  In general the composition of Fourier integral operators (FIOs) need not be
an FIO. Motivated by the problem of linearized seismic inversion in the
presence of cusp caustics for the background sound speed, we consider FIOs
whose canonical relations have certain two-sided cusp degeneracies, and show
that the resulting compositions have wave-front relations in the union of the
diagonal and an open umbrella, the simplest type of singular Lagrangian
manifold.
