  In this article, we study an inverse boundary value problem for the
time-dependent convection-diffusion equation. We use the nonlinear Carleman
weight to recover the time-dependent convection term and time-dependent density
coefficient uniquely. Nonlinear weight allows us to prove the uniqueness of the
coefficients by making measurements on a possibly very small subset of the
boundary. We proved that the convection term and the density coefficient can be
recovered up to the natural gauge from the knowledge of the Dirichlet to
Neumann map measured on a very small open subset of the boundary.
