Improving the forecasting accuracy of monthly runoff time series of the Brahmani River in India using a hybrid deep learning model

Runoff modeling plays a vital role in hydrology and water resource management. Runoff refers to the portion of rainfall or snowmelt that flows over the land surface and eventually enters streams, rivers, and other water bodies. Precise prediction of daily runoff is essential for various applications, including flood forecasting, water supply planning, irrigation management, and environmental impact assessments (Halwatura & Najim 2013; Liu et al. 2020). The accurate prediction of monthly runoff in river basins holds paramount importance for effective water resource management, flood forecasting, and sustainable development initiatives. In the context of the Brahmani river basin at Jenapur station, India, accurate forecasting plays a pivotal role in enabling informed decision-making processes for sustainable water resource allocation and management. Traditionally, hydrologists have relied on empirical and physical models to estimate runoff (Scharffenberg et al. 2010). These models often require extensive data inputs, such as precipitation, temperature, soil properties, and land cover information (Legesse et al. 2003). While these models have been successful to some extent, they often face challenges in capturing the complex nonlinear relationships and spatial-temporal dynamics inherent in hydrological processes (Gain et al. 2021). The existing literature has emphasized the challenges associated with achieving precise runoff predictions, underscoring the necessity for advanced forecasting techniques that can accommodate the complex interplay of hydrological processes. While traditional methods have yielded valuable insights, their limitations in capturing intricate nonlinear relationships within the hydrological system have prompted the exploration of more sophisticated modeling approaches. In the past few years, deep learning (DL) models have gained significant prominence as a robust tool for analyzing intricate data patterns and making precise predictions across diverse fields (Miotto et al. 2018). DL, a subfield of machine learning, involves training artificial neural networks (ANNs) with multiple hidden layers to learn hierarchical representations of data (Ongsulee 2017). These techniques have shown remarkable success in computer vision, natural language processing, and speech recognition (Torfi et al. 2020). Applying DL techniques to daily runoff modeling holds great promise for improving prediction accuracy and capturing intricate hydrological processes (Ji et al. 2021). DL models can automatically learn features and representations from raw data, enabling them to effectively handle nonlinear relationships, temporal dependencies, and spatial patterns that influence runoff (Hu et al. 2018). Moreover, DL techniques have the potential to leverage the vast amounts of hydrological data available today, including remotely sensed data, weather station records, and historical runoff measurements (Zhu et al. 2023).

Traditional hydrological modeling approaches have been widely used for predicting and understanding daily runoff (Mao et al. 2021). These models are typically based on empirical or physical methods that aim to simulate the hydrological processes and relationships governing runoff generation (Legesse et al. 2003). Some commonly used traditional modeling approaches are discussed below.

Empirical models: These models are constructed on statistical relationships between observed runoff and meteorological variables (Thornton et al. 1997). They assume that historical patterns and correlations can be used to predict future runoff. Examples include regression models, autoregressive models, and ANNs with a simplified architecture (Abbott et al. 1986; Furundzic 1998; Jain & Kumar 2007).

Conceptual models: These models are built on simplified representations of hydrological processes (Atkinson et al. 2012). They divide a catchment into hydrological components, such as rainfall excess, soil moisture, and groundwater storage (Skøien et al. 2003). These models utilize conceptual equations and parameters to simulate the flow and storage dynamics. Examples include the soil conservation service-curve number method and the variable infiltration capacity model (Kheimi & Abdelaziz 2022).

Distributed physical models: These models simulate the hydrological processes at a spatially distributed level within a catchment. They account for spatial variations in topography, soil properties, land use, and meteorological inputs (Abbott et al. 1986). These models typically employ mathematical equations representing the physical principles of hydrological processes, such as rainfall–runoff transformation, infiltration, evapotranspiration, and flow routing (Ramírez 2000). Examples include the soil and water assessment tool, the hydrological modeling system, and the MIKE Système Hydrologique Européen (MIKE-SHE) models.

Rainfall–runoff models: These models focus on the transformation of rainfall into runoff without explicitly considering the detailed physical processes (Jakeman & Hornberger 1993). These models often use lumped or semidistributed approaches and rely on calibration against observed runoff data (Ruelland et al. 2008). Examples include the tank model, the rational method, and the unit hydrograph method (Suryoputro et al. 2017).

Statistical time series models: These models focus on analyzing the statistical properties of hydrological variables, such as rainfall and runoff (Efstratiadis et al. 2014). They utilize techniques like autoregressive integrated moving average, seasonal decomposition of time series, and Fourier analysis to capture temporal patterns and forecast future runoff (Zhou et al. 2021).

These traditional hydrological modeling approaches have been widely used and have contributed significantly to our understanding of runoff processes. However, they often face limitations in capturing the complex spatial-temporal dynamics and nonlinear relationships present in hydrological systems (Krause et al. 2015). DL techniques provide an alternative and potentially more accurate approach by leveraging the power of neural networks to learn intricate patterns and relationships directly from data (Kautz et al. 2017; Berisha et al. 2019).

DL techniques have revolutionized various fields by enabling machines to learn and make predictions from complex data patterns (Rajula et al. 2020). DL is a subfield of machine learning that utilizes ANNs with multiple layers to automatically learn hierarchical representations of data (Xie et al. 2017). These techniques have shown immense potential in domains such as computer vision, natural language processing, and speech recognition (Hirschberg & Manning 2015). In recent years, their application to hydrology and runoff modeling has gained attention, offering new avenues for improved prediction accuracy and understanding of hydrological processes (ASCE Task Committee on Application of Artificial Neural Networks in Hydrology 2000; Shen et al. 2018).

The potential of DL techniques in runoff modeling stems from their ability to effectively capture intricate spatial-temporal patterns, handle nonlinearity, and extract features directly from raw data (Tahmasebi et al. 2020; Ha et al. 2021). Unlike traditional hydrological models that rely on predefined equations and assumptions, DL models can learn and adapt to the data, allowing them to capture complex relationships that may be challenging to formulate explicitly (Reichstein et al. 2019).

Nonlinear modeling: DL models excel at capturing nonlinear relationships between input variables and runoff (Ditthakit et al. 2023). They can learn complex patterns and interactions, even in the presence of high-dimensional data (Advani et al. 2013).

Temporal dependency: DL models can effectively model temporal dependencies, considering the sequential nature of runoff data. They can capture short-term and long-term dependencies, enabling accurate predictions of runoff dynamics over time (Li et al. 2021b).

Spatial patterns: DL models have the potential to capture spatial patterns by incorporating spatially distributed data, such as rainfall, soil properties, and topography (Zandi et al. 2022). This enables them to account for the heterogeneity within a catchment and improve the accuracy of runoff predictions (Santikari & Murdoch 2018).

Data integration: DL techniques can integrate various data sources, including remotely sensed data, meteorological observations, and historical runoff measurements. This integration enhances the representation of hydrological processes and improves the model's generalization capabilities (Goetz et al. 2011).

Feature learning: DL models can learn relevant features from raw input data (Yuan et al. 2019). They can extract meaningful representations, reducing the need for manual feature engineering and potentially capturing important but previously unknown relationships in runoff processes (Kim et al. 2021).

Scalability: DL models can handle large datasets efficiently and scale well with increasing data volume (Al-Jarrah et al. 2015). This capability is particularly valuable in hydrology, where vast amounts of data are available from weather stations, remote sensing platforms, and other sources. By leveraging these advantages, DL techniques offer new opportunities for improved runoff modeling, including more accurate predictions, enhanced understanding of underlying procedures, and better management of water resources (Tahmasebi et al. 2020).

In this study, we introduce a novel hybrid DL model, Fourier transform long short-term memory (FT-LSTM), designed to enhance the prediction accuracy of monthly discharge time series. By integrating the Fourier transform with LSTM, our model aims to capture both temporal dependencies and periodic patterns present in the hydrological data, thus improving the overall forecasting performance. The Brahmani River basin at the Jenapur station in India is selected as the typical study site. The primary objective of this research is to assess the effectiveness of the FT-LSTM model in comparison to traditional DL models, including LSTM, recurrent neutral network (RNN), and gated recurrent unit (GRU) across various lag periods. In this paper, we present a comprehensive analysis of the performance of the FT-LSTM model, demonstrating its efficacy in achieving enhanced forecasting accuracy and providing valuable insights into water resource management and hydrological applications. By addressing the existing research gaps and leveraging the potential of advanced DL techniques, this study contributes to the growing body of knowledge in the field of hybrid DL models for hydrological forecasting.

The daily runoff data from January 1990 to December 2020 of the Jenapur station were obtained from the Central Water Commission (CWC), Bhubaneswar. Guo et al. (2011) found that various elements, such as temperature, evaporation, and rainfall, have an impact on the creation process of streamflow. As rainfall, a natural phenomenon with significant temporal and spatial variability, is the primary determinant of streamflow, this research focuses on creating a streamflow forecasting model based on rainfall.

To prevent the model from overfitting and compare its performance in predicting streamflow, it is therefore clear and quick to divide the input data into training (train LSTM), validation (assess LSTM), and testing (confirm the findings).

The back values that will be used for testing and training must be taken into account when using time series data; therefore, the data were divided into three groups: training (70%), validation (15%), and testing (15%). From 1 January 1990 to 31 December 2010, the learning period was in effect. From 1 January 2010 to 31 December 2015, the validation period was in effect. From 1 January 2015 to 31 December 2020, the testing period was in effect.

FT is a mathematical technique used to transform a time domain signal into its frequency domain representation. The Fourier smoothing filter is one that can be applied to achieve a smoothing effect on a time series. The Fourier smoothing filter involves performing the FT on a time series, attenuating or removing high-frequency components (i.e., noise or rapid changes), and then applying the inverse Fourier transformation to obtain the smoothed time series. It is important to note that the Fourier smoothing filter is a linear filter and assumes that the noise or unwanted variations in the time series are represented by high-frequency components. Therefore, the effectiveness of Fourier smoothing depends on the characteristics of the specific time series and the desired smoothing effect.

In our study, the application of the Fourier Transform methodology played a crucial role in enhancing the predictive capabilities of the proposed FT-LSTM model. Fourier transform was initially employed to decompose the time series data into its frequency components. Subsequently, the low-pass filter was utilized to retain the lower-frequency components, which are indicative of the significant temporal trends and variations, while minimizing the influence of higher-frequency noise and fluctuations. By effectively harnessing the Fourier transform methodology in conjunction with the low-pass filtering technique, the FT-LSTM model was able to capitalize on the inherent temporal patterns within the monthly discharge time series, thereby enhancing the accuracy and robustness of the predictive capabilities for hydrological forecasting in the Brahmani River at the Jenapur station.

More details on FT for time series smoothing can be found in the research paper by Kimball (1974).

The achievements witnessed in recent times regarding the application of DL have been particularly notable in sequential prediction tasks. These tasks include statistical language modeling, analysis of chaotic time series, ecological modeling for dynamic systems control, and applications in finance and marketing. The success of DL in these domains has inspired researchers to explore its potential for time series forecasting in hydrology events (Duriez et al. 2017; Chattopadhyay et al. 2020; Pan et al. 2022)

The capability of RNNs to capture dependencies between previous terms is crucial for accurate weather pattern predictions (Banerjee et al. 2022). RNNs are particularly valuable in this context as they are ANNs specifically designed for building prediction models using long-term time series datasets (Kim et al. 2018; Chimmula & Zhang 2020). Depending on the specific application, RNN's architecture offers various variations, including the many-to-one model (used to predict the current time step based on all previous inputs), the many-to-many model (employed to predict multiple future time steps simultaneously based on all previous inputs), and other models (Sahoo et al. 2019; Bose et al. 2022). The selection of the final structure is determined by the formulation of the problem, which is influenced by the observed phenomena.

The GRU is a type of RNN architecture that has gained significant popularity in the field of DL, as shown in Figure 6. It was introduced by Cho et al. (2014) as an alternative to the traditional LSTM units. GRUs are designed to address some of the limitations of LSTMs while achieving comparable performance in many tasks. Like LSTMs, GRUs are capable of capturing long-range dependencies in sequential data, making them well-suited for tasks such as speech recognition, natural language processing, and machine translation. One of the key advantages of GRUs over LSTMs is their simpler architecture. While LSTMs have separate input and output gates and a forget gate, GRUs combine the functionality of these gates into two gates: an update gate and a reset gate. This results in a more streamlined structure with fewer parameters, making GRUs computationally more efficient and easier to train. The update gate in a GRU determines how much information from the previous time step should be passed to the current time step. It controls the trade-off between incorporating new input and retaining past information. The reset gate, on the other hand, decides how much of the past information should be forgotten.

The GRU's update gate allows it to dynamically adapt to the input and the context, controlling the flow of information through the network. This adaptability is beneficial when dealing with long sequences where retaining relevant information over long periods is crucial. Due to their simpler architecture, GRUs are generally faster to train and require fewer computational resources compared with LSTMs. However, the performance of GRUs and LSTMs can vary depending on the specific task and dataset. It is often recommended to experiment with both architectures to determine which one works best for a given problem. The GRU is a recurrent neural network architecture that addresses some of the complexities of LSTMs while achieving comparable performance. With its simplified design and efficient computational properties, the GRU has become a popular choice for modeling sequential data in various domains of DL.

Numerous RNN and LSTM models for runoff forecasting have been constructed using sequential raw data. The LSTM layer is a standard component in many recent machine learning software packages, including TensorFlow and Keras (Xiang et al. 2020). Keras is a Python-based high-level neural network library that can run on top of Theano or TensorFlow. TensorFlow is a well-known machine learning package that is commonly used for constructing DL models. Nonetheless, TensorFlow has several limits. Keras, on the other hand, is a user-friendly and easily available high-level application programming interface (API) built on top of TensorFlow (Tang 2016).

To build this model, the monthly runoff data were taken. The datasets were categorized into three sets: training, validation, and testing sets. Researchers have used diverse datasets to generate training and testing sets in various research investigations. In this study, 70% of the available dataset was allotted for training sets, with the remaining 30% allocated for validation and testing sets (15% each). The model's input generally consists of different time lags of inputs and their combinations that are shown in Table 1. Five cases of input variable (X₁, X₃, X₆, X₉, and X₁₂) were designed with lags ranging from 1 to 12 months before the current time for runoff forecasting. It was a 3D array with the dimensions of number of samples (372); number of time steps (lag) with values of 1, 3, 6, and 12; and an output dimension of 1. In all cases, the model was trained using 70% of the sample data. The model was compiled using the mean square error loss function and the Adam optimizer, and this was the final phase in the model's construction. The model was trained for 500 epochs. A scatter plot between observed vs estimated runoff during training and testing for all the models is shown in (Figure 7).

Table 1

The optimal input value for the Brahmani River Basin

⁠, the predicted runoff; ⁠, the 1-month lagged; ⁠, the 3-month lagged; ⁠, the 6-month lagged; ⁠, the 9-month lagged; ⁠, the 12-month lagged; f, the model type.

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Input selection is considered to be the most important factor for model development. Hence, various machine learning methods were adopted for determining the input parameters, including the parsimonious method, the autocorrelation function (ACF), and the PACF (Figure 2) (Abdel-Aal & Al-Garni 1997; Kumar et al. 2019).

LSTM was chosen due to its ability to capture long-range dependencies and retain information over extended time intervals. Its effectiveness in handling sequential data and mitigating the vanishing gradient problem makes it a popular choice for time-series forecasting tasks, including hydrological modeling. RNNs, similar to LSTM, were included as a comparative model to assess the performance of the FT-LSTM model against a more basic recurrent architecture. The comparison helps in understanding the benefits of incorporating specialized mechanisms, such as memory cells and gating units, as present in the LSTM and FT-LSTM models. GRU, known for its simplified gating mechanism compared with the LSTM, was selected to provide insights into the impact of different gating structures on the predictive accuracy of monthly runoff. Comparing GRU with LSTM and FT-LSTM aids in evaluating the contributions of more intricate memory management in capturing the temporal dynamics of the Brahmani River Basin's runoff data.

Considering these limitations and the rationale behind the selection of comparative models, a comprehensive analysis was conducted to ensure a fair and insightful comparison of the FT-LSTM model's performance with respect to traditional and other DL architectures.

Table 2 presents the error metrics for runoff prediction in the Brahmani River Basin at the Jenapur station, considering different lag periods (1, 3, 6, and 12) for four forecasting models: LSTM, RNN, GRU, and FT-LSTM. When comparing the models, it is evident that the FT-LSTM consistently outperforms the other models in terms of error metrics across all lag periods. For example, at lag 12, the FT-LSTM model achieves an MAE of 487.65 (m³/s), which is significantly lower than the MAEs of the other models: LSTM (795.57 m³/s), RNN (841.06 m³/s), and GRU (829.69 m³/s). Similarly, the RMSE for FT-LSTM at lag 12 (592.58 m³/s) is substantially lower compared with LSTM (1,028.22 m³/s), RNN (1,074.98 m³/s), and GRU (1,102.37 m³/s).

The superiority of the FT-LSTM model is consistent across all lag periods, as indicated by consistently lower MAE and RMSE values. In addition, the FT-LSTM model demonstrates higher NSE and R² values, suggesting a better fit between predicted and actual runoff values. This indicates that the FT-LSTM model is better at explaining the variance in the actual runoff data. For instance, at lag 12, the FT-LSTM model achieves an NSE of 0.92 and an R² of 0.93, which are notably higher compared with the values obtained by LSTM, RNN, and GRU. The accuracy percentage of FT-LSTM is notably higher by 10.75%, 18.27%, and 11.82% when compared with LSTM, RNN, and GRU, respectively. The runoff volumes of observed and estimated values from different models is given in Figure 8. This indicates that the FT-LSTM model provides more accurate and reliable runoff predictions. The results of FT-LSTM were also compared with other models of previous studies to provide its reliability and accuracy, shown in Table 3.

The consistent and superior performance of the FT-LSTM model across all lag periods demonstrates its effectiveness in accurately forecasting runoff levels in the Brahmani River Basin at the Jenapur station. The FT-LSTM model's feature transformation capabilities likely contribute to its ability to capture complex patterns in the data and make more accurate predictions.

Data availability and quality: Variations in data availability and quality across different models could impact the comparative analysis. Insufficient data points or inconsistencies in data quality might affect the performance assessment, particularly in cases where certain models are more sensitive to data characteristics.

Model complexity and interpretability: Comparing the FT-LSTM model with traditional models and other DL architectures might pose challenges in terms of model complexity and interpretability. Ensuring a fair comparison while considering the interpretability of results remains crucial, especially in scenarios where simpler models might be favored due to their ease of understanding and implementation.

Computational requirements: DL models often require significant computational resources, which could pose constraints on the practical implementation and scalability of the models. A thorough evaluation of the computational requirements and efficiency is essential to understand the practical feasibility of deploying these models in real-world hydrological forecasting scenarios.

The purpose of this study is to improve the prediction accuracy of monthly runoff time series in the Brahmani River Basin at the Jenapur station. To achieve this, we compared the performance of the following three DL models: LSTM, RNN, GRU, and a new hybrid model called FT-LSTM. From the results, it was found that the FT-LSTM model outperformed other models in terms of prediction accuracy (Table 1). This shows that integrating the Fourier transform into his LSTM architecture can effectively improve the predictive ability of the model for the monthly runoff time series. The superior performance of the FT-LSTM model can be attributed to its ability to capture the inherent periodic patterns and seasonality of runoff data. By using Fourier transform as a preprocessing step, the model was able to decompose the time series into frequency components, allowing LSTM components to effectively learn complex time dependencies. The results of this study suggest that the FT-LSTM model can be expected to improve the accuracy of monthly discharge predictions in the Brahmani River Basin at the Jenapur station. The model can capture both long-term dependencies and cyclical variations in runoff data, making it a valuable tool for water resource management, flood forecasting, and other hydrological applications.

All relevant data are available from an online repository or repositories: https://github.com/bibhuatGitHUb/Accurate-prediction-of-monthly-runoff.

The authors declare there is no conflict.