  We consider initial boundary value problems of time-fractional
advection-diffusion equations with the zero Dirichlet boundary value
$\partial_t^{\alpha} u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d
\partial_i(a_{ij}(x)\partial_j) + \sum{j=1}^d b_j(x)\partial_j + c(x)$. We
establish the uniqueness for an inverse problem of determining an order
$\alpha$ of fractional derivatives by data $u(x_0,t)$ for $0<t<T$ at one point
$x_0$ in a spatial domain $\OOO$. The uniqueness holds even under assumption
that $\OOO$ and $A$ are unknown, provided that the initial value does not
change signs and is not identically zero. The proof is based on the
eigenfunction expansions of finitely dimensional approximating solutions, a
decay estimate and the asymptotic expansions of the Mittag-Leffler functions
for large time.
