A Machine Learning-Based Method for Modeling TEC Regional Temporal-Spatial Map

Abstract

In order to achieve the high-accuracy prediction of the total electron content (TEC) of the regional ionosphere for supporting the application of satellite navigation, positioning, measurement, and controlling, we proposed a modeling method based on machine learning (ML) and use this method to establish an empirical prediction model of TEC for parts of Europe. The model has three main characteristics: (1) The principal component analysis (PCA) is used to separate TEC’s temporal and spatial variation characteristics and to establish its corresponding map, (2) the solar activity parameters of the 12-month mean flux of the solar radio waves at 10.7 cm (F10.7₁₂) and the 12-month mean sunspot number (R₁₂) are introduced into the temporal map as independent variables to reflect the temporal variation characteristics of TEC, and (3) The modified Kriging spatial interpolation method is used to achieve the spatial reconstruction of TEC. Finally, the regression learning method is used to determine the coefficients and harmonic numbers of the model by using the root mean square error (RMSE) and its relative value (RRMSE) as the evaluation standard. Specially, the modeling process is easy to understand, and the determined model parameters are interpretable. The statistical results show that the monthly mean values of TEC predicted by the proposed model in this paper are highly consistent with the observed values curve of TEC, and the RRMSE of the predicted results is 12.76%. Furthermore, comparing the proposed model with the IRI model, it can be found that the prediction accuracy of TEC by the proposed model is much higher than that of the IRI model either with CCIR or URSI coefficients, and the improvement is 38.63% and 35.79%, respectively.

1. Introduction

The ionosphere is the part of the upper atmosphere between 60 km to 1000 km above the ground. It is affected by many factors, such as solar radiation, geomagnetic disturbance, and atmospheric activity. Furthermore, with the change in geographical location and solar activity, the ionosphere presents the inherent periodic variation of day, month, season, year, and complex synoptic and climatological changes [1]. In general, when observational radio waves transmitted by space geodetic techniques such as global navigation satellite system (GNSS) [2] and very long baseline interferometry (VLBI) [3] pass through the ionosphere, their phase velocity and group velocity due to the collision of free electrons in the ionosphere will produce a certain degree of interference, which is approximately proportional to the total electron content (TEC) along the signal propagation path [4]. The resulting delay effect can be up to tens of meters [5], thus mastering TEC is very crucial for these systems such as GNSS and VLBI, etc. Supposing we can master the distribution of electron content in the ionosphere, we will not only be quantitatively able to study the influence of the ionosphere on the propagation of radio wave signals but also able to summarize the variation laws of the ionosphere at different time and space scales. Furthermore, we will achieve advanced prediction and real-time processing of ionospheric disturbances and reduce the attenuation and variation in the signal propagation process.

Considering the importance of accurately reflecting the distribution of electron content in the ionosphere for improving applications such as satellite communication, navigation, positioning [6], measurement, and disaster prevention [7,8], TEC observation, research, and modeling have always been a hot topic of the exploration for decades. TEC can be directly measured by the ionospheric sounder or indirectly inverted by carrier phase delay during the transmission of the radio signal [9]. In the absence of measurement conditions or other extreme cases, TEC can be calculated by physical models [10,11], mathematical models [12,13], and empirical models [14,15]. Among them, the international reference ionosphere (IRI) is the most common empirical model and is also a well-known recommended international standard, which is often used as a comparison model to verify other models [16,17]. The IRI model has been steadily improved with newer data and better modeling techniques by the joint working group of the Committee on Space Research and the International Union of Radio Science (URSI) [18]. For some ionospheric parameters such as TEC and the critical frequency of the F2 layer (foF2), there are two sub-models with the corresponding coefficients provided by the URSI and the Consultative Committee of International Radio (CCIR). The latest version is IRI-2016 [18], which determines the electron density by extending the E and F-layer’s anchor points of peak electron densities. In particular, the maximum electron density is calculated by the NeQuick model [19].

The recent research has the following properties in modeling TEC. On the one hand, machine learning (ML) and deep learning (DL) methods have been introduced into the modeling of ionospheric parameters and played a good role in improving the accuracy of empirical models [20]. For example, Kaselimi et al. [21] propose a recurrent neural network (RNN) model based on deep learning to predict TEC. Some ionospheric TEC prediction models based on long short-term memory (LSTM) were proposed to predict TEC [22,23], and some ionospheric TEC maps have been proposed based on bidirectional LSTM algorithms in different regions [24,25]. A TEC prediction model based on a multi-step auxiliary LSTM algorithm was established and can effectively alleviate the increasing error with prediction time [26]. The TEC forecasting methods based on generative adversarial networks (GAN) [27,28] and conditional GAN [29] were proposed and improved the IRI [30]. At the same time, Han et al. have modeled ionospheric TEC by four different ML methods such as artificial neural network (ANN), LSTM networks, adaptive neuro-fuzzy inference system, and gradient boosting decision tree (GBDT) [31]. In addition, the nearest neighbor method [32], prophet model [33], and decision tree [34] based on ML are also introduced into the modeling of ionospheric TEC, providing new modeling ideas and excellent prediction results. On the other hand, compared with the global model, the regional model has made great progress and provided higher prediction accuracy. Therefore, to achieve higher accuracy prediction of specific regions, TEC prediction models focusing on local regions have gradually become a research hotspot in recent years. Recent achievements have finished in Japan [35], China [36], Korean Peninsula [37], African Region [38], Antarctic Region [39], the low latitude region [2], the sea region [31], etc. For example, a European TEC prediction model [40] based on thin-plate splines (TPS) interpolation was self-calibrated, making the model better accurate during geomagnetic storms. El-Diasty [41] combined a neural network (NN) algorithm and wavelet decomposition algorithm to establish a short-term forecast model of TEC, which can adapt to various ionospheric environments more flexibly and has good robustness. A regional TEC model was established using the single-layer model and proved superior to the IRI model [42]. Constantly updated data measurement methods, data processing techniques, and modeling methods are designed to improve TEC prediction accuracy for military and civilian applications.

The TEC estimating methods can divide into the Long-term Prediction Method (LTPM) and the Short-term Forecast Method (STFM). LTPM is often used to predict the monthly statistics of VTEC, and STFM refers to forecasting TEC for several hours or days in the future. Unlike STFM, which pays more attention to the diurnal variation of TEC, LTPM pays more attention to the periodic variation trend of TEC monthly median value [43]. Considering the previous investigation, the prediction accuracy of the IRI model in some parts of Europe was low due to the complex dynamic change characteristics of the ionosphere in Europe, which needed to be urgently improved [42]. Compared with previously used ML methods such as RNN [21], LSTM networks [26], traditional NN [35], etc., the proposed ML-based modeling method is to make the model interpretable and simple. To improve the TEC long-term prediction accuracy in Europe, we propose an ML-based modeling method and use this method to establish an empirical prediction model of TEC for parts of Europe. This paper is organized as follows. First, Section 2 will specifically describe model creation and parameters determination. Then, the validation of the model and comparison with other models will be presented in Section 3. Finally, Section 4 will summarize the article and propose the direction for further improvement in the follow-up work.

2. Modeling

2.1. Modeling Method

Modeling methods of the TEC model can be divided into traditional and intelligent methods. Most traditional methods are based on mathematical principles; that is, different mathematical functions or spatial interpolation methods are used to characterize the temporal-spatial dynamic variations of TEC. However, with the rapid development of artificial intelligence, many intelligent methods are beginning to emerge, the methods based on ML are introduced into the model of TEC and organic combination with traditional methods to promote the leapfrog development of the TEC model. ML is interdisciplinary and builds a probabilistic statistical model based on data and uses the model to predict and analyze data [44]. Compared with the black box algorithm of the NN [45], the data processing ideas of ML are easy to understand, and the determined model parameters can be interpreted [46]. Therefore, ML is widely applied in the modeling of various ionospheric parameters. As shown in Figure 1, ML is a process of determining the three elements (model, algorithm, and strategy) and solving the following four problems:

• What kind of data are needed for training and predicting? The premise of ML is that the data used for training models have the same statistical characteristics, and the modeling process is to find the appropriate map to describe such statistical characteristics. This paper selects the monthly mean values of TEC (hereinafter referred to as TEC) from eight ionospheric observation stations in parts of Europe as the modeling dataset. The validation dataset consists of six stations in the same region, of which three are used for modeling, and three are not involved in the modeling process. The details are described in Section 2.2.
• How to choose a suitable model? This requires us to have an in-depth understanding of the purpose of modeling, the principles and applicable scope of different models, their advantages and disadvantages, and the characteristics of the training dataset before modeling. In this paper, the principal component analysis (PCA) function set is selected as the hypothesis space to construct the temporal and spatial characteristic map of TEC. PCA is a commonly used data analysis method known as eigenvector analysis [47]. It transforms the original data into a set of linearly independent ordered basis functions and correlation coefficients through a linear transformation. It can extract the main eigen components of the data and is often used for dimensionality reduction in high-dimensional data.

In 1901, Person [48] first proposed PCA as an effective method to establish an empirical model. And then, Dvinskikh [49] introduced it into the modeling of the ionospheric parameter empirical model for the first time in 1988. Subsequently, many empirical models of ionospheric parameters, such as foF2 [50], the propagation factor at 3000 km of the F2 layer [51], and TEC [52], were modeled using the PCA method. PCA can not only reduce the number of parameters involved in the modeling process but also reduce the amount of calculation and storage of intermediate data to achieve high accuracy and fast convergence [53]. Therefore, we apply PCA to decompose TEC orthogonally into temporal and spatial variables controlled by independent and unrelated processes so that TEC can be calculated through the orthogonal temporal and spatial map. The flow chart of modeling and validation is shown in Figure 2.

The nonlinear map of geographical location (latitude and longitude), solar activity, month, and universal time to TEC (unit is TECU) constructed based on PCA theory can be expressed as the harmonic function:

$$\mathrm{TEC}\left(\lambda ,\phi ,p,r,m,t\right)={\displaystyle \sum _{n=1}^N{W}_n\left(\lambda ,\phi \right)}\cdot \hat{\mathrm{T}}\mathrm{EC}\left(p,r,m,t\right),$$

(1)

where λ is the geographical latitude, φ is the geographical longitude, p is the average solar radio flux of 10.7 cm in 12 months (F10.7₁₂), r is the average number of sunspots in 12 months (R₁₂), m is the month, t is the given universal time, N is the number of ionospheric observation stations used for modeling, $\hat{\mathrm{T}}\mathrm{EC}\left(p,r,m,t\right)$ is the temporal map of TEC, and W_n(λ, φ) is the weight coefficients of the spatial map of TEC (details are provided in Section 2.3).

• How to determine the parameters of the model? ML is a supervised learning method [44] that obtains the target model and the corresponding algorithm through training. The least-squares (LS) regression analysis algorithm is one of the criteria used to determine the parameters of the regression model. Due to its advantages of simple calculation, short modeling time, easy understanding, and being highly sensitive to outliers, this algorithm has been widely used in parameter prediction and time series modeling in many fields. And LS shows good performance; therefore, this is also the reason why we choose it for modeling.
• How to evaluate the performance of the model? First, we must develop a unified standard to determine the optimal model and its correlation coefficients from the hypothesis space. Therefore, we calculate the following parameters as the evaluation standard:

  (a)

  Root mean square error (RMSE):

  $$\mathrm{RMSE}=\sqrt{\frac{1}{H}{\displaystyle \sum _{h=1}^H{\left({\mathrm{TEC}}_o{}^h-{\mathrm{TEC}}_p{}^h\right)}^2}},$$

  (2)

  (b)

  Relative root mean square error (RRMSE):

$$\mathrm{RRMSE}=\sqrt{\frac{1}{H}{\displaystyle \sum _{h=1}^H{\left(\frac{{\mathrm{TEC}}_o{}^h-{\mathrm{TEC}}_p{}^h}{{\mathrm{TEC}}_o{}^h}\right)}^2} }\times 100,$$

(3)

where TEC_o^h is the hth observed value of TEC, TEC_p^h is the hth predicted value of TEC, and H is the total statistical amount.

2.2. Modeling Dataset

As previously mentioned, ML is a data-driven modeling approach. In simple terms, the process of ML can be summarized as determining the learning model, strategy, and algorithm based on the determined dataset, finally finding the optimal model through training, then using the proposed model to predict the unknown data. Therefore, the selection of appropriate modeling data is an important prerequisite to ensuring the accuracy of modeling.

The data used for modeling and validation in this paper are obtained from the National Oceanic and Atmospheric Administration (NOAA) and the Global Ionospheric Radio Observatory (GIRO). This paper focuses on exploring an ML-based method for modeling the TEC temporal-spatial map, and the TEC provided by NOAA and GIRO is input data to test this method’s performance. These data have good credibility and were selected for modeling, as reported by [54]. The data information is listed in

Table 1. Modeling and validation stations data information.

No.	Station	Latitude (°N)	Longitude (°E)	Duration of Valid Data	Data Volume	Data Sources		Receiver [57,58]	Including in Modeling
						NOAA [55]	GIRO [56]
1	Athens	38.00	23.50	2004–2021	4948	√	√	DPS-4	Yes
2	Chilton	51.60	−1.30	2005–2009	1417	√	√	DPS-1	No
3	Dourbes	50.10	4.60	2004–2021	5160	√	√	DGS-256	Yes
4	El Arensillo	37.10	−6.70	2004–2017	3834	√	√	DGS-256	Yes
5	Fairford	51.70	−1.50	2004–2021	4726	√	√	DGS-256	Yes
6	Nicosia	35.10	33.20	2013–2016	1546	-	√	DPS-4D	No
7	Pruhonice	50.00	14.60	2004–2021	5136	√	√	DPS-4	Yes
8	Rome	41.80	12.50	2006–2021	4584	√	√	DPS-4	Yes
9	Roquetes	41.00	0.00	2004–2021	5160	√	√	DGS-256	Yes
10	San Vito	40.60	17.80	2004–2021	4779	√	√	DGS-256	Yes
11	Sopron	47.63	16.72	2019–2021	759	-	√	DPS-4D	No

. We can download the required data by selecting the corresponding time, observed station, and data type, through the official websites of NOAA [55] and GIRO [56]. The downloaded data can be saved as a text file in which recorded 24 h of TEC data for each day of the year at a station; that is, each line gives the TEC at one moment. Specifically, part of the data of the Pruhonice, Roquetes, Rome, etc., stations are obtained through NOAA, part of the data of the Athens, Fairford, Dourbes, etc., stations, and all the data of the Sopron and Nicosia stations used for validation are obtained through GIRO.

Figure 3 shows the geographical distribution of the 11 ionospheric stations used for modeling and validation in this paper in parts of Europe. The monthly mean valid data volume of TEC is 42,049 in total. To ensure the data continuity and the model’s validity, we selected that the continuous period of each station’s valid data is longer than one solar activity cycle, which covers at least one low solar activity year and one high solar activity year (as shown in Figure 4). As mentioned above, the observation data of different periods at the same station are obtained from NOAA and GIRO, respectively. Considering that the TEC data provided by GIRO has a longer continuous period and a larger amount, the missing periods in GIRO were supplemented by NOAA and then merged into a group for modeling and validation. For some stations whose valid data continuous period is less than one solar activity cycle, such as the Sopron, Nicosia, and Chilton, we take them as validation stations and not as modeling stations to participate in the modeling process. For the collected data, we first preprocessed them in the following two steps:

(1)

Considering that the observation period of different stations includes 15 min, 30 min, and 60 min, to ensure the uniformity of data format and avoid introducing errors, we choose 60 min as the sampling period to ensure data consistency.

(2)

In order to meet the modeling requirements of the Long-term regional ionospheric TEC prediction model, the monthly 24-h TEC mean values of each TEC observation station are calculated, and one observation station corresponds to a standardized modeling data file. After the above processing, we selected 8 stations for modeling and 6 stations for validation. Among the validation stations, three participated in modeling and three did not.

2.3. Modeling Determination

2.3.1. Modeling of Temporal Characteristics

It is well known that photoionization is the main mechanism of ionospheric plasma generation, and solar radiation’s energy mainly determines photoionization’s intensity. The main parameters used to describe solar activity are the flux of the solar radio waves at 10.7 cm (F10.7) and sunspot number (R), which show similar periodic variation due to the influence of the corona and chromosphere. The 12-month mean F10.7 (F10.7₁₂) is used to characterize the energy of extreme ultraviolet radiation (EUV) in the empirical model of solar radiation intensity [59]. The 12-month mean R (R12) reflects the average intensity of solar activity. Specifically, in a month, F10.7₁₂ has 24 data points, corresponding to the 24 h of a day; that is, each hour corresponds to a value of F10.7₁₂. The R₁₂ and TEC are consistent with F10.7₁₂. Studies have shown that F10.7₁₂ and R12 can better reflect photoionization intensity than daily F10.7 and R [60]. In addition, Sivavaraprasad [61] also used F10.7₁₂ and R₁₂ to establish a TEC prediction model suitable for low latitude areas of the equator, which was proved to have a great prediction effect, further illustrating the correctness and applicability of using F10.7₁₂ and R₁₂ to establish a long-term TEC prediction model.

Figure 5 compares F10.7₁₂, R₁₂, and TEC in the same period using the data of the Fairford station at UT 0:00 from 2004 to 2020. It is evident that TEC keeps the same variation cycle with F10.7₁₂ and R₁₂; that is, rising or falling at the same time points. However, the variation of TEC in detail is slightly different from that of F10.7₁₂ and R₁₂. The value of TEC increased significantly in high solar activity years, showing prominent annual variation characteristics, which are also drawn in [62,63]. Furthermore, we calculated the correlation coefficients between TEC and different solar activity parameters in 12 months and 24-h, which are given in Figure 6. The circle size corresponds to the correlation coefficients value. It can be seen that the corresponding correlation coefficients of the three solar activity parameters and TEC are all concentrated above 0.8, indicating that the three solar activity parameters have an obvious correlation with TEC. However, there are still some differences in the correlation coefficients of different parameters at the same time points. For example, at 19h in February, the correlation coefficient between F10.7₁₂ and TEC is significantly greater than that of R₁₂.

Considering the coupling effect of two solar activity parameters on TEC in different aspects, F10.7₁₂ and R₁₂ are introduced into the temporal model together as independent variables. This paper focuses on the prediction and analysis of TEC, so we assume that the prediction of solar activity parameters is known [64,65]. The time resolution of the temporal model is 1h. Given specific geographic coordinates and universal time, we can establish a temporal map from solar activity parameters and periodic oscillations to TEC, as shown in (4):

$$\begin{array}{l}\mathrm{TEC}=\hat{\mathrm{T}}\mathrm{EC}(p,r,m,t)\\ ={\displaystyle \sum _{k=0}^K}{\displaystyle \sum _{l=0}^L[c_{k,l}^p{p}^l\cdot \cos (2\mathsf{\pi }km/12)}+s_{k,l}^p{p}^l\cdot \sin (2\mathsf{\pi }km/12)+,\hfill \\ c_{k,l}^r{r}^l\cdot \cos (2\mathsf{\pi }km/12)+s_{k,l}^r{r}^l\cdot \sin (2\mathsf{\pi }km/12)]\hfill \end{array}$$

(4)

where p is F10.7₁₂, r is R₁₂, m is the month, t is the given universal time, K represents the upper bound of the time cycle, with values of 1, 2, 3, and 4 respectively representing annual, semi-annual, seasonal, and monthly cycle. L represents the periodic variation of the solar activity; that is, the l-order harmonic of the solar activity index is taken for regression. The values of K and L are determined by regression analysis. In addition, the coefficients $c_{k,l}^p{,s}_{k,l}^p{,c}_{k,l}^r{,s}_{k,l}^r$ are determined by the LS regression.

2.3.2. Modeling of Spatial Characteristics

TEC shows the irregular and uneven distribution in space, and the existing ionospheric observation methods cannot achieve global observation. Therefore, we can obtain several observed samples from ionospheric observation stations in different regions. Furthermore, these observations can reflect part of the characteristics of ionospheric TEC spatial distribution and can predict the characteristics of TEC in the unknown regions accordingly.

Considering the modeling data volume, the spatial distribution of observations, and the final purpose of modeling, we selected the modified Kriging interpolation method as the basic algorithm for spatial characteristics modeling. And then, it is further optimized according to the modeling data to obtain a better prediction result. The specific process is shown in Figure 7.

Firstly, the covariance function is generated by using the regionalization variables that satisfy second-order stationary and eigen hypothesis. Then, we can determine the variation function using the ionospheric distance between the observation and reconstruction stations. Namely, we can estimate the spatial autocorrelation between TEC samples. Secondly, with the minimum variance as the standard, the weight coefficients of different observation stations are solved to determine the Kriging interpolation function. Finally, the determined Kriging interpolation function is used to calculate the TEC of the reconstruction stations, and the continuous TEC data plane is generated.

In summary, the general expression of TEC spatial reconstruction based on modified Kriging [66] can be written as follows:

$$\tilde{\mathrm{T}}\mathrm{EC}={\displaystyle \sum _{n=1}^N{W}_n\cdot \mathrm{TEC}(n),}$$

(5)

where $\tilde{\mathrm{T}}\mathrm{EC}$ is the reconstruction value, N is the total number of modeling stations, TEC(n) is the TEC observation value of the nth observation station, and W_n is the weight coefficients of the nth observation station. To ensure the unbiased reconstruction result, the weight coefficients should meet the following requirements:

$${\displaystyle \sum _{n=1}^N{W}_n=1,}$$

(6)

the weight coefficients are appropriately selected to make the reconstruction variance less than any linear combination of observation values. The minimum reconstruction variance is obtained by the appropriate variation function, whose equation is defined as:

$$\gamma (d_{nj})=\frac{1}{2}E[{(\mathrm{TEC}(n)-\mathrm{TEC}(j)]}^2=\frac{1}{2}E[{(\mathrm{TEC}(n)-\mathrm{TEC}(n+d_{nj})]}^2,$$

(7)

where d_nj is the ionospheric distance between the nth and jth station, TEC(n) is the observation at the nth station, and TEC(j) is the observation at the jth station whose distance is d_nj from the nth station. According to (7), the variation function reveals the variation pattern in the entire spatial region, and the commonly used variation functions are Gaussian, exponential, sinusoidal, and linear models [67].

In conclusion, we can use the following equation to determine the optimal weight coefficients:

$${\displaystyle \sum _{n=1}^N\gamma (d_{nj})\times W_n=\gamma (d_{j0})-\Phi , j=1,2\dots N,}$$

(8)

where d_nj is the ionospheric distance between the nth and jth station, γ(d_nj) is a variation function expressed as the ionospheric distance between the nth and jth station, d_j0 is the ionospheric distance between the jth and reconstruction station, γ(d_j0) is a variation function expressed as the ionospheric distance between the jth and reconstruction station, W_n is the weight coefficients of the nth station, N is the total number of modeling stations, and Φ is the Lagrange multiplier.

Combining (8) and the above, it can be seen that the variation function γ(d_nj) of the TEC spatial reconstruction is the determination of the ionospheric distance between observation stations.

The spatial distance between two observed stations differs from the mathematically defined distance. The spatial resolution of the model can be accurate to 1 degree by multiplying 1 degree in geographic coordinates. Therefore, based on the linear variation model, this paper adopts the modified Euclidean distance with a scale factor (SF) as the ionospheric distance [68]; that is, the distance between any two observed stations can be expressed as:

$$d_{nj}=\sqrt{{({\phi }_n-{\phi }_j)}^2+{\left[SF\cdot ({\lambda }_n-{\lambda }_j)\right]}^2,}$$

(9)

where (φ_n,λ_n), and (φ_j,λ_j) represent the longitude and latitude coordinates of the nth and the jth ionospheric observed station, respectively; SF is the scale factor of correlation distance, and is related to the region located and reflects the correlation distance between ionospheric parameters in a region to some extent. Therefore, the goal of spatial reconstruction is to determine the optimal value of SF so that the reconstruction value is as close as possible to the observation value.

In the following subsection, we will discuss how to determine SF through cross-validation with RRMSE as the evaluation standard and then give the determined spatial reconstruction model.

2.4. Parameter Determination

2.4.1. Temporal Model Determination

Figure 8 shows the correlation coefficients between the observed and predicted TECs with F10.7₁₂, R₁₂, and dual-parameter (F10.7₁₂+R₁₂) regression for 24 h in the specific year of 2008, 2011, and 2014. It can be seen that the predicted TECs obtained by dual-parameter regression are highly correlated with the observed TECs compared with those obtained by single-parameter regression of F10.7₁₂ and R₁₂. And the correlation coefficients of dual-parameter regression are closer to 1, which further confirms the correctness of introducing the dual-parameter into the model. In particular, the correlation coefficients are higher during periods of high solar activity than during other periods of solar activity, which has also been mentioned in [69,70].

Figure 9 shows the temporal regression results under different solar activity parameters, and the RMSEs and RRMSEs with F10.7₁₂, R₁₂, and the dual-parameter. It is easy to see that the RRMSE of temporal regression with dual-parameter at the same time is less than that of single-parameter regression under the same conditions. Moreover, the regression results coincide with the TEC observations, in which some outliers were filtered out. Therefore, we simultaneously introduce F10.7₁₂ and R₁₂ into the temporal regression model.

Figure 10a shows the RRMSE of TEC prediction when K and L take different values in the case of dual-parameter is introduced into the temporal map. When L is fixed, the value of K goes from 1 to 2 and then to 3, and the regression error decreases significantly, whereas K goes from 3 to 4, and the regression error does not change significantly. Therefore, we take K equal to 3 as the upper bound of the time cycle. When K is 3, the dual-parameter regression error corresponding to different L and the time required for regression are shown in Figure 10b. L increased from 3 to 5, the corresponding regression error decreased significantly, and regression time has not increased dramatically. In contrast, L rose from 5 to 7, the corresponding regression error decreased slightly, and the regression time increased sharply. Therefore, considering the regression error and regression time comprehensively, we take L equal to 5.

In conclusion, the temporal map (2) of TEC can be written as:

$$\begin{array}{l}\mathrm{TEC}=\hat{\mathrm{T}}\mathrm{EC}(p,r,m,t)\\ ={\displaystyle \sum _{k=0}^3}{\displaystyle \sum _{l=0}^5[c_{k,l}^p{p}^l\cdot \cos (2\mathsf{\pi }km/12)}+s_{k,l}^p{p}^l\cdot \sin (2\mathsf{\pi }km/12)+,\hfill \\ c_{k,l}^r{r}^l\cdot \cos (2\mathsf{\pi }km/12)+s_{k,l}^r{r}^l\cdot \sin (2\mathsf{\pi }km/12)].\hfill \end{array}$$

(10)

The undetermined coefficients in the temporal map can be determined by using the LS method with the RRMSE as the evaluation standard. Figure 11 shows the temporal regression coefficients of the Dourbes station as an example. It can be seen that the temporal regression coefficients increase significantly at noon when the solar radiation is the strongest compared with other times.

2.4.2. Spatial Model Determination

Considering the modeling data and spatial distribution of the modeling stations, we select the cross-validation of 8 modeling stations to determine the spatial reconstruction model. The specific process is shown in Figure 12. First, the TEC observations of the reconstruction station are removed from the modeling dataset. Then, the remaining observed TEC of the modeling stations (represented as the reconstruction dataset in Figure 12) were used to carry out spatial interpolation for the reconstruction stations, and the cycle was carried out successively until the TEC reconstruction of all stations was completed. And RRMSE was used as the evaluation standard to determine the optimal SF.

Figure 13 shows the RRMSE of the spatial reconstruction results against different SF. The minimum reconstruction error is 13.49%, and the corresponding SF is 1.15. Accordingly, the ionospheric distance between two ionospheric observation stations can be expressed as:

$$d_{nj}=\sqrt{{({\phi }_n-{\phi }_j)}^2+(1.15\cdot {({\lambda }_n-{\lambda }_j)}^2)},$$

(11)

where (φ_n,λ_n) and (φ_j,λ_j) represent the longitude and latitude coordinates of the nth and the jth ionospheric observation station, respectively.

After the definition of the ionospheric distance is determined, the Lagrange extreme algorithm can be applied to solve the variation function (8) and spatial interpolation (5), then the spatial reconstruction weight coefficients of each observation station can be determined. Figure 14 shows the spatial reconstruction weight coefficients corresponding to the 8 modeling stations used in this paper. It can be seen that different modeling stations have different weights on reconstructed TECs of the same reconstruction station. Specifically, the closer the spatial distance between the modeling station and the reconstruction station, the greater the corresponding weight coefficients of these stations.

To sum up, we determine the temporal and spatial reconstruction model of TEC, as well as the harmonic function composed by them. To validate the correctness and effectiveness of the proposed model, we will take RMSE and RRMSE as the evaluation standard to test the model by comparing the observed TECs of the validation stations and the predicted TECs calculated by using the empirical model determined by the above process. Furthermore, the proposed model is compared with the IRI model to prove its superiority.

3. Validation

As described in Section 2, the Nicosia, Chilton, and Sopron stations, which have less than 11 years of continuous data and are not involved in modeling, are used as validation stations. Meanwhile, to satisfy the comprehensiveness of validation, the Roquetes, San Vito, and Dourbes stations are selected from the modeling stations for validation. From the perspective of space, the geographical location distribution of validation stations is shown in Figure 15.

Considering the distribution of the modeling region, we took 10° east longitude as the dividing line, adhering to the principle of taking 3 stations for each side. From the perspective of time, the stations used for validation cover the four seasons of spring, summer, autumn, and winter in low and high solar activity years. The selection of validation stations fully ensures the non-overlapping of spatial and temporal distribution and the comprehensiveness of validation. The details are listed in

Table 2. Validation stations data information.

No.	Station	Latitude (°N)	Longitude (°E)	Solar Activity Epoch	Year	Season	Month	Including in Modeling
1	San Vito	40.60	17.80	High	2015	Spring	May	Yes
2	Roquetes	41.00	0.00	High	2014	Summer	September	Yes
3	Dourbes	50.10	4.60	Middle	2004	Autumn	October	Yes
4	Nicosia	35.10	33.20	Middle	2016	Winter	February	No
5	Sopron	47.63	16.72	Low	2020	Summer	August	No
6	Chilton	51.60	−1.30	Low	2009	Spring	June	No

.

For a given geographic latitude and longitude coordinates, year, month, universal time, F10.7₁₂, and R₁₂, the TECs for any time and location can be predicted using the process shown in Figure 16. The specific process is as follows:

(1)

Using the temporal regression map (10) to calculate the time component $\hat{\mathrm{T}}\mathrm{EC}$.

(2)

Using the temporal regression value $\hat{\mathrm{T}}\mathrm{EC}$ and the given geographical latitude and longitude coordinates, the final predicted $\stackrel{\mathrm{~}}{\mathrm{T}}\mathrm{EC}$ value can be obtained by the modified Kriging interpolation method (5) for spatial reconstruction.

The predicted TECs of the 6 validation stations calculated by the proposed model (marked as Prop., black) are compared with the observed TECs (marked as Obs., blue) and the predicted TECs calculated by the IRI model with the CCIR coefficients (marked as CCIR, green) and the URSI coefficients (marked as URSI, red), respectively. Figure 17 shows the scatter plots of monthly mean values of TEC for a given year and month calculated using three models during three periods of solar activity.

To better evaluate the performance of the models, the statistical RMSEs and RRMSEs of predicted TECs for different models are listed in

Table 3. RMSE and RRMSE of different models in TEC prediction.

Statistical Analysis Item		RMSE (TECU)			RRMSE (%)			Difference between CCIR and Prop. (%)	Difference between URSI and Prop. (%)
		CCIR	URSI	Prop.	CCIR	URSI	Prop.
Station	San Vito	3.10	2.17	2.00	18.56	14.03	9.11	9.45	4.92
	Roquetes	3.80	4.63	2.17	32.30	42.25	11.74	20.56	30.51
	Dourbes	3.68	3.54	1.08	43.45	44.27	12.58	30.87	31.69
	Nicosia	4.92	3.86	2.46	98.65	78.18	21.71	76.94	56.46
	Sopron	2.01	2.09	0.51	42.30	48.57	7.04	35.26	41.53
	Chilton	1.56	1.83	0.53	32.34	41.60	8.74	23.60	32.86
Solar activity	High	3.47	3.61	2.08	26.34	31.48	10.50	15.84	20.98
	Middle	4.34	3.70	1.90	76.23	63.53	17.75	58.48	45.78
	Low	1.80	1.97	0.52	37.65	45.22	7.94	29.71	37.28
Average		3.37	3.20	1.65	51.39	48.55	12.76	38.63	35.79

. At the end of Table 3, we give the average prediction error when using different models to predict any station; that is, the error of different models when predicting six validation stations is calculated again to obtain the average value.

Combined with Figure 17 and Table 3, we can draw the following four conclusions:

(1)

The TEC curve predicted by the IRI model and the model proposed in this paper has a highly fitting variation trend with the observed TEC curve. That is, there is an apparent variation trend of the day, month, season, and year as well as periodic variation with the solar activity. Furthermore, the maximum TEC value in high solar activity years is obviously higher than that in low solar activity years, which further confirms the close relationship between the variation of TEC and solar activity.

(2)

The prediction curve of the proposed model is closer to the TEC observation curve than that of the IRI model. Especially in Roquetes and Dourbes, the prediction effect of the proposed model is significantly better than that of the IRI model.

(3)

In Nicosia, the prediction error of both the IRI model and the proposed model is the maximum. The RRMSE of the IRI model using CCIR coefficients and URSI coefficients and the proposed model are 98.65%, 78.18%, and 21.71%, respectively, in predicting the monthly mean values of TEC of Nicosia station. The main reason is that the station is located at the junction of Asia and Europe, and the dynamic characteristics of the ionosphere are complex. In addition, the valid observation data provided by the observation station are insufficient, or the observation data provided by the limited ionospheric observation station are not accurate due to the limitation of observation technology, which significantly impacts the modeling accuracy.

(4)

When the IRI model was used to predict TEC of Sopron and Chilton stations, it could be found from the results that both CCIR coefficients and URSI coefficients showed double peaks that did not conform to the law of diurnal variation of TEC. This is one reason that the prediction error of the IRI model is higher than that of the proposed model.

To make the data more intuitive, Figure 18 shows the predicted RMSEs and RRMSEs of three models in six stations during three solar activity epochs, respectively. By comprehensively analyzing Figure 18 and Table 3, we can draw the following three conclusions:

(1)

Compared with the IRI model with CCIR and URSI coefficients, the predicted accuracy of the proposed model is improved by 38.63% and 35.79%, respectively, which is a significant improvement in parts of Europe. Similarly, other study [42] have also provided several sets of comparison graphs showing that the IRI model does have a significant deviation in TEC prediction, which further demonstrates the effectiveness of the PCA decomposition method in ionospheric parameters modeling and the necessity of the proposed model.

(2)

The prediction error of the proposed model is the minimum at the low solar activity epoch, which is 7.94%, and the maximum at the middle solar activity epoch, which is 17.75%, both of which are lower than the IRI models. On the other hand, the IRI model with CCIR and URSI coefficients has the best predictions during the high solar activity year, and the prediction errors are 26.34% and 31.48%, respectively.

(3)

The IRI model is an empirical model applicable to the global scope, whereas the proposed model is a regional empirical model focusing on the parts of Europe region. The prediction accuracy of the regional model is better than that of the global model in specific regions. Namely, the regional model has better spatial adaptability and pertinence in ionospheric parameter prediction.

The temporal resolution of the proposed model is 1h, and the spatial resolution can be accurate to 1° × 1° in geographic coordinates. Similarly, Zhang et al. used a 2.5°× 5° spatial grid and 1 h temporal resolution interpolated updated IG12 index to drive IRI-2016 to generate ionospheric TEC values consistent with IGS-TEC [17]. Figure 19 shows the spatial distribution of TECs predicted by the IRI model based on CCIR and URSI coefficients and the proposed model over parts of Europe at 06:00 UT in May 2009 (low solar activity year). It can be seen that TECs predicted by the three models show a similar spatial distribution. In particular, compared with the IRI model based on CCIR coefficients, the variation of the proposed model is closer to that of the IRI model based on URSI coefficients, which is consistent with the numerical validation results mentioned above. Furthermore, it can be seen that the distribution of TEC contour lines at high latitudes is relatively scattered; that is, the TECs in this region are relatively gentle. While the TEC contour lines at low latitudes are more concentrated, indicating that the TEC variations in this region are more intense. In addition, the numerical range of TECs predicted by the three models is basically the same, and the changes in details are slightly different. Compared with the sampling values of the modeling stations, the predicted value of the model proposed in this paper is closer to the sampling value at a given station, which proves the validity and correctness.

Figure 20 shows the spatial distribution of TECs predicted by the IRI model based on CCIR and URSI coefficients and the proposed model over parts of Europe at 06:00 UT in June 2014 (high solar activity year). In common with Figure 19, the TECs predicted by the three models show a similar spatial distribution; that is, the TECs located in the northwest region are smaller, while the TECs in the southeast region are larger. On the whole, the TECs predicted by the IRI model based on the URSI coefficients are the largest, followed by the IRI model based on the CCIR coefficients, and the predicted TECs of the model proposed in this paper are the smallest, which is consistent with the results mentioned in [42]; that is, when the IRI model predicts regional TEC, there will be an overall numerical deviation. Compared with the sampling values of the modeling stations, the predicted values of the proposed model are closer to the sampling values at a given station, and this model effectively improves the accuracy of the spatial TEC prediction model.

4. Conclusions

This paper proposes an ML-based modeling method and uses this method to establish an empirical prediction model of TEC for parts of Europe. It has been proved that the monthly mean values curve of TEC predicted by the model proposed in this paper is highly consistent with the observed curve of TEC. Furthermore, it is found that the prediction accuracy of the proposed model is significantly better than that of the IRI model with CCIR and URSI coefficients by comparing the observed TECs and the predicted TECs calculated by the IRI model and the proposed model, respectively. The improvement is 38.63% and 35.79%, respectively, which confirms the effectiveness of the proposed model.

This model is proposed for parts of Europe. Later we hope that it can have better effects and provide early warning for natural disasters and sudden magnetic storms by collecting more observed data, expanding the application scope, and improving the prediction accuracy. We will further explore the modeling methods for other regions to improve and enhance this model, which is compared with different models to validate the proposed model’s effectiveness and correctness based on more collected data such as GNSS TEC. In addition, we will finish the STFM of TEC based on this paper and make more efforts to improve the temporal and spatial resolution of the model and achieve a more accurate and detailed prediction of TEC. At the same time, we hope that this modeling idea can be applied to modeling other ionospheric parameters, such as the peak height of electron density in the F2 layer (hmF2), the maximum electron density in the F2 layer (NmF2), etc., and can provide a reference for modeling in other directions.