The Florida Current, which flows through the Straits of Florida, transports a significant fraction of the heat which must be transferred from low to high latitudes in order for the earth to remain in thermal balance. While ocean acoustic tomography is an attractive candidate for monitoring this heat transport, its use is dependent on being able to identify individual ray paths. All published investigations of tomography have been concerned with deep-ocean propagation where experience has shown that paths can be identified. Propagation in the Straits is quite different from deep-ocean propagation, however. The sound speed is such that a ray path which travels modest distances necessarily experiences bottom bounces. It is not clear, a priori, that paths can be identified in the Straits. We report the results of an investigation of this question. Using a fast, specialized ray-tracing program, we analyzed the fluctuations in ray path geometry resulting from small perturbations in the bathymetry. We found that the topographic features amplify these fluctuations to such a degree that identification of individual rays becomes impossible after only a few bottom bounces. Consequently, there is no adequate interpretive tool for relating an acoustic signal recorded more than a few tens of kilometers from a source to the heat transport, i.e., for carrying out the tomographic inversion. We propose a way of overcoming this identification problem based on the use of acoustic profilers.