The scattering of linear acoustic radiation by a periodic layered structure is a fundamental model in the geosciences as it closely approximates the propagation of pressure waves in the earth's crust. In this contribution, the authors describe new algorithms for (1) the forward problem of prescribing incident radiation and, given the known structure, determining the scattered field, and (2) the inverse problem of approximating the form of the structure given prescribed incident radiation and measured scattered data. Each of these algorithms is based upon a novel statement of the problem in terms of boundary integral operators (Dirichlet–Neumann operators), and a boundary perturbation algorithm (the method of operator expansions) for their evaluation. Detailed formulas and numerical simulations are presented to demonstrate the utility of these new approaches.