RMO analysis in ADCIGs

When the migration velocity is correct and the image is focused at zero subsurface offset, the transformation to angle domain does not change the depth of the image point ($z_\xi=z_{\widehat{\gamma}}$), and the reflections are imaged at the same depth for all aperture angles ($\frac{\partial z_{\widehat{\gamma}}}{\partial \widehat{\gamma}}$). In contrast, when the reflections are not focused at zero offset, the transformation to the angle-domain maps the events at different depths for each different angle. The variability of the depth with the aperture angle is described by the RMO function, which we want to measure and quantify as a function of the perturbation in anisotropic parameters encountered along the propagation paths. Below, we briefly present the approach proposed in (3), which consists in approximating the RMO function by the first-order Taylor series expansion about the correct migration velocity.

We use the same notation as in (3): the VTI velocity function parameterization is ${\bf V}=(V_V,V_H,V_N)$, which is equivalent to Thomsen's parameterization ${\bf V}=(V_V,\epsilon,\delta)$.V_V is the velocity of a vertical ray, V_H is the velocity of a horizontal ray ($V_H=V_V\sqrt{1+2\epsilon}$)and V_N is the NMO velocity ($V_N=V_V\sqrt{1+2\delta}$).

We define the perturbations as a three-component vector = $\left(\rho_V_V,\rho_V_H,\rho_V_N\right)$, where each component is a multiplicative factor for each migration velocity. It generates a perturbed velocity field, $_\rho{\bf V}$ defined by $_\rho{\bf V}=\left({_\rho}V_V,{_\rho}V_H,{_\rho}V_N\right) =\left(\rho_V_VV_V,\rho_V_HV_H,\rho_V_NV_N\right)$.

From the analytic expression of the impulse response (derived from figure 1) and some geometric interpretation of the transformation to the angle domain, the first-order derivatives of the imaging depth in the angle domain with respect to perturbations in the anisotropic parameters can be written as follows (7):

$$\begin{eqnarray} \frac{\partial z_{\widehat{\gamma}}}{\partial \rho_{i}} = -\fra... ...(\alpha_x-\gamma)} \frac{\partial S_r}{\partial \rho_{i}} \right),\end{eqnarray}$$ (3)

where $\gamma$ is the phase aperture angle, $\tilde{\gamma}$ is the group aperture angle, $\widehat{\gamma}$ is the arctan of the offset dip, S_s and S_r are the slowness along the source and receiver rays.

Similarly, since residual moveout $\Delta z_{\rm RMO}$ is defined as the difference between the reflector movement at finite angle $\widehat{\gamma}$ and the reflector movement at normal incidence z₀, from equation 3, we can express the first-order derivatives of the residual moveout with respect to perturbations in the anisotropic parameters:

$$\begin{eqnarray} \frac{\partial \Delta z_{\rm RMO}}{\partial \rho_{i}} = \left. ... ...hat{\gamma}}}{\partial \rho_{i}} \right\vert _{\widehat{\gamma}=0}\end{eqnarray}$$ (4)

The residual moveout $\Delta z_{\rm RMO}$ is eventually approximated by the first-order Taylor series expansion about the correct migration velocity (${bf \rho}={\bf 1}$). The linearized expression is the following:  

$$\Delta z_{\rm RMO}=\left.\frac{\partial \Delta z_{\rm RMO}}{... ..._{\widehat{\gamma},{\bf \rho}={\bf 1}} \left(\rho_V_N-1\right).$$ (5)