We are concerned with inverse boundary problems for first order perturbations of the Laplacian, which arise as model operators in the acoustic tomography of a moving fluid. We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded domain in $\mathbb{R}^n$, $n\ge 3$, determines the first order perturbation of low regularity up to a natural gauge transformation, which sometimes is trivial. As an application, we recover the fluid parameters of low regularity from boundary measurements, sharpening the regularity assumptions in the recent results of [A. D. Agaltsov, Bull. Sci. Math., 139 (2015), pp. 937--942] and [A. Agaltsov and R. Novikov, J. Inverse Ill-Posed Probl., 24 (2016), pp. 333--340]. In particular, we allow some fluid parameters to be discontinuous.