Extended Abstract
Introduction
Satellites in geodesy receive and transport important information. Among those, satellites with Low Earth Orbit (LEO), which are at altitudes less than 1000 km, have a significant role in the advancement of geophysical sciences such as earth’s potential field. Many parameters have an impact on the precision and accuracy of their information. Atmospheric friction is one of the most principal forces on satellites, which may cause deviation and falling of satellite on a short period. From the beginning of aerospace missions, many efforts have been done to determine atmospheric friction by geodesists, e.g., empirical models of atmosphere neutral density. Because of the complex nature of atmosphere behavior and also data limitations, these models may have low accuracy. So, there is a need for methods to improve the accuracy of empirical models by means of combining observations of atmospheric density to predict its future state.
 
Materials & Methods
Along with the extension of computer science, new reliable algorithms have been introduced which are able to predict a time series; Artificial Intelligent (AI) and Neural Networks (NN) are the best of these methods. These simple algorithms are inspirations of the human brain and its ability to learn and have been used in many different scientific fields. In these techniques without any requirement for constructing complex modeling, the relation between input and output will be provided only using weight and bias vectors during the training procedure. Simple Neural Networks are memoryless meaning that the value of time-series in previous can’t be used for predicting the future value of time series and therefore some important dependency of signal values with time will be lost. A Recurrent Neural Network (RNN) has been implemented to overcome this issue. RNN’s can store some important information of the values of the time series in the previous steps in a chain-like structure and using this information for predicting the next value of time series that will improve the accuracy of prediction. In this study, the Long Short-Term Memory (LSTM) Neural Network which is a kind of Recurrent Neural Network’s has been implemented to predict the scale for correcting atmospheric density of numerical models. The data of Grace Accelerometer observation in the 6 first month of the year 2014 have been used for training the LSTM for univariate training. Also, the LSTM has been trained in multi-variants mode once with using the coefficient of atmospheric correction expansion up to degree 2 and once with using sun geomagnetic information along with information of k_p index.
 
Results & Discussion
After training the LSTM network, by using the estimated parameters of the model, the zero degrees coefficient of harmonic expansion for a scale factor of correcting atmospheric density has been predicted in periods of 7, 14, 30, 60, and 90 days. The results of the univariate model show that the lower RMSE (Root Mean Square Error) is obtained about 0.054 in the period of prediction of about 14 days. Also, the results show that the multi-variants model with input data of sun geomagnetic information and k_p index has lower RMSE values in considered prediction periods compared to the other modes and the lowest RMSE is about 0.03 and belongs to the prediction of about 7 days. For evaluation of LSTM parameters in the obtained results, the predictions have been implemented with various Window sizes. The results show that by increasing windows size, the RMSE of the prediction will be reduced and the lowest RMSE was for prediction of 7 days with a window size of about 90 days. For the purpose of more evaluation, with the predicted atmospheric densities correction coefficient, the orbit of GRACE satellites has been propagated and the calculated position and velocity of satellites have been compared with the real orbit data. The results show that the lower RMSE will be provided with the prediction of 7 days with an RMSE for position and velocity of about 50 meters and 0.15 m/s respectively.
 
Conclusion
In this study, due to the complex nature of the atmosphere, the LSTM Neural Network has been used for modeling and predict the zero-order scale for correcting atmospheric densities harmonic expansion. For training the network, the data of Grace Satellites Accelerometer in the 180 days of the year 2014 have been used. The LSTM has been in univariate and multi-variant models. In the multi-variants model, once with using the coefficient of atmospheric correction expansion up to degree two and once with using sun geomagnetic information along with information of k_p index the network have been trained. The period of prediction was considered of about 7, 14, 30, 60, and 90 days.
The results show that the LSTM is capable to predict the correction coefficient in considered periods with a mean RMSE of about 0.05 for zero-order degree. Also, the results show that the lowest RMSE was for the 7 and 14 days of prediction and by increasing the window size of LSTM the RMSE will be decreased. The results of calculating the position of GRACE satellites position and velocity using predicted correction coefficients with real data show that the lowest RMSE was for prediction of 7 days for implemented method.