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):  

$$\begin{array} {lllll} \bold 0 &\approx& {\bf r_d} &=& \frac... ...&\approx& \epsilon {\bf r_p} &=& \epsilon \bold p, \end{array}$$ (40)

where $\frac{d}{ds}$ is the derivative along the track and minimizing  

$$g(\bold p) = \vert{\bf r_d}\vert _{\ell^1}+\epsilon^2\Vert{\bf r_p}\Vert^2 .$$ (41)

The purpose of the derivative is to remove the drift, which is assumed to have a zero frequency component. The derivative poses two types of problem, however. First, it attenuates the bathymetry as well, which has frequency components very close to zero. Second, it creates more bad data points for the high frequency noise. Both effects are illustrated in Figure b where almost all the details inside the lake have disappeared after the minimization of equation (). In addition, the map is more noisy due to the aggravating effect of the derivative on bad data points.

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. Based on Brown's idea 2001 and following Chapter , I propose introducing an operator that will adaptively model and subtract the systematic shift within the inversion scheme.

In the next section, I show that by incorporating a modeling operator for the drift in the data, the ship tracks can be effectively removed without any loss of resolution in the estimated depth map.