Water-bottom multiple

The water-bottom multiple from a dipping reflector has exactly the same kinematics as a primary from a reflector with twice the dip Alvarez (2005), that is,  

$$t_m=\sqrt{\left(\frac{2\hat{Z}_D}{V_1}\right)^2+\left(\frac{... ...)^2}=\sqrt{t_m^2(0)+\left(\frac{2h_D}{\hat{V}_{NMO}}\right)^2},$$ (31)

where $\varphi$ is the dip of the reflector, $\hat{Z}_D$ is the perpendicular depth to the equivalent reflector with twice the dip (at the CMP location) and $\hat{V}_{NMO}$ is the NMO velocity of the equivalent primary $\hat{V}_{NMO}=V_1/\cos(2\varphi)$.

Following the same procedure as for the flat water-bottom, we compute the coordinates of the image point using equations 2-4 and noting that in this case $\alpha_r=\alpha_s+4\varphi$,

$$\begin{eqnarray} h_\xi&=&h_D-\frac{V_1}{2}\left[t_{s_1}\sin\alpha_s+t_{r_1}\sin(... ...phi)+\rho(\tilde{t}_{s_2}\sin\beta_s-\tilde{t}_{r_2}\sin\beta_r)).\end{eqnarray}$$ (32) (33) (34)

where, according to equations 7-10,

$$\begin{eqnarray} \sin\beta_s&=&\rho\sin(\alpha_s+\varphi)\cos\varphi-\sqrt{1-\rh... ...ha_s+3\varphi)}\cos\varphi-\rho\sin(\alpha_s+3\varphi)\sin\varphi.\end{eqnarray}$$ (35) (36) (37) (38)

The traveltimes of the individual ray segments are computed by repeated application of the law of sines as shown in Appendix C:

$$\begin{eqnarray} t_{s_1}&=&\frac{\tilde{Z}_s}{V_1\cos(\alpha_s+\varphi)}=\frac{\... ...)\cos\alpha_s}{V_1\cos(\alpha_s+3\varphi)\cos(\alpha_s+4\varphi)},\end{eqnarray}$$ (39) (40) (41) (42)

where $\tilde{Z}_D$ is the perpendicular depth to the reflector at the CMP location and is given by (Appendix C):  

$$\tilde{Z}_D=\frac{V_1t_m(0)\cos\varphi}{2[1+\cos(2\varphi)]}.$$ (43)

Notice that this is not the same as $\hat{Z}_D$ in equation 31, which corresponds to the perpendicular depth to the equivalent reflector whose primary has the same kinematics as the water-bottom multiple.

The traveltime of the refracted ray segments are given by equations 5 and 6 with

$$\cos\alpha_r=\sqrt{1-\sin^2(\alpha_s+4\varphi)},\quad\mbox{and}\quad \cos\alpha_s=\sqrt{1-\sin^2\alpha_s}.$$ (44)

In order for equation 32-34 to be useful in practice, we need to express them entirely in terms of the known data coordinates, which means that we need to find an expression for $\alpha_s$ in terms of $(t_m,h_D,m_D,\varphi)$. In Appendix C it is shown that  

$$\alpha_s=\sin^{-1}\left[\frac{2h_D\cos(2\varphi)}{V_1t_m}\right]-2\varphi.$$ (45)

We now have all the pieces to compute the image space coordinates, since $\tilde{t}_{s_2}$ and $\tilde{t}_{r_2}$ can be computed from equations 5 and 6 using equations 35-45.

Appendix D shows that equations 32-34 reduce to the corresponding equations for the non-diffracted multiple from a flat water bottom when $\varphi=0$, as they should.

Figure  shows the zero subsurface offset section from a migrated non-diffracted multiple from a dipping water-bottom. The overlaid curve was computed with equations 32-34. The dip of the water-bottom is 15 degrees and intercepts the surface at CMP location zero. The CMP range of the data is from 2000 to 3000 m and the surface offsets from 0 to 2000 m. The multiple was migrated with the same two-layer model described before. Notice how the multiple was migrated as a primary. Since the migration velocity is faster than water-velocity, the multiple is over-migrated and appears much steeper and shallower than it should (recall that it would be migrated as a reflector with twice the dip is the migration velocity were that of the water.)

 

image3 Figure 11 image section at zero subsurface offset for a non-diffracted multiple from a dipping water-bottom. The overlaid curve was computed with equation 34 and 33.

Figure  shows the SODCIG at CMP 1500 m in i Figure . Just as for the flat water-bottom, the multiple energy is mapped to negative subsurface offsets since $\rho\gt 1$. The overlaid curve is the moveout computed with equations 32-34.

 

odcig3 Figure 12 SODCIG from a non-diffracted multiple from a dipping water-bottom. The overlaid residual moveout curve was computed with equation 32 and 33.

The aperture angle is given by equation 11 with  

$$\beta_r=\sin^{-1}(\rho\sin(\alpha_s+3\varphi))-\varphi \quad... ...and}\quad \beta_s=\sin^{-1}(\rho\sin(\alpha_s+\varphi))-\varphi$$ (46)

and $\alpha_s$ given by equation 45. The image depth in the ADCIG is given by equation 12 with $\beta_s$ and $\beta_r$ given by equation 46 and $h_\xi$ and $z_\xi$ given by equations 32 and 33. Figure  shows the ADCIG corresponding to the SODCIG in Figure . Notice that the apex is at zero aperture angle.

 

adcig3 Figure 13 ADCIG corresponding to the SODCIG shown in Figure . The overlaid residual moveout curve was computed with equation 33,  11,  12, and 46.