# Response Surface Models Using the Wavelet Technique for Reservoir Inflow Prediction

Authors: Adnan Bashir; Muhammad Ahmed Shehzad; Aamna Khan; Muhammad Nabeel Asghar; Muhammad Aslam; Ramy Aldallal; Mutua Kilai; Mohamed S. Mohamed
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Reliable streamflow prediction is vital to improving river operations, flood avoidance, water supply, and water resources management. Recently, response surface models have been launched in reservoir inflow prediction due to their potential to model composite nonlinear behaviour. Authors develop a hybrid model, wavelet quadratic response surface for reservoir inflow prediction in Chenab river basin, Pakistan. Wavelet transform has extensive applications in the field biomedical, engineering, and hydrology. Discrete wavelet transform technique discloses the structure of nonstationary signals. A proper and careful selection of mother wavelet ensure the best performance of wavelet transform. The choice of a suitable wavelet function participates in implementing the wavelet function used in response surface based models for reliable prediction. The performance of the proposed model is checked on different performance indices for model evaluation. The new developed model, wavelet quadratic response surface, depicts excellent results than other studied models.


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## Body

## 1. Introduction

The forecasting of stream flow is an important issue for hydrologists, policymakers, water resource management, and to achieve the requirement of water. Forecasting streamflow is a challenging task, especially to forecast with increasing lead time [1]. Floods are natural disasters producing severe socio-economic consequences to society with massive harm up to billions of dollars. The intensity and frequency of unexpected flood events have been increased with the change in climate conditions [2]. Precise prediction of reservoir inflow dynamics has a significant contribution to water resource planning, management, and policymaking. It attains an essential position for preventing adequate flood mitigation in the presence of recent changing aspects of weather due to global warming [3].Diversified factors of artificial neural network (ANN) model for accuracy, prediction, scalability, and performance comparison make it preferred choice for the researchers of several different fields [4–10]. Sun et al. [11] studied four familiar models for streamflow prediction by combining them with wavelet transform. The hybrid models for short-term forecasting of river water flow of two Chinese river basins (Heihe & Pearl) are wavelet transform (WT-AR), wavelet transform autoregressive moving average (WT-ARMA), wavelet transform linear regression (WT-LR), and wavelet transform ANN (WT-ANN). The accuracy of models is checked by using statistics: RMSE, MAE, & correlation coefficient. Better performance of ANN and WT-ANN models for the prediction of the Northern basin, however, the performance of WT-ARMA was found more reliable for the southern basin. This study claims higher reliability of wavelet-based hybrid models in comparison with traditional models. The authors also suggested that the wavelet technique is a suitable choice for long lead time prediction. Alexander et al. [12] discussed the WANN model for river flow prediction and proof that coupling the wavelet technique to the ANN model produces a more reliable result for river stage prediction. Artificial neural network with backpropagation (ANN - BP) including support vector regression (SVR), and extreme machine learning (EML) are studied by Parisouj et al. [13] for streamflow prediction.Kumar and Sahay [14] introduced wavelet genetic programming (WGP) by coupling DWT and GP techniques for streamflow forecasting. Comparison of developed model with simple GP and autoregressive models indicated better forecasting ability of WGP model, as compared to the traditional GP and AR models. This study examined different wavelet transformation (WT) techniques as pre-processing tools applied in ANNs, support vector machine (SVM), and adaptive neuro-fuzzy inference system (ANFIS) models for river flow prediction. WT techniques applied in this study are continuous WT (CWT), discrete WT (DWT), and discrete-continuous WT (DCWT) a combination of DWT and CWT). CWT technique increases the prediction accuracy of the model only includes the rainfall and temperature data but this technique is not good for streamflow data while the DWT had excellent results for streamflow data. In the proposed DCWT model, CWT applied on meteorological data (temperature and rainfall) and DWT had an application with streamflow data and combination of these two techniques improved the forecasting capability when coupled with the ANN model [15]. Hadid et al. [16] examined a new model, piecewise auto-regressive exogenous (PWARX) for river flood prediction. This method is an alternative among data assignments and parameter estimation. The proposed model can deal with nonlinearities. Wavelet SVR (WSVR) and wavelet ANN (WANN) models applied by Qasem [17] for forecasting evaporation rates. They claim that the WSVR model had reliable prediction results. The typhoon flood discharge forecasting model using the fuzzy time series (FTS) analysis approach, proved to be more effective in flood forecasting [18]. Three models: ARMA, ANFIS and SVR were coupled with wavelet technique to form Wavelet-ANFIS, Wavelet-ARMA, and Wavelet-SVR models. Wavelet-based models performed better for lake-level prediction, as compared to the single models used in the study. However, the overall performance of wavelet-SVR models is good for the prediction of Lake water level [19].Jiang et al. [20] combined discrete wavelet transform with the projection pursuit autoregressive (PPAR) model to develop a new prediction model. The proposed model wavelet-based PPAR (PPAR-WA) is applied for runoff forecasting in the LixianJiang reservoir. Results indicate that the PPAR-WA model produces high accurate results. An experimental study was conducted by Supatmi et al. (2019) for flood event prediction and they claim that hybrid neuro-fuzzy inference system (HN-FIS model had good forecasting accuracy with the lowest error than the existing models. Models dealing with nonstationary data set have less reliability than static data set. Additional research is necessary to study such models that have a better ability to handle nonstationary data. DWT applications have a powerful tool for evaluating the nonstationary time series data.Numerous studies linked wavelet transform theory, for the prediction of hydrological data, with other data-driven techniques to produce consistent results as compared to the simple models, wavelet technique performed better, mainly due to its noise removing potential from raw data, especially data with nonstationary behaviour [21–25]. In the last few decades, hybrid models have essential applications in predicting hydrological data. They applied EMD based ARMA model with a wavelet technique for nonstationary time series data [26]. Baydarogluo et al [27] explained hybrid support SVR models for monthly flow prediction in the Kizihrmak river in Turkey. SVR is the branch SVMs, useful to transforms observed data into high dimension matrix, by applying kernel function and finally perform a linear regression to transformed data. Hybrid models are formulated by using wavelet transform (WT), SSA, and chastic approach (CA). WT decomposes signal into a series of wavelet components. CA produces phase space using trajectories. Overall performance of SVR-WT hybrid model yield better results than other forecasting hybrid models for streamflow prediction. Solgi et al. [28] introduced SVR and gene expression programming (GEP) models for river flow in Iran. The Wavelet technique is used to remove noise from the data set by combining it with traditional models. They claim that the WSVR model performs better than other models.Nourani et al. [29] studied a hybrid model, wavelet artificial intelligence (WAI) for hydrological processes. WAI model has excellent ability in optimization, prediction and adaption process. The wavelet regression (WR) model introduced by Kisi [30] for river stage prediction. He elucidated that WR model is a better choice than the ANN model. Wavelet- Volterra model developed for streamflow Prediction [31]. A hybrid model, wavelet adaptive neuro-fuzzy inference system for monsoon flow prediction introduced by Sahay and Sehgal [32]. A new hybrid model, wavelet gene expression programming (WGEP) studied to predict runoff by applying rainfall data [33]. This study revealed that the WGEP model showed superior results to the simple GEP model for prediction purposes.This study is an effort to develop a new hybrid model for reliable daily streamflow prediction. However, many researchers work on daily streamflow prediction but very rare researchers were applied response surface (RS) models to predict river streamflow. Yu et al. [34] introduced RS models, which are based on first-order polynomial, second-order polynomial and nonlinear methodologies for hydrologic prediction. RS methods based on higher-order polynomial functions were investigated for river streamflow prediction [35]. In the present study, we introduce wavelet technique due to nonstationary behaviour of river streamflow data. In this research study, we coupled wavelet technique with the quadratic response surface (QRS) model to introduce a model, wavelet quadratic response surface (WQRS) for river streamflow prediction on Chenab river basin, Pakistan. The performance of the proposed model matches with four other models: multiple linear regression (MLR), first-order response surface (FORS), wavelet first-order response surface (WFORS) and QRS by using different performance indices. This study discloses that the new hybrid model WQRS performs with high accuracy for reservoir inflow prediction as compared with other models applied in this study on the same data set.


## 2. Methodology

### 2.1. Wavelet Analysis

Wavelet analysis (WA) is a mathematical function that describes the time-scale presentation of signals. Due to time-scale joint presentation, WA has better prediction potential for non-stationary data. WA reveals discontinuities, breakdown points, and trends features of data [36, 37]. A wavelet is a small wave and denoted by the mother wavelet. The mother wavelet function plays the key role to produce a wavelet family by stretching and scaling the mother wavelet [10].In WA, low-frequency data are captured by long intervals, however, short intervals capture high-frequency data. A wavelet can be defined as “a small wave of limited duration having a zero average value” [38]. In wavelet family, two orthogonal functions are used, including scaling function or father function, denoted by ϕη and wavelet function or mother function, expressed by ψη.Wavelet functionψη can be defined by mathematical form as(1)∫−∞+∞ψηdt=0.Wavelet functionψη associated with the non-dimensional time parameter η, mathematically described as(2)ψη=ψτ,s=s−1/2ψt−τs,where t is time, τ is time step describing the complete iteration is performed in window function, s is wavelet scale.Mallat [39] proposed a new dyadic network that constitutes into two filters: a high pass filter (detail component) and low pass filter. The low pass filter (approximation component) constitutes a high scale and lower frequency component. High pass filter (detail component) has a low scale and high-frequency component.The dyadic arrangement is easy to handle and the most efficient method for a practical purpose.Xi Occurs at discrete-time i, Xi symbolizes discrete time series. DWT can be defined as(3)Wm,n=2−m/2∑i=0N−1xiψ2−mi−n.Wm,n Indicated by the wavelet coefficient of the DWT. The scale parameter denoted by S=2m and τ=2mn represents the location parameter. Xi Stands for finite time series while i=0,1,2,…,N−1. N is an integer that is equal to the power of 2 (N=2M) [29, 40].


### 2.2. Response Surface Models

RS models comprise multivariate polynomial models that utilize the relationship between response and predictors using a mathematical equation group. Consideringy, the response variable, depends upon predictors X1,X2,…Xn. The general mathematical equation is given as follows.(4)y=fx1,x2,…,xn+ε,where f symbolize the unknown response function and ε stands for error term (means the source of variation not accounted for f). The first-order polynomial’s general form based on the RS model explains the following equation that represents the linear relationship between response and predictors X1,X2,…Xn.(5)y=β°+∑i=1kβixi+∑∑i=j<1kβijxixj+ε.However, the first-order polynomial model requires a high linear relationship between input and output variables for reliable prediction. On the other hand, in case of a low correlation between input and output variables, it lacks a fitted model for hydrological prediction. Consequently, a second-order polynomial based on the RS model is used in such a situation.(6)y=β°+∑i=1kβixi+∑i=1kβiixi2+∑i=1k−1∑j=2kβijxixj+ε.Here the input variables are X1,X2,…Xn that affect the response variable y. βo,βi,βij are unknown parameters and ε is random error.Generally, the first-order RS model and second-order RS model includes the cross-terms in the models. The inclusion of cross-terms in the first-order RS model and second-order RS model (Quadratic response surface) produces an appropriate prediction for river time series data.


### 2.3. MLR Model

In MLR model, a single response variable has a linear relationship with two or more independent variables.Mathematically, the MLR model can be defined as,(7)y=β°+∑i=1kβixi+ε.


### 2.4. Performance Metrics

#### 2.4.1. Root Mean Square Error

The mathematical form of RMSE is given below(8)RMSE=∑i=1nOi−Pi2n.Here Oi is the observed flow, Pi is the predicted flow and n is the number of observations.


#### 2.4.2. Mean Absolute Error

Mathematically, It is indicated as,(9)MAE=1n∑i=1nOi−Pi.


#### 2.4.3. Nash-Sutcliffe Coefficient of Efficiency

Mathematically, it is defined as(10)NSE=1−∑i=1nOi−Pi2∑i=1nOi−O¯2.The range of NSE lies from−∞ to 1. If NSE is equal to one, it means a perfect match between model predictive discharge data and model observed discharge data. If NSE’s efficiency has zero value, model prediction results are the same as observed data. If the NSE value has less than zero implies model predictive results are not satisfactory.


#### 2.4.4. Coefficient of Persistence

It is denoted by(11)CP=1−∑i=2nPi−Oi2∑i=1N−1Oi+1−Oi2.The CP statistic has ranged from 0 to 1. Model performance improves as the value of CP statistic closer to 1.


## 2.1. Wavelet Analysis

Wavelet analysis (WA) is a mathematical function that describes the time-scale presentation of signals. Due to time-scale joint presentation, WA has better prediction potential for non-stationary data. WA reveals discontinuities, breakdown points, and trends features of data [36, 37]. A wavelet is a small wave and denoted by the mother wavelet. The mother wavelet function plays the key role to produce a wavelet family by stretching and scaling the mother wavelet [10].In WA, low-frequency data are captured by long intervals, however, short intervals capture high-frequency data. A wavelet can be defined as “a small wave of limited duration having a zero average value” [38]. In wavelet family, two orthogonal functions are used, including scaling function or father function, denoted by ϕη and wavelet function or mother function, expressed by ψη.Wavelet functionψη can be defined by mathematical form as(1)∫−∞+∞ψηdt=0.Wavelet functionψη associated with the non-dimensional time parameter η, mathematically described as(2)ψη=ψτ,s=s−1/2ψt−τs,where t is time, τ is time step describing the complete iteration is performed in window function, s is wavelet scale.Mallat [39] proposed a new dyadic network that constitutes into two filters: a high pass filter (detail component) and low pass filter. The low pass filter (approximation component) constitutes a high scale and lower frequency component. High pass filter (detail component) has a low scale and high-frequency component.The dyadic arrangement is easy to handle and the most efficient method for a practical purpose.Xi Occurs at discrete-time i, Xi symbolizes discrete time series. DWT can be defined as(3)Wm,n=2−m/2∑i=0N−1xiψ2−mi−n.Wm,n Indicated by the wavelet coefficient of the DWT. The scale parameter denoted by S=2m and τ=2mn represents the location parameter. Xi Stands for finite time series while i=0,1,2,…,N−1. N is an integer that is equal to the power of 2 (N=2M) [29, 40].


## 2.2. Response Surface Models

RS models comprise multivariate polynomial models that utilize the relationship between response and predictors using a mathematical equation group. Consideringy, the response variable, depends upon predictors X1,X2,…Xn. The general mathematical equation is given as follows.(4)y=fx1,x2,…,xn+ε,where f symbolize the unknown response function and ε stands for error term (means the source of variation not accounted for f). The first-order polynomial’s general form based on the RS model explains the following equation that represents the linear relationship between response and predictors X1,X2,…Xn.(5)y=β°+∑i=1kβixi+∑∑i=j<1kβijxixj+ε.However, the first-order polynomial model requires a high linear relationship between input and output variables for reliable prediction. On the other hand, in case of a low correlation between input and output variables, it lacks a fitted model for hydrological prediction. Consequently, a second-order polynomial based on the RS model is used in such a situation.(6)y=β°+∑i=1kβixi+∑i=1kβiixi2+∑i=1k−1∑j=2kβijxixj+ε.Here the input variables are X1,X2,…Xn that affect the response variable y. βo,βi,βij are unknown parameters and ε is random error.Generally, the first-order RS model and second-order RS model includes the cross-terms in the models. The inclusion of cross-terms in the first-order RS model and second-order RS model (Quadratic response surface) produces an appropriate prediction for river time series data.


## 2.3. MLR Model

In MLR model, a single response variable has a linear relationship with two or more independent variables.Mathematically, the MLR model can be defined as,(7)y=β°+∑i=1kβixi+ε.


## 2.4. Performance Metrics

### 2.4.1. Root Mean Square Error

The mathematical form of RMSE is given below(8)RMSE=∑i=1nOi−Pi2n.Here Oi is the observed flow, Pi is the predicted flow and n is the number of observations.


### 2.4.2. Mean Absolute Error

Mathematically, It is indicated as,(9)MAE=1n∑i=1nOi−Pi.


### 2.4.3. Nash-Sutcliffe Coefficient of Efficiency

Mathematically, it is defined as(10)NSE=1−∑i=1nOi−Pi2∑i=1nOi−O¯2.The range of NSE lies from−∞ to 1. If NSE is equal to one, it means a perfect match between model predictive discharge data and model observed discharge data. If NSE’s efficiency has zero value, model prediction results are the same as observed data. If the NSE value has less than zero implies model predictive results are not satisfactory.


### 2.4.4. Coefficient of Persistence

It is denoted by(11)CP=1−∑i=2nPi−Oi2∑i=1N−1Oi+1−Oi2.The CP statistic has ranged from 0 to 1. Model performance improves as the value of CP statistic closer to 1.


## 2.4.1. Root Mean Square Error

The mathematical form of RMSE is given below(8)RMSE=∑i=1nOi−Pi2n.Here Oi is the observed flow, Pi is the predicted flow and n is the number of observations.


## 2.4.2. Mean Absolute Error

Mathematically, It is indicated as,(9)MAE=1n∑i=1nOi−Pi.


## 2.4.3. Nash-Sutcliffe Coefficient of Efficiency

Mathematically, it is defined as(10)NSE=1−∑i=1nOi−Pi2∑i=1nOi−O¯2.The range of NSE lies from−∞ to 1. If NSE is equal to one, it means a perfect match between model predictive discharge data and model observed discharge data. If NSE’s efficiency has zero value, model prediction results are the same as observed data. If the NSE value has less than zero implies model predictive results are not satisfactory.


## 2.4.4. Coefficient of Persistence

It is denoted by(11)CP=1−∑i=2nPi−Oi2∑i=1N−1Oi+1−Oi2.The CP statistic has ranged from 0 to 1. Model performance improves as the value of CP statistic closer to 1.


## 3. Study Area and Data Set

The Chenab river basin, the second-largest river in Pakistan was selected due to its socioecological importance. River Jhelum joins the Chenab river at the Trimmu gauging station, and river Ravi meets the Chenab river at the Ahmed Pur Siyaal, Punjab. River Sutlej merged into river Chenab (near the Uch Sharif site) to shape the Punjnad gauging station. The Chenab river joins the Indus river (at Minthankot point) before falling into the Arabian sea. The total length of River Chenab is 960 km, running between east longitudes71.1o E to 77.3o E and north latitude 29.2o N to 32.4o N. In almost every year, floods are a severe issue in the Chenab river delta region during the monsoon season. The training data set (daily Chenab river basin inflow data from Ist July to 30 September for the year 2005 to 2009) is used to train the model while the testing data set (daily Chenab river basin inflow data Ist July to 30 September for the year 2010) is used to develop and evaluate the performance of models. Summary statistics of five stations of Chenab river are represented in Table 1 and geographical location of Chenab river with other rivers are located in Figure 1.Table 1
Summary of descriptive data of stream inflow at different stations of Chenab river basin.
Stream flow (m3 s−1)MaralaKhankiQadrabadTrimmuPunjnadTraining data setAverage65872.9246864.4850543.9047235.6336523.95Standard dev.33651.3645266.6044143.9349900.0745126.30Range273976416066375906322432310952Maximum285567422740381906328926316192Minimum115916674600064945240Testing data setAverage82525.4168807.4370487.8891933.4181646.89Standard dev37218.5553944.6948004.8280989.7982306.97Range194551326423292172316178309209Maximum215650334170309172328926316192Minimum21099774717000127486983Figure 1
Map of river Chenab in Pakistan.


## 4. Model Development and Application

A crucial phase in water resource modelling is developing a satisfactory prediction model that constitutes relevant input variables. However, there is no rigid definition of appropriate input variables.Shabri and Suhartono [41] used linear correlation analysis (LCA) and stepwise linear regression (SLR) for the selection of an ideal number of input variable, and concluded that both methods (LCA and SLR) are suitable for input variable selection. In this research study, the LCA method is used to determine the appropriate input variables for reservoir inflow prediction, as recommended in previous research. By using such a technique, a smaller number of input variables are obtained for model formulation.Input stations (Khanki, Marala, and Qadrabad) are strongly correlated as shown in Table2. In comparison, these three stations have a weak relationship with the input station Trimmu and Punjab. While input stations Marala and Khanki have a strong correlation with the dependent variable Qadrabad, which is our forecast site.Table 2
Linear correlation analysis between gauging stations.
MaralaKhankiQadrabadTrimmuPunjnudMarala0.880.800.520.45Khanki0.910.490.43Qadrabad0.480.43Trimmu0.95PunjnudIn geographical observation, the selected significant input stations make logic as river Jhelum meet with the Chenab river at site Trimmu. The Ravi and Sutlej rivers join the Chenab river approximately at site Punjnad. Due to the inclusion of rivers at Trimmu and Punjnad cites, the first three gauging stations have a weak correlation with Trimmu and Punjnad cites but have a strong correlation with each other. So, the forecast site to be selected in this study is the Qadrabad gauging station.In Table2, the correlation coefficient between both regressors is 0.88, which is the sign of multicollinearity.One such method for eliminating multicollinearity between predictors is principal component analysis (PCA). It would be beneficial to produce new variables that are linear combinations of the original variable and are mutually orthogonal. These new variables are called principal components and scores obtained from these components provides input to models under study [42]. PCA has been employed to split the interrelationships among independent statistical variables [43–45]. This study combines principal component scores with MLR, FORS, QRS, WFORS, and WQRS models to obtain the regression equation for Qadrabad gauging station.


### 4.1. Time Series Decomposition via DWA

DWA uses different resolution levels to decompose the signal of each input variable into approximation and detail components. A high pass filter of time series data represents the detail component, while a low pass filter denotes the approximation component. Such sub-series represent other periodic elements. Correlating significant features with time series data produced significant results. Numerous wavelet families exist, and each family has different vanishing moments. Commonly, all mother wavelets are classified into two categories: orthogonal wavelet family and nonorthogonal wavelet family. The wavelets with orthogonal bases are assumed to be the best mother wavelets for hydrological variables. The major orthogonal wavelets are Haar, Daubechies (db), Coiflets (Coif), and Symlets (sym) [12]. Haar, Daubechies, Symlets, and Coiflets are wavelet functions with distinct vanishing moments that are commonly used in hydrological modelling. Wavelet functions are distinguished by their distinct properties, which include vanishing moments and region of support. In the wavelet technique, the region of support is proportional to the wavelet length’s span, which determines the signal’s localization [46, 47].To test 1-d lead-time prediction using various vanishing moments, thirteen wavelet FORS models are constructed using various mother wavelet functions. In comparison to other wavelet functions, the sym15 wavelet function has better performance in Table3. So, sym15 mother wavelet function is used for the purpose of transformation.Table 3
Performance of different WFORS models with numerous mother wavelet functions.
WaveletVanishing momentNSE (m3 s−1)RMSE (m3 s−1)MAE (m3 s−1)CP (m3 s−1)Haar10.95600.38500.26880.9596coif110.92830.46000.26270.9288coif330.95320.32700.20460.9271coif550.95800.28600.18860.9250db440.94760.37700.25190.9418db880.96320.29400.19950.9470db12120.96410.27000.19260.9514db16160.95270.29500.20630.9551db20200.95780.32500.22910.9465sym550.96090.34700.20450.9357sym10100.97690.23800.15940.9536sym15150.97760.22500.14690.9521Sym20200.96810.23900.15380.9467


### 4.2. Selection of Decomposition Level

The next step is the selection of suitable decomposition levels. Decomposition of DWT depends upon levels or stages, and decision about the total number of levels depends upon the length of data. In the first level, the original signal decomposes into two parts; one is approximation denoted byA1, and the other is detail denoted by D1. At level two, approximation A1 further decomposes into two components: approximation A2 and detail D2. In the next third level, DWA decomposes the signal into approximation A3 and detail D3. In the third level, DWA decomposes the signal into approximation A3 & detail (D1, D2, D3) and so on.The following formula is used to choose the number of levels of the original signal [48–50].(12)L=intLogN,where L indicates the number of decomposition levels and N represents the signal length. This study used three levels for decomposing the data while Figure 2 represents wavelet components with three levels.Figure 2
Original signal decomposes into wavelet components.


### 4.3. Coupled WFORS Model and Ensemble Prediction

The WFORS model is derived by using discrete wavelet components. The significant components are used to develop the WFORS model. A similar procedure adopts for the WQRS model.


## 4.1. Time Series Decomposition via DWA

DWA uses different resolution levels to decompose the signal of each input variable into approximation and detail components. A high pass filter of time series data represents the detail component, while a low pass filter denotes the approximation component. Such sub-series represent other periodic elements. Correlating significant features with time series data produced significant results. Numerous wavelet families exist, and each family has different vanishing moments. Commonly, all mother wavelets are classified into two categories: orthogonal wavelet family and nonorthogonal wavelet family. The wavelets with orthogonal bases are assumed to be the best mother wavelets for hydrological variables. The major orthogonal wavelets are Haar, Daubechies (db), Coiflets (Coif), and Symlets (sym) [12]. Haar, Daubechies, Symlets, and Coiflets are wavelet functions with distinct vanishing moments that are commonly used in hydrological modelling. Wavelet functions are distinguished by their distinct properties, which include vanishing moments and region of support. In the wavelet technique, the region of support is proportional to the wavelet length’s span, which determines the signal’s localization [46, 47].To test 1-d lead-time prediction using various vanishing moments, thirteen wavelet FORS models are constructed using various mother wavelet functions. In comparison to other wavelet functions, the sym15 wavelet function has better performance in Table3. So, sym15 mother wavelet function is used for the purpose of transformation.Table 3
Performance of different WFORS models with numerous mother wavelet functions.
WaveletVanishing momentNSE (m3 s−1)RMSE (m3 s−1)MAE (m3 s−1)CP (m3 s−1)Haar10.95600.38500.26880.9596coif110.92830.46000.26270.9288coif330.95320.32700.20460.9271coif550.95800.28600.18860.9250db440.94760.37700.25190.9418db880.96320.29400.19950.9470db12120.96410.27000.19260.9514db16160.95270.29500.20630.9551db20200.95780.32500.22910.9465sym550.96090.34700.20450.9357sym10100.97690.23800.15940.9536sym15150.97760.22500.14690.9521Sym20200.96810.23900.15380.9467


## 4.2. Selection of Decomposition Level

The next step is the selection of suitable decomposition levels. Decomposition of DWT depends upon levels or stages, and decision about the total number of levels depends upon the length of data. In the first level, the original signal decomposes into two parts; one is approximation denoted byA1, and the other is detail denoted by D1. At level two, approximation A1 further decomposes into two components: approximation A2 and detail D2. In the next third level, DWA decomposes the signal into approximation A3 and detail D3. In the third level, DWA decomposes the signal into approximation A3 & detail (D1, D2, D3) and so on.The following formula is used to choose the number of levels of the original signal [48–50].(12)L=intLogN,where L indicates the number of decomposition levels and N represents the signal length. This study used three levels for decomposing the data while Figure 2 represents wavelet components with three levels.Figure 2
Original signal decomposes into wavelet components.


## 4.3. Coupled WFORS Model and Ensemble Prediction

The WFORS model is derived by using discrete wavelet components. The significant components are used to develop the WFORS model. A similar procedure adopts for the WQRS model.


## 5. Results and Discussion

After developing the prediction models, the performance indices: root mean square error (RMSE), Nash-Sutcliffe coefficient of efficiency (NSE), persistence index (CP), and mean absolute error (MAE) are used to identify the best prediction model at Qadrabad gauging station of the Chenab river basin, Pakistan. The input variables considered in the study are Marala and Khanki gauging stations. The observed data is used as input to MLR, FORS, and QRS models in the first step. In the second step, the empirical data decomposes into wavelet components using the sym15 wavelet function at level third. The decomposed sub-series’ useful components are used as input to the FORS model and the QRS model to produce WFORS and WQRS models. This research utilized several RS-based models: FORS, WFORS, QRS, and WQRS.Figure3 depicts a scatter plot of observed and predicted water discharge of MLR, FORS, WFORS, QRS, and WQRS models to predict 1 to 3 days lead time. It is observed that the prediction of the WQRS model is in good agreement with observed data points. The WQRS model is closer to form a straight line, which gives an idea about the strong relationship between prediction values of the WQRS model and observed data points for 1-d ahead prediction. Similarly, for 2-d and 3-d lead time, the performance of the model remains stable. For an extreme data point, model prediction is also in good agreement with observed data points. These RS-based models are compared to each other and with the traditional MLR model to evaluate models’ performance. Results are described in Table 4.Figure 3
Scatter plots of observed discharge streamflow versus predicted streamflow of MLR, FORS, WFORS, QRS and WQRS models for testing data set (a) 1 day ahead (b) 2 day ahead (c) 3 day ahead.
(a)(b)(c)Table 4
Performance metrics for MLR, FORS, WFORS, QRS and WQRS models.
ModelsDaysNSERMSE (m3 s−1)CPMAE (m3 s−1)MLR1 day0.84260.40100.73980.20822 day0.84290.40100.73990.20643 day0.84300.40100.74020.2059FORS1 day0.86210.37800.77190.19412 day0.86220.37700.77190.19233 day0.86220.37700.77190.1922WFORS1 day0.97610.22500.95210.14702 day0.97790.22300.95260.14373 day0.97820.22200.95290.1415QRS1 day0.89130.33900.82010.19522 day0.89130.33900.82020.19423 day0.89140.33900.82020.1936WQRS1 day0.97960.21700.95630.13542 day0.97990.21600.95650.13283 day0.98000.21500.95670.1316Comparing the applications of RS-based models (FORS, WFORS, QRS, WQRS) with each other and with the traditional MLR model. The RMSE for the MLR model is 0.4010 m3/s for one day ahead prediction and remains the same for the next two days ahead prediction. NSE has the same value 0.8426 for 1-d ahead forecast on the MLR model. For the remaining two days, there is a slight increase in NSE value. The value of MAE for 1-d ahead prediction of the MLR model is 0.2080; for the remaining two days, MAE shows a slightly decreasing trend that means the performance of the model remains stable. CP for the MLR model is 0.7398 for 1-d lead time. Compared with the MLR model, the FORS model shows better results with NSE prediction value 0.8621 for one day ahead of prediction time. For 2 and 3 days ahead prediction, NSE has the same values 0.8622 for both days. At the same time, RMSE shows a value of 0.3780 m3/s for one day ahead prediction of the FORS model, which shows improved performance of the FORS model and the MLR model. On the other hand, performance indices MAE, NSE, and CP have improved results for the FORS model. Overall, the FORS model performs well in comparison with the MLR model.As discussed in the previous section, the wavelet technique is useful to make the trend apparent and obvious to predict the river stage. The results of this article represent that wavelet analysis gets better prediction accuracy. The ability of the wavelet technique to denoise time series has a good impact on the prediction accuracy of the model. In the WFORS model, the NSE value for the first day ahead prediction is 0.9761, and for the remaining two days, it is 0.9779 & 0.9782 respectively. RMSE for 1-d ahead forecast is 0.2250 m3/s, and it will be decreased for the remaining two days. The WFORS model also has improved performance on other performance indices CP and MAE. Obviously, by applying the wavelet technique to the original time series data, results of WFORS are better than traditional MLR, QRS, and simple FORS model.QRS model shows good results, then the MLR model and FORS model on all four performance indices. By conjecturing the wavelet technique with the QRS model, the WQRS model develops in this way. As apparent in Table4, the WQRS model expresses excellent results regarding performance indices RMSE = 0.2170 m3/s, NSE = 0.9796, CP = 0.9563 and MAE = 0.1354 m3/s for 1-d ahead prediction and a more reasonable choice to choose this model for reservoir inflow prediction. NSE values remain the same for all remaining two days prediction. As can be observed from the results, models coupled with the DWT technique prove good results compared to models MLR, FORS, and QRS. The overall performance of the WQRS model is superb than other models on all performance indices.For one day to three day lead time prediction, the observed and predicted water flow data are in better agreement as illustrate by different performance metrics used in the article. MLR, FORS, WFORS, QRS, and WQRS models improve on all performance indices for one to three days ahead of prediction. So, the results show a reliable forecast for one to three day lead time.


## 6. Conclusion

This study explores the RS models’ effectiveness when coupled with the DWT technique to check the daily reservoir inflow prediction. In this study, independent variables are selected using the correlation technique and by observing the geographic location. Three variables are selected for model formulation: Trimmu, Khanki, and Qadarabad for river Chenab, Pakistan, to determine daily streamflow prediction. The Qadarabad gauging station is chosen as the dependent variable, while Trimmu and Khanki gauging stations are selected as independent variables. The RS models with first and second-order provide weak performance for daily reservoir inflow prediction. The accuracy improves significantly when the RS model is coupled with DWT. Choosing useful input variables from data, DWT is applied to these input variables. Discrete wavelet components obtain after decomposing the data by using DWT technique. Useful wavelet components are used to provide input to the model. DWT technique gains better performance of the model by capturing useful information on different resolution levels. Then, the WFORS model is developed by using Daubechies, Haar, Symlets, and Coiflets wavelet function to check one day ahead prediction on performance indices NSE, RMSE, MAE, and CP. By choosing Symlets wavelet function with vanishing moment 15 for wavelet-based models under study: WFORS and WQRS. Comparing all developed models in the study, namely FORS, WFORS, QRS, WQRS, and MLR. WQRS model performs well on performance indices then remaining models used in the study and the proposed WQRS model is the most promising choice to predict daily reservoir inflow prediction.



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