Non-diffracted multiple

The traveltime of the water-bottom multiple is given by Alvarez (2005)  

$$t_m=\frac{4}{V_1}\sqrt{\left(\frac{h_D}{2}\right)^2+Z_{wb}^2}=\sqrt{t_m^2(0)+\left(\frac{2h_D}{V_1}\right)^2},$$ (13)

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

 

mul_sktch4 Figure 4 Imaging of water-bottom multiple for a flat water-bottom. Notice that $m_D=m_\xi$ and that the apparent reflector at the image point is flat.

From Figure  it is clear that due to the symmetry of the problem, t_s1=t_s2=t_r1=t_r2=t_m/4 and $\alpha_s=\alpha_r$, which in turn means $\beta_s=\beta_r$. Furthermore, from Equations 5 and 6 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 corresponding traveltimes of the multiple. Equation 2 thus simplifies to  

$$h_\xi=\frac{h_D}{2}(1-\rho^2),$$ (14)

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 claim that the trace is imaged exactly as a primary since it is mapped to zero subsurface offset irrespective of its surface offset. It should also be noted that, since usually sediment velocity is faster than water velocity, then $\rho^2\gt 1$ and therefore the multiples are mapped to subsurface offsets with the opposite sign with respect to the sign of the surface offset h_D when migrated with sediment velocity.

From Equation 3, the depth of the image point can be easily computed as  

$$z_\xi=Z_{wb}+\frac{\rho}{2}\sqrt{h_D^2(1-\rho^2)+4Z_{wb}^2},$$ (15)

which for migration with the water velocity reduces to $z_\xi=2Z_{wb}$,which shows that the multiple is migrated as a primary at twice the water depth. Finally, from Equation 4, the horizontal position of the image point reduces to  

$$m_\xi=m_D.$$ (16)

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 obviously a direct consequence of the symmetry of the raypaths of the multiple reflection in this case.

Equations 14-16 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 15 we get  

$$z_\xi=\frac{z_\xi(0)}{1+\rho}+\rho\sqrt{\left(\frac{z_\xi(0)}{1+\rho}\right)^2+\frac{h_\xi^2}{1-\rho^2}}\quad\quad (\rho\ne 1)$$ (17)

which shows that the moveout is an hyperbola (actually half of an hyperbola since we already established that $h_\xi\le 0$ if $h_D\ge 0$).

 

odcig1 Figure 5 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 17.

Figure  shows an SODCIG for a non-diffracted 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 14 correspond to larger surface offsets) map to shallower depths (for the normal situation of $\rho\gt 1$), 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 the 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  

$$\gamma=\sin^{-1}\left[\frac{2\rho h_D}{V_1t_m}\right].$$ (18)

The depth of the image can be easily computed from Equation 12. In particular, if the data are migrated with the velocity of the water, $\rho=1$, and therefore $z_{\xi\gamma}=2Z_{wb}$ which means a horizontal line in the ($z_{\xi_\gamma},\gamma$) plane. 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 12 can be expressed in terms of the data space coordinates using Equations 14 and 15 and noting that  

$$\tan\gamma=\tan\beta_s=\frac{\rho\sin\alpha_s}{\sqrt{1-\rho^... ...\rho^2h_D^2}}=\frac{\rho h_D}{\sqrt{4Z_{wb}^2+h_D^2(1-\rho^2)}}$$ (19)

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.

 

adcig1 Figure 6 Angle domain common image gather corresponding to the SODCIG shown in Figure . Overlaid is the residual moveout curve computed with equation 20.

As we 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 14 and 15 into equation 12, using Equations 13, and 18 to eliminate h_D and simplifying we get  

$$z_{\xi_\gamma}=Z_{wb}\left[1+\frac{\cos\gamma(\rho^2-\tan^2\... ...^2-\tan^2\gamma(1-\rho^2))}{\sqrt{\rho^2-\sin^2\gamma}}\right].$$ (20)

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). Figure  shows the ADCIG corresponding to the SODCIG shown in Figure . Notice that the migrated depth at zero aperture angle is the same as that for zero sub-surface offset in Figure . For larger aperture angles, however, the migrated depth increases as indicated in equation 20 and as seen in the schematic of Figure .