The possibility of using topography in a state estimation context as a control parameter is explored in a linear barotropic shallow water model. Along with its adjoint, the model is used to systematically assess the influence of the depth field on the modeled circulation in a steady-state. Sensitivity of the flow field to the topography is greater in a partially blocked zonal channel than in a subtropical gyre. Hypothetical surface elevations are used to represent the types of data actually available. In neither case can all the details of the topography be recovered, showing that the relationship between topography and elevation does not have a unique inverse, and that many details of the topography are irrelevant to the particular physics under consideration.