New formulation for predicting total dissolved gas supersaturation in dam reservoir: application of hybrid artificial intelligence models based on multiple signal decomposition

Abstract

Total dissolved gas (TDG) concentration plays an important role in the control of the aquatic life. Elevated TDG can cause gas-bubble trauma in fish (GBT). Therefore, controlling TDG fluctuation has become of great importance for different disciplines of surface water environmental engineering.. Nowadays, direct estimation of TDG is expensive and time-consuming. Hence, this work proposes a new modelling framework for predicting TDG based on the integration of machine learning (ML) models and multiresolution signal decomposition. The proposed ML models were trained and validated using hourly data obtained from four stations at the United States Geological Survey. The dataset are composed from: (i) water temperature (T_w), (ii) barometric pressure (BP), and (iii) discharge (Q), which were used as the input variables for TDG prediction. The modelling strategy is conducted based on two different steps. First, six singles ML model namely: (i) multilayer perceptron neural network, (ii) Gaussian process regression, (iii) random forest regression, (iv) random vector functional link, (v) adaptive boosting, and (vi) Bootstrap aggregating (Bagging), were developed for predicting TDG using T_w, BP, and Q, and their performances were compared. Second, a new framework was introduced based on the combination of empirical mode decomposition (EMD), the variational mode decomposition (VMD), and the empirical wavelet transform (EWT) preprocessing signal decomposition algorithms with ML models for building new hybrid ML models. Hence, the T_w, BP, and Q signals were decomposed to extract the intrinsic mode functions (IMFs) by using the EMD and VMD methods and the multiresolution analysis (MRA) components by using the EWT method. Then after, the IMFs and MRA components were selected and regraded as new input variables for the ML models and used as an integral part thereof. The single and hybrid prediction models were compared using several statistical metrics namely, root mean square error, mean absolute error, coefficient of determination (R²), and Nash–Sutcliffe efficiency (NSE). The single and hybrid models were trained several times with high number of repetitions, depending on the kind of modeling process. The obtained results using single models gave good agreement between the predicted TDG and the situ measured dataset. Overall, the Bagging model performed better than the other five models with R² and NSE values of 0.906 and 0.902, respectively. However, the extracted IMFs and MRA components using the EMD, VMD and the EWT have contributed to an improvement of the hybrid models’ performances, for which the R² and NSE were significantly increased reaching the values of 0.996 and 0.995. Experimental results showed the superiority of hybrid models and more importantly the importance of signal decomposition in improving the predictive accuracy of TDG.

Graphical abstract

1 Introduction

Nowadays, high dam’s reservoir has become more numerous and play an important role for the production of hydropower energy (Qin et al. 2022). Another important role of high dams is the creation of artificial flood over the spillways of hydropower stations (Wang et al. 2019b), which is the major cause of the formation of total dissolved gas (TDG) along the river downstream of high dams (Yuan et al. 2018; Huang et al. 2021). TDG can be defined as the quantity of dissolved air available in water, and it is considered a natural phenomenon resulting from the interaction between air and water near surface water (Li et al. 2023a). From a computational point of view, the formation of TDG (i.e., N₂, O₂ and CO₂) persisted if pressure of TDG become superior to the atmospheric pressure (Yuan et al. 2022). Elevated TDG concentration affects the dynamics of aquatic life (Cheng et al. 2021), and thus an important ecological indicator that can be utilized for the evaluation of the state of the aquatic life downstream of high dam’s reservoirs (Yuan et al. 2023). However, water managers and decision makers are conducting an ongoing risk assessment of the elevated TDG concentration leading to the conclusion that; TDG supersaturation should not be exceeded the amount of 110% of saturation. If not, this will lead to many serious problems designated as the ‘’gas-bubble trauma in fish’’, i.e., “GBT”, and a downward correction of the elevated concentration become imminent (Zeng et al. 2020). Downstream of the high dam’s reservoirs, fish suffer from elevated concentration of TDG. Among other ecological indicators that can help in increasing TDG concentration, water temperature, barometric pressure and discharge are particularly important as they influence the fluctuation of TDG (Li et al. 2022; Chen et al. 2023). Furthermore, due to the limited in situ sampling sites for controlling TDG, knowledge on the formation, fluctuation, and spatiotemporal distribution of TDG along the rivers downstream of high dams is relatively sparse, except for some locations in USA and China. Nevertheless, at Columbia and Snake rivers, USA, increased in situ stations and continuous monitoring of TDG have enabled researchers to gain insight about formation and behaviors of TDG supersaturation.

The literature revealed many researchers have done several works on TDG concentration prediction (Li et al. 2009, 2022; Lu et al. 2019; Zhang et al. 2023). More precisely, previous investigations were mainly focused on understanding two important phenomena: (i) how TDG supersaturation is generated near downstream of dams, and (ii) how TDG can be dissipated to avoid the GBT (Ma et al. 2019). Furthermore, for achieving these two objectives, available research papers were focalized and mainly oriented toward the application of numerical models and simulation approaches. Understanding the fluctuation, dissipation and the overall behavior of TDG is of high importance for the protection and control of the fish and many other ecological phenomena’s in rivers (Yuan et al. 2018). The complexity of TDG supersaturation formation depends mainly on the environmental factors responsible for the increase of the TDG rates, and recent recognition of the environmental causes are good steps towards a more comprehensive framework for the TDG simulation.

Nowadays, the role of numerical models seems to be crucial, and they significantly helped in improving our understanding of TDG formation and dissipation. Several numerical models have been developed and applied for predicting TDG dissipation (Lin et al. 2022). Among the proposed models, it is worth to point out: the 1-D unsteady TDG model proposed by (Ma et al. 2016), the depth averaged 2-D for the dissipation of TDG (Shen et al. 2016), the numerical 2-DTDG model for analyzing the link between water temperature and TDG (Feng et al. 2013), a basic prediction water renewal model for TDG proposed by (Peng et al. 2022a), the two phases flow model proposed by (Wang et al. 2019a). Several other models are available in the literature. For example; a study proposed a mass transfer for modelling TDG using water depth as predictor, and the unsteady 3D two-phase (Politano et al. 2007, 2009, 2012, 2017). Despite the benefits of numerical and fluid mechanic models for TDG simulation, they also have some disadvantages.

However, they have been faced to several particular difficulties mainly related to the high number of variables needed for calibrating the numerical models (Zhang et al. 2022). Certain TDG conditions can be ambiguous, hard to understand, and the environmental factors that contribute to their control and diagnosis cannot be accurately determined using single models. To solve the forgoing limitation, multi-resolution signal decomposition is an ensemble of methods that help to alleviate these underlying problems. Among the disadvantages of the numerical and fluid mechanic models, it can be highlighted: (i) the need of very large number of variables for model calibrations, (ii) the difficulties to use the calibrated models outside of the calibration sites, (iii) the need for a large dataset based on the laboratory experiment for which, the operator’s intervention is required and the equipment shall be placed at the most adequate measuring point to ensure safe and timely access data.

Over the past few years, the exploration of newly developed modelling strategy based on the application of machine learning (ML) models has been established and successfully applied for modelling TDG concentration (Heddam 2017; AlOmar et al. 2020; Wang et al. 2022). Among the proposed models, extreme learning machine (ELM) and support vector regression models proposed by (AlOmar et al. 2020), ELM optimized genetic algorithm (GA-ELM) developed by (Wang and Sheng 2022), multilayer perceptron neural network (MLPNN) used by (Han et al. 2019), the generalized regression neural network (GRNN) proposed by (Heddam 2017). The parallel chaos search based incremental extreme learning machine (PC-ELM) developed by (Heddam 2023), dynamic evolving neural-fuzzy inference system (DENFIS) (Heddam and Kisi 2021), the kriging interpolation method (KIM) and response surface method (RSM) (Heddam and Kisi 2020), and the least squares support vector machine (LSSVM) used by (Keshtegar et al. 2019). All above reported models were successfully applied for modelling TDG and the obtained results were found to be very promising. However, previous ML approaches for TDG prediction have been faced to several particular difficulties mainly related to the high number of variables needed for calibrating the numerical models (Zhang et al. 2022). Certain TDG conditions can be ambiguous, hard to understand, and the environmental factors that contribute to their control and diagnosis cannot be accurately determined using single ML models. Multiresolution signal decomposition is an ensemble of methods that help to alleviate these underlying problems.

The aim of this paper is to continue the work already undertaken in the context of TDG prediction. Hence, this work seeks to propose new formulation for TDG estimation aiming at improving the prediction accuracy of TDG, and at the same time introducing a new modelling framework, that contributes to the improvement of standalone ML methods by reducing the error calculated between measured and predicted TDG data. For this purpose, we make use of different multiresolution signal decomposition techniques namely; the empirical mode decomposition (EMD), the variational mode decomposition (VMD), and the empirical wavelet transform (EWT) techniques. The basic steps involved in our new modelling framework can be summarized as follow. Initially, the EMD, VMD, and the EWT are performed on the available input variables. The Tw, BP, and Q input variables are then decomposed into various intrinsic mode functions (IMFs) by using the EMD and VMD methods and the multiresolution analysis (MRA) components by using the EWT method. After decomposition, the IMFs and the MRA were used as new input variables for the ML approaches. Compared to the single models, the use of the EMD, VMD, and EWT have the advantages of extracting and combining the multitude of nonlinear information available in the original signal, producing new input variables with greater useful information, making the establishment of an appropriate relationship between TDG and the Tw, BP, and Q more practical. For convenience, we refer to Tw, BP, and Q as the input variables for the single models, while the IMFs and MRA typically represent the input variables of the hybrid models.

2 Materials and methods

2.1 Study area and data

In this study, TDG (% saturation), water temperature (Tw: °C), barometric pressure (BP: mmHg), and discharge (Q: kcfs) data, were collected from four United States geological survey (USGS) stations. All data are available at (https://or.water.usgs.gov/cgi-bin/grapher/table_setup.pl). These stations were selected tacking into account continuous data availability. Supporting details characteristics of the stations are reported in Table 1. According to Table 1, we can see that, the period of record varied between stations, and all available data have been used, however, the presence of the incomplete days with missing values in data records make the length of data varied between stations. All data were available at hourly time step and composed from TDG, Tw, BP, and Q. The study area and locations of in situ measured data are shown in Fig. 1, in which the different stations are represented by different colors. For each station, we split the dataset into training (70%) and validation (30%). Thus, TDG was predicted using three input variables namely, Tw, BP and Q, and all variables were standardized using the Z-score method (Eq. 1) by subtracting the mean and dividing by the standard deviation.

$${Z}_{n}=\frac{{x}_{n}-{x}_{m}}{{\sigma }_{x}}$$

(1)

where: Z_n is the normalized value of the variable n; x_n is the measured value of the variable n; x_m and σ_x are the mean value and standard deviation of the variable x.

Table 1 Dataset presentations covering the period of the study

Fig. 1

The location of the studied sites “USGS stations” at the north-west of United States

Table 2 lists a brief descriptive statistic for all data and for all stations. In the Table 2, the mean value (X_mean), the maximal value (X_max), the minimal value (X_min), the standard deviation (S_x), the coefficient of variation (C_v), and the coefficient of correlation calculated between TDG and the three input variables are summarized. Finally, it is noted that, TDG was predicted according to two different scenarios: (i) modelling TDG using three input variables (i.e., Tw, BP and Q), and (ii) the Tw, BP and Q variables were decomposed into several IMFs and MRA using the EMD, VMD and EWT.

Table 2 Summary statistics of total dissolved gas concentration and input variables

2.2 Machine learning methods

In the present study, six ML models were developed for predicting TDG supersaturation namely: (i) multilayer perceptron neural network (MLPNN), (ii) Gaussian process regression (GPR), (iii) random forest regression (RFR), (iv) random vector functional link (RVFL), (v) adaptive boosting (AdaBoost), and (vi) Bootstrap aggregating (Bagg). The theoretical description of these models is given below.

2.2.1 Multilayer perceptron neural network (MLPNN)

Artificial neural network (ANN) was inspired from the structure of the human brain. Thus, the basic component of the brain, i.e., the biological neurons were simulated to be artificial neurons arranged in several layers and dealing with specific tasks (Salman and Kadhum 2022). In the present study, the famous and well-known multi-layer perceptron neural network (MLPNN) was used for modelling total dissolved gas (TDG). The MLPNN includes several layers: (i) input layer with three input variables, i.e., BP, T_w and Q, each one designated as x_i, (ii) one or more hidden layers arranged in parallel and possess an ensemble of neurons, and (iii) on output layer with only one neuron, i.e., the TDG. Similar to the biological neuron, the information is disseminated rapidly from the input to the output layer using an ensemble of parameters called the weight factors and biases, which need to be updated during the training process. Consequently, we can summarize the mathematical formulas of the MLPNN as follow:

$${{\text{A}}}_{{\text{j}}}=\left(\sum_{{\text{i}}=1}^{{\text{N}}}{{\text{W}}}_{{\text{ij}}}{{\text{x}}}_{{\text{i}}}\right)+{{\text{b}}}_{{\text{j}}}$$

(2)

where, A_j is the output of the hidden neuron j, W_ij is the weight linking the input variable x_i to the hidden neuron j, and b_j is bias of the hidden neuron j. Before passing the A_j to the next layer, the sigmoid activation function is used as follow:

$${{\text{k}}}_{{\text{j}}}={\text{f}}\left({{\text{A}}}_{{\text{j}}}\right)={\text{sigmoid}}\left({{\text{A}}}_{{\text{j}}}\right)=\frac{1}{1+{{\text{e}}}^{-{{\text{A}}}_{{\text{j}}}}}$$

(3)

Finally, the output of the MLPNN is calculated as follow:

$${\text{y}}=\left(\sum_{{\text{j}}=1}^{{\text{N}}}{{\text{W}}}_{{\text{jk}}}{{\text{k}}}_{{\text{j}}}\right)+{{\text{b}}}_{{\text{o}}}$$

(4)

where, y is the output of the MLPNN model, W_jk is the weight linking the hidden layer to the output layer, and b₀ is bias of the single output neuron. More details about the MLPNN can be found in large number of published papers.

2.2.2 Gaussian process regression (GPR)

The Gaussian Process Regression (GPR) proposed by (Williams and Rasmussen 2006) can be formulated as follow (Ouyang et al. 2022; Zhao et al. 2023):

$${\text{f}}\left({\text{x}}\right)\sim {\text{GP}}\left({\text{m}}\left({\text{x}}\right),{\text{k}}\left({\text{x}},{{\text{x}}}^{\mathrm{^{\prime}}}\right)\right)$$

(5)

In the above equation, x refers to the input vector; m(x) refers to the mean function, while ${\text{k}}\left({\text{x}},{{\text{x}}}^{\mathrm{^{\prime}}}\right)$ corresponds to the covariance (kernel) function. For example, the Radial basis function (RBF) can be expressed as follow (He and Zhou 2022):

$${{\text{k}}}_{{\text{SE}}}\left({{\text{x}}}_{{\text{i}}},{{\text{x}}}_{{\text{j}}}\right)={\upsigma }_{{\text{f}}}^{2}{\text{exp}}\left(-\frac{{\left({\text{x}}-{{\text{x}}}^{\mathrm{^{\prime}}}\right)}^{2}}{2{{\text{l}}}^{2}}\right)$$

(6)

In the above equation, ${\upsigma }_{{\text{f}}}^{2}$ is the variance of the dataset (i.e., the signal), and l, is the length scale of the uncertainty fluctuations. The GPR is derived from the standard linear model as follow:

$${\text{y}}={\text{f}}\left({\text{x}}\right)+\upepsilon$$

(7)

In the above equation, ε refers to the noise having a mean zero and variance ${\upsigma }_{{\text{f}}}^{2}$ (He and Zhou 2022; Ouyang et al. 2022; Zhao et al. 2023).

2.2.3 Random forest regression (RFR)

Random forest regression (RFR) is an ensemble of classification and regression tree (CART) models built on a serie of trees with a training process made using the concept of Bagging by random smpaling with replacement, tacking into account the statbility and the imporvement in the terms of accuracy (Breiman 2001; Takoutsing and Heuvelink 2022). As the model is based on improving the performances of week learners and making a final decision using averaging or majority voting, the RFR use the ‘’out-of-bag: OOB’’ for quantifying the calculted error and for ranking the variables in terms of importance using the permutation strategy. Building a RFR model can be achieved according to the follwing steps: (i) start by extracting an ensemble of subset randomly form the original dataet, (ii) growing a tree for each subset, (iii) repeat this until the constrution of k decision tree (DT), and (iv) the final response is than calculted based on averaging (AVG) the response of all DT (Zong and Zhang 2019; Giri et al. 2023). However, it is important to note that, RFR works only with two parameters: the total number of trees and the number of predictors at each node of the subset (Weiqi et al. 2022). Furthermore, the remaining paprt of the data not included in the subset, i.e., the OOB is used for chacking the regression or the classificiation performances (Zong and Zhang 2019). Finally, we can summarize the adavatage of the RFR as follow: (i) can not be affected by the nonlinearity between variables, (ii) high capability for avoinding over fitting problem, (iii) the gerated trees are uncorrelated, and (iv) the predicotors can be ranked in terms of their contribution (Chong et al. 2019).

2.2.4 Random vector functional link (RVFL)

The random vector functional link neural network (RVFL) can be viewed as an improved version of the original single layer neural network with an important difference (Pao et al. 1992; Pao et al. 1994): in addition to the transition from the input to the output layer through the hidden layer, there is a direct link between the input and output layer. Regarding the model parameters, the weights between the input neurons and the hidden neurons (also called enhancements neurons) were randomly assigned and remain unchangeable during the training process (W_ij), while the remaining weight (W_jk) should be updated during the training of the model. Given a set of data point (x_i, i = 1, …, N) with the corresponding output (y_i), the response of the enhancements, neurons can be calculated as follow (δ_j):

$${\updelta }_{{\text{j}}}\left[{{\text{W}}}_{{\text{j}}}{{\text{x}}}_{{\text{i}}}+{\upbeta }_{{\text{j}}}\right]=\frac{1}{1+{{\text{e}}}^{-\left({{\text{W}}}_{{\text{ij}}}{{\text{x}}}_{{\text{i}}}+{\upbeta }_{{\text{j}}}\right)}}, {\upbeta }_{{\text{j}}}\in \left[0,{\text{S}}\right], {{\text{W}}}_{{\text{j}}}\in \left[-{\text{S}},+{\text{S}}\right],\,\mathrm{ j}=\mathrm{1,2},\dots , {{\text{N}}}_{{\text{h}}}$$

(8)

In Eq. 8, Wj refers to the weight between the input and the hidden neurons, β_j is the bias of the hidden neuron j, and S refers to the scale factor determined during the training process. The final output can be calculated as follow (Jiao et al. 2023; Nabih et al. 2023):

$${\text{Y}}={\text{Bw}},{\text{w}}\in {{\text{R}}}^{{\text{N}}+{\text{P}}},\mathrm{ and \,B}=\left[{{\text{B}}}_{1}{{\text{B}}}_{2}\right]$$

(9)

In Eq. 9, B₁ corresponds to the input data, and B₂ corresponds to the output of the enhancements neurons. Finally, the weight w is calculated using the Moore–Penrose pseudo inverse as follow (Jiao et al. 2023; Nabih et al. 2023):

$${\text{w}} = {\text{B}}^{\dag } {\text{Z}}$$

(10)

More details about the RVFL can be found in (Pao et al. 1992; Pao et al. 1994; Jiao et al. 2023; Nabih et al. 2023).

2.2.5 Adaptive boosting (AdaBoost)

The adaptive Boosting (AdaBoost) developed by (Freund and Schapire 1997), is one of the most widely reported ensemble ML method. Similar to the ensemble models, AdaBoost is a series of weaker learners (i.e., trees) and each one take into account one subset. An important point to note is that, the data that are hardly predicted (i.e., complex) should weighted stronger, and during the training process of the AdaBoost, the weights are being reshuffled and consequently they are increased for the poor learned dataset: this is a ‘‘sequential procedure’’ (Truong et al. 2022; Saha et al. 2023; Zhu et al. 2023). This process is very important as it helps the poor learners improve their performances during the training and at the end, we can obtain a robust model by weighted averaging of the response of poor regressor. From a mathematical point of view, the AdaBoost model can be expressed as follow (Truong et al. 2022):

$${\updelta }_{{\text{M}}}\left({\text{x}}\right)=\sum_{{\text{m}}=1}^{{\text{M}}}{\mathrm{\varphi }}_{{\text{m}}}\left({\text{x}}\right)$$

(11)

where, δ_M(x) is the form training of the AdaBoost model, M corresponds to the number of iterations, φ_m(x) is the weak learner (Truong et al. 2022; Saha et al. 2023; Zhu et al. 2023).

2.2.6 Bootstrap aggregating (Bagg)

The bootstrap aggregating, i.e., Bagging (Bagg) is used for combining an ensemble of weak learners for composing a strong model which can help in decreasing the variance and avoiding the overfitting (Li et al. 2023b). However, an important point to note is, the Bagging model should be handled as being a suite of multiple similar learners arranged in parallel, and the final response is calculated as the average (Khozani et al. 2019). For simplicity, the initial dataset is divided into an ensemble of sub-training set using bootstrap sampling with replacement, this cause to have some sample available multiple time in the training set while other none, which involve looking to repeat this several times. From a mathematical point of view, if the training dataset is designated as δ and composed of an ensemble of pairs as follow (Nancy Jane et al. 2023; Sun 2023):

$$\updelta =\left[\left({{\text{x}}}_{1},{{\text{y}}}_{1}\right) ,\left({{\text{x}}}_{2},{{\text{y}}}_{2}\right) ,\dots , \left({{\text{x}}}_{{\text{m}}},{{\text{y}}}_{{\text{m}}}\right)\right]$$

(12)

Each individual subset is affected by an equal (1/M) weight, and consequently a specific weak learner is attributed to one subset, and based on this the error is calculated. The procedure of updating the weight is then started. If the example is correctly predicted or classified, its weight is reduced and vice versa, until the end of the training. The final response is calculated as the average (AVG) (Nancy Jane et al. 2023; Sun 2023).

2.3 Signal decomposition methods

In the present study, three signal decomposition algorithms were used, namely, the variational mode decomposition (VMD) (Dragomiretskiy and Zosso 2014), the empirical mode decomposition (EMD) (Huang et al. 1998), and the empirical wavelet transform (EWT) (Gilles 2013). The three algorithms were used for decomposing the input variables, i.e., T_w, BP, and Q, into several subcomponents. Thus, in EMD and VMD, the components are called intrinsic mode functions (IMF), while the components of the EWT are called multiresolution analysis (MRA) components (Bokde et al. 2020). An example of BP decomposition is provided in Fig. 2. According to Fig. 2, the BP signal depicted at the top of the Figure was decomposed into nine IMFs using the VMD and EMD, and nine MRA using the EWT. Furthermore, the extracted subcomponent were arranged from high frequency to low frequency. The number of extracted IMF is determined by trial and error, and in our present study, nine subcomponent was found to be sufficient and their aggregation have helped in providing excellent predictive accuracies. This process of decomposition make a very complicated signal very simpler (Rezaie-Balf et al. 2020). When the whole process of decomposition is finished, the obtained IMFs and MAR were aggregated and used as input variables of the models. The example trend shown in Fig. 3 is considered to demonstrate the decomposition process. If we consider that each input variable, i.e., the BP, T_w, and Q was decomposed into nine subcomponents, then, in total twenty-seven new input variables are used by the ML models. Theoretical description of these algorithms is given below. The flowchart of the overall modelling framework is depicted in Fig. 4.

Fig. 2

Barometric pressure (BP) signal decomposition using the EMD, VMD, and EWT

Fig. 3

The schematic diagram of the preprocessing signal decomposition with the aggregation of the intrinsic mode function (IMFs) with machine learning approaches

Fig. 4

Flowchart of the proposed modelling framework

According to Fig. 4, our investigation were oriented toward a deeply comparison between standalone single models and hybrid models. Thus, our study originality can be summarized as follow:

• To the best of the author’s knowledge, this is the first study in the literature focused on the application of machine leaning combined with signal decomposition for modelling TDG in river.

• The primary objective of the present paper is to demonstrate whether signal decomposition can be presented as a good and robust tool for predicting TDG.

• Fewer variables are used for modelling TDG, i.e., T_w, BP, and Q.

• Three robust signal decomposition are selected and compared, i.e., VMD, EMD and EWT.

2.3.1 Variational mode decomposition (VMD)

Variational mode decomposition (VMD) is a preprocessing signal decomposition developed by (Dragomiretskiy and Zosso 2014). The VMD is highly recognized by its capability to find the suitable nombre of modal decompositions tacking into account the actual situation (Qiao et al. 2022). The VMD is used for decomposing a specific signal into a series of intrinsic model function (IMF) components having a particular bandwidth and starting by the construction of a variational problem (Xiong et al. 2022). From a mathematical point of view, the VMD decomposes a signal f(t) into M narrowband IMFs as follow:

$$f\left(t\right)=\sum_{m=1}^{M}{u}_{m}\left(t\right)$$

(13)

In the above equation, the IMFs have the following characteristics (Netsanet et al. 2022):

• Each IMF has an envelope ${\mathrm{\varnothing }}_{m}\left(t\right)$ and phase ${A}_{m}\left(t\right)$ as follow:

  $${u}_{m}\left(t\right)={A}_{m}\left(t\right)cos\left({\varnothing }_{m}\left(t\right)\right)$$

  (14)

• The envelopes of all IMF are positive and will progress more slowly.

• An instantaneous frequency (${{\varnothing }{\prime}}_{m}\left(t\right)$) is attributed to each IMF and it should be concentrated around a central frequency (${w}_{m}$).

More details about the VMD can be found in (Dragomiretskiy and Zosso 2014).

2.3.2 Empirical mode decomposition (EMD)

Empirical mode decomposition is a preprocessing signal developed by (Huang et al. 1998). The EMD is used for analyzing nonlinear and smooth signal, and it works by decomposing an original signal into a series of intrinsic mode function (IMF) components. Each component (i.e., each IMF) becomes himself a new signal with respect to a particular frequency, without requiring prior knowledge, and it works by supposing that: (i) signal and noise are ‘’uncorrelated’’ and (ii) denoising the signal can be done by discarding lower order IMF signals (Li et al. 2023b). Suppose we have an original signal:

$${{\text{X}}}_{{\text{t}}}={\text{Signal}}\left({\text{t}}\right)+{\text{Noise}}\left({\text{t}}\right)=\sum_{{\text{k}}=1}^{{\text{K}}}{{\text{IMF}}}_{{\text{k}}}+{{\text{r}}}_{{\text{t}}}$$

(15)

where K denotes the number of extracted IMF and r_t is the residual. The EMD works by applying a ‘’sifting’’ process as follows (Li et al. 2023b; Shamaee and Mivehchy 2023):

• The mean envelope ${\updelta }_{{\text{avg}}}\left({\text{t}}\right)$ is calculated as follow:

  $${\updelta }_{{\text{avg}}}\left({\text{t}}\right)=\frac{{\updelta }_{{\text{max}}}\left({\text{t}}\right)+{\updelta }_{{\text{min}}}\left({\text{t}}\right)}{2}$$

  (16)

  where ${\updelta }_{{\text{max}}}\left({\text{t}}\right),{\updelta }_{{\text{min}}}\left({\text{t}}\right)$ are the maximum and minimum envelopes of the initial signal.

• Thus, the calculated mean envelope is then extracted from the original signal as follow:

  $$\upbeta \left({\text{t}}\right)={\text{X}}\left({\text{t}}\right)-{\updelta }_{{\text{avg}}}\left({\text{t}}\right)$$

  (17)

• Stop, control and check that the conditions are correctly satisfied. If ok, the first IMF is therefore obtained, if none, repeat:

  $$\upgamma \left({\text{t}}\right)=\upbeta \left({\text{t}}\right)$$

  (18)

• The first residual is therefore calculated and considered to become the original signal (X(t)):

  $${{\text{r}}}_{1}\left({\text{t}}\right)=\mathrm{ X}\left({\text{t}}\right)-\upgamma \left({\text{t}}\right)$$

  (19)

This process of decomposition and extraction shall continue to grow until the function become a “monotonic function”. Finally, we obtain the function as follow:

$${\text{x}}\left({\text{t}}\right)=\sum_{{\text{k}}=1}^{{\text{K}}}\upgamma \left({\text{t}}\right)+{\text{r}}\left({\text{t}}\right)$$

(20)

More details about the EMD can be found in (Huang et al. 1998).

2.3.3 Empirical wavelet transform (EWT)

The empirical wavelet transform (EWT) was developed by (Gilles 2013). This algorithm of decomposition is based on ‘’revolutionary’’ rather than the ‘’stochastic volatility’’ available in the data (Karbasi et al. 2022). The EWT can be used as robust decomposition algorithm for any nonlinear and non-stationary signal because it can select the ideal value of frequency (Rout et al. 2022). Briefly, using the EWT we can obtain, and ensemble of subcomponents called the multiresolution analysis (MRA) components, to make a reasonable extraction efficiency. This can be achieved by dividing any signal X (t) into suite of MRA in the range of ‘’frequency’’ domain, and keep by building wavelet ‘’band-pass’’ filters for each sub-interval (Ren et al. 2022). From a mathematical point of view, the EWT extract the sub-components using two particular functions: the empirical wavelet functions (${\widehat{\gamma }}_{n}\left(\omega \right)$) and the empirical scale function ${\widehat{\theta }}_{n}\left(\omega \right)$, expressed as follow (Peng et al. 2022b; Wang and Sheng 2022):

$${\widehat{\gamma }}_{n}\left(\omega \right)=\left\{\begin{array}{l}1 \\ cos\left[\frac{\pi }{2}\alpha \left(\frac{1}{2{\sigma }_{n}}\left|\omega \right|-{\omega }_{n}{+\sigma }_{n}\right)\right] \\ 0 \end{array}\begin{array}{l}if\, \left|\omega \right|\le {\omega }_{n}{-\sigma }_{n} \\ if \,{\omega }_{n}{-\sigma }_{n}\le \left|\omega \right|\le {\omega }_{n}{+\sigma }_{n}\\ otherwise\end{array}\right.$$

(21)

$${\widehat{\theta }}_{n}\left(\omega \right)=\left\{\begin{array}{l}\begin{array}{ll}1& if {\omega }_{n}{+\sigma }_{n}\le \left|\omega \right|\le {\omega }_{n+1}{-\sigma }_{n-1}\end{array}\\ \begin{array}{ll}cos\left[\frac{\pi }{2}\alpha \left(\frac{1}{2{\sigma }_{n+1}}\left|\omega \right|-{\omega }_{n+1}{+\sigma }_{n+1}\right)\right]& if\,{\omega }_{n+1}{-\sigma }_{n+1}\le \left|\omega \right|\le {\omega }_{n+1}{+\sigma }_{n+1}\end{array}\\ \begin{array}{ll}sin\left[\frac{\pi }{2}\alpha \left(\frac{1}{2{\sigma }_{n}}\left|\omega \right|-{\omega }_{n}{+\sigma }_{n}\right)\right] & {if \,\omega }_{n}{-\sigma }_{n}\le \left|\omega \right|\le {\omega }_{n}{+\sigma }_{n} \end{array} \\ \begin{array}{ll}1& otherwise\end{array} \end{array}\right.$$

(22)

The function α(x) ∈ L^k ([0, 1]) is an arbitrary function and expressed as follow (Gilles 2013):

$$\alpha \left(x\right)=\left\{\begin{array}{c}0\, x\le 0\\ 1\, x\ge 1\end{array}\right. and \alpha \left(x\right)+\alpha \left(1-x\right)=1\forall\, x\in \left[\mathrm{0,1}\right]$$

(23)

where $\left(\omega \right)$ is the n-th maxima of the Fourier spectrum (Peng et al. 2022b; Ren et al. 2022; Wang and Sheng 2022). More details about EWT can be found in (Gilles 2013).

2.4 Performance assessment of the models

The performances of all models developed in the present study were evaluated using root mean square error (RMSE), mean absolute error (MAE), coefficient of determination (R²), and Nash–Sutcliffe efficiency (NSE) (Yaseen 2021), calculated as follow:

$$RMSE=\sqrt{\frac{1}{N}{\sum }_{i=1}^{N}[({TDG}_{obs}{)}_{i}-({TDG}_{est}{)}_{i}{]}^{2}},(0\le RMSE<+\infty )$$

(24)

$$MAE=\frac{1}{N}{\sum }_{i=1}^{N}|({TDG}_{obs}{)}_{i}-({TDG}_{est}{)}_{i}| , (0\le MAE<+\infty )$$

(25)

$${R}^{2}={\left[\frac{\sum_{i=1}^{N}\left({TDG}_{obs,i}- \overline{{TDG }_{obs}}\right) \left({TDG}_{est,i}- \overline{{TDG }_{est}}\right) }{\sqrt{\sum_{i=1}^{N}({{TDG}_{obs,i}- \overline{{TDG }_{obs}})}^{2} \sum_{i=1}^{N}({{TDG}_{est,i}- \overline{{TDG }_{est}})}^{2} } }\right]}^{2}$$

(26)

$$NSE=1-\left[\frac{{\sum }_{i=1}^{N}[{TDG}_{obs}-{TDG}_{est}{]}^{2}}{{\sum }_{i=1}^{N}[{TDG}_{obs}-\overline{{TDG}_{obs}}{]}^{2}}\right],(-\infty <NSE\le 1)$$

(27)

where ${\overline{TDG} }_{obs}$ and ${\overline{TDG} }_{est}$ are the mean measured and mean forecasted TDG, respectively, and TDG_obs and TDG_est specifies the observed and forecasted total dissolved gas for ith observations, and N shows the number of data points.

2.5 Models development

In this study, hybrid predictive models based on the EMD, VMD and EWT algorithms and ML methods are developed to predict the TDG in rivers. To build the models, different scenarios are considered. First, the six ML models, i.e., the MLPNN, RVFL, RFR, GPR, AdaBoost and Bagg were applied using three input variables (i.e., T_w, BP and Q). Second, the same models were combined with the EMD, VMD and EWT and further compared. The models’ accuracies are assessed using model efficiency indices, including MAE, RMSE, R and NSE. Tables 3, 4, 5 and 6 show the performances of the developed models for the four stations. It is noteworthy that the models designated as MLPNN, RVFL, RFR, GPR, AdaBoost and Bagg correspond to the single models without decomposition. The MLPNN_EWT, RVFL_EWT, RFR_EWT, GPR_EWT, AdaBoost_EWT and Bagg_EWT correspond aux hybrid models based on the EWT signal decomposition. Similarly, the MLPNN_VMD, RVFL_VMD, RFR_VMD, GPR_VMD, AdaBoost_VMD and Bagg_VMD correspond aux hybrid models based on the VMD signal decomposition. Finally, the MLPNN_EMD, RVFL_EMD, RFR_EMD, GPR_EMD, AdaBoost_EMD and Bagg_EMD correspond aux hybrid models based on the EMD signal decomposition. Details of the obtained results are depicted and discussed hereafter, and only model’s performances during the validation stage were highlighted and compared.

3 Experimental results

3.1 USGS 14019240 station modeling results

Table 3 presents the results of TDG prediction using all models at the USGS 14019240 station. The values stated in Table 3 clearly demonstrated that using single models, the RFR and Bagg achieved the high prediction efficiencies among all the developed models based on the four performances metrics (RMSE≈2.706, MAE≈1.922, R²≈0.901, NSE≈0.900). In general, the two models MLPNN and GPR appear to have performed better than the RVFL and AdaBoost models based on the different efficiency indices and the differences between the two is negligible. However, it is worth nothing that the RVFL was the poorest one and they worked with a moderate degree of accuracy (RMSE≈4.223, MAE≈3.345, R²≈0.757, and NSE≈0.756). In overall, obtained results reported in Table 3 confirm the good level of efficiency of the developed single ML models without including the signal decomposition algorithms and more importantly, using only fewer inputs variables, i.e., T_w, BP, and Q. Figure 5 displays the scatterplot of measured and predicted TDG data for the USGS 14019240 station. In accordance with the depicted data points, it is evident that the two models RFR and Bagg appear to predict TDG more accurately and with more precision than the other models, and this is reflected by the data, which are less scattered, while the RVLF model was characterized by a high-scattered data. According to Table 3 and Fig. 5 (i.e., the scatterplot), there is a remarkable improvement of the model’s performances gained using the three signals decomposition algorithms. Using the EWT algorithm, it is clear that all models show their numerical performances improved by an increase of the R² and NSE values and a decrease of the RMSE and MAE indices. Using the R²-values as a basis for comparison, it is clear that the MLPNN_EWT, Bagg_EWT, GPR_EWT, and RFR_EWT model’s performances were similar with negligible difference, and in the case of Bagg_EWT, and RFR_EWT, the equality between the two is obvious. The comparison between single and hybrids models is further highlighted using Taylor diagram (Fig. 6), for which the superiority of the hybrid models is obvious.

Table 3 Performances of different models at the USGS 14019240 station

Fig. 5

Scatterplot of measured versus predicted TDG concentration using all models for the USGS 14019240 station

Fig. 6

Taylor diagram for models comparison using the validation dataset: USGS 14019240 station

Although the hybrid models exhibit a statistically similar performance (e.g., statistically similar values for the RMSE, MAE, R² and NSE), the GPR_EWT model seems to be slightly more accurate showing the biggest R² (≈0.994) and NSE (≈0.993) values, and the lowest RMSE (≈0.707) and MAE (≈0.477) values. Overall, the EWT algorithm helped in improving the performances of MLPNN_EWT, AdaBoost_EWT, Bagg_EWT, GPR_EWT, and RFR_EWT by ≈72.816%, ≈52.954%,≈65.950%,≈77.612%, and ≈66.112% in terms of RMSE metric, and by ≈73.092%, ≈56.671%,≈69.388%,≈79.033%, and ≈69.615% in terms of MAE metric, respectively. However, the improvement of the RVFL_EWT model is less than the other models and does not exceed the ratios of ≈8.217% and ≈5.441% in terms of RMSE and MAE metrics, respectively. Using the VMD algorithm as reported in Table 3 and Fig. 5 (i.e., the scatterplot), it is clear that, for all models, the improvements gained are less than those obtained from the EWT. Furthermore, using the VMD, it clear that, only the Bagg_VMD and the RFR_VMD have guaranteed great improvement (e.g., ≈50.797%, ≈50.702% for the RMSE, and ≈51.125%, ≈51.145% for the MAE). No improvement was recorded for the GPR_VMD model, while using the RVFL_VMD, a decrease in terms of models performances was recorded. More precisely, using the EWT algorithm, the performances of the RVFL model were decreased by ≈4.780% and ≈8.079% in terms of RMSE and MAE metrics, the performances of the GPR model remains constant, the performances of the MLPNN_VMD and AdaBoost_VMD were improved by ≈34.758%, ≈23.183% for the RMSE and by ≈23.183%, ≈23.220% for the MAE, respectively. Finally, in the case of the EMD signal decomposition algorithm as reported in Table 3 and Fig. 5 (i.e., the scatterplot), the performances evaluation metrics RMSE and MAE showed an improvement (i.e., decrease) by more than ≈60% and ≈65% for all models except the RVFL_EMD for which the improvement rate has note exceeded ≈35% and ≈36%, respectively. Furthermore, the higher improvement rates were achieved using the GPR_EMD model with ≈72.293% and ≈73.714% in term of RMSE and MAE, and it is clear that, the GPR_EMD model was the only one for which high R² and NSE values were obtained, i.e., ≈0.995 and ≈0.990, respectively. It can be seen that, beyond the RVFL model for which a negligible improvement was gained using the signal decomposition algorithms, all other models were significantly improved and the percentage of improvement is higher for the EMD than that for the VMD and EWT. Furthermore, there is not yet any clear superiority concerning all performances metrics.

3.2 USGS 13341000 station modeling results

Table 4 presents the results of TDG prediction using all model’s at the USGS 13341000 station. The single RVFL model tend to have moderate performances having an R² and NSE values of approximately ≈ 0.656 and ≈ 0.626, respectively, while the RVFL_VMD tend to have the worst performances with R², NSE, RMSE, and MAE of approximately ≈ 0.335, ≈ 0.336, ≈ 3.050 and ≈ 2.373, respectively. More in depth analysis of the numerical results reported in Table 4 revealed that, the Bagg and the RFR models show the same numerical performances, and they were more accurate compared to the other models exhibiting an R² and NSE values of approximately ≈ 0.906 and ≈ 0.902. Furthermore, the RMSE and MAE values for the single models ranged from ≈ 1.156 to ≈ 2.269 and from ≈ 0.818 to ≈ 1.726, respectively, while the R² and NSE values were in the range of ≈ 0.656 to ≈ 0.906 and ≈ 0.626 to ≈ 0.903, respectively. Among all three decomposition algorithms, it is clear that, the most accurate simulation of the TDG was obtained using the EWT algorithm, followed by the EMD, while the VMD was ranked last. The comparison between the singles and hybrids models was also conducted using the scatterplot (Fig. 7) and the Taylor diagram (Fig. 8). It is clear from the figures that, the hybrid models exhibited high numerical performances.

Table 4 Performances of different models at the USGS 13341000 station

Fig. 7

Scatterplot of measured versus predicted TDG concentration using all models for the USGS 13341000 station

Fig. 8

Taylor diagram for models comparison using the validation dataset: USGS 13341000 station

Comparison of the models one by one revealed that, the GPR_EWT was the most accurate in terms of R², NSE, RMSE, and MAE, with values of ≈ 0.978, ≈ 0.978, ≈ 0.548 and ≈ 0.392, respectively. The GPR_EWT was slightly higher than the GPR_EMD who provided the values of ≈ 0.976, ≈ 0.976, ≈ 0.579 and ≈0.401, respectively, while the RVFL_VMD model was the worst among all hybrid models for which the values of ≈ 0.335, ≈ 0.336, ≈ 3.050 and ≈ 2.373 were obtained. In this regard, it is important to note that, the performances of the RVFL model were significantly decreased using the EMD, VMD, and EWT algorithms, and the AdaBoost benefits less from the these three algorithms. Regarding the RVFL model, it is clear that, the VMD algorithm leads to a slightly decrease of the model’s performances for which the R² and NSE values were dropped from (≈ 0.656 and ≈ 0.626) to (≈ 0.335 and ≈ 0.336), while the RMSE and MAE were raised from (≈ 2.269 and ≈ 1.726) to (≈ 3.050 and ≈ 2.373), respectively. Among all hybrid model’s as stated in Table 4, it is clear that, the best predictive accuracies was gained using the hybrid GPR_EWT for which the R², NSE, RMSE, and MAE were immediately increased to reach their extreme values with improvement rates of ≈ 10.020%, ≈10.123%, ≈ 57.552%, and ≈ 57.804%, respectively. The GPR_EWT was followed by the MLPNN_EWT with R², NSE, RMSE, and MAE values of ≈ 0.972, ≈ 0.972, ≈ 0.616, and ≈ 0.454, respectively. Thus, the improvement observed using the EWT was more obvious, followed by the EMD and the VMD in the last round.

3.3 USGS 14019220 station modeling results

Table 5 presents the results of TDG prediction using all model’s at the USGS 14019220 station. As clearly stated in Table 5, as we combine the signal decomposition with the ML models, an improvement can be seen in all models except: the AdaBoost_EWT, the GPR_VMD and the RVFL_VMD. More precisely, using the EWT algorithm, the R² and NSE values of the AdaBoost were dropped from (≈ 0.827 and ≈ 0.836) to (≈ 0.815 and ≈ 0.810), while the RMSE and MAE were raised from (≈ 1.745 and ≈1.377) to (≈ 1.881 and ≈ 1.340). According to Table 5, using single model’s it is clear that the Bagg and the RFR were the most accurate model’s exhibiting the higher R² and NSE values of ≈ 0.901 and ≈ 0.899, respectively, and the lowest RMSE and MAE value of ≈ 1.360 and ≈ 1.021, respectively. Among all single model’s, the RVFL was found to be the poorest one having the lowest numerical performances with R² ≈ 0.773, NSE ≈ 0.772, RMSE ≈ 2.060 and MAE≈1.629, respectively.

Table 5 Performances of different models at the USGS 14019220 station

Further comparison between the hybrid model’s based on the EWT algorithm revealed that, the GRP_EWT was the most accurate model having the greatest numerical performances (R² ≈ 0.992, NSE ≈ 0.992, RMSE ≈ 0.394, MAE ≈ 0.297), respectively, slightly higher than the MLPNN_EWT, while the RVFL_EWT (R² ≈ 0.808, NSE ≈ 0.808, RMSE ≈ 1.889, MAE ≈ 1.502) was the less accurate model. Using the VMD algorithm, it is clear from the results reported in Table 5 that, the Bagg_VMD and the RFR_VMD were the most accurate model’s compared to the all other and they exhibited high numerical performances of approximately (R² ≈ 0.972, NSE ≈ 0.972, RMSE ≈ 0.723, MAE ≈ 0.507). Finally, using the EMD algorithm, it is clear that the GPR_EMD was the most accurate with the biggest numerical performances of approximately (R² ≈ 0.984, NSE ≈ 0.984, RMSE ≈ 0.540, MAE ≈ 0.370).

The Scatterplot (Fig. 9) and Taylor diagram (Fig. 10) have helped in providing a concise and solid comparison between the singles and hybrids models. According to Fig. 9, it is clear that the comparison between measured and predicted data demonstrated that the hybrids models exhibited less scattered data compared to the single models.

Fig. 9

Scatterplot of measured versus predicted TDG concentration using all models for the USGS 14019220 station

Fig. 10

Taylor diagram for models comparison using the validation dataset: USGS 14019220 station

3.4 USGS 13352950 station modeling results

Table 6 presents the results of TDG prediction using all models at the USGS 13352950 station. The performances of the singles and hybrid models were assessed through a comparative analysis based on the performances metrics reported in Table 6. First, the models were arranged from the highest (R² and NSE) to the lowest (RMSE and MAE), and consequently the models were arranged as follow: RFR(1), Bagg(2), GPR(3), MLPNN(4), AdaBoost(5) and RVFL(6). It is clear that, the RFR and Bagg revealed the best performances by presenting the lowest RMSE and MAE values of ≈ 1.914 and ≈ 1.400, respectively, and the highest R² and NSE values of ≈ 0.891 and ≈ 0.891, respectively. The RVFL model presented the worst results compared to all other models with R², NSE, RMSE, and MAE values of ≈ 0.724, ≈ 0.722, ≈ 3.061 and ≈ 2.410, respectively. Moreover, according to the previous results, we can conclude that, obtained results demonstrated that RFR and Bagg would be more feasible for modelling TDG and possess more capability of nonlinear mapping, followed by the GPR and the MLPNN models.

Table 6 Performances of different models at the USGS 13352950 station

The previous statement was further confirmed by analyzing the results obtained using the VMD, EMD and EWT signal decomposition. According to Table 6, the VMD, EMD and EWT further improve the model’s performances, which lead to a great improvement with the involvement of these techniques in overcoming some limitations of the standalone models, especially, the capabilities in capturing the nonlinearity present in the input variables. As results of this, we can see that, the R² and NSE metrics reached extreme values of ≈ 0.996 and ≈ 0.995 (i.e., the maximal values obtained at the 13,352,950 station) obtained using the GPR_EWT accompanied by an improvements rate of ≈ 80% and ≈ 80.44% in terms of RMSE (≈ 0.405) and MAE (≈ 0.303) reduction compared to the single GPR model. Notably, the EWT is the sole algorithm for which the performances of all-single models were improved. We can note that the MLPNN_EWT was slightly lower than the GPR_EWT with negligible difference in terms of model’s performances, while the Bagg_EWT and RFR_EWT worked equally. By further analysis of the obtained results, the following conclusions can be drawn. First, the GPR_EMD yielded the best performances (equally with the MLPNN_EMD) compared to the other models, while the AdaBoost_EMD was the worst model. More precisely, the R² and NSE values of ≈ 0.992 and ≈ 0.990 were obtained using the GPR_EMD and MLPNN_EMD against ≈ 0.839, and ≈ 0.828 obtained using the AdaBoost_EMD. Second, the improvement gained using the VMD was the least significant and two models have shown their performances decreased compared to the single models, i.e., the GPR_VMD and the RVFL_VMD, for which the R² and NSE values were dropped from (≈ 0.880 and ≈ 0.879) to (≈ 0.859 and ≈ 0.858), and from (≈ 0.724 and ≈ 0.722) to (≈ 0.694 and ≈ 0.693), respectively. Yet, it is important to note that, high accuracy was obtained using the MLPNN_VMD, RFR_VMD, and Bagg_VMD, for which the R² and NSE values have raised the values of ≈ 0.978 and ≈ 0.970, respectively. The scatterplot of measured and calculated TDG using all models are depicted in Fig. 11. The Taylor diagram in Fig. 12 is presented for further highlighting the superiority of the hybrid models compared to the single models.

Fig. 11

Scatterplot of measured versus predicted TDG concentration using all models for the USGS 13352950 station

Fig. 12

Taylor diagram for models comparison using the validation dataset: USGS 13352950 station

4 Summary and remarks

It is clear from the above discussed results that the proposed hybrid models based on signal decomposition are valuable for modelling TDG concertation. Meanwhile, the superiority, robustness and effectiveness of the VMD, EMD and EWT algorithms were justified and validated by a deeply comparison of the singles and hybrid models performances. For further discussion and comparison of the efficiency of the proposed modelling framework based on signal decomposition, previous models available in the literature were compared with our models and the comparison is presented hereafter.

A research study used the ELM and SVR models for predicting TDG concentration using several input variables namely, T_w, BP, Q and gage height (GH) (AlOmar et al. 2020). However, in their study, they have forecasted the TDG one hour in advance (t + 1), using the input variables measured at the previous lag time (t − 1). From the obtained results, the SVR was slightly more accurate and exhibiting an R-value of ≈ 0.992, compared to the values of ≈ 0.990 obtained using the ELM model. In comparison to our results, it is clear the SVR and ELM have provided the same performances obtained in our present study, however, the inclusion of the TDG measured at (t − 1) have certainly contributed to the improvement of the ELM and SVR models performances. A research used the SVR, ELM, genetic algorithm ELM (GA-ELM), and GA-SVR for modelling TDG measured at three dams’ reservoir stations (Wang and Sheng 2022). They used a large number of input variables namely, Q, BP, T_w, discharge per unit width (q), upstream and downstream water level difference (ΔH), bubble pressure(F) in stilling basin, and retention time (TS). Form the obtained results, it was found that, GA-SVR (RMSE ≈ 1.43, MAE ≈ 0.95) was more accurate compared to GA-ELM (RMSE ≈ 1.48, MAE ≈ 1.31). However, the two models, i.e., the GA-ELM and GA-SVR were significantly less accurate compared to the models proposed in the present study, for which an RMSE and MAE of ≈ 0.394 and ≈ 0.297 were obtained using the GPR combined with the EWT, EMD and VMD algorithms. Another study used the MLPNN model for modelling TDG using sensor depth (SD), Q, BP, T_w, spill from dam (SFD), and water elevation (WL) (Han et al. 2019). It was found that; the MLPNN was more accurate compared to the MLR model exhibiting R, NSE, RMSE, and MAE values of ≈ 0.970, ≈ 0.930, ≈ 2.05 and ≈ 1.22, respectively, which were less accurate compared to models developed in the present study (R ≈ 0.996, NSE ≈ 0.992, RMSE ≈ 0.394 and MAE ≈ 0.297).

Recently, a scientific research introduced a new modelling strategy using the parallel chaos search based incremental extreme learning machine (PC-ELM) for predicting hourly TDG using only water temperature (T_w) as predictor (Heddam 2023). The PC-ELM model was tested using data from four station located at Snake River, USA, and operated by the USGS. For improving the performance of the PC-ELM, the T_w was combined with the periodicity, i.e., the year, month, day, and hour number. It was found that; the PC-ELM was slightly more accurate compared to the standalone ELM, exhibiting an R² value of approximately ≈ 0.965, however, the PC-ELM is less accurate compared to the hybrid models reported in our present study (R² ≈ 0.990). In an investigation a research article compared between adaptive neuro-fuzzy inference systems (ANFIS) and dynamic evolving neural-fuzzy inference system (DENFIS) for modelling hourly TDG measured in two USGS stations (Heddam and Kisi 2021). The two ANFIS and DENFIS models were developed using Q, BP, T_w, and SFD. The ANFIS model was found to be more accurate with R, NSE, RMSE and MAE values of ≈ 0.977, ≈ 0.954, ≈ 1.084 and ≈ 0.773, compared to the values of ≈ 0.968, ≈ 0.936, ≈ 1.271 and ≈ 0.868, obtained using the DENFIS model. By comparison with the present research, the DENFIS and ANFIS models were less accurate compared to the hybrid models proposed in our study. Another study used the generalized regression neural network (GRNN) with the MLR model for predicting TDG using a large number of predictors namely, Q, BP, T_w, SFD, SD (Heddam 2017). Good performances were obtained using the GRNN model with R, NSE, RMSE and MAE values of ≈ 0.946, ≈ 0.895, ≈ 0.995 and ≈ 0.593, respectively, however, they are significantly lower than the values obtained in this study. In another study, the authors compared between kriging interpolation method (KIM), response surface method (RSM), and the MLPNN models for predicting TDG using Q, BP, T_w, and SFD (Heddam and Kisi 2020). It was found that he proposed KIM was more accurate compared to the MLPNN and RSM, and excellent performances were obtained with R, NSE, RMSE and MAE values of ≈ 0.973, ≈ 0.941, ≈ 1.462 and ≈ 1.122, respectively, always less than the values obtained using the hybrid models proposed in the present study. Finally, in a research the authors applied the high-order response surface method (H-RSM), M5Tree, least squares support vector machine (LSSVM), and multivariate adaptive regression spline (MARS) for modelling TDG, showing the superiority of the H-RSM model with R, NSE, RMSE and MAE values of ≈ 0.965, ≈ 0.931, ≈ 1.456 and ≈ 1.022, respectively [31].

5 Conclusion

Total dissolved gas (TDG) produced at high dam reservoir was regressed against water temperature, barometric pressure and discharge measurements from four USG stations to explore whether ML algorithms were able to accurately predict TDG. In addition to this, the objective was to determine how signal decomposition approaches and conventional ML could be combined to improve the predictive accuracy. The results showed that the modelling using single models without signal decomposition results in TDG estimates with a good predictive accuracy for all models. However, the RVFL and AdaBoost have provided moderate results. In addition, the performances of the models varied from one station to another and from one model to another and no general conclusion could be drawn. The second part of the study was mainly motivated by the requirement of high-quality TDG estimations, which is crucial to efficiently control water resources and aquatic life. We looked at where could make the greatest gains on reducing the predictive errors between measured and predicted TDG. For answering this query, we have deeply analyzed the potential that can be gained from the signal decomposition algorithms. Three algorithms were then tested, i.e., the VMD, EMD, and EWT. Thus, hybrid models were used and compared to the single models, providing consistent estimations for the major’s cases and for all stations. We argued that it was essential to introduce new robust tools for better predicting of TDG, which is successfully done in the present study, the combined ML and signal decomposition reveals reliability and relevance of the TDG estimations. Future research may be focused on the application of the proposed models for large dataset and by testing other ML models; in addition, testing the models with other input variables could be an innovative idea.