We propose an algorithm to compute an approximate singular value decomposition of least squares operators related to linearized inverse medium problems with multiple events. Such factorizations can be used to accelerate matrix-vector multiplications and to precondition iterative solvers. We describe the algorithm in the context of an inverse scattering problem for the low-frequency time-harmonic wave eqation with broadband and multi-point illumination. This model finds many applications in science and engineering (e.g., seismic imaging, non-destructive evaluation, and optical tomography).