The choice of in equation () is also critical. I tried different values by starting from a very small number and increasing it slowly. I then chose the smallest value that removed enough tracks in the final image (). Nemeth et al. (2000) demonstrates that the noise (the tracks) and signal (the depth) can be separated in equation () if the two operators and do not model similar components of the data space.
Figures a and c display a comparison of the estimated with or without the attenuation of the vessel tracks. It is delightful that Figure c is essentially track-free without any loss of details compared to Figure a. The difference plot in Figure d between the two results corroborates this and does not show any geological feature. A close-up of Figure is displayed in Figure . The differences between the proposed techniques are clearly visible.
Comparing Figure c and Figure b, we see that the drift-modeling strategy [equation ] works much better than the noise-filtering strategy [equation ]. One possible explanation for the difference between the two results is that the modeling approach is more adaptive than the filtering of the residual. Indeed, by introducing the modeling operator, we basically look for the best that models the drift of the data on each track at each point. The price to pay is an increase of the number of unknowns in equation (). The reward is a surgically removed acquisition footprint. Notice that we can identify the ancient shorelines in the west and east parts of the lake very well.
To better understand what is done, Figures and show some segments of the input data (), the estimated noise-free data (), the estimated secular variations () and the residual () after inversion. The estimated noise-free data in Figures b and b show no remaining spikes. The effect of the track attenuation is more difficult to see because the amplitude of the drift is much smaller than the amplitude of the measurements. Notice in Figure c that the estimated drift seems to have reasonable amplitudes: the average drift is around 15 cm for an accuracy of about 10 cm for the measurements. We also observe that the estimated drift is relatively constant throughout Figure c. Now, looking at the estimated drift for another portion of the data (Figure c), notice that the drift has more variance than in Figure c and oscillates between 0 to 2 m, which is too much. In addition, the estimated drift seems to follow the bathymetry of the lake in Figure a. Decreasing would attenuate the drift component with the effect of increasing the tracks in the final image, however.
Looking closely at the residual (Figure d), the drift is large where the data are noisy (Figure a). It is possible that the day of acquisition was very windy, which is not a rare weather condition for the Sea of Galilee Volohonsky et al. (1983). Thus, the wind forces the water to pile-up on one side of the lake which can explain the lower water level on the other side. A rapid calculation shows that the seiche period for the Sea of Galilee is roughly 40 mn (assuming a lake length of 20 km and an average depth of 30 m), which is well within a day of data acquisition. In addition, the strong wind in the middle of the lake induces noisy measurements because of the waves and of the erratic movement of the ship. It is also possible that the depth sounder was not working properly that day and had problems to correctly measure the deepest part of the lake. These causes could probably explain the shape and amplitude of the estimated drift in Figure c, but we can't be absolutely sure. It is very unfortunate that no daily logs of the survey were kept in order to better interpret these results, especially for such a noisy dataset.