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Fomel (2002) shows that a
possible solution to the slope estimation problem is obtained by
minimizing the non-linear function
| |
(13) |

where is the operator convolving the data
with the 2-D filter *C*_{n}(*Z*_{t},*Z*_{h}) and is the known data.
Fomel (2002) proposes iterating with a Gauss-Newton
algorithm:
| |
(14) |

where the step is estimated with a conjugate gradient method.
One problem with this approach is that it converges well only if we are close
enough to the true solution. Another problem stems from the fact that
no line search in the gradient direction is implemented. Therefore,
this method might oscillate close to the solution.
To make the dip estimation more robust and also to incorporate the
possibility to bound-constrain the dips, the L-BFGS-B method
is used instead of the Gauss-Newton approach. For the L-BFGS-B method,
similar to BFGS, we need to estimate the function and its gradient.
Incorporating a regularization operator that penalizes differences
between adjacent dip values, the objective function becomes

| |
(15) |

and the gradient is
| |
(16) |

The dips can be then estimated reliably. In the next section, the
L-BFGS-B method is illustrated on different examples. These examples
illustrate that the bound constrained optimization improves the dip
values when events with different local slopes overlap.

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Stanford Exploration Project

10/23/2004