= Solution
Take $e^{-i\omega t}$ time dependence and measure angles from the upward normal. Write $p_i=k\sin\theta_i$, $q_i=k\cos\theta_i>0$ and choose incident amplitude $A_i$. Separate the flat Dirichlet reflection from the rough correction:
$$
\psi^{[0]}=A_i e^{ip_ix-iq_iz}-A_i e^{ip_ix+iq_iz},\qquad
\psi=\psi^{[0]}+\psi_s^{[1]}+O(h^2).
$$
<Taylor expansion> of the boundary condition about $z=0$ gives
$$
0=\psi^{[0]}(x,0)+h(x)\partial_z\psi^{[0]}(x,0)+\psi_s^{[1]}(x,0)+O(h^2).
$$
Since $\partial_z\psi^{[0]}(x,0)=-2iq_iA_ie^{ip_ix}$, the induced trace is
$$
\boxed{\psi_s^{[1]}(x,0)=2iq_iA_ih(x)e^{ip_ix}.}
$$
Use $\widehat h(Q)=\int h(x)e^{-iQx}dx$ and $\beta(\xi)=\sqrt{k^2-\xi^2}$ with the outgoing branch: $\beta>0$ for $|\xi|<k$ and $\operatorname{Im}\beta>0$ outside. The <outgoing angular spectrum> solves the <Helmholtz equation> with that trace, giving the <first-order rough-surface scattered field>
$$
\boxed{\psi_s^{[1]}(x,z)=\frac{iq_iA_i}{\pi}\int_{\mathbb R}\widehat h(\xi-p_i)e^{i\xi x+i\beta(\xi)z}d\xi.}
$$
If “<scattered field>” includes the specular reflection, add $-A_i e^{ip_ix+iq_iz}$ to this rough correction. <Fourier transforms> of an infinite random surface are generalized objects; finite-window transforms give an ordinary <integral> and are also needed for its far-field intensity. The expansion assumes a fixed regular surface shape scaled to small height. Highly oscillatory profiles can require additional control of slopes and spectral moments; small height alone is not a uniform error estimate for every profile.
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