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:
where is mass density, is the second time derivative of , is the elastic stiffness tensor Cijkl, 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:
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.