Vertical normal modes provide a decomposition of the vertical structure of oceanic variables. With a view to the assimilation of sea surface altimeter data we consider the relationship between density and pressure, using vertical normal modes. In theory these modes form a complete set, and subject to the normal limitations of the theory, pressure fields can be derived from density fields. In practice, using theoretical and numerical models, we find that results show an extreme sensitivity to the barotropic mode, making the decomposition unusable for many purposes. This work warns against using a density normal mode analysis in regions where the barotropic mode is thought to be important. ¿ American Geophysical Union 1996