The three-dimensional elliptic Radon transform (eRT) averages distributions over ellipsoids of revolution. It thus serves as a linear model in seismic imaging where one wants to recover the earth’s interior from reflected wave fields. As there is no inversion formula known for the eRT, approximate formulas have to be used. In this paper we suggest several of those, microlocally analyze their
properties, provide and implement an adapted algorithm whose performance we test by diverse numerical experiments. Our previous results of [Inverse Problems, 34 (2018), 014002 & 114001] are thus generalized to three space dimensions.