To proceed, we use isotropic elastic WKBJ (ray-valid) Green's tensors for P waves of the form:

(16) |

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:

(17) |

(18) |

and

(19) |

where is the compressional wave velocity. In the gradient equation (19) for we have used the fact that

(20) |

which follows from the ray eikonal equation defining the traveltime :

(21) |

(Cervený et al., 1977), and the farfield approximation that

(22) |

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

(23) |

where and are the Lamé parameters, and is the second-order identity matrix .The expression for the divergence of the stress field becomes

(24) |

where

(25) |

and

(26) |

where we have used the identities:

(27) |

and

(28) |

for any scalar and second order tensor (Ben-Menahem and Singh, 1981, p.953). Now, using the identity

(29) |

for any two vectors and , and the fact that

(30) |

we rewrite (25) as

(31) |

where we have used (18) and the definition:

(32) |

Furthermore, we compress the dot products in (26) to yield:

(33) |

Recalling the elastic wave equation

(34) |

and substituting (31) and (33) for the stress divergence term, we arrive at the following expression for the body force :

(35) |

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

(36) |

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:

(37) |

which is the background wavefield, and

(38) |

which is the scattered wavefield.

To obtain the reflected field at the receiver position , recall the volume integral representation (15) for P waves:

(39) |

We substitute the WKBJ Green's tensor for the wavefield and as defined by (16) and (14):

(40) |

and

(41) |

where

(42) |

(43) |

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 *u*_{z}^{<<208>>P}.
Substituting (35), (16) and (41) into (39)
we obtain:

(44) |

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).

11/16/1997