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Now, we show our formulation of the regridding problem.
Let be an abstract vector containing as components the
water depth over a 2D spatial mesh and 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
 
(1) 
where is a 2D linear interpolation operator
and is the data residual.
This fitting goal simply requires that the gridded data take on appropriate
values where the data 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.

 
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.

 
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
 
(2) 
where and is the model space
residual. We then minimize the misfit function
 
(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 apriori model . Since we do not have
any apriori model, we simply choose the gradient operator 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: Preconditioning for accelerated convergence
Up: Guitton and Claerbout: Galilee
Previous: Introduction
Stanford Exploration Project
7/8/2003