  We consider the inverse problem to determine a smooth compact Riemannian
manifold $(M,g)$ from a restriction of the source-to-solution operator,
$\Lambda_{\mathcal{S,R}}$, for the wave equation on the manifold. Here,
$\mathcal{S}$ and $\mathcal{R}$ are open sets on $M$, and
$\Lambda_{\mathcal{S,R}}$ represents the measurements of waves produced by
smooth sources supported on $\mathcal{S}$ and observed on $\mathcal{R}$. We
emphasise that $\overline{\mathcal{S}}$ and $\overline{\mathcal{R}}$ could be
disjoint. We demonstrate that $\Lambda_{\mathcal{S,R}}$ determines the manifold
$(M,g)$ uniquely under the following spectral bound condition for the set
$\mathcal{S}$: There exists a constant $C>0$ such that any normalized
eigenfunction $\phi_k$ of the Laplace-Beltrami operator on $(M,g)$ satisfies
  \begin{equation*}
  1\leq C\|\phi_k\|_{L^2(\mathcal{S})}.
  \end{equation*}
  We note that, for the Anosov surface, this spectral bound condition is
fulfilled for any non-empty open subset $\mathcal{S}$. Our approach is based on
the paper [18] and the spectral bound condition above is an analogue of the
Hassell-Tao condition there.
