  In this article, we characterize the strength of the reconstructed
singularities and artifacts in a reconstruction formula for limited data
spherical Radon transform. Namely, we assume that the data is only available on
a closed subset $\Gamma$ of a hyperplane in $\mathbb{R}^n$ ($n=2,3$). We
consider a reconstruction formula studied in some previous works, under the
assumption that the data is only smoothened out to a finite order $k$ near the
boundary. For the problem in the two dimensional space and $\Gamma$ is a line
segment, the artifacts are generated by rotating a boundary singularity along a
circle centered at an end point of $\Gamma$. We show that the artifacts are $k$
orders smoother than the original singularity. For the problem in the three
dimensional space and $\Gamma$ is a rectangle, we describe that the artifacts
are generated by rotating a boundary singularity around either a vertex or an
edge of $\Gamma$. The artifacts obtained by a rotation around a vertex are $2k$
orders smoother than the original singularity. Meanwhile, the artifacts
obtained by a rotation around an edge are $k$ orders smoother than the original
singularity. For both two and three dimensional problems, the visible
singularities are reconstructed with the correct order. We, therefore,
successfully quantify the geometric results obtained recently by J. Frikel and
T. Quinto.

Classical Analysis and ODEs