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## A new fitting goal

Now, we show our new idea of removing the tracks by adaptively subtracting them within our inversion scheme. Building on Nemeth et al. (2000), we introduce a modeling operator for the ship tracks inside our fitting goal in equation (6) as follows:
 (9)
where is a drift modeling operator (leaky integration), is a new variable of our inversion, and a balancing constant between gridding and noise modeling. We then minimize the misfit function
 (10)
where estimates the interpolated map of the lake. Again, we set , and we do not iterate to completion. Note that we invert for and at the same time. Our hope is that by introducing the new variable for the drift, we will obtain a cleaner depth map. For the operator , we choose a leaky integration operator such that is the portion of data value that results from drift. This choice is consistent with the derivative used in the abandoned strategy: since we can not properly filter the tracks with , we have to define a new operator, intuitively close to the inverse of the derivative, in order to model the drift in our new approach [equation (9)]. The leaky integration seems to be a good candidate. Consistent with the way we use a rough variable to represent the smooth water depth , we now represent (for the purpose of speeding iteration) by a rougher function .The operator has the following recursive form
 (11)
The parameter controls the decay of the integration. For , leaky integration represents causal integration. The operator is then appropriate to model the secular variations implied by the different season and human conditions during the data acquisition. We simply have to choose a value of that best represents the variations between the different tracks. We have roughly 200 data points per track (Figure ). With , we have , which represents a amplitude decay for one track. This seems to be a reasonable decay for what we are trying to model, i.e., the drift. We keep this value of for our results in the next section. We show that the operator removes most of the vessel tracks present in Figure .

The choice of in equation (9) is also critical. We tried different values by starting from a very small number and increasing it slowly. We 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 (9) if the two operators and do not model similar components of the data space. The parameter helps us to mitigate the possible crosstalk. A similar approach has been used by Guitton (2002) to successfully remove ground-roll on common midpoint gathers.

We display in Figure a comparison of the estimated with or without the attenuation of the vessel tracks. It is delightful that Figure b is track-free without any loss of details compared to Figure a. The difference plot in Figure c between the two results corroborates this and does not show any geological feature.

Comparing Figure b and Figure , we see that the drift-modeling strategy (equation 9) works much better than the noise-filtering strategy (equation 8). One possible explanation for the difference between the two results is that our 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 (9). 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 we are doing, we show in Figures and some segments of the input data (), the estimated secular variations () and the residual () after inversion. We can notice in Figure b 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, which is satisfying to us. We also observe that the estimated drift is relatively constant throughout Figure b. Now, if we look at the estimated drift for another portion of the data (Figure b), we notice that the drift has more variance than in Figure b and oscillates between 0 to 2 m, which is a lot. In addition, the estimated drift seems to follow the bathymetry of the lake in Figure a.

Looking closely at the residual (Figure c), we notice that 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. 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 b, 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 our results, especially for such a noisy dataset.

compressecul
Figure 7
(a) Input data at some location of the measurements. (b) Estimated drift after inversion (). (c) Data residual after inversion.

compressecul14
Figure 8
(a) Input data at different location than in Figure . (b) Estimated drift after inversion (). (c) Data residual after inversion.

We have shown that we can effectively subtract the tracks without any loss of resolution by introducing a noise modeling operator within our inversion scheme. In the next section, we go one step further and attempt to improve our result by using a prediction-error filter (PEF) instead of the helix derivative as a preconditioner in equation (9). We show that the PEF helps us to improve the details inside the lake, however, shoreline resolution is degraded.

comp-L1-HD
Figure 9
(a) estimated with equation (6) with the tracks. (b) estimated with equation (9) without the tracks. (c) Difference between (a) and (b): no geological feature has leaked in.

Next: Approximating the model covariance Up: Attenuation of the ship Previous: Abandoned strategy for attenuating
Stanford Exploration Project
7/8/2003