A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations

Appendix B: The Fréchet derivative operator

We continue the derivation of the Fréchet derivative matrix from equation 12. Using the same notations as introduced in equation 10, after discretization equation 12 becomes

$$\mathbf{J} \equiv \frac{\partial \mathbf{f}}{\partial \mathb... ...\mathbf{w}} - \mbox{diag}(\mathbf{v_d} \star \mathbf{v_d})\;.$$ (34)

Because $v_d$ is in time-domain $(t_0,x_0)$, we need to apply the chain-rule for its derivative with respect to $w$, i.e.,

$$\frac{\partial \mathbf{v_d}}{\partial \mathbf{w}} = \frac{\p... ...{x_0}} \frac{\partial \mathbf{x_0}}{\partial \mathbf{w}}\;,$$ (35)

where $\mathbf{t_0}$ is the discretized column vector of $t_0 (z,x)$. According to equation 7 and after denoting another matrix operator $\mathbf{L}_{t_0} = \nabla \mathbf{t_0} \cdot \nabla$, we find

$$\frac{\partial \mathbf{x_0}}{\partial \mathbf{w}} = - (\nab... ...L}_{x_0} \frac{\partial \mathbf{t_0}}{\partial \mathbf{w}}\;.$$ (36)

Another differentiation of equation 6 leads to

$$\frac{\partial \mathbf{t_0}}{\partial \mathbf{w}} = \frac{1}... ...cdot \nabla)^{-1} \equiv \frac{1}{2} \mathbf{L}_{t_0}^{-1}\;.$$ (37)

Finally, by inserting equations B-2 through B-4 into B-1, we complete the derivation of the Fréchet derivative matrix:

$$\mathbf{J} = - \mathbf{L}_{x_0} \mathbf{L}_{t_0}^{-1} \mathbf{L}_{x_0} \mathbf... ...f{w}) \frac{\partial \mathbf{v_d}}{\partial \mathbf{t_0}} \mathbf{L}_{t_0}^{-1}$$

$$+ \mbox{diag}(\mathbf{v_d} \star \mathbf{w}) \frac{\partial \mathbf... ...}_{x_0} \mathbf{L}_{t_0}^{-1} - \mbox{diag}(\mathbf{v_d} \star \mathbf{v_d})\;.$$ (38)

A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations