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An earlier strategy to remove the ship tracks is to filter the
residual during the inversion as follows Fomel and Claerbout (1995):
| |
(8) |

where is the derivative along the track. The purpose
of the derivative is to remove the drift from the field data while
preserving the geological features. One consequence of the derivative
is that it creates more glitches and spiky noise at the track ends and
at the bad data points. In addition, the use of the derivative might
induce a loss of resolution in the final image.
Both effects are illustrated in Figure where we
display the estimated after inversion with the fitting goals
in equation (8). We can see that the
tracks have been attenuated, as expected. However, we lost important
geological features in the middle of the lake and on the sea shores.
In addition, the map is more noisy because of the aggravating effect
of the derivative on bad data points. We do not fully understand why
this approach works badly. One possible explanation is that the
conditioning of our problem with the four operators ,
, and worsens, making the
optimization very difficult. Theoretically, we could estimate a
prediction-error filter (PEF) from the data residual and use it
as a data-residual weight within the inversion. Unfortunately, the PEF
would probably mix more glitches with good data because it would
spread further in space than the derivative.
Recently, Brown (2001) proposed estimating
systematic errors between tracks by analyzing measurements at points
where the acquisition swaths cross. This approach has the advantage
of preserving the resolution of the depth map compared to the
derivative along the tracks. Brown (2001) uses this idea as a preprocessing
step, however. The comparison of Figure b and Figure
teaches us two lessons. First, the filtering approach with the derivative along the tracks
does not produce a good image of the bathymetry
Fomel and Claerbout (1995). Second, based on Brown's idea 2001,
we propose introducing an operator
that will adaptively model and subtract the systematic shift
within the inversion scheme. In the next section, we show that by
incorporating a modeling operator for the drift in the data, we
can effectively remove the ship tracks without any loss of
resolution in the estimated depth map.

**ds
**

Figure 6 Estimated after
inversion with the fitting goals in equation (8). The
derivative removes the tracks but creates a noisy image with a
loss of resolution.

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Stanford Exploration Project

7/8/2003