The satellite tracks are much like feathered marine geophone cables, sail lines, or shot lines in a 3D seismic survey. Any method that hopes to succeed on 3D seismic data should be able to deal with this toy problem.
Early work on this dataset at SEP Ecker and Berlioux (1995); Lomask (1998, 2002) has mainly dealt with the systematic errors present in the dense dataset Ecker and Berlioux (1995), or with ways in which to use information in the dense portion of the data to regularize the missing bins in the northern, sparse portion of the data Lomask (1998, 2002). In the latter, it is assumed that the statistics of the data are stationary over both regions. More recent work has started to deal with the interpolation of only the sparse tracks Curry (2004); Lomask (2004).
In this paper, only the lower half of the dataset (30 to 40 degrees (S) latitude) is examined, so that interpolation of the sparse tracks can be compared with the dense tracks. Here, a different approach is presented to dealing with the sparse track problem, where a pair of prediction-error filters (PEFs) are estimated directly on the two sets of tracks. This pair of filters is estimated in the data space, so that the issues of missing data and irregular geometry are no longer present. Once these filters have been estimated, they can be used in tandem to regularize the missing portions of the model space.
Extension of this method to incorporate non-stationary PEFs is quite straightforward. The similarities between the Madagascar data and a Colombian 2D seismic data line are noteworthy enough that this method should be applicable to 2D land data, where the two sets of tracks correspond to positive and negative offsets. The geometry of a 3D cross-swath land seismic survey also has similarities to the Madagascar data, where when data predicted by reciprocity is added irregular crossing tracks are present in cmp_x, offset_x space.