 Free-surface anisotropic reflection amplitudes

Select the three downgoing or depth-decaying elastic modes for a solid below a plane free surface. For each mode use its reduced traction column $b=(Tq+R)a$. Zero total boundary traction gives $BA=-b_0$ for an incident mode of reduced traction $b_0$. Invertibility gives the displayed amplitudes. A singular matrix instead permits an outgoing homogeneous surface mode, so a radiation-limit or solvability analysis is necessary.