SEABEAM: Filling the empty bins with a laplacian

Figure  shows a day's worth of data collected at sea by SeaBeam, an apparatus for measuring water depth both directly under a ship, and somewhat off to the sides of the ship's track. The data is measurements of depth h(x,y) at miscellaneous locations in the (x,y)-plane.

 

seabin90 Figure 8 Depth of the ocean under ship tracks. Empty bins are displayed with an average depth $\bar h$.

The locations are scattered about, according to various aspects of the ship's navigation and the geometry of the SeaBeam sonic antenna. Figure was made by binning with bin2() and equation (). The spatial spectra of the noise in the data could be estimated where tracks cross over themselves. This might be worth while, but we do not pursue it now.

Here we focus on the empty mesh locations where no data is recorded (displayed with the value of the mean depth $\bar h$). These empty bins were filled with module mis2 . Results are in Figure .

 

prcfill
Figure 9 The ocean bottom restoring missing data with a helix derivative.

In Figure the left column results from 20 iterations while the right column results from 100 iterations.

The top row in Figure shows that more iterations spreads information further into the region of missing data.

It turned out that the original method strictly honoring known data gave results so similar to the second method (regularizing) that the plots could not be visually distinguished. The middle row in Figure therefore shows the difference in the result of the two methods. We see an outline of the transition between known and unknown regions. Obviously, the missing data is pulling known data towards zero.

The bottom row in Figure shows that preconditioning spreads information to great distances much quicker but early iterations make little effort to honor the data. (Even though these results are for $\epsilon=0$.) Later iterations make little change at long distance but begin to restore sharp details on the small features of the known topography.

What if we can only afford 100 iterations? Perhaps we should first do 50 iterations with preconditioning to develop the remote part of the solution and then do 50 iterations by one of the other two methods to be sure we attended to the details near the known data. A more unified approach (not yet tried, as far as I know) would be to unify the techniques. The conjugate direction method searches two directions, the gradient and the previous step. We could add a third direction, the smart direction of equation (14). Instead of having a $2\times 2$ matrix solution like equation () for two distances, we would need to solve a $3\times 3$ matrix for three.

Figure has a few artifacts connected with the use of the helix derivative. Examine equation () to notice the shape of the helix derivative. In principle, it is infinitely long in the horizontal axis in both equation () and Figure . In practice, it is truncated. The truncation is visible as bands along the sides of Figure .

As a practical matter, no one would use the first two bin filling methods with helix derivative for the roughener because it is theoretically equivalent to the gradient operator $\bold \nabla$ which has many fewer coefficients. Later, in Chapter we'll find a much smarter roughening operator $\bold A$called the Prediction Error Filter (PEF) which gives better results.