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Evaluation of the image shift as a function of $\alpha$ and $\gamma$

The final step is to take the derivative of the impulse response of equation (34) and use the relationships of these derivatives with $\tan{\alpha}$ and $\tan \gamma$:
      \begin{eqnarray}
\frac{\partial z}{\partial x} & = \tan{\alpha} = & 
\sqrt{\frac...
 ...idetilde{h_0}}}
{\frac{x_h^2}{\widetilde{h_0}^2} -1}.
\\ \nonumber\end{eqnarray} (35)
(36)
Substituting equations (35) and (36) into the following relationships:
      \begin{eqnarray}
\Delta_z I_{x_h}=
\bar{z}-z
& = & - \widetilde{h_0}\tan{\gamma}...
 ... \widetilde{h_0}\tan{\gamma}\tan{\alpha}\cos{\alpha},
\\ \nonumber\end{eqnarray} (37)
(38)
and after some algebraic manipulation, we prove the thesis. B This appendix demonstrates equations (19) in the main text: that for energy dipping at an angle $\alpha$in the (z,x) plane, the wavenumber kn along the normal to the dip is linked to the wavenumbers kz and kx by the following relationships:  
 \begin{displaymath}
k_{n}=
\frac{k_z}{\cos \alpha}=
\frac{k_{x}}{\sin \alpha}.\end{displaymath} (39)
For energy dipping at an angle $\alpha$the wavenumbers satisfy the well-known relationship  
 \begin{displaymath}
\tan\alpha = \frac{k_{x}}{k_z},\end{displaymath} (40)
where the positive sign is determined by by the conventions defined in Figure [*]. The wavenumber kn is related to kx and kz by the axes rotation  
 \begin{displaymath}
k_{n}=
k_z\cos\alpha + k_{x}\sin \alpha.\end{displaymath} (41)
Substituting equation (40) into equation (41) we obtain
\begin{displaymath}
k_{n}=
\frac{k_z}{\cos\alpha}
\left(\cos^2\alpha + \tan\alph...
 ...(\cos^2\alpha + \sin^2 \alpha \right) =
\frac{k_z}{\cos\alpha},\end{displaymath} (42)
or,
\begin{displaymath}
k_{n}=
\frac{k_{x}}{\sin\alpha}
\left(\cot\alpha\sin\alpha\c...
 ...cos^2\alpha + \sin^2 \alpha \right) =
\frac{k_{x}}{\sin\alpha}.\end{displaymath} (43)
C In this appendix we derive the equations for the ``kinematic migration'' of the reflections from a sphere, as a function of the ratio $\rho$ between the true constant slowness S and the migration slowness $S_\rho=\rho S$.For a given $\rho$we want to find the coordinates of the imaging point ${I}_{\gamma}(z_\gamma,x_\gamma)$ as a function of the apparent geological dip $\alpha_{\rho}$ and the apparent aperture angle $\gamma_{\rho}$.Central to our derivation is the assumption that the imaging point ${I}_{\gamma}$ lies on the normal to the apparent reflector dip passing through $\bar{I}$,as represented in Figure [*].

The first step is to establish the relationships between the true $\alpha$ and $\gamma$and the apparent $\alpha_{\rho}$ and $\gamma_{\rho}$.This can be done through the relationships between the propagation directions of the source/receiver rays (respectively marked as the angles $\beta$ and $\delta$in Figure [*]), and the event time dips, which are independent on the migration slowness. The true $\beta$ and $\delta$ can be thus estimated as follows:
\begin{eqnarray}
\beta=
\arcsin\left(\rho \sin {\beta_\rho}\right)=
\arcsin\left...
 ...[
\rho 
\sin \left({\alpha_{\rho}}+{\gamma_{\rho}}\right)
\right];\end{eqnarray} (44)
(45)
and then the true $\alpha$ and $\gamma$ are:
\begin{displaymath}
\alpha=
\frac
{\beta + \delta}
{2},
\;\;\;\; 
{\rm and}
\;\; \;\;
\gamma=
\frac
{\delta - \beta}
{2}.\end{displaymath} (46)
Next step is to take advantage of the fact that the reflector is a sphere, an thus that the coordinates $(\hat{z},\hat{x})$ of the true reflection point are uniquely identified by the dip angle $\alpha$as follows:
\begin{displaymath}
\hat{z}=\left(z_c - R \cos \alpha\right),
\;\;\;\; 
{\rm and}
\;\; \;\;
\hat{x}=\left(x_c + R \sin \alpha\right),\end{displaymath} (47)
where (zc,xc) are the coordinates of the center of the sphere and R is its radius.

The midpoint, offset, and traveltime of the event can be found by applying simple trigonometry (see Sava and Fomel (2002)) as follows:
   \begin{eqnarray}
{x_h}_{\rm surf}
&=&
\frac{\sin \gamma \cos \gamma}{\cos^2\alph...
 ...rac{\cos \alpha \cos \gamma}{\cos^2\alpha - \sin^2\gamma}
\hat{z}.\end{eqnarray} (48)
(49)
(50)

The coordinates of the point $\bar{I}({\bar{z}},{\bar{x}})$,where the source and the receiver rays cross, are:
      \begin{eqnarray}
\lefteqn{
{\bar{z}}=
}
&&
{x_h}_{\rm surf}
\frac
{{\cos^2}{\alp...
 ...o}}}
{\sin {\gamma_{\rho}} \cos {\gamma_{\rho}}}
{x_h}_{\rm surf};\end{eqnarray} (51)
(52)
and the corresponding traveltime ${{t_{D}}_\rho}$ is:  
 \begin{displaymath}
{{t_{D}}_\rho}=
2 {\rho S} 
\frac{\cos {\alpha_{\rho}} \cos ...
 ...ho}}}{\cos^2{\alpha_{\rho}} - \sin^2{\gamma_{\rho}}}
{\bar{z}}.\end{displaymath} (53)

Once we have the traveltimes tD and ${{t_{D}}_\rho}$,the normal shift ${\bf \Delta n}_{\rm tot}$ can be easily evaluated by applying equation (15) (where the background velocity is $S_\rho$and the aperture angle is $\gamma_{\rho}$), which yields:  
 \begin{displaymath}
{\bf \Delta n}_{\rm tot}=
-\frac
{\left({{t_{D}}_\rho}- {t_{D}}\right)}
{2 \rho S\cos\gamma_{\rho}}{\bf n}.\end{displaymath} (54)

We use equation (54), together with equations (51) and (52), to compute the lines superimposed onto the images in Figure [*]. D In this Appendix we derive the expression for the residual moveout (RMO) function to be applied to ADCIGs computed by wavefield continuation. The derivation follows the derivation presented in Appendix C. The main difference is that in this appendix we assume the rays to be stationary. In other words, we assume that the apparent dip angle $\alpha_{\rho}$ and aperture angle $\gamma_{\rho}$are the same as the true angles $\alpha$ and $\gamma$.This assumption also implies that the (unknown) true reflector position $(\hat{z},\hat{x})$ coincides with the point $\bar{I}({\bar{z}},{\bar{x}})$where the source and the receiver ray cross.

Given these assumptions, the total traveltime through the perturbed slowness function $S_\rho$ is given by the following expression:
\begin{displaymath}
{{t_{D}}_\rho}=
2 \rho {S} 
\frac{\cos \alpha \cos \gamma}{\cos^2\alpha - \sin^2\gamma}
\bar{z},\end{displaymath} (55)
which is different from the corresponding equation in Appendix C [equation (53)]. The difference in traveltimes $({t_{D}}_\rho-{t_{D}})$,where tD is given by equation equation (50), is thus a linear function of the difference in slownesses $[(\rho-1)S]$;that is,
\begin{displaymath}
{t_{D}}_\rho-{t_{D}}=
2 \left(\rho-1\right) 
{S} 
\frac{\cos \alpha \cos \gamma}{\cos^2\alpha - \sin^2\gamma}
\bar{z}.\end{displaymath} (56)

As in Appendix C, the normal shift ${\bf \Delta n}_{\rm tot}$ can be evaluated by applying equation (15) (where the background velocity is $S_\rho$and the aperture angle is $\gamma$), which yields:  
 \begin{displaymath}
{\bf \Delta n}_{\rm tot}
=
\frac{1-\rho}{\rho}
\frac{\cos \alpha}{\cos^2\alpha - \sin^2\gamma}
\bar{z}\; {\bf n}.\end{displaymath} (57)
The RMO function (${\bf \Delta n}_{\rm RMO}$) describes the relative movement of the image point at any $\gamma$ with respect to the image point for the normal-incidence event $(\gamma =0)$.From equation (57), it follows that the RMO function is:
   \begin{eqnarray}
\lefteqn{
{\bf \Delta n}_{\rm RMO}=
{\bf \Delta n}_{\rm tot}\le...
 ...t(\cos^2\alpha - \sin^2\gamma\right)\cos\alpha}
\bar{z}\; {\bf n}.\end{eqnarray}
(58)
The true depth $\bar{z}$ is not known, but at normal incidence it can be estimated as a function of the migrated depth z0 by inverting the following relationship:
\begin{displaymath}
z_0=
\left(
\frac{1-\rho}{\rho \cos\alpha} +1
\right)
\bar{z},\end{displaymath} (59)
as:  
 \begin{displaymath}
\bar{z}=
\left[
\frac{\rho\cos\alpha}{1-\rho\left(1-\cos\alpha\right)}
\right]
z_0.\end{displaymath} (60)
Substituting relation (60) in equation (58) we obtain the result:  
 \begin{displaymath}
{\bf \Delta n}_{\rm RMO}=
\frac{1-\rho}
{1-\rho\left(1-\cos\...
 ...gamma}{\left(\cos^2\alpha - \sin^2\gamma\right)}
z_0\; {\bf n},\end{displaymath} (61)
which for flat reflectors $(\alpha=0)$simplifies into:  
 \begin{displaymath}
{\bf \Delta n}_{\rm RMO}=
\left(1-\rho\right)\tan^2\gamma
\; z_0\; {\bf n}.\end{displaymath} (62)

In Figure [*], the solid lines superimposed into the images are computed using equation (61), whereas the dashed lines are computed using equation (62).

 


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Next: About this document ... Up: REFERENCES Previous: Evaluation of the impulse
Stanford Exploration Project
7/8/2003