Transformation to dip-dependent Common Image Gathers

Two related observations are at the basis of the proposed method. The first one is that the HOCIG and the VOCIG are just particular cases of offset-domain gathers. In general, the offset can be oriented along any arbitrary direction. The second one, is that the offset direction aligned with the apparent geological dip of the imaged event has the unique property of affording the sharpest focusing of the event. The goal of our method is to transform both HOCIGs and VOCIGs into an equivalent set of CIGs (DDOCIGs), for which the effective offset is aligned with the local apparent dips. After the transformation, the DDOCIGs obtained from the HOCIGs and the VOCIGs can be appropriately averaged to obtain a single set of DDOCIGs that contain accurate information for all the geological dips.

Figure  illustrates the geometry of the offset-domain CIGs for a single event recorded at the surface for the source location S and receiver location R. The crucial assumption of our geometric construction is that the traveltime along the source ray summed with the traveltime along the receiver ray is the same for all the offset directions and equal to the recording time of the event $\left( \left\vert S-S_0\right\vert+ \left\vert R-R_0\right\vert= \left\vert S-S... ...\vert= \left\vert S-S_{z_h}\right\vert+ \left\vert R-R_{z_h}\right\vert \right)$.

In this sketch, the migration velocity is assumed to be lower than the true velocity, and thus the reflections are imaged too shallow and above the point where the source ray crosses the receiver ray (SR). The line passing through SR, and bisecting the angle formed by the source and receiver ray, is oriented at an angle $\alpha$ with respect to the vertical direction. The angle $\alpha$ is the apparent geological dip of the event after imaging. It would correspond to the true geological dip if the migration velocity were correct. Half of the angle formed between the source and receiver ray is the aperture angle $\gamma$.

When HOCIGs are computed, the end point of the source ray (S_xh) and the end point of the receiver ray (R_xh) are at the same depth. The imaging point I_xh is in the middle between S_xh and R_xh and the imaging offset is h_x=R_xh-S_xh. Similarly, when VOCIGs are computed, the end point of the source ray (S_zh) and the end point of the receiver ray (R_zh) are at the same horizontal location. The imaging point I_zh is in the middle between S_zh and R_zh and the imaging offset is z_h=R_zh-S_zh. When the offset direction is oriented along the apparent geological dip $\alpha$(what we called the optimally focusing offset direction), the end point of the source ray is S₀ and the end point of the receiver ray is R₀. The imaging point I₀ is in the middle between S₀ and R₀ and the imaging offset is h₀=R₀-S₀. It is easy to demonstrate that both I_xh and I_zh lie on the line passing through $S_0, I_0~\rm{and}~R_0$.The demonstration is based on the assumption that $\left\vert S_{x_h}-S_0\right\vert=\left\vert R_{x_h}-R_0\right\vert$and $\left\vert S_{z_h}-S_0\right\vert=\left\vert R_{z_h}-R_0\right\vert$.

The offsets along the different directions are linked by the following simple relationship, which can be readily derived by trigonometry applied to Figure ; that is,

$$\begin{eqnarray} h_x& = &\frac{h_0}{\cos\alpha}, \ z_h& = &\frac{h_0}{\sin\alpha}.\end{eqnarray}$$ (65) (66)

Also the shift of the imaging points I_xh and I_zh can be easily expressed in terms of the offset h₀ and the angles $\alpha$ and $\gamma$ as:

$$\begin{eqnarray} \Delta I_{x_h}& = \left(I_{x_h}-I_0\right) & =-h_0\tan{\gamma}\... ... \left(I_{z_h}-I_0\right) & =h_0\frac{\tan{\gamma}}{\tan{\alpha}}.\end{eqnarray}$$ (67) (68)

Notice the dependence of $\Delta I_{x_h}$ and $\Delta I_{z_h}$ on the aperture angle $\gamma$.This dependence causes events with different aperture angles to be imaged at different locations, even if they originated at the same reflecting point in the subsurface. This phenomenon is related to the well known ``reflector-point dispersal'' in common midpoint gathers. In this context, this dispersal is a consequence of using a wrong imaging velocity, and we will refer to it as image-point dispersal.

The fact that all three imaging points are aligned along the apparent geological dip allows our transformation to remove the image-point dispersal, and it is crucial to the effectiveness of DDOCIGs. In other words, to transform one set of CIGs into another set we just need to transform the offset axis; the image is then automatically shifted along the apparent geological dip by the right amount. Appendix A demonstrates this fact.

The proposed CIG transformation is a simple dip-dependent non-uniform stretching of the the offset-axis according to the relationships in equations () and (). The transformation is easily implemented in the wavenumber (k_z,k_x) domain, by taking advantage of the well known relationship $\tan{\alpha}=k_{x}/k_z$.

After both the HOCIGs and the VOCIGs are transformed, they can be merged together. A simple scheme to merge them is a weighted average, where the weights w_xh and w_zh are set to

$$\begin{eqnarray} w_{x_h}= \cos^2{\alpha}, \ w_{z_h}= \sin^2{\alpha} .\end{eqnarray}$$ (69) (70)

 

Mig-all-zo-both
Figure 8 Images of the synthetic data set obtained with a) correct velocity, b) too low velocity by 4%.