Grid-Search-Optimized, Gated Recurrent Unit-Based Prediction Model for Ionospheric Total Electron Content

Abstract

Accurately predicting the ionosphere’s Total Electron Content (TEC) is significant for ensuring the regular operation of satellite navigation and communication systems and space weather prediction. To further improve the accuracy of TEC prediction, this paper proposes a TEC prediction model based on the grid-optimized Gate Recurrent Unit (GRU). This model has the following main features: (1) it uses statistical learning methods to interpolate the missing data of TEC observations; (2) it constructs a sliding time window by using the multi-dimensional time series features of two types of solar activity indices to support modeling; (3) It adopts grid search combined with optimization of network depth, time step length, and other hyperparameters to significantly enhance the model’s ability to extract the characteristics of the ionospheric 11-year cycle and seasonal variations. Taking the EGLIN station as an example, the proposed model is verified. The experimental results show that the root mean square error of the GRU model during the period from 2019 to 2020 was 0.78 TECU, which was significantly lower than those of the CCIR, URSI, and statistical machine learning models. Compared with the other three models, the RMSE error of the GRU model was reduced by 72.73%, 72.64%, and 57.38%, respectively. The above research verifies the advantages of the proposed model in predicting TEC and provides a new idea for ionospheric modeling.

1. Introduction

The ionosphere is a plasma-rich region containing substantial ions and free electrons, spanning approximately 60 km above the Earth’s surface to the top of the magnetosphere. As a critical medium influencing satellite–ground electromagnetic-wave propagation, it induces signal delays, attenuation, and scattering in radio communication and navigation systems [1].

TEC represents the cumulative electron density along a signal transmission path. It serves as a key parameter for measuring the spatial and temporal inhomogeneities and disturbance characteristics of the ionosphere, as well as the signal delay [2]. TEC can be directly measured using observed instruments [3]. Current satellite-based TEC retrieval methods exhibit an inherent latency of several hours. Therefore, ionospheric modeling is essential to enable near-real-time TEC estimation, thereby enhancing the timeliness of space environment monitoring [4]. High-precision ionospheric models effectively mitigate ionospheric effects in signal propagation [5], providing error compensation solutions for satellite–ground wireless communication systems and advancing the understanding of coupling mechanisms between solar activity and geospace environments.

Given the complexity of ionospheric dynamics, diverse modeling approaches have been developed in academia. The International Reference Ionosphere (IRI) model has emerged as the international standard for predicting ionospheric parameters, garnering significant scientific attention [6,7]. The IRI model integrates multi-source data, including solar activity indices and geomagnetic parameters, and generates predictions for quiet-time critical frequencies, altitude information, and TEC values. Jointly developed through international scientific collaboration and subject to continuous refinement, the IRI has evolved over a dozen iterations, spanning versions from IRI-78 to IRI-2020 [8,9]. Researchers worldwide have developed numerous regionally corrected models within the IRI framework to enhance regional applicability. These observation-driven empirical models, characterized by structural simplicity and computational efficiency, are widely employed for ionospheric TEC prediction.

In recent years, with the development of intelligent algorithms, machine learning has significantly advanced the innovation of ionospheric modeling. Researchers have introduced nonlinear methods such as Artificial Neural Networks (ANNs) and empirical orthogonal functions into predicting and analyzing parameters, such as TEC, critical frequency of the ionospheric F2 layer (foF2), and peak height. Some of these achievements have been incorporated into the international standard system. Typical achievements include the following: Liu et al. [10] established a European regional prediction model by introducing statistical machine learning; Jia et al. [11] proposed an improved radial basis function neural network classification method by integrating partial least squares and genetic algorithms; Hu et al. [12] enhanced the prediction accuracy of foF2 by using the particle swarm optimization algorithm to improve the backpropagation neural network; Zhang et al. [13] employed an encoder–decoder structure combined with convolutional optimization to the long short-term memory network to achieve the prediction of TEC; Wang et al. [14] developed a global ionospheric model based on an ANN optimized by genetic algorithms; Weng et al. [15] innovatively combined adaptive genetic algorithms with BP networks to construct a combined prediction model; Xia et al. [16] utilized GPU-accelerated support vector machines to achieve hourly step TEC prediction; and Yu et al. [17] established a long-term TEC prediction model using a support vector regression model. These studies have effectively improved the prediction accuracy of ionospheric parameters through algorithm innovation or data fusion.

While these methods demonstrate notable advancements, they exhibit inherent limitations for capturing long-term ionospheric dependencies: ANN and BP architectures often face challenges in modeling extended temporal sequences due to gradient vanishing issues; SVR models, despite their robustness, scale poorly with high-dimensional solar-terrestrial datasets; and LSTM networks, though effective for temporal patterns, incur substantial computational overhead from their complex gating mechanisms. In contrast, the Gated Recurrent Unit (GRU) provides a balanced solution—preserving LSTM’s capacity for long-range dependency modeling while reducing parameter complexity and training latency through its streamlined dual-gate design—making it particularly suited for TEC sequences governed by decadal solar cycles and seasonal variations.

To further enhance the accuracy for predicting the long-term variation characteristics of TEC, this paper proposes a prediction model based on grid-optimized gated recurrent units. It combines a data filling strategy to optimize the modeling process. The structure of this paper is as follows: Section 2 provides an introduction to GRU and grid optimization methods; Section 3 introduces the ionospheric TEC data and solar activity index data used in this paper; Section 4 conducts data preprocessing and applies a GRU for modeling to complete the training of the model; Section 5 compares the established GRU model with the IRI model and Statistical Machine Learning (SML) model to complete the testing of the GRU model; and Section 6 summarizes this paper.

2. Methodology

This section systematically elaborates on the core mechanism of the model design from a theoretical perspective. The GRU models the dependency of time series data through gating dynamics. At the same time, the grid search method achieves joint optimization of hyperparameters based on the global traversal property of the parameter space. Combining these two methods provides an approach for ionospheric TEC prediction.

2.1. GRU Network

The GRU network was proposed in 2014 [18] as an improved Recurrent Neural Network (RNN) architecture. Its core objective is to address the gradient vanishing/explosion problem of traditional RNNs through structural simplification. The design concept of GRU originated from the Long Short-Term Memory Network (LSTM). The core innovation is the simplified gating mechanism. GRU uses a dual gating structure, merging the input and forget gates of the LSTM into one update gate and introducing a separate reset gate. It eliminates the cell state, transmitting information only through the hidden state. This reduces model complexity, cutting parameters to about one-third of the LSTM, but it still enables modeling of long-range dependencies. The simpler structure also allows for faster training convergence and higher computational efficiency [18]. These features make GRU well-suited for capturing complex and long-term temporal dependency features in the ionospheric TEC sequence, including the 11-year solar cycle and seasonal variations. These are the main focuses of this prediction model.

Figure 1 shows the network structure of the GRU network. At a specific time, t, the input of the GRU neuron consists of the following two parts: the current network input, x_t, and the cell state, h_t−1, from the previous time; h_t is the output of the current neuron. In this paper, the input vector is composed of the TEC sequence (TEC_t) and the solar activity indicators related to the TEC within the sliding time window. The solar activity indicators related to TEC include the relative sunspot number and F_10.7. In this paper, the solar activity index is processed as a twelve-monthly smoothed value, denoted as R₁₂ and F₁₂. Its specific form is the following:

$${\mathit{x}}_{\mathit{t}}={[{\mathbf{TEC}}_{\mathit{t}},{\mathit{R}}_{12}(\mathit{t}),{\mathit{F}}_{12}(\mathit{t})]}^{\mathit{T}}$$

(1)

For detailed descriptions of the GRU’s internal gating mechanisms (update gate, reset gate, candidate hidden state, and hidden state update), we refer readers to the foundational work by Cho et al. [18].

2.2. Grid Search Method

The grid search method is one of the most commonly used methods for exploring the hyperparameter configuration space [19]. This method can be regarded as an exhaustive search or brute-force search strategy, where it evaluates all possible combinations of hyperparameters within the given configuration grid to find the optimal solution [20]. The core mechanism lies in performing a Cartesian product operation on the pre-defined finite set of parameter values defined by the user, evaluating the performance of each combination.

When there are k hyperparameters and each parameter has n candidate values, the computational complexity of the grid search will increase exponentially at a rate of $\mathrm{O}(n^k)$ [21]. This characteristic makes it more suitable for optimization scenarios in low-dimensional configuration spaces. In the current research scenario, the hyperparameters to be optimized for the GRU model mainly include the following four categories: time step, hidden layer unit number, learning rate, and training epoch. Their low-dimensional nature and discrete value characteristics keep the computational complexity of the grid search within a controllable range, enabling efficient identification of the optimal parameter combination.

3. Data

Based on the architecture proposed in Section 2, this subsection introduces the data used for conducting experiments on the above architecture and the reasons for selecting these data. The research data consist of the following three parts: the TEC of the ionosphere, closely related solar 10.7 cm radio flux (F_10.7), and number of sunspots (R).

3.1. TEC

The TEC used in this study was sourced from the Eglin Observatory in Florida, USA. The TEC data were generated by the Digisonde-Portable-Sounder-4D using the autoscaling algorithm. The equipment was manufactured by Lowell Digisonde International in Lowell, MA, USA. And the TEC was obtained from the Global Ionospheric Radio Observation Database, and the URL is as follows: https://giro.uml.edu/didbase/ (accessed on 10 October 2023) [22]. Figure 2 shows that this station is located at a 30°29′ north latitude and a 86°30’ west longitude. The reason for choosing the Eglin station as the location for model validation is that this station has an 18-year-long, continuous, and publicly available TEC data record from 2004 to 2022. The relatively complete data are beneficial for the GRU model to capture the long-term trend in the TEC data. Furthermore, in the mid-latitudes, the TEC of the ionosphere is usually at a moderate level, with its value significantly higher than that in high-latitude regions but lower than the notable equatorial anomaly peak. Compared to more dynamic equatorial or polar regions, the thickness variation in the ionosphere in the mid-latitudes is relatively more stable [23]. Therefore, it serves as an ideal initial testing platform for evaluating the core predictive capability of the proposed grid-search optimization GRU method under typical conditions. It is particularly important to note that the achieved Root Mean Square Error (RMSE) performance is applicable only to this mid-latitude region. Similar error levels may not directly apply to the equatorial region and high-latitude/polar regions. The upper-right corner shows the data distribution map of Eglin station, with data missing in 2004 and 2009.

This study focuses on analyzing the long-term trend of TEC. Therefore, the original TEC data were processed into monthly median data: the daily 24 h observation data were calculated as the monthly median hourly, generating a monthly average TEC sequence with hourly time resolution. This processing strategy effectively suppresses short-term disturbance noise through median filtering, while retaining the long-term correlation features such as solar activity cycles and seasonal variations [24]. The monthly median value of the mathematical expression of the TEC is as follows:

$${\mathrm{TEC}}_{month}(h,m)median={\mathrm{TEC}}_d(h,m)|h\in [0,23],m\in [1,12],d\in [1,D]$$

(2)

where h represents the hour, and m represents the month. D represents the actual number of observation days in the current month, and TEC_d(h,m) is the observation value at the h-th hour of the d-th day. By analyzing the data from 2004 to 2021, it was found that there were data gaps in some periods. The median TEC data for the month are shown on the left side of Figure 3, and TECU is the unit of TEC, where 1TECU = 1 × 10¹⁶ electrons/m².

3.2. F10.7

F_10.7 is the core input parameter of the ionosphere parameter prediction model, with the unit being the solar flux unit (sfu,1 sfu = 10⁻²² W·m⁻²·Hz⁻¹) [25]. Its physical meaning is the solar radio radiation intensity at the 10.7 cm wavelength (2.8 GHz). The F_10.7 data are provided by the Space Weather Prediction Center of the National Oceanic and Atmospheric Administration, and the URL is as follows: https://www.ngdc.noaa.gov/stp/space-weather/solar-data/ (accessed on 10 October 2023). F_10.7 quantifies the ionization efficiency of neutral gas in the ionosphere due to solar electromagnetic radiation, directly influencing TEC’s spatial distribution and temporal evolution. Studies have shown that F_10.7 is significantly positively correlated with the peak electron density of the ionosphere and TEC [26]. To characterize temporal variations in ionospheric parameters and develop predictive models, the twelve-monthly smoothed value of F_10.7 is a key indicator for quantifying solar activity dynamics in ionospheric prediction studies [27].

3.3. Sunspot

The sunspot relative number (R) is a classic parameter used to characterize the intensity of magnetic field activity on the surface of the Sun. It was proposed and continuously released by the Royal Observatory of Belgium in 1849 [28]. R data are provided by the Belgian Solar Influences Data Analysis Center. The standardized sequence released by this institution, and the URL is as follows: https://www.sidc.be/silso/datafiles (accessed on 10 October 2023) is calculated using Equation (3) [29], as follows:

$$R=k\left(10g+s\right)$$

(3)

where g represents the number of visible solar surface sunspots, s represents the total number of individual sunspots, and k is the correction coefficient of the observation station. The long-term variation in the R value directly reflects the periodic oscillation of the solar magnetic field activity. Further, it regulates the total solar radiation and the intensity of EUV/XUV radiation [30]. Since EUV radiation is the primary energy source for ionizing the ionosphere, a significant nonlinear correlation exists between R and TEC [2]. Analogous to the application of F_10.7, the 12-month smoothed R value functions as a complementary solar activity metric in ionospheric modeling research, particularly for capturing long-term cyclical variations essential for predictive accuracy [31].

The calculation formula for the twelve-monthly smoothed value of the sun activity parameter is as follows:

$${MS}_{12}={\displaystyle \frac{1}{12}}\left[{\displaystyle {\sum }_{i=n-5}^{n+5}{\overline{MS}}_i+\frac{1}{2}({\overline{MS}}_{n-6}+{\overline{MS}}_{n+6})}\right]$$

(4)

where MS denotes the solar activity parameter requiring 12-month smoothing, and $\overline{MS}$ indicates its monthly mean value. The variable n specifies the target month for calculating the moving average, with i being its sequential index. Post-smoothing, the transformed R and F_10.7 indices are labeled R₁₂ and F₁₂, respectively.

TEC exhibits periodic daily and annual cycles. Its level and variation range are highly correlated with the level of solar activity [32]. Figure 3 demonstrates the temporal evolution of R₁₂, F₁₂, and monthly averaged TEC (derived from hourly median values) during 2004–2022. While R₁₂ and F₁₂ exhibit analogous macro-trends, nuanced differences emerge in their temporal profiles. Notably, both solar indices maintain positive correlations with TEC variations.

Our parameter selection gives priority to F_10.7 and R, because they are widely used solar indices in ionospheric research, and they are directly related to solar ultraviolet radiation and magnetic activity cycles. For instance, papers by Maruyama et al. [33] and Yang et al. [34] both use F_10.7 and R to predict TEC.

4. Modeling

This section describes the modeling process as shown in Figure 4. First, data preprocessing was performed (Figure 4a), including using statistical learning methods to impute missing TEC observation data. Then, a GRU model was constructed (Figure 4b), with hyperparameters determined by the grid-search optimization method, using the RMSE of the validation set as the evaluation criterion. Finally, the optimal parameter combination was used to complete the model building.

4.1. Modeling Preparation

This study divides and constructs the training, validation, and test sets based on the 11-year periodic characteristics of solar activity. The training set selects data from 2004 to 2014, totaling 132 months; the validation set covers data from 2015 to 2018, totaling 48 months; and the test set uses data from 2019 to 2020, totaling 24 months.

Deep learning models such as GRU have strict requirements for the integrity of the input sequence. Missing data may cause deviations in the update of network weights and affect the continuous learning of temporal features [35]. The rate of missing TEC data in the observation is relatively low. The missing rate of the entire dataset at each time point is lower than 3.03%, as shown in Figure 5. Therefore, the statistical machine learning method proposed by Liu et al. [10] is considered to fill in the missing values.

According to Reference [10], the solar activity index (R, F_10.7) has a multi-dimensional nonlinear mapping relationship with TEC. Based on the SML theory, using trigonometric basis functions can characterize TEC’s temporal correlation, such as annual and semi-annual cycles. Moreover, a polynomial is introduced to describe the nonlinear effect of the solar activity index [10]. Specifically, as follows:

$$\mathrm{TEC}(I_{12},m)={\displaystyle \sum _{k=0}^k{\displaystyle \sum _{j=0}^j[{\alpha }_{k,j}{F_{12}}^j\cdot \cos (2\mathsf{\pi }km/12)+{\beta }_{k,j}{F_{12}}^j\cdot \sin (2\mathsf{\pi }km/12)]}}$$

(5)

where I₁₂ denotes the solar activity parameter, m corresponds to the month index (integer values), K and J define the maximum expansion orders for harmonic and polynomial components, respectively, with ${\alpha }_{k,j}$ and ${\beta }_{k,j}$ as the associated coefficients. Model calibration is achieved through least squares optimization to determine these parameters. I₁₂, K, and J are selected based on the RMSE as the evaluation index, as shown in the following formula:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left({\mathrm{T}\hat{\mathrm{E}}\mathrm{C}}_i-{\mathrm{TEC}}_i\right)}^2}}$$

(6)

where $\mathrm{T}\widehat{\mathrm{E}}{\mathrm{C}}_i$ represents the model predicted value, TEC_i represents the actual observed value, and N represents the total data quantity.

To ensure the accuracy of data filling, we trained the model using the data from the TEC training dataset collected in this paper and determined the final SML-based model using the RMSE of the validation dataset with the measured TEC. Figure 6 shows the RMSE values of the model under different parameter combinations.

The experimental results in Figure 7 indicate optimal parameter combinations. For an R₁₂ index with K = 3 and J = 1, the SML model achieves a minimum RMSE of 1.64 TECU. Similarly, using the F₁₂ index with K = 3 and J = 2 yields an improved performance with the RMSE reduced to 1.40 TECU. Therefore, in the final model, F₁₂ is used, K is selected as 3, and J is chosen as 2, as shown in the following equation:

$$\mathrm{TEC}\left(F_{12},m\right)={\displaystyle \sum _{k=0}^3{\displaystyle \sum _{j=0}^2\left[{\gamma }_{k,j}{F_{12}}^j\cdot \cos (2\mathsf{\pi }km/12)+{\beta }_{k,j}{F_{12}}^j\cdot \sin (2\mathsf{\pi }km/12)\right]}}$$

(7)

where m represents the corresponding month.

The CCIR and URSI models are widely used sub-models within the IRI model, serving as standard benchmarks for ionospheric parameter predictions. The CCIR model utilizes spherical harmonic functions combined with universal time harmonic functions to describe the global distribution of parameters such as foF2 and M(3000)F2. Primarily based on historical data (with observation stations predominantly located in the Northern Hemisphere), it relies on statistical methods to establish a global-scale parametric representation and holds distinct value for studying long-term variation trends. However, this dependency limits its accuracy in capturing fine structures, particularly in equatorial anomaly zones. In contrast, the URSI model integrates more modern and globally balanced datasets, significantly improving coverage in the Southern Hemisphere and equatorial regions. Although both models demonstrate comparable global accuracy, URSI outperforms CCIR in low-latitude regions [9]. The URSI utilizes data from modern global observation networks and employs complex algorithms to resolve physical mechanisms within the ionosphere, providing greater accuracy for short-term predictions, particularly at mid- and low latitudes.

We chose these models as references because they represent current international standards and have been applied in many comparative studies. For instance, studies by Yu et al. [17], Sezen et al. [36], and Liu et al. [37] all used these models for comparison. The model data were obtained from NASA’s Instant Run platform, and the URL is as follows: https://kauai.ccmc.gsfc.nasa.gov/instantrun/iri/ (accessed on 21 April 2025), where inputting specific coordinates, time parameters, and data types yields corresponding outputs. As delineated by Zhen et al. (2024) [38], these two model types play complementary roles in ionospheric research. Their synergistic application, therefore, enables comprehensive coverage of scientific needs—from physical mechanism analysis to long-term evolution studies—ultimately enriching the understanding of the ionosphere.

The comparison results are shown in Figure 7. The RMSEs of the predicted values of the CCIR and URSI relative to the measured values were 3.01 TECU and 3.06 TECU, respectively, which are greater than 1.40 TECU. This proves the accuracy of the SML-based method in filling in missing data.

Figure 8 presents a periodic characteristic comparison analysis of the data sequences before and after interpolation. The results show that the dataset filled by the SML-based model is highly consistent with the original observations regarding diurnal variation patterns and seasonal fluctuation characteristics. This also proves the reliability of filling data with the SML method.

In the end, this study employed min–max normalization to map TEC, R₁₂, and F₁₂ to the interval [0, 1] to accelerate the gradient convergence of the GRU network and eliminate the influence of feature scale differences [31].

4.2. Model Training

While building the GRU prediction model, we first established the basic framework of the neural network. The model consists of an input layer, a GRU layer, and a fully connected output layer. The GRU layer serves as the core processing unit and captures the temporal dependency features in the TEC sequence. This network uses the sigmoid and tanh functions as activation functions, takes the mean square error as the loss function, and uses the Adam optimizer for optimization [1]. The model input is a time series of the timestep length, predicting the value of the next timestamp. The time step unit is in months. Although the dataset contains data for each hour (as mentioned above, TEC represents the monthly median, and the solar activity index is the twelve-monthly smoothed value), when inputting the data into the model, we organized it into monthly time series samples by hour for input.

Subsequently, we constructed four sets of data input schemes for a horizontal comparison:

• Only the original TEC observation values (OBS);
• OBS combined with the R₁₂ index;
• OBS combined with the F₁₂ index;
• OBS combined with both the R₁₂ and F₁₂ dual indices.

We set the fixed batch size to 64. Then, we conducted the grid-search optimization method separately under four feature input schemes, and the obtained hyperparameters are as follows:

• Number of training epochs: the search range is from 20 to 26;
• Learning rate: the search range is from 0.001 to 0.005, with a step size of 0.001;
• Number of hidden layer units: the search range is 120 to 136, with a step size of 4;
• Time step: The search range is 12 and 24;
• Number of GRU layers: the search range is 1 and 2.

To quantitatively evaluate the training effect of the GRU neural network model, we used the RMSE to assess the prediction results. We selected the input data and parameter group corresponding to the minimum RMSE value in the validation set as the final model. Taking the model with input as OBS, R₁₂, and F₁₂, and a single-layer GRU structure and a 12 h time step as an example, Figure 9 shows the RMSE between the predicted values and the observed values within the parameter search range.

The other three input combinations were also optimized using the grid search method, and the results are shown in

Table 1. Optimization results and performance comparison of hyperparameters for different input combinations.

Model Input	Epoch	Learning Rate	Hidden Size	Timestep	Layer Number	Minimum RMSE (TECU)
OBS	22	0.004	128	12	1	1.35
OBS & R12	22	0.003	120	12	1	1.20
OBS & F12	22	0.003	124	12	1	1.16
OBS & R12 & F12	25	0.003	120	12	1	1.15

. Among them, the prediction performance of the combination of inputs OBS, R₁₂, and F₁₂ was the best (minimum RMSE = 1.15 TECU). Based on this, OBS, R₁₂, and F₁₂ were finally selected as the input, and the optimal parameter combination is as follows:

• Number of training epochs = 25;
• Learning rate = 0.003;
• Number of hidden units = 120;
• Time step = 12;
• Number of GRU layers = 1.

5. Discussion

To conduct a comprehensive evaluation of the model’s performance established in this paper, the model was tested using the test dataset, and the obtained results were compared and analyzed with those of the IRI (including CCIR and URSI models) and SML-based models. It should be noted that, unlike the comparison models, such as SML and GRU, which can ingest hourly-updated measurements, the IRI, as an empirical model, lacks the capability of real-time data access. This disparity in data temporality may introduce a systematic asymmetry in performance benchmarking. Nevertheless, the GRU’s capability to integrate real-time data holds significant operational value for space weather forecasting.

Figure 10 presents the observed values of the TEC, while Figure 11 shows the prediction errors (predicted–observed) of the CCIR, URSI, SML-based, and GRU models. All times are expressed in LT without daylight-saving adjustments. The following can be seen from the figure:

(1)

All models demonstrate capability in tracking long-term TEC trends yet exhibit varying degrees of predictive accuracy across temporal segments.

(2)

Comparative analysis reveals distinct error patterns, as follows: The CCIR and URSI models systematically overestimate TEC measurements, particularly from LT = 20 to LT = 1 the next day. The SML approach shows elevated predictions between 22:00 and 10:00 LT. The proposed GRU model maintains prediction stability across all time segments without notable systematic bias.

(3)

In the predictions of different hours, the GRU model demonstrated the highest prediction accuracy, with a significant advantage.

To comprehensively evaluate the predictive performance of the model, in addition to using the RMSE as the evaluation metric, the Mean Absolute Error (MAE) and Relative Root Mean Squared Error (RRMSE) are further introduced as supplementary indicators. The specific calculation formulas are as follows:

$$\mathrm{MAE}={\displaystyle \frac{{\displaystyle \sum _{i=1}^N\left({\mathrm{T}\hat{\mathrm{E}}\mathrm{C}}_i-{\mathrm{TEC}}_i\right)}}{N}}$$

(8)

$$\mathrm{RRMSE}={\displaystyle \frac{\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left({\mathrm{T}\hat{\mathrm{E}}\mathrm{C}}_i-{\mathrm{TEC}}_i\right)}^2}}}{\overline{\mathrm{TEC}}}}$$

(9)

where $\mathrm{T}\widehat{\mathrm{E}}{\mathrm{C}}_{\mathrm{i}}$ represents the model’s predicted value, $\mathrm{T}\mathrm{E}{\mathrm{C}}_{\mathrm{i}}$ represents the observed value, N represents the total amount of the data being analyzed, and $\overline{\mathrm{T}\mathrm{E}\mathrm{C}}$ represents the arithmetic mean of the observed values, that is, the following:

$$\overline{\mathrm{TEC}}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\mathrm{TEC}}_i}$$

(10)

Figure 12 presents the hourly RMSEs, MAEs, and RRMSEs of the predicted values and observations for the four models. The results show the following:

(1)

Throughout the entire testing period from 2019 to 2020, the relative magnitudes of the error indicators of the four models remained consistent. In other words, at the same point in time, regardless of the error indicator, the order of the error magni-tudes of the four models is consistent.

(2)

For the 24 h prediction, there are significant differences in error among the various models. From the perspectives of the RMSE and MAE, the errors of the CCIR and URSI models are relatively large, from LT = 11 to LT = 18. Analyzing the RRMSE, the SML-based model exhibited larger errors from LT = 23 to LT = 5 the next day, the CCIR from LT = 19 to LT = 23, and the URSI from LT = 19 to LT = 0 the next day.

(3)

Considering the overall performance of the 24 h prediction throughout the day, the GRU model’s RMSE, MAE, and RRMSE are significantly lower than those of the other models, indicating that the GRU model has higher prediction accuracy and better performance.

(4)

Further comparing the errors between the predicted values and the observed values of the GRU model showed the following: from LT = 17 to LT = 9 the next day, the RMSE and MAE are small, while from LT = 10 to LT = 16, the RMSE and MAE are large. This indicates that the absolute error of the GRU model predictions is significantly higher during LT = 10–16. However, the RRMSE performance is relatively consistent per hour because it reflects the relative differences, and the overall performance of the GRU model is relatively balanced.

Figure 13 shows the four models’ RMSEs, MAEs, and RRMSEs compared with observations for different seasons (spring: March–May; summer: June–August; autumn: September–November; winter: December–February). The results indicate the following:

(1)

The SML-based model exhibits greater prediction errors in spring than other models due to inadequate representation of noontime ionospheric disturbances triggered by enhanced solar radiation [39].

(2)

CCIR shows large errors in autumn/winter because it ignores frequent magnetic storms in winter [40].

(3)

The GRU model achieves the smallest prediction errors across all seasonal metrics. However, its error is relatively higher in summer. This is because of physical processes not captured by the model (such as rapid electron loss at night via recombination), which represent a common challenge for purely data-driven methods like GRU to learn [41].

Figure 14 presents the statistics of MAE and RMSE between the predicted values and OBS for each model. From this figure, we can see the following:

(1)

Regarding the RMSE, the GRU model has the smallest RMSE of 0.78 TECU. Compared with the SML-based model, CCIR model, and URSI model, it was reduced by 1.05 TECU, 2.08 TECU, and 2.07 TECU, respectively;

(2)

Regarding the MAE, the GRU model has the smallest MAE of 0.50 TECU. Compared with the SML-based model, CCIR model, and URSI model, it was reduced by 1.11 TECU, 1.74 TECU, and 1.73 TECU, respectively;

(3)

Regarding the RRMSE, the GRU model has the smallest RRMSE of 20.34%. Compared with the SML-based model, CCIR model, and URSI model, it was reduced by 27.49%, 54.45%, and 54.27%, respectively;

(4)

Overall, the GRU model significantly outperforms the RMSE, MAE, and RRMSE comparison models. The error reduction ranges from 1.05 to 2.08 TECU and 27.49% to 54.45%, demonstrating comprehensive performance advantages.

6. Conclusions

This paper proposes a prediction model of ionospheric TEC based on a grid-optimization GRU. This model was tested and evaluated using data from Eglin station as a case study. A detailed comparison and analysis were conducted against existing models, including SML-based, CCIR, and URSI models. The results show that the GRU model has significant advantages in the prediction of TEC. Specifically, the RMSE of the GRU model was 72.73%, 72.64%, and 57.38% lower than those of the CCIR, URSI, and SML models, respectively. Additionally, the MAE was reduced by 77.73% and 77.58% compared to the CCIR and URSI models, respectively, and by 68.94% compared to the SML-based model. These results indicate that the GRU model can effectively capture the long-term trend of TEC and shows good predicting effects at different seasonal and hourly scales. Similarly, these results can demonstrate that the GRU model has broad applicability and practical value and can be extended to the prediction of other ionospheric parameters (such as foF2 and hmF2). In the future, we plan to further research and verify the GRU model using multi-station data to evaluate its TEC prediction performance across different stations.