Indian Journal of Agricultural Research

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Research Article

Indian Journal of Agricultural Research, volume 54 issue 1 (february 2020) : 27-34

Comparative Study between Wavelet Artificial Neural Network (WANN) and Artificial Neural Network (ANN) Models for Groundwater Level Forecasting

Anandakumar1,*, B. Maheshwara Babu1, U. Satishkumar1, G.V. Srinivasa Reddy1
1Department of Soil and Water Engineering, College of Agricultural Engineering, Raichur-584 104, Karnataka, India.
Cite article:- Anandakumar, Babu Maheshwara B., Satishkumar U., Reddy Srinivasa G.V. (2019). Comparative Study between Wavelet Artificial Neural Network (WANN) and Artificial Neural Network (ANN) Models for Groundwater Level Forecasting . Indian Journal of Agricultural Research. 54(1): 27-34. doi: 10.18805/IJARe.A-5079.

ABSTRACT

Groundwater level fluctuation modeling is a prime need for effective utilization and planning the conjunctive use in any basin.The application of Artificial Neural Network (ANN) and hybrid Wavelet ANN (WANN) models was investigated in predicting Groundwater level fluctuations. The RMSE of ANN model during calibration and validation were found to be 0.2868 and 0.3648 respectively, whereas for the WANN model the respective values were 0.1946 and 0.1695. Efficiencies during calibration and validation for ANN model were 0.8862 per cent and 0.8465 per cent respectively, whereas for WANN model were found to be much higher with the respective values of 0.9436 per cent and 0.9568 per cent indicating substantial improvement in the model performance. Hence hybrid ANN model is the promising tool to predict water table fluctuation as compared to ANN model. 

INTRODUCTION

In meeting the global demands of water for various sectors ground water plays a major role. Groundwater usage in India accounts over 65 per cent for irrigation and 85 per cent for drinking water supplies especially in rural areas. Due to competition between the users, over-exploitation of ground water is rising as serious problem in many parts of the country, which can be tackled by sustainable utilization and precise management of groundwater resources. Prediction of groundwater level well very much helps in proper and prior management of this important natural resource.
       
Most of ground water models are oriented on relevant partial differential equation solved by finite difference or finite element method. But, in recent past Artificial Neural Network (ANN) models are being applied increasingly to simulate the hydrological processes, because of their better performance over the traditional modeling techniques viz., empirical models, statistical models (autoregressive, autoregressive moving average models) and physical based models. ANN has the capability of mapping non-linear system data and identifying the input and output relationship of a process without adequate knowledge of the underlying principles. Although ANN is considered as flexible tool for modelling hydrological time series, it showed some inaccuracy when applied to hydrologic process involving high non-stationary signal with seasonality varying from 1 day to several decades. In such situations, pre-processing of time and space data may be an effective approach to overcome these drawbacks.
       
Wavelet transform is the improved version of Fourier transform. Even though Fourier transform is a powerful tool for analyzing the components of a stationary signal but failed for analysing the non stationary signal. But, wavelet transform allows the components of a non-stationary signal to be analysed. Wavelet transform helps in better understanding of the whole hydrological process by disintegrating non-stationary time series into sub-series at different scales (levels). Therefore, the combination of ANN with wavelet transform as a hybrid wavelet-ANN (WANN) model that can explain concurrently spatial and temporal information of the signal creates an effective tool for prediction of hydrological processes (Vahid et al., 2013). A number of recent hydrological studies have implemented wavelet analysis (e.g., Kisi, 2009; Nourani et al., 2011; Nourani and Parhizkar, 2013; Maheswaran and Khosa, 2012; Sang, 2012 and Tiwari and Chatterjee, 2011).

MATERIALS AND METHODS

Study area
 
The observation well located in the National Institute of Hydrology (NIH) campus, Roorkee, Uttarakhand, was selected for the study. Roorkee is located in Hardwar district at 29° 51' N and 77° 53' E latitude and longitude respectively. Daily rainfall, water table depth and maximum, minimum and mean temperature data of Roorkee were collected from July 2008 to April 2013. Evapotranspiration was calculated using Hargreaves temperature model. The Hargreaves temperature equation is one of the simplest, less data intensive and most accurate equations used to estimate evapotranspiration (ETo) in mm/day and expressed as,

                                   …(1)
 
Where ETo = evapotranspiration in mm/day, Tmean, Tmax and Tmin = mean, maximum and minimum air temperatures (°C) respectively and Ra = extraterrestrial radiation (mm/day) (Bhabagrahi et al., 2012).
 
ANN Model Development
 
An ANN is a parallel information processing architecture which contains number of inter connected processing elements called nodes resembles the neurons in the brain. Each node combines number of inputs and produces an output, which is then transmitted to many different locations, including other nodes (Azhar et al., 2007). ANN is characterized by its architecture i.e., pattern of connection between nodes, its method of determining the connection weights and the activation function (Fausett, 1994). In this study feed-forward neural network architecture was used in predicting monthly water table depths. Most important step in the ANN model development is the selection of significant input variables. Generally, all the potential input variables are not equally informative, because some input variables may be correlated, noisy, or may not have any correlation with the output variable being modelled (Maier and Dandy, 2000). Hence, statistical methods like cross-correlation, auto-correlation and partial auto-correlation techniques were used for selection of significant input variables.
 
Feed-forward Neural Network (FNN): Feed-forward means all the inter connections between the layers propagate forward to the next layer, flow of information is only in forward direction. In ANN, the type of node being used determines the method in which the total input is calculated and the way the node calculates its output as a function of its net input (Eq. 2). Each nodes acts as a simple processing element that responds to the weighted inputs it receives from other/previous nodes.
 
The net input xj to node j is the weighted sum of all the incoming signals as given in Eq. 2.
 
           .................(2)
 
Where,
xj = net input coming to node j; wij = weight between node i and node j; yi = activation function at node i.
 
The log sigmoidal activation function (Eq. 2) , were used between input and the hidden layers.
 
 .....................(3)
 
       
Typical feed forward neural network with input output combinations were presented in Fig 1. The main advantage of feed forward neural network is that they are easy to handle and can approximate any input-output map.
 

Fig 1: Feed-forward Neural Network.


 
Training with Algorithm
 
Determining the best values of all the weights and updating the weights to reduce the error is called training the ANN. Weights are usually randomly set in the initial iterations, are then adjusted so that the next iteration will produce near value between the desired and the actual output. The training phase can consume a lot of time. Bayesian regularization algorithm was used in this study in order train the given network more efficiently. The advantage of this algorithm is that whatever the size of the network, the function won’t be over-fitted.

Network architecture
 
The network geometry indicates number of nodes in input, hidden and output layer. Network geometry is generally problem oriented. Numbers of nodes in the input layer were decided based on the desired inputs and number of hidden neurons in the network, which were responsible for capturing the dynamic and complex relationship between various input and output variables, were identified by various trials. Network was trained for each set of hidden neurons with the input datasets in batch mode to minimize the mean square error at the output layer. MATLAB 2012a software was used for this analysis.
 
Wavelet transformation
 
In analysing the non-stationary time series, wavelet analysis will be the more effective tool than the Fourier transform. Mainly wavelet analysis consists of breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet. Wavelet analysis can be majorly used to decompose an observed time series (such as rainfall, evapotranspiration and groundwater levels) into various components so that the new time series can be used as inputs for WANN model. In wavelet analysis, signal-cutting problem can be eliminated by the use of a fully scalable modulated window. Generally wavelet transformation is classified into continuous and Discrete Wavelet Transformation (DWT). In the present study discrete wavelet transformation were used. It transforms a time series using a set of basis functions called wavelets. The main purpose of transformation is to reduce the size of data and/or to decrease noise in the data set. The advantage of DWT over Fourier transforms is temporal resolution, it captures both frequency and location information. The signals of the time series was divided into high and the low frequency parts in case of one-dimensional DWT. This splitting were done using signal decomposition equation as indicated in Eq. 4

                    …(4)
 
       
The DWT of any signal x was calculated by passing it through series of low pass and high pass filters. Initially the samples were passed through a low pass filter (Eq. 5) with impulse response ‘g’ resulting in a convolution of the two. Then again the samples were passed through a series of high pass filters (Eq. 6) to analyse the high frequencies.
 
                    …(5)
 
 
           …(6)
 
               
Discrete wavelet transform permits easy and fast de-noising of a noisy signal hence it can be implemented over conventional wavelet transformation. In this study Haar wavelet and Daubechies wavelets (Fig 2) were used for decompose the input signal time series.
 

Fig 2: Different type of wavelet [Haar and Daubechies (Db2)] used in the study.


 
        
The performance during calibration and validation was evaluated by using statistical parameters like coefficient of efficiency (Eq. 7) and root mean square error (Eq. 8) as follows. Coefficient of efficiency (CE)
 
  …(7)
 
Root mean square error (RMSE)
 
 …(8)
 
Where,
Yj = Observed water table depth, Xj = Predicted water table depth, n = Number of observations.

RESULTS AND DISCUSSION

The auto-correlation function (ACF), partial auto-correlation function (PACF) and cross-correlation function (CCF) were used to find out the significant lag values of input variables. The auto-correlation and partial auto correlation analysis of the monthly average water table depth indicated that the monthly average water table depth at (t-1) lag got higher values than the other lag periods (Fig 3 and Fig 4). Cross correlation analysis of monthly average water table depth at (t) with monthly total rainfall (Fig 5) at (t-2) and monthly total evapotranspiration (Fig 6) at (t-1) gave significant correlation compared to other lags. Hence the final inputs for the model were rainfall (R) at (t-2) lag, evapotranspiration (ET) at (t-1) lag and water table depth of observation well (H) at (t-1) lag. One node in the output layer predicted water table depth of the observation well (H) for one month ahead at (t).Final input equation given by
 
                  H (t) = f {R (t-2), ET (t-1), H (t-1)}            … (10)
 

Fig 3: Auto correlation of water table depth.


 

Fig 4: Partial auto correlation of water table depth.


 

Fig 5: Cross correlation between Evapotranspiration and water table depth.


 

Fig 6: Cross correlation between Rainfall and water table depth.


 
Training of ANN model
 
The ANN models were trained using Bayesian regularization algorithm. The whole data set was divided into two sets for the training and validation of the ANN model. The data from July 2008 to April 2014 (58 Months) was considered for the analysis. Out of 58 months dataset, 56 month data set was available for analysis considering 2 month time lag for rainfall series. In that 35 sets (63 per cent) of data were used for calibration (training) and 21 sets (37 per cent) of data were used for validation. Number of neurons in the hidden layer was found by a trial and error procedure, started with one hidden neuron initially and increased up to 20.
       
The transfer functions of hidden and output layers were considered as log sigmoid and pure linear respectively. The performance of the ANN model during calibration and validation with the input combination derived from statistical procedure is given in Table 1.
 

Table 1: Results of ANN model during calibration and validation.


       
The model ANNGWL10 with ANN structure 3-10-1 was the best among all the structures because the performance of the structure in terms of all the statistical parameters was the best among all ANN structure trained as given in the table. Even ANN structure 3-20-1 gave better results, but the difference between the results of these two structures were negligible and also after crossing number of neurons 10 the performance of the model was fluctuating (decreasing and then it was increasing) it might have led to the over fitting and a large ANN structure.
 
Development of WANN model
 
The wavelet transform was used to decompose rainfall, evapotranspiration and water level depth time series at different significant decomposition level. In order to have a comprehensive overview on decomposition level, initially the Eq. 11 was employed which gave minimum level of decomposition (Nourani et al., 2009).
 
                                   L =  int [log (N)]                           …(11)
 
Where L = decomposition level and N = number of time                             series data.
        
In the present study number of time series data i.e., N=56, Hence initial level of decomposition L is taken as 2.The above equation (Eq. 10) is not suitable in all the cases, Hence another two decomposition levels (i.e., 3 and 5) also were examined for identifying the significant decomposition level, in which decomposition level 2 led to better results.
 
Both evapotranspiration and water table depth time series were decomposed at same level of decomposition (i.e. level 2) using Daubechies-2 (db2) mother wavelets, due to proportional relationship between amount of evapotranspiration and water table depth. Also db2 wavelet structure/pattern as shown in Fig 2 was similar to the evapotranspiration and water table depth signals so that it could capture the signal features especially peak points efficiently and led to comparatively good results.
 
Harr mother wavelet was chosen for decomposition of rainfall time series signals according to the formation of main signal. As the Haar wavelet has a pulsed shape, it could properly capture the signal features of rainfall time series and might yield comparatively high efficiency (Nourani et al., 2009). Rainfall time series was decomposed at different decomposition levels from decomposition level 1 to 5 and the decomposition level 3 gave better results. Fig 7, 8 and 9 shows approximation and details sub-series of rainfall time series decomposed by the Haar mother wavelet at level 3, evapotranspiration and water table depth time series decomposed by Daubechies mother wavelet at level 2 and water table depth time series.
 

Fig 7: Approximation and details sub-signals of Rainfall time series decomposed using Harr mother Wavelet at decomposition level 3.


 

Fig 8: Approximation and details sub-signals of evapotranspiration time series decomposed using Daubechies mother wavelet at level 2.


 

Fig 9: Approximation and details sub-signals of water table depth time series decomposed using Daubechies mother wavelet at level 2.


 
The wavelet decomposed signals were used as input for WANN model and the results are tabulated in the Table 2. The WANNGWL20 model with WANN structure 3-10-1 was the best among the all the other model. Even though there was no much difference in results among the structures from WANNGWL2 to WANNGWL10, but least RMSE recorded was selected as best. More over the performance was increasing gradually, there was no fluctuation. Hence model with WANN structure 3-10-1 was selected as the best WANN model.
 

Table 2: Results of WANN model during calibration and validation.

 
 
The performance of best ANN and best WANN model for the prediction of water level depth at Roorkee during calibration and validation were analyzed using scatter plots as indicated in Fig 10, 11, 12 and 13 respectively. The scatter plots clearly demonstrated the potentiality of the developed ANN and WANN models in the prediction of water level depth. All the outliers in case of ANN model are brought near to the best fit line in case of WANN. The correlation graph between the observed and computed monthly water table depths by ANN and WANN for calibration and validation period were indicated in Fig 14 and 15 respectively, the plots clearly demonstrate ANN and WANN both performing well but WANN plot both during calibration and validation following the almost similar trend as observed water table. Hence WANN is considered as more accurate tool for water table prediction modeling. 
 

Fig 11: Scatter plot of the best ANN model.


 

Fig 12: Scatter plot of the best WANN model.


 

Fig 13: Scatter plot of the best WANN model


 

Fig 14: Observed and computed monthly water table depths.


 

Fig 15: Observed and Computed monthly water table depths.

CONCLUSION

The analysis of the performance of the both ANN and WANN models clearly represents the RMSE of ANN model during calibration and validation was found to be 0.2868 and 0.3648 respectively, whereas for the WANN model, RMSE value during calibration and validation was 0.1946 and 0.1695 respectively and also the ANN model efficiency during calibration and validation was 0.8862 and 0.8465 respectively, whereas the WANN model efficiency during calibration and validation was 0.9436 and 0.9568 respectively, indicated a substantial improvement in the model performance. In addition, comparison of time series plots and the scatter plots showed that the water level depth values estimated by the WANN model were more precise than those found by the ANN clearly indicated that Wavelet-ANN is a promising tool for water table depth prediction.

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