Abstract: Wave propagation problems arise in many areas of science and engineering, including seismology, medical imaging, telecommunications, and wireless technology. In this talk, I will present a dual-pairing summation-by-parts (DP-SBP) numerical framework for forward and adjoint wave propagation problems. By combining this framework with simultaneous approximation terms for the weak imposition of initial and boundary conditions, we derive discrete energy estimates and show adjoint consistency for inverse wave propagation problems. Numerical experiments in one and two spatial dimensions will be presented to verify the theoretical analysis and demonstrate convergence of numerical errors.