Kinematics of water-bottom Multiples in image space

 

mul_sktch1 Figure 1 Water-bottom multiple. The subscript s refers to the source and the subscript r to the receiver.

The propagation path of a water-bottom multiple, as shown in Figure , consists of four segments, such that the total travel-time for the multiple is given by

 

t_m=t_s1+t_s2+t_r2+t_r1, (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 CMP gather and h_D is the half-offset between source and 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.

 

mul_sktch2 Figure 2 Imaging of water-bottom multiple in SODCIG. The subscript $\xi$ refers to the image point.

As illustrated in the sketch of Figure , at any given depth the image space coordinates of the migrated multiple are given by:

$$\begin{eqnarray} x_{s_\xi}&=&m_D-h_D+V_1(t_{s_1}\sin\alpha_s+\rho\tilde{t}_{s_2}... ...rho(\tilde{t}_{s_2}\sin\beta_s-\tilde{t}_{r_2}\sin\beta_r)\right),\end{eqnarray}$$ (2) (3) (4)

where V₁ is the water velocity, $\rho=V_2/V_1$ with V₂ the sediment velocity, and $\alpha_s$, $\alpha_r$ are the acute takeoff angles of the source and receiver rays with respect to the vertical. The traveltime of the refracted ray segments $\tilde{t}_{s_2}$ and $\tilde{t}_{r_2}$can be computed from two 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 extrapolated time equals the traveltime of the multiple. As shown in Appendix A, the traveltimes of the refracted rays are given by

$$\begin{eqnarray} \tilde{t}_{s_2}&=&\frac{t_{r_1}\cos\alpha_r-t_{s_1}\cos\alpha_s... ...+\rho(t_{s_2}+t_{r_2})\cos\beta_s}{\rho(\cos\beta_s+\cos\beta_r)}.\end{eqnarray}$$ (5) (6)

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

$$\begin{eqnarray} \sin\beta_s&=&\rho\sin(\alpha_s+\varphi)\cos\varphi-\sqrt{1-\rh... ...lpha_r-\varphi)}\cos\varphi-\rho\sin(\alpha_r-\varphi)\sin\varphi.\end{eqnarray}$$ (7) (8) (9) (10)

Equations 2-10 are valid for any water-bottom multiple, whether from a flat or dipping water-bottom. They even describe the migration of source- or receiver-side diffraction multiples, since no assumption has been made relating $\alpha_r$ and $\alpha_s$ or the individual traveltime segments.

 

mul_sktch3 Figure 3 Imaging of water-bottom multiple in ADCIG. The subscript $\xi$ refers to the image point. 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 . The half-aperture angle is given by  

$$\gamma=\frac{\beta_r+\beta_s}{2},$$ (11)

which is the same equation used for converted waves Rosales and Biondi (2005). The depth of the image point ($z_{\xi_\gamma}$) is given by (Appendix B)  

$$z_{\xi_\gamma}=z_\xi-h_\xi\tan\gamma.$$ (12)

Equations 2-12 formally describe the image coordinates in terms of the data coordinates. They are, however, of little practical use unless we can relate the individual traveltime segments (t_s1, t_s2, $\tilde{t}_{s_2}$, t_r2, $\tilde{t}_{r_2}$, t_r1), and the angles $\alpha_s$ and $\alpha_r$ (which in turn determine $\beta_s$ and $\beta_r$) to the known data space parameters (m_D, h_D, t_m, V₁, $\varphi$ and $\rho$). This may not be easy or even possible analytically for all situations, but it is for some simple but important models that I will now examine.