$\ell^1$ norm

Spikes and erratic noise glitches can be suppressed with an approximate $\ell^1$ norm. One main problem with the Galilee data is the presence of outliers in the middle of the lake and at the track ends. We could attenuate these spikes by editing or applying running median filters. However, the former involves human labor while the latter might compromise small details by smoothing and flattening the signal. Here we formulate the estimation to eliminate the drastic effect of the noise spikes. We introduce a weighting operator that deemphasizes high residuals as follows:  

$$\begin{array} {lllll} \bold 0 &\approx& \bold r_d &=& \bold... ... \bold 0 &\approx& \bold r_p &=& \epsilon \bold p \end{array}$$ (23)

with a diagonal matrix $\bf{W}$:

$${\bf W} = {\bf diag} \left( \frac{1}{(1+r_i^2/\bar{r}^2)^{1/4}} \right)$$ (24)

where r_i is the residual for one component of $\bold r_d$and $\bar r$ is a prechosen constant. This weighting operator ranges from $\ell^2$ to $\ell^1$, depending on the constant $\bar r$.We take $\bar{r}=10$ cm because the data was given to us as integer multiples of 10 cm. (A somewhat larger value might be more appropriate).

 

antoine2
Figure 18 Estimated ${\bf p}$ in a least-squares sense (left) and in an $\ell^1$ sense (right). Pleasingly, isolated spikes are attenuated. Some interesting features are shown by the arrows: AS points to few ancient shores, O points to some outliers, T points to few tracks, and R points to a curious feature.

Figure displays ${\bf p}$ estimated in a least-squares sense on the left and in a $\ell^1$ sense on the right (equation (23) with a small $\bar r$). Most of the glitches are no longer visible. One obvious glitch remains near (x,y)=(205,238). Evidently a north-south track has a long sequence of biased measurements that our $\ell^1$ cannot overcome. Some ancient shorelines in the western and southern parts of the Sea of Galilee are now easier to identify (shown as AS). We also start to see a valley in the middle of the lake (shown as R). Data outside the lake (navigation errors) have been mostly removed. Data acquisition tracks (mostly north-south lines and east-west lines, one of which is marked with a T) are even more visible after the suppression of the outliers.

 

antoine3
Figure 19 East-west cross sections of the lake bottom (${\bf h}={\bf H^{-1}p}$). Top with the $\ell^2$ solution. Bottom with the $\ell^1$ approximating procedure.

Figure shows the bottom of the Sea of Galilee (${\bf h}={\bf H^{-1}p}$)with $\ell^2$ (top) fitting and $\ell^1$ (bottom) fitting. Each line represents one east-west transect, transects at half-kilometer intervals on the north-south axis. The $\ell^1$ result is a nice improvement over the $\ell^2$ maps. The glitches inside and outside the lake have mostly disappeared. Also, the $\ell^1$ norm gives positive depths everywhere. Although not visible everywhere in all the figures, topography is produced outside the lake. Indeed, the effect of regularization is to produce synthetic topography, a natural continuation of the lake floor surface.

We are now halfway to a noise-free image. Figure shows that vessel tracks overwhelm possible fine scale details. Next we investigate a strategy based on the idea that the inconsistency between tracks comes mainly from different human and seasonal conditions during the data acquisition. Since we have no records of the weather and the time of the year the data were acquired we presume that the depth differences between different acquisition tracks must be small and relatively smooth along the super track.