SEABEAM: FILLING THE EMPTY BINS WITH A LAPLACIAN

Figure 7 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 7 Depth of the ocean under ship tracks.

The locations are scattered about, according to various aspects of the ship's navigation and the geometry of the SeaBeam sonic antenna. Figure 7 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. More interesting are the empty mesh locations where no data is recorded. Here I left empty locations with a background value equal to the mean depth .Supposing the roughening operator to be the Laplacian operator and using module mis2 led to the result in Figure 8. After many iterations, both regularization and preconditioning lead us to the same result. After a small number of iterations, we see that regularization has filled the small holes but it has not reached out far away from the known data. With preconditioning, it is the opposite.

 

prcfill
Figure 8 Views of the ocean bottom after filling. (We'll return to this data in the next chapter to do a better job.) (Fomel)