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:
Definition of the misfit via the Huber function results in a nonlinear optimization problem because any residual component ri 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.