Relating the zero-offset and migration slopes

The chain rule of differentiation leads to the equality  

$$p_x = \frac{\partial t}{\partial x} = -p_{\tau} \frac{\partial \tau}{\partial x},$$ (24)

where $p_{\tau}= -\frac{\partial t}{\partial \tau}$. It is convenient to transform equality (A-1) to the form  

$$\frac{\partial \tau}{\partial x} = -\frac{p_x}{p_{\tau}}.$$ (25)

Using the expression for $p_{\tau}$ from the main text, we can write equation (A-2) as a quadratic polynomial in p_x² as follows

a p_x⁴ +b p_x² +c=0,

where

$$a=-2v^2 \eta,$$

$$b= (\frac{\partial \tau}{\partial x})^2 v^2 (1+2 \eta)+1,$$

and

$$c=-(\frac{\partial \tau}{\partial x})^2.$$

Since $\eta$ can be small (as small as zero for isotropic media), we use the following form of solution to the quadratic equation

$$p_x^2 = \frac{2 c}{-b \pm \sqrt{b^2-4ac}}$$

Press et al. (1992). This form does not go to infinity as $\eta$ approaches . We choose the solution with the negative sign in front of the square root, because this solution complies with the isotropic result when $\eta$ is equal to zero.