  In this article, we study the limited angle problem for the weighted X-ray
transform. We consider the approximate reconstructions by applying two filtered
back projection formulas to the limited data. We prove that each resulted
operator can be decomposed into the sum of three Fourier integral operators
whose symbols are of types $(\varrho,\delta) \neq (1,0)$. The first operator,
being a pseudo-differential operator, is responsible for the reconstruction of
visible singularities. The other two are responsible for the generation of the
artifacts. The theory of Fourier integral operators then implies, in
particular, the continuity of the reconstruction operator and geometry of the
artifacts. We then extend the technique developed by the author in [Inverse
Problems 31 (2015) 055003] to obtain more refined microlocal estimates for the
strength of the artifacts.
