The Stability for an Inverse Problem of Bottom Recovering in Water-Waves

In this article we deal with a class of geometric inverse problem for bottom detection by one single measurement on the free surface in water-waves. We found upper and lower bounds for the size of the region enclosed between two different bottoms, in terms of Neumann and/or Dirichlet data on the free surface. Starting from the general water-waves system in bounded domains with side walls, wemanage to formulate the problemin terms of the Dirichlet to Neumann operator and thus, as an elliptic problem in a bounded domain with Neumann homogeneous condition on the rigid boundary. Then we study the properties of the Dirichlet to Neumannmap and analyze the calledmethod of size estimation. © 2020 IOP Publishing Ltd.