This paper studies the mathematical foundation of time-dependent three-dependent three-dimensional (3-D) computerized ionospheric tomography (CIT) for reconstructing ionospheric electron density,N_e,from ground-based GPS beacon obser|vations,After simplifying the relation between N_e and time, the time-dependent 3-D inversion in consideration is reduced to a 3-D tomography with incomplete projections. To see clearly the effects of the incompleteness on the quality of reconstruction under 3-D condition,the fromula of 3-D parallelbeam tomography is deduced theoretically. After establishing the mathematical foundation,simulations based on actual GPS ray paths with the help of the IRI-90 model are performed, and reasonable time-dependent 3-D distribution images of N_e are obtained when taking proper layout of the network and allowing variable resolutions. The quality of the reconstruction is rather good when compared with the images from the IRI90 model directly. Therefore, results in this paper demonstrate that imaging of the ionospheric electron density distribution from GPS beacon observations is reasonable in theory and feasible in pratice.
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