 Reference-plane validity of a rough-surface angular spectrum

An upward angular spectrum obtained from a horizontal field trace is exact in the homogeneous half-space above that reference plane. For a rough boundary $z=f(x)$, choose the plane above $\sup f$. A trace at $z=0$ is not physically defined in the medium if any part of the surface lies above it. Using such a mean-plane trace requires an analytic continuation convention; convergence of the continued spectrum down to the rough boundary is an additional hypothesis. Small-height boundary perturbation uses the reference plane formally order by order and is distinct from an exact expansion throughout an arbitrary rough domain.