Fuzzy chain model fundamentals and hydrograph estimations

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2017Theses (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 2017Su, insan ve diğer canlıların hayatlarını sürdürmeleri için ihtiyaç duydukları hava, toprak ve ateş (enerji) gibi birincil maddelerdendir. Bu sebepledir ki, tarihte kurulmuş şehirler ve medeniyetler çoğunlukla su kaynaklarını merkeze almak suretiyle gelişmişlerdir. Ancak tarihte ve günümüzde canlılar için son derece gerekli suyun faydalı yönlerinin yanı sıra zararlı yönleri de vardır ki, bunların en önemlisi özellikle şiddetli yağışlar sonrası ortaya çıkan taşkın zararları ve yağışların az olması durumunda ortaya çıkan kurak süreler ve canlılar üzerindeki etkileridir. Taşkın (sel), özellikle şiddetli yağış sonrasında akarsuyun olağan şartlar altında akışını sürdürdüğü yatağından taşmak suretiyle akışına devam ettiği olaya denir. Su basması ise taşkınla birlikte meskûn bölgenin sular altında kalması olayıdır. Bu tür afet durumlarından korunmanın birinci kuralı ilgili havzaya ait mecranın uygun bir şekilde tasarlanıp inşa edilmesidir. Diğer bir kural ise ortaya çıkacak yağışa göre gerçek zamanlı ve en doğru tahminleri ortaya koyabilmektir. Bu tez çalışmasında ikinci kural için adımlar atılmıştır ve en doğru hidrograf tahmini adına literatüre önemli katkılar yapılmıştır. Hidrolojinin önemli konularından tepe (pik) debi ve hidrograf tahmini, şehirleşme ile birlikte daha önemli bir hal almıştır. Mevcut modeller özellikle de orta ve büyük boyuttaki havzalar için iyi sonuçlar verirken küçük havzalar için tahminler ölçümlerle pek örtüşmemektedir. Bu çalışmada İstanbul'un Avrupa yakasında bulunan ve Ayamama ismiyle bilinen küçük bir havza üzerinde 2013 ve 2014 arası iki yıllık ölçümler yapılarak üretilen dört farklı modelin ayarlaması yapılmış ve tahminler üretilmiştir. Hidrograf tahmini gerçek girdilerinin fazlalığı sebebiyle tahmin edilmesi son derece zor ve karmaşık bir meseledir. Karmaşık olayların modellenmesi için önemli bir araç konumundaki Bulanık Mantık günümüzde artık yaygın bir şekilde kullanılmaktadır. Bu düşünceyle tez çalışması için bulanık zincir modeli (Mamdani-ANFIS) önerilerek iyi bilinen ve yaygın kullanılan bir modelle (SCS-Snyder) kıyaslanmıştır. Bulanık zincir modeli Mamdani ve Sugeno (ANFIS) isimli iki halkadan meydana getirilmiştir. Birinci halka da üç girdi ile toprak nemi çıktısı ikinci halkada da dört girdi ile tepe debi çıktısı elde edilmiştir. Mamdani bulanık çıkarım yapısında sözellik tam anlamıyla hâkimiyet kurduğundan modelin girdi ve çıktısına ait herhangi bir veri takımı bulunmasa da tecrübeye dayalı cümleler ve sözler sayısallaştırılabilmektedir, çünkü girdiler gibi çıktılar da bulanıklaştırılmaktadır. Bulanık zincir modelinin en önemli özelliği birinci aşamada toprak nemini tahmin etmesidir, ancak neme dair veri takımı mevcut değildir. Dolayısıyla, toprak nemi tahmin edilirken ilgili girdilerle Mamdani çıkarım yapısı kullanılmıştır. Sugeno bulanık çıkarım yapısında ise girdiler her ne kadar bulanıklaştırılsa bile çıktı için bir takım kesin sayısal ifadelerin belirlenmesi gerekmektedir. Dolayısıyla bu çıkarım yapısında hem girdi hem de çıktı değerleri ölçülmelidir. Bu bilgiler düşünülerek, önerilen bulanık zincir modelin ikinci halkasında ANFIS (Sugeno) yazılımı kullanılmıştır. Ek olarak, aynı veriler kullanılarak tezde ikinci bir bulanık zincir modeline de yer verilmiştir. Bulanık zincir modelinin eğitilip çalıştırılması için fazladan üç farklı girdiye daha ihtiyaç duyulmaktadır. Bu durum her ne kadar hassas sonuçlar verse de, modelin başka havzalarda kullanımını zorlaştırmaktadır. Bunun için daha az veriyle çalışmaya imkan tanıyan ve tek aşamalı bulanık modeller de önerilerek tezde sunulmuştur. Ortalama mutlak hata sonuçlarına göre bulanık zincir modeli en iyi uç (pik) debi tahminlerini, bulanık modeller yaklaşık tahminleri ve SCS-Snyder de çoğunlukla ölçümden uzak tahminleri üretmişlerdir. Bu aşamadan sonra hidrograflar çizdirilerek görsel kıyaslamalar da yapılmıştır. Hidrograflar çizdirilirken SCS tarafından önerilmiş boyutsuz birim hidrograf koordinatlarından faydalanılmıştır. Bu tez çalışmasında BZM temellerinin kullanılabilirliği farklı bir uygulama ile ayrıca gösterilmiştir. Arap yarımadasının en büyük ülkesi Suudi Arabistan'da bulunan bazı meteorolojik ölçüm noktaları üçerli öbekler halinde dikkate alınarak Buharlaşma+Terleme (BT) tahmini yapılacaktır. BT zeminin toprak ve bitki ile değişik oranlarda örtülmesi durumunda suyun sıvı halden gaz haline geçişine denmektedir. İki olayı birden içeren BT birçok araştırmacı tarafından bir takım denklemler önerilerek hesaplanmıştır. Ancak, birçok girdiyi içeren Penman-Monteith (PM) yöntemi hem kurak hem de nemli bölgeler için en hassas yöntem olduğu ortaya koyulmuştur. PM hassas sonuç üretmesi sebebiyle birçok araştırmacının BT hesaplaması için kullandığı vazgeçilmez bir denklemdir. Günümüzde PM denklemi yoğun bir şekilde kullanılırken yapay zekâ yöntemleri de kullanarak BT tahmini konusunda yol alınmaktadır. Farklı yapay zekâ yöntemleri bulanık mantık, yapay sinir ağı, genetik algoritma ve bunların türevleri kullanılmak suretiyle farklı girdi türleri ile BT modelleri kurulmuş ve tahminler üretilmiştir. Gerçekleştirilen tüm çalışmalar ilgili ölçüm noktasının kendi verisi dikkate alınarak o nokta için BT tahmini ve hesabı yapılmıştır. Bu çalışmada ilk defa iki noktanın verilerinden bu iki noktaya yakın olan üçüncü bir ölçüm noktası için BT tahmini yapılacaktır. Bu maksat için tezin uç debi tahmininde önerilen BZM'ne benzer bir zincirleme akış diyagramı geliştirilmiştir. Buna ek, iki noktaya ait PM ile hesaplanan BT'ler ile üçüncü noktann BT'sini tahmin etmek için bulanık model geliştirilecektir. BT hesabı için PM denklemi kullanılırken önerilen tahmin modelleri için de Sugeno bulanık mantık ve BZM ilkelerinden faydalanılacaktır. Sonuçta, BT tahmini için biri bulanık model biri de BZM toplam iki model üretilmiştir ve birbiri ile kıyaslanmıştır.In this study, four different fuzzy models are suggested; the first two are fuzzy chain models (FCM) that appear the first time in this thesis and the next two are fuzzy system models (FSM). The comparisons of these models with the well-known classical one indicate better validity and reliability of the fuzzy models for peak discharge estimations. The application is presented for a small basin, Ayamama basin, within Istanbul City, Turkey. This basin lies on the European side of Istanbul and its basin area is 40.1 km2. Its upstream is forest area, whereas the downstream portion is covered by urbanization areas. For potential flood model applications, small sub-basins are chosen at the upper part of the Ayamama Basin. Since the beginning of 2013 up to the present, Istanbul Metropolitan Municipality Disaster Coordination Center, (IBB-AKOM, in Turkish) has worked on flood forecasting for the basin, but with the use of classical methods. Available reliable data are taken into account including 23 records over two years (2013-2014), 15 of which are used to train the models and the remaining 8 events for testing. In this thesis, four fuzzy system models are proposed and their comparisons are given with frequently used classical SCS-Snyder method. In the literature, there are many methods for hydrograph production or peak discharge estimation. One of the most important hydrograph methods, Snyder is combined with SCS loss method methodology within Flood Hydrograph Package, HEC-1 (Hydrologic Engineering Center) software that runs under Watershed Modeling System (WMS) software. The Snyder method has two geomorphologic parameters for hydrograph description; one for time to peak discharge from the center of hyetograph, which is referred as the peak time (tp) and the other for the peak discharge estimation itself. These parameters are dependent on the drainage basin geomorphologic features such as the drainage area, main stream length, slope and the distance between the centroid projection on the main stream and the outlet point. In this study, they are calculated using the observed rainfall-runoff records. Excess rainfall is calculated according to SCS loss method. For this purpose, the SCS Curve Number and initial abstraction rate expressing the losses are calibrated. After the peak discharge and delay time calculations, the hydrograph is generated from the dimensionless unit hydrograph values. There are always uncertainties in any drainage basin hydrograph and its peak discharge calculations, but the fuzzy logic inference system can alleviate the uncertainties through fuzzification procedure. Fuzzy logic plays the major role in approximate reasoning through computers based on the fuzzy logic rules. There are several fuzzy inference systems (FIS), with slightly different outputs. In Mamdani FIS both input and output variables are in the form of fuzzy membership functions (MF), where linguistically expressions in the form of rule base are dominant. However, in the Sugeno FIS, although input is in the form of fuzzy MFs, but the output is crisp either as a constant or a first order linear function. The Adaptive Neuro-Fuzzy Inference System (ANFIS) uses mainly Sugeno FIS and gets support from the artificial neural network (ANN) in training phase. It allows estimation in the best possible way providing compliance between input and output variables. The sum of the error should be equal to zero or very close to zero and the sum of the square error should be the smallest possible. ANFIS trains the data by taking into account these two points and prepares convenient solutions. FCMs include Mamdani and Sugeno FISs (ANFIS) in the form of two chain rings. Mamdani approach is used as the first part of the chain with the output soil moisture data. Both input and output data should be known for the execution of FCM steps. Additionally, FMs include only Sugeno FIS. FCM and FM are used for peak discharge estimation by taking into consideration the following variables; Average temperature, T (Co), during period between previous and present precipitation events, Interval duration, DB, (hour), between previous and present precipitation events, Total average rainfall, RP, (mm), during previous precipitation event, Soil moisture, S, (%), as the output whose inputs are T, DB, and RP in Mamdani FIS, Total rainfall in Olimpiyat, RO, (mm), during present precipitation event at the station, Total rainfall in 212 AVM, R212, (mm), during present precipitation event at the station, Rainfall duration, RD, (minute), of present precipitation event, Peak discharge, QP, (m3/s), as the output whose inputs are all the variables mentioned in the previous items for the ANFIS software. FCMs have been formed from two chain rings as Mamdani and Sugeno (ANFIS). Although lack of data set for both input and output variables in the Mamdani FIS, meaningful words and expressions related to the concerned event can be quantifiable by fuzzy sets in the system. In the same manner, if only the output data set is missing, then they can be quantified using fuzzification and corresponding fuzzy sets. Hence, in FCMs, Mamdani approach is used for the soil moisture as an output and three data sets as inputs. Consequently, the inputs in Mamdani FIS are used in the first chain model to obtain the soil moisture data as a percentage. It is necessary to determine any output as crisp numerical expressions in the Sugeno FIS, even if the inputs are fuzzy. Hence, both input and output data can be measured in this FIS. Besides, all data sets being introduced into the models are measured in order to use ANFIS software design according to Sugeno. Subsequently, ANFIS software is used for second chain model. On the other hand, soil moisture account is not taken into consideration for other fuzzy models, FM I and FM II, with one phase and they are executed by the ANFIS software, because the remaining inputs and output are already obtained. When precipitation events with the same characteristics occur at different times, the classical models yield constant peak discharge, but FCMs estimate peak discharge differently, because they take into account the soil moisture variability. The reason for production of FCM II working with average rainfall is to demonstrate how the results change compared to FCM I. Contributions of each rainfall station are considered separately through the ANFIS software and the results are more accurate. Soil moisture percentages are calculated according to the average rainfall via Thiessen method. FCMs need three extra data sets (T, DB, and RP). The first chain Mamdani model is not considered, because usually detailed measurements are not available. Estimates are obtained by decreasing the data sets and entering the remaining data sets to ANFIS software. In this case, FM I and FM II peak discharge estimations appear as a constant because the soil moisture is not taken into consideration. In this thesis, the simplest forms of the hydrograph are obtained from the triangular flood hydrograph and the dimensionless unit hydrograph. For the construction of triangular hydrograph three time instances are important, namely, the precipitation (tpr), the peak flow (tpe) and recession limb (td) durations. While peak discharge is estimated by FCM and FM, the duration of rainfall and its amount are obtained by meteorological predictions, the remaining two durations (tpe and td) should be determined for hydrograph completion. During the training stage, tpe is related to tpr, and td to tpe through two constants, cpe and cd, as follows. tpe = tpr+cpe (1) td = tpe+cd (2) On the other hand, dimensionless unit hydrograph (DUH) developed by United States Department of Agriculture, Natural Resources Conservation Service (USDA-NRCS) provides a basis for generation of the hydrograph provided that the peak discharge and its time of occurrence are known. The following points are helpful for rational and logical rule deductions. The applications of the classical model and the proposed ones as two FMs and two FCMs are presented for the upper part of the Ayamama Basin, Istanbul. As mentioned before, 15 of the measured 23 events are used for training stage for parameter adjustment and model calibration, whereas the remaining 8 events are for testing (cross-validation). SCS-Curve Number (SCS-CN) and initial abstraction (IA) for SCS loss method, lag time to peak (tp) and peak discharge coefficient (cp) for the Snyder; totally 4 parameters are adjusted for the classical model through HEC-1 software coupled with the Watershed Modeling System (WMS) software. FCM I and FCM II have two phases. The soil moisture value must be digitized prior to estimation with ANFIS. Therefore, moisture values for each event are determined as percentages by Mamdani FIS. Relationships between three inputs (T, DB, and RP) and soil moisture have been considered rationally and logically for determination of the rules. For the first chain model, Soil Moisture values are determined with logical relationship considerations. Four inputs versus peak discharge are trained by ANFIS software in the FCM I and three inputs versus peak discharge in FCM II. Finally, FCMs are ready to estimate and then forecasts have been achieved as already. The estimation errors are calculated according to the well-known mean absolute error, (MAE). It is obvious that FCM I are capable of producing peak discharge estimation better than the others. Another way of comparison among the three methodologies is the scatter diagram, which shows the scatter of the measurement values versus model predictions. The significance of this graph is that 1:1 (45o) straight-line is the indicator of the perfect model estimations. The closer are the model scatter points to this straight-line the better is the model. The scatter of all the model results indicates clearly that the FCM I points have the closest positions to the 1:1 straight-line, and hence, FCM I model outperforms the others. The triangular flood hydrographs for 8 events are obtained with FCMs and FMs. For each event, observed and estimated hydrographs from five different models are compared visually on the same graph. In this way, a total of 8 cases are presented in Figures for the case of triangular hydrograph approach. FCM I hydrographs are closer to observation hydrograph. Another figure is for the dimensionless unit hydrograph based calculations, which shows the hydrographs in a better curvature form and again the FCM I results are far better than other methodology. The proposed FCM I methodology is useful especially in cases of soil moisture component incorporation. The availability of FCM bases is also shown with a different application. Some meteorological stations' measurements in Saudi Arabia the largest country of Arabian Peninsula will be used for Evapotranspiration (ET) estimations. A set equations are proposed by many researchers to calculate ET. ET calculation models use meteorological measurements as input variables, but others consider several inputs and produce ET values. Penman-Monteith (PM) method is the most sensitive one for all regions with many input parameters. Recently, artificial intelligence (AI) techniques are entering into the scene of ET estimation. Among the AI methods are fuzzy logic (FL), artificial neural networks (ANN), genetic algorithms (GA) and their derivatives with different input types for ET estimation. In this study, some stations in the Kingdom of Saudi Arabia (KSA) are considered as two groups in three parts. Initially, ET is estimated by using two nearby stations data set for the third station in the same area. For this purpose, a regional chain flow chart is developed similar to FCM. The third station ET estimations are obtained by the suggested regional fuzzy model (RFM) based on the two stations' ET values that are calculated by PM equation, which produces the most accurate results of the ET estimation. Sugeno FL system inference principles are the bases of the proposed models. RFM and regional fuzzy chain model (RFCM) model results are compared with the PM calculations and they yield almost the same result within practically acceptable limits of error. From Saudi Presidency of Meteorology and Environment (PME), 6 meteorology stations are considered in this study. Each station has daily records, which are used for estimations. Herein, station groups are chosen at two different elevations, namely, at the mountainous and sea levels for testing the proposed model reliability. Aseer region Bisha, Abha and Khamis Mushait stations are on the mountainous region, whereas Ad-Dhahran, Al-Dammam and Al-Ahsaa stations in Ash-Sharqia region are approximately at the sea level. In this study, the data with the shortest length is considered for each part such as the 14-year (1999-2012) data for Ash-Sharqiya region and 35-year (1978-2012) for the Aseer region. The proposed methodology of this paper has three steps. In the first stage, PM (FAO-56) equation is used to calculate ET values. The next stage is practiced by many researchers and the classical model is based on the FL principles for ET estimations with four data types. Finally, in case of missing data at any station, RFM and RFCM approaches are proposed in the first stage for ET estimation by using the data from other stations in the same region. In this study, two alternative fuzzy models are proposed using ANFIS software, and hence, ET estimations are obtained. The classical fuzzy model (CFM) with four-input-one-output are explained and its application is performed for the data at hand. So far, researchers have estimated ET for any station by benefiting from its own data with the model based on fuzzy logic (FL) principles. This model is extremely useful and produces accurate results, if there is complete data set for the station. However, FL model is also for the cases where there is no or lack of data and thus uncertainty. Herein, two-input-one-output Sugeno model called RFM is set up to estimate ET because the third station's ET values are estimated with the other two stations' ET values. For RFCM, three stations are taken into consideration in the same region similar to RFM but RFCM has two steps. In order to estimate the third station ET values, first of all (first ring of chain), two-input-one-output Sugeno models are established and each parameter is obtained. For the second ring of RFCM, four-input-one-output Sugeno model is set up and the model is retrained. Data from two different regions of the KSA are used for PM and ET calculations by means of three FL inference system models two of which are suggested in this thesis. Ash-Sharqia region in the western part of the KSA has rather flat topography and the measurements from the 1999-2012 period measurements are used in the model development. Proposed ANFIS system modeling based models is used for calibration (training) data from 1999-2007 period for 9 years and the validation (testing) data from 2008-2012 period for 5 years. From the three meteorology stations, Ad-Dhahran station is selected for estimations. In Al-Aseer mountainous region in the southern KSA, the measurements are used from 1978-2012 periods. ANFIS model training and testing periods are 1978-1998 and 1999-2012 periods, respectively. In this region, Khamis Mushait meteorology station is selected for estimations. As a result, CFM approach uses the station data, and hence, fewer errors are expected at Ad-Dhahran and Khamis Mushait locations with 5% and 4% errors, respectively. If at any station in a region there is not enough data then from the two adjacent stations, the ET estimation can be achieved at the third station. RFM is comparatively simpler and its use is shown for the estimation of ET values at a missing data station from the