To evaluate the least-squares reflectivity solution (10), one requires representations for the vector wavefields and .

A vector wavefield must satisfy the elastodynamic wave equation operator :

subject to initial and boundary conditions:

(11) |

where is mass density, is the second time derivative of
, is the elastic stiffness tensor *C*_{ijkl}, and
is a body force density. The surface *S* bounds the solution domain
volume *V*, and the () symbol is a second order inner contraction.
An integral solution can be obtained for using Betti's Theorem (the
vector equivalent of Green's Theorem for scalars),
and by assuming zero initial conditions:

(12) |

as demonstrated in Aki and Richards (1980, p.29). The unit normal to the
surface *S* is denoted as . Note that the integral solution (12)
is the elastodynamic vector equivalent of the Kirchhoff-Rayleigh-Sommerfeld
integral equation for scalar waves.

11/17/1997