To begin, I define a forward model for a ``generalized'' PP reflectivity, , which combines elements of Zoeppritz plane wave reflection (e.g., Aki and Richards, 1980), and Rayleigh-Sommerfeld diffraction (e.g., Goodman, 1968). I define a differential volume element as
for , and zero otherwise. Here, w is angular frequency, is compressional velocity, is the Zoeppritz PP plane wave reflection coefficient and () is the angle between the incident source (receiver) ray and the specular ray in Figure . The specular ray is defined as the reflection raypath for which the incident angle is equal to the angle of reflection, with respect to the normal to the reflecting surface. The specular angle is related to and in that , where is the angle between an arbitrary source ray and the reflector normal, and is the angle between an arbitrary receiver ray and the reflector normal. Frazer and Sen (1985, eqn. 9b) derive an analogous expression to (1) for elastic Kirchhoff-Helmholtz surface integrals.
Equation (1) can be integrated over the subsurface volume to forward model the reflected scalar wavefield:
where () is the shot (receiver) coordinate, and is the total traveltime from source to reflection point to receiver. As (Ar) is the WKBJ amplitude factor along the source (receiver) leg of the raypath to the subsurface reflection point , and incorporates the amplitude effects of source (receiver) directivity, geometric spreading, transmission loss, and intrinsic high-frequency Q attenuation. All of these quantities are readily available by raytracing through the background migration velocity model. The scalar U may be the wave pressure measured in a hydrophone, the magnitude of the displacement vector measured at a geophone, or an individual (e.g. vertical) component of the displacement vector if the appropriate component directivity is introduced into Ar.
It is important to note that the volume integration (2) models a single trace for any arbitrary source and receiver location. There is no ``preferred'' acquisition geometry for this synthesis; it works for shot gathers as well as random trace orientations. In particular, (2) is perfectly adequate for constant offset survey data, which will be the focus of this work.