We now test our proposed method to get rid of the outliers. We then use the fitting goals in equations (4) and (6) to produce depth images of the Sea of Galilee. Equation (4) is referred as the norm solution and equation (6) as the norm solution.
In Figure a, we show estimated with the norm. Although appears to be a variable of mathematical interest only, in fact, the solution is so smooth that we have difficulty viewing it. We could view the two components of but it happens that is a roughened version of . Hence it is more convenient to view than the two images and .We can see a lot of spurious noise everywhere in the map of Figure a. In addition, we can see the vessel tracks in the north part of the map. This first result is obtained after 1000 iterations which means that we essentially simulate a least-squares solution without damping. Therefore, all the noise in the data is inverted for and mapped in the final model. With noisy data, it is common practice to use a damped least-squares to minimize the effects of the noise. We can easily simulate a damped least-squares solution by decreasing the number of iterations, as explained in the preceding section. In Figure b, we show the roughened map of the Sea of Galilee after 50 iterations, thus recreating the solution of a damped least-squares problem. Most of the noise has been attenuated but glitches are still present in the northern part of the lake and the norm should be utilized.
Figure c displays estimated with the norm (e.g., equation (6) with a small ). 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. In addition, we also start to see a ``valley'' in the middle of the lake that probably represents the on-going rifting in this area. The data outside the sea have been also partially removed.
Figures a,b,c show the bottom of the Sea of Galilee () after inversion. The result is a great improvement over the maps with or without damping. The glitches inside and outside the sea have disappeared. It is also pleasing to see that the 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.
We have shown that the combined utilization of preconditioning and IRLS removes the spikes in the depth map of the Sea of Galilee. In the next section, we propose removing the ship tracks by introducing an operator in equation (6) that will model the coherent noise created by different weather and human conditions during the acquisition of the data.