  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, we manage to formulate the problem in 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 Neumann map and analyze the called method of
size estimation.
