Improving the estimation with a 3-D RMO function

Biondi and Tisserant (2004) extend the 2-D RMO function (equation (1)) to a 3-D function that includes the azimuth of the reflector ($\eta$), and the azimuth of the reflection ($\phi$):  

$$\Delta n_{RMO}= \frac {\rho-1} {\cos \alpha} \frac {\sin^2\g... ...n^2\alpha\cos^2\left(\eta-\phi\right)-\sin^2\gamma\right)} z_0.$$ (3)

Equation (1) is a special case of equation (3), where the azimuth of the acquisition and the reflection are the same ($\eta=\phi$): the 2-D RMO function implicitly assumes dip-reflections just as the 2-D offset-to-angle transformation does. In the case of a flat reflector ($\alpha=0$), the RMO functions are the same in 2-D and 3-D, because horizontal planes have no defined azimuth and do not generate 3-D reflection.

Figure 5 synthesizes all the velocity analysis done with equation (3) on two CIGs taken from the model. CIG₇₀₀ (top) and CIG₄₂₅ (bottom) stacked over $\phi$ are represented in Figure 5a.

 

stk_3D_700_450 Figure 5 Velocity error analysis by semblance using a 3-D RMO function on CIG700 on the top row, and on CIG425 on the bottom row. a) CIG after stack over $\phi$.b) Semblance analysis of the left panel using the 3-D RMO function at $\phi=0$.c) $\phi$ semblance analysis. d) Semblance analysis of the first panel using the 3-D RMO function with a $\phi(z)$ law. All semblance panels are squared to increase resolution.

Figure 5b is the semblance analysis of the stack using the 3-D RMO function (equation (3)), with $\eta=45^\circ$ (the true azimuth of the reflectors) and assuming $\phi=0^\circ$ in the absence of $\phi$information. For the first CIG, the comparison with the semblance panel obtained with the 2-D RMO function (Figure 4c) shows a more accurate resolution of the velocity error when the reflector azimuth is provided.

 

pfan_700_450 Figure 6 Reflection azimuth analysis using a 3-D RMO function on a) CIG700 b) CIG425. All semblance panels are squared to increase resolution.

We perform a semblance analysis to determine the reflection azimuth. The procedure consists in scanning all the slices at constant $\phi$ of the (z,$\gamma$,$\phi$) cube using equation (3), with the velocity error obtained from a previous analysis performed on the stack with the $\phi=0$ assumption (Figure 5b). Then, only the trace corresponding to the measured velocity error is taken from each semblance panel at constant $\phi$. All the traces displayed side by side form the $\phi$ semblance panel displayed in Figure 6. The peaks of semblance give the reflection azimuth for each event. The value of $\phi$ ranges from $0^\circ$ to $7^\circ$ for the first CIG, and from $6^\circ$ to $16^\circ$ for the second CIG. They are consistent with the $\phi$ values read on Figure 3. Once again, the reflection azimuths are comparable at similar reflector dips.

Figure 5c adds the reflection azimuth information picked in Figure 6 to the velocity error analysis. This extra information does not improve the resolution of the velocity error for the first CIG, where the range of $\phi$ is limited and satisfies the $\phi=0$ assumption because of the limited dips. Note that the peaks of semblance are not centered perfectly on $\rho=1.03$ but are closer to $\rho=1.025$. This underestimated velocity error is due to the straight rays assumption that over-corrects the move-out.

For the second CIG, where the $\phi=0$ assumption is not valid, we expect some changes. Paradoxically, the estimation seems less accurate for the $60^\circ$ reflector when the value of $\phi$ is used ($\rho=1.02$) instead of setting $\phi$ to ($\rho=1.03$). The reason is when the $\phi=0$ approximation is not valid, the RMO function over-estimates the velocity error. Simultaneously, the RMO function under-estimates the velocity error because of the straight rays approximation. Hence the errors cancel out and make the estimation apparently more accurate with the $\phi=0$ assumption than with the correct value of $\phi$.

The 3-D analysis of the RMO in the ADCIGs improves the resolution of the velocity error, even if the knowledge of the reflection azimuth does not bring visible improvement. The precise estimation of the reflection azimuth still holds for tomographic or amplitude-versus-angle studies.