  In this paper, we consider a problem inspired by the real-world need to
identify the topographical features of ocean basins. Specifically we consider
the problem of estimating the bottom impermeable boundary to an inviscid,
incompressible, irrotational fluid from measurements of the free-surface
deviation alone, within the context of dispersive shallow-water wave models.
The need to consider the shallow-water regime arises from the ill-posed nature
of the problem and is motivated by prior work. We design an algorithm using
which, both fluid velocities and the bottom-boundary profile, may be accurately
recovered assuming an a priori relatively inaccurate guess for the bottom
boundary. We achieve this by considering two separate inverse problems: one to
deduce the bottom-boundary from velocities and the surface deviation, and
another to recover the velocities from the surface deviation and an approximate
bottom-boundary. The former is a classic inverse problem that requires the
inversion of an ill-conditioned matrix while the latter employs the observer
framework. Combining the two inverse problems leads to our reconstruction
algorithm. We emphasise the role played by model selection and its impact on
algorithm design and the accuracy of the reconstruction.
