Block copolymer structures have been directed to assemble on chemically patterned surfaces with the domain interfaces oriented perpendicular to the substrate. Such methods have been pursued for lithographic applications to achieve long-range order in the assembled structures and, potentially more important, provide nanometer-level control over the interfaces between structures. The chemically striped surfaces used for the directed assembly of lamellae are patterned by top-down lithographic techniques and thus often have rough edges between the regions of different chemistry. Here we quantitatively characterize, using experiments and molecular-level simulations, the propagation of line edge roughness from the chemically patterned surfaces into the interfaces between domains of block copolymer lamellae as a function of the wavelength, amplitude, and geometry of the roughness. Two geometries of surface pattern roughness are considered with oscillatory neighboring interfaces that are either in-phase or out-of...