Kuruyan Akarsuların Günlük Akımlarının Modellenmesi

Abstract

Hidrologların ilgisini bugüne kadar çok az çeken kurak ve yan kurak bölge akımlarının modellenmesi, dünyadaki su kaynaklarının özellikle miktar ve kalite bakımından çok konuşulduğu şu günlerde büyük bir önem taşımaktadır. Kurak bölge akımlarının tipik özelliği yılın belli dönemlerinde kurumalarıdır. Bu yüzden sulak bölge akımlarına yönelik yapılan modelleme çalışmaları kurak bölge akımlarına uygulanamamaktadır. Ayrıca kurak bölge akımlarının modellenmesi sulak bölge akımlarının modellenmesinden daha fazla parametre içeren modeller ile mümkün olabilmektedir. Günlük akımların modellenmesinde kullanılan verilerin çokluğu hesaplan artırdığından bilgisayar teknolojisinin yaygın kullanıma girmesi ile ilk kez 1960'lı yılların ortalarından itibaren bu konuya yönelimler başlamıştır. Kurak bölge günlük akımları ise pek incelenmemiştir. Literatürde gözlenen eksiklik ve günümüz koşullarında konunun önemi gözönünde bulundurularak zaman zaman kuruyan akarsuların günlük akımlarının modellenmesi amacı ile bir model geliştirilmiştir. Model aşağıdaki adımlardan oluşmaktadır: 1. Akım Oluşan Günlerin Belirlenmesi: Model önce herhangi bir günde akımın oluşup oluşmayacağını belirlemektedir. Bu amaçla 2-durumlu bir Markov zinciri kullanılmıştır. 2. Akımda Artma Olan Günlerin Belirlenmesi: Akımın oluşacağının tahmin edildiği bir günde akım artmış veya azalmıştır. Bu durum için de 2-durumlu bir Markov zinciri kullanılmıştır. Kuruyan akarsularda akım değeri ya artmakta, ya azalmakta ya da sıfır olmakta, yani akarsu kurumaktadır. Kuruyan akarsuların bu üç durumunu içeren 3-durumlu bir Markov zinciri kullanılarak akım değerinin arttığı, azaldığı veya akarsuyun kuruduğu günler belirlenebilir. 3. Artma Miktarının Belirlenmesi: Akımda artma olan günlerde artma miktarının belirlenmesi amacı ile 2-parametreli gamma dağılımı kullanılmış, böylece hidrografin yükselme eğrisi oluşturulmuştur. 4. Çekilme Eğrisi Şeklinin belirlenmesi: Akım olan ancak akım değerinin bir önceki güne göre azaldığı bir günde hidrografin çekilme eğrisi gözlenir. Çekilme eğrisinin benzeştirilmesi için gözlenen verilerden en uygun çekilme eğrisi şekli belirlenmiş, bunun için kullanımı çok yaygın eksponansiyel çekilme denklemi kullanılmıştır. Bu şekilde özetlenebilen modelle türetilen serilerin gözlenen seriye benzerliğine akım serisinin çok sayıda karakteristiğinin gözlenen ve türetilen değerlerinin karşılaştırılması ile karar verilmiştir. Bu model kullanılarak akarsu yapılarının hidrolojik çalışmalarında gerekli olan uzun akım serilerinin türetilmesi, böylece planlanan akarsu yapısının maksadını daha güvenli ve ekonomik bir şekilde gerçekleştirmesi mümkün olacaktır.

Water scarcity is the most important issue for arid and semi-arid areas which cover more than a third of the earth's surface, and is the major problem in the development of these areas. Therefore, planning and management of water resources in such regions have importance in order to meet human and environmental water demands. For such purposes, the analysis of water availability in the form of streamflow is extremely important. Streamflows in such areas are typically intermittent. Efforts to benefit from streams, which flow into the sea and lakes under the effect of gravity have begun with industrialization and growth of the world's population. Some projects with or without construction have been developed on streams for various purposes. Required data have started to be collected by flow gauging stations for these projects. However, flow records don't have sufficient length. Therefore, the way of obtaining suitable mathematical equations (models) and then generation of synthetic data has been followed. Only a few attempts have been done in the past by hydrologists to model streamflow processes in arid regions. Models developed for flows of humid regions cannot be applied to flows of arid regions, because of their intermittent character. Models to be developed for intermittent flows should have larger number of parameters and be more complex than the ones developed for flows of humid regions. A model can be defined as a simple presentation of a complex system or as an equation which contains physical principles about the system. A model is called a hydrological model when it is used for a hydrological system. In this study it is mainly aimed to develop a model generating daily flows of intermittent streams. The developed model consists of the steps explained below. Determination of Days on which Flow Occurs or not Intermittency, the most important feature of flows of arid and semi-arid regions of the world, means that streams sometimes have flow and sometimes have not. In such streams, determination of days on which flow occurs or not has a prior importance. Days on which flow occurs or not have been determined in the first step of the developed model. For this purpose a 2-state Markov chain has been used. The chain uses transition probabilities in the matrix given below. XXVH p = p p Ml MO p p / 01 -'oo. (1) Here, state 1 represents occurrence of flow in the stream while 0 represents non occurrence of the flow. P is the probability of transition from one state to another. By this 2-state (1-0) Markov chain, the occurrence of flow can be determined by using calculated transition probabilities from historical records and generated uniform random numbers. Determination of Days on which Flow Increases or Decreases In a day which has flow, the flow either increases or decreases. Similar to determination of days with and without flow, determination of days with an increase in flow and a decrease in flow can be done by a 2-state (W-D) Markov chain, whose transition probability matrix is given below. Increase in the flow corresponds to the ascension curve of the hydrograph while decrease in the flow corresponds to the recession curve of the hydrograph. PT, 1 WW DW WD DD. (2) Here, state W corresponds to an increase in the flow while D corresponds to a decrease in the flow. By this 2-state (W-D) Markov chain, days with an increase or a decrease can be determined by using calculated transition probabilities from historical records and generated uniform random numbers as in the previous Markov chain. Instead of this two-step algorithm given above for determination of days on which flow occurs or not, and later determination of days on which an increase or a decrease occurs in the flow, a one step algorithm can be used alternatively. In an intermittent stream flow either increases, or decreases, or is zero. By using these three states, days on which flow occurs or not, and later ascension curve of the hydrograph and recession curve of the hydrograph can be simply determined by a 3-state Markov chain. In this alternative way it is determined whether flow occurs or not, and whether an increase or a decrease occurs if flow occured. Such an investigation can be done by a 3-state (a-r-z) Markov chain. Matrix of the transition probabilities of the chain can be written as P = Par m, P" (3) States a, r and z given in the matrix correspond to an increase in the flow (ascension curve of the hydrograph), a decrease in the flow (recession curve of the hydrograph) and zero flow (days on which flow does not occur), respectively. In the matrix, P is the probability of transition from one state to another. When we look at the matrix given in (3) we can see that Pzr=0 which means that the flow cannot recess after a day xxvm without flow and that Paz=0 which means that recession in the flow after a day with an increase in the flow occurs very rarely. In that case the number of parameters is reduced from 6 to 4 which is equal to the total number of parameters of the two (0-1 and W-D) 2-state Markov chains, given before. In order to take periodicity into consideration parameters have been calculated in a monthly time interval which results in 48 parameters for each alternative. Determination of Ascension Curve of the Hydrograph Determination of quantities of increments occurring in the flow is done in this step of the model. For this it is necessary to select an appropriate probability density function which fits to the increments in the flow. In this study it has been assumed that the increments in the flow on the ascension curve of the hydrograph are 2-parameter gamma distributed. The shape and scale parameters of the distribution have been estimated for each month by the well known method of moments. The total number of parameters for this step of the model is 24. By using estimated parameters of the 2- parameter gamma distribution, 2-parameter gamma distributed random variables have been generated by means of a simple computer software. Generated random variables on the ascension curve of the hydrograph have been ranked, so it has been provided that the greatest increment is the one that is closest to the peak of the hydrograph. Calculation of Recession Curve of the Hydrograph Recession curve of the hydrograph is assumed to be deterministic rather than probabilistic. In order to find the best fitting recession equation, over 1000 observed recession curves have been investigated. Exponential recession equation given as and based on the linear reservoir theory has been accepted. Here, a is the recession coefficient, t is time, Qt is flow t days after the peak and Q0 is the peak flow value. Recession curves have been divided into two, those which have peak flow value greater than historical monthly mean flow value, and those which have peak flow value smaller than historical monthly mean flow value. Two different recession coefficients have been used along the recession curve. All recessions observed in the historical series have been taken into consideration in order to determine recession coefficients (24 coefficients, 2 for each month). In order to group recession curves into two, historical monthly mean flow values are needed. So, required parameter number for this step of the model is 36. By taking determined recession coefficients, flow values on recession curves have been calculated and compared with those taken from the historical records. Calculated recession curves have been found very similar to the ones from historical records. Model Parameters The developed model is based on transition probabilities and has 108 parameters. In order to decrease the total number of parameters of the model, Fourier series have been fitted to the recession coefficients. In the Fourier expansion two harmonics have xxix been used. In that case the total number of parameters of the model decreases to 94. After that Fourier series have been applied to transition probabilities of the 3 -state Markov chain. When the Fourier series have been fitted to the transition probabilities of the 3 -state Markov chain it has been found that the probability Pzz took negative values or values greater than 1. For this reason Fourier series have not been fitted to this probability, but have been fitted to other probabilities (Pa,,, Pra, ?")? After fitting of Fourier series with two harmonics to the transition probabilities of the 3 -state Markov chain, the total number of parameters of the model decreases to 73. Simulations With this process, simulation of daily flows of intermittent streams have been performed. Flows have been generated by using the process given above step by step. Ten simulations for the case of the two 2-state Markov chains and thirty simulations for the case of the 3 -state Markov chain have been done. In the first ten simulations, done by the 3 -state Markov chain, recession curves have been calculated by determined recession coefficients, in the second ten simulations Fourier series applied recession coefficients have been used. In the last ten simulations, done by the 3-state Markov chain, Fourier series fitted probabilities of the chain and Fourier series fitted recession coefficients have been used. Results Results obtained from forty simulated series of the same length as the observed one, are given below and explained with the help of some graphs. Figures 1-4 show the average of daily flows obtained from ten simulations. In Figure 1, calculated 0-1 Markov chain's transition probabilities, calculated W-D Markov chain's transition probabilities, and determined recession coefficients have been used. From the figure it can be said that the daily averages of the flow have been well preserved. In Figure 2, calculated a-r-z Markov chain's transition probabilities and determined recession coefficients have been used. Daily average flow has been preserved well, as in the previous case. In Figure 3, calculated a-r-z Markov chain's transition probabilities and Fourier series fitted recession coefficients have been used. Daily flows are very similar to the observed ones in that case, too. Figure 4 shows results obtained from simulations, done by using Fourier series fitted transition probabilities of the a-r-z Markov chain and Fourier series fitted recession coefficients. In that case the overall average has been preserved but small monthly differences are observed. XXX Observed ---Average of 10 Simulated Series Observed (Fourier-5) -Average of 10 Simulated Series (Fourier-5) Day Figure 1. Observed and simulated daily average flows as average often simulations done by using 0-1 Markov chain, W-D Markov chain, and estimated recession coefficients 10 T Observed --Average of 10 Simulated Series Observed (Fourier-5) ^-Average of 10 Simulated Series (Fourier-5) -sfiiasgissss Figure 2. Observed and simulated daily average flows as average often simulations done by using a-r-z Markov chain, and estimated recession coefficients XXXI 10 T Observed ?""Average of 10 Simulated Series Observed (Fourier-5) -Average of 10 Simulated Series (Fourier-5) Day Figure 3. Observed and simulated daily average flows as average often simulations done by using a-r-z Markov chain, and Fourier series fitted recession coefficients 10 t Observed -Average of 10 Simulated Series Observed (Fourier-5) -Average of 10 Simulated Series (Fourier-5) Hay « Ot M W) o\ o cs m ~ t» M r« >or»co5**© Figure 4. Observed and simulated daily average flows as average often simulations done by using Fourier series fitted a-r-z Markov chain, and Fourier series fitted recession coefficients xxxu Conclusions Conclusions obtained from the study can be grouped as conclusions related to the state of the stream, those related to the ascension curve of the hydrograph, those related to the recession curve of the hydrograph, those related to the flow characteristics, and those related to the model structure. The main conclusion about determination of the state of the stream is usability of Markov chains. In an intermittent stream, flow either increases, or decreases, or is zero. These three states of an intermittent stream can be determined by two 2-state Markov chains or by a 3 -state Markov chain which is more practical. It can be assumed that flow increments on the ascension curve of the hydrograph is 2- parameter gamma distributed. Estimation of the shape and scale parameters of the distribution can be made by the classical method of moments. It is accepted that recession of the flow has an exponential decay with parameters changing from month to month. As given in Figures 1-4, the model is successful in preserving most of the characteristics of the stream. The well preserved characteristics are daily mean, standard deviation, coefficients of skewness and first order correlation, zero flow percentage, monthly and yearly average flows, and D-day yearly maximum flows. Only D-day yearly minimum flows have been underestimated probably due to the used recession structure. An important feature of the daily flows, existence of rapid rises followed by slow recessions, has been obtained in the simulated series. Additionally, seasonality has been preserved. The developed model is based on transition probabilities and has 108 parameters. The required number of model parameters can be decreased by fitting of Fourier series to the recession coefficients and to the transition probabilities. The model can be used for non-intermittent (perennial) streams with less number of parameters.