Akarsu Askı Maddesi Debisi Tahmini İçin Nonlineer Bir Sistem Modeli

Abstract

Akarsu havzalarının yağışlara karşı davranışı sözel ve nitel olarak oldukça iyi anlaşılmış olsa da, bu davranışların kantitatif anlamda yeterince aydınlığa kavuşturulmuş olduğu söylenemez. Bu nedenle akarsu havzaları, kara-kutu yaklaşımıyla analize uygun sistemlere iyi bir örnek oluşturmaktadır. Bu da çeşitli tahmin problemlerinin çözümünde kolaylıklar sağlar. Belli bir periyottaki girdilerin, çıktıları etkilediği yani hafızalı bir nonlineer kara-kutu sisteminin davranışı, matematikte fonksiyonel dizi olarak bilinen entegral denklemi ile yaklaşık olarak temsil edilebilmektedir. Bu çalışmada, hafızalı ve nonlineer bir sistem olarak havza ele alındığında, davranışı gözden geçirilmiş, lineer ve nonlineer bazı kara-kutu- sistem modelleri, özellikleri tartışılarak karşılaştırılmış ve akarsuların taşıdığı askı maddesi konsantrasyonunun veya askı maddesi yükünün haldeki ve etkin geçmişteki yağışlardan yararlanılarak tahmin edilebildiği nonlineer kara-kutu modelleri geliştirilmiştir. İki boyutlu birim askı maddesi eğrisi adı verilen modeller, etkin yağışlar kullanmakta ve birim sediment eğrisi modellerinde olduğu gibi sadece şiddetli sağanaklarda değil bütün yağış şiddetlerinde geçerli olup havzanın davranışını bütün olarak temsil etmektedirler. Orijinal hali ile model esasen bir fonksiyonel dizinin yalnızca ikinci teriminden ibaret olup havza davranışının lineer kabul edilebildiği hafıza bölümünde lineer, geri kalan bölümünde ise ikinci derece bir fonksiyonel kullanılmak suretiyle tadil edilmiştir. Daha yüksek dereceden modellerin güçlükleri dolayısıyla şimdiye kadar zaten, hep ikinci dereceden modeller kullanılmıştır. Fakat ikinci dereceden bir modelin uygun olduğu hakkındaki düşünceler de sadece uygulama sonuçlarına dayanmaktadır. Gerek orijinal gerekse tadil edilmiş modelde sistem davranış fonksiyonları birim etkin yağıştan meydana gelen askı madde debisinin zamansal dağılımım temsil etmektedirler. Bunlar bir ve iki boyutlu birim askı maddesi eğrisi olarak yorumlanan ve fiziksel bakımdan anlamlı olup havza davranışını temsil için gerek ve yeter sayıda fonksiyonlardır. Yeni modellerin literatürden sağlanan datalar kullanılarak kapsamlı uygulamaları gerçekleştirilmiş, lineer ve ikinci dereceden fonksiyonel dizi modeller ve Williams'in modelinin sonuçlan ile bir karşılaştırması yapılmış ve sonuçta tadil edilerek geliştirilmiş modelin hepsinden daha iyi tahminler verdiği görülmüştür. Bu modelin daha iyi sonuçlar vermesinin sebebi, yeterince uyum kabiliyetine sahip ve daha az, ancak yeter sayıda davranış fonksiyonu ihtiva etmesidir. Dolayısıyla lineer bir model bileşeni ile temsil edilebilecek olan geçmişin, fonksiyonel dizi ve orijinal modelde olduğu gibi nonlineer modelle temsili gereksiz uyum kabiliyetini hasıl etmekte bu da tahminlerde olumsuz rol oynamaktadır.

The prediction of river sediment load constitutes an important issue in hydraulic and sanitary engineering. It is a well-known fact that all reservoirs are designed to contain a volume known as "the dead storage" to accommodate the sediment income that will accumulate over a specified period. The underestimation of sediment yield results in insufficient reservoir capacities while the overestimation will lead to over-capacity reservoirs. Only the appropriate reservoir design is sufficient to justify every effort to determine sediment yield accurately but in sanitary engineering the prediction of river sediment load has an additional significance, especially if the particles transport pollutants. The real time distribution of the sediment concentration is needed in this case and the sediment concentration forecast is necessary for controlling the pollution level in rivers and reservoirs. Classical approach of hydromechanics has not yet succeeded in modeling the complete process of sediment transport in rivers for reasons that particle movements in turbulent flow as well as the properties of the particles are all random. The properties of the riverbed are irregular and hence can also be considered as random. Moreover, all the processes affect each other such that the flow causes erosion and transportation of particles; the particles transported in turn affect the flow as well as the rate of erosion. Inadequacy of the classical approach compelled engineers to search for practical ways for sediment prediction. Sediment rating curve which is assumed to define the relationship between the flow and sediment discharge for a given stream is used for prediction. The rating curves generally enables the designer to estimate roughly the mean monthly and mean annual sediment rates. Therefore, the relationship between the hydrographs and sediment graphs were investigated. Methods were proposed for computing sediment graphs by Rendon Herrero [21,22] in 1974 and 1978, Renard and Laursen [23] in 1975, and Bruce and his colleagues [24] in 1975. However, all these models have received very limited testing and have same shortcomings. A more satisfactory sediment graph model was developed by Williams [25] in 1978. All these models are capable of determining only storm sediment graphs. XI! Müftüoğlu [29] developed non-linear discrete rainfall-suspended sediment concentration models in 1978 but was unable to test them due to lack of data and hence the models remained as blue prints since then. The objective of this thesis is to derive the continuous forms of these models and to refine the derivation of their discrete forms and then realize their comparative applications. A linear discharge - suspended load model is also proposed and tested. CATCHMENT AS A SEDIMENT PRODUCING BLACK-BOX SYSTEM During and after rainfalls and/or snowmelts, the water partly infiltrates and run downhill over the land, forming what is known as sheet flow. The water accumulating in rills and galleys reaching larger channels form the streams, which also runs temporarily or permanently. Sheet flows wash immense quantities of rock fragments and soil particles downslopes to the streams. Streams transport even larger quantities of sediment, for they scour additional material out of their valleys. The sediment moving in water in suspension, called suspended load, is composed of the particles washed away from catchment and the fine material scoured out of the channel network. As mentioned in the introduction the irregularities of the properties of the catchment and randomness in its operation make it forbiddingly difficult to apply the classical hydromechanic analysis. Therefore, the black - box analysis is applied, as an investigation of the input-output relationship of the catchment system without reference to its inner properties and the physical laws governing its operation. Since the sediment load at a certain time is produced by the effects of rainfalls of a finite past period the catchment is a causal system with a finite memory, and consequently its response can be analyzed by means of a functional equation, the input and the output of the system being the effective precipitation and the suspended sediment concentration. Black-box approach is based on the following simplifications concerning the system, its input and output: Since all its properties especially its credibility vary in time, the catchment is a variable system. Nevertheless it is not practically possible to determine the variation laws and introduce them into the same model. Therefore, the system is assumed as a time-invariant system, at least in certain periods during which the variations are negligible. Although the precipitation as well as the evaporation, and in consequence the effective precipitation is unevenly distributed over the catchment with random patterns the effective precipitation is usually taken as a single, evenly distributed input and hence represented by a single value input to the system. Such input is usually referred to as "lumped" input. The sediment concentration is also distributed unevenly over the gauging cross-section. However, it is also taken as a lumped output having the cross - sectional average of the concentration in the section.