Two hybrid Artificial Intelligence approaches for modeling rainfall–runoff process
Highlights
► We developed two Artificial Intelligence-based rainfall–runoff models. ► The models include ANN, Wavelet and SARIMAX concepts. ► The models examined for two distinct watersheds with different climatologic regimes. ► The models’ performances are different at different space–time scales. ► The inputs should be selected according to the seasonal and periodic patterns.
Introduction
Accurate modeling of hydrological processes such as rainfall–runoff can provide important information for the urban and environmental planning, land use, flood and water resources management of a watershed. It also plays an important role in mitigating the impact of drought on water resources systems. Therefore, numerous hydrological models have been developed in order to simulate this complex process and a comprehensive classification of these models was presented by Nourani et al. (2007). Due to the large number of obscure parameters involved in the rainfall and runoff physical relationship in a watershed, black box lumped modeling may have some advantages over fully distributed modeling (Nourani and Mano, 2007).
Conventional black box time series models such as Auto Regressive Integrated Moving Average (ARIMA) or Seasonal ARIMA with exogenous input (SARIMAX) are widely used for hydrological time series forecasting (Salas et al., 1980, Tankersley et al., 1993). However, they are basically linear models assuming that data are stationary, and have limited ability in capturing that which is non-stationary and non-linearity in hydrologic data.
Recently, Artificial Intelligence (AI) techniques have shown great ability in modeling and forecasting non-linear hydrological time series. AI techniques offer an effective approach for handling large amounts of dynamic, non-linear and noisy data, especially when the underlying physical relationships are not fully understood. This makes them well suited to time series modeling problems of a data-driven nature. In general, the application of AI technique does not require a prior knowledge of the process. Numerous papers have already been presented on the successful application of ANNs for modeling the rainfall–runoff process as a non-linear complex phenomenon (e.g., Hsu et al., 1995, Dawson and Wilby, 1998, Tokar and Johnson, 1999, Sajikumara and Thandaveswara, 1999, Sudheer et al., 2000, Lallahem and Maina, 2003, Senthil Kumar et al., 2004, Jain et al., 2004, Antar et al., 2006).
The model and data used for simulation of the rainfall–runoff process usually contain uncertainties. For example, an averaged value of the pointy measured rainfalls by the rain gauges over a watershed is usually assigned to the whole of the watershed. Using this constant real number as the watershed rainfall in the ANN input layer can be a source of uncertainty. In such uncertain situations, fuzzy numbers and the fuzzy system may be employed in the estimation of uncertainties in real world problems. The hybrid ANN and fuzzy system is a research focus, which can make use of the advantages of both ANN and fuzzy system namely ANFIS (Adaptive Neuro-Fuzzy Inference System) (Rajaee et al., 2009). ANFIS is capable of combining the benefits of both these fields in a single framework. There are a few studies on the application of ANFIS in rainfall–runoff modeling (e.g., Gautam and Holz, 2001, Nayak et al., 2004, Tayfur and Singh, 2006, Jothiprakash et al., 2009).
In spite of the suitable flexibility of ANN and ANFIS in modeling a hydrologic process such as rainfall–runoff, sometimes there is a shortage when signal fluctuations are highly non-stationary and the physical hydrologic process operates under a large range of scales varying from 1 day to several decades. In such a situation, ANN and ANFIS models may not be able to cope with non-stationary data if pre-processing of the input and/or output data is not performed (Cannas et al., 2006).
To overcome the above-mentioned shortage, the combination of ANN and ANFIS with other approaches as hybrid models may be an appropriate choice. The basic idea of model combination in forecasting is to use each model’s unique features to capture different patterns in the data. Both theoretical and empirical findings suggest that combining different methods can be an efficient way to improve forecasting (Zhang, 2003). One of such hybrid models is the ARIMA-ANN model which was developed in the last years and is still being used in engineering and financial time series forecasting. This model was first proposed and evaluated by Zhang (2003) in order to forecast a univariate time series without considering seasonal effect. This technique has enjoyed a few applications in hydrological time series modeling which usually show highly non-stationary seasonal behavior (e.g., Mishra et al., 2007).
Furthermore, the wavelet-ANN is another reliable hybrid model used in time series forecasting problems. Recently, wavelet transform analysis has become a popular analysis tool due to its ability to elucidate simultaneously both spectral and temporal information within the signal. This overcomes the basic shortcoming of Fourier analysis, which is that the Fourier spectrum contains only globally averaged information. Therefore, a data pre-processing can be performed by time series decomposition into its subcomponents using wavelet transform analysis. Wavelet transforms provide useful decompositions of the main time series, so that wavelet-transformed data improve the ability of a forecasting model by capturing useful information on various resolution levels. The wavelet decomposition of a non-stationary time series into different scales provides an interpretation of the series structure and extracts significant information about its history, using few coefficients. For these reasons, this technique is largely applied to time series analysis of non-stationary signals (Nason and Von Sachs, 1999, Adamowski, 2008a, Adamowski, 2008b). Hence, a hybrid wavelet-ANN model which uses multi-scale signals as input data may present more probable forecasting than a single pattern input.
The wavelet-ANN conjunction model was first presented by Aussem et al. (1998) for financial time series forecasting. Zhang and Dong (2001) proposed a short-term load forecast model based on ANN and the multi-resolution wavelet decomposition. In hydrology, Wang and Ding (2003) applied a wavelet-network model to forecast shallow groundwater level and daily discharge. Kim and Valdes (2003) proposed a conjunction model based on dyadic wavelet transform and ANNs to forecast droughts for the Conches river basin in Mexico; they used ANN to forecast sub-signals from wavelet decomposition and also to reconstruct the main signal from the forecasted sub-signals. In both cited studies, “a trous” algorithm accompanied by three-layered feed forward neural networks was used in order to predict the hydrological time series. Nourani et al. (2009a) predicted the monthly precipitation time series of a watershed by a combined wavelet-ANN model. Partal and Cigizoglu (2008) and Kisi (2008) used the neuro-wavelet technique for forecasting daily suspended sediment and monthly river flow, respectively. Partal and Cigizoglu (2008) decomposed a daily sediment time series into many components (sub-signals) using wavelet transform and then composed a new time series by adding the dominant sub-signals and used this time series in the ANN. They used the linear correlation coefficient between the main time series and sub-signals in order to determine the dominant components. However in a non-linear process, two time series may have a weak linear correlation but strong non-linear relation. Cannas et al. (2006) investigated the effects of data pre-processing on ANN model performance using continuous and discrete wavelet transforms; the results showed that networks trained with pre-processed data performed better than networks trained on undecomposed, noisy raw signals. Anctil and Tape (2004) decomposed a rainfall time series by wavelet into three sub-series depicting the rainfall–runoff processes: short, intermediate and long wavelet periods, then three multi-layer networks were trained for the wavelet sub-series in order to estimate the runoff values. Their results showed that short wavelet period fluctuations are thus the key to any further improvement in ANN rainfall–runoff forecasting models. Since this model uses more than one ANN network, simultaneous optimization of the networks could be a difficult and time consuming procedure.
As a preliminary study to the current research, the authors used the wavelet-ANN (WANN) combined approach for modeling Lighvanchai watershed rainfall–runoff process on a daily time scale (Nourani et al., 2009b).
Considering the ability of ANFIS in modeling hydrological processes which usually involve some degree of uncertainty, the hybrid wavelet-ANFIS (WANFIS) approach is proposed for multivariate hydrological modeling in the current paper. The univariate daily precipitation forecasting model proposed by Partal and Kisi (2007) is the sole application of a wavelet and nero-fuzzy conjunction model in the hydrological literature but it differs from the proposed multivariate WANFIS model in employing wavelet analysis.
In this paper, two new multivariate black box models based on AI techniques are proposed for the rainfall–runoff modeling of two watersheds located in Azerbaijan, Iran which have different climatologic characteristics. In the first model, considering the existence of seasonality in the time series, a seasonal ARIMA model with exogenous inputs (i.e., rainfall and runoff values), SARIMAX, is combined with the ANN approach to construct the hybrid SARIMAX-ANN model. A multivariate wavelet-ANFIS model is introduced as the second model. In this model the daily and monthly rainfall and runoff time series of the watersheds are decomposed into sub-signals in various resolution levels and periodicity scales; then these sub-signals are entered into the ANFIS model to reconstruct the forecasted runoff time series. One of the main aims of this study is to investigate the effect of simulation time scale, and also the watershed’s climatologic conditions, on the model’s performance. Finally, in order to evaluate the models’ abilities, the results are compared with those of individual ANN, ANFIS and SARIMAX models as well as the previously presented hybrid WANN model.
The remainder of this paper is organized as follows. In the next three sections, the concepts of wavelet transform, ANNs and ANFIS are briefly reviewed, respectively. Section 5 presents the formulations and structures of the proposed hybrid models. In Sections 6 Efficiency criteria, 7 Study area, the efficiency criteria and study area are introduced and in Section 8 the models performances are evaluated, discussed and compared with some other conventional methods. Concluding remarks are in the final section of the paper.
Section snippets
Wavelet transform
The wavelet transform has increased in usage and popularity in recent years since its inception in the early 1980s, yet still does not enjoy the wide spread usage of the Fourier transform. Fourier analysis has a serious drawback. In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place but wavelet analysis allows the use of long time intervals where we want more precise
Artificial Neural Networks (ANNs)
ANN is widely applied in hydrology and water resource studies as a forecasting tool. In ANN, feed forward back-propagation (BP) network models are common to engineers. It has proved that BP network model with three-layer is satisfied for the forecasting and simulating in any engineering problem (Hornik, 1988, ASCE, 2000, Nourani et al., 2008).
As shown in Fig. 2, three-layered feed forward neural networks (FFNNs), which have been usually used in forecasting hydrologic time series, provide a
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 (Jang et al., 1997). 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 (Lin et al., 2005). Neuro-fuzzy
Hybrid SARIMAX-ANN (SANN) Model
For construction of the SANN model, the proposed methodology by Zhang (2003) in developing the ARIMA-ANN model is followed in this paper but with considering two other components. Firstly, because of the seasonality characteristics of the hydrological time series and secondly in order to employ two variables (i.e., rainfall and runoff) in the modeling, SARIMAX-ANN is proposed and used for rainfall–runoff modeling instead of ARIMA-ANN model which considers just autoregressive property of the
Efficiency criteria
The model that yields the best results in terms of determination coefficient and root mean squared error on the training and verifying steps can be determined through trial and error process. For this purpose the following measures of evaluation have been used to compare the performance of the different models (Nourani, 2009):where R2, RMSE, N, , and are determination coefficient, Root Mean Squared Error,
Study area
The data used in this paper are from Lighvanchi and Aghchai watersheds, located in northwest Iran at Azerbaijan province. These watersheds are main sub-branches of the Ajichai and Aras Rivers which discharge to Urmieh and Caspian Lake, respectively (Fig. 7).
The time series data for 12 years (from 1995 to 2007) were used in the modeling process (the first 8 years for training and the rest 4 years for verification). The statistic characteristics of rainfall and runoff for both watersheds in daily
The ANN model
At first, a Multi-Layer Perceptron (MLP) feed forward ANN model without any data pre-processing was used to model the watersheds’ rainfall–runoff process. This kind of ANN model accompanied by back propagation training algorithm is widely used in hydrologic modeling (ASCE, 2000). In this study four structures were examined for the ANN model and the results have been shown in Table 2. Each MLP was trained with 3–20 hidden neurons in a single hidden layer using the Levenberg–Marquardt training
Concluding remarks
The data pre-processing technique warrants further investigation. In fact it should be noted that in general, and in the Lighvanchai and Aghchai basins in particular, rainfall and runoff time series are characterized by high non-linearity, non-stationary and seasonality behavior. Neural network and SARIMAX models may simply be unable to cope with these different features if pre-processing of the input and/or output data is not performed.
In this study the wavelet transform, ANN, SARIMAX and
Acknowledgements
The research was financially supported by a grant from Research Affairs of Univ. of Tabriz.
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