Anisotropic residual moveout for flat reflectors

The kinematic formulation of the generalized impulse response presented in the previous section enables a simple analysis of the residual moveout (RMO) in ADCIGs caused by errors in anisotropic velocity parameters. For the sake of simplicity, at the present, 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.

A VTI velocity function, either group or phase, is described by the following vector of three velocities ${\bf V}=(V_V,V_H,V_N)$,as for example used in equations 5, or by the corresponding vector of three slownesses ${\bf S}=(S_V,S_H,S_N)$used in equation 6. I define the perturbations as one multiplicative factors for each of the velocities and one multiplicative factor for all velocities; that is, the perturbed velocity $_\rho{\bf V}$ is defined as:

$$_\rho{\bf V}= \left({_\rho}V_V,{_\rho}V_H,{_\rho}V_N\right)= \rho_V\left(\rho_V_VV_V,\rho_V_HV_H,\rho_V_NV_N\right).$$ (30)

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

For flat reflectors, the transformation to angle domain maps an image point at coordinates $(z_\xi,h_\xi)$ into an image point with coordinates $(z_\gamma,\widetilde{\gamma})$according to the following mapping:

$$\begin{eqnarray} \widetilde{\gamma} &=& \arctan \left. \frac{\partial z_\xi}{\pa... ...\vert _{m_\xi=\widebar m_\xi}= z_\xi-h_\xi\tan \widetilde{\gamma}.\end{eqnarray}$$ (31) (32)

The partial derivative of the angle-domain depth $z_\gamma$with respect to the i-th component in the perturbation vector can be expressed as follows:

$$\begin{eqnarray} \frac{\partial z_\gamma}{\partial \rho_{i}} &=& \frac{\partial ... ...de{\gamma}} \frac{\partial \widetilde{\gamma}}{\partial \rho_{i}}.\end{eqnarray}$$ (33)

In Appendix B I demonstrate that the terms multiplying the partial derivatives with respect to the angles are zero, and equation 34 simplifies into:  

$$\frac{\partial z_\gamma}{\partial \rho_{i}} = \frac{\partial... ...{\partial L}{\partial S} \frac{\partial S}{\partial \rho_{i}},$$ (34)

where

$$\frac{\partial z_\gamma}{\partial L}= \frac{\partial z_\xi}{... ...ilde{\gamma} = \cos \gamma+ \sin \gamma\tan \widetilde{\gamma},$$ (35)

and

$$\frac{\partial L}{\partial S\left(\gamma\right)}= -\frac{z_\xi}{S\left(\gamma\right)\cos \gamma},$$ (36)

 

• Uniform scaling of velocity
• Arbitrary scaling of velocity
• Conversion of depth errors into traveltime errors