Robust error measures such as the *l ^{1}* norm have found a number of
uses in geophysics. As measures of data misfit, they show considerably
less sensitivity to large measurement errors than does the mean square
(

A simple choice of robust measure is the *l ^{1}* norm: denoting
the residual (misfit) components by ,

These drawbacks of the *l ^{1}* norm have led to various proposals which
combine robust treatment of large residuals with Gaussian
treatment of small residuals. In the work reported here, we use
a hybrid

This paper describes the application of the Huber misfit function to
velocity analysis. Estimation of RMS velocity (or slowness) can be
posed as a linear inverse problem through the *velocity
transform* described in the next section. Definition of the misfit
*via* the Huber function (or any other robust error measure)
results in a nonlinear optimization problem for the velocity
model. This nonlinearity would seem to compare unfavorably with the
least squares (*l ^{2}*) treatment of the same problem, which leads to a
linear system (the normal equation) and so can be solved by efficient
iterative methods such as conjugate gradient. We show that
use of an appropriate nonlinear optimization method gives a
Huber-based solution with comparable efficiency to that
of conjugate gradient least squares solution. Thus the
noise rejection properties of the Huber misfit function come at
no appreciable premium in computational effort.

huber
Error measure proposed by Huber Huber (1973).
The upper part above is the Figure 1 l norm while the lower part is the ^{1}l norm.
^{2} |

In the work reported here we have used a version of the Limited Memory BFGS algorithm Nocedal (1980) as implemented in the Hilbert Class Library Gockenbach et al. (1999). Other nonlinear iterative optimizers could be used; we have solved the same examples with nonlinear conjugate gradient methods Fletcher (1980) and obtained comparable results. We note that specially adapted Huber minimizers have been suggested Ekblom and Madsen (1989). One of our questions in beginning this work was whether a standard quasi-Newton method, as opposed to a special solver, would perform satisfactorily in Huber estimation.

The second section of the paper explains the velocity transform and formulates a linear inverse problem for velocity analysis. The third and fourth sections present synthetic and field data examples.

4/20/1999