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Next: Appendix C: Analytical expressions Up: Li & Fomel: Time-to-depth Previous: Appendix A: Ill-posedness of

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

\begin{displaymath}
\mathbf{J} \equiv \frac{\partial \mathbf{f}}{\partial \mathb...
...\mathbf{w}}
- \mbox{diag}(\mathbf{v_d} \star \mathbf{v_d})\;.
\end{displaymath} (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.,

\begin{displaymath}
\frac{\partial \mathbf{v_d}}{\partial \mathbf{w}} = \frac{\p...
...{x_0}}
 \frac{\partial \mathbf{x_0}}{\partial \mathbf{w}}\;,
\end{displaymath} (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
\begin{displaymath}
\frac{\partial \mathbf{x_0}}{\partial \mathbf{w}} =
- (\nab...
...L}_{x_0} \frac{\partial \mathbf{t_0}}{\partial \mathbf{w}}\;.
\end{displaymath} (36)

Another differentiation of equation 6 leads to
\begin{displaymath}
\frac{\partial \mathbf{t_0}}{\partial \mathbf{w}} = \frac{1}...
...cdot \nabla)^{-1}
\equiv \frac{1}{2} \mathbf{L}_{t_0}^{-1}\;.
\end{displaymath} (37)

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

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


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Next: Appendix C: Analytical expressions Up: Li & Fomel: Time-to-depth Previous: Appendix A: Ill-posedness of

2015-03-25