Fitting Goals

All four data sets were merged into one and sorted into quadruplets of (x,y,z,w). The x and y components are longitude and latitude in degrees, respectively. The z value is sea-surface height above a reference ellipsoid in millimeters. Lastly, the w component is a weighting value that is nonzero for any suspicious z value and at the track-ends.

I used fitting goals that are essentially the same as those applied to the Sea of Galilee ().

$$\begin{eqnarray} \ {\bf W} \frac{d}{dt}[{\bf L}{\bf m} -{\bf d}] &\approx& {\bf 0} \\ \ \epsilon {\bf A m} &\approx& {\bf 0} \end{eqnarray}$$ (113) (114)

The first goal is to find a map, ${\bf m}$, that when sampled into data-space by linear interpolation, has a derivative that is equal to the derivative along the tracks of the observed data, ${\bf d}$. The weighting term ${\bf W}$, will throw out any derivative values influenced by noisy values or track-ends.

The next goal applies a regularization operator, ${\bf A}$, which insures that the model with infilled missing data is smooth. I regularized with the 2D gradient, (${\nabla_x},{\nabla_y}$), and the 2D Laplacian, $\nabla^2$. Finally, I began testing 2D prediction-error filters (PEFs), which were estimated on the dense data.

The models in this paper have $400 \times 400$ bins which seem to have enough resolution to make detailed images while not slowing down the solver too much. The entire merged data set consists of 537979 samples.