Extreme learning machine-based spherical harmonic for fast ionospheric delay modeling

Due to the growing real-time data of a global navigation satellite system (GNSS) applications and strict requirements for positioning users, especially under the condition of abnormal ionospheric activity, the fast and high-accuracy model is increasingly becoming important. In the past few years, the spherical harmonic (SH) model is one of the commonly-used methods, which has been widely used for global and regional modeling of ionospheric delay. In this paper, in order to further improve the accuracy of the SH model whilst without extra time consumption, we explored how the extreme learning machine (ELM) and the SH model can be combined for accurate and real-time modeling of ionospheric delay. Feasibility of our proposed method (denoted as ELM-SH model) was evaluated, where the available ionospheric data was firstly used to produce an SH model, while the ELM was deployed to compensate for the error of the SH model. Afterward, we tested the ELM-SH model based on the ionospheric data obtained from the Continuously Operating Reference Stations (CORS) in Jiangsu Province, China. Notably, experiment results demonstrate that compared with the conventional SH model, our proposed ELM-SH method can greatly improve the precision with over 37.09% on Root Mean Square Error (RMSE). The ELM-SH achieves competitive accuracy against the traditional backpropagation algorithm (BP) based method, but with a very short modeling time (within 10-2 s). This will be significant for real-time ionosphere modeling for those real-time GNSS positioning applications.