General formulation

The propagation path of a first-order water-bottom multiple generated by a planar dipping reflector, as shown in Figure 1, consists of four segments, such that the total travel-time for the multiple is given by

$$t_m=t_{s_1}+t_{s_2}+t_{r_2}+t_{r_1},$$ (1)

where the subscript $s$ refers to the source-side rays and the subscript $r$ refers to the receiver-side rays. The data space coordinates are $(m_D,h_D,t_m)$ where $m_D$ is the horizontal position of the common-midpoint (CMP) gather and $h_D$ is the half-offset between the source and the receiver.

mul-sktch1 Figure 1. Water-bottom multiple. The subscript $s$ refers to the source and the subscript $r$ to the receiver.

Wave-equation migration maps the CMP gathers to SODCIGs with coordinates $(m_\xi,h_\xi,z_\xi)$ where $m_\xi$ is the horizontal position of the image gather, and $h_\xi$ and $z_\xi$ are the half subsurface-offset and the depth of the image, respectively. As illustrated in the sketch of Figure 2, at any given depth the spatial coordinates of the downward-continued source and receiver rays are given by:

$$x_{s_\xi} = m_D-h_D+V_1(t_{s_1}\sin\alpha_s+\rho\tilde{t}_{s_2}\sin\beta_s),$$ (2)

$$x_{r_\xi} = m_D+h_D-V_1(t_{r_1}\sin\alpha_r+\rho\tilde{t}_{r_2}\sin\beta_r),$$ (3)

where $V_1$ is the water velocity, $\rho=V_2/V_1$ with $V_2$ the sediment velocity, $\alpha_s$, $\alpha_r$ are the takeoff angles of the source and receiver rays with respect to the vertical and $\beta_s$ and $\beta_r$ are the angles of the refracted source and receiver rays, respectively. The coordinates of the migrated multiple in the image space are given by:

$$h_\xi = \frac{x_{r_\xi}-x_{s_\xi}}{2}=h_D-\frac{V_1}{2}\left[t_{s_1}\sin\... ...sin\alpha_r+\rho(\tilde{t}_{s_2}\sin\beta_s+\tilde{t}_{r_2}\sin\beta_r)\right],$$ (4)

$$z_\xi = V_1(t_{s_1}\cos\alpha_s+\rho\tilde{t}_{s_2}\cos\beta_s)=V_1(t_{r_1}\cos\alpha_r+\rho\tilde{t}_{r_2}\cos\beta_r),$$ (5)

$$m_\xi = \frac{x_{r_\xi}+x_{s_\xi}}{2}=m_D+\frac{V_1}{2}\left(t_{s_1}\sin\... ...sin\alpha_r+\rho(\tilde{t}_{s_2}\sin\beta_s-\tilde{t}_{r_2}\sin\beta_r)\right),$$ (6)

The traveltime of the refracted ray segments $\tilde{t}_{s_2}$ and $\tilde{t}_{r_2}$ can be computed from the two imaging conditions: (1) at the image point the depth of both rays has to be the same (since we are computing horizontal subsurface offset gathers) and (2) $t_{s_2}+t_{r_2}=\tilde{t}_{s_2}+\tilde{t}_{r_2}$ which follows immediately from equation 1 since at the image point the total extrapolated time equals the traveltime of the multiple. As shown in Appendix A, the traveltimes of the refracted rays are given by

$$\tilde{t}_{s_2} = \frac{t_{r_1}\cos\alpha_r-t_{s_1}\cos\alpha_s+\rho(t_{s_2}+t_{r_2})\cos\beta_r}{\rho(\cos\beta_s+\cos\beta_r)},$$ (7)

$$\tilde{t}_{r_2} = \frac{t_{s_1}\cos\alpha_s-t_{r_1}\cos\alpha_r+\rho(t_{s_2}+t_{r_2})\cos\beta_s}{\rho(\cos\beta_s+\cos\beta_r)}.$$ (8)

The refracted angles are related to the takeoff angles by Snell's law: $\sin(\beta_s+\varphi)=\rho\sin(\alpha_s+\varphi)$ and $\sin(\beta_r-\varphi)=\rho\sin(\alpha_r-\varphi)$, from which we get

$$\sin\beta_s = \rho\sin(\alpha_s+\varphi)\cos\varphi-\sqrt{1-\rho^2\sin^2(\alpha_s+\varphi)}\sin\varphi,$$ (9)

$$\sin\beta_r = \rho\sin(\alpha_r-\varphi)\cos\varphi+\sqrt{1-\rho^2\sin^2(\alpha_r-\varphi)}\sin\varphi,$$ (10)

$$\cos\beta_s = \sqrt{1-\rho^2\sin^2(\alpha_s+\varphi)}\cos\varphi+\rho\sin(\alpha_s+\varphi)\sin\varphi,$$ (11)

$$\cos\beta_r = \sqrt{1-\rho^2\sin^2(\alpha_r-\varphi)}\cos\varphi-\rho\sin(\alpha_r-\varphi)\sin\varphi.$$ (12)

mul-sktch3
Figure 2. Imaging of water-bottom multiple in SODCIG and ADCIG. The subscript $D$ refers to the data space while the subscript $\xi$ refers to the image space. The points $(x_{r\xi },z_{r\xi })$ and $(x_{s\xi },z_{s\xi })$ represent the end points of the source and receiver ray after migration and must be at the same depth at the image point (for horizontal ADCIGs). The coordinates $(m_\xi ,\gamma _\xi ,z_{\gamma \xi })$ correspond to the image point in the angle domain. The coordinates $(m_\xi ,h_\xi ,z_{r\xi })$ correspond to the image point in the subsurface offset gather. The line AB represents the apparent reflector at the image point.

In ADCIGs, the mapping of the multiples can be directly related to the previous equations by the geometry shown in Figure 2. The half-aperture angle is given by

$$\gamma=\frac{\beta_r+\beta_s}{2},$$ (13)

which is the same equation derived for converted waves by (). The depth of the image point in ADCIGs ( $z_{\xi_\gamma}$) is given by (Appendix B)

$$z_{\xi_\gamma}=z_\xi-h_\xi\tan\gamma.$$ (14)

Equations 4-6 describe the transformation performed by wave-equation migration between CMP gathers $(m_D,h_D,t)$ and SODCIGs $(m_\xi,h_\xi,z_\xi)$. Equations 7-12 relate the traveltimes and angles of the refracted segments to parameters that can in principle be computed from the data (traveltimes, takeoff angles, reflector dips and velocities). Equations 13 and 14 provide the transformation from SODCIGs to ADCIGs. These equations are valid for any first-order water-bottom multiple, whether from a flat or dipping water-bottom. They even describe the migration of source- or receiver-side diffracted multiples with the diffractor at the water bottom, since no assumption has been made relating $\alpha_r$ and $\alpha_s$ or the individual traveltime segments. These equations, however, are of little practical use unless we can relate the individual traveltime segments ($t_{s_1}$, $t_{s_2}$, $t_{r_2}$, $t_{r_1}$), and the angles $\alpha_s$ and $\alpha_r$ to the known data space coordinates ($m_D$, $h_D$, $t_m$) and the model parameters ($V_1$, $\varphi$ and $\rho$). This may not be easy or even possible analytically for all situations, but it is for the simple but important case of a specular multiple from a flat water-bottom.