Now that expressions for the incident and reflected vector wavefields have been obtained in (21) and (24), the least-squares estimation of the specular reflection coefficient can be evaluated from (10):

(26) |

First, the incident wave scalar can be evaluated given (5) and (21):

(27) |

Next, the reflected wave scalar can be evaluated given (5), (24) and (25):

(28) |

Note that (28) requires knowledge of , the local normal stress of the displacement field at each receiver, and , the local stress tensor at each receiver. For marine seismic data, the pressure is measured directly by hydrophones. However, for both land and marine data acquisition, the local receiver stress tensor is not routinely measured. As suggested by Keho (1986), I use a farfield WKBJ approximation of the displacement vector wavefield gradient and divergence terms such that

(29) |

(30) |

(31) |

In this case, (28) can be approximated for land acquisition as:

(32) |

Substituting (27) and (32) into (26) and performing
the *t* integrations, one obtains

(33) |

(34) |

where is now evaluated at the total ray traveltime
from each source at to any subsurface point
and back up to each receiver position . The primed coordinates
indicate that the polarization vectors should be evaluated at the
receiver positions . The subscripts on
and indicate that the Lamé parameters should be
evaluated at the receiver positions . Note that the symbol
signifies a convolution of the surface data with the estimated
*P*-wave source wavelet *w _{1}*.
The autocorrelation integral is

(35) |

and is related to the energy of the incident *P* wavefield.
Combining (33)-(35) yields:

(36) |

Equation (36) gives the least-squares elastic wavefield integral solution for specular reflectivity. It can be slightly simplified further by noting that

(37) |

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

11/17/1997