results originate from the research project "Assessment of Climate Change Impact on Serbia's Water Resources" (TR-37005), 2011–2017, of the Serbian Ministry of Education and Science. The authors express their gratitude to the ministry for its financial assistance and support.

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