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In this paper we consider a tank containing fluid and we want to estimate thehorizontal currents when the fluid surface height is measured. The fluid motionis described by shallow water equations in two horizontal dimensions. We builda simple non-linear observer which takes advantage of the symmetries of fluiddynamics laws. As a result its structure is based on convolutions with smoothisotropic kernels, and the observer is remarkably robust to noise. We prove theconvergence of the observer around a steady-state. In numerical applicationslocal exponential convergence is expected. The observer is also applied to theproblem of predicting the ocean circulation. Realistic simulations illustratethe relevance of the approach compared with some standard oceanographytechniques.

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