Huber separation

To get a better ($\tau,v$) model we decided to replace the linear iterative solver used by Lumley et al. (1994) with the Fletcher-Reeves non-linear conjugate gradient conditions Polak (1997) and the Dennis-Schnabel line search method Dennis and Schnabel (1983). We replaced the L₂ function, with a Huber functional Huber (1973) that is less sensitive to large outliers. The Huber functional is L₂ until some cutoff value and then smoothly switches to L₁ (Figure 5). The idea is compromise between the convergence speed of L₂ and the less sensitive nature to outliers with the L₁. Guitton and Symes1999 showed that the Huber functional does a better job of localizing energy in (t,v) space. For multiples this mean that the primary and multiple trains are better separated.

 

huber Figure 5 The L2, Huber, and L1 functionals.

To show the advantage of the Huber method over a straight L₂ we took a multiple contaminated CMP gather, Figure 6, and iterated on fitting goals (5). Figure 7 shows the envelope of the ($\tau,v$) representation of both the Huber and L₂ approach. Note how the Huber result shows a more compact representation of the primary and multiple trains.

 

mobil Figure 6 A multiple infested gather from the Mobil AVO dataset.

 

compare
Figure 7 The envelope of the tau-velocity space representation of the Mobil AVO gather (Figure 6) using both an L₂, left, and a Huber, right, functional. Note how the Huber gather shows more energy and better isolation of the primary train. Further, the L₂ approach show significantly more energy at high, unreasonable, velocities.