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Alternatively, we can use the L-BFGS-B algorithm for imposing very
tight constraints on the picked values of the tau field. The L-BFGS-B
algorithm seeks to find a vector of model parameters such
that we minimize
| |
(18) |

where
| |
(19) |

with *l*_{i} and *u*_{i} being the lower and upper bounds for the model
, respectively. In this case, *l*_{i} and *u*_{i} are called simple
bounds. For the flattening technique, we want to minimize Guitton et al. (2005b); Lomask (2003b)
| |
(20) |

The L-BFGS-B algorithm combines a quasi-Newton update of the Hessian
(second derivative) with a trust-region method. It has been
successfully applied for flattening Guitton et al. (2005b) and
dip estimation Guitton (2004).
Incorporating the initial field is trivial with the
L-BFGS-B method: we simply set the bounds where an a-priori value
exists:

| |
(21) |

| (22) |

where is a small number (). Note that the
L-BFGS-B algorithm allows us to optionally activate the constraints for
every point of the model space. Note that in equation
20, the objective function incorporates a smoothing
in the vertical direction of the field as well.
Although not shown in this paper, the results of the L-BFGS-B algorithm
are comparable to the Gauss-Newton approach. However at this stage,
the L-BFGS-B algorithm converges faster than the Gauss-Newton technique with
preconditioning.

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Stanford Exploration Project

4/6/2006