Radio wave absorption in the ionosphere is a function of electron density, collision frequency, radio wave polarisation, magnetic field and radio wave frequency. Several studies have used multi-frequency measurements of cosmic radio noise absorption to determine electron density profiles. Using the framework of statistical inverse problems, we investigated if an electron density altitude profile can be determined by using multi-frequency, dual-polarisation measurements. It was found that the altitude profile cannot be uniquely determined from a “complete” measurement of radio wave absorption for all frequencies and two polarisation modes. This implies that accurate electron density profile measurements cannot be ascertained using multi-frequency riometer data alone and that the reconstruction requires a strong additional a priori assumption of the electron density profile, such as a parameterised model for the ionisation source. Nevertheless, the spectral index of the absorption could be used to determine if there is a significant component of hard precipitation that ionises the lower part of the D region, but it is not possible to infer the altitude distribution uniquely with this technique alone.