Comments on the result

From observating the fitted drift between figure 6 (b) and 6 (d) , the Huber solver is not doing significantly better than the least-squares solver. One possible explanation is that the underlying Taylor series assumption failed while trying to solve for the stepping coeficient $\alpha$ and $\beta$ . Recall that the formulas for the Huber norm are

$\displaystyle C(r)$	$\displaystyle =$	$\begin{displaymath}\begin{cases} \vert r\vert-\vert r_t\vert/2 \quad\quad &\vert... ...{2 r_t} \quad \quad &\vert{r}/{r_t}\vert < 1 \end{cases} \notag\end{displaymath}$	(5)
$\displaystyle C'(r)$	$\displaystyle =$	$\begin{displaymath}\begin{cases} \text{sgn} ({r}/{r_t} ) \quad& \vert{r}/{r_t}\vert \ge 1 \\ {r}/{r_t} \quad &\vert{r}/{r_t}\vert < 1 \end{cases}\end{displaymath}$	(6)
$\displaystyle C''(r)$	$\displaystyle =$	$\begin{displaymath}\begin{cases} 0 \quad\quad &\vert{r}/{r_t}\vert \ge 1 \\ {1}/{r_t} \quad\quad &\vert{r}/{r_t}\vert < 1 \end{cases} \notag\end{displaymath}$	(7)

Notice that the second derivative vanishes if the residual falls to the threshold value $ r_t$

. This could lead to failure of the Huber solver, because the second derivatives are used in the denonimator when solving for the step size $\alpha$ and $\beta$ in the conjugate-direction scheme (Claerbout (2009)).

We are delighted to see that hybrid solver gives a resonable result for the full Galilee problem as shown in figure 5(c), we can hardly describe the model solution as ``blocky." This might be because we have used a small threshold value for the model-fitting goal. For example, a threshold value of 0.30 percentile means we would like to see blocks about 3 to 4 points long. A higher threshold value for the model-fitting goal (equation 4) can increase blockiness; however our present solver has restricted us to use the same threshold value for both the model-fitting and the data-fitting goals (equation 3). One possible improvement for the future is to separate the thresholds for these goals.