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Computation of Common Image Gathers by wavefield continuation

In this section we briefly revisit the method for computing Common Image Gathers (CIG) by wavefield-continuation migration. The following development assumes that both the source wavefield and the receiver wavefield have been numerically propagated into the subsurface. The analytical expressions represent wavefields in the time domain, and thus they appear to implicitly assume that the wavefields have been propagated in the time domain. However, all the considerations and results that follow are independent of the specific numerical method that was used for propagating the wavefields. They are obviously valid for reverse-time migration when the wavefields are propagated in the time domain Baysal et al. (1983); Biondi and Shan (2002); Etgen (1986); Whitmore (1983). They are also valid when the wavefields are propagated by downward continuation in the frequency domain, if there are no overturned events. The results presented in this paper are valid even when source-receiver migration is used instead of shot-profile migration, if the conditions are satisfied for these two apparently dissimilar methods to be equivalent Biondi (2003).

The conventional imaging condition for shot-profile migration is based on the crosscorrelation in time of the source wavefield (S) with the receiver wavefield (R). The equivalent of the stacked image is the average over sources (s) of the zero lag of this crosscorrelation; that is:  
 \begin{displaymath}
I\left(z, {x}\right) =
\sum_s 
\sum_t 
S_s\left( t,z,{x}\right)
R_s\left( t,z,{x}\right),\end{displaymath} (1)
where z and x are respectively depth and the horizontal axes, and t is time. The result of this imaging condition is equivalent to stacking over offsets with Kirchhoff migration.

The imaging condition expressed in equation (1) has the substantial disadvantage of not providing prestack information that can be used for either velocity updates or amplitude analysis. Equation (1) can be generalized Biondi and Shan (2002); Rickett and Sava (2002); de Bruin et al. (1990) by crosscorrelating the wavefields shifted horizontally with respect to each other. The prestack image becomes a function of the horizontal relative shift, which has the physical meaning of a subsurface half offset (xh). It can be computed as  
 \begin{displaymath}
I\left(z,{x},{x}_h\right) =
\sum_s 
\sum_t 
S_s\left( t,z,{x}- {x}_h\right)
R_s\left( t,z,{x}+ {x}_h\right).\end{displaymath} (2)
A section of the image cube taken at constant horizontal location x is a Horizontal Offset Common Image Gather, or HOCIG. The whole image cube can be seen as a collection of HOCIGs.

 
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Figure 1
Geometry of an ADCIG for a single event migrated with the wrong (low in this case) velocity. The propagation direction of the source ray forms the angle $\beta$ with the vertical, and the propagation direction of the receiver ray forms the angle $\delta$ with the vertical; $\gamma$ is the apparent aperture angle, and $\alpha$ is the apparent reflector dip. The source ray and the receiver ray cross at $\bar{I}$.Notice that in this figure $\beta,\delta$ and $\alpha$ are positive, but $\gamma$ is negative.

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Sava and Fomel (2002) presented a simple method for transforming HOCIGs into ADCIGs by a slant stack transformation applied independently to each HOCIG Schultz and Claerbout (1978):  
 \begin{displaymath}
I_{{\gamma_{x}}}\left(z,{x},\gamma \right) =
{\rm SlantStack}
\left[
I\left(z,{x},{x}_h\right) 
\right];\end{displaymath} (3)
where $\gamma$ is the aperture angle of the reflection, as shown in Figure [*]. This transformation from HOCIG to ADCIG is based on the following relationship between the aperture angle and the slope, $\partial z/\partial x_h$,measured in image space:  
 \begin{displaymath}
\left.\frac{\partial z}{\partial x_h}\right\vert _{t,{x}}
=\tan\gamma 
=-\frac{k_{x_h}}{k_z};\end{displaymath} (4)
where kxh and kz are respectively the half-offset wavenumber and the vertical wavenumber. The relationship between $\tan \gamma$ and the wavenumbers also suggests that the transformation to ADCIGs can be accomplished in the Fourier domain by a simple radial-trace transform Sava and Fomel (2002).

Sava and Fomel (2002) demonstrated the validity of equation (4) based only on Snell's law and on the geometric relationships between the propagation directions of the source ray (determined by $\beta$ in Figure [*]) and receiver ray (determined by $\delta$ in Figure [*]). Its validity is thus independent of the focusing of the reflected energy at zero offset; that is, it is valid regardless of whether the image point coincides with the intersection of the two rays (marked as $\bar{I}$ in Figure [*]). In other words, it is independent of whether the correct migration velocity is used. The only assumption about the migration velocity is that the velocity at the imaging depth is locally the same along the source ray and the receiver ray. This condition is obviously fulfilled when the reflected energy focuses at zero offset, but it is, at least approximately, fulfilled in most practical situations of interest. In most practical cases we can assume that the migration velocity function is smooth in a neighborhood of the imaging point, and thus that the velocity at the end point of the source ray is approximately the same as the velocity at the end point of the receiver ray. The only exception of practical importance is when the reflection is caused by a high-contrast interface, such as a salt-sediment interface. In these cases, our results must be applied with particular care. When the migration velocity is correct, $\alpha$ and $\gamma$ are respectively the true reflector dip and the true aperture angle; otherwise they are the apparent dip and the apparent aperture angle. In Figure [*], the box around the imaging point signifies the local nature of the geometric relationships relevant to our discussion; it emphasizes that these relationships depend only on the local velocity function.

When the velocity is correct, the image point obviously coincides with the crossing point of the two rays $\bar{I}$.However, the position of the image point when the velocity is not correct has been left undefined by previous analyses Prucha et al. (1999); Sava and Fomel (2002). In this paper, we demonstrate the important result that in an ADCIGs, when the migration velocity is incorrect, the image point lies along the direction normal to the apparent geological dip. We identify this normal direction with the unit vector ${\bf n}$ that we define as oriented in the direction of increasing traveltimes for the rays (see Figure [*]).

Notice that the geometric arguments presented in this paper are based on the assumption that the source and receiver rays cross, even when the data were migrated with the wrong velocity. This assumption is valid in 2-D except in degenerate cases of marginal practical interest (e.g. diverging rays). In 3-D, this assumption is more easily violated, because the two rays are not always coplanar. This discrepancy between 2-D and 3-D geometries makes the generalization to 3-D of the results presented in this paper less than trivial. Therefore, we consider the 3-D generalization beyond the scope of this paper.

As will be discussed in the following and exemplified by the real-data example in Figure [*]a, the HOCIGs, and consequently the ADCIGs computed from the HOCIGs (Figure [*]a), have problems when the reflectors are steeply dipping. At the limit, the HOCIGs become useless when imaging almost vertical reflectors using either overturned events or prismatic reflections. To create useful ADCIGs in these situations we introduce a new kind of CIGs Biondi and Shan (2002). This new kind of CIG is computed by introducing a vertical half offset (zh) into equation (1) to obtain:  
 \begin{displaymath}
I\left(z,{x},z_h\right) =
\sum_s 
\sum_t 
S_s\left( t,z - z_h,{x}\right)
R_s\left( t,z + z_h,{x}\right).\end{displaymath} (5)
A section of the image cube computed by equation (5) taken at constant depth z is a Vertical Offset Common Image Gathers, or VOCIG.

As for the HOCIGs, the VOCIGs can be transformed into an ADCIG by applying a slant stack transformation to each individual VOCIG; that is:  
 \begin{displaymath}
I_{{\gamma_{z}}}\left(z,{x},\gamma \right) =
{\rm SlantStack}
\left[
I\left(z,{x},z_h\right) 
\right].\end{displaymath} (6)
This transformation is based on the following relationship between the aperture angle and the slope $\partial {x}/\partial z_h$measured in image space:  
 \begin{displaymath}
-\left.\frac{\partial {x}}{\partial z_h}\right\vert _{t,z}
=\tan\gamma
=\frac{k_{z_h}}{k_{x}}.\end{displaymath} (7)
Equation (7) is analogous to equation (4), and its validity can be trivially demonstrated from equation (4) by a simple axes rotation. However, notice the sign differences between equation (7) and equation (4) caused by the conventions defined in Figure [*].

Notice that our notation distinguishes the result of the two transformations to ADCIG $\left(I_{\gamma_{x}}~{\rm and}~I_{\gamma_{z}}\right)$,because they are different objects even though they are images defined in the same domain $\left(z,{x},\gamma \right)$.One of the main results of this paper is the definition of the relationship between $I_{\gamma_{x}}$ and $I_{\gamma_{z}}$,and the derivation of a robust algorithm to ``optimally'' merge the two sets of ADCIGs. To achieve this goal we will first analyze the kinematic properties of HOCIGs and VOCIGs.

 
cig-gen-v6
Figure 2
Geometry of the three different kinds of offset-domain (horizontal, vertical and geological-dip) CIG for a single event migrated with the wrong velocity. Ixh is the horizontal-offset image point, Izh is the vertical-offset image point, and I0 is the geological-dip offset image point.

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Next: Kinematic properties of Common Up: Biondi and Symes: ADCIGs Previous: Biondi and Symes: ADCIGs
Stanford Exploration Project
7/8/2003