In satellite ray tomography the measurements are samples of Radon tran
sform of the ionospheric electron density. Regardless of the practical
numerical method applied, the goal of tomographic inversion is to cal
culate the inverse of the Radon transform. From mathematical point of
view it is clear that provided a function satisfies certain mathematic
al conditions, it has a unique Radon transform, which also has a uniqu
e inverse. In satellite radiotomography, however, the sampling of the
Radon transform is seriously restricted by the small number of receive
rs. No unique inverse can be calculated from such an incomplete set of
samples. It turns out that an infinite number of electron density fun
ctions can be defined which are completely invisible for a given exper
imental setup, not only in practice but also in a strict mathematical
sense. These functions are not necessarily horizontally stratified, bu
t they may contain strong horizontal gradients, in which case their in
visibility is not associated with the lack of horizontal rays. In this
paper a method of constructing such functions is presented and exampl
es 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 ma
y be partly invisible to transmitters on the ground.