Inversion of the Spherical Mean Transform with Sources on a Hyperplane

The object of this study is an integral operator $\mathcal{S}$ which averages functions in the Euclidean upper half-space ${\mathbb{R}}_{+}^n$ over the half-spheres centered on the topological boundary $\partial $\mathbb{R}$_{+}^{n}$. By generalizing Norton’s approach to the inversion of arc means in the upper half-plane, we intertwine $\mathcal{S}$ with a convolution operator $\mathcal{P}$. The latter integrates functions in ${\mathbb{R}}^n$ over the translates of a paraboloid of revolution. Our main result is a set of inversion formulas for $\mathcal{P}$ and $\mathcal{S}$ derived using a combination of Fourier analysis and classical Radon theory. These formulas appear to be new and are suitable for practical reconstructions.