To proceed, we use isotropic elastic WKBJ (ray-valid) Green's tensors for P waves of the form:
where A<<860>>P and are the ray-valid P-wave amplitude and traveltime from the ``source'' location to the ``observation'' point .The unit vector is the direction parallel to P-wave propagation, and is perpendicular to the wavefronts .Given the definition of in (16), we derive the following useful quantities:
where is the compressional wave velocity. In the gradient equation (19) for we have used the fact that
which follows from the ray eikonal equation defining the traveltime :
(Cervený et al., 1977), and the farfield approximation that
since and so .
Now, we choose to get the P-wave reflectivity response at the subsurface point from the source location .Under the previous assumption of linear elastic isotropy for , the stress-strain relationship is isotropic such that
where and are the Lamé parameters, and is the second-order identity matrix .The expression for the divergence of the stress field becomes
where we have used the identities:
for any scalar and second order tensor (Ben-Menahem and Singh, 1981, p.953). Now, using the identity
for any two vectors and , and the fact that
we rewrite (25) as
where we have used (18) and the definition:
Furthermore, we compress the dot products in (26) to yield:
Recalling the elastic wave equation
and substituting (31) and (33) for the stress divergence term, we arrive at the following expression for the body force :
Equation (35) is a body force equivalent for surface reflectivity excitations, and is clearly dependent on material property contrasts (gradients): , and .
At this point, it is convenient to make an analogy to Born elastic scattering (Beydoun and Mendes, 1989) in that we can write
where is the gradient with respect to a reflectionless smooth background model, and is the stress divergence field associated with the true model material property contrasts: , and , which give rise to the scattered wavefield. The two stress divergence terms can be associated with (31) and (33) such that:
which is the background wavefield, and
which is the scattered wavefield.
To obtain the reflected field at the receiver position , recall the volume integral representation (15) for P waves:
We substitute the WKBJ Green's tensor for the wavefield and as defined by (16) and (14):
The latter is evaluated at , and all other compact quantities are evaluated at .The factor is required for multi-component displacement data. For example, if 3-component data are available, or if pressure data are recorded (marine), then and . Or, if only vertical component data are available, then is just the cosine of the incident arrival ray direction with respect to the normal to the recording surface at , and is the recorded vertical component uz<<208>>P. Substituting (35), (16) and (41) into (39) we obtain:
where we have compacted the notation: .Equation (44) defines how to forward model non-Zoeppritz, non-geometric P-P reflections and diffractions created by gradients in the material properties , and . We note that the concept of a single ``reflecting surface'' is generalized to three such surfaces, each defined by , and , which are not necessarily identical! We also note that the scattering contribution from the parameter gradients involves no plane-wave assumption, and therefore is more general than the Zoeppritz coefficients. Finally, since and are arbitrary incident and emergent directions at the reflecting ``surface'', they do not satisfy Snell's Law in general, and therefore create a total scattered response which is a hybrid combination of geometric plane-wave reflection (Aki and Richards, 1980), and Rayleigh-Sommerfeld diffraction (Goodman, 1968).