Anisotropic residual moveout for flat reflectors

The generalization of kinematic anisotropic migration and the analysis of the kinematics of the offset-to-angle transformation presented in the previous section enables a simple analysis of the residual moveout (RMO) in ADCIGs caused by errors in anisotropic velocity parameters. In this section I derive the RMO function by linearizing the relationship of the imaging depth in the angle domain with respect to perturbations in the anisotropic parameters. The linearization is evaluated around the correct migration velocity function; that is, when the image in the subsurface-offset domain is well focused at zero offset.

As in the previous section, I limit my analysis to reflections from flat interfaces. However, a generalization of the flat-events analysis to dipping events should be conceptually straightforward, though not necessarily simple from the analytical point of view. Furthermore, I derive relationships assuming that the velocity perturbations are limited to a homogeneous half-space above the reflector. The same relationships can be easily adapted to the case of a homogeneous layer above the reflector by transforming the depth variable into a relative depth with respect to the top of the layer under consideration. At the end of this section I present the fundamental relationship for broadening the application of the theory to heterogeneous media. This relationship links the traveltime perturbations to the reflector movements and it can be used in a ray-based tomographic velocity-update procedure.

A VTI velocity function, either group or phase, is described by the following vector of three velocities ${\bf V}=(V_V,V_H,V_N)$,or by the corresponding vector of three slownesses ${\bf S}=(S_V,S_H,S_N)$used in equation 36. I define the perturbations as the combination of one multiplicative factor for each of the velocities and one multiplicative factor for all velocities; that is, the perturbed velocity $_\rho{\bf V}$ is defined as:

$$_\rho{\bf V}= \left({_\rho}V_V,{_\rho}V_H,{_\rho}V_N\right)= \rho_V\left(\rho_V_VV_V,\rho_V_HV_H,\rho_V_NV_N\right).$$ (10)

The velocity-parameter perturbations is thus defined by the following four-components vector = $\left(\rho_V,\rho_V_V,\rho_V_H,\rho_V_N\right)$.

 

cig-2d-aniso-delta1-flat-v1 Figure 3 Linearized perturbations of the image-point locations (both in the subsurface-offset domain and the angle domain) caused by changes in the ray length L, as evaluated using the first term in equation 11.

 

cig-2d-aniso-delta2-flat-v1 Figure 4 Linearized perturbations of the image-point locations (both in the subsurface-offset domain and the angle domain) caused by changes in the aperture angle $\gamma$,as evaluated using the second term in equation 11. Notice that the image point in the angle domain does not move, no matter how large the corresponding movement in the subsurface-offset domain is.

Differentiating, the expression for the depth of the image point in the angle domain $z_\gamma$(equation 9) with respect to the i-th component in the perturbation vector, we obtain the following:

$$\begin{eqnarray} \frac{\partial z_\gamma}{\partial \rho_{i}} &=& \frac{\partial ... ...de{\gamma}} \frac{\partial \widetilde{\gamma}}{\partial \rho_{i}}.\end{eqnarray}$$ (11) (12) (13)

In Appendix B I demonstrate that the terms multiplying the partial derivatives with respect to the angles are zero, and equation 13 simplifies into:  

$$\frac{\partial z_\gamma}{\partial \rho_{i}} = \frac{\partial... ...{\partial L}{\partial S} \frac{\partial S}{\partial \rho_{i}},$$ (14)

where

$$\frac{\partial z_\gamma}{\partial L}= \frac{\partial z_\xi}{... ...ilde{\gamma} = \cos \gamma+ \sin \gamma\tan \widetilde{\gamma},$$ (15)

and

$$\frac{\partial L}{\partial S\left(\gamma\right)}= -\frac{z_\xi}{S\left(\gamma\right)\cos \gamma},$$ (16)

and consequently  

$$\frac{\partial z_\xi}{\partial \rho_{i}} = -\frac{z_\gamma\l... ...t)}{S\left(\gamma\right)} \frac{\partial S}{\partial \rho_{i}}.$$ (17)

Figures  and  graphically illustrate the image perturbations related to the first two terms in equation 11. Figure  shows the movement of the image points (both in the subsurface-offset domain and the angle domain) caused by changes in the ray length L. Figure  provides a geometrical explanation of why the second term in equation 11 vanishes. It shows that perturbations in the aperture angle $\gamma$ cause the subsurface-offset domain image point to move along the tangent to the wavefront (tilted with the phase angle $\widetilde{\gamma}$). Since this movement is constrained along the tangent, the image point in the angle domain does not move, no matter how large the movement in the subsurface-offset domain is.

 

• RMO function with uniform scaling of velocity
• RMO function with arbitrary scaling of velocity
• Conversion of depth errors into traveltime errors in heterogeneous media
• Synthetic-data examples of RMO function in ADCIGs