Neural Network Forecast of the Total Electron Content of the Ionosphere and Optimization

The information about the state of the ionosphere is very important because the ionosphere critically affects the propagation of radio signals which are widely used in GNSS applications, HF/ VHF/UHF radio communications, satellite control and operations, navigation timing, direction finding and positioning, spectrum management, radar detection, disaster warning and others [1]. The main parameter describing the state of the ionosphere is the total electron content (TEC) of the ionosphere, measured by GPS receivers using transmitters on satellites. The state of the ionosphere can be monitored and predicted using this parameter. In recent years, neural network methods have become prevalent in forecasting ionospheric parameters [2]. First of all, these are the classical LSTM [3] and GRU [4] methods, which, on the one hand, are used as refer ence methods for comparison with newly developed methods, and on the other hand, when combined with other architectures, they provide an increase in forecast accuracy. The most effective was the combination with bidirectional architectures. Papers [5-8] can be cited as examples.

In [5], a method combining random forest with a Bi-LSTM neural network is proposed. The random forest algorithm is used for regression analysis to select the most important input variables. The Bi-LSTM method was used for the prediction of TEC 1 h ahead on a dataset of 9 Chinese stations located near and along the 110- 120° meridians in 2021. The training dataset included values from DOY 001 to DOY 273; the validation dataset, from DOY 274 to DOY 334 in 2021; the test dataset, from DOY 335 to DOY 365. The results obtained are presented as RMSE for three latitude ranges: (1) 10°- 25° E, (2) 25°-40° E, (3) 40°-55° E. The RMSE were: (1) 2.56 TECU, (2) 1.34 TECU, (3) 0.93 TECU. The paper [6] concerns ТЕС map prediction and proposes an improved model, Mixed Convolutional Neural Network (CNN) - Bidirectional Long Short-Term Memory (BiLSTM), for predicting TEC in China. The training array included the longest (GIM)-TEC maps from 1998 to 2023 in China. The model utilizes historical TEC map data for the previous 72 hours, along with inputs including ap, Kp, Dst, AE, disturbance index, F10.7 index, and time factor, to predict the future 1-24 hours of TEC maps. The main focus was on the forecast during the disturbance. Results for 1h, 2h, 24 h ahead are presented. From full 702 storms, the train set comprised 552 geomagnetic storms, the validation set consisted of 100 storms, and the remaining 50 storms (27 minor and moderate storms and 23 strong storms) were used as the test set. The model performs well in short-term predictions, accurately capturing the occurrence, evolution, and classification of ionospheric storms. When the predicted length increases, the accuracy gradually decreases, and some erroneous predictions may occur. RMSE increases with the increase of predicted length and the enhancement of geomagnetic activity. The RMSE exhibits strong geographic dependence, with the maximum values occurring in the EIA region around magnetic latitudes of 20° to 30°. Under strong geomagnetic conditions, the RMSE reaches approximately 4 TECU, 8 TECU, and 10 TECU for 1 h, 12 h, and 24 h predicted lengths, respectively. Under quiet conditions, the RMSE is approximately 1 TECU lower than under moderate geomagnetic conditions and 2 TECU lower than under strong geomagnetic conditions. The paper [7] describes in detail all the stages of the Temporal Convolutional Network (TCN) method and compares its results with the results of LSTM, GRU, BiLSTM. The study uses TEC data of the global CODE map with 2h resolution for 6 mid-latitude stations and 6 low-latitude stations in the period 2017-2021. The data is divided into two parts (training, test) by the ratio of 9:1 and use the past 24 data to predict and verify the next data. It is noted that such parameters of the method as loss, learning rate, optimizer, timesteps, batch size, input dimension and others were selected by the grid search method. Among the input parameters, Dst index, latitude, longitude, SN, F10.7, Kp index, Ap index were used. Data for January 2017 - December 2020 were used for training, and data from 2021 were used for testing. It was found that the results depend on latitude. The mean average error of TCN (1.2385 TECU) is lower in most areas compared with LSTM (1.2727 TECU), GRU (1.2602 TECU) and BiLSTM (1.2767 TECU). The paper [8] proposes a method that is a combination of attention-based bidirectional long short-term memory and gated recurrent unit (Bi-LSTM GRU) to predict TEC and signal delay in the ionosphere. The training array included data from 2009-2017, the test array was from 2018 in Bangalore, India. The results are presented for ionospheric delay and 24 h and showed the advantage of the presented method over LSTM, Bi-LSTM, GRU. Moreover, for Bi- LSTM GRU, an experiment was conducted with the attention mechanism, which showed a better result than without this mechanism.

It should be noted that a common feature of many models and methods is manual assignment of hyperparameters. In the works [9-10], attention was drawn to the role of optimization procedures. The general approach is presented in [9], and this problem is considered in [10] in relation to TEC prediction. The paper [10] notes the growing popularity of deep learning models, but their requirement for a large number of hyperparameters is considered a drawback due to the complexity of their selection. This process can be accomplished using optimization algorithms, in particular, the Beluga Whale Optimization (BWO) algorithm that can be used to optimize hyperparameters of deep learning models. The paper [10] analyzed the drawbacks of BWO and proposed an improved BWO algorithm, named FAMBWO (Firefly Assisted Multi-Strategy Beluga Whale Optimization). This method was compared with 11 state-of-the-art swarm intelligence optimization algorithms on 30 benchmark functions, and the results showed that the improved algorithm had faster convergence speed and better solutions on almost all benchmark functions. This algorithm was then used to optimize the MA-BiLSTM model for TEC prediction and obtained significantly better results than the MA-BiLSTM model optimized by other algorithms. The aim of this work is to use the method [10] to optimize the most efficient BiTCN method from [11].

The experimental data included the TEC values of the JPL map calculated from the IONEX files (https//cddis.nasa.gov/archive/ gnss/products/ionex/) for the Juliusruh station (54.6° N, 14.6° E) and 2015. The values of the indices of solar and geomagnetic activity – proton density Np, planetary 10Kp index, disturbance storm time Dst, and solar flux F10.7, solar wind velocity Vsw were taken from SPDF OMNIWeb Service (http://omniweb.gsfc.nasa.gov/ form/dx1.html). Methods: (1) LSTM-FRF, (2) LSTM-RFF, (3) GRUFRF, (4) GRU-RFF, (5) TCN, (6) BiLSTM-FRF, (7) BiLSTM-RFF, (8) BiGRU-FRF, (9) BiGRU-RFF, (10) BiTCN are described in detail in [11], the additions related to the BiTCN optimization procedure are as follows.

All convolutional layers of the model were parameterized, and the following parameters were allocated for each layer: dilation factor (DF), number of filters, filter kernel size (Fsz). Due to the requirements for the tensor shape, the filter number was identical for all convolutional layers, so the final number of parameters was equal to 9 (one dilation factor parameter and filter kernel size parameter per Convolution1D Layer plus a single filter number parameter). Sine and cosine DOY and HOD components were included as additional indices for the initial model and optimized versions. Data preparation procedure is identical to [11]. Beluga Whale Optimization algorithm is applied during training for validation loss at the final training epoch (Figure 1).

Calculations were carried out for widely used forecast advance times: 2h, 1d, 7d. The results consisted of a comparison of the metrics: mean absolute error (MAE), mean absolute percentage error (MAPE), root-mean-square error (RMSE), averaged over half a year during a single step of 2h, 24h and 7 days advance time forecasts for different models. A comparison of the initial and optimized models is shown in Figure 2 for BiTCN_new, BiTCN-BWO_init, BiTCN-BWO_corr procedures.

It is evident that the forecast accuracy deteriorates with increasing advance time. Initial estimates for BiTCN-BWO_init were actually worse than estimates for BiTCN_new, where parameters were determined via trial and error. It was caused by the fact that BWO algorithm was not effective in determining the number of filters parameter. Therefore, we kept all the BWO estimates for parameters but corrected the number of filters via trial and error, again (BiTCN-BWO_corr). This allowed one to decrease the errors further, making the model more effective than BiTCN_new and BiTCN-BWO_init. Figure 3 compares the new models with the previously obtained results for bidirectional models [11].

It is evident that the corrected method improved the accuracy of the TEC forecast. It can be assumed that combination of such optimization with other methods and models can provide a much greater improvement.

Methods aimed at improving the accuracy of the forecast constantly appear in publications. This paper shows that the optimization algorithms proposed in [10] can improve the accuracy, but in order to recommend these algorithms for widespread use, it is necessary to weigh the advantages and disadvantages of these algorithms. The advantages of BWO include a good initial approximation for manual optimization. The method is scalable to a large number of parameters and reveals non-obvious dependencies. The disadvantages include a much longer calculation time, and the need for repeated verification, since not all parameters are optimized equally effective.

The research was carried out at Southern Federal University with the financial support of the Ministry of Science and Higher Education of the Russian Federation (State contract GZ0110/23- 10-IF).

No conflict of interest.

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