This paper applies variational data assimilation to inundation problems governed by the shallow water equations with wetting and drying. The objective of the assimilation is to recover an unknown time-varying wave profile at an open ocean boundary from inundation observations. This problem is solved with derivative-based optimisation and an adjoint wetting and drying scheme to efficiently compute sensitivity information. The capabilities of this approach are demonstrated on an idealised sloping beach setup in which the profile of an incoming wave is reconstructed from wet/dry interface observations. The method is robust to noise in the observations if a regularisation term is added to the optimisation objective. Finally, the method is applied to a laboratory experiment of the Hokkaido-Nansei-Oki tsunami, where the wave profile is reconstructed with a relative L∞ error of less than 1%.