A simple choice of robust measure is the norm: denoting
the residual (misfit) components by (*N* being
the number of data points), the
norm misfit function of the residual vector is . This function is not
smooth: it is singular where any residual component vanishes. As a result,
numerical minimization is difficult. Various approaches based for example
on a linear programming viewpoint Barrodale and Roberts (1980) or iterative
smoothing Scales et al. (1988), have been used with
success but require considerable tuning. Moreover, the singularity
implies that small residuals are ``taken as seriously'' as large
residuals, which may not be appropriate in all circumstances.

These drawbacks of the norm have led to various proposals which combine robust treatment of large residuals with gaussian treatment of small residuals. These proposals are known as ``hybrid '' methods. For example, Bube and Langan (1997) apply an iteratively reweighted least-squares (IRLS) method to minimize a hybrid objective function on a tomography problem. More recently, Zhang et al. (2000) use an IRLS procedure to locate bed boundaries from electromagnetic data.

In this chapter, a hybrid error measure (or norm) proposed by Huber (1973) is presented:

(1) |

Figure 1

Definition of the misfit *via* the Huber function
results in a nonlinear optimization problem because any residual
component *r*_{i} close to the threshold can oscillate
between the and norm. In the first section following
this introduction, I propose solving
the optimization problem with a quasi-Newton method called
limited-memory BFGS
Broyden (1969); Fletcher (1970); Goldfarb (1970); Nocedal (1980); Shanno (1970).
In the second section, this method is tested to estimate the root mean
square (rms) velocity (or slowness) of noisy common midpoint (CMP) and
shot gathers
for synthetic and field seismic data. There the Huber norm is compared
with the norm with and without regularization.

5/5/2005