A novel application of a neuro-fuzzy computational technique in event-based rainfall–runoff modeling

https://doi.org/10.1016/j.eswa.2010.04.015Get rights and content

Abstract

Intelligent computing tools based on fuzzy logic and Artificial Neural Networks (ANN) have been successfully applied in various problems with superior performances. A new approach of combining these two powerful AI tools, known as neuro-fuzzy systems, has increasingly attracted scientists in different fields. Although many studies have been carried out using this approach in pattern recognition and signal processing, few studies have been undertaken to evaluate their performances in hydrologic modeling, specifically rainfall–runoff (R–R) modeling. This study presents an application of an Adaptive Network-based Fuzzy Inference System (ANFIS), as a neuro-fuzzy-computational technique, in event-based R–R modeling in order to evaluate the capabilities of this method for a sub-catchment of Kranji basin in Singapore. Approximately two years of rainfall and runoff data which from 66 separate rainfall events were analyzed in this study. Two different approaches in the selection criteria for calibration events were adopted and the performance of an ANFIS R–R model was compared against an established physically-based model called Storm Water Management Model (SWMM) in R–R modeling. The results of this study show that the selected neuro-fuzzy-computational technique (ANFIS) is comparable to SWMM in event-based R–R modeling. In addition, ANFIS is found to be better at peak flow estimation compared to SWMM. This study demonstrates the promising potential of neuro-fuzzy-computationally inspired hybrid tools in R–R modeling and analysis.

Introduction

One of the most important problems in hydrology is the modeling of the rainfall–runoff (R–R) process. This is an important subject because of its vast applications in different hydrologic problems like flood forecasting, design of spillways and waterways, water quality modeling, and urban planning and management. Rainfall–runoff is a process which is highly affected by a variety of non-linear factors like rainfall characteristics, watershed morphology, soil moisture, etc. However, any effort to model the R–R relationship would be confronted with difficulties including being highly non-linear, time-varying, spatially distributed, and stochastic. In addition, the deficiencies in data like missing data, noisy data, and in some cases having insufficient data present a major problem in R–R modeling.

To-date many methods and approaches have been introduced to model the R–R relationship. These methods can be categorized into two main groups: physically-based models and system theoretic models. Physically-based models are designed to approximate the general internal sub-processes and physical mechanisms which govern the hydrologic cycle. They usually incorporate simplified forms of physical laws and are generally non-linear, time-varying, and deterministic, with parameters that are representative of watershed characteristics. Some examples for this group are Storm Water Management Model (SWMM), flood hydrograph package of Hydrologic Engineering Center of US Army Corps of Engineers (HEC-1), and Sacramento Soil Moisture Accounting Model (SAC-SMA). Although physically-based models help us in understanding the physics of hydrological processes, they require sophisticated mathematical tools, and usually require significant user expertise.

On the other hand, system theoretic or black-box models apply a different approach to identify a direct mapping between rainfall and runoff, without the need for a detailed consideration of the physical processes. Linear time series models like Autoregressive Moving Average with Exogenous Inputs (ARMAX) and other linear and non-linear regression models, Artificial Neural Networks (ANN), and neuro-fuzzy systems are examples of this group. Although these kinds of models are fast and their results are often comparable with physically-based models, they do not give us any information about the hydrologic process in the system.

Artificial neural networks, as generalizations of mathematical models of human cognition or neural biology, are massively parallel distributed processors made up of simple processing units known as neurons which are capable of storing experiential knowledge and making it available for use (Haykin, 1998). Mathematically, an ANN is often viewed as a universal approximator. It has the ability to identify the relationship from given patterns and solve large-scale complex problems such as pattern recognition, non-linear modeling, classification, association, and control. The early applications of ANNs in R–R modeling were by Halff et al., 1993, Hjelmfelt and Wang, 1993, and since then the ANN technique has been used in many different applications in hydrology including stream flow forecasting (Chua & Holz, 2005), groundwater modeling (Nayak, Satyaji Rao, & Sudheer, 2006), water quality modeling (May & Sivakumar, 2008), and R–R forecasting (Kang, Kang, Park, Lee, & Yoo, 2006). Fuzzy set theory has been used in many fields of application, such as pattern recognition, data analysis, system control, etc. since its development by Zadeh (1965). Recently the research focus on neural networks has shifted from a black box approach to a semantic-based fuzzy neural architecture (Ang & Quek, 2005). Neuro-Fuzzy System (NFS) can be defined as an integration of the fuzzy systems and neural networks (Lin & Lee, 1996). In other words, it is a fuzzy system that uses a learning algorithm derived from or inspired by neural network theory to determine its parameters (fuzzy sets and fuzzy rules) by processing data samples. It has the significant advantage of reduced training time in comparison with ANNs due to its smaller dimension and being initialized with parameters relating to the problem domain (Maguire, Roche, McGinnity, & McDaid, 1998). Recently, neuro-fuzzy systems have become popular, taking advantage of the low-level learning ability of neural networks and the high-level reasoning ability of fuzzy systems such as traffic analysis and forecasting (Cho et al., 2009, Quek et al., 2006), computational finance (Tan, Quek, & Ng, 2007), diabetic insulin regulation (Ting & Quek, 2009) and in Antiforgery (Quek & Zhou, 2002) using a set of thinning algorithm with established performance criteria (Zhou, Quek, & Ng, 1995).

Fuzzy models that assume local model presentations with local function dynamics at the consequent or rule-layer of the models are known as Takagi–Sugeno–Kang (TSK) models. In this model, the output is calculated by performing fuzzy interpolations of simpler functional models in the neighboring fuzzy partitions. The ability of accurate modeling of a system, globally or locally, is the significant advantage of TSK-models (Quah & Quek, 2006). Specifically, the accurate global learning ability of TSK-models motivates various practical applications of such models in non-linear system estimation (Yen & Langari, 1999). One of the main criteria to categorize existing TSK-models is locality of learning. This criterion depends on the model’s learning objective function, which is a minimization problem of the global or the local learning errors (Quah & Quek, 2006). Adaptive Network-based Fuzzy Inference Systems (ANFIS) is an example of such TSK-models in which the global parameter tuning has been considered by means of minimization of the global error of the model (Jang, 1993). As a matter of fact, ANFIS has been found to be an appropriate tool in non-linear mapping problems between input and output data such as R–R modeling.

In general, the concept of fuzzy theory and application of fuzzy-based systems, especially neuro-fuzzy systems has found expression in many papers relating to hydrology since the late 1990s. Previous studies with neuro-fuzzy systems have shown that these techniques have the potential to be an effective tool in R–R modeling. For example, Yu and Yang (2000) applied Fuzzy Multi-Objective Function (FMOF) for continuous R–R model calibration to improve conventional objective functions like root mean square error (RMSE) and the mean absolute percent error (MPE). Hundecha, Bardossy, and Theisen (2001) developed a fuzzy logic based R–R model. Xiong, Shamseldin, and O’Conner (2001) applied the first-order Takagi–Sugeno neuro-fuzzy system as a tool for non-linear combination of the forecasts of R–R models and compared it with the simple average method (SAM), the weighted average method (WAM), and neural network method (NNM). Their study showed that the Takagi–Sugeno method was as efficient as the other methods and due to its simplicity and efficiency, was recommended for use as a tool for flood forecasting. Nayak, Sudheer, Rangan, and Ramasastri (2004) applied ANFIS for modeling the discharge in the Baitarani River in India and compared their model with ANN and Auto Regressive Moving Average (ARMA) models. ANFIS was reported to out-perform ARMA but was similar in performance with the ANN model; although ANFIS was much better in peak estimation compared to ANN. Nayak, Sudheer, Rangan, and Ramasastri (2005) applied a neuro-fuzzy model for short-term flood forecasting. In this study, the ability of the ANFIS in flood forecasting was compared with ANN and FIS models and it was confirmed that ANFIS captured the inherent non-linearity in the R–R process better than other models. Vernieuwe et al. (2005) made a comparison of data driven Takagi–Sugeno models of R–R dynamics. The authors developed three different methods for constructing rule-based models of the Takagi–Sugeno in R–R modeling and then compared them to each other. Aqil, Kita, Yano, and Nishiyama (2007) did a comparative study of ANN and neuro-fuzzy systems in continuous modeling of the daily and hourly behavior of runoff. Their study focused on 3-year continuous rainfall and runoff data for the Cilalawi River, one of the tributaries of the Citarum River, in Java province, Indonesia. They developed two different versions of ANN trained with Levenberg-Marquardt and Bayesian Regularization algorithms and compared them with a model based on neuro-fuzzy systems. These results indicate that the neuro-fuzzy model out-perform the other two models. More recently, Nasr and Bruen (2008) developed neuro-fuzzy models to account for the temporal and spatial variations in a lumped R–R model. Basically, each sub-model may perform differently under different temporal and spatial conditions. The authors proposed and compared three combination methods which can use any lumped catchment model. It was found in this study that the neuro-fuzzy combined-sub-models of Simple Linear Model (SLM) and the Soil Moisture and Accounting Routing (SMAR) modeled the temporal and spatial variation in catchment response better than the lumped version of each model.

It can be inferred from the above-mentioned studies that Neuro-Fuzzy Systems (NFS) have good potential in R–R modeling. In addition, comparisons made between neuro-fuzzy systems and other models show that neuro-fuzzy systems are capable of providing good predictions. However, the fact remains that there is still a need for more studies in order to evaluate thoroughly the capabilities of these systems in R–R modeling. In this regard, more studies comparing NFS with traditional physically-based models is required before NFS models can be used with confidence. The objective of the present study therefore, is to: (i) evaluate the capabilities of an Adaptive Network-based Fuzzy Inference System (ANFIS) in event-based R–R modeling for a sub-catchment in the Kranji reservoir catchment, Singapore, and (ii) compare its performance with a physically-based model. The physically-based model chosen for this study is the Storm Water Management Model (SWMM) which incorporates the Kinematic Wave (KW) routing procedure.

Section snippets

Fuzzy systems

Fuzzy logic is based on the concept of fuzzy sets. A fuzzy set is defined as a set with no crisp or clear boundary. Unlike two-valued Boolean logic, fuzzy logic is multi-valued and it deals with degrees of membership and degrees of truth. Fuzzy logic uses any logical value from the set of real numbers between 0 (completely false) and 1 (completely true) which is known as its membership value and the function that represents such values is called a membership function.

Membership functions can be

Study site and data

The total area of the catchment under study is about 5.6 km2, and the land use consists of 32.6% (about 1.8 km2) high-density residential area and the parts of the catchment that are undeveloped are mainly covered by vegetation. The in situ soils of the site are mainly clayey soils. Runoff in the study site is served mainly by a concrete-lined drainage system. CP1 (Fig. 5) is the final discharge point of the catchment. The runoff discharging from CP1 is conveyed through a concrete-lined channel

Model calibration

Tan et al. (2008) calibrated a SWMM model using 10 representative events (Events 7, 8, 10, 11, 13, 22, 30, 32, 36, and 53) selected based on a correlation between the direct runoff volume and total rainfall volume. After calibrating the SWMM model with these 10 events, a set of nine calibration parameters was obtained which was found to give reasonable estimates to the runoff hydrograph for all 66 events. The average values of these parameters are summarized in Table 1 and the SWMM model

Results and discussion

The SWMM1, ANFIS1, SWMM2, and ANFIS2 models were employed to model all the 66 storm events listed in Appendix A and individual values of Ef and EP were calculated. The average values of Ef and EP are presented in Table 4 as a measure of the gross performance of the SWMM1, ANFIS1, SWMM2, and ANFIS2 models. It can be observed from Table 4 that the SWMM models are not sensitive to the change in calibration events since the overall performance of SWMM1 and SWMM2 are similar (Note that the number of

Conclusions and recommendations

The following can be concluded from this study:

  • (i)

    SWMM and ANFIS have different sensitivity to any changes in calibration events. Selecting calibration events with regard to the correlation between rainfall and runoff volume, gave acceptable results in SWMM but not for ANFIS. On the other hand, selecting training events based on the shape of hydrograph was found to be a good strategy in ANFIS. Only two rainfall inputs were found to be needed by the ANFIS model. In addition, it was concluded that

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