$\ell^1$ norm

One main problem with the Galilee data is the presence of outliers in the middle of the lake. These spikes could be attenuated by editing or applying running median filters. However, the former involves human interpretation and the later might compromise small details by flattening and distorting the signal Claerbout and Fomel (2002). Therefore, inversion appears to be the best compromise by eliminating the spikes while honoring the data in an automated way. I introduce in equation () the Huber norm defined in Chapter and minimize  

$$g(\bold p) = \vert{\bf r_d}\vert _{Huber}+\epsilon^2\Vert{\bf r_p}\Vert^2$$ (39)

For the Huber threshold [equation ()], I selected $\alpha=10$ cm, which corresponds to the measurement error of the sounder. Figure a shows ${\bf p}$ estimated with the $\ell^2$ norm [equation ()] after 50 iterations, which simulates a least-squares solution with damping. Note that the scale bar is not displayed whenever ${\bf p}$ is plotted because its values are of little interest for this analysis. Although ${\bf p}$ appears to be a variable of mathematical interest only, in fact, the solution ${\bf h}$ is so smooth that we have difficulty viewing it. We could view the two components of $\nabla \bold h$ but it happens that ${\bf p}$ is a roughened version of ${\bf h}$. In addition, it is more convenient to view ${\bf p}$ than the two images $\partial \bold h/\partial x$and $\partial \bold h/\partial y$ because it is only a single component vector. We can see considerable spurious noise in the map of Figure a. In addition, we can see the vessel tracks in the north part of the map.

Figure b displays ${\bf p}$ estimated with the $\ell^1$ norm [equation (]. Most of the glitches are attenuated showing vessel tracks only. Some ancient shorelines in the west part and south part of the Sea of Galilee are now easy to identify (shown as ``AS'' in Figure b). In addition, we also start to see a ``valley'' in the middle of the lake (shown as ``R'' in Figure b). This feature is also present in Figure a where no attempts were made to remove the spikes. Therefore, this can be either a geological feature that represents the on-going rifting in this area or a track. The next section will prove that this valley is not a processing artifact or some noise not accounted for in our inversion scheme. The data outside the sea have been also partially removed. The tracks (shown as ``T'' in Figure ) are still clearly visible after the attenuation of the outliers because they do not fit the model of the noise we are trying to remove.

Figures a,b show the bottom of the Sea of Galilee (${\bf h}={\bf H^{-1}p}$)after inversion. Each line represents one east-west line of the interpolated data every 500 meters. The $\ell^1$ result is a great improvement over the $\ell^2$ maps. The glitches inside and outside the sea have disappeared. It is also pleasing to see that the $\ell^1$ norm gives us positive depths everywhere. Although not everywhere visible in Figure , it is interesting to notice that we produce topography outside the lake. Indeed, the effect of regularization is to produce synthetic topography which is a natural continuation of the lake floor surface.

I have shown that the combined utilization of preconditioning and the Huber norm removes the spikes in the depth map of the Sea of Galilee. In the next section, I propose removing the ship tracks by introducing an operator in equation () that will model the coherent noise created by different weather and human conditions during the acquisition of the data.

 

fig2
Figure 3 (a) ${\bf p=Hh}$ estimated with equation () in a least-squares sense after 50 iterations, which simulates a least-squares solution with damping. (b) ${\bf p=Hh}$estimated with equation () in a $\ell^1$ sense. The spikes have been correctly attenuated. Some interesting features are shown by the arrows: AS points to few ancient shores, O points to some outliers, T points to a few tracks and R points to a ridge.

 

fig3
Figure 4 (a) View of the bottom of the lake (${\bf h}={\bf H^{-1}p}$) with the $\ell^2$norm after 50 iterations, which simulates a least-squares solution with damping. (b) View of the bottom of the lake with the $\ell^1$ norm. Note that with the $\ell^1$ norm, the spikes have been attenuated. Each line represents one east-west track every 500 meters.