An expression for the reflected wavefield can be obtained from the surface integrals of the general representation theorem (12):

(22) |

in which I have assumed a free-surface boundary condition, such that the
traction vanishes on *S*. Since the reconstruction
of the subsurface reflected wavefield requires reverse time propagation, the
conjugate is used. Using the WKBJ form (14), with
, the conjugate
transpose Green's function can be evaluated as:

(23) |

where the primed polarization vectors are evaluated at the
receiver positions , and the unprimed polarization vectors are evaluated at the subsurface positions .Substituting (23) into (22) and performing the *t*' integration
yields:

(24) |

In the case that the background propagation model is isotropic, which has been implicitly assumed in the definition of the Green's function (although this need not be the case), then the term can be written as

(25) |

where and are the Lamé parameters. Thus, the solution for the reconstructed reflected vector wavefield in an isotropic ray-valid heterogeneous background medium is given by the surface integral (24) and the constitutive equation (25).

11/17/1997