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In the lower uniform half-space the causal vertical wavenumber branch has $\mathrm{Im}{k}_{\beta ,b}>0$. The downward factor $e^{i{k}_{\beta ,b}(z-z_b)}$ decays as $z\to +\infty$, whereas the upward factor grows there and represents an incoming wave from the lower infinity. The outgoing radiation condition therefore retains only the downward factor and givesFor each uniform layer $j$ of thickness $h_j$, work upwards from its known bottom load $Z_{j,b}$. With $q_j={\mu }_j{p}_{\beta ,j}$ and ${\delta }_j=\omega p_{\beta ,j}{h}_j$, part (iv) givesvelocity and traction are continuous across a bonded interface, so this top impedance becomes the bottom load of the next shallower layer. Repeat up to $z_a$ to obtain $Z_{\mathrm{in}}=Z(z_a)$. Poles can be handled by carrying $(V,T)$ or the reciprocal ratio. Since the lower load has positive real part, part (iii) actually ensures that the outgoing whole-stack impedance stays finite with positive real part at all shallower depths in the upper frequency half-plane.

The upper half-space has directional impedance $q_a={\mu }_a{p}_{\beta ,a}$. At $z_a$, an incident velocity $V_0$ and reflected velocity $R{V}_0$ giveTaking their ratio and solving establishes reflection from a layered SH impedance:Thus no individual deeper reflection coefficients are needed after $Z_{\mathrm{in}}$ is known: it contains all underlying interfaces and their coherent multiples. A matched load gives $R=0$; the real-frequency limits of zero traction and zero velocity give $R=+1$ and $R=-1$. For real frequency and propagating upper and lower half-space waves, the usual energy-flux balance provides a further check on the computed reflection and transmission. A stable implementation for thick layers uses the ratio update or its decaying-exponential version instead of multiplying growing transfer matrices.