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Regularization example

We chose an environmental dataset (Claerbout, 2002) 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 (Fomel and Claerbout, 1995).

Figure 9 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 10 and 11. 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 5). It is clearly seen in the cross-section plots in Figure 11. 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 9.
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 10.
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 11.
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.
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next up previous [pdf]

Next: Conclusions Up: Application of spectral factorization: Previous: Finite differences and spectral

2014-02-15