We consider the wave equation , in , , where the metric is known outside an open and bounded set with smooth boundary . We define a source as a sum of point sources, , where the points , form a dense set on . We show that when the weights are chosen appropriately, determines the scattering relation on , that is, it determines for all geodesics which pass through the travel times together with the entering and exit points and directions. The wave contains the singularities produced by all point sources, but when for some , we can trace back the point source that produced a given singularity in the data. This gives us the distance in between a source point and an arbitrary point . In particular, if is a simple Riemannian manifold and is conformally Euclidian in , these distances are known to determine the metric in . In the case when is nonsimple, we present a more detailed analysis of the wave fronts yielding the scattering relation on .
"An inverse problem for the wave equation with one measurement and the pseudorandom source." Anal. PDE 5 (5) 887 - 912, 2012. https://doi.org/10.2140/apde.2012.5.887