Consider the elastodynamic wave equation for a displacement vector field and second order stress tensor field due to a body force vector excitation :
as per Aki and Richards (1980, p.19, eqn. 2.17). Consider a second displacement and stress field and , with body force , which also satisfy the elastodynamic wave equation:
The two sets of elastic wavefield can be related by the Divergence Theorem:
We assume that the scattered (reflected + diffracted) wavefield within the volume is due to a volumetric body force equivalent of reflection surface excitations. This means that, although reflections are created by reflector surface discontinuities in material properties as shown later, we represent those surface excitations as equivalent volumetric body forces, as opposed to conventional surface tractions. In this case, we can write the forward theory explicitly as a mapping from interior points in the volume to an recording surface . More importantly, the inverse theory is an explicit map from the recorded surface data on to the equivalent body force functions at all within , rather than to arbitrary reflection surfaces within . This distinction is subtle yet critical for a rigorous solution to the angle-dependent reflectivity inverse problem.
Under the body force equivalence assumption, the surface integral in (3) vanishes (no volume boundary effects):
For any vector and second order tensor ,
where is a second-order inner contraction of with , and is equivalent to their dot product in this specific case. Using identity (5), (4) can be expanded as:
We next assume a linear elastic stress-strain relationship such that
where is the fourth-order elastic stiffness tensor Cijkl. Due to the symmetries inherent in (Aki and Richards, 1980, p.20),
after Frazer and Sen (1985, p.123), for example.
Thus, and so (6) becomes:
The difference between the two dot-product equations and can be written as
Substituting (10) into (9) yields
as per Aki and Richards (1980, p.27, eqn. 2.35) with the surface integral terms set to zero. We specify such that
where selects an arbitrary component of the vector wavefield at the receiver location . The delta function form (12) of implies that is by definition a solution to the Green's function problem:
where is the second order Green's tensor. In this case, substituting (12) into (11) and integrating out the delta function reveals:
Equation (15) is our volume integral representation of the reflected wavefield at , due to a body force equivalent of reflective surface excitation, weighted against the source wavefield at each subsurface point .