Multiple prediction beyond 2-D

First, I will list a summarization of the key points in this approach and then explain them in more details:

1.

There are many missing traces in 3-D multiple prediction.

2.

Instead of interpolating traces, this approach finds good proxies for the missing ones.

3.

To be qualified as ``good'', a proxy must have the same offset and either a similar CMP location or a similar azimuth angle.

4.

Using those proxies for the missing traces, we hope to predict multiples with first-order accuracy in multi-streamer geometry.

In order to better understand the approach, let's assume we have a multi-streamer acquisition system, as shown in Figure 1, with one shotline and seven streamers. Supposing that we want to predict the multiple from source S₀ to receiver R₄, we need to consider the contributions from all the possible multiple reflection points between S₀ and R₄ by cross-convolution. For instance, we need to collect all the traces with sources located at S_i (i=1,...,7) and receiver located at R₄. In Figure 1, the thin solid line represents the corresponding trace collected in the survey, and the thin dashed line stands for a missing trace in the survey. The challenge is to find appropriate proxies for such missing traces in the survey.

 

multi-streamer Figure 1 A typical multi-streamer geometry. The $\stackrel{\longrightarrow}{S_4R_1}$ path is a good proxy for the absent out-of-plane shot $\stackrel{\longrightarrow}{S_1R_4}$, since the two paths share the same midpoint and have the same offset. Similarly, $\stackrel{\longrightarrow}{S_2R_4}$ is replaced by $\stackrel{\longrightarrow}{S_4R_2}$, $\stackrel{\longrightarrow}{S_3R_4}$ by $\stackrel{\longrightarrow}{S_4R_3}$, $\stackrel{\longrightarrow}{S_5R_4}$ by $\stackrel{\longrightarrow}{S_4R_5}$, $\stackrel{\longrightarrow}{S_6R_4}$ by $\stackrel{\longrightarrow}{S_4R_6}$, and $\stackrel{\longrightarrow}{S_7R_4}$ by $\stackrel{\longrightarrow}{S_4R_7}$.

There is one well-known geophysical concept that can help us meet the challenge, common-midpoint (CMP), which assumes that traces with the same CMP location and the same offset contain the same information about one location in the earth. Although the common-midpoint assumption is a first-order approximation when the structure is not strictly flat, I will demonstrate that it is useful in our search for the substituting traces.

For example, we try to find the best proxy for $\stackrel{\longrightarrow}{S_1R_4}$ from the collected dataset. Not surprisingly, we find that the real trace $\stackrel{\longrightarrow}{S_4R_1}$ shares the same CMP location with the virtual trace $\stackrel{\longrightarrow}{S_1R_4}$ and has the same offset as well. The only difference is the azimuth angle. Therefore, trace $\stackrel{\longrightarrow}{S_4R_1}$ is a proxy for trace $\stackrel{\longrightarrow}{S_1R_4}$ in the multiple prediction with first-order accuracy. Similarly, we find other substituting traces for other virtual traces.

The central streamer in Figure 1 is a special case, in which we can always find the substituting traces for the virtual ones with the same CMP location and offset. Working with other streamers (e.g. Figure 2), we cannot find proxies with the same CMP location and offset. However, we can relax the definition of a substitute trace by giving up the requirement that the proxy share the same CMP location. Then we can find another group of proxies for the missing traces, as shown in Figure 2. Since the cross-line spreading aperture is usually smaller than the in-line aperture, this extension might be acceptable in many real applications.

 

multi-streamer-1 Figure 2 While working with other streamers, we have to give up the requirement that the proxy share the same CMP location, and find a group of proxies with reasonable accuracy. $\stackrel{\longrightarrow}{S_1R_1}$ can be replaced by $\stackrel{\longrightarrow}{S_4R_4}$, $\stackrel{\longrightarrow}{S_2R_1}$ by $\stackrel{\longrightarrow}{S_4R_3}$, and $\stackrel{\longrightarrow}{S_3R_1}$ by $\stackrel{\longrightarrow}{S_4R_2}$. Each pair shares the same offset and azimuth angle.

Figure 3 shows two types of 3-D multiples' geometries, $\stackrel{\longrightarrow}{S_0M_1R_1}$ and $\stackrel{\longrightarrow}{S_0M_2R_2}$, which are generated by different earth structures. For example, $\stackrel{\longrightarrow}{S_0M_1R_1}$ is mainly caused by cross-line dip or scattering reflector and $\stackrel{\longrightarrow}{S_0M_2R_2}$ by 1-D or in-line dip reflector. The difference between these two types is that the source, the multiple reflection, and the receiver locations are aligned in the second type, whereas they are not in the first one. The approach discussed here is fully applicable to the second type of 3-D multiple. There is no kinematic approximation at all. The approximation error happens only when we deal with the first type of 3-D multiple.

 

multiple-type Figure 3 Two different types of 3-D multiple. Type one, the source S, the multiple reflection point M1, and the receiver R1 cannot be aligned by a single straight line, is probably caused by cross-line dip or scattering reflector. Type two, the source S, the multiple reflection point M2, and the receiver R2 are aligned by a single straight line $\stackrel{\longrightarrow}{S_0M_2R_2}$, happens when the structure is approximately 1-D or in-line dip reflector. The approach discussed here has no approximation for the second type.