Regularization example

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 $\lambda$) are shown in Figures  and . Interpolation with the minimum-phase Laplacian ($\lambda=0$) 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 ($\lambda=1$) 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.

 

mesh 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.

 

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Figure 7 The Sea of Galilee dataset after missing data interpolation with helical preconditioning. Different plots correspond to different values of the tension parameter. An east-west derivative filter was applied to illuminate the surface.

 

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Figure 8 Cross-sections of the Sea of Galilee dataset after missing-data interpolation with helical preconditioning. Different plots correspond to different values of the tension parameter.

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.