*preprint*

**Inserted:** 28 sep 2020

**Year:** 2020

**Abstract:**

We prove global Lipschitz stability and conditional local H\"older stability for inverse source and coefficient problems for a first-order linear hyperbolic equation, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.