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Evaluation of imaging condition

While we have maintained our coordinate system thus far in parallel with a shot-receiver migration strategy, we will now detail the evaluation of the imaging condition with the order of operations normally associated with shot-profile migration. In fact, the convolution of the source and receiver grids associated with the imaging condition (and gives rise to imaging condition aliasing) can be performed/implied before the migration which is common to resorted mh-coordinate migrations or at each depth step within the migration. As we have maintained the distinctiveness of the source and receiver grids to this point, the convolution of their axes must now be evaluated.

Calculating Common Image Gather (CIG) offsets involves the evaluation of an imaging condition at all acceptable values of $\r$ and $s_{\xi}$. This gives rise to two new image space variables: the horizontal image coordinate for the earth model, $x_{\xi}$, and the subsurface horizontal offset coordinate, $h_{\xi}$. These variables have much similarity to the data space variables midpoint and offset. In a strictly v(z) medium, these axes overlay. However, in more complicated media, the midpoint variable is somewhat misleading. This is because the wavefield continuation extrapolates energy from a midpoint on the surface to a different midpoint as the wavefields are successively downward-continued. Thus, mixing these two ideas is inappropriate. Source and receiver coordinates and horizontal and sub-surface offset coordinates are related through transforms $\r=x_{\xi}+h_{\xi}$ and $s_{\xi}=x_{\xi}-h_{\xi}$.The new coordinate, $h_{\xi}$, a derived parameter with a magnitude equal to an integer multiple of $\Delta x_{\xi}$, is naturally represented as the product of an integer multiplication factor h and horizontal image space discretization interval $\Delta x_{\xi}$ (i.e. $h_{\xi}=h\Delta x_{\xi}$) that will most commonly be unity.

Using these definitions, the image cube may be constructed by applying the general correlation imaging condition to wavefield W,  
 \begin{displaymath}
I(x_{\xi},z_{\xi},h_{\xi})=\sum_{\omega}\left[\d(\r-h_{\xi})...
 ...\xi}\d(s_{\xi}+h_{\xi})\right]\d(\r-x_{\xi})\d(s_{\xi}-x_{\xi})\end{displaymath} (17)
Note that this expression reduces to the familiar zero subsurface offset form when $h_{\xi}=0$, 
 \begin{displaymath}
I(x_{\xi},z_{\xi})=\sum_{\omega} W(\r,s_{\xi},\omega, z_{\xi}) \d (\r-x_{\xi}) \d (s_{\xi}-x_{\xi}).\end{displaymath} (18)
The convolution arguments applied to wavefield W, in equation (18), yield
   \begin{eqnarray}
I(x_{\xi},z_{\xi},h_{\xi})= & \sum_{\omega}\left[\d(\r-h\Delta ...
 ...a x_{\xi}=x_{\xi},s_{\xi}+h\Delta x_{\xi}=x_{\xi},\omega,z_{\xi}).\end{eqnarray}
(19)
Before continuing with this development, it is useful here to stop and interpret the meaning of equations (18) and (20). The imaging condition itself builds the image-space coordinate axes x and h during the convolution expressed above. The arguments within the wavefield W of equation (20) are the equations of a line. This line, $x_{\xi}$, defines the axis for surface location of the image, and is independent of any assumptions about surface midpoints during the experiment. This is one reason [*]we have avoided using the midpoint variable, m. These two coordinates indeed share many traits, though the midpoint concept is an arbitrary, while intuitive and convenient, coordinate transform. Surface location, $x_{\xi}$, is a rigorous development required by the imaging process.

Continuing our derivation, we now reintroduce the lattice in equation (16) to the imaging condition which yields,
\begin{displaymath}
I(x_{\xi},z_{\xi},h_{\xi})=\sum_{\omega}H(\r,s_{\xi},\omega,...
 ...\Delta x_{\xi},x_{\xi}=s_{\xi}+h\Delta x_{\xi},\omega,z_{\xi}) \end{displaymath} (20)
However, $h\Delta x_{\xi}$ is an integer shift by $\Delta x_{\xi}$ and is defined only at known points on the lattice allowing the index of the Shah function to be shifted to yield,
\begin{displaymath}
I(x_{\xi},z_{\xi},h_{\xi}) = {\cal L}_{FL}(\r=x_{\xi},s_{\xi...
 ...um_{\omega} {\cal L}_{FL}(\omega) H(\r,s_{\xi},\omega,z_{\xi}) \end{displaymath} (21)
Expanding lattice ${\cal L}_{FL}$ into its components ${\cal L}$ and FL,
\begin{multline}
I(x_{\xi},z_{\xi},h_{\xi})=\\ {\cal L}(\r=x_{\xi},s_{\xi}=x_{\x...
 ...ga}{\cal L}_{FL}(\omega) H(\r,s_{\xi},\omega) FL(\omega) \nonumber\end{multline}
and applying a Fourier transform over coordinates $\r$, $s_{\xi}$, and $z_{\xi}$ yields,
 \begin{multline}
I(k_{x_{\xi}},k_{z_{\xi}},k_{h_{\xi}})=\left[{\cal L}(k_{r_{\xi...
 ...mega}{\cal L}_{FL}(\omega) H(\r,s_{\xi},\omega) FL(\omega)\right].\end{multline}
The Rect functions of coordinates $k_{r_{\xi}}$ and $k_{s_{\xi}}$ are collapsed back to a single Rect function in $k_{x_{\xi}}$, where the frequency limit is given by min$(B_\r,B_s_{\xi})$. The min function arises because the maximum grid-spacing along either shot or receiver axis alone dictates the aliasing criteria for the kx-axis. This also allows for simplified calculations in the particular case. Generally, however, the bracketed expression in equation ([*]) is
 \begin{multline}
{\cal L}(k_{r_{\xi}}=k_{x_{\xi}},k_{s_{\xi}}=k_{x_{\xi}},k_{z_{...
 ...a_2\Delta s_{\xi}}) \d(k_{z_{\xi}}-\frac{u_4}{a_4\Delta z_{\xi}}).\end{multline}
The summations of the delta functions over u1 and u2 collapse to a single summation over the variable with the lowest common factor (lcf),
\begin{displaymath}
{\cal L}(k_{x_{\xi}},k_{z_{\xi}})=\sum_{u=-{\rm min}(B_\r,B_...
 ...Delta s_{\xi})})
\d(k_{z_{\xi}}-\frac{u_4}{a_4\Delta z_{\xi}}).\end{displaymath} (22)
The horizontal image coordinate is being sampled at a spacing ${\rm lcf}(a_1\Delta r_{\xi},a_2\Delta s_{\xi})$.Thus, for aliasing to be absent the following condition must hold,  
 \begin{displaymath}
B_x_{\xi}={\rm min}(B_\r,B_s_{\xi})=\frac{1}{{\rm lcf}(2a_1\...
 ...a s_{\xi})}={\rm lcf} 
(\frac{N_\r}{a_1},\frac{N_s_{\xi}}{a_2})\end{displaymath} (23)
where $N_\r$ and $N_s_{\xi}$ are the Nyquist frequencies defined by fundamental sampling interval $\Delta r_{\xi}$ and $\Delta s_{\xi}$. Thus, the alias-free wavefield is given by the following geometry  
 \begin{displaymath}
{\cal L}(k_{x_{\xi}},k_{z_{\xi}})=
\sum_{u=-{\rm min}(\frac{...
 ...Delta s_{\xi})})
\d(k_{z_{\xi}}-\frac{u_4}{a_4\Delta z_{\xi}}).\end{displaymath} (24)

Notice that for the simplified case of zero-offset migration, the pre-supposed notion that there are no operator aliasing artifacts introduced can be shown conclusively within the presentation above. Without two different sampling intervals, be they source/receiver or orthogonal surface coordinates, there are no choices for the min operator in equation ([*]) nor the max operator of equation (25). Instead, the sole variable available, surface location x, dictates the sampling of the model space.

Notice that for the simplified case of zero-offset migration, the pre-supposed notion that there are no operator aliasing artifacts introduced can be shown conclusively within the presentation of the above results. Without two possibly different sampling intervals, for source and receiver grids, there are no choices for the $\mbox{min}$operator in equation (24) nor the $\mbox{max}$ operator of equation (25). Instead, the sole variable available, surface location, dictates the sampling of the model space. This does not however release zero-offset migrations from the ramifications of image condition aliasing, as the implied correlation of the source wavefield associated with source-receiver migrations is still present.


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Stanford Exploration Project
5/23/2004