Moveout-based wave-equation migration velocity analysis

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

This section provides the derivation of the direct $\frac{\partial{\alpha}}{\partial{s}}$ operator. The approach is to define the auxilary function that links moveout parameter $\alpha(z,x)$ and the slowness $s$ .

$$J_{aux} = \int dz_w \int d\gamma \, I(z+z_w+\theta(\alpha,\beta,\gamma),\gamma;z,x,s_0)I(z,\gamma;z,x,s) \; \; ,$$ (11)

in which

$$\theta(\alpha,\beta,\gamma) = \alpha \tan^2{\gamma} + \beta = \alpha g(\gamma) + \beta h(\gamma) .$$

Notice that here the moveout between the initial image $I(s_0)$ and the new image $I(s)$ is characterized by both curvature $\alpha$ and constant shift $\beta$ . The $\beta$ parameter does not affect the gather flatness; therefore there is no need to put it into the objective function in () , however $\beta$ is essential to describe the change of image's kinematics.

Since $\alpha,\beta$ maximize eq (11),

$$\{ \begin{array}{c} \frac{\partial{J_{aux}}}{\partial \alpha} = 0 \\ \\ \frac{\partial{J_{aux}}}{\partial \beta} = 0 \end{array} \, .$$ (12)

We differentiate the equation (12) with respect to $\alpha,\beta$ and $s$ , which gives

$$\left[ \begin{array}{cc} \frac{\partial^2{J_{aux}}}{\partial{\alp... ...\ \\ \frac{\partial{J_{aux}}}{\partial{\beta}\partial{s}} \end{array} \right] .$$ (13)

Now we need to invert a Jacobian to get $d\alpha/ds$ . We define the following:

$$\frac{\partial^2{J_{aux}}}{\partial{\alpha}^2} = \int dz_w \int d\gamma \, \ddot{I}(z+z_w+\theta,\gamma;z,x,s_0) g^2(\gamma) I(z+z_w,\gamma;z,x,s) = E_{11}(z,x)$$

$$\frac{\partial^2{J_{aux}}}{\partial{\alpha}\partial{\beta}} = \int dz_w \int d\gamma \, \ddot{I}(z+z_w+\theta,\gamma;z,x,s_0) g(\gamma)h(\gamma) I(z+z_w,\gamma;z,x,s) = E_{12}(z,x)$$

$$\frac{\partial^2{J_{aux}}}{\partial{\beta^2}} = \int dz_w \int d\gamma \, \ddot{I}(z+z_w+\theta,\gamma;z,x,s_0) h^2(\gamma) I(z+z_w,\gamma;z,x,s) = E_{22}(z,x)$$ (14)

Let the inverse of matrix $E$ to be matrix $F$ :

$$F = \left[ \begin{array}{cc} F_{11} & F_{12} \\ F_{12} & F_{22} \... ...\begin{array}{cc} E_{11} & E_{12} \\ E_{12} & E_{22} \end{array} \right]^{-1}$$

Then

$$\frac{\partial{\alpha}}{\partial{s}}\vert _{s=s_0} = -\int dz_w \... ...t{I}(z+z_w,\gamma;z,x,s_0) \frac{\partial{I(z+z_w,\gamma;z,x,s)}}{\partial{s}}.$$ (15)

There is another way to define the relation between $\alpha$ and $s$ , leading to the indidirect $\frac{\partial{\alpha}}{\partial{s}}$ operator using a weighted least-squares fitting formula. Suppose we have the locations of one event in the ADCIGs at location ( $\gamma_{i},z_i)$ , and we introduce a moveout formula $\theta(\gamma)={z_0}+\alpha \tan^2{\gamma}$ . Now we define the best fitted intercept and curvature ( $z_0$    and $\alpha$ ) values as follows:

$$(z_0,\alpha) = \arg\min_{z_0,\alpha} \, \sum_{i}\{ (z_i - \alpha \tan^2{\gamma_{i}} - z_0)^2 w_i^2 \},$$ (16)

where $w_i$ is the energy of the event at angle $\gamma_i$ , serving as a weight for the least-squares fitting. We denote $\overline{x} = \sum_{i}x_{i} w^2_i$ (the weighted average of quantity $x$ ), then

$$\alpha = \frac{\overline{1}\,\overline{(z \tan^2{\gamma})} - \ove... ...z}} {\overline{1}\,\overline{\tan^4{\gamma}} - (\overline{\tan^2{\gamma}})^2}.$$

It is easy to find $\Delta\alpha$ if there is a pertubation of $b_i$ on $z_i$ :

$$\frac{\partial{\alpha}}{\partial{b_i}} = \frac{w_i^2(\overline{1}... ...)} {\overline{1}\,\overline{\tan^4{\gamma}}-(\overline{\tan^2{\gamma}})^2} \, .$$ (17)

Finally,

$$\frac{\partial{\alpha}}{\partial{s}} = \sum_{i}{\frac{\partial{\alpha}}{\partial{b_i}} \frac{\partial{b_i}}{\partial{s}}},$$ (18)

where $\frac{\partial{\alpha}}{\partial{b_i}},\frac{\partial{b_i}}{\partial{s}}$ are defined respectively in equations (17) and (10).

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Moveout-based wave-equation migration velocity analysis