Dix inversion constrained by L1-norm optimization

Dix inversion by an $L1/L2$ hybrid method

Extending the discussion in last section, many authors (Claerbout, 2009; Bube and Langan, 1997) generalize the optimization problem of Dix inversion. Claerbout (2009) points out that any arbitrary norm ${P}$ can be used as a penalty function. Then the optimization problem can be written as:

$$0 \approx \sum_{i} {P}(\sum_{j}C_{i,j}u_j-d_i),$$ (9)

where ${P}()$ is a convex function of a scalar, $C_{i,j}$, $u_j$, and $d_i$ are elements in the causal integration operator $C$, the model $u$ and the known data $d$.

We have special interests in an $L1/L2$ hybrid norm, because instead of a sharp transition, this norm provides a smooth transition between $L1$ and $L2$. This can be shown in the formulation of the hybrid norm:

$$P =  \sigma^2(\sqrt{1+r^2/\sigma^2}-1)$$ (10)

where $r$ is the data residual and $\sigma$ is the same threshold as in last section, and thus can be chosen according to the same physical explanation. The hybrid norm approaches the least squares limit as $\sigma \to \infty$ and approaches the $L1$ limit as $\sigma \to 0$.

The first and the second derivative of this hybrid norm with respect to the residuals are given in equation 11 and equation 12. We can see the penalty function transits from $L2$ to $L1$ smoothly at $r=\sigma$ since the first derivative is continuous.

$$P' =  \frac{r}{\sqrt{1+r^2/\sigma^2}}$$ (11)

$$P'' =  \frac{1}{(1+r^2/\sigma^2)^{3/2}}$$ (12)

Claerbout (2009) also proposed a new method based on Taylor's series to search the plane spanned by the gradient and the previous step. He embedded the new iterative bivariate solver in a conjugate direction solver, hoping for significant savings by expending more effort to find a better next step. In this experiment, we use the solver coded by Maysami and Mussa (2009), who adapt Claerbout's theory. For more information, refer to these two papers.

Dix inversion constrained by L1-norm optimization