Forward problem

The theory underlying this study is more fully discussed in Lumley and Beydoun (1991) and Lumley (1993). I assume a forward model for a ``generalized'' PP reflectivity, $R_{_{P\!P}}$, which combines elements of plane wave reflection and Rayleigh-Sommerfeld diffraction theory. I define a differential volume element of $R_{_{P\!P}}$ 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 . $\Theta$ is the specular angle of reflection. Equation (1) can be integrated over the subsurface volume to 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 obtained by raytracing through the background migration velocity model.

 

geom
Figure 1 Ray geometry for generalized reflection and diffraction.