Water-Level Forecasting Based on an Ensemble Kalman Filter with a NARX Neural Network Model ^†

Abstract

It is fundamental, yet challenging, to accurately predict water levels at hydrological stations located along the banks of an open channel river due to the complex interactions between different hydraulic structures. This paper presents a novel application for short-term multivariate prediction applied to hydrological variables based on a multivariate NARX model coupled to a nonlinear recursive Ensemble Kalman Filter (EnKF). The proposed approach is designed for two hydrological stations of the Atrato river in Colombia, where the variables, water level, water flow, and water precipitation, are correlated using a NARX model based on neural networks. The NARX model is designed to consider the complex dynamics of the hydrological variables and their corresponding cross-correlations. The short-term two-day water-level forecast is designed with a fourth-order NARX model. It is observed that the NARX model coupled with EnKF improves the robustness of the proposed approach in terms of external disturbances. Furthermore, the proposed approach is validated by subjecting the NARX–EnKF coupled model to five levels of additive white noise. The proposed approach employs metric regressions to evaluate the proposed model by means of the Root Mean Squared Error (RMSE) and the Nash–Sutcliffe model efficiency (NSE) coefficient.

1. Introduction

Accurate modeling of hydrologic variables is crucial for effective flood forecasting and the design and operation of water resource systems [1]. A common method to describe the hydrological model is by using artificial neural networks (ANN) [2]. In [3], an ANN with a multi-layer perceptron structure is used to design a flood prediction model for short-term forecasts based on water-flow hydrological variables. In [4], an ANN is used to develop a flood monitoring system with flow and water-level measurements. Also, in [5], a convolutional neural network (CNN) is used to predict time series variables such as the water level in a flood model. For example, in [6], water-level forecasting is presented using a nonlinear autoregressive model with exogenous inputs (NARX) and extreme learning machine (ELM) neural networks, where a reduced number of exogenous variables is used. Also, in [7], a multivariate hydrological model is presented based on a NARX model, where water level, water flow, and water precipitation variables are correlated, and a short-term forecast is obtained. In addition, [8] presented a NARX structure for groundwater-level forecasts, where short-term and mid-term forecasts are achieved. And, in [9], a system for predicting flood levels is developed based on real-time sensor data.

The ANN can also be used to improve the performance of hydrological models based on physical variables. Such is the case of the research published, which explores how LSTM can be applied in the field of hydrology to predict water levels [10], which is crucial for water resource management, flood prevention, and infrastructure planning. Other published research explores an innovative approach that integrates neural networks and Kalman smoothing techniques [11] to model and predict water levels in a specific river, which is crucial for water resource management, infrastructure planning, and flood prevention.

The hydrological models can also be improved by using optimal filters for data assimilation, such as the Kalman Filter (KF), the Unscented Kalman Filter (UKF), or the Ensemble Kalman Filter (EnKF). The Ensemble Kalman Filter (EnKF) has made a contribution to the prediction of water levels [12]. The EnKF is used to improve the performance of hydrological models; the coupled model [13], a UKF, and a recurrent neural network are combined for long-term flood forecasting. In [14], the EnKF improves water-level simulation in complex river networks. In [15], the EnKF is used to improve the forecasting capability of flood models. Moreover, in [16], the improvement of an ANN is presented by using the KF for water-level forecasting, and in [17], the ANN is improved by using the EnKF by using a hybrid algorithm for water-level forecasting. Finally, in [18], the UKF and EnKF are tested for improvement of a lowland hydrologic model, and in [19], the UKF and support vector regression (SVR) are used to improve the predictions on a lake. The scientific community has employed modeling methods or techniques that can be used to improve water management and reduce water losses in the system.

This paper presents a novel application for multivariate short-term forecasting for hydrological variables based on a multivariate NARX model coupled to an EnKF. The importance of the hybrid NARX–EnKF model for water-level prediction lies in its ability to handle the complexity and uncertainty of hydrological systems. The NARX model (Nonlinear Autoregressive Model with Exogenous Inputs) is effective in capturing the nonlinear relationships between multiple hydrological variables, such as water level, flow, and precipitation. This allows the model to make accurate short- and medium-term predictions, considering the internal dynamics of the system. To this end, two hydrological stations are considered; these stations are located in the Atrato River in Colombia, where the hydrological variables of water level, water flow, and water precipitation are considered. It is worth noting that the hydrological stations are monitored by the Institute of Hydrology, Meteorology, and Environmental Studies (IDEAM) of Colombia. A multivariable NARX structure is used to describe the hydrological model by using data measurements. The data include measurements that are sampled every 12 h during a period of 789 days. In order to evaluate the performance of the proposed approach, a comparison analysis is developed by considering several additive noise disturbances and the NARX forecasting with and without the EnKF. The performance is evaluated in terms of the Mean-Square Error (RMSE) and the Nash–Sutcliffe model efficiency coefficient (NSE).

2. Theoretical Framework

2.1. NARX Neural Network Model for Hydrological-Level Forecasting

The dynamics of the hydrological variables are defined by considering the following:

$$\begin{array}{c}\hfill {\mathit{y}}_k=f\left({\mathit{x}}_k\right)+{\mathit{ϵ}}_k\end{array}$$

(1)

where n is the order of the NARX model, $f(.)$ the nonlinear function, and ${ϵ}_k$ the additive noise at time instant k, where ${\mathit{x}}_k$

$$\begin{array}{c}\hfill {\mathit{x}}_k=\left[\begin{array}{c}{y}_{k-1}\\ ⋮\\ y_{k-n}\\ u_{k-1}\\ ⋮\\ u_{k-n}\end{array}\right]\end{array}$$

(2)

Consider the following state-space nonlinear model of hydrological activity:

$$\begin{array}{cc}\hfill {\mathit{x}}_{k+1}& =\left[\begin{array}{c}f\left({\mathit{x}}_k\right)\\ \left[\begin{array}{ccccc}I& 0& \dots & 0& 0\\ 0& I& \dots & 0& 0\\ 0& 0& \dots & 0& 0\\ 0& \dots & I& 0& 0\\ 0& \dots & 0& I& 0\end{array}\right]\end{array}\right]+\left[\begin{array}{c}0\\ ⋮\\ 0\\ I\\ ⋮\\ 0\end{array}\right]{\mathit{u}}_k+{\mathit{\eta }}_k\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\mathit{x}}_{k+1}& =\mathit{F}({\mathit{x}}_k,{\mathit{u}}_k)+{\mathit{\eta }}_k\hfill \end{array}$$

(4)

In order to consider the NARX model of (1), the inputs are selected as $u[k-j]$ and $y[k-j]$, with $j=1,\dots ,n$, which correspond to a $n-th$ order model. In this work a $4-$th order model ($n=4$) is considered according to [7], where an analysis of the order selection is performed and the lowest estimation error is obtained for the $3-rd$ order model or higher. Therefore, by considering the variables described in (1) and (2), the proposed NARX model consists of 24 inputs and 2 outputs.

In order to approximate the nonlinear function of (1), the nonlinear function $\mathit{f}(.)$ is approximated by using a neural-network structure $f^{*}(.)$ where the NARX model can be defined as follows:

$$\begin{array}{c}\hfill y\left[k\right]=f^{*}\left({\mathit{x}}_k\right)+{\mathit{ϵ}}_k\end{array}$$

(5)

To this end, 24 input activation functions with one hidden layer and 2 outputs are considered. A feed-forward network is selected as a candidate for the NARX model in order to speed up the training process. The training of the NARX model is performed offline by considering the data sample.

For better understanding, a flowchart is presented using a NARX neural-network structure, coupled with an EnkF filter and decoupled as shown in Figure 1.

2.2. EnKF Forecasting

The state estimation ${\mathit{x}}_k$ can be obtained by applying the EnKF in two stages [20]. First, the forecast stage [21], where an ensemble [22] of q forecasted state estimates is computed

$$\begin{array}{c}\hfill {\mathit{x}}_k^{f_i}=\mathit{F}({\overline{\mathit{x}}}_k^a,{\mathit{u}}_k)+{\mathit{\omega }}_k^i\end{array}$$

(6)

The state transition function in the NARX–ENKF model is given by the following:

$$\mathbf{F}({\overline{\mathbf{x}}}_k^a,{\mathbf{u}}_k)$$

where ${\mathit{x}}_k^a$ represents the a posteriori state estimate at time step k, obtained after data assimilation [23] using the ENKF. ${\mathit{u}}_k$ is the external input to the system, which may include variables such as precipitation, river discharge, or water level, where $i=1,\dots ,q$,${\mathit{\omega }}_k^i\in {\mathbb{R}}^{n\times 1}$ is a zero-mean random variable with normal distribution and covariance ${\Omega }_k\in {\mathbb{R}}^{n\times n}$. The sample error covariance matrix computed from ${\mathit{\omega }}_k^i$ converges to ${\Omega }_k$ as $q\to \infty$. The forecasted measurement is given by the following:

$$\begin{array}{c}\hfill {\mathit{y}}_k^{f_i}={\mathbf{f}}^{*}\left({\mathit{x}}_k^{f_i}\right)\end{array}$$

(7)

where $f_i$ denotes the $i-$th member of the forecast ensemble. Then, the ensemble mean for states ${\overline{\mathit{x}}}_k^f\in {\mathbb{R}}^{n\times 1}$ is defined by

$$\begin{array}{c}\hfill {\overline{\mathit{x}}}_k^f={\displaystyle \frac{1}{q}}\sum _{i=1}^q{\mathit{x}}_k^{f_i}\end{array}$$

(8)

and the ensemble mean for measurements ${\overline{\mathit{y}}}_k^f$ is defined by

$$\begin{array}{c}\hfill {\overline{\mathit{y}}}_k^f={\displaystyle \frac{1}{q}}\sum _{i=1}^q{\mathit{y}}_k^{f_i}\end{array}$$

(9)

The ensemble error matrix around the ensemble mean is defined as

$$\begin{array}{c}\hfill {\mathit{E}}_k^f=\left[\begin{array}{ccc}{\mathit{x}}_k^{f_1}-{\overline{\mathit{x}}}_k^f& ···& {\mathit{x}}_k^{f_q}-{\overline{\mathit{x}}}_k^f\end{array}\right]\end{array}$$

(10)

where ${\mathit{E}}_k^f\in {\mathbb{R}}^{n\times q}$, and the ensemble output error is

$$\begin{array}{c}\hfill {\mathit{E}}_{y_k}^f=\left[\begin{array}{ccc}{\mathit{y}}_k^{f_1}-{\overline{\mathit{y}}}_k^f& ···& {\mathit{y}}_k^{f_q}-{\overline{\mathit{y}}}_k^f\end{array}\right]\end{array}$$

(11)

where ${\mathit{E}}_{y_k}^f\in {\mathbb{R}}^{d\times q}$, and where the forecast covariances are approximated as

$$\begin{array}{cc}\hfill {\mathit{P}}_k^f& ={\displaystyle \frac{1}{q-1}}{\mathit{E}}_k^f{\left({\mathit{E}}_k^f\right)}^{\top }\hfill \\ \hfill {\mathit{P}}_{x{y}_k}^f& ={\displaystyle \frac{1}{q-1}}{\mathit{E}}_k^f{\left({\mathit{E}}_{y_k}^f\right)}^{\top }\hfill \\ \hfill {\mathit{P}}_{y{y}_k}^f& ={\displaystyle \frac{1}{q-1}}{\mathit{E}}_{y_k}^f{\left({\mathit{E}}_{y_k}^f\right)}^{\top }\hfill \end{array}$$

where ${\mathit{P}}_k^f\in {\mathbb{R}}^{n\times n}$, ${\mathit{P}}_{x{y}_k}^f\in {\mathbb{R}}^{n\times d}$, and ${\mathit{P}}_{y{y}_k}^f\in {\mathbb{R}}^{d\times d}$.

The second stage is the analysis stage, where an ensemble of perturbed observations ${\mathit{y}}_k^i$ is obtained as follows:

$$\begin{array}{c}\hfill {\mathit{y}}_k^i={\mathit{y}}_k+{\mathit{\upsilon }}_k^i\end{array}$$

(12)

where ${\mathit{\upsilon }}_k^i\in {\mathbb{R}}^{d\times 1}$ is a zero-mean random variable with normal distribution and covariance ${\mathbf{{\rm Y}}}_k\in {\mathbb{R}}^{d\times d}$. The sample error covariance matrix computed from ${\mathit{\upsilon }}_k^i$ converges to ${\mathbf{{\rm Y}}}_k$ as $q\to \infty$. Also, the EnKF gain matrix is approximated as

$$\begin{array}{c}\hfill {\mathit{K}}_k={\mathit{P}}_{x{y}_k}^f{\left({\mathit{P}}_{y{y}_k}^f\right)}^{-1}\end{array}$$

(13)

where it is worth noting that ${\left({\mathit{P}}_{y{y}_k}^f\right)}^{-1}$ is an inverse of $d\times d$, where $d<<n$.

Therefore, an ensemble of data assimilation cycles is obtained as follows:

$$\begin{array}{c}\hfill {\mathit{x}}_k^{a_i}={\mathit{x}}_k^{f_i}+{\mathit{K}}_k\left({\mathit{y}}_k^i-{\mathit{f}}^{*}\left({\mathit{x}}_k^{f_i}\right)\right)\end{array}$$

(14)

where $a_i$ denotes the $i-$th analysis ensemble member. And finally, the ensemble mean of the analysis stage is computed as

$$\begin{array}{c}\hfill {\overline{\mathit{x}}}_k^a={\displaystyle \frac{1}{q}}\sum _{i=1}^q{\mathit{x}}_k^{a_i}\end{array}$$

(15)

where ${\overline{\mathit{x}}}_k^a$ is the estimated activity at sample k.

The use of a coupled NARX–EnKF (Nonlinear Autoregressive Model with Exogenous Inputs–Ensemble Kalman Filter) model to predict water levels is crucial because of its ability to improve prediction accuracy and handle uncertainty effectively. The NARX model captures complex nonlinear relationships between hydrological variables, while the EnKF allows real-time data assimilation, continuously adjusting the model state with new observations. This results in more accurate and robust predictions, especially in dynamic environments with noisy data and variable conditions.

In addition, the combination of NARX and EnKF is particularly useful in complex hydrologic systems with multiple stations and interrelated variables. This approach improves the management and planning of disaster risks, such as floods, facilitating the implementation of early-warning systems and the optimization of resources. Overall, the NARX–EnKF hybrid model is presented as a key tool for improving the accuracy of hydrological forecasts, as it allows us to effectively adapt to external disturbances and optimize decision-making in water management. Its ability to reduce forecast uncertainty makes it a valuable resource for planning and preparing for extreme events, thus contributing to more efficient and resilient water resources management. where $y_{L_1}\left[k\right]$, $y_{L_2}\left[k\right]$ correspond to the two outputs of the level variable of the two stations; $u_{F_1}\left[k\right]$, $u_{F_2}\left[k\right]$, $u_{P{T}_1}\left[k\right]$, and $u_{P{T}_2}\left[k\right]$ are the four inputs of the neural-network system corresponding to the two stations of the multivariable system, ie, the $u_{F_j}\left[k\right]$ represents the $j-th$ two inputs of the flow variable of the two stations and $u_{P{T}_j}\left[k\right]$ represents $j-th$ two more inputs of the rainfall variable of the two stations already mentioned, thus obtaining a multivariable system with four inputs and two outputs.

3. Results

To validate the proposed approach, a comparative analysis is performed between the NARX model coupled and decoupled with the nonlinear recursive filter EnKF using a nonlinear autoregressive state-space model with exogenous variables. This analysis is applied to an order 4 system for two hydrological stations with 24 inputs and 2 outputs that is trained online. Five noise perturbations are incorporated as variance parameters to evaluate the robustness of the model in river-level prediction. Finally, a visual comparison between the real and estimated signals for the nonlinear model is presented, considering the five noise parameters and comparing the performance of the NARX model with and without the EnKF filter.

In this work, the evaluation of prediction performance is the key to assessing the quality of the model and optimizing efficiency. In this study, the Root Mean Squared Error (RMSE) and the Nash–Sutcliffe model efficiency coefficient (NSE) [24] are the indicators used to evaluate the prediction performance of the outputs of the feedforward NARX neural-network model that adds noise to the inputs and outputs of the 24-input, 2-output model for a fourth-order system. In turn, the NARX and LSTM model is coupled to an Ensemble Kalman recursive, nonlinear EnKF filter to improve the estimation of the NARX and LSTM models for the three noise variance parameters ($0.001$, $0.05$, $0.1$, $0.2$, $0.4$).

$$\begin{array}{c}\hfill \begin{array}{c}\hfill RMSE={\displaystyle \frac{1}{n}}{\Sigma }_{i=1}^n{(y_{0_i}-y_i)}^2\end{array}\end{array}$$

(16)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill NSE=1-{\displaystyle \frac{{\Sigma }_{i=1}^n{(y_{0_i}-y_i)}^2}{{\Sigma }_{i=1}^n(y_{0_i}-{\overline{y_0)}}^2}}\end{array}\end{array}$$

(17)

The feedforward network structure for the NARX neural-network approach considers a hidden layer with 20 nodes. The NARX-rectified linear unit (ReLU) activation function is selected for the proposed approach, where a ReLU is a piecewise linear function that will output the input directly if it is positive or otherwise, zero. A NARX model combined with an Ensemble Kalman Filter (EnKF) is essential for hydrological forecasting because of its ability to handle external disturbances such as white noise, thus improving the accuracy of the estimates. The EnKF continuously adjusts NARX model predictions, allowing dynamic adaptation to rapid changes in hydrological and meteorological conditions. This hybrid approach also facilitates uncertainty assessment and the constant optimization of model parameters, providing more reliable and accurate predictions. These features are particularly valuable for risk management and early-warning flood systems, improving decision-making in water resource management and disaster mitigation.

The implementation of the NARX model and the proposed EnKF filter based on neural networks is performed in an algorithm developed in matlab under the sim function for its machine learning training. It should be noted that the members with which the training of the NARX is performed, with the filter and without the EnKF filter, is 80 members.

In Figure 2 the Taylor diagram is shown for the two models, which are NARX and NARX coupled to an EnKF filter. There we can observe the analysis for the standard deviation of the two models, the Centered Root Mean Square Difference and Correlation. As a result we observe how the NARX model coupled to the EnKF filter presents a better correlation coefficient in terms of the estimation of the two models.

In

Table 1. Parameters of the NARX model and NARX–EnKF.

Non-Linear	# Input	# Hidden	# Output	Learning	Epochs
Models	Nodes	Nodes	Nodes	Rate
ENKF	4	80	2	–	12
NARX	4	40	2	0.02	12

, NARX receives, as input, the relevant hydrological variables (e.g., water level, flow, and precipitation) and generates an initial prediction of the future state. ENKF then adjusts this prediction by incorporating real-time observations, reducing model uncertainty. The table configuration shows that the NARX network has 4 input nodes, 40 hidden nodes, and 2 output nodes, with a learning rate of 0.02 and a training of 12 epochs. On the other hand, ENKF employs 80 hidden nodes, suggesting that its processing is more complex and needs higher representation capacity to improve state estimation. This hybrid coupling allows the improvement of the accuracy of water-level prediction by combining the nonlinear learning capability of NARX with the statistical correction of ENKF, which makes the system more robust to data variability and changing hydrological conditions. The NARX–ENKF model subjected to white-noise perturbation demonstrates higher adaptability and robustness compared to the model in [11]. The incorporation of the Ensembled Kalman Filter allows for improved data assimilation and reduced impact of white noise, resulting in more accurate and stable predictions.The combination of NARX and ENKF was found to maintain superior stability under conditions of high uncertainty, reducing the variance of the predictions.

In addition,

Table 2. Root Mean Square Estimation (RMSE) with Gaussian noise for NARX neural network coupled with EnKF and without EnKF.

NARX Coupled with EnKF
Variance Parameter	RMSE Output 1	RMSE Output 2	Total
0.01	0.0158	0.0016	0.0174
0.05	0.0231	0.0075	0.0306
0.1	0.0389	0.0215	0.0604
0.2	0.0636	0.0327	0.0963
0.4	0.1238	0.1515	0.2753
NARX without EnKF
0.01	0.0250	0.0026	0.0276
0.05	0.0265	0.0081	0.0346
0.1	0.0486	0.0158	0.0644
0.2	0.0901	0.0659	0.1560
0.4	0.2484	0.2116	0.4600

shows the RMSE estimates of the NARX model without the EnKF filter and with the EnKF filter for the two outputs of the hydrologic variable level for the two hydrologic stations already described. It should be noted that the EnKF is an implementation of the Bayesian Monte Carlo update problem, given the probability density function of the modeled system state.

In Table 2 a metric regression analysis of the data for the Root Mean Squared Error (RMSE) estimates of a NARX neural network with noise coupled to an EnKF and with decoupled noise without an EnKF is shown.

From Table 2, it is observed that the total estimation error of the feedforward NARX neural network is greater than the estimation error of the NARX neural network coupled to the nonlinear recursive filter EnKF. It can be observed that the estimation of the two-station hydrological model used in this research presents good estimation of the model outputs; this can be observed in the calculated estimation error which is quite low and well below 1. Also, it is observed that the total estimation error of the feedforward neural network is larger than the estimation error of the NARX neural network decoupled to the EnKF nonlinear recursive filter; it can be observed that the estimation of the two-station hydrological model used in this research presents good estimation, but it is not as good as the NARX neural network coupled with the EnKF.

A highlight in the results is the analysis of the Nash–Sutcliffe model efficiency (NSE) coefficient, which measures the quality of the predictions, which are optimal when the value is 1. In this study, it was observed that for the NARX neural network without the EnKF filter, as the noise of the variance parameter increases, the NSE moves away from 1, indicating that the estimation of the real signal becomes less accurate compared to the estimated one. This shows the influence of noise on the quality of the model predictions as shown in

Table 3. Nash–Sutcliffe model efficiency coefficient (NSE) with Gaussian noise for the NARX neural network coupled with EnKF and decoupled without EnKF.

NARX Coupled with EnKF
Variance Parameter	NSE Output 1	NSE Output 2
0.01	0.9918	0.9994
0.05	0.9917	0.9992
0.1	0.9916	0.9991
0.2	0.9913	0.9987
0.4	0.9892	0.9970
NARX without EnKF
0.01	0.9953	0.9996
0.05	0.9936	0.9978
0.1	0.9908	0.9949
0.2	0.9770	0.9814
0.4	0.9317	0.9401

. Such is the case that for output 2 of the model proposed in the calculation of the NSE with a noise in the variance parameter of 0.4; a value of $0.4$ is obtained and a value of $0.9401$ is obtained. The opposite case is seen with the NARX model coupled to the EnKF filter, where it is observed how the NSE coefficient conserves optimal values close to 1 at the moment of performing the mathematical calculation of the NSE. We can observe in Table 3 that for a noise of variance $0.4$, a value of $0.9970$ is obtained. In this case, the filtering of the NARX model contributes to an estimation improvement of $5.7\%$ with respect to the NARX model without filtering. Therefore, the EnKF filter contributes favorably to the NARX feedforward network model in terms of giving robustness to the hydrological model in terms of an external perturbation. The NARX model coupled with the Ensemble Kalman filter (EnKF) and subjected to external white-noise perturbations has significant and promising future scopes. This approach improves the accuracy and robustness of hydrological predictions under variable conditions. With advances in data processing and machine learning, the model could be integrated into real-time monitoring systems, providing more accurate and adaptive predictions. Its application could benefit watershed management, water infrastructure planning, and natural disaster mitigation. Thanks to its adaptive capacity and resistance to external disturbances, it is ideal for early-warning systems, helping to increase resilience and sustainability in the face of extreme hydrological events.

4. Discussions and Conclusions

Daily water-level forecasting is of great importance for the safety of people and the utilization of water resources. Taking into account the NARX–EnKF hybrid or coupled hydrologic model, a two-day forward water-level forecast model based on an ANN model has been proposed. The predictions of the coupled model are globally satisfactory since the regression metrics indicate that, for the hybrid model, the estimation of the estimated signal, real signal with noise, and real signal without noise, is very good compared to the neural-network model NARX not coupled to the EnKF filter by subjecting the two models, NARX coupled with EnKF and NARX not coupled to EnKF, with five external white-noise perturbations. This can be observed in the metric regressions taken to validate the proposed study, such as the numerical calculation of the estimation error RMSE, where the closer it is to zero, the better the estimation and the NSE coefficient are, which indicates that the closer its value is to 1, the better the estimation. This can be observed in Table 2 and Table 3.

A hydrologic model with 24 inputs and 2 outputs was developed using NARX–EnKF to improve predictions. Experiments showed that the combination of predictions and measurements increases accuracy, especially when uncertainty is high. The EnKF–NARX model with external white noise showed significant improvement over traditional methods, being robust to perturbations and able to generalize well to new data. Although there is room for improvement, such as adjusting the variance of the white noise, the results support the effectiveness of the model for predicting dynamic systems.

Future work could explore the integration of the NARX–EnKF model with advanced approaches such as convolutional neural networks to capture spatial patterns in hydrological data or attention models to improve the interpretation of temporal dependencies. In addition, a combination with particle filters or ensemble variational methods would allow for more robust estimation under conditions of high uncertainty. It would also be valuable to investigate hybrid models that integrate physical equations with artificial intelligence to improve accuracy in different hydrological scenarios. Automated optimization using Bayesian algorithms and reduction in dimensionality using sing techniques such as autoencoders could make the system more efficient and adaptable to large volumes of data. Finally, the implementation of the model on cloud-computing platforms and edge devices would facilitate its application in real-time monitoring systems, optimizing water management in vulnerable watersheds and improving response to extreme events.