Background

The Madagascar regularization problem has been approached using the following fitting goals Lomask (2002):

$$\begin{eqnarray} \ {\bf W} \frac{d}{dt}[{\bf L}{\bf m} -{\bf d}] &\approx& {\bf 0} \nonumber \\ \ \epsilon {\bf A m} &\approx& {\bf 0} . \end{eqnarray}$$ (1)

In these fitting goals: W corresponds to a weight for ends of tracks and spikes in the data, $\frac{d}{dt}$ is a derivative along each track used to eliminate low frequency variations along each track, L is a linear interpolation operator that moves from values on a regular grid to the data points, m is the desired gridded model, d are the data points along the tracks, A is a regularization operator, and $\epsilon$ is a trade-off parameter between the two fitting goals.

The regularization operator (A) typically is a Laplacian, a prediction-error filter (PEF), or a non-stationary PEF Crawley (2000). When using a PEF, it first must be estimated on some training data, using a least-squares fitting goal,  

$$\bold{W(DKa + d)} \approx \bold 0 ,$$ (2)

in which W is a weight to exclude equations with missing data, D is convolution with the data, K constrains the first filter coefficient to 1, a is the unknown filter, and d is a copy of the data.

 

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Figure 1 (a) sparse tracks. (b) dense tracks.

In order to set a benchmark for how effective any interpolation of the sparse tracks is, a PEF is estimated using fitting goal (2) on the output of fitting goals (1) when using the dense tracks of the Madagascar dataset and a Laplacian regularization operator. The PEF is then used to interpolate the sparse tracks in the same area, using the same fitting goals (1), this time with the regularization being convolution with a PEF. This result provides an upper bound to what an ideal interpolation of the data would be if we already knew the answer. Any new result should be much better than the Laplacian regularization shown in Figure b.

 

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Figure 2 (a) dense data regularized with a Laplacian. (b) sparse data regularized with a Laplacian. (c) sparse data regularized with a PEF trained on the top.