Akım serilerinin kaotik analizi karadeniz havzası uzerine bir uygulama

Abstract

Kendiliğinden doğada var olan sistemlere, doğal sistemler denir. Doğal sistemler, insan yapılı sistemlere göre daha karmaşık ve analizi zor sistemlerdir. Çevredeki değişimlere tepkisi bakımından ise sistemler, dinamik ve statik sistemler olarak ayrılabilir. Doğal sistemler, çevredeki değişikliklere uğrayan sistemler olduğundan çoğunlukla dinamik sistem sınıfında incelenirler. Dinamik sistemler, kararlı ve kararlı olmayan davranışlar gösterebilmektedir, sistem davranışlarının belirlenmesi için doğrusal ve doğrusal olmayan yöntemler kullanılmaktadır Doğal sistemler, kontrol altına alınabilen sistemler değillerdir. Bu sebeple, sistemlerin doğru analiz edilerek, karakterlerin en etkin şekilde belirlenmesi çok önemlidir. Sistemlerin karakterlerinin doğru belirlenerek modellenmesi ve öngörülmesi, iklim değişimi ve küresel ısınmanın etkisi ile artan önem kazanmaktadır. Su canlıların en temel ihtiyacıdır. Bu sebeple, bu ihtiyaca yönelik hizmet veren Su havzalarının doğru analizi ve etkin yönetimi çok önemlidir. Akarsu akımlarının yeterli ve doğru bir şekilde modellenebilmesi için akımı oluşturan süreçler hakkında ayrıntılı ve yeterli bilgiye sahip olmak gerekir. Literatürde, akarsu akımlarını meydana getiren süreçlerin birbirleriyle doğrusal olmayan etkileşim içinde oldukları kabul gören bir varsayımdır. Ancak araştırmacılar arasında bu ilişkinin türü hakkında bir uzlaşma sağlanamamıştır. Akarsu akımlarının stokastik doğrusal olmayan yapıda olabileceğini söyleyen geleneksel araştırmacılar ile deterministik doğrusal olmayan yapıda olabileceğini söyleyen kaotist hidrolojistler arasındaki tartışma sürmektedir Bu çalışma, kaos teorisinin temel matematiksel açıklamalarından, Türkiye Karadeniz sahilinde yer alan 4 havza üzerindeki, akım gözlem istasyonlarına ait günlük akım serilerinin analizinin uygulanmasına kadar bir çok konuyu kapsamaktadır. Çalışma literatürdeki örneklerden farklı olarak, faz uzayının kurulumu esnasında, dalgacık analizi uygulamasını da içermektedir. Dalgacık analizi ile serilerin içerdiği gürültünün giderimini de çalışma kapsamında incelemektedir. Literatürde, serilerin içerdiği gürültünün kaotik analiz kapsamındaki etkisi belirtilmiş olmasına rağmen, Türkiye'de kaotik analiz kapsamında Dalgacık Analizi kullanımına Hidroloji literatüründe şuana kadar yapılan incelemelerde rastlanmamıştır. Doğal serilerin içerdiği gürültünün frekansının belirli olmaması nedeniyle Dalgacık Analizi uygulanırken, uygun seviyede gürültü giderimi yapılması için, bilgi (enformasyon) kriteri kullanılmış ve seviye, dalgacık entropisi kullanılarak belirlenmiştir. Ayrılan parçalar, yaklaşım (A) ve detay (D), üzerinde her bir istasyon için kaotik analiz, öngörü ve sistem modelinin belirlenmesi uygulamaları yapılmış ve serilerin orijinal halleri ile karşılaştırılarak; gürültünün bu tip uygulamalarda sistem üzerindeki etkisi inceleme altına alınmıştır. Çalışmada kullanılan, DSİ'ye ait 22 adet akım gözlem istasyonlarının hepsi uzun kayıtlara (30 yıl ve üzeri) sahip olduğu için, elde edilen sonuçlar kullanılan seriler ile havzaların karakterini belirleyecek uzunluktadır. Çalışmadan elde edilen başarılı sonuçlar, kullanılan yöntemlerin gerek ileriye gerek geriye yönelik akım tahminleri için başarılı sonuçlar sağlamasının yanı sıra, eksik istasyonların verilerinin tamamlanması konusunda da uygulanabilirliğinin bir göstergesidir. Su havzalarının doğru analizinin yapılması için, her geçen gün ilerleme gösteren bilim ve teknolojinin sunduğu imkanlar değerlendirmeye alınmalıdır. Günümüzde, gelişimi halen devam eden kaotik analiz ile Su havzalarının karakterlerinin belirlenmesi ve davranışlarının modellenmesi Hidroloji alanında çalışan araştırmacıların ilgisini çekmiş ve her geçen gün çekmeye devam etmektedir. Çalışmada kullanılan veri, içme suyu temini, sulama ve enerji üretimikarşılanması gibi birçok amaca hizmet veren günlük nehir akımı serileridir. Veri setleri üzerinde uygulanan Kaotik Analiz, başarılı sonuçlar elde etmemizi sağlamıştır. Çalışmadan elde edilen sonuçlar neticesinde, kaotik analizin diğer veri kümeleri üzerinde de uygulanması önerilmektedir. Türkiye'de bu güne kadar yapılan Hidrolojik araştırmalarda, havza bazında kaotik analiz araştırmasına rastlanmamıştır. Havza istasyonların tamammının incelenerek, havzaların karakterlerinin belirlenmesi Su kaynakları yönetimi için büyük önem arzetmektedir. Özellikle, günlük akım verileri ile havza analizi ve analiz neticesinde ileriye dönük tahminlerin yapılması, Su kaynaklarının etkin kullanımı için büyük fayda sağlayacaktır. Bu nedenle, çalışmanın Türkiye'de yapılan Su ve diğer doğal kaynaklarla ilgilenen araştırmacılara örnek teşkil edecek niteliktedir.

Natural systems are the systems those exist in nature and independent from all human involvement. Natural systems are more complex and difficult to analyze than the human or social systems. Systems can be categorized according to their responses to the environmental impacts as statics and dynamic. Since natural systems are strongly responsive to the environmental impacts, they are considered in the dynamic system category. Dynamic systems can exhibit both stable and/or unstable behavior and they can be determined by non-linear and linear methods. Natural systems are not controllable thus, proper analyses are an asset to identify the system behavior. The accurate analysis of natural system characteristics are getting more important by the effect of global warming and climate change. Water is an indispensable need for all living organisms. Therefore, accurate analysis and effective management of river systems are very important. For achieving an accurate analysis, it is necessary to have sufficient information about the forming processes of river flows. The dynamics of forming process of river systems is assumed to have nonlinear interaction with each other. But, in literature, there is not any consensus between the researchers for the forming processes of river flow systems. The approaches on whether the river flows can be analyzed by stochastic or dynamic is still a continuing debate. According to the theory of dynamical systems, the evolution of the system can be represented by a trajectory in the phase space. The coordinates of the phase space indicate the state variables, which are necessary to demonstrate the evolution of the system. Transforming a time series into the geometry of a single moving point along a trajectory, where each of its points corresponds to a state of the system, carries out the identification. An attractor can be in the forms of; a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure. If the variable is scalar then the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory. If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable. Once some sequences have entered the attractor, nearby points diverge from each other but never depart from the attractor. Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then appear to become random. Chaos depends on the structure of the system as it is based on occurrence of certain parameters. Chaos occurs often in unstable, complex and nonlinear systems. Complex systems are the systems where various number of elements interact with one another by many degrees of freedom. A non-linear system is the system in which the rules of alternation may also alter during the alternation process by the environmental effects, so the system may give unexpected responses. There are some points to determine a system as chaotic those are listed in perspective: time series should exhibit irregular oscillations, autocorrelation function should exhibit a exponential decay, power spectrum of small frequency band should have wide band noise structure, a random Poincare section view, at least one Lyapunov exponent must be positive, a chaotic set must have fractal dimension. However; those criteria are not robust such as, autocorrelation function analysis are not reliable if the data set do not have normal probability distribution function or power spectrum analysis do not always present the evidence whether the sampled data set is deterministic. Further more, the power spectrum may have a wide range of noise band in small frequencies in both stochastic and deterministic data sets. Therefore, the fractal dimension analysis and Lyapunov exponents are considered the most reliable evidence of chaos.This study covers many issues from brief Mathematical description of chaos theory to the implementations of those methods on daily gauge river discharge data of 4 basins on Karadeniz Coast, Turkey. As a contribution to the literature, Wavelet Analysis was also implemented on the data to analyze the noise effect on phase space reconstruction. Since the certain noise frequency in natural data can not be determined, the information criteria was use to determine the proper level for Wavelet Analysis. Beside the original data, Approximation (A) and Detail (D) components of the Wavelet Analysis were both examined in chaotic analysis to observe the effect of noise on the phase system. The data of 22 gauge stations of General Directorate of State Hydraulic Works (DSİ), were gathered for the analysis. The data length of the stations have long records (30 years and more), the results are considered to be reasonable to determine the characteristics of the systems. As the first step of chaotic analysis, phase space system parameters; delay time (T) and embedding dimension (m) were determined to reconstruct the phase space. Delay time (T) was calculated by using Average Mutual Information (AMI) function which is a nonlinear form of autocorrelation function. Embedding dimension (m) was calculated by using False Nearest Neighbor (FNN) algorithm. After the reconstruction of phase space, the dimension of the attractor occurred on the phase was calculated by using Corellation Dimension algorithm. The package program TISEAN 3.0.1 (Time Series Analysis) was used. The program which was written by Hegger et. al. 1999, is the most popular program in literature. The results signify the chaotic behavior of the observed system that all of the data set have fractal dimensions. In addition, it is also observed that the approximation (A) components have less amount of dimension than the original data set, which is a proof of the effect of noise component on systems. The increase of the dimension makes it difficult to monitor the attractor on the phase space. As a conclusion remark, the main characteristic of the system is hidden by the noise. Lyapunov exponents method is another reliable criteria to determine the chaotic behavior. Rosenstein et. al. (1993) algorithm was used for calculating Lyapunov exponents. Although the exponents have a very small amount (less than 1), a positive exponent is considered to be enough to determine the chaotic character (Khatibi, 2012). Thus, the data sets in the study were proven to exhibit the chaotic behavior.In the last part of the case study, Local Approximation Method were examined to predict the data. The method based on the chaotic dynamics of the systems. Both the original data and approximation component (A) were used for prediction. The results of both series, show good performance of prediction (R2>0.9), as a matter of fact; the approximation component exhibited better performance than the original data sets. As an explanation in details for each basins, all approximation components of the basin date have super prediction performance (R2>0.99) though, in the original data set Doğu Karadeniz basin was observed to have less prediction performance than the other basins. Doğu Karadeniz basin have a different physical hydrological characteristics. By reason of the river regime in the basin having big variation over time, the results signify the fact that, Local Approximation method is not very successful to catch the peaks in the data set. Auto Regressive eXegenous (ARX) model was also used in the study to compare the prediction performance of the Local Approximation Model. As it was in Local Approximation model, again the A component of the data sets exhibit a super prediction performance while the original data have less prediction performance. The performance of ARX methods were also less than the Local Approximation. Thus, the Local Approximation Method is a better prediction model than ARX model since it is believed that the Local Model detects the dynamics of the system better than ARX. Apart from the prediction performance on original data sets, both models exhibited the same performance on approximation (A) component (R>0.99). ARX models are based on the nonlinear dynamics in the systems where Local Model base on chaotic dynamics. But, both models were observed to be sensitive on the noise component in the data sets. Therefore, it is strongly recommended to de-noise the data sets to obtain a better performance. In conclusion, the successful results of the case study prove that the both methods can achieve good results for forward or backward predictions. Beside, they can also be applicable for estimating the missing data in the data sets. Data is the most important part of accurate analysis of natural systems. Considering the data collection is a World wide issue for researches, new developed methods should be used for missing data estimation and data prediction. Therefore, the study is a good example for further researches on data analysis of natural systems. The progress on the development of new methods in science and technology should be taken into account to perform more accurate analysis of Hydrological Systems. Nowadays, most of the hydrologists around the World are drawn into the new era of Chaotic Analysis. Unfortunately this era is not very popular among hydrologists in Turkey. However, considering the ongoing water shortage in Turkey, water systems must be well analyzed to build up effective management policies. Daily river data are highly important in drinking water supply , irrigation and energy production. Our main motivation for this study was the lack of nonlinear analysis researches in Turkey on daily river flows. The successful results obtained from the study have convinced us that the chaotic analysis is very useful to determine main characteristics of the river basins. It also provides a better modeling. After all, it is believed this study will be a novel approach for further studies in Turkish River Basin Systems.