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Abandoned strategy for attenuating tracks

An earlier strategy to remove the ship tracks is to filter the residual during the inversion as follows Fomel and Claerbout (1995):  
 \begin{displaymath}
\begin{array}
{lllll}
 \bold 0 &\approx& \bold r_d &=& \bold...
 ...&\approx& \epsilon \bold r_p &=& \epsilon \bold p,
 \end{array}\end{displaymath} (8)
where $\frac{d}{ds}$ 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 ${\bf p}$ 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 $\bold W$, $\frac{d}{ds}$, $\bold B$ and $\bold {H^{-1}}$ 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
ds
Figure 6
Estimated ${\bf p}$ 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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next up previous print clean
Next: A new fitting goal Up: Attenuation of the ship Previous: Attenuation of the ship
Stanford Exploration Project
7/8/2003