A common computation in seismology is the calculation of reflection and transmission coefficients that describe the partitioning of energy when a seismic wave strikes a boundary between elastic materials. The coefficients depend on the seismic wave speeds and densities on either side of the boundary and on the incident wave horizontal slowness, or ray parameter. The formulas for the values can be found in Table 3.1 of Lay and Wallace (1995) (note that an error in the sign of the second term in the computation of b) or in equations 5.38 and 5.39 of Aki and Richards (1980). For a welded elastic boundary, an incident P or SV wave results in four waves, two reflected and two transmitted or refracted waves. Thus for each case we need four coefficients – notation for each is illustrated in the figure below.
Notation used to identify reflection and transmission coefficients. We must consider two cases, that for an incident P wave and that for an incident SV wave. R identifies “Reflected” waves, T represents “Transmitted” or “refracted” waves.