Moveout-based wave-equation migration velocity analysis

Details of the $\frac{\partial{b}}{\partial{s}}$ sensitivity kernel calculation

This section illustrates how to calculate the sensitivity kernel of the image shift parameter $b(\gamma,x)$ : $\frac{\partial{b}}{\partial{s}}$ . Since $b$ maximizes the auxillary objective function,

$$J_{aux}(b) = \int dz_w \int d\gamma \, I(z+z_w+b,\gamma;z,x,s_0) I(z+z_w,\gamma;z,x,s) \; ,$$ (7)

we have

$$\frac{\partial{J_{aux}}}{\partial b} = 0 .$$ (8)

To find the relation between ${b}$ and $s(x)$ , we differentiate equation (8) with respect to $b$ and $s$ , which yields

$$\frac{\partial^2{J_{aux}}}{\partial{b}^2} \frac{\partial{b}}{\partial{s}} = -\frac{\partial{J_{aux}}}{\partial{b}\partial{s}} ,$$ (9)

in which we can find

$$\frac{\partial{J_{aux}}}{\partial b} = \int dz_w \dot{I}(z+z_w+b,\gamma;z,x,s_0) I(z+z_w,\gamma;z,x,s),$$

$\dot{I},\ddot{I}$ indicate the first and second derivatives in $z$ (depth). Let

$$\frac{\partial^2{J_{aux}}}{\partial{b^2}} = \int dz_w \, \ddot{I}(z+z_w+b,\gamma;x,z,s_0) I(z+z_w,\gamma;x,z,s) = E(x,z).$$

Then substituting the above two equations into eq. (9) leads to

$$\frac{\partial{b}}{\partial s} = -\frac{ \int dz_w \, \dot{I}(z+z... ...ma;z,x,s_0)\frac{\partial{I(z+z_w,\gamma;z,x,s)}}{\partial{s}}}{E(\gamma,z,x)},$$ (10)

which is eq. (2).

Moveout-based wave-equation migration velocity analysis