Symmetry-preserving observers for some water tank problems: theory and application to a shallow water model

In this paper we consider a tank containing fluid and we want to estimate the horizontal currents when the fluid surface height is measured. The fluid motion is described by shallow water equations in two horizontal dimensions. We build a simple non-linear observer which takes advantage of the symmetries of fluid dynamics laws. As a result its structure is based on convolutions with smooth isotropic kernels, and the observer is remarkably robust to noise. We prove the convergence of the observer around a steady-state. In numerical applications local exponential convergence is expected. The observer is also applied to the problem of predicting the ocean circulation. Realistic simulations illustrate the relevance of the approach compared with some standard oceanography techniques.