Modeling Ionospheric TEC Using Gradient Boosting Based and Stacking Machine Learning Techniques

Accurately predicting and modeling the ionospheric total electron content (TEC) can greatly improve the accuracy of satellite navigation and positioning and help to correct ionospheric delay. This study assesses the effectiveness of four different machine learning (ML) models in predicting hourly vertical TEC (VTEC) data for a single‐station study over Ethiopia. The models employed include gradient boosting machine (GBM), extreme gradient boosting (XGBoost), light gradient boosting machine (LightGBM) algorithms, and a stacked combination of these algorithms with a linear regression algorithm. The models relied on input variables that represent solar activity, geomagnetic activity, season, time of the day, interplanetary magnetic field, and solar wind. The models were trained using the VTEC data from January 2011 to December 2018, excluding the testing data. The testing data comprised the data for the year 2015 and the initial 6 months of 2017. The RandomizedSearchCV algorithm was used to determine the optimal hyperparameters of the models. The predicted VTEC values of the four ML models were strongly correlated with the GPS VTEC, with a correlation coefficient of ∼0.96, which is significantly higher than the corresponding value of the International Reference Ionosphere (IRI 2020) model, which is 0.87. Comparing the GPS VTEC values with the predicted VTEC values based on diurnal and seasonal characteristics showed that the predictions of the developed models were generally in good agreement and outperformed the IRI 2020 model. Overall, the ML models used in this study demonstrated promising potential for accurate single‐station VTEC prediction over Ethiopia.


Introduction
The ionosphere, which is the upper portion of Earth's atmosphere, comprises ionized plasma that undergoes variations in its composition due to factors such as latitude and longitude, local time, season, solar and geomagnetic activity, and other factors.The electromagnetic wave propagating through this dynamic environment suffers a range delay whose magnitude depends on the frequency of the wave and the amount of the total electron content (TEC) (Hajra et al., 2016;Shi et al., 2022).TEC, which is the total number of electrons within a unit-square-meter column along a path through the ionopsphere, is a significant parameter characterizing ionospheric variability (Tang et al., 2022).For long-range radio communications, surveying, navigation, and other space weather-related operations, it is necessary to understand the changes in TEC.
Over the years, to better understand and mitigate the effect of ionospheric delay, the ionospheric community and others have investigated the ionospheric delay error in trans-ionospheric signals (Davies & Hartmann, 1997;Liu et al., 2020).There are, however, challenges associated with ionospheric studies due to the lack of observational data on the necessary time and spatial scales.This led to the development of global ionospheric models like NeQuick (Hochegger et al., 2000;Nava et al., 2008) and International Reference Ionosphere (IRI) models (Bilitza, 2001;Bilitza et al., 2011Bilitza et al., , 2017)).Despite their ability to improve several aspects of ionospheric modeling, different research findings (e.g., Habarulema et al., 2007Habarulema et al., , 2009;;Nigussie et al., 2013;Okoh et al., 2018;Tebabal et al., 2018) have highlighted the shortcomings of these global models in the African sector.Several neural network (NN)-based single station and regional models have been developed in the African region to fill these gaps and improve prediction accuracy (e.g., Habarulema et al., 2007Habarulema et al., , 2009Habarulema et al., , 2011;;Habyarimana et al., 2020;Okoh et al., 2016Okoh et al., , 2018;;Tebabal et al., 2018Tebabal et al., , 2019)).It was found from these studies that NN-based models are very promising at capturing the overall dynamics of ionospheric variations compared to global models.However, this does not imply that the NN models always provide accurate predictions in all cases as compared to other machine learning (ML) techniques.NN algorithms tend to overfit small data sets due to their need for large amounts of data to fully exploit their potential (Natras et al., 2022).
In recent years, gradient-boosting decision tree (GBDT)-based ML techniques like extreme gradient boosting (XGBoost) (Chen & Guestrin, 2016) and light gradient boosting machine (LightGBM) (Ke et al., 2017) have been created.These methods have been successfully used for modeling and forecasting of ionospheric TEC (e.g., Han et al., 2021;Natras et al., 2022;Zhukov et al., 2021).The findings have demonstrated that GBDT techniques are efficient and accurate in ionospheric modeling.Thus, adopting GBDT algorithms for ionospheric modeling is efficient in regions where there is a shortage of ionospheric TEC data, as these techniques are effective for both small and large data sets.Apart from that, GBDT-based algorithms are not expensive because they don't necessitate a significant amount of computational resources and training time compared to NN algorithms (Bentéjac et al., 2021).As a result, GBDT algorithms are relatively simple to optimize compared to NN techniques.
In previous studies conducted in the African region, ML methodologies other than GBDT-based algorithms were primarily used for modeling ionospheric vertical TEC (VTEC).Therefore, this study has been conducted to test the effectiveness of GBDT-based algorithms and their stacked integration to model VTEC.For this purpose, gradient boosting machine (GBM), XGBoost, LightGBM, and a stacked (STK) model of these three algorithms with a linear regression model are employed for a single-station VTEC study over Ethiopia.Additionally, the present study has incorporated model input parameters that represent the influence of the interplanetary magnetic field (IMF) and solar wind, which have not been utilized in prior neural network-based models in the African region.In order to validate their predictive capability, the performances of the ML models used in this study are also assessed by comparing them with the International Reference Ionosphere (IRI 2020) global ionospheric model.

Data and Data Preparation
In this study, we used hourly VTEC data obtained from a dual-frequency GPS receiver located over Addis Ababa, Ethiopia (ADIS, with geo lat: 9.035°N and geo long: 38.766°E).The values were derived using the calibration technique of Ciraolo et al. (2007) from 2011 through 2018.This data is publicly available at the Global Navigation Satellite System (GNSS) TEC calibration service provided by the International Center for Theoretical Physics (ICTP).To mitigate errors caused by multipath effects, only VTEC values obtained at elevation mask angles greater than 30°were considered.The calibrated VTEC data sampling was 30-s and was then averaged to hourly values.
Missing data is common when processing raw data from GNSS stations.As a result, there are many missing VTEC values at the ADIS GPS station.Therefore, to fill these gaps, the nearby GPS stations VTEC data from Ambo (ABOO, with geo lat: 8.992°N and geo long: 37.809°E) and Nazret (NAZR, with geo lat: 8.568°N and geo long: 39.291°E) was utilized.Figure 1 shows the Video Imputation with SoftImpute, Temporal-Smoothing and Auxiliary data (VISTA) TEC map (Sun et al., 2022), as well as the positions of the ADIS, ABOO, and NAZR GPS stations.
The fundamental principle used to obtain ionospheric TEC values from GPS observations is that GPS signals with varying frequencies encounter distinct ionosphere time delays as they pass through the same part of the ionosphere.A GPS signal with frequency f will experience an ionospheric time delay t, which can be given by Klobuchar (1996) as: where c is the speed of light in free space.Dual-frequency GPS receivers make use of two frequencies, L1 (1575.42MHz) and L2 (1227.60MHz), in order to compensate for the delay caused by the ionosphere.This particular receiver, operating at frequencies f 1 and f 2 , calculates the discrepancy in time delay between the two frequencies, given by Therefore, the measured time delay (Δt) between the L1 and L2 frequencies is utilized for the computation of the TEC along the path of the ray.The slant TEC (STEC) measurements made here are the sum of the actual slant TEC, the GPS satellite differential delay b S (satellite bias), and the receiver differential delay b R (receiver bias).Thus, the VTEC can be given by (Ciraolo et al., 2007;Rama Rao et al., 2006) where S(ɛ) is the obliquity factor (mapping function) with zenith angle, θ z at the ionospheric pierce point, defined by (Mannucci et al., 1993;Rama Rao et al., 2006) where R E is the mean radius of the Earth in km, h S is the height of the ionospheric pierce point, and ɛ is the elevation angle in degrees.
The variability of VTEC is modeled as a function of known physical and geophysical parameters.Several of these factors have been well documented, including solar and geomagnetic activity, seasonal changes, and diurnal variations (e.g., Habarulema et al., 2007Habarulema et al., , 2009;;Maruyama, 2007;Tebabal et al., 2018Tebabal et al., , 2019)).The variation associated with season and time of the day is effectively represented by the day number of the year (DOY) and the hour of the day (HR), respectively.The measure of solar activity is represented by the sunspot number and solar radio flux at 10.7 cm wavelength (F10.7 index).The planetary amplitude (ap index) and the disturbance storm time (Dst index) were used as inputs for geomagnetic activity.The solar wind plasma speed (SW speed) and the north-south component of the interplanetary magnetic field (IMF Bz) based on the Geocentric Solar Ecliptic System (GSE) were also used as input variables to repersent the effects of solar wind and IMF, receptively.The explicit list of the input parameters used for the models are presented in Table 1.
The parameter F10.7 was not used directly as an input parameter; instead, we used its 28-and 81-day moving averages.The data for the input variables was obtained from the Goddard Space Flight Center Space Physics Data Facility.Subsequently, we prepared our data set at intervals of 1 hr.

Modeling Techniques
This study utilized ML algorithms for the purpose of creating models.ML allows computers to recognize patterns, make predictions, and make decisions by analyzing and adapting to data rather than relying on explicit instructions (Zhou, 2021).Ensemble learning, a technique that combines multiple models to solve computational problems and enhance prediction accuracy, is employed.Bagging, boosting, and stacking are the most commonly used ensemble ML algorithms, and this study used models based on boosting and stacking methodologies (Sagi & Rokach, 2018;Yang, 2017).Boosting is a technique that improves the predictions of weak learners by adding them sequentially.This involves training a new weak learner model based on the errors of the previously learned models (Natekin & Knoll, 2013).In tree-boosting ensembles, decision trees are often used as weak learners.Decision trees are supervised learning techniques that can be used for tasks like classification and regression.In regression trees, the goal is to make predictions of continuous values, and the accuracy of these predictions is assessed by calculating the sum of squared differences between the predicted values and the actual values (Hastie et al., 2009;Rokach & Maimon, 2005).Gradient tree boosting is a boosting ensemble technique that uses a combination of decision trees and an additive model to minimize a loss function (Brownlee, 2016).

Gradient Boosting Machine (GBM)
A gradient-boosting machine uses a learning method that fits new models in succession to improve the accuracy of the response variable estimation.The GBM algorithm aims to construct predictive models using back-fitting and non-parametric regressions.Rather than creating just one model, the GBM begins by generating an initial model and continuously adjusts new models by minimizing the loss function to achieve the most accurate model (He et al., 2019).The algorithm's main concept is to create new base learners that are highly correlated with the negative gradient of the loss function and associated with the entire ensemble.The loss functions used can be any differentiable functions, but to provide a clearer understanding, if the error function is the classic squared-error loss, the training process will lead to successive fitting of errors (Natekin & Knoll, 2013).

Extreme Gradient Boosting (XGBoost)
XGBoost is a scalable and enhanced implementation of gradient-boosted decision trees.
XGBoost is an opensource library that was initially developed by Tianqi Chen in 2014 and now has contributions from many developers.XGBoost is highly scalable because of various system and algorithmic optimizations.These include a unique tree learning algorithm for sparse data, a weighted quantile sketch procedure for handling instance weights, and parallel computing for faster learning.XGBoost also allows data scientists to process large amounts of data on a desktop using out-of-core computation (Brownlee, 2016;Chen & Guestrin, 2016).XGBoost uses a term called objective function, which is the sum of the loss function and a regularization term.This term plays a crucial role in reducing overfitting by promoting smoother learning of weights.Like GBM, XGBoost constructs a successive extension of the objective function through the reduction of a loss function.XGBoost uses Taylor expansion of the loss function up to the second order to discover the best solution, which is then used to balance the complexity of the model and the decline of the objective function in order to prevent overfitting (Fafalios et al., 2020).

Light Gradient Boosting Machine (LightGBM)
LightGBM is a high-performing implementation of the gradient-boosting decision tree algorithm developed by Microsoft in 2017.It is designed to handle large data sets and improve prediction accuracy.It does this by using a leaf-wise tree growth approach, which focuses on nodes with the highest change in loss.Additionally, it incorporates techniques such as gradient-based one-side sampling (GOSS) and exclusive feature bundling (EFB) to enhance efficiency.The GOSS method selectively keeps instances with large gradients and drops instances with small gradients to better estimate information gain.This approach is more effective than random sampling, particularly when the range of information gained is wide.In sparse, high-dimensional data, features that do not occur together can be combined into one feature bundle to reduce the number of features using the EFB technique (Ke et al., 2017).

Stacking Ensemble Technique
A stacking technique is a type of ensemble ML algorithm that uses meta-learning techniques to find the best way to combine predictions from multiple base models.It involves two stages: training the base models and training the meta model.In the first stage, the original data is split into a training set and a testing set, and the training set is trained using k-fold cross-validation.In the second stage, the predictions from the base models are reassembled in the original order and used to create a new training set for the meta model.The predictions from the testing sets of the base models are combined to form the testing set for the meta model.Finally, the meta model is trained using this new data set (Lu et al., 2023).In this study we have applied stacking ensemble learning, using the linear regression, GBM, XGBoost, and LightGBM as the base models and the LR as the a final meta model.Linear regression is a modeling technique that linearly combine explanatory variables to predict a response variable (Hastie et al., 2009).

Development of the Models
We have used 8-year (2011-2018) GPS VTEC data at an hourly interval with 58% of the data from ADIS and 36% the data from ABOO and NAZR GPS stations.After utilizing VTEC data from ABOO and NAZR stations to complete the missing values, there is still a 6% absence in the overall data.We dropped the missing values and utilized the available VTEC data to train and test the ML models since we discovered that the use of any mathematical data imputation technique impacts the models' performance.Once the data was prepared, it was divided into training and testing sets.The testing set, which accounted for 20% of the total available data, consisted of data from 2015 and the first half of 2017.The remaining data was used to train the models and determine the best parameters.Selecting appropriate inputs is crucial for designing an effective ML model.It determines the model's ability to learn and generalize the relationship between the inputs and the target.In order to align the input variables with the VTEC, an approximation function can be used.This function establishes a nonlinear connection between the input data and the VTEC prediction output based on the input variables.As the function is not known, it is estimated by optimizing the ML algorithms for the purpose of VTEC prediction, as explained by Hastie et al. (2009).The steps we followed to develop the ML models are shown in Figure 2.  (3,4,5,6,7,8,9,10,11,12,13,14,15) subsample = 1 (0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1)learning_rate = 0.1 (0.01,0.03,0.04,0.05,0.06,0.08,0.1,0.15,0.2) n_estimators = 200 (100,150,200,250,300,350,400,450,500,600) XGBoost max_depth = 6 (3,4,5,6,7,8,9,10,11,12,13,14

Model Optimization
ML algorithms contain a set of parameters, called hyperparameters, that cannot be predicted from data and must instead be customized for a particular learning problem.Hyperparameters specify the complexity and construction of the model.Depending on the data and the problem, the optimal values of hyperparameters may differ, and they are often discovered by testing with varying combinations and analyzing how well each model performs.
Hyperparameters of boosting ensemble ML techniques that frequently require optimization include the maximum depth of the tree (max_depth), minimum loss reduction to create a new tree split (gamma), the number of trees used in ensemble learning (n_estimators), the fraction of samples to be used for fitting the individual base learners (subsample), the learning rate to reduce the gradient step (learning_rate), and the maximum number of leaves in each weak learner (num_leaves).
The RandomizedSearchCV algorithm was applied to the training data to determine the optimal hyperparameters for the three gradient boosting models used in this study.RandomizedSearchCV is an algorithm that combines random search with cross-validation to randomly select combinations of hyperparameters to train the model.Cross-validation is an approach to measuring the performance of a model by training it on a particular portion of input data and then testing it on a subset of input data that has not been used before (Rahmadayana & Sibaroni, 2021).The negative of the root mean-square error (RMSE) was used in the RandomizedSearchCV technique.
The selected optimal hyperparameters and the range of values used to search for the best hyperparameters are provided in Table 2.

Model Evaluation
To evaluate the effectiveness of the models, we have utilized various statistical parameters such as residual error (r i ), root-mean-square error (RMSE), mean absolute error (MAE), standard deviation of residual errors (σ), and correlation coefficient (R).These parameters are widely recognized and utilized to determine the performance of a model (Shi et al., 2022;Xiong et al., 2021).The equations are as follows: where N is the number of data points, VTEC Obs i is the GPS VTEC value, VTEC Pred i is the predicted VTEC value of the ith data point, VTEC Obs is the mean of GPS VTEC values, VTEC Pred is the mean of predicted VTEC, and ri is the mean of the residuals for i = 1, 2,…, N.

Performance of the Models
We utilized gradient boosting and stacking ML techniques to investigate their effectiveness in VTEC modeling.The data set for the years 2011-2014, 2016, 2018, and the second half of 2017 was used in developing the ML models.Once the establishment of the models was done, we saved the model hyperparameters for future applications.The data for 2015 and the first 6 months of 2017 was then used to test the models.The scatter plots of the predicted VTEC by each model against the GPS VTEC for the testing data are shown in Figure 3.The horizontal axis represents the observed values, while the vertical axis represents the predicted values.A scatter-fitted red-colored solid line is defined by y = f(x), where y represents the predicted value and x represents the GPS VTEC value.As seen from the figure, it is evident that the modeled VTEC values have a strong correlation with the observational values of GPS VTEC, with a correlation coefficient of R ≈ 0.96.This high correlation indicates that the ML models can precisely represent most of the variations and are capable of explaining about 92% of the variability of GPS VTEC.To enable further comparisons, we presented a contour plot in Figure 4 that highlights the differences in error distribution between GPS VTEC and predicted values for the testing periods.The x-axis represents the DOY, while the y-axis represents universal time coordinates (UTC).The maximum differences between GPS VTEC and predicted VTEC by GBM, XGBoost, LightGBM, and Stacked models are 27.5, 28.0, 29.5, and 27.0 TECU, respectively.The LightGBM and Stacked models showed 72% and 75% of data points, respectively, with absolute residual errors within the range of 0 and 5 TECU.Meanwhile, both GBM and XGBoost models demonstrated 73% of data points.Therefore, it can be concluded that the stacked model is slightly better at reducing errors when compared to other GBDT-based techniques.
Table 3 is a summary of the R, RMSE, MAE, and σ values computed using GPS VTEC and modeled VTEC for both the ML models (from 4 algorithms) and the IRI 2020 model.The RMSE, MAE, and σ values for ML models are considerably smaller than those obtained from the IRI 2020 global model, indicating superior performance.These statistical analyses confirm that the ML models are well-trained with the training data and exhibit accurate predictions for new data sets.Although there is no significant difference between statistical values of the four ML  The results of the current study indicate that the performance level of the ML models developed is comparable to, or in most cases, better than, other existing single-station and regional models applied in the low-latitude African region.The models showed exceptional prediction accuracy with minimal error.The testing data produced an RMSE value of approximately 5.3 TECU for the three GBDT models, while the stacked model achieved an RMSE value of 5.1 TECU.In contrast, previous studies by Tebabal et al. (2018) on a single-station feed neural network-based model over Arba Minch, Ethiopia, yielded R and RMSE values of 0.95 and 6.0 TECU, respectively, which are less favorable than those obtained in our study.Similarly, in another single-station neural network-based model over Mbarara, Uganda, by Habyarimana et al. (2020), an RMSE value of 5.7 TECU was found, which exceeded the RMSE values achieved in the present study.Okoh et al. (2016) reported that the RMSE values for a neural network-based model over Nigeria ranged from 5.4 to 12.6 TECU, which is significantly higher than the values obtained in our study.Another regional neural network-based model over Ethiopia, developed by Tebabal et al. (2019), reported RMSE values ranging from 3.8 to 6.5 TECU, which are comparable to the values obtained in this study.

Day-To-Day Variability
In this section, the day-to-day variations between the observed and predicted VTEC values are presented.The performance of the models was tested on both quiet and disturbed days based on the unseen data.The quiet and disturbed days for the analysis were chosen based on the Dst-index values.

Quiet Time
The performance of the models under geomagnetically undisturbed conditions was evaluated by comparing the predicted VTEC values with GPS VTEC and IRI 2020 model predictions on selected quiet days ( 20 nT ≤ Dst ≤ 20 nT) in September and December 2015, and March and June 2017.Figure 6 shows the comparison for the quiet days of September 2015, December 2015, March 2017, and June 2017.In the figure, the red solid lines represent GPS VTEC, while the blue, black, lime (yellow-green), dark violet, and orange-colored lines depict VTEC values predicted by the GBM, XGBoost (XGB), LightGBM (LGBM), STK, and IRI 2020 models, respectively.The results suggest that the ML models consistently exhibit better agreement with GPS VTEC predictions on these selected quiet days.In each instance, the VTEC predictions by the ML models closely align with GPS VTEC, outperforming the IRI 2020 model, as shown in the plots for the quiet days.

Disturbed Conditions
To assess how well the models can predict VTEC during geomagnetic disturbances, we compared the predicted VTEC values of the four ML and IRI 2020 models with the GPS VTEC.We analyzed the comparisons between predicted VTEC and GPS VTEC during intense ( 200 nT < Dst ≤ 100 nT), moderate ( 100 nT < Dst ≤ 50 nT), and weak ( 50 nT < Dst ≤ 30 nT) geomagnetic storms.We selected three intense storm days on 23 June 2015 (minimum Dst = 198 nT), 7 October 2015 (minimum Dst = 130 nT), and 28 May 2017 (minimum The plots show that the ML models effectively captured the variations in VTEC and were closely aligned with GPS VTEC measurements, unlike the IRI 2020 model, which had less accurate VTEC predictions in most instances.During the intense storm period from 21 to 25 June 2015, the ML models slightly overestimated GPS VTEC but were still well captured by the IRI 2020 model in the early stages of the storm.The ML models accurately predicted VTEC during the intense storm day and the subsequent recovery phase, while the IRI 2020 model underestimated it.During the intense storm period from 7 to 9 October 2015, the storm caused an enhancement in VTEC the following day; however, the models slightly underestimated the enhancement in GPS VTEC.This may be because the models may not have obtained the necessary information from the training data to

Seasonal Analysis
To compare the predictive performance of different models in predicting seasonal variations of VTEC, we used the 24-hr monthly average VTEC data for selected months in various seasons as the testing data.In 2015, we selected March, June, September, and December, while March and June were selected in 2017.Figure 11 present the comparisons of 24-hr monthly mean variations of GPS VTEC and VTEC predicted by the GBM, XGB, LGBM, STK, and IRI 2020 models for the selected months.In the plots, GPS VTEC is represented by the redcolored line, while VTEC predicted by the models is represented by the blue, black, lime, dark violet, and orangecolored lines, respectively.As shown in the plots, the ML models successfully predicted VTEC values that align with GPS VTEC.However, the IRI 2020 model predictions showed significant deviations from GPS VTEC during the selected months.

Conclusions
This paper compares the performance of four ML models for estimating ionospheric VTEC.The models used include GBM, XGBoost, Light-GBM, and a stacked algorithm that combines the three models with a linear regression algorithm.A total of 8 years (2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018) of data was used for this single-station study.The ADIS GPS station data was supplemented with data from nearby GPS stations, ABOO and NAZR, to handle missing values.The data for years 2011-2014, 2016, 2018, and the second half of 2017 was utilized in the development of the ML models.Testing was conducted on the data set for 2015 and the first 6 months of 2017.In developing and testing models, the ML input data includes factors affecting VTEC variation such as time of day, season, solar and geomagnetic activity, and solar wind.The RandomizedSearchCV algorithm was used to determine the optimal hyperparameters of the models.A comparative analysis was conducted to validate the performance of ML models against the global model.The correlation between GPS VTEC and predicted VTEC for the four ML models showed almost identical results, with an R value of approximately 0.96, while the global model had a correlation of 0.87.
An error analysis between the model-predicted VTEC values and the GPS VTEC for the testing data showed that the ML models have significantly outperformed the IRI 2020 model in predicting VTEC.The VTEC predictions of the four ML and IRI 2020 models were compared with the GPS VTEC at selected quiet and geomagnetically disturbed conditions.The ML models have predicted VTEC with good accuracy and outperformed the IRI 2020 model in the selected quiet and disturbed conditions.The seasonal predictive performances of the models were also evaluated by comparing the 24-hr monthly average predicted VTEC values with the GPS VTEC for selected months at different seasons on the testing data.The VTEC values predicted by the ML models are in good agreement with the GPS VTEC, greatly outperforming the IRI 2020 model in the selected months with far smaller RMSE values.In general, the stacking algorithm applied in this study slightly reduced errors and slightly enhanced the predictive performance of the three gradient-boosting-based models in some instances.The findings in this study suggest that using GBDT algorithms and their stacked combination can accurately predict ionospheric VTEC in the African low-latitude region while also being computationally efficient.A. Tebabal's research is supported by the TWAS-DFG cooperation visits program 2020 and the Air Force Office of Scientific Research, United States, under award number FA8655-22-1-0001.We are thankful to the anonymous reviewers for their insightful comments and constructive feedback, which greatly enhanced the quality of this work.

Figure 1 .
Figure 1.Locations of ADIS, ABOO, and NAZR GPS stations on the global VISTA TEC map.

Figure 2 .
Figure 2. Diagram of the development of the VTEC machine learning models.

Figure 3 .
Figure 3. Scatter plots for hourly GPS VTEC and corresponding modeled VTEC values using different ML algorithms on the testing data.

Figure 4 .
Figure 4. Contour plots of residual errors between the GPS VTEC and the VTEC predicted by the models for years 2015 (left panel) and 2017 (right panel).

Figure 5 .
Figure 5. Contour plots of the GPS VTEC and the VTEC predicted by the models for years 2015 (left panel) and 2017 (right panel).

Figure 6 .
Figure 6.Comparison of predicted VTEC with the GPS VTEC for five quiet days of the months of September 2015 (first row), December 2015 (second row), March 2017 (third row), and June 2017 (fourth row).

Figure 7 .
Figure 7. Plots of Dst index and comparison of predicted VTEC with the GPS VTEC for 5-day periods with the day with intense storm considered at the center.

Figure 8 .
Figure 8. Plots of Dst index and comparison of predicted VTEC with the GPS VTEC for 5-day periods with the day with moderate storm considered at the center.

Figure 9 .
Figure 9. Plots of Dst index and comparison of predicted VTEC with the GPS VTEC for 5-day periods with the day with weak storm considered at the center.

Figure 10 .
Figure 10.Comparison of the RMSE values of the machine learning models and IRI 2020 model for the periods considered to compare the performance of the models in geomagnetic disturbance.

Figure 11 .
Figure 11.Comparison of seasonal variations of VTEC using the monthly 24 hr average VTEC of the models with the GPS VTEC for selected months in 2015 and 2017.

Figure 12 .
Figure 12.Comparison of RMSE values of the ML and IRI 2020 models for the selected months at different seasons in 2015 and 2017.

Table 1
Input Parameters Used for the Models

Table 2
Hyperparameters Used to Build GBM, XGBoost, and LightGBM Models

Table 3 R
, RMSE, MAE, and σ Values of the Machine Learning Models and IRI 2020 Model for Testing Data