Flood Development Process Forecasting Based on Water Resources Statistical Data

The Ukrainian Carpathians is the territory with a great threat of floods. This is due to natural and climatic conditions of this region, which is characterized by mountainous terrain, high density of hydrological network and a significant amount of precipitation. Amount of precipitation here ranges from 600 mm on plains to 1,600 mm on mountain tops. The main factors of floods occurrence are excessive precipitation, low water permeability of soil and a high proportion of low-permeability rocks (flysch layers with a predominance of clay layers). Therefore, catastrophic floods in the region were also observed in previous centuries, when the anthropogenic impact on the environment, including forest ecosystems, was not comparable with the current one. Any flood is characterized by a period of development, a period of its critical (maximum) intensity and a period of decline. In the present paper, based on the use of methods for approximating the curves and the results of experimental studies of flood waters, a method of mathematical description and forecasting of the flood development is suggested. The recommended direction of further research may be related to the development of experimental means to determine the parameters that affect the process of flood formation.


Introduction
The Ukrainian Carpathians is the territory with a great threat of floods.This is due to natural and climatic conditions of this region characterized by mountainous terrain, high density of hydrological network and a significant amount of precipitation.Amount of precipitation here ranges from 600 mm on plains to 1600 mm on mountain tops.The main factors of flood occurrence are excessive amount of precipitation, low water permeability of soil and a high proportion of low-permeability rocks (flysch layers with a predominance of clay layers).Therefore, catastrophic floods in the region were also observed in previous centuries, when the anthropogenic impact on the environment, including forest ecosystems, was not comparable with the current one (Figure 1) (Mandryk et al., 2017;Arkhypova et al., 2019).The rivers of the Carpathian which have flood hazard area belong to transboundary waters.In the basins of Carpathian rivers, some part of which belong to neighboring countries, the European Union Water and Flood Directives are implemented in accordance with the Association Agreement between Ukraine and the European Union.These directives are implemented with the funding from European Union (Prykhodko et al., 2020;Kinash et al., 2019).
The estimation of the level of flood waters is an urgent scientific and technical problem in the context of complex geo-climatic processes, which lead to catastrophic floods, taking place in European countries (such as the floods of Ukraine in 1927, 1941, 1969, 1974, 1980, 1998, 2001, 2008, 2010and 2020, of Poland and Slovakia in 2012, of the Czech Republic and Great Britain in 2013, and of Germany in 2002).The stated problem is addressed by a wide range of scientists studying and solving the applied problems by using precise models of filtrating flows (Oliynyk and Panchuk, 1992;Oliynyk and Steyer, 2012) and empirical dependences (Mandryk, Pukish and Zelmanovych, 2017;Maslova and Susidko, 2006;Sosedko, 1980;Leontiev, 2009), which are established on the basis of experimental data analysis (Zasidko et al.,2019; Oleg Mandryk, Andriy Oliynyk, Roman Mykhailyuk, Lidiia Feshanych Pietrzak et al., 2018).However, the above-mentioned works do not suggest the idea of the possibility to build the express method for forecasting the phenomenon of flood development.
In the present paper, based on the use of methods for approximating the curves and the results of experimental studies of flood waters, a method of mathematical description and forecasting of the flood development is suggested.Presented method gives next possibility: • to apply the mathematical methods for studying of real processes (floods, seasonal phenomena of emergency water distribution; • to offer the functional analytical structure based on the statistical information about the floods in concrete region, which give the possibility to predict the flood's duration and intensity; • to restore the ordinary differential equation which describe the flood process development using the information about one's solution; • to choose the factors that have the significant influence on the flood's characteristics; and

Mathematical modeling of the flood development process
When analyzing the real phenomena of flood situations, it is possible to depict schematically the relationship between time ( t ) and the level of flood waters ( l ) (Figure 2).
Obviously, many functions meet such conditions, and, therefore, the problem of constructing the flood development model from a mathematical point of view is ill-defined (Leontiev, 2009) to regulate.It is needed to develop certain algorithms using additional information about type of the function that would satisfy the conditions (1) (Tikhonov and Arsenin, 1979;Zorich, 1981;Filippov, 2007).
The simplest of the known functions is the following one: ( ) e − = t f t t , (2) the graph of which is shown in Figure 3.
However, application of function (2) to describe flood phenomena is connected with some problems.It cannot be changed, and its type cannot be selected according to certain results of experimental studies and analysis of statistical data on floods.To solve this problem, the following function can be suggested:  that allows to obtain a whole range of curves, which, however, are topologically (according to their spatial location) similar, and that does not allow to introduce many parameters on which the level of flood water rising depends.That is why, the following two-parameter model is proposed to determine the function of type (1) of the form (Figure 5).e , 0, 0 for which all conditions (1) are determined: ( e e ) e ( ) 0 according to the L'Hospital's Rule (Zasidko et al., 2019).(5) Then, by substituting in the determinant (5) of the function 1 () yx, according to the properties of determinants, the equation ( 5) is satisfied identically.Applying similar values, it is possible to obtain differential equations, the solutions of which are the corresponding functions: for the function (2), equation (5) requires the form: (8) For each of the equations ( 6) -( 8), especially for equation ( 8), there can be set the initial conditions of the form: 00 () = y t y , (9) setting the conditions in the form of ( 9) allows to take into account the initial level of flooding: 00 0 () 00 0 It is also important to establish the physical content of coefficients n and a , which may be functions of the form: ... , ; ... , , where the values are soil permeability, air humidity, water intake humidity, relief features, etc., and the variable is time.
The question arises how to determine the numerical values of parameters ( 10) for different types of floods and how to choose the variables related to (10), which most significantly influence these dependencies.It is also necessary to define the components of the equation ( 8).Function ( ) yt represents the level of flood waters at some point of time, which varies in proportion to itself with an alternating-sign coefficient of the form: where n and a are empirically determined coefficients.When , we obtain: that is, the linear regression equation.Using the known formulas (Zamikhovskii et al., 2014) for linear regression coefficients, we obtain '   (1 ) 0 Oleg Mandryk, Andriy Oliynyk, Roman Mykhailyuk, Lidiia Feshanych from where, using communication formulas, is obtained the following: (15) that is, parameters n and a can be determined unambiguously, that allows us to talk about the constructed regulable algorithm for the ill-defined problem of restoring the function under conditions (1).To determine the factors influencing the process, the method of (Pietrzak et al., 2018) associative analysis is used, which allows to identify variables and parameters that affect the process of increasing flood waters.Suppose, according to the results of experimental research and analysis of statistical data, a quantitative characteristic of some parameter i x has been established, the range of change of which can be divided into two segments that correspond to approximately equally probable values i x .Simultaneously, the following table is arranged: where f stands for the value of the level of flood waters, 0 f stands for some average value that divides the range of flood waters change into intervals in which the values f are distributed approximately equally in number., , , A B C D are the numbers of comparison results that correspond to the specified values f and i x .
The following values are calculated: Obviously, the total number of experiments is equal to either 12 + nn or 34 + nn .The contingency ratio is calculated according to the formulas: , then the relationship between the values is considered to be proved and significant; it should be studied in more detail, but if 0, 3   , then the relationship between the specified values can be considered insignificant.

Methods of forecasting
The following method of estimating the level of flood waters and forecasting their development has been proposed.Suppose that by studying the floods that took place in the region under study with the help of methods ( 12) -( 15) the ratio: e , 1,..., where k is the number of the studied floods that has been obtained.18)), along with corresponding graphs, we choose the possible level of flooding and its intensity.
The scheme for estimating the level and duration of floods is estimated, for example, as shown in the graph (Figure 6).The values % a and % n are determined on the basis of statistical analysis.

Conclusion
The method of estimating the flood water level and forecasting of flood phenomena has been suggested.Compared with the existing methods, the presented one is the method allows to offer the function analytical structure based on the statistical information about the floods in concrete region, which give the possibility to predict the flood's duration and intensity, to restore the ordinary differential equation which describe the flood process development using the information about one's solution.It is possible to change the order of differential equation to describe the processes more in detail, to apply the method of associative analysis for the choosing the factors which have the significant influence on the flood's characteristics.From the one side it is the method with the strict mathematical justification but from the second side it is quite simply in

Figure 2 :
Figure 2: Diagram of the relationship between time and the level of flood waters

Figure 3 :
Figure 3: The graph of the function ( ) e − = t f t t ) which obviously also satisfies the stated conditions (Figure4Oleg Mandryk, Andriy Oliynyk, Roman Mykhailyuk, Lidiia Feshanych Figure 4: The graph of the function Figure 5: Family of graphs for the function and for the function of type (4), the form: N is the number of observations, which during intense floods can be quite significant due to the fact that flood water levels are monitored regularly, estimation of parameters n and a can be done by using a linear regression apparatus.For this purpose, function (4) can be written (applying logarithm problem) as: the method(Pietrzak et al., 2018) of associative analysis (16), for each of the values , 1,..., M = s xs the contingency ratio (16) is calculated and the level of connection between the corresponding s x and f is determined.Thus, the number of values , i 1,..., M = i x that influence f , is reduced, and in the future only MM , that influence the process are considered.The presented results allow to optimize the experimental research procedurethe number of which it is necessary to develop methods of experimental evaluation and control, is reduced.When studying the possibility of flooding in the given region, j are determined.We choose those formulas (17) that correspond to the determined * s j .Applying the corresponding dependences i f (17) determined for * s j (in the simplest case * s j is the only one and ** = s jj , determined based on (

Figure 6 :
Figure 6: Scheme for estimating the flood level and duration [Explanation to figure 6: m y stands for critical increase in the level of flood waters, t for duration of a flood.]  the formation and development of floods, are determined.They are water and physical properties of soils (water permeability, water intensity), air humidity, wind direction and speed, etc.It is considered that based on experimental studies, these values are known for each of the floods m xx , that affect 12 , ,..., x Oleg Mandryk, Andriy Oliynyk, Roman Mykhailyuk, Lidiia Feshanych