Reconstruction of stratified steady water waves from pressure readings on the ocean bed

Consider a two-dimensional stratified solitary wave propagating through a body of water that is bounded below by an impermeable ocean bed. In this work, we study how such a wave can be reconstructed from data consisting of the wave speed, upstream and downstream density profile, and the trace of the pressure on the bed. First, we prove that this data uniquely determines the wave, both in the (real) analytic and Sobolev regimes. Second, for waves that consist of multiple layers of constant density immiscible fluids, we provide an exact formula describing each of the interfaces in terms of the data. Finally, for continuously stratified fluids, we detail a reconstruction scheme based on approximation by layer-wise constant density flows.