First-order rough-surface scattered field (${\psi }_s^{[1]}=-\mathcal{E}[h{\partial }_z{\psi }^{[0]}{|}_0]$)

Taylor expansion of the Dirichlet boundary condition gives the first-order boundary trace $-h{\partial }_z{\psi }^{[0]}$ at the mean plane. The outgoing angular spectrum then propagates it. A zero-mean surface gives zero mean for this linear rough correction, while the complete scattered field still contains its nonzero flat reflection.