We develop efficient high-order algorithms for simulation of the statistical properties of quantities of interest (QoI) in a class of stochastic configurations comprising multiple wave propagation components. In particular, the moments of the QoI play an important role in quantifying random uncertainty in the multiple structure wave scattering and absorption models. The stochastic nature of configurations governing the models may include randomness in location and orientations of the obstacles, their shapes, and their material properties.