Physical surface trace and reference-plane trace

For normal incidence on a Dirichlet rough surface, the scattered trace on the moving boundary is exactly $-e^{-ikh(x)}$. Its trace on the mean plane is instead $-1+2ikh+2kh\mathcal{B}h+\cdots$. The two traces are related by a normal Taylor expansion and must not be identified. Their ensemble averages can therefore have different quadratic coefficients without inconsistency.