The general theory

In the case of zero-offset reflection, the ray travel distance, l, from the source to the reflection point is related to the two-way zero-offset time, t, by the simple equation

 

l = v_g t, (3)

where v_g is the half of the group velocity, best expressed in terms of its components, as follows:

$$v_g = \sqrt{v_{gx}^2 + v_v^2 v_{g\tau}^2}.$$

Here v_gx denotes the horizontal component of group velocity, v_v is the vertical P-wave velocity, and $v_{g\tau}$ is the v_v-normalized vertical component of the group velocity. Under the assumption of zero shear-wave velocity in VTI media, these components have the following analytic expressions:  

$$v_{gx} = {\frac{{v^2}\,{p_x}\,\left( -1 - 2\,\eta + 2\,\eta ... ...left( 1 + 2\,\eta \right) \,{{{p_x}}^2} + {{{p_{\tau }}}^2}}},$$ (4)

and  

$$v_{g\tau} = {\frac{\left(1- 2\,{v^2}\,\eta \,{{{p_x}}^2} \ri... ...eft( 1 + 2\,\eta \right) \, {{{p_x}}^2} + {{{p_{\tau }}}^2}}},$$ (5)

where p_x is the horizontal component of slowness, and $p_{\tau}$is the normalized (again by the vertical P-wave velocity v_v) vertical component of slowness. The two components of the slowness vector are related by the following eikonal-type equation Alkhalifah (1997):  

$$p_{\tau} = \sqrt{1 - {\frac{{v^2}\,{{{p_x}}^2}} {1 - 2\,{v^2}\,\eta \,{{{p_x}}^2}}}}.$$ (6)

Equation (6) corresponds to a normalized version of the dispersion relation in VTI media.

If we consider v and $\eta$ as imaging parameters (migration velocity and migration anisotropy coefficient), the ray length l can be taken as an imaging invariant. This implies that the partial derivatives of l with respect to the imaging parameters are zero. Therefore,  

$$\frac{\partial l}{\partial v} = \frac{\partial v_{g}}{\partial v} t+ v_{g} \frac{\partial t}{\partial v} = 0,$$ (7)

and  

$$\frac{\partial l}{\partial \eta} = \frac{\partial v_{g}}{\partial \eta} t+ v_{g} \frac{\partial t}{\partial \eta} = 0.$$ (8)

Applying the simple chain rule to equations (7) and (8), we obtain  

$$\frac{\partial t}{\partial v} = \frac{\partial t}{\partial \... ...partial t} {\partial \tau} \frac{\partial \tau}{\partial \eta},$$ (9)

where $\frac{\partial t}{\partial \tau} = - p_{\tau}$, and the two-way vertical traveltime is given by

$$\tau = v_{g\tau} t.$$

Combining equations (7-9) eliminates the two-way zero-offset time t, which leads to the equations

$$\frac{\partial \tau}{\partial v} = \frac{\partial v_{g}}{\partial v} {\frac{\tau } {{p_{\tau }}\,{v_{g\tau }}{v_{g}}}},$$ (10)

and

$$\frac{\partial \tau}{\partial \eta} = \frac{\partial v_{g}}... ...rtial \eta} {\frac{\tau } {{p_{\tau }}\,{v_{g\tau }}{v_{g}}}}.$$ (11)

After some tedious algebraic manipulation, we can transform equations (5) and (6) to the general form  

$$\frac{\partial \tau}{\partial v} = \tau F_v\left(p_x,v,\eta \right),$$ (12)

and  

$$\frac{\partial \tau}{\partial \eta} = \tau F_{\eta}\left(p_x,v,\eta \right).$$ (13)

Since the residual migration is applied to migrated data, with the time axis given by $\tau$ and the reflection slope given by $\frac{\partial \tau}{\partial x}$, instead of t and p_x, respectively, we need to eliminate p_x from equations (12) and (13). This task can be achieved with the help of the following explicit relation, derived in Appendix A,  

$$p_x^2 = {\frac{2\,{{{{\tau }_x}}^2}} {1 + {v^2}\,\left( 1 + 2\,\eta \right) \, {{{{\tau }_x}}^2} + {S}}},$$ (14)

where ${\tau }_x$=$\frac{\partial \tau}{\partial x}$, and

$$S = \sqrt{-8\,{v^2}\,\eta \,{{{{\tau }_x}}^2} + {{\left( 1 ... ...\,\left( 1 + 2\,\eta \right) \,{{{{\tau }_x}}^2} \right) }^2}}.$$

Inserting equation (14) into equations (12) and (13) yields exact, yet complicated equations, describing the continuation process for v and $\eta$. In summary, these equations have the form  

$$\frac{\partial \tau}{\partial v} = \tau f_v\left(\frac{\partial \tau}{\partial x},v,\eta \right)$$ (15)

and  

$$\frac{\partial \tau}{\partial \eta} = \tau f_{\eta}\left(\frac{\partial \tau}{\partial x},v,\eta \right).$$ (16)

Equations of the form (15) and (16) contain all the necessary information about the kinematic laws of anisotropy continuation in the domain of zero-offset migration.