VTI migration velocity analysis using RTM

Lagrangian augmented functional method

We are now going to use the recipe with the augmented functional that Plessix (2006) provides to derive the image-space DSO gradient. First, let us form the Lagrangian augmented functional, ${\mathcal L}$ :

$${{\mathcal L}({\bf p},{\bf q},I_{\bf h},{\bf\lambda},{\bf\mu},\gamma_{\bf h},c) = }$$

$$\sum_{\bf h} \frac{1}{2} \left < {\bf h}~ I_{{\bf h}},~ {\bf h}~ I_{{\bf h}} \right >$$

$$+ \left < \bf\lambda,~ {\bf f} - {\bf L}(c) {\bf p} \right >$$

$$+ \left < \bf\mu,~ {\bf f~'} - {\bf L^*}(c) {\bf q} \right >$$

$$+ \sum_{\bf h} \left < \gamma_{\bf h},~ ({\bf S}_{+{\bf h}} {\bf p})^* {\bf M}^* ({\bf S}_{-{\bf h}} {\bf q}) - I_{\bf h} \right >$$ (24)

Then the adjoint state equations are obtained by taking the derivative of ${\mathcal L}$ with respect to state variables ${\bf p}$ , ${\bf q}$ and $I_{\bf h}$ :

$$\frac{\partial {\mathcal L}}{\partial {\bf p}} = - {\bf L^*} (c) ... ...{\bf S}_{+{\bf h}})^* {\bf M}^*({\bf S}_{-{\bf h}} {\bf q}) \gamma_{\bf h} = 0,$$ (25)

$$\frac{\partial {\mathcal L}}{\partial {\bf q}} = -{\bf L} (c) \bf... ... ({\bf S}_{-{\bf h}})^* {\bf M} ({\bf S}_{+{\bf h}} {\bf p})\gamma_{\bf h} = 0,$$ (26)

$$\frac{\partial {\mathcal L}}{\partial I_{\bf h}} = -\gamma_{\bf h} + {\bf h}^2 I_{\bf h} = 0, \forall ~{\bf h}.$$ (27)

Equation 25, 26, 27 are the adjoint-state equations. Variables $\lambda=(\lambda_x,\lambda_y,\lambda_z,\lambda_V,\lambda_H)^T$ and $\mu=(\mu_x,\mu_y,\mu_z,\mu_V,\mu_H)^T$ are the adjoint-state fields and the solution of the adjoint-state equations 25 and 26. Variable $\gamma_{\bf h}$ is the scaled image slice at the subsurface offset ${\bf h}$ .

Now the gradient of the objective function 15 with respect to velocity is:

$$\nabla_c J = \left < \bf\lambda,~ -\frac{\partial {\bf L}}{\partial c} {\bf p}... ...ht > + \left < \bf\mu,~ -\frac{\partial {\bf L}^*}{\partial c} {\bf q} \right >$$

$$= \left ( -\frac{\partial {\bf L}}{\partial c} {\bf p} \right )^* \... ...bda + \left ( -\frac{\partial {\bf L}^*}{\partial c} {\bf q} \right )^* \bf\mu,$$ (28)

If we combine equations 25, 26, and 27 with equation 28, we will arrive at the same solution as in the previous section.

VTI migration velocity analysis using RTM