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Generating Constraint Equations

Generating ADCIGs for forward-scattered wavefields requires specifying reflectivity as a function of either the source-side, $\gamma_s$, or receiver-side, $\gamma_r$, reflection angles. Either choice, though, requires isolating one angle from a system with 6 free parameters: $\beta_s, \beta_r, \gamma_s, \gamma_r,\alpha, $ and $\gamma$. Hence, solving for, say, $\gamma_r$ requires specifying 6 constraint equations.

Three constraint equations are specified by geometric relationships (c.f. Figure [*]). The first constraint equation is a local conservation of reflection angle given by,  

2 \gamma = \gamma_s+ \gamma_r.
\end{displaymath} (8)
The second and third constraint equations derive from a global conservation of reflection angle arguments that relate the S and R plane-wave angles, geologic dip, and the source- and receiver-side reflection angles through Biondi (2005),  

\beta_s= \alpha - \gamma = \alpha - \frac{\gamma_s+\gamma_r}{2},
\end{displaymath} (9)

\beta_r= \alpha + \gamma = \alpha + \frac{\gamma_s+\gamma_r}{2}.
\end{displaymath} (10)
Snell's Law provides a fourth physical constraint equation by relating the source- and receiver-side reflection angles with the local propagation slownesses,  

s_s \; {\rm sin} \;\gamma_s= s_r \; {\rm sin} \;\gamma_r,
\end{displaymath} (11)
which can be rewritten using Equation (10) as,  

{\rm tan}\;\gamma_r= \frac{ {\rm sin}\;2\gamma }{ \frac{s_r}{s_s} + {\rm
\end{displaymath} (12)

Constraint equations (8-11) do not incorporate physical observables measured from the generated image volume. However, we can calculate image-space dips in both the horizontal subsurface half-offset, $\frac{\partial z}{\partial h_x}$, and midpoint, $\frac{\partial z}{\partial x}$, directions. Thus, the final two constraint equations relating measured dips to free parameters can be obtained by taking the appropriate partial derivatives of the parametric hyper-plane surface in Equation (7),  

\left. \frac{\partial z}{\partial h_x}\right\vert _{x,h_z}...
 ...\rm cos} (\alpha + \gamma)+s_s \;{\rm cos} (\alpha - \gamma)},
\end{displaymath} (13)

\left. \frac{\partial z}{\partial x}\right\vert _{z,h_z}= ...
 ...\rm cos} (\alpha + \gamma)+s_s \;{\rm cos} (\alpha - \gamma)}.
\end{displaymath} (14)
We rewrite Equations (13) and (14) using the trigonometric angle addition and subtraction rules,  

\left. \frac{\partial z}{\partial h_x}\right\vert _{x,h_z}...
 ... \gamma+ (s_r+s_s)\;{\rm sin} \; \alpha\;{\rm sin} \; \gamma},
\end{displaymath} (15)
\left. \frac{\partial z}{\partial x}\right\vert _{z,h_z}= -\...
 ... \gamma- (s_r+s_s)\;{\rm sin} \; \alpha\;{\rm sin} \; \gamma},
\end{displaymath} (16)
which we rearrange to yield,
\left. \frac{\partial z}{\partial h_x}\right\vert _{x,h_z}= ...
 ... \; \alpha}{\phi - {\rm tan} \; \alpha\; {\rm tan} \; \gamma},
\end{displaymath} (17)
where $\phi$ is a ``normalized difference'' of slownesses given by, $\phi = \frac{s_r - s_s}{s_r + s_s}$. Solving for ${\rm tan} \; \gamma$ and ${\rm tan} \; \alpha$ leads to,
{\rm tan} \; \gamma= \frac{\phi \;\frac{\partial z}{\partial...
 ...}}{\frac{\partial z}{\partial x}\; {\rm tan} \; \gamma- \phi},
\end{displaymath} (18)
where the parameters held constant during partial differentiation are no longer explicitly written. These two expressions can be manipulated to specify independent equations for reflection angle, $\gamma$,  

{\rm tan}^2 \gamma \left[\phi \frac{\partial z}{\partial x...
 ...ial z}{\partial h_x}+ \frac{\partial z}{\partial x}\right]= 0,
\end{displaymath} (19)
and true geologic dip, $\alpha$,  

{\rm tan}^2 \alpha \left[\phi \frac{\partial z}{\partial h...
 ...ial z}{\partial x}+ \frac{\partial z}{\partial h_x}\right]= 0.
\end{displaymath} (20)
When source and receiver propagation slownesses are equal (i.e., $\phi = 0$), these quadratic equations reduce to,  

\frac{\partial z}{\partial h_x}= - {\rm cot}\; \gamma\;\;\...
\frac{\partial z}{\partial x}= - {\rm cot}\; \alpha. 
\end{displaymath} (21)
which is similar to the expressions derived for the backscattered case save for a $\pi/2$ phase rotation (i.e., ${\rm tan}\;x = {\rm
 cot} (\pi/2 - x)$). Finally, the solution for the receiver-side reflection angle, $\gamma_r$, is obtained from angle $\gamma$ through the relation specified in Equation (12).

In Equations (13) and (14), we differentiated with respect to variables, x and hx. This choice was one step in the development of horizontal ADCIGs. Equally, we can create vertical ADCIGs by developing two constraint equations from partial derivatives with respect to vertical variables, z and hz, holding horizontal variables x and hx constant. Vertical ADCIGs are then generated through introduction of these functions into Equations (15-20). Biondi and Symes (2004) detail situations where it is more advantageous to use vertical ADCIGs than their horizontal counterparts. In particular, vertical ADCIGs provide better spatial resolution for scenarios where the wavefield propagation direction is oriented at steep angles to the geologic dip-field.

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