I will discuss the inverse problem of exploration seismology, in which one tries to determine the sound speed of the subsurface of the Earth from measurements made at the surface. This is a difficult nonlinear problem, but a linearized, high frequency version is potentially accessible to techniques from microlocal analysis, and is in fact an impetus to developing new microlocal techniques. I will discuss the underlying symplectic geometry in the presence of caustics (i.e., conjugate points), and new results concerning composition and estimates for Fourier integral operators we have obtained. This is joint work with Raluca Felea and Malabika Pramanik.