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The condition number of the target-oriented Hessian matrix can be high, making the solution of the non-stationary least-squares filtering problem in equation (5) unstable.
One solution is adding a smoothing regularization operator to equation (5):
| ![\begin{eqnarray}
{\bf H}\hat{{\bf m}}-{\bf m}_{mig}&\approx&0, \nonumber\\
\epsilon{\bf I}\hat{{\bf m}}&\approx&0,
\end{eqnarray}](img12.gif) |
|
| (6) |
where the choice of the identity operator (
) as regularization operator is customary. A more sophisticated regularization scheme could involve applying a smoothing operator in the reflection angle (or offset ray-parameter) dimension Kuehl and Sacchi (2001); Prucha et al. (2000) or, more generally, in the reflection and azimuth angles as proposed by Valenciano and Biondi (2005).
Next: Numerical results: Sigsbee model
Up: Inversion setting
Previous: Linear least-squares inversion
Stanford Exploration Project
10/31/2005