  Working in the time domain, we show that the location of the singularities
and the principal symbol of a potential that is conormal to nested submanifolds
$S_2 \subset S_1 \subset \mathbb{R}^n$, for $n \geq 3$, can be recovered from
the backscattering as well as from the restriction of the far-field pattern to
more general determined sets of scattering data. This extends the work of
Greenleaf and Uhlmann where the potentials considered are conormal to a single
submanifold $S \subset \mathbb{R}^n$. We utilize the microlocal analysis of the
wave operator $\square=\partial_t^2 - \triangle_x$ and multiplication by a
nested conormal distribution in order to study their action on spaces of
conormal-like distributions.
