I now demonstrate the analogous derivation for the specular reflection coefficient. Again, the least-squares solution is given by (10):

(38) |

For a reflection, the incident *P*-wave scalar is the same
as (27):

(39) |

However, the reflected *S _{1}*-wave scalar needs to be evaluated
given (4), (24) and (25):

(40) |

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

(41) |

(42) |

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 *w _{1}*. The

(43) |

(44) |

Equation (44) gives the least-squares elastic wavefield integral solution for specular reflectivity. It also can be slightly simplified further by noting that

(45) |

which follows from the *S*-wave eikonal equation in (15),
where is the
*S*-wave velocity evaluated at each receiver position .

11/17/1997