GNSS radio tomography of the ionosphere: The problem with essentially incomplete data

We discuss the specific features of the spatiotemporal radio tomography of ionospheric electron density based on the data from high orbiting global navigational satellite system (GNSS). The main peculiarities are four-dimensionality of the problem and essential incompleteness of input data. The approach suggested for the solution of this problem is based on the search for the solution with minimal Sobolev's norm (the most smooth solution), which provides, in particular, smooth extension of the solution to the area where no input data is available. Methods and algorithms for the solution of the problem are developed, and the resolution of the proposed technique is estimated. Examples of global and regional radio tomographic imaging based on GNSS data are presented. The obtained results are compared with the data of ionosondes and with the results of radio tomography based on the radio transmissions from low-orbiting satellites.