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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.
mul_sktch1
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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

 
tm=ts1+ts2+tr2+tr1, (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 (mD,hD,tm) where mD is the horizontal position of the CMP gather and hD 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.
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view

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.
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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  
 \begin{displaymath}
\gamma=\frac{\beta_r+\beta_s}{2},\end{displaymath} (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)  
 \begin{displaymath}
z_{\xi_\gamma}=z_\xi-h_\xi\tan\gamma.\end{displaymath} (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 (ts1, ts2, $\tilde{t}_{s_2}$, tr2, $\tilde{t}_{r_2}$, tr1), and the angles $\alpha_s$ and $\alpha_r$ (which in turn determine $\beta_s$ and $\beta_r$) to the known data space parameters (mD, hD, tm, 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.


next up previous print clean
Next: Flat water-bottom Up: Alvarez: Multiples in image Previous: Introduction
Stanford Exploration Project
11/1/2005