Uniqueness and stability in bottom detection through surface measurements of water waves

This paper investigates the geometric inverse problem of recovering the bottom shape from surface measurements of water waves. Using the general water-waves system on a bounded subdomain of the fluid domain, we address this inverse problem, focusing on the identifiability and the stability issues. We establish uniqueness and derive logarithmic stability estimates in the determination of the bathymetry on any fixed smooth, bounded, open domain ${\mathcal O}\subset {\mathbb R} ^d$, $d=1,2$, from the knowledge of the free surface, its first time derivative, and the trace of the velocity potential on the free surface, at a given instant $t_0$ within $\mathcal O $, together with the knowledge of the bottom along $\partial \mathcal O$. No further assumptions are required for uniqueness. For stability, we impose only a \textit{local fatness} condition on the region between the bottom profiles, allowing us to adapt the size estimates method.