Imaging with multiples using linearized full-wave inversion

One-layered model

We construct a one-layered model (Figure 5 a) with ocean-bottom geometry. The only sharp interface in the migration velocity is the seabed. Figure 5 b shows the synthetic data. The labels $d_1$ , $d_2$ and $d_3$ correspond to the first, second, and third order events as shown in figure 2 b and c. Note that we used equation 2 to generate the synthetic data. Hence, internal multiples are absent. Figure 5 c shows the migration image $m_{mig}$ . The migration image is made up of signal $m_{sig}$ and crosstalk artifacts $m_{xtalk}$ . In the figure, the label $A$ indicates spurious reflectors generated by migrating the primary signal ($d_1$ ) as if it were a multiple. $B$ is the correct reflector in the image. $C$ is an artifact generated by migrating the multiple signal ($d_2$ or $d_3$ ) as if it were a primary reflection. In equation form, they are denoted as follows:

$$m_{mig} = m_{signal} + \left[m_{xtalk}\right] = m_B + \left[m_A+m_C\right]$$ (5)

$$m_A = {\mathbf L'}_2 {\mathbf d_1} + {\mathbf L'}_3 {\mathbf d_1} + {\mathbf L'}_4 {\mathbf d_1} + ...$$

$$m_B = {\mathbf L'}_1 {\mathbf d_1} + {\mathbf L'}_2 {\mathbf d_2} + {\mathbf L'}_3 {\mathbf d_3} + ...$$

$$m_C = {\mathbf L'}_1 {\mathbf d_2} + {\mathbf L'}_1 {\mathbf d_3} + {\mathbf L'}_2 {\mathbf d_3} + ...$$

where $m_A$ , $m_B$ ,and $m_C$ correspond to the parts of the image labeled with $A$ , $B$ , and $C$ in Figure 5 c. ${\mathbf L'}_1$ , ${\mathbf L'}_2$ , and ${\mathbf L'}_3$ are migration operators that correspond to different orders of reflection events. Figure 5 d shows the inversion result. Notice that the artifacts are removed from the image. In conventional imaging, if there were residual multiple energy in the data, then artifacts of type $C$ would show up in the image. Treating those as real signal would negatively affect the interpretation of the sub-surface.

Onelayerv2
Figure 5. (a) Original one layered model, (b) synthetic data, (c) migration image and (d) inversion image.

Imaging with multiples using linearized full-wave inversion