Specular multiple from flat water-bottom

The traveltime of the first-order water-bottom multiple is given by

$$t_m=\frac{4}{V_1}\sqrt{\left(\frac{h_D}{2}\right)^2+Z_{wb}^2}=\sqrt{t_m^2(0)+\left(\frac{2h_D}{V_1}\right)^2},$$ (15)

which is simply the traveltime of a primary at twice the depth of the water-bottom $Z_{wb}=\frac{V_1t_m(0)}{4}$.

From the symmetry of the problem, $t_{s_1}=t_{s_2}=t_{r_1}=t_{r_2}=t_m/4$ and $\alpha_s=\alpha_r$, which in turn means $\beta_s=\beta_r$. Furthermore, from Equations 7 and 8 it immediately follows that $\tilde{t}_{s_2}=t_{s_2}$ and $\tilde{t}_{r_2}=t_{r_2}$ which says that the traveltimes of the refracted rays are equal to the traveltimes of the corresponding segments of the multiple. Equation 4 thus simplifies to

$$h_\xi=\frac{h_D}{2}(1-\rho^2),$$ (16)

which indicates that the subsurface offset at the image point of a trace with half surface offset $h_D$ depends only on the velocity contrast between the water and the sediments. In particular, if the trace is migrated with the water velocity, i.e. $\rho=1$, then $h_\xi=0$ which proves the property that the multiple is imaged exactly as a primary. It should also be noted that, since usually sediment velocity is faster than water velocity, then $\rho^2>1$ and therefore the multiples are mapped to subsurface offsets with the opposite sign to that of the surface offset $h_D$ when migrated with sediment velocity.

From Equation 5, the depth of the image point can be easily computed as

$$z_\xi=Z_{wb}+\frac{\rho}{2}\sqrt{h_D^2(1-\rho^2)+4Z_{wb}^2},$$ (17)

which for migration with the water velocity reduces to $z_\xi=2Z_{wb}$, showing that the multiple is migrated as a primary at twice the water depth as is intuitively obvious. Finally, from Equation 6, the horizontal position of the image point reduces to

$$m_\xi=m_D.$$ (18)

This result shows that the multiple is mapped in the image space to the same horizontal position as the corresponding CMP even if migrated with sediment velocity. This result is a direct consequence of the symmetry of the raypaths of the multiple reflection in this case. For dipping water bottom or for diffracted multiples this is not the case (, ).

Equations 16-18 give the image space coordinates in terms of the data space coordinates. An important issue is the functional relationship between the subsurface offset and the image depth, since it determines the moveout of the multiples in the subsurface-offset-domain common-image-gathers (SODCIGs). Replacing $h_D=2h_\xi/(1-\rho^2)$ and $Z_{wb}=z_\xi(0)/(1+\rho)$ in Equation 17 we get

$$z_\xi=\frac{z_\xi(0)}{1+\rho}+\rho\sqrt{\left(\frac{z_\xi(0)... ...\rho}\right)^2+\frac{h_\xi^2}{1-\rho^2}}\quad\quad (\rho\ne 1)$$ (19)

which shows that the moveout is a hyperbola (actually, for off-end geometry, half of a hyperbola, since we already established that $h_\xi\le 0$ if $h_D\ge 0$).

odcig1 Figure 3. Subsurface offset domain common image gather of a water-bottom multiple from a flat water-bottom. Water velocity is 1500 m/s, water depth 500 m, sediment velocity 2500 m/s and surface offsets from 0 to 2000 m. Overlaid is the residual moveout curve computed with Equation 19.

Figure 3 shows an SODCIG for a specular water-bottom multiple from a flat water-bottom 500 m deep. The data was migrated with a two-layer velocity model: the water layer of 1500 m/s and a sediment layer of velocity 2500 m/s. Larger subsurface offsets (which according to Equation 16 correspond to larger surface offsets) map to shallower depths for the usual situation of $\rho>1$, as we should expect since the rays are refracted to increasingly larger angles until the critical reflection angle is reached. Also notice that the hyperbola is shifted down by a factor $(1+\rho)$ with respect to its image point when migrated with water velocity.

In angle-domain common-image-gathers (ADCIGs), the half-aperture angle reduces to $\gamma=\beta_s=\beta_r$, which in terms of the data space coordinates is given by

$$\gamma=\sin^{-1}\left[\frac{2\rho h_D}{V_1t_m}\right].$$ (20)

The depth of the image can be easily computed from Equation 14. In particular, if the data are migrated with the velocity of the water, then $\rho=1$, and therefore $z_{\xi\gamma}=2Z_{wb}$, which means a horizontal line in the ( $z_{\xi_\gamma},\gamma$) plane at twice the depth of the water-bottom. Equivalently, we can say that the residual moveout in the ( $z_{\xi_\gamma},\gamma$) plane is zero, once again corroborating that the water-bottom multiple is migrated as a primary if $\rho=1$. Equation 14 can be expressed in terms of the data space coordinates using Equations 16 and 17 and noting that

$$\tan\gamma=\tan\beta_s=\frac{\rho\sin\alpha_s}{\sqrt{1-\rho^... ...^2\alpha_s}}=\frac{\rho h_D}{\sqrt{4Z_{wb}^2+h_D^2(1-\rho^2)}}$$ (21)

If $\rho=1$ this expression simplifies to $\tan\gamma=\frac{h_D}{2Z_{wb}}$, which is the aperture angle of a primary at twice the water-bottom depth.

As I did with the SODCIG, it is important to find the functional relationship between $z_{\xi_\gamma}$ and $\gamma$ since it dictates the residual moveout of the multiple in the ADCIG. Plugging Equations 16 and 17 into equation 14, using Equations 15, and 20 to eliminate $h_D$ and simplifying we get

$$z_{\xi_\gamma} = Z_{wb}\left[1+\frac{\cos\gamma(\rho^2-\tan^2\gamma(1-\rho^2))}{\sqrt{\rho^2-\sin^2\gamma}}\right]$$ (22)

$$= \frac{z_{\xi_\gamma}(0)}{1+\rho}\left[1+\frac{\cos\gamma(\rho^2-\tan^2\gamma(1-\rho^2))}{\sqrt{\rho^2-\sin^2\gamma}}\right].$$ (23)

Once again, when the multiple is migrated with the water velocity ($\rho=1$) we get the expected result $z_{\xi_\gamma}=z_{\xi_\gamma}(0)$, that is, flat moveout (no angular dependence). The residual moveout in ADCIGs is therefore given by

$$\Delta n_{\mbox{RMO}}=z_{\xi_\gamma}(0)-z_{\xi_\gamma}=\left... ...\gamma)}{\sqrt{\rho^2-\sin^2\gamma}}\right]\frac{z_0}{1+\rho}.$$ (24)

This equation reduces to that of Biondi and Symes (2004) when $\gamma$ is small (Appendix C), which is when we can neglect ray bending at the multiple-generating interface. Panel (a) of Figure 4 shows the ADCIG corresponding to the SODCIG shown in Figure 3. Notice that the migrated depth at zero aperture angle is the same as that for the zero sub-surface offset in Figure 3. For larger aperture angles, however, the migrated depth increases as indicated in equation 23. The continuous line corresponds to Equation 24 whereas the dotted line corresponds to the tangent-squared of Biondi and Symes (2004). For this model, the departure of the straight ray approximation can be more than 5% for large aperture angles as illustrated in panel (b). The relative error represents the different between the two approximations divided by that the more accurate of equation 24.

adcig1
Figure 4. Panel (a) is an ADCIG for a water-bottom multiple from a two flat-layer model. The dotted curve corresponds to the straight ray approximation whereas the solid curve corresponds to the ray-bending approximation. Panel (b) is the relative error between the two approximations.