Multiple prediction beyond two dimensions

One of key issues in 3-D demultiple is that there are many missing traces. Instead of interpolating the missing traces, the critical points of my approach are:

1.

This approach tries to find proxies for the missing traces.

2.

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

3.

Those proxies for the missing traces are used to predict multiples with first-order accuracy in multi-streamer geometry.

In order to better understand the approach, let's assume that 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 a 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.

 

multi-streamer
Figure 1 A multi-streamer geometry. The $\stackrel{\longrightarrow}{S_4R_1}$ path is a 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, the 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 substitute traces.

For example, for the virtual trace $\stackrel{\longrightarrow}{S_1R_4}$,the real trace $\stackrel{\longrightarrow}{S_4R_1}$ shares the same CMP location 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 can find substitutes for other virtual traces.

The central streamer in Figure 1 is a special case, in which we can always find substitute traces for the virtual ones with the same CMP location and offset. When we try to predict other streamers' multiple reflections, though, as in Figure 2, we are not so lucky to 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 Figure 2 illustrates. Since the cross-line spreading aperture is usually smaller than the in-line aperture, this extension may be acceptable in many real applications.

 

multi-streamer-1
Figure 2 While working with streamers not in the middle, 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.

There are some limitations to the method discussed in this paper. Before addressing those limitations, I would first classify the surface multiple reflections into two categories. Figure 3 depicts two types of geometries for the surface multiples, $\stackrel{\longrightarrow}{S_0M_1R_1}$ and $\stackrel{\longrightarrow}{S_0M_2R_2}$. The difference between these two categories is that, source S₀, surface multiple reflection position M₁, and receiver R₁ are aligned together on the surface; whereas S₀, M₂, and R₂ can not be aligned together. The embedded physical reason is that, multiple $\stackrel{\longrightarrow}{S_0M_1R_1}$ is mainly caused by 1-D earth's structures or in-line dip reflectors (2.5-D), and multiple $\stackrel{\longrightarrow}{S_0M_2R_2}$ by cross-line dips or scattering reflectors.

The definition of proxies in my proposal guarantees that the method in this paper is fully applicable to the multiples like $\stackrel{\longrightarrow}{S_0M_1R_1}$ without kinematic approximations. The approximation errors occur only when we deal with the multiples like $\stackrel{\longrightarrow}{S_0M_2R_2}$. In other words, when there are strong cross-line dips or scattering reflectors, my approach will introduce the approximation errors inevitably.

 

multiple-type Figure 3 Two types of surface multiples. $\stackrel{\longrightarrow}{S_0M_2R_2}$, in which the source S0, the multiple reflection point M2, and the receiver R2 can be aligned, is more likely caused by cross-line dips or scattering reflectors. $\stackrel{\longrightarrow}{S_0M_1R_1}$, in which S0, M1, and R1 are aligned together, occurs when the earth's structures are approximately 1-D or there are only in-line dip reflectors.