Plane-wave ODCIGs

In this section, we examine the ODCIG volumes generated by forward-scattered wavefields. For simplicity, we illustrate these concepts using plane-wave S and R wavefields. We also assume that S and R propagate at constant, though not necessarily equal, slownesses (i.e., reciprocal of velocity). These idealizations allow us to generate an analytic surface in ODCIG space for both P-P diffracted and P-S converted waves. We specify planar S and R wavefields in constant velocity media using source and receiver ray parameter vectors, $\mathbf{w}_s$ and $\mathbf{w}_r$, defined by,

$$\mathbf{w}_s=\left[p_s,q_s\right]= s_s \;\left[\;{\rm sin}\;... ...]= s_r \;\left[\;{\rm sin}\;\beta_r,{\rm cos}\;\beta_r\right],$$ (3)

where p_s and p_r are the source and receiver horizontal ray parameters, q_s and q_r are the source and receiver vertical ray parameters, and s_s and s_r are the source and receiver wavefield propagation slownesses, respectively. Also, we use a convention where angles are defined clockwise positive with respect to the vertical depth axis.

Forward-scattered S and R wavefields must satisfy the causality arguments illustrated in Figure , which requires a negative sign in the source and receiver extrapolation operators. Using the aforementioned assumptions, we generate the following extrapolated S and R wavefield volumes,  

$$S(\mathbf{x_s};t) = \delta( t+\mathbf{w}_s\cdot \mathbf{x_... ...R(\mathbf{x_r};t) = \delta( t+\mathbf{w}_r\cdot \mathbf{x_r}).$$ (4)

Applying a Fourier transform over the t-axis of both S and R yields,  

$$S(\mathbf{x_s};\omega) = {\rm exp} \left(-i \omega\mathbf{... ...{\rm exp} \left(-i \omega\mathbf{w}_r\cdot\mathbf{x_r}\right).$$ (5)

Evaluating the imaging condition in Equation (2) with the wavefields in Equation (5) generates the following image-space volume,

$$\begin{eqnarray} I(\mathbf{x},\mathbf{h}) = & \sum_{\omega} \delta(\mathbf{x_s... ...) + \mathbf{h}\cdot(\mathbf{w}_s+\mathbf{w}_r) \right]. \nonumber \end{eqnarray}$$ (6)

The non-zero $\delta$-function argument,  

$$x \left(p_r-p_s\right)+ z \left(q_r-q_s \right)- h_x \left(p_r+p_s \right)- h_z \left(q_r+q_s \right)=0,$$ (7)

represents an analytic forward-scattered ODCIG hyper-plane surface. Importantly, this surface interrelates source and receiver plane-wave angles, propagation slownesses and image-space variables, $\mathbf{x}$ and $\mathbf{h}$. In the next section, we manipulate this formula to generate constraint equations that help isolate the receiver-side contribution to the total reflection angle.