SEABEAM: FILLING THE EMPTY BINS

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 $\bar h$.After you study the water-depth map for a while, you come to realize that at many locations, your human intuition suggests some obvious extensions of the ocean floor ridges (which are white) and we seek mathematical methods to guess what humans can guess, namely, a background depth $\bar h(x,y)$ more plausible than $\bar h$.Theoretically this amounts to finding and using the model spectrum. What we actually do is use a Prediction Error Filter (PEF) to fill in the missing data. Supposing the PEF to be the Laplacian operator $\nabla^2$and using module mis2 led to the result in Figure 8.

 

sealap90 Figure 8 Depth of the ocean under ship tracks interpolated by minimizing the energy of the Laplacian of the water depth.

The problem we see with the Laplacian operator as an interpolator is that it smears things uniformly in all directions. From Figure 8 we see that we need an anisotropic interpolation oriented along the regional trends. To find a better PEF we apply module pef to the image of Figure 7. After binning the data and finding this PEF, we do a second stage of linear-least-squares optimization with mis2 , as we did for Figure 6, and we obtain the pleasing result in Figure 9.

 

seapef90
Figure 9 Depth of the ocean interpolated with a PEF.