I chose an environmental Galilee dataset Claerbout (1999); Fomel and Claerbout (1995) for a simple illustration of smooth data regularization. The data were collected on a bottom sounding survey of the Sea of Galilee in Israel Ben-Avraham et al. (1990). The data contain a number of noisy, erroneous and inconsistent measurements, which present a challenge for the traditional estimation methods.
Figure shows the data after a nearest-neighbor binning to a regular grid. The data were then passed to an interpolation program to fill the empty bins. The results (for different values of ) are shown in Figures and . Interpolation with the minimum-phase Laplacian () creates a relatively smooth interpolation surface but plants artificial ``hills'' around the edge of the sea. This effect is caused by large gradient changes and is similar to the sidelobe effect in the one-dimensional example (Figure ). It is clearly seen in the cross-section plots in Figure . The abrupt gradient change is a typical case of a shelf break. It is caused by a combination of sedimentation and active rifting. Interpolation with the helix derivative () is free from the sidelobe artifacts, but it also produces an undesirable non-smooth behavior in the middle part of the image. As in the one-dimensional example, intermediate tension allows us to achieve a compromise: smooth interpolation in the middle and constrained behavior at the sides of the sea bottom.
Figure 6 The Sea of Galilee dataset after a nearest-neighbor binning. The binned data is used as an input for the missing data interpolation program.
Smooth surfaces are rarely encountered in the practice of seismic exploration. In the next section, I develop a regularization operator suitable for characterizing more typical models of seismic data.