Linearized approximations

Although the exact expressions might be sufficiently constructive for actual residual migration applications, linearized forms are still useful, because they give us valuable insights into the problem. The degree of parameter dependency for different reflector dips is one of the most obvious insights in the anisotropy continuation problem. Perturbation of a small parameter provides a general mechanism to simplify functions by recasting them into power-series expansion over a parameter that has small values. Two variables can satisfy the small perturbation criterion in this problem: The anisotropy parameter $\eta$ ($\eta <<1$) and the reflection dip ${\tau }_x$ ($\tau_x v <<1$or p_x v <<1).

Setting $\eta=0$ yields equation (19) for the velocity continuation in elliptical anisotropic media and  

$$\frac{\partial \tau}{\partial \eta} =\frac{{v^4}\,\tau \,{{{... ...t) \, \left( {{{v_v}}^2} + {v^4}\,{{{{\tau }_x}}^2} \right) }.$$ (27)

which represents the case when we initially introduce anisotropy into our model.

Because p_x (the zero-offset slope) is typically lower than ${\tau }_x$ (the migrated slope), we perform initial expansions in terms of y=p_x v. Applying the Taylor series expansion of equations (12) and (13) in terms of y and dropping all terms beyond the fourth power in y, we obtain  

$$\frac{\partial \tau}{\partial v}={\frac{v\,\tau \,{{{p_x}}^2... ...t( 1 + 6\,\eta \right) \,{{{v_v}}^2} \right) }{{{{v_ v}}^4}}},$$ (28)

and  

$$\frac{\partial \tau}{\partial \eta}=\frac{{v^4}\,\tau \,{{{p... ...^4}\,\left( 4\,{v^2} - 3\,{{{v_v}}^2} \right) } {{{{v_v}}^2}}.$$ (29)

Although both equations are equal to zero for p_x=0, the leading term in the velocity continuation is proportional to p_x², whereas the the leading term in the $\eta$ continuation is proportional to p_x⁴. As a result the velocity continuation has greater influence at lower angles than the $\eta$ continuation. It is also interesting to note that both leading terms are independent of the size of anisotropy ($\eta$).

Despite the typically lower values of p_x, expansions in terms of ${\tau }_x$ are more important, but less accurate. For small ${\tau }_x$,$p_x \approx \tau_x$, and, therefore, the leading-term behavior of ${\tau }_x$ expansions is the same as that of p_x As a result, we arrive at the equation  

$$\frac{\partial \tau}{\partial v}={\frac{v\,\tau \,\left( 2\,... ...}}^2} \right) } {v\,{{{v_v}}^2}}} \right) \,{{{{\tau }_x}}^4},$$ (30)

and  

$$\frac{\partial \tau}{\partial \eta}=\frac{{v^4}\,\tau \,\lef... ...} - 3\,{{{v_v}}^2} \right) \,{{{{\tau }_x}}^4}} {{{{v_v}}^2}}.$$ (31)

Most of the terms in equations (B-4) and (B-5) are functions of the difference between the vertical and NMO velocities. Therefore, for simplicity and without a loss of generality, we set v_v=v and keep only the terms up to the eighth power in ${\tau }_x$. The resultant expressions take the form  

$$\frac{\partial \tau}{\partial v}= v\,\tau \,{{{{\tau }_x}}^2... ...3\,\eta + 144\,{{\eta }^2} \right) \,\tau \, {{{{\tau }_x}}^8}$$ (32)

and  

$$\frac{\partial \tau}{\partial \eta}={v^4}\,\tau \,{{{{\tau }... ...54\,\eta + 156\,{{\eta }^2} \right) \,\tau \,{{{{\tau }_x}}^8}.$$ (33)

Curiously enough, the second term of the $\eta$ continuation heavily depends on the size of anisotropy ($\sim20\eta$). The first term of equation (B-6) ($\sim \tau_x^2$) is the isotropic term; all other terms in equations (B-6) and (B-7) are induced by the anisotropy.