Numerical Results

I tested the GDM method on the 2D SEG/EAGE salt model (see Figure a) and associated synthetic data computed by FD solutions to the 2D acoustic wave equation. The data include shot gathers generated by 323 shots with a peak-frequency of 13 Hz. There are 176 receivers in each shot gather, and shot and receiver intervals are 48.8 m and 24.4 m, respectively. There are 1001 time samples with a time interval of 0.008 s in each trace. To reduce the computation time, I only focus on the small region below the salt shown in Figure b.

Figure a shows a typical Green's function generated by the wave equation solver. I first mute everything but the early-arrivals of the Green's function (Zhou and Schuster, 2002), and the migration kernel is calculated by convolving only the traces with the early-arrivals (Figure b), which only employs a few periods of the wavefront. The GDM image of the early-arrivals is shown in Figure a. For comparison, I also implement the GDM using the entire wavefield (Figure a) instead of just using the early-arrivals. The migration result is shown in Figure b. The wavefront image (Figure a) and the entire waveform image (Figure b) are almost identical. The reason is that the direct waves, or the early-arrivals in the Green's function are the strongest events, the amplitude of which is about two orders-of-magnitude larger than that of the multiples. Therefore, after the dot-product and stacking processes, the contributions of the weak-amplitudes (i.e., later arrivals) of the multiples in the migration image are concealed by the relatively strong-amplitudes of the direct waves and primaries. This can be mathematically demonstrated in the following way.

Figure: Velocity model used in the synthetic tests.

Figure: Synthetic Green's function and its separation in time. a) A typical bandlimited Green's function generated from the 2D SEG/EAGE salt model. The dashed box shows b) the early-arrivals. c) The Green's function only containing multiples.

Figure: Migration results using the GDM method. a) Migration image constructed from migration kernel formed by convolving the early-arrivals. b) Migration image using all of the arrivals to form a migration kernel. c) Migration image using only multiples in the migration kernel. d) The optimal stack of a) and c).

The Green's function shown in Figure a can be divided into two parts, direct waves (Figure b) and multiples (Figure c):

$$\tilde{g}({\bf {s}},t\vert{\bf {x}},0) = \tilde{g}_s^D+\tilde{g}_s^M,$$

$$\tilde{g}({\bf {r}},t\vert{\bf {x}},0) = \tilde{g}_r^D+\tilde{g}_r^M,$$ (14)

where the superscripts $D$ and $M$ stand for direct waves and multiples. And the subscripts $s$ and $r$ denote the source-side and receiver-side Green's function, respectively. Plugging equation  into , I get

$$\tilde{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t) =\tilde{g}_s... ...g}_r^M + \tilde{g}_s^D \ast \tilde{g}_r^M + \tilde{g}_s^M \ast \tilde{g}_r^D.$$ (15)

It is clear that the generalized diffraction-stack migration using the early-arrivals of the Green's function is nothing but the dot-product of the recorded shot gathers with the first term of the migration kernel in equation . The direct waves are strong compared to multiples, so the other three terms in equation , especially the second term, is very small compared to the first term. Therefore, when I apply the GDM using the entire wavefield of the Green's function, I get nearly the same result as applying the GDM using early-arrivals only.

Inspired by the above analysis, I calculate the GDM images separately, one using the direct-wave migration kernel and the other using the multiple migration kernel; and then compute the optimal weighting factor before stacking the two images together. Here I only consider the first two parts of equation  and neglect the last two terms. This is illustrated by rewriting equation  as

$$m_{mig}({\bf {x}})' = \sum_{s}\sum_{r}\sum_{t}\bigg[\tilde{g}_s^D \ast \tilde{g}_r^D + ... ...igg(\tilde{g}_s^M \ast \tilde{g}_r^M\bigg)\bigg]~d({\bf {r}},t\vert{\bf {s}},0)$$

$$= \sum_{s}\sum_{r}\sum_{t}\bigg(\tilde{g}_s^D \ast \tilde{g}_r^D\bigg)~d({\bf {r}},t\vert{\bf {s}},0)$$

$$+ \beta\sum_{s}\sum_{r}\sum_{t}\bigg(\tilde{g}_s^M \ast \tilde{g}_r^M\bigg)~d({\bf {r}},t\vert{\bf {s}},0),$$ (16)

where $\beta$ is the optimal weighting factor for stacking migration images in a small window. Hence, the contribution from both the direct waves and multiples are equalized which in theory should result in a more detailed migration image.

Figure c shows the GDM result using the Green's function only containing multiples (Figure c); this migration kernel is generated by muting the early-arrivals in the dashed box shown in Figure a and keeping all multiple arrivals. This indeed represents the second term in equation . Comparing with Figures a and b, I get a migration image with better illumination of the subsalt structure and inevitably more noise as well. After estimating the optimal factor $\beta$ , I stack the two images from the primary reflection migration (Figure a) and the migration of multiples (Figure c) following equation ; and I finally get the optimal stacked image shown in Figure d. Note that more noise is introduced in Figure d using this method, but such artifacts can be attenuated with a least-squares migration algorithm.