  We consider the Boussinesq-Peregrine (BP) system as described by Lannes
[Lannes, D. (2013). The water waves problem: mathematical analysis and
asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water
regime, and study the inverse problem of determining the time and space
variations of the channel bottom profile, from measurements of the wave profile
and its velocity on the free surface. A well-posedness result within a Sobolev
framework for (BP), considering a time dependent bottom, is presented. Then,
the inverse problem is reformulated as a nonlinear PDEconstrained optimization
one. An existence result of the minimum, under constraints on the admissible
set of bottoms, is presented. Moreover, an implementation of the gradient
descent approach, via the adjoint method, is considered. For solving
numerically both, the forward (BP) and its adjoint system, we derive a
universal and low-dissipation scheme, which contains non-conservative products.
The scheme is based on the FORCE-{\alpha} method proposed in [Toro, E. F.,
Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred
schemes for hyperbolic equations in conservative and non-conservative form.
Journal of Computational Physics, 416, 109545]. Finally, we implement this
methodology to recover three different bottom profiles; a smooth bottom, a
discontinuous one, and a continuous profile with a large gradient. We compare
with two classical discretizations for (BP) and the adjoint system. These
results corroborate the effectiveness of the proposed methodology to recover
bottom profiles.
