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Application of Artificial Intelligence Methods in Geosciences and Hydrology Edited by: Ata Allah Nadiri ISBN: 978-1-63278-061-4 DOI: http://dx.doi.org/10.4172/978-1-63278-061-4-062 Published Date: December, 2015 Printed Version: December, 2015

Published by OMICS Group eBooks 731 Gull Ave, Foster City, CA 94404, USA.

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To my parents, my wife, my family, my teachers, my students, and the readers of this book

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Preface Soft computing refers to a consortium of computational methodologies including artificial neural networks (ANNs), Fuzzy Logic (FL), and so forth. All methods have their roots in Artificial Intelligence (AI). Nowadays, accurate and acceptable solution method to complex and nonlinear problems in geoscience and hydrology is AI techniques. Also, a combination of one or more of methodologies has resulted in emergence of new categories like a Neuro-Fuzzy (NF) technique is more powerful than individual methods. Considering excess application and promotion of AI methods in geoscience and hydrology during last decade, it would be an essential task to infer the different new application of the AI methods. It was therefore our objective to present the complex and nonlinear problems in this field such as classification and prediction that could be solved by AI methods accurately. This book attempts to 1) emphasis on learning the design, implementation and application of individual and hybrid AI techniques through geosciences, environmental studies, hydrology, and hydrogeology, 2) Assess where we are in term of our research activity in this field, 3) serve as an important benchmark for future estimate of process.

Signature

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About Editor

Dr. Ata Allah Nadiri is assistant professor in Earth Sciences Department of the University of Tabriz at East Azarbajan from 2013. He received his B.S., M.S. and PhD degrees in 2004, 2007, and 2013 respectively from University of Tabriz. His M.S. and PhD projects focused on application of Artificial Intelligence (AI) in groundwater modeling. He was first rank student in B.S. and M.S. steps and member of Iranian National Elites Foundation from 2007. He also received the best student of geology of Iran award in 2010. Dr. Nadiri awarded a scholarship in 2011 from Ministry of Science, Research and Technology of the Islamic Republic of Iran and Iranian National Elites Foundation to carry out part of his PhD research work under supervision of professor Frank T-C Tsai at the Louisiana state university. He served as a consultant and supervisor of different Water Resources Authorities of Iran. Many journal articles were published and reviewed and several research projects were done by Dr. Nadiri.

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Acknowledgement I wish to thank Professor Asghar Asghari Moghaddam, Professor Frank T-C Tsai and Professor Vahid Nourani, who supported me during M.S. and PhD steps. I wish to thank my friends and colleagues Professor Vahid Nourani, Dr. Ali Kadkhodaie, Dr. Rasoul Mirabbasi and Dr. Farnaz Daneshvar Vousoughi to accept my invitation as an author that was their help and support for writing and preparation of this Book. We are thankful to the reviewers of chapters, who helped in the modification and revision of the manuscript of the book. I thank OMICS Group editors for their unfailing cooperation.

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Introduction to the eBook To date, there are numerous articles that admire the advantages of artificial intelligence (AI) as computational tools and gain impressive performances over conventional techniques. Also, a sequence of unacceptable failures has been reported associated with this technology; however these tend to be considerably less presented. In addition to its relative novelty, it is this tempting mixture of successes and failures that add to the attraction of AI. We would basically have not much interest in a problem if the end result is always known. Researcher claim to be drawn to AI model, because their high abilities in: 1) parameter prediction as a universal approximator, 2) Parameter estimation, 3) pattern recognition, 4) data clustering and so forth. Geologist and hydrologist have been adopted AI technologies when they have been proven and acceptable. In this respect, AI models don’t have along enough history in solving geology and hydrology problems. The AI model as a black box or gray box, have some advantages over conventional techniques, because they don’t require the prescription of a mathematical functional form. Based on majority of application of AI models, they could classify as a lumped models. Most of geology and hydrology systems are inherently complex and nonlinear that could be often simulated and predicted by simple structure of AI models. The AI models are now being used for predicting behavior of complex system that could not revealed by existing methods, precisely. The last decade has seen significant activity in AI model in various hydrology and geology related areas such as hydraulic conductivity estimation, groundwater predicting, water quality estimation and distribution, groundwater vulnerability, rainfall- runoff modeling, stream flow forecasting. AI models have also been suggested for use in control application and designing optimal strategies in groundwater remediation. Obviously, AI models are new technology, and their high capability for solving geology and hydrology problems must be investigated further. While many geologist and hydrologist have heard of IA methods in literatures and other, only a small fraction of them are aware of different applications of AI methods. The main motivation for this book is to present the position of hydrologist and geologist in this area today. To achieve this aim, this book provides contributions from researchers in this field. Recently, some review articles discussed concept and application of AI models in Hydrology and Earth sciences. However, the scope of this book is greater than them. Organization of this book includes 5 chapters. Chapter 1 introduces a supervised Committee Machine with Artificial Intelligence (SCMAI) method to predict hydraulic conductivity in Tasuj Plain aquifer, Iran. The SCMAI predicts hydraulic conductivity by a nonlinear combination of five individual AI models, Sugeno fuzzy logic, Mamdani fuzzy logic, Larsen fuzzy logic, artificial neural network, and neuro-fuzzy. The results show that all of these models have similar fitting to the hydraulic conductivity data in the Tasuj aquifer. To improve the results SCMAI employs an ANN model to re-predict hydraulic conductivity based on the five AI model predictions. The result shows higher performance of the SCMAI method than individual AI models.Investigation of capability of an Adaptive Neuro-Fuzzy Inference System (ANFIS) model conjugated with data preprocessing methods for predicting monthly Groundwater Level (GWL) was discussed in VI

eBooks Chapter 2 In proposed method, Self Organizing Map (SOM)-based clustering method as spatial pre-processor and Wavelet Transform (WT) as temporal data pre-processor are linked to the ANFIS concept and the main time series of three variables (rainfall, runoff and GWL) are decomposed into some multi-frequency time series by WT. The obtained results of the models show that the proposed models can predict GWL more accuracy because using SOM-based clustering method decreased the dimensionality of the input variables and on the other hand, the application of the WT to GWL data increased the performance of the ANFIS model by revealing the dominant periods of the process. On the other hand, the application of the wavelet transform to GWL data increased the performance of the ANFIS model up to 15.3% in average by revealing the dominant periods of the process. Chapter 3 introduces Fuzzy inference system to show that how well log and seismic data can be related to lithology, rock types, fluid content, porosity, shear wave velocity and other reservoir properties. For this purpose, Fuzzy inference system adopted for simulating a set of predefined inputs to a desired output. Petrophysical parameters, such as water saturation and porosity, are very important data for hydrocarbon reservoir evaluation and characterization. In this chapter, some of the recent applications of fuzzy inference systems in petroleum industry are reviewed. Chapter 4 attempts to provide an overview of Artificial Intelligence (AI) methods for forecasting groundwater quality. In this section, among many available AI based algorithms, the Artificial Neural Networks (ANNs), Fuzzy models and their hybrids are discussed for water quality prediction. The ability of AI model can be compared with conventional methods to predict water quality. Chapter 5 introduces five Artificial Intelligence (AI) methods, Sugeno fuzzy logic(SFL), Mamdani fuzzy logic, Larsen fuzzy logic (LFL), artificial neural network (ANN), and neurofuzzy (NF) to predict groundwater vulnerability in Maraghaeh-Bonab aquifer, Iran. The results show all of these models have better fitting to the groundwater vulnerability than general DRASTIC model. Also, all AI model is applicable to DRASTIC optimization, but the best model was NF model due to reap SFL and ANN model advantages.

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Chapter Title

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Supervised Committee Machine with Artificial Intelligence Model for Predicting the Hydraulic Conductivity of the Aquifer

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Conjunction of SOM-based clustering method and hybrid wavelet-ANFIS approach for groundwater level prediction

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Application of Fuzzy Inference System to Estimating Rock Properties from Well Logs and Seismic Data for groundwater level prediction

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Application of Artificial Intelligence methods for groundwater quality prediction

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Comparison of Ability of Artificial Intelligence Methods to Assess Groundwater Vulnerability

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ISBN: 978-1-63278-061-4-01 DOI: http://dx.doi.org/10.4172/978-1-63278-061-4-062

Supervised Committee Machine With Artificial Intelligence Model For Predicting the Hydraulic Conductivity of the Aquifer Ata Allah Nadiri Department of Earth Sciences, Faculty of Science, University of Tabriz, 29 Bahman Boulevard, Tabriz, East Azarbaijan, Iran

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*Corresponding author: Department of Earth Sciences, Faculty of Science, University of Tabriz, 29 Bahman Boulevard, Tabriz, East Azarbaijan, Iran. E-mail: [email protected]

Abstract The research introduces a supervised committee machine with artificial intelligence (SCMAI) method to predict hydraulic conductivity in Tasuj Plain aquifer, Iran. Hydraulic conductivity is the vital parameter in ground water management. Both Labratovary and field estimation methods of this parameter are time consuming and costly. Moreover, heterogenity and anisotropy of aquifers may lead to difficulties in accurately predicting hydraulic conductivity. The SCMAI predicts hydraulic conductivity by a nonlinear combination of individual AI models through an artificial intelligent system. Five AI models, namely, Sugeno fuzzy logic, Mamdani fuzzy logic, Larsen fuzzy logic, artificial neural network, and neuro-fuzzy are employed to predict hydraulic conductivity. The results show that all of these models have similar fitting to the hydraulic conductivity data in the Tasuj aquifer. To improve the results SCMAI employs an ANN model to re-predict hydraulic conductivity based on the five AI model predictions. The result shows higher performance of the CMAI method than a committee machine with the linear combination of AI model predictions. The results also show significant fitting improvement to individual AI models.

Keywords Artificial Neural Network; Fuzzy Logic; Hydraulic Conductivity;Neuro-fuzzy

Introduction Artificial Intelligence (AI) is an alternative approach for estimating hydrogeological and hydrochemical parameters such as hydraulic conductivity, porosity, soil water retention relationship, ion concentration and storage coefficient. The literature shows the success of benefiting AI models such as Artificial Neural Network (ANN) [1-7], Fuzzy Logic (FL) [812,5] and Neurofuzzy (NF), which reap ability of both FL and ANN [13-15]) for estimating hydrogeological and hydrochemical parameters .Estimation of hydraulic conductivity is vital for managing groundwater resources, contaminant transport, and designing remediation plans which have been extensively studied for decades. Mostly, different numerical and physical models were adopted for hydraulic conductivity [16]. AI methods provide flexibility, generality and less complexity to incorporate different types of input data for estimation in Application of Artificial Intelligence Methods in Geosciences and Hydrology Edited by: Ata Allah Nadiri

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comparison to physical-based models that requires detailed characterization of interested systems [5]. Hydraulic conductivity are not clear-cut and most of the times are associated with uncertainties [5]. The inherent uncertainty of hydraulic conductivity makes it important to consider alternative plausible AI methods to make estimation and increase accuracy. Generally, more than one AI model provides a similar acceptable fit to the observations. This study emphasizes the importance of using multiple AI methods for hydraulic conductivity predicting than a single AI method. Each AI method has its own advantages. The fuzzy models tend to be robust to parameter changes, and are also tolerant to imprecision and uncertainty [8,5,17]. Besides, the ANN model represents non-linear relationships and learns these relationships directly from the data being modeled (Palani, Obviously, the NF model takes advantage of the FL and ANN in modeling.None of AI methods can dominate other AI methods. Therefore, usage of multi-model interface has many advantages. So, Committee Machine With Artificial Intelligence Intelligent (CMAI) which is an artificial intelligence multi-model interface, and applied in different disciplines [18-20,15] can be utilized for hydraulic conductivity estimation. The CMAI uses the results of AI models in order to arrive at overall decision that is superior to that of any of single AI models acting alone [21,22]. Combination of multiple AI models to estimate hydraulic conductivity was suggested beneifiting a Supervised Committee Machine With Artificial Intelligence (CMAI) method [17,18-20]. CMAI makes prediction by linearly combining the outputs of individual AI models through a set of weights. There are two methods to determine weights for CMAI [19,23]: simple averaging using equal weights and weighted averaging using optimized weights. Kadkhodaie-Ilkhchi et al. (2009) [20] and Labani et al. (2010) [23] used a Genetic Algorithm (GA) to optimize weights and found that weighted averaging performed superior than simple averaging for parameter estimation. Instead of linearly combining AI models, this study introduces a Supervised Committee Machine With Artificial Intelligence (SCMAI) method that replaces linear combination with an Artificial Neural Network (ANN). In the SCMAI, the ANN receives individual model predictions as input and derives new predictions. The advantage of the SCMAI is the nonlinear combination of the AI models under supervision such that the SCMAI may perform better for hydraulic conductivity prediction in complex aquifer systems. This study tests the SCMAI for predicting hydraulic conductivity in the Tasuj Palin, north of West Azarbaijan, northwest of Iran. The Tasuj plain aquifer is a heterogeneous unconfined aquifer. The Tasuj plain aquifer, one of the endangered aquifers around Urmia Lake (the third largest saline lake in the world), where groundwater level has considerably declined due to large groundwater withdrawals. The continuation of groundwater level declination at its current rate could cause desiccation of Urmia Lake and an ecosystem disaster to its surrounding area. Therefore, water resources monitoring, management and protection are essential to sustain the Tasuj plain aquifer. For this purpose, a fundamental understanding hydrogeological conditions and it parameters for an aquifer system is important. In this study, five artificial intelligence models, Sugeno fuzzy logic (SFL), Mamdani Fuzzy Logic (MFL), Larsen Fuzzy Logic (LFL), Artificial Neural Network (ANN), and neuro-fuzzy (NF) are adopted to test the SCMAI and CMAI methods. However, the SCMAI and CMAI can include any number of AI models for multimodel analysis.

Study Area The Tasuj basin shown in Figure 1 is located about 100 km northwest of the city of Tabriz, in the northwest region of Iran. The basin is a subbasin of the Urmia Lake basin. The Tasuj basin is about 559.32 km2. This includes 302.67 km2 of the Tasuj plain and 256.65 km2 of Mishu Mountain. The study area is surrounded by Urmia Lake (south), Mishu Mountain (north), Salmas Plain (west) and Shabestar Plain (east). The study area 2

contains the city of Tasuj and 15 other villages. Agriculture is the main economic activity in the are [15]. Based on de Martonne (1925)[24] and Emberger (1930)[25], the prevailing climate in the Tasuj plain is semiarid-cold. Average annual precipitation is about 232.7 mm [26]. The highest and lowest precipitation occurs in the spring and summer seasons, respectively. The mean daily temperature at the Tasuj climatological station (1,411m amsl) varies from –11.33 oC (in January) to 33 oC (in August). The annual temperature average is 14 oC. In general, average monthly relative humidity at the Tasuj climatological station is fairly high ranging from 54% (in July) to 68% (in March). In the Tasuj basin, only a few seasonal rivers originate from Mishu Mountain. These seasonal rivers can flood the Tasuj plain in the wet season.

Figure 1: The study area and locations of piezometers [27]..

Hydrogeology and Geology Of the Tasuj Plan The Tasuj basin has geological formations from the Precambrian to the Quaternary age. The geological map is shown in Figure 2. There are several faults in the Tasuj basin shown in Figure 2. The most important faults include the Tasuj fault, the Gzeljeh fault and the Angoshtejan fault. The Tasuj fault is a long, deep, and an active reverse fault, extending from northeast to southwest. The north-south trending Gzeljeh and Angoshtejan faults are the important faults in the area. The Gzeljeh fault cuts the moderately consolidated conglomerate and sandstone unit and probably has a dominant role in the recharge of the Tasuj plain aquifer [15]. The Tasuj plain aquifer is a heterogeneous unconfined aquifer. Groundwater in the Tasuj plain aquifer was withdrawn through 147 water wells, 70 springs and 70 qanats. In 2009, groundwater was withdrawn 36.81 million cubic meters [28]. Despite enacting restrictive rules such as limiting drilling new pumping wells and reducing pumping rate, 24 springs and 14 qanats became dry in the recent years due to over-extraction.The geo-electrical survey shown in Figure 3 was conducted in the plain by the Abkav Consulting Engineering Co. (1973)[29] to delineate thickness of the Tasuj plain aquifer. It was found that the bedrock of the plain is Miocene formation, which generally dips south towards Urmia Lake. 3

The aquifer thickness (B) and transverse resistance ( Rt ) were estimated at 63 points along the geo-electrical profiles. The maximum thickness is 182 m and the minimum thickness is 44m.The B and Rt distributions were obtained by an ordinary kriging method. The alluvial aquifer is thick along profiles 2, 4, 5, and 7 and this thickness decreases towards Urmia Lake .Based on seven pumping tests carried out in the plain by the East Azerbaijan Regional Water Authority (2001) [28], the Till village, the city of Tasuj, and alluvial terraces have high transmissivity (150-300 m2/d). Low transmissivity zone (25-50 m2/d) is in the southwest of the plain due to thin aquifer thickness. Thirty piezometers were installed in the Tasuj plain to monitor groundwater level. Their locations are distributed over the entire region shown in Figure 1. Groundwater level discernibly declined in the last decade. Due to the agricultural, industrial and drinking water demands, high withdrawal rates around the city of Tasuj produced the maximum cone of depression in this area. The maximum groundwater level decline rate was 3.7 meters per year recorded in the Cheshmekonan piezometer. Groundwater flow direction in the Tasuj plain aquifer is mainly towards Urmia Lake.

Figure 2: Geological map and Till piezometer logging [17].

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Figure 3: Thickness of Tasuj aquifer and location of profiles of geo-electrical survey [5].

Physical Relationship Between K and Input Data In the Tasuj plain, hydraulic conductivity was determined in 132 locations by constant and step drawdown pumping tests. The maximum K is 29.74 m/day and the minimum K is 0.13 m/day [5]. The mean and the standard deviation of K are 2.35 and 3.30 m/ day, respectively. Hydraulic conductivity in an unconfined aquifer is related to electrical resistivity (ρ) of the aquifer which is transverse resistance (Rt) divided by the thickness of the aquifer (B). Therefore, hydraulic conductivity relates to Rt and B [30-31] .Since the relationship between hydraulic conductivity and ρ depends on the salinity of formation water [32], Electrical Conductivity (EC), which responds to the salinity of formation water, can also be related to hydraulic conductivity. The EC was measured at 132 water wells and the EC distribution was derived by ordinary kriging. In addition to Rt, B, and the EC, the distance of each estimation point to the origin of the coordinate system (O) as another input to the AI models was also considered to take into account the geological and geomorphological effects. In order to estimate hydraulic conductivity by the AI models, B ,Rt, EC and the distance of locations to the origion (O) are used in the input vector. Theory and Methodology Artificial Neural Network (ANN) Artificial neural networks are imitating human brain by using mathematical methods and have been proven to be extremely beneficial tools for simulating, predicting and forecasting hydrological variables [33-36]; The most widely used neural network is the Multi-Layer Perceptron (MLP) [21,37]. In the MLP as a feed forward ANN, the neurons are organized in layers, and each neuron is connected only with neurons in the other layers. A typical three-layer feedforward ANN is shown in Figure 4. The input signal propagates through the network in a forward direction, layer by layer. The mathematical functioning of a three-layer feed forward ANN is given as[6,17,38]. 5

= O j f1 (b j + ∑ W ji I i ) i

= Ok Nor = .K f 2 (b K + ∑ W ji OJ )

(1) (2)

i

where f1 and f 2 are activation functions of the hidden layer and output layer, respectively, I i is the ith input, O j is the jth output, W ji and Wkj are weights that control the strength of connections between two layers, and the biases bj and b K are used to adjust the mean value for input layer and hidden layer, respectively. The activation function for the hidden layer is typically a continuous and bounded nonlinear transfer function such as sigmoid and log sigmoid functions. The activation function for the output layer is usually a linear function. This study employed the hyperbolic tangent sigmoid (Tansig) for f1 and linear (Purelin) for f 2 . In the ANN training step, the Levenberg-Marquardt Algorithm (LM) was used as a supervised learning algorithm to estimate the weights W ji and Wkj [34,38-40].

Figure 4: Schematic structure of typical feedforward model.

Fuzzy Logic (FL) In this study, three FL models were used for predicting hydraulic conductivity in Tasuj Plain. Fuzzy logic or fuzzy set theory [41] can handle problems associated with inherent uncertainty. Therefore, fuzzy sets are more suitable to describe things that are inherently imprecise, such as hydrogeologic parameters. Fuzzy sets include partial membership ranging between 0 to 1 [5]. Each fuzzy set is represented by a Membership Function (MF) which has ambiguous boundaries and gradual transitions between the defined sets which render them amenable to overcome the inherent uncertainty [17]. The membership functions may have different shapes, such as triangular, trapezoidal, Gaussian, etc. An FL model consists of three main parts: fuzzification, inference engine (fuzzy rule base), and defuzzification. In the fuzzification step, the four crisp inputs change to fuzzy sets for constructing the inference engine. The inference engine consists of rules. Each rule, in turn, is formed from multiple inputs to a single output. When the antecedents of fuzzy rules include more than one rule, fuzzy operators are used to connect them. The most commonly used fuzzy operators are AND which supports min (minimum) and prod (product), OR (maximum) and NOT [17]. The consequences of a fuzzy rule assign an entire fuzzy set to the output through the process, which is called implication. The input to the implication process is a single number given by the antecedent, and the output is a fuzzy set. Since decisions are based on testing all of the rules in an FL model, the rules must be combined via aggregation processes in order to make a decision. The process of transforming the aggregation result into a crisp output is termed defuzzification. The most 6

commonly used defuzzification methods are centroid, bisector, middle of maximum (the average of the maximum value of the output set), largest of maximum, and smallest of maximum. The most important task to construction of fuzzy model is to identify a model structure which is composed of the optimal number of rules and cluster of the data. To perform structure identification, many studies used different clustering methods [42,43]. The idea of fuzzy clustering is to find natural groupings of data, within a large dataset, hence revealing patterns that can represent one specific part of the system behavior [44]. The most appropriate clustering methods are the Fuzzy C-Means (FCM) [45] and Subtractive Clustering (SC) [43,44]. The FCM method categorizes data points that populate multidimensional spaces into a specific number of clusters. The FCM clustering starts with an initial guess for the cluster centers, which intends to mark the mean location of each cluster. Additionally, FCM assigns every data point a membership grade for each cluster.By iteratively updating the cluster centers and the membership grades for each data point, FCM iteratively moves the cluster centers to the correct location within a data set. During this iteration an objective function is minimized and shows the distance from any given data point to a cluster center weighted by the membership grade of that data point . The FCM output is a list of cluster centers and several membership grades for each data point. Therefore, the MFL fuzzy rules are extracted through the FCM. In this fashion, the model matrices of data pass through the FCM algorithm and the cluster centers are calculated. In the FCM algorithm, the number of clusters is defined by the user. Choosing the optimum number of clusters is accomplished by measuring the performance of the model during systematically changing the number of the clusters from 1 to the number of the model data points [17]. The Subtractive Clustering (SC) method was introduced by Li et al. (2001)[44]. The important parameter in subtractive clustering, which controls the number of clusters and the fuzzy if-then rules is the cluster radius. The important parameter in the SC method is the cluster radius, which controls the number of clusters and fuzzy if-then rules . Decreasing the cluster radius will increase the number of clusters and lead to smaller clusters. This will create more rules and complicate the system behavior and may lead to a low performance of the model. In contrary, a large cluster radius produces large clusters in the data and results in few rules [43], which may not be sufficient to cover the entire domain. Searching for the optimal cluster radius can be accomplished by systematically varying cluster radius value from 0 to 1 until Minimal Root Mean Squared Error (RMSE) is met. Based on the type of the output membership function and fuzzy operators, FL may be constructed by the methods proposed by Mamdani, Sugeno, and Larsen [46-48]. In the Mamdani FL (MFL) method, the output membership functions are fuzzy sets. After the aggregation process, there is a fuzzy set for each output variable that needs defuzzification [47,49]. The SC method is adapted to SFL model construction. It has been improved as an efficient and useful way to cluster the data and determine the number of membership functions and rules in recent research .The forst model is MFL. For kˆ estimation, a fuzzy ifthen rule,i , can be expressed as: i ) and ( Rt belongs to MFRt ), and (B Rule i: If (O belongs to MFBi ) and (EC belongs to MFEC i i belongs to MFK ),then (K belongs to MFK − )

i



i

where K output variable of hydraulic conductivity, MFO is the membership function of i th th the i cluster of input o , MFEC is the membership function of the i cluster of input EC , and so forth. The basic Information about FCM method presented in elsewhere [42,50,51]. The Larsen FL method (LFL) is similar to the Mamdani FL method. The major difference between them is that LFL uses the product operator for the fuzzy implication which scales the output fuzzy set. Sugeno FL (SFL) system is similar to the Mamdani method in many aspects. However, 7

the main difference between them is that output membership functions are either linear or constant in the Sugeno method which are called zero and first order Sugeno FL model, respectively (Takagi and Sugeno, 1985; Sugeno, 1985). The most capable clustering algorithm in this method is a subtractive clustering method which can be used for extraction of clusters and fuzzy If-Then rules [43]. The details of Subtractive Clustering (SC) could be found in Chiu (1994)[43]. The efficiency of an SFL model is related to the clustering radius parameter of subtractive clustering which controls the number of clusters and fuzzy If-Then rules. For kˆ estimation in this study, a fuzzy if-then rule i can be expressed as: i i ) and ( MFRi belongs to MFRt ), and (B Rule i: If (O belongs to MFoi ) and (EC belongs to MFEC belongs to MFBi ), then K i = mi o + ni EC + Pi R t + q i B+ Ci t

th

where K i is the output of rule i, MFOi is the membership function of the i cluster of th i input O , MFEC is the membership function of the i cluster of input EC , and so forth. mi , pi , qi and Ci are coefficients to be determined by linear least-squares estimation. The final output is the weighted average of all rule outputs (aggregation) as follows: ∑W K = K (3) ∑W where Wi is the firing strength of rule i, which is obtained via “and” (minimize) operator. i

i

i

i

i

Neuro-Fuzzy (NF) Neuro-fuzzy modeling is a combination technique for describing the behavior of a system using fuzzy inference rules within a neural network structure. The NF inference system consists of a given input/output data set and an SFL whose MF parameters are tuned using a hybrid algorithm . The most compatible method for construction of NF model is Sugeno method using subtractive clustering. For this study, the NF architecture of a five-layer MLP network is considered in the hydraulic conductivity estimation. In each layer, the NF operations are as follows: Layer 1: Generate membership function of input data. The output of neuron i is defined by Oil = µ ji (X)

(4) where j is the number of inputs and i is the membership function index. X = {O, R, EC , B} is a set of input. µ ji (X) is a fuzzy set associated with neuron i given a membership function. We used a generalized Gaussian function to develop membership functions. Layer 2: Calculate firing strength

wi

for the ith rule via multiplication:

2 O= w= µli (X) µ2i (X) µ3i (X) µ4i (X) µ5i (X) µ6i (X) i i

(5)

Layer 3: Compute the normalized firing strengths for the i neuron: th

3 O= w= i i

wi

= i 1,......., 6

∑w

i

i

(6)

Layer 4: Compute the contribution of the ith rule in the model output based on the first order SFL method: Oi4 = wi K i = wi (mi d + n i EC+ pi R i + q i B+ Ci )

For

i= 1,...., 6

(7)

Layer 5: Calculate the final output as the weighted average of all rule outputs (aggregation): O 5= k= wK (8) i

∑ i

i

i

The NF parameters in equation (7) and membership function parameters are estimated using hybrid algorithm in this study, which is a combination of the gradient descent and least-squares method . 8

SCMAI Model The Committee machine with artificial intelligence approach combines artificial intelligence model results to reap advantages of all AI models to produce the final output. Previous works recommend two methods of the simple averaging and the weighted averaging for construction of SCMAI model [19,23].This study introduces a supervised committee machine with artificial intelligence (SCMAI) model that employs an ANN as a supervised combiner of AI models to replace simple averaging or weighted averaging. The SCMAI model consists of four artificial intelligence models shown in Figure 5 and includes two major steps. In the first step, hydraulic conductivity is estimated using the artificial intelligence models including MFL, LFL, SFL, ANN and NF. In the second step, a supervised artificial neural network is constructed as a nonlinear and supervised combiner. The mathematical expression of the SCMAI model is:

ki = AI (O, EC, R t , B)



(9)

i



(10)

= OK k= SCIM

f 2 (b k + ∑ Wkj Oi )

(11)

= Oj

i ) f1 (b j + ∑ W ji K

ki

i

is the output of the each AI model which has been used as i input, f1 and f 2 where are activation functions for the hidden layer and output layer, respectively, O j is the jth output of nodes in hidden layer, W ji and Wkj are weights that control the strength of connections between two layers, and the biases bj and bK are used to adjust the mean value for hidden layer and output layer, respectively. The activation functions are hyperbolic tangent sigmoid  SICM . In the (Tansig) for f1 and linear (Purelin) for f 2 . The output OK of the SCMAI model is K ANN training step, the LM algorithm was adopted as a learning algorithm to estimate the weights W ji and Wkj and biases . th

Figure 5: Schematic structure of SCMAI model.

Discussion Artificial Intelligence System Artificial neural network (ANN) A three layer network as a multilayer perceptron network with Levenberg-Marquardt (LM) algorithm which is called MLP-LM structure was used for K estimation. The data set 9

4 3.5 3 2.5 2 1.5 1 0.5 0

4

Measured

Measured

was divided into three groups including training (109 data points), and testing (23 data points). Four inputs EC, Rt, B, and O were used in the first layer. The number of neurons in the hidden and the output layers were three and one neurons, respectively. The schematic diagram of MLP-LM model designed in this study is shown in Figure 4. RMSE and R2 values of the training step were 1.41 m/d, 0.81, respectively (Table 1). A comparison between measured and predicted K values in the test step is shown in Figure 6(a).

0

2

3 2 1 0

4

0

2

Calculated

B)SFL

A)ANN

B)SFL

4

4

3

3

Measured

Measured

4

Calculated

2 1 0

2 1 0

0

2

4

0

2

Calculated

4

Calculated D)LFL

C)MFL

Measured

4 3 2 1 0 0

2

4

Calculated E)NF

Figure 6: Comparison of measured and calculated hydraulic conductivity in test step.

Model ANN MFL LFL SFL NF

Step

R2

RMSE

Train

0.81

1.41

Test

0.73

1.62

Train

0.83

1.31

Test

0.725

1.71

Train

0.79

1.5

Test

0.652

1.8

Train

0.83

1.31

Test

0.725

1.7

Test

0.91

0.9

Train

0.856

1.1

Table 1: R and RMSE values in training and test steps of five models. 2

FL model In this study, three different methods of FL such as MFL, LFL and SFL were adopted for hydraulic conductivity. FCM and SC clustering methods are three powerful fuzzy clustering techniques which could be used for the construction of a fuzzy rule base[44]. Therefore, three LFL, MFL and SFL model were constructed using FCM and SC clustering methods, respectively. For Mamdani Fuzzy Logic (MFL) and Larsen Fuzzy Logic (LFL) models, a FCM clustering 10

method was used for extraction of clusters and fuzzy if–then rules. The input data is O, B, Rt, and EC for MFL model based on the Gaussian membership function. Results showed that the optimum number of clusters for hydraulic conductivity is 10. The R2, and RMSE of MFL and LFL models have been shown in Table 1 and Figure 6(b,d). The first step in construction of SFL model is data clustering. The subtractive clustering method was applied for data clustering. Radius clustering was selected based on the minimum RMSE. Choosing a value of 0.4 for clustering radius was associated with the lowest RMSE (1.31 m/d) and this generates six fuzzy if-then rules. Thus, the SFL model was established by six Gaussian membership functions (clusters) for input and output data resulting in six rules. A Gaussian membership function (x) shows jth input data (x) and ith membership function which is described as a following:  (ci − x) 2 2σ i2 

µ= exp  − i (x)

  

(12)

where σ and c are the parameters of normal distribution showing the standard deviation and mean of data, respectively. These Gaussian membership functions are constructed from the mean and standard deviation values of the clusters. Figure 7 shows one of the generated membership functions of inputs as an example, which is fitted to the extracted input clusters.

Figure 7: The generated membership functions of Rt.

In the fuzzy model, after training the model using 109 data sets, the input matrix of the ˆ values were calculated. The test model (23 data sets) was exposed to the SFL model and K error was measured using RMSE and determination coefficient (R2). The R2 and RMSE of training stage are 0.83 and 1.31m/d, respectively. K values were calculated using test step input data that are exposed to the SFL model. The efficiency of the model is 1.7 m/d and 0.727 based on the RMSE and determination coefficient (R2), respectively (Figure 6(d)).

Neuro-fuzzy(NF) In this study, an NF model was developed for hydraulic conductivity estimation. The NF model like fuzzy model need to classify (four inputs and one output). The same clusters of input and outputs and rules were used for NF construction. Hybrid algorithm which is a combination of the least-squares method and the back propagation gradient descent method was applied to optimize and adjust Gaussian membership function parameter and the coefficients of output linear equation [52]. After 3 epochs of training, no change was found in the model performance. RMSE and R2 of training and testing steps of NF model are shown in Table 1. Based on the results of different AI models (Table 1), all of them are applicable to estimation of hydraulic conductivity. Therefore, using advantages of these individual models, the multi-model such as Intelligence Committee Machine (CMAI) can now be considered to obtain minimum RMSE for estimation of hydraulic conductivity.

SCMAI For Predicting The K Value The SCMAI method shown in Figure 5 adopts a simple ANN method to re-estimate 11

hydraulic conductivity obtained by SFL, LFL, MFL, ANN, and NF in the training step (109 sample data). The MLP-LM structure based on equations (10) and (11) is employed in the ANN model to be used in the SCMAI. The ANN model has 5 inputs in the first layer for Kˆ via SFL, LFL, MFL, ANN, and NF and the target Kˆ SCMAI . The number of neurons in the hidden layer was two. The LM algorithm was used to optimize 12 weights and 4 biases in the ANN. After 70 epochs, the RMSE of 0.11m/day is obtained. Then, the SCMAI model is tested against 23 data sets. The RMSE and R2 for SCMAI predictions are 0.22m/day and 0.95, respectively. Comparing the error measure values in Table 1, it is seen that SCMAI outperforms individual AI models with low errors. These results imply that SCMAI model which uses a nonlinear combination shows high effectiveness for predicting the hydraulic conductivity value in the heterogeneous unconfined aquifer in Tasuj Plain.

Comparison of SCMAI To CMAI For Estimating Hydraulic Conductivity Here, we compare the SCMAI model against the CMAI model. For the simple averaging method, CMAI estimates hydraulic conductivity using SFL, LFL, MFL, ANN, and NF with equal weights as follows:

 SCMAI = üüüüüüü  SFL + K K

 LFL + K

 MFL + K

 ANN + K

 NF K

(13)

For the weighted averaging method, optimal weights, wi are determined by minimizing the Mean Squared Error (MSE): m

1

∑ m (w K + w K + w K + w K + w K − K ) where m is the number of training data (109 samples). The weights, wi MSE =

i =1

1

SFL

2

LFL

3

MFL

4

1 and 0. The sum of weights is unity,

NF

5

ANN

i

2

∑ i Wi = 1 .

(14) range between

A GA optimizer (MATLAB 2010) is used to minimize the MSE. The initial population size is set to 23. The maximum number of generations can go up to 160. The probability for crossover operation is 0.8 and the mutation function is Gaussian. After optimal weights are obtained by GA, the CMAI model estimates hydraulic conductivity by the following equation:

 SCMAI = 0.18 K  SFL + 0.15 K  LFL + 0.17 K  MFL + 0.23K  ANN + 0.27 K  NF K

(15)

The order of the weights in the equation(15) is consistent with the ranking of the AI models in Table 1. The NF model receives the highest weight 0.27 and the LFL model received the lowest weight of 0.15. The performance result of the SCMAI and CMAI is shown in Table 2 for K data for the testing stage. The R2 and RMSE for CMAI with simple averaging (equation(13)) are 0.82 and 0.42, respectively, and for CMAI with weighted average are 0.86 and 0.30 m/d, respectively. Table 2 concludes that CMAI and SCMAI performed better than individual AI models. CMAI with weighted averaging performs better than CMAI with simple averaging, which agrees to Kadkhodaie-Ilkhchi et al. (2009)[20] and Labani et al. (2010) [23]. The SCMAI model which takes advantages of a nonlinear combination of the AI models performs better than the linear combination in CMAI for estimation of hydraulic conductivity in the Tasuj Plain. Figure.8 shows the distribution of K in Tasuj Plain which estimated by SCMAI model. CMAI-SA CMAI-WA SCMAI

Train

0.89

0.31

Test

0.82

0.42

Train

0.92

0.29

Test

0.86

0.30

Train

0.99

0.11

Test

0.95

0.22

Table 2: R2 and RMSE values of different CMAI models in training and test steps.

12

Figure 8: Distribution of estimated K values by SCMAI.

Conclusions This study introduces a supervised committee Machine With Artificial Intelligence (SCMAI) algorithm, which combines the outcomes of AI models, to better predict the hydraulic conductivity of Tasuj aquifer. In SCMAI, the ANN receives predictions of four individual artificial intelligence models; Sugeno Fuzzy Logic (SFL), Mamdani Fuzzy Logic (MFL), neuro-fuzzy, and Artificial Neural Network (ANN) estimations as input and derives a new estimation. Following conclusions can be drawn from this study: 1. Adopting SFL, MFL, MLP-LM, and NF as AI models was successful to estimate hydraulic conductivity. The results of these models are within the same order. It can be stated that AI models are applicable for the estimation of hydraulic conductivity in the heterogeneous and unconfined Tasuj aquifer. 2. The CMAI and SCMAI models could improve the efficiency of AI models for hydraulic conductivity estimates. 3. The SCMAI is more capable than CMAI in predicting hydraulic conductivities of the heterogeneous and unconfined aquifer, Tasuj plain, as a case study. 4. Most of the aquifers in nature are heterogeneous and complex. Therefore, the presented method (SCMAI) can be used for prediction of different hydrogeological parameters such as hydraulic conductivity, porosity, water content and etc., in various case studies.

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27. Nadiri AA (2012) Comparison of efficiency of numerical and artificial intelligence models in aquifer management (case study: Tasuj plain). Ph.D. Disertation, University of Tabriz. 28. East Azerbaijan Regional Water Authority (2001) Studying of Groundwater resources and mathematical modeling of Tasuj Plain using GIS. The first volume, p. 212 (in Persian). 29. Abkav Consulting Engineering Co (1973) Geophysical studies reports of Tabriz, Tasuj and Shabestar plains. East Azerbaijan Regional Water Authority (in Persian). 30. Maillet R (1947) The fundamental equations of electrical prospecting. Geophysics 12, 529-556. 31. Harb N, haddad K, farkh S (2010) Calculation of transverse resistance to correct aquifer resistivity of groundwater saturated zones: implications for estimating its hydrogeological properties. Lebanese Science Journal 11(1), 105-115. 32. Putvance DT (2000) On the electrical-hydraulic conductivity correlation in aquifers. Water Resources Research 36(10), 2905-2913. 33. Nadiri AA (2007) Water level evaluation in Tabriz underground area by artificial neural networks. M.S. Theses, University of Tabriz: 178p. 34. Nourani V, Asgharimogaddam A, Nadiri AA (2008b) An ANN-based model for spatiotemporal groundwater level forecasting. Hydrological Process 22, 5054-5066. 35. Piotrowski , A.P., Napiorkowski, J.J, (2011) Optimizing neural networks for river flow forecasting – Evolutionary Computation methods versus the Levenberg–Marquardt approach. Journal of Hydrology 407, 12–27. 36. Siou L K, Johannet A, Borrell V, Pistre S (2011) Complexity selection of a neural network model for karst flood forecasting: The case of the Lez Basin (southern France). Journal of Hydrology 403, 367–380. 37. Haykin S (1999) Neural Networks, a Comprehensive Foundation. Macmillan College Publishing Co., New York, USA. 38. ASCE Task Committee on Application of Artificial Neural Networks in Hydrology (2000) Artificial neural network in hydrology, part I and II.Journal of Hydraulic Engineering5, 115-137. 39. Daliakopoulos IN, Coulibaly P, Tsanis IK (2005) Groundwater level forecasting using artificial neural networks. Journal of Hydrology 309, 229-240. 40. Nourani V, Asgharimogaddam A, Nadiri AA, Sing VP (2008a) Forecasting Spatiotemporal water levels of Tabriz Aquifer. Trend in Applied sciences Research 3(4), 319-329. 41. Zadeh LA (1965) Fuzzy sets. Information and Control 8, 338–353. 42. Bezdek JC, Ehrlich R and Full W (1984) The fuzzy c-means clustering algorithm. Computers & Geosciences 10, 191-203. 43. Chiu S (1994) Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems 2, 267–278. 44. Li H Philip, ChCL, Huang HP (2001) Fuzzy Neural Intelligent Systems: Mathematical Foundation and the Applications in Engineering, CRC Press, Inc., Boca Raton, FL. 45. Bezdec JC(1981) Pattern Recognition with Fuzzy Objective Function Algorithms. New York, Plenum Press. 46. Sugeno M (1985 ) Industrial Application of Fuzzy Control. North-Holland, New York. 269 pp. 47. Mamdani EH (1976 ) Advances in the linguistic synthesis of fuzzy controllers. International Journal of Man-Machine Studies 8, 669–678. 48. Mamdani EH, Assilian S (1975) An experimental in linguistic synthesis with a fuzzy logic control. International Journal of Man-Machine Studies 7, 1–13. 49. Mamdani EH (1977) Applications of fuzzy logic to approximate reasoning using linguistic synthesis. IEEE Transactions on Computers 26, 1182–1191. 50. Arrell, K E Fisher, P F Tate, N J and Bastin L (2007) A fuzzy c-means classification of elevation derivatives to extract the morphometric classification of landforms in Snowdonia, Wales. Computers & Geosciences 33, 1366-1381. 15

51. Kannan SR, Ramathilagam S and Chung, PC (2012) Effective fuzzy c-means clustering algorithms for data clustering problems. Expert Systems with Applications 39, 6292-6300. 52. Zounemat-Kermani M, Teshnehlab M (2008) Using adaptive neuro-fuzzy inference system for hydrological time series prediction. Applied Soft Computing, 8(20), 928-936.

16

eBooks

ISBN: 978-1-63278-061-4-02 DOI: http://dx.doi.org/10.4172/978-1-63278-061-4-062

Conjunction of SOM-Based Clustering Method and Hybrid Wavelet-ANFIS Approach For Groundwater Level Prediction Vahid Nourani1* and Farnaz Daneshvar Vousoughi2 Professor, Dept. of Water Resources Engineering, Faculty of Civil Eng., Univ. of Tabriz, Iran

1

Ph.D. Candidate, Dept. of Water Resources Engineering, Faculty of Civil Eng., Univ. of Tabriz, Iran

2

*Corresponding author: Vahid Nourani Professor, Dept. of Water Resources Engineering, Faculty of Civil Eng., Univ. of Tabriz, Iran ,Tel.: +98 914403 0332; fax: +98 413 334 4287. E-mail: [email protected]

Abstract The knowledge of groundwater table fluctuations is important in agricultural lands as well as in the studies related to groundwater utilization and management levels. This research demonstrates the potential use of Artificial Intelligence (AI)-based techniques for predicting Monthly Groundwater Level (GWL). In this paper, an Adaptive Neuro-Fuzzy Inference System (ANFIS) model conjugated with data pre-processing methods was used for GWL modeling at 2 scenarios of Ardabil plain. In proposed method, Self Organizing Map (SOM)-based clustering method as spatial pre-processor and Wavelet Transform (WT) as temporal data pre-processor are linked to the ANFIS concept and the main time series of three variables (rainfall, runoff and GWL) are decomposed into some multi-frequency time series by WT. Afterwards, these time series are imposed as input data in each cluster to the ANFIS to predict the GWL one and multi time step ahead. The performance of the ANFIS model was compared to the newly proposed combined WT–ANFIS (WANFIS) model and also the conventional linear forecasting method of Auto Regressive Integrated Moving Average With Exogenous Input (ARIMAX). GWL predictions were investigated under two different scenarios. The obtained results of the models show that the proposed models can predict GWL more accuracy because using SOM-based clustering method decreased the dimensionality of the input variables and on the other hand, the application of the WT to GWL data increased the performance of the ANFIS model by revealing the dominant periods of the process. On the other hand, the application of the wavelet transform to GWL data increased the performance of the ANFIS model up to 15.3% in average by revealing the dominant periods of the process.

Key words Adaptive Neural-Fuzzy Inference System; Ardabil plain; Data pre-processing; Entropy; Groundwater level modeling, Wavelet Application of Artificial Intelligence Methods in Geosciences and Hydrology Edited by: Ata Allah Nadiri

1

Introduction In many watersheds, groundwater is often one of the major sources of water supply for domestic, agricultural and industrial users. In many such regions, groundwater has been withdrawn at rates far in excess of recharge, leading to harmful environmental side effects such as major water-level declines, drying up of wells, reduction of water in streams and lakes, water-quality degradation, increased pumping costs, land subsidence, and decreased well yields [1]. Groundwater systems possess features such as complexity, nonlinearity, being multi-scale and random, all governed by natural and/or anthropogenic factors, which complicate the dynamic predictions. Therefore, different models have been developed to simulate this complex process. Models based on their involvement of physical characteristics generally fall into three main categories: black box, conceptual, and physical-based models [2]. Although physical-based and conceptual models are the main tools for representing hydrological variables and understanding the physical processes in a system, they have practical limitations and when accurate prediction is more important than conceiving the physics, black box modeling can be a good alternative [2]. The complexity and uncertainty of the groundwater process caused black box models such as Adaptive Neural-Fuzzy Inference System (ANFIS) are widely used by hydrogeologists. There have been a few surges in reports on the applications the conjunction of the Artificial Neural Network (ANN) and Fuzzy Inference System (FIS) models as an ANFIS model for groundwater level (GWL) prediction. The seasonal GWL are predicted using the ANFIS and Radial Basis Function (RBF) based on previous seasonal rainfall and GWL by Amutha and Porchelvan (2011) [3]. It is observed that the ANFIS model is able to capture the dynamics of the surface water and ground water interactions better when compared to RBF and thus able to predict the seasonal GWL accurately. Mayilvaganan and Naidu (2011a) [4] compared ANNs and Fuzzy Logic (FL) techniques in GWL prediction of a watershed. It was observed that ANNs perform significantly better than FLs. Mayilvaganan and Naidu (2011b) [5] used ANN, Mamdani Fuzzy Inference Systems (MFIS) and ANFIS to predict the GWL of Thurinjapuram watershed, Tamilnadu. Simulation results reveal that ANFIS is an efficient and promising tool in compare to ANN. In any data-driven model like ANFIS, the data quality can affect the modeling performance and if irrelevant and redundant information or noisy and unreliable data are used, the knowledge discovery during the training phase of ANFIS will be more difficult. Therefore to improve the quality of the data and consequently, of the modeling results, the raw data should be pre-processed so as to improve the efficiency and ease of the training process. A groundwater system as a spatially and temporally varying hydrologic process is usually represented by spatio-temporal data and therefore, implication of appreciate time-space data pre-processing techniques can enhance the efficiency of any data driven-based modeling (e.g. ANFIS) of GWL. The Wavelet Transform (WT) is an appropriate temporal pre-processing method that can be utilized to extract a variety of features from the time series, such as short-term and long-term fluctuations, by decomposing the time series into different sub-components. The wavelet decomposition of a non-stationary time series into various scales provides an interpretation of the time series structure and extracts significant information about its history. As a result of these merits, the WT has been applied to analysis of non-stationary hydrological signals [6]. The use of wavelet-based decomposed sub-signals as inputs into an ANFIS model helps the ANFIS model to distinguish the dominant sub-signals by applying relatively strong weights [7-9]. Different components of the hydrologic cycle, including precipitation, groundwater, river flow, and sedimentation, have since been modeled using hybrid approach, reviewed by Nourani et al. (2014a)[10]. In the field of GWL modeling, Investigating the ability of a joint Wavelet And Neural-Fuzzy (NF) model to perform one,two- and three-day-ahead groundwater depth forecasting, Kisi and Shiri (2012)[11] found 2

that the joint model outperformed the NF model, particularly for two- and three-day-ahead forecasts. Moosavi et al. (2013)[12] compared several data-driven models (i.e.,ANN, ANFIS, WTANN (WANN) and WT-ANFIS (WANFIS) models) for forecasting GWL at a monthly scale. The comparison of results demonstrated that the WANFIS model outperformed the other models since it could handle both uncertainty and seasonality involved in the process. Through a comparative study, [13] investigated the optimum structures of WANN and WANFIS models for GWL forecasting. Their research revealed that transfer functions of ANN and membership function types of ANFIS besides the mother wavelet type are the most important factors in the performance of WANN and WANFIS models, respectively. Comparison of optimal WANN and WANFIS demonstrated the better performance of WANFIS. On the other hand, a clustering technique may be employed as a spatial data preprocessing method to improve the performance of the hydrological modeling. In the context of ANN-based hydrological modeling, clustering is usually performed for classification of the data, stations or zones into homogeneous classes [14,15] and/or for optimization of the model structure by selecting dominant and relevant inputs [16]. Clustering techniques identify structure in an unlabeled data set by objectively arranging data into homogeneous groups, where the within- group-object dissimilarity is minimized, and the between group-object dissimilarity is maximized [17]. Therefore the training of ANFIS, using such homogeneous data can lead to better outcomes. Among the various clustering methods, the conventional clustering methods (e.g. K-Mean) require the number of clusters to be specified in advance, and their results are usually relevant to linear characteristics [18]. As a result recently Self-Organizing Map (SOM) an unsupervised ANN-based clustering technique, has found several hydrological applications [16-21]. The SOM operates as an effective tool to convert complex, nonlinear, statistical relationships between high-dimensional data items into simple, geometric relationships on a low-dimensional display so as to allow the number of clusters to be determined by inspection [25]. The SOM based classification is attractive due to its topology preserving properties for solving various problems that traditionally have been the domain of conventional statistical and operational research techniques. In the field of GWL modeling [22-24] applied the hybrid SOM-ANN model. Their results showed that the SOM-ANN model can perform more precisely than a sole ANN model. This study proposes a new method of GWL modeling based on coupling ANFIS and data pre-processing techniques. The proposed method is tested for predicting GWLs simultaneously with a large number of piezometers for one and multi-step-ahead time steps via two different scenarios over the Ardabil plain in northwestern Iran. Mutual Information (MI), as a non-linear correlation measure, is also used to select dominant hydrological parameters and their lags as inputs of ANFIS. In the next sections, the concepts of ANFIS, WT and SOM are briefly reviewed, respectively. Then, the efficiency criteria are introduced and the following sections describe the study area and data sources and the proposed methodology. In section 3, the results obtained using the proposed methodology are presented and discussed. Concluding Remarks will be the final section of the paper.

Materials and Methods The Adaptive Neuro-Fuzzy Inference System (ANFIS) Each fuzzy system contains three main parts, fuzzifier, fuzzy data base and defuzzifier. Fuzzy data base contains two main parts, fuzzy rule base, and inference engine. In fuzzy rule base, rules related to fuzzy propositions are described [25]. Thereafter, analysis operation is applied by fuzzy inference engine. There are several fuzzy inference engines which can be employed for this goal, which Sugeno and Mamdani are the two of well-known ones [26]. 3

Neuro-fuzzy simulation refers to the algorithm of applying different learning techniques produced in the neural network literature to fuzzy modeling or a Fuzzy Inference System (FIS)[27]. This is done by fuzzification of the input through membership functions, where a curved relationship maps the input value within the interval of [0 1]. The parameters associated with input as well as output membership functions are trained using a technique like back propagation and/or least squares. Therefore, unlike the Multi-Layer Perceptron (MLP), where weights are tuned, in ANFIS, fuzzy language rules or conditional (if–then) statements, are determined in order to train the model [28]. The ANFIS is a universal approximator and as such is capable of approximating any real continuous function on a compact set to any degree of accuracy [25]. The ANFIS is functionally equivalent to fuzzy inference systems [25]. Specifically the ANFIS system of interest here is functionally equivalent to the Sugeno first-order fuzzy model[25]. The general construction of the ANFIS is presented in Figure. 1. Figure. 1a shows the fuzzy reasoning mechanism for the Sugeno model to derive an output function f from a given input vector [x, y]. The corresponding equivalent ANFIS construction is shown in Figure. 1b.

Figure 1: (a) Sugeno’s fuzzy if–then rule and fuzzy reasoning mechanism. (b) Equivalent ANFIS structure.

According to this figure, it is assumed that the FIS has two inputs x and y and one output f. For the first order Sugeno fuzzy model, a typical rule set with two fuzzy if–then rules can be expressed as :

Rule (1) : If µ ( x ) is A1 and µ ( y ) is B1 : f1 = p1 x + q1 y + r1 Rule ( 2 ) : If µ ( x ) is A2 and µ ( y ) is B2 : f 2 = p2 x + q2 y + r2 where A1, A2 and B1, B2 are the membership functions for inputs x and y, respectively; p1, q1, r1 and p2, q2, r2 are the parameters of the output function. The functioning of the ANFIS is as follows: Layer 1: Each node in this layer produces membership grades of an input variable. The output of the ith node in layer k is denoted as Qki. Assuming a generalized bell function as the membership function, the output Q1i can be computed as (Jang and Sun, 1995) [29]: = Qi1 µ= Ai ( x )

1

1 + ( ( x − ci ) / ai )

2 bi



where {ai, bi, ci} are adaptable variables known as premise parameters. 4

(1)

Layer 2: Every node in this layer multiplies the incoming signals (Jang and Sun, 1995) [29]: 2 Q= w= µAi ( x ) .µBi ( x )= i 1, 2 i i

(2)

Layer 3: The ith node of this layer calculates the normalized firing strengths as (Jang and Sun, 1995) [29]:

Qi3 = w ?i = wi / ( w1 + w2 )i =1, 2

(3)

Layer 4: Node i in this layer calculate the contribution of the ith rule towards the model output, with the following node function 4 Q= wi ( pi x + qi y + r= wi fi i i)



(4)

is the output of layer 3 and {pi, qi, ri} is the parameter set.

where

Layer 5: The single node in this layer calculates the overall output of the ANFIS as [25,30]: = Qi5

= wf ∑ i

i

i

∑w f ∑w i

i

i

i

i



(5)

The learning algorithm for ANFIS is a hybrid algorithm, which is a combination of the gradient descent and least-squares method [31-33]. The parameters for optimization are the premise Parameters {ai, bi, ci} and the consequent parameters {pi, qi, ri}. In the forward pass of the hybrid learning approach, node outputs go forward until layer (4) and the consequent parameters are identified by the least-squares technique. In the backward pass, the error signals propagate backward and the premise parameters are updated by gradient descent. More information for ANFIS and hybrid algorithm can be found in related literature [25,34].

Wavelet Transform (WT) The WT has enlarged in occupation and popularity in recent years since its inception in the early 1980s, but the widespread usage of the Fourier transform has yet to occur. Fourier analysis has a serious disadvantage. In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of signal, it is impossible to tell when a particular event took place, but wavelet analysis allows the use of long time intervals where more precise low-frequency information and shorter regions are necessary where high-frequency information is wanted. In the field of earth sciences, Grossman and Morlet (1984) [35], who did particular work on geophysical seismic signals, introduced the WT application. The most recent hydrological contributions have been cited by Labat (2005) [36] and Sang (2013)[37]. This paper will not delve into the theory behind wavelets and only the main concepts of the transform are briefly presented; recommended literature for the wavelet novice includes Mallat (1998)[38] and Labat et al. (2000)[39]. The time-scale WT of a continuous time signal, x(t), is defined as [38]: + ∞

* t −b (6) ∫ g ( a ) x(t ). dt a −∞ Where * corresponds to the complex conjugate and g(t) is called wavelet function or mother wavelet. The parameter a acts as a dilation factor, while b corresponds to a temporal translation of the function g(t), which allows the study of the signal around b. The main property of WT is to provide a time-scale localization of process, which derives from the compact support of its basic function. The WT searches for correlations between the signal and wavelet function.

T ( a, b) =

1

In real hydrological problems, the time series are usually in the discrete format rather continues and therefore, the discrete WT in the following form is usually used (Mallat, 1998) [38]: 5

g m,n (t) =

1 a 0m

g* (

t − nb 0a 0m ) a 0m



(7)

where m and n are integers that control the wavelet dilation and translation respectively; a0 is a specified fined dilation step greater than 1; and b0 is the location parameter and must be greater than zero. The most common and simplest choice for parameters are a0 = 2 and b0 = 1. This power-of-two logarithmic scaling of the dilation and translation is known as the dyadic grid arrangement. The dyadic wavelet can be written in more compact notation as (Mallat, 1998)[38]: = g m,n (t) 2− m / 2 g(2− m t − n)

(8)

For a discrete time series, xi, the dyadic WT becomes (Mallat, 1998)[38]: N −1

Tm ,n = 2 −m / 2 ∑ g (2 −m i − n) xi i =0

(9)

where Tm,n is wavelet coefficient for the discrete wavelet of scale a=2m and location b=2mn. Eq.5 considers a finite time series, xi, i = 0,1,2, ... , N-1; and N is an integer power of 2: N =2M. This gives the ranges of m and n as, respectively, 0 < n 1;

2.

choose termination threshold ε > 0;

3.

initialize prototypes pk;

4.

repeat

5.

update memberships using (4);

6.

update prototypes using (3);

7.

until change in memberships drops below ε;



For the sake of simplicity, let us assume that it will never happen that a prototype matches a data object perfectly, or in other words the Euclidean distance between any pair of data object and prototype never vanishes. Then, we have no need for considering special cases as in (4), Ij is always empty. In practice, we can add a small constant η>0 near the floating point precision to all distance values to guarantee this. We reformulate the membership degrees between data objects and prototypes as a function (11) Where q=2/ (m-1) and P := (R DIM ) . If p denotes the tuple of prototypes (p1,…, pc) then we have uk,j =uk (xj ; p). c

Fuzzy Model Descriptions In order to construction of MFIS and LFIS for estimating Sw, fuzzy rule base was generated through FCM derived input and output cluster centres. Each cluster centre was used to generate a Gaussian membership function in each rule. That is, each rule is represented by a Gaussian MF which is constructed from centre and standard deviation of corresponding cluster. So, number of membership functions and if-then rules for each input and output dataset is equal to number of the clusters. As mentioned, number of the FCM derived clusters for water saturation was equal to 31. Considering four inputs and one output, 31 by 5 MFs were generated participating in 31 fuzzy rules (Table 2.5). To connect antecedents of each rule min operator was used. As mentioned, fuzzy rule base structure for MFIS and LFIS is similar. Their main difference is in implication method. In MFIS, min operator was used for implication, whereas in LFIS product operator was used for this purpose. For the both techniques, centroid defuzzification method was applied. In SFIS, input MFs are of Gaussian type. They were constructed using the cluster centres obtained from subtractive clustering (43 clusters for Sw). But, output membership functions are linear equations constructed from inputs. For example, output MF1 of Sw model, which is the consequent of rule no. 1, is constructed from four seismic attributes as below: Output MF1= γ 1*Time + γ 2*Average frequency + γ 3*Filter15/20-25/30 + γ 4*Dominant frequency + γ5 (12) In this equation, parameters γ1, γ2, γ3 and γ4 are coefficients corresponding to input seismic attributes. Parameter γ5 is the constant of each equation. These parameters are obtained by linear least squares estimation. With these explanations in order to estimate Sw there will be 43 by 5 output MF parameters (Table 2.6).

Construction of CFIS In this part of research, a CFIS was constructed for the overall prediction of petrophysical data by integrating the results of predicted data from SFIS, MFIS \and LFIS each of them has a weight factor showing its contribution in overall prediction. At the first step, outputs of the three fuzzy inference systems were averaged for predicting target data, namely each of them has the weight value of 0.333. This output will be used as one of the experts of the CFIS. In the next step, a genetic algorithm-pattern search tool was used to obtain optimal combination of the weights for constructing CFIS. The fitness function for GA-PS was defined as below: 23

Table 2.5: Gaussian membership function parameters derived by FCM for predicting Sw (Kadkhodaie et al., 2009b)[18]. l

MSECMIS = ∑1 / k ( β 1O 1i + β 2 O 2i + β 3O 3i + β 4 O 4i − L i ) 2 i =1





(10)

This function shows the MSE of CFIS for training step predictions where β 1 , β 2 , β 3 and β 4 are the weight coefficients corresponding to the outputs of Sugeno, Mamdani, Larsen and simple averaging method, respectively. Oi and Li are output and target values, respectively. k is the number of test data (76 samples). Parameters of GA-PS are described as following: Population is of a double vector type. Initial population size is 25 which specifies how many individuals are in each generations. Initial range is [0, 1]. This parameter specifies the range of the vectors in the initial population. The selection function was chosen as stochastic uniform which chooses parents for the next generation based on their scaled values from the fitness scaling function. The crossover function is scattered that creates a random binary vector and selects the genes where the vector is [1] from the first parent, and the genes where the vector is [0] from the second parent, and combines the genes to form a child. The value of crossover fraction is 0.78. This parameter specifies the fraction of the population that could be seen in the crossover children. Mutation function is Gaussian that adds a random number, or mutation, from a Gaussian distribution, to each entry of the parent vector. Parameters controlling the mutation are specified as the scale value of 1 and shrink value of 1. The scale value controls the standard deviation of the mutation at the first generation. This parameter is multiplied by the range of the initial population. Shrink value controls the rate at which the average amount of mutation decreases. The standard deviation decreases linearly so that its final value equals 1. Hybrid function was chosen as pattern search. This is another minimization function that runs after genetic algorithm terminates. Stopping generation of GA was chosen as 100. After 100 generations, change in the fitness function values over Stall generations was insignificant and the mean fitness value for water saturation was fixed in 0.00915. Finally, CFIS was constructed using the GA-PS 24

derived coefficients for the results of SFIS, MFIS, LFIS and simple averaging method. Final estimation of water saturation was done through Eq. (13).

Sw CFIS = 0.303*Sw SFIS +0.127 *Sw MFIS +0.098 *Sw LFIS +0.472 *Sw Average



(13)

Table 2.6: Gaussian and linear membership function parameters derived by subtractive clustering and gradient descent methods for predicting Sw using SFIS [18].

Performance of the CFIS model was compared to that of a Probabilistic Neural Network (PNN). The results (figure 2.24) show that CFIS method performs better than neural network, best individual fuzzy model and simple averaging method. 25

Using the methodology of CFIS, 3D cube of seismic data and their attributes were converted to water saturation volume. A map water saturation distribution estimated from CFIS model is shown in figure 2.25. As shown, water saturation in the central and North West sector of the reservoir is low (< 50%), which corresponds to the hydrocarbon-bearing area.

Figure 2.24: Graphical comparison between measured and predicted water saturation for test samples using SFIS (a), MFIS (b), LFIS (c) and CFIS (d) [18].

Figure 2.25: Map showing distribution of CFIS estimated water saturation for Top Ghar reservoir [18].

Rock Types Classification and Estimation Lithofacies typing is useful in well correlation and can be important for building a 3D model of the field by geostatistical or stochastic techniques. These models can be used for volumetrics, well placing and reservoir engineering. Using fuzzy logic for lithofacies prediction makes no assumptions and retains the possibility that a particular facies type can give any log reading although some are more likely than others. This error or fuzziness has been measured and used to improve the facies prediction. 26

The Iran offshore gas field, the Iranian part of the world’s largest non-associated gas accumulation is located in the Persian Gulf between Qatar and Iran, some 100 km from shore. The Upper Permian to Lower Triassic Dalan and Kangan Formations are two huge, condensate rich and gas bearing reservoirs over the field. In a case study from the Iranian offshore gas field by Kadkhodaie et al. (2006)[9], reservoir rock types were determined based on core porosity and permeability data and a fuzzy centre means clustering (FCM). Then, the FCM clustering derived rock types were estimated from well log data using fuzzy mathematics. Gaussian function was used to estimate relative probability or “fuzzy possibility” that a data value belongs to each rock type. Each log data value may belong to any of FCM clustering derived rock types to a degree that can be calculated from Gaussian membership function using Eq. (14).

P(x) =

e

1 − (x − c) 2 / ( σ ) 2 2



(σ ) 2 π

(14)

Each rock type has its own mean and standard deviation, namely, for n number of rock types; there are n pairs of c and σ rock type. For example, the fuzzy possibility that a neutron log data belongs to rock type 1 is obtained by substituting σ rock type1 and σ rock type1 in Eq. (15): P(N) =

e

− ( NPHI − Crock

(σ rock

type1 )

type1

2

/ ( σ rock

type1

)2



) 2π

(15)

The ratio of the fuzzy possibility for each rock type with the fuzzy possibility of the mean or most likely observation is obtained by de-normalizing Eq. (2.16). The fuzzy possibility for mean of neutron in rock type 1 is obtained by substituting NPHI by c rock type1 in Eq. (16):

P(c rock

type1

)=

e

− (Crock

type1 − C rock

(σ rock

type1 )

type1

2

/ ( σ rock

type1

)2



) 2π

(16)

The relative fuzzy possibility R(Nrock type 1) of a neutron porosity (NPHI) belonging to rock

type 1 compared to the fuzzy possibility of measuring the mean value c rock divided by Eq. (16):

R(N RT1 ) = e

− ( NPHI − Crock

type1 )

2

/ ( σ rock

type1

)2







type1

is Eq. (15)

(17)

Each value derived from Eq. (2.17) is now indicated to possible rock types. To compare the relative fuzzy possibilities of this equation among rock types, Eq. (17) is multiplied by a coefficient named relative occurrence of each rock type in the reservoir interval. For rock type 1, it is notified by n rock type1 : F(NPHI rock ( n rock

type1

type1

)e

)=

− ( NPHI − Crock

type1 )

2

/ ( σ rock

type1

)2



(18)

The obtained fuzzy possibility from Eq. (18) is based on neutron log data only. This process should be repeated for other logs such as sonic (DT), density (RHOB), ... at this point. This will give F(DTrock type1), F(RHOBrock type1), … for rock type 1. These fuzzy possibilities are combined harmonically to give a final fuzzy possibility: 27

1 1 = C rock type1 F(NPHI rock

type1 )

+

1 F(DTrock

type1

)



(19)

This process is repeated for other rock types and all derived fuzzy possibilities are combined harmonically. Then, the rock type with the highest combined fuzzy possibility is taken as most possible rock type at that point. A comparison between FCM clustering derived and fuzzy predicted rock types versus depth for the test well that was not used to model construction (the test well) is shown in figure 2.26.

Figure 2.26: A comparison between clustering derived and fuzzy predicted rock types versus depth for the test well in Kangan Formation, Iranian offshore gas field [9].

28

In a similar study from the Southern North Sea, Cuddy (1998)[1] studied the lithofacies estimation using the fuzzy logic model in the Permian Rotliegendes Sandstone (Viking). The Viking field was developed in 1972 and to date has produced 2.8 Tcf of gas. Consideration has recently been given to tying back several smaller satellite pools. As part of the feasibility study, 13 exploration and production wells, drilled between 1969 and 1994, have been re-evaluated using fuzzy logic. The reservoir was deposited in a desert by aeolian, fluvial, and lacustrine processes. Three major lithofacies associations have been recognised from core studies: Aeolian Dune: Aeolian sandstones have the best permeability by virtue of their better sorting and lack of detrital clays. Clean aeolian dune sandstones give the highest porosities in the reservoir, with an average around 16 pu. Dune base sandstones (wind ripple) give a lower average porosity of 12-14 pu, as they are less well sorted.Sabkha: Sandy sabkha has good porosity but the presence of detrital clay enhances compaction effects and thus reduces primary porosity. Muddy sabkha porosities and permeability are very low with no reservoir potential. Fluvial: The fluvial sandstones often have poorer permeabilities (