In satellite ray tomography the measurements are samples of Radon transform of the ionospheric electron density. Regardless of the practical numerical method applied, the goal of tomographic inversion is to calculate the inverse of the Radon transform. From mathematical point of view it is clear that provided a function satisfies certain mathematical conditions, it has a unique Radon transform, which also has a unique inverse. In satellite radiotomography, however, the sampling of the Radon transform is seriously restricted by the small number of receivers. No unique inverse can be calculated from such an incomplete set of samples. It turns out that an infinite number of electron density functions can be defined which are completely invisible for a given experimental setup, not only in practice but also in a strict mathematical sense. These functions are not necessarily horizontally stratified, but they may contain strong horizontal gradients, in which case their invisibility is not associated with the lack of horizontal rays. In this paper a method of constructing such functions is presented and examples of them are shown, some of which could be reasonable in a realistic ionosphere. An experimental result and a simulation is also presented, which indicates how a wave-like traveling ionospheric disturbance may be partly invisible to transmitters on the ground.¿ 1997 American Geophysical Union