How can one determine the shape of one's surroundings by listening to ambient signal sources? Traditional remote-sensing ideas place strong requirements on the topology and geometry of the sensorium in order to exploit linearity, often by using integral transform methods. While useful for understanding the propagation of low frequency waves, in practice they are difficult to use. I will motivate a change in focus away from global, spectral methods to local, sheaf methods. New results following from this change in focus suggest a two-stage method for recovering first topology and then geometry. These theoretical results supply a new family of practical algorithms for addressing a class of distributed sensing problems.