  We study ray transforms on spherically symmetric manifolds with a piecewise
$C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is
injective on the space of $L^2$ functions on such manifolds. We also prove
injectivity results for broken ray transforms (with and without periodicity) on
such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low
regularity, we introduce and study a class of generalized Abel transforms and
study their properties. This low regularity setting is relevant for geophysical
applications.
