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In this section, the proposed algorithm is used on a geophysical inverse
problem which is velocity estimation with noisy data. Some possible
applications of a robust solver are, for example, tomography
Bube and Langan (1997) and deconvolution of noisy data
Chapman and Barrodale (1983). My goal is to demonstrate that the Huber
function with the L-BFGS method gives a robust estimate of the model
parameters when outliers are present in the data. The velocity
estimation problem with the Huber norm has potential applications when
multiples need to be separated from the signal
in the velocity domain Kostov and Nichols (1995); Lumley et al. (1995).
More conventional multiple attenuation techniques using the
parabolic radon transform Herrmann et al. (2000); Kabir and Marfurt (1999)
can also benefit from using the Huber norm.
The ``velocity domain'' representation of seismic data using
the hyperbolic radon transform (HRT) is an alternative to the standard
common midpoint (CMP) gather. Transformation of CMP data into
the velocity domain (producing a velocity model or panel
of the data) exhibits clearly the moveout inherent in the data and
therefore, forms a convenient basis for velocity analysis as a linear
inverse problem.

Thorson and Claerbout (1985) were the first to define the
forward and adjoint operators of the HRT, formulating it as an inverse
problem in which the velocity domain is the unknown space.
In their approach the forward operator **L** maps the model
space (velocity domain) into the data space (CMP gathers). This
transformation is a superposition of hyperbolas in
the data space. The adjoint operator , the HRT,
maps the data space into the model space. This
transformation is a summation over hyperbolic trajectories in the data space
(related to the velocity stack as defined by Taner and Koehler (1969)).
With *d*(*t*,*x*) being a CMP gather and the corresponding
velocity model, the forward operation is

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(4) |

and the adjoint transformation
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(5) |

where *x* is the offset (*x*_{min} and *x*_{max} being the offset
range), *s* the slowness (*s*_{min} and *s*_{max} being the range
of slownesses investigated), the two-way zero offset travel
time, and *w*_{o} a weighting function that compensates to
some extent for geometrical spreading and other effects Claerbout and Black (1997).
Having defined the forward operator **L** and its adjoint , the inverse problem can be posed. Inverse theory helps
us to find a velocity panel which synthesizes a given CMP
gather *via* the operator **L**. In equations, given data
**d** (CMP gather), we want to solve for the model **m** (velocity panel)

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(6) |

which leads in a least-squares sense to the linear system
(``normal equations'')
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(7) |

This system is easy to solve if , i.e.,
if **L** is close to unitary. Unfortunately, **L** is far from
an unitary operator Kabir and Marfurt (1999); Sacchi and Ulrych (1995). In addition, the
number of equations and unknowns may be large, making an iterative
data-fitting approach reasonable.
Consequently, with *E* being a misfit measurement function, our goal
is to iteratively calculate the model that minimizes
the misfit function

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(8) |

One possibility for *E* is the norm (least-squares
inversion). The misfit function is then usually minimized with
conjugate-gradient. Another possible approach is of course
by taking the Huber norm along with the L-BFGS method introduced
in the preceding section. The two norms are compared in the next section
for velocity estimation problems. The results show that the Huber norm
gives the expected behavior when outliers (non-gaussian noise)
are present in the data.