Forward problem

To begin, I define a forward model for a ``generalized'' PP reflectivity, $R_{_{P\!P}}$, 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 $R_{_{P\!P}}$ as

 

$$R_{_{P\!P}}= \frac{iw}{\alpha} \grave{P}\!\acute{P}\cos\phi_s\cos\phi_r \;,$$ (1)

for $\vert \phi_s, \phi_r \vert \le \pi/2$, and zero otherwise. Here, w is angular frequency, $\alpha$ is compressional velocity, $\grave{P}\!\acute{P}$ is the Zoeppritz PP plane wave reflection coefficient and $\phi_s$ ($\phi_r$) 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 $\Theta$ is equal to the angle of reflection, with respect to the normal to the reflecting surface. The specular angle $\Theta$ is related to $\phi_s$ and $\phi_r$ in that $\Theta = \vert\theta_s-\phi_s\vert = \vert\theta_r-\phi_r\vert$, where $\theta_s$ is the angle between an arbitrary source ray and the reflector normal, and $\theta_r$ 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.

 

geom
Figure 1 Ray geometry for generalized reflection and diffraction.

Equation (1) can be integrated over the subsurface volume to forward model the reflected scalar wavefield:

 

$$U({\bf x}_r,{\bf x}_s,w) = \int_V A_s A_r R_{_{P\!P}}e^{iw\tau} dV \;,$$ (2)

where ${\bf x}_s$ (${\bf x}_r$) is the shot (receiver) coordinate, and $\tau$ is the total traveltime from source to reflection point to receiver. A_s (A_r) is the WKBJ amplitude factor along the source (receiver) leg of the raypath to the subsurface reflection point ${\bf x}$, 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 A_r.

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.