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Attenuation of the noise bursts and glitches

Now, we show our formulation of the regridding problem. Let $\bold{h}$ be an abstract vector containing as components the water depth over a 2-D spatial mesh and $\bold{d}$ be an abstract vector whose successive components represent depth along the vessel tracks. One way to grid irregular data is to minimize the length of the residual vector $\bold r_d(\bold h)$ 
 \begin{displaymath}
\bold 0 \approx \bold r_d = \bold B \bold h - \bold d\end{displaymath} (1)
where $\bold B$ is a 2-D linear interpolation operator and $\bold r_d$ is the data residual. This fitting goal simply requires that the gridded data $\bold{h}$ take on appropriate values where the data $\bold{d}$ was collected. The bin size is 60 by 50 meters. We display a simple binning (without interpolation or inversion) of the raw data (Figure [*]) in Figure [*]. A coarser mesh would avoid the empty bins but lose resolution. As we refine the mesh for more detail, the number of empty bins grows as does the care needed in devising a technique for filling them. The black lines in Figure [*] are the ship tracks. Notice that some data points are outside the contour of the water. These must represent navigation errors. Figure [*] displays the ship tracks only. The straight lines in the north part of the lake are due to positioning errors. The tracks match almost exactly with the black lines in Figure [*].

 
galileedatabinned
Figure 2
Simple binning of the raw data in Figure [*]. The ship tracks and empty bins are visible and need to be accounted for in the inversion process.
galileedatabinned
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seetrack
Figure 3
Ship tracks for the Sea of Galilee dataset. The north part of the lake (top) has many navigation glitches which show up as long straight lines. Notice that very is no track going all the way from the north (top) to the south (bottom). Most of the track stop in the middle of the lake.
seetrack
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Unless data is collected everywhere, and depending on how we parameterize the grid, the regridding will leave holes on the mesh. We can get rid of the holes by adding some regularization, like  
 \begin{displaymath}
\begin{array}
{lllll}
 \bold 0 &\approx& \bold r_d &=& \bold...
 ...x& \epsilon \bold r_h &=& \epsilon \nabla \bold h 
 \end{array}\end{displaymath} (2)
where $\nabla=\left ( \frac{\partial}{\partial x},
\frac{\partial}{\partial y}\right)$ and $\bold r_h$ is the model space residual. We then minimize the misfit function  
 \begin{displaymath}
f(\bold h) = \Vert\bold r_d\Vert^2+\epsilon^2\Vert\bold r_h\Vert^2\end{displaymath} (3)
to estimate the interpolated map of the lake. In theory Tarantola (1987), the regularization operator (squared) should be the model covariance operator given an a-priori model ${\bf h_0}$. Since we do not have any a-priori model, we simply choose the gradient operator $\nabla$as a way of saying that the bottom of the lake is smooth. However, as pointed out by Harlan (1995), the regularization and the data fitting goal in equation (2) contradict each other. One equation tends to add details in the final map whereas the second one (the regularization) tends to smooth it. We can more easily balance these two goals by preconditioning the problem Fomel (2001).


 
next up previous print clean
Next: Preconditioning for accelerated convergence Up: Guitton and Claerbout: Galilee Previous: Introduction
Stanford Exploration Project
7/8/2003