An automated algorithm uses recordings of acoustic energy across a spatially-distributed array to derive information about multiply scattered acoustic waves in heterogeneous media. The arrival time and scattering-order of each recorded diffracted acoustic wave, and the exact sequence of diffractors encountered by that wave, are estimated without requiring an explicit model of the medium through which the wave propagated. Individual diffractors are identified on the basis of their unique single-scattering relative travel-time curves (move-outs) across the array, and secondary (twice-scattered) waves are detected using semblance analysis along temporally offset primary move-outs. This information is sufficient to estimate travel times and scattering paths of all multiply diffracted waves of any order, and for these events to be identified in recorded data. The algorithm is applied to synthetic acoustic data sets from a variety of media, including different numbers of point-diffractors and a medium with strong heterogeneity and non-hyperbolic move-outs.