I now demonstrate the analogous derivation for the specular reflection coefficient. Again, the least-squares solution is given by (10):
For a reflection, the incident P-wave scalar is the same as (27):
However, the reflected S1-wave scalar needs to be evaluated given (4), (24) and (25):
In deriving (40) I have taken the farfield WKBJ approximation of the displacement vector wavefield gradient and divergence terms given in (30)-(31).
Substituting (39) and (40) into (38) and performing the required t integrations, one obtains
Now is now evaluated at the total incident P plus reflected S ray traveltime . The subscripts on and indicate that the Lamé parameters should be evaluated at the receiver positions , and the symbol still signifies a convolution of the surface data with the estimated P-wave source wavelet w1. The P-wave autocorrelation integral is the same as the case:
Combining (41)-(43) yields:
Equation (44) gives the least-squares elastic wavefield integral solution for specular reflectivity. It also can be slightly simplified further by noting that
which follows from the S-wave eikonal equation in (15), where is the S-wave velocity evaluated at each receiver position .