DISCUSSION

The method of inverse linear interpolation with IRLS enhancement was able to produce a fairly clear image of the Sea of Galilee bottom, showing large-scale, presumably geological structures. The lessons learned from the Sea of Galilee project include the following:

• If a model (interpolation result) contains sharp edges, it can be preferable to use a gradient filter instead of a Laplacian one for smoothing purposes.
• The iteratively reweighted least-squares approach is capable of removing non-Gaussian noise influence on the result of inversion.
• The conjugate-gradient solver can behave differently in nonlinear and piece-wise linear implementations of nonlinear problems.

The main cause of the regular noise in the Galilee data set is the inconsistency among repeated measurements. An alternative approach could be applied to the problem of the data track inconsistencies. A well-known example of such an approach is the seismic mis-tie resolution technique Harper (1991). Combining a technique of this kind with the inverse linear interpolation is a possible direction for future research.

Another interesting untested opportunity is preconditioning. As Bill Harlan has pointed out, the two equations in the system (1) - (2) contradict each other, since the first one aims to build details in the model, while the second tries to smooth them out. Applying the model preconditioning method could change the contradictory nature of the algorithm and speed up its convergence. The inversion problem takes the form  

$${\bf 0 \approx W (d - LBx)}$$ (9)

 

$${\bf 0 \approx \epsilon \, x} \,\,\,,$$ (10)

where ${\bf m=Bx}$, and ${\bf B}$ is a smoothing operator (an approximate inverse of the roughening operator ${\bf A}$ in (2)). Possible advantages of preconditioning deserve special investigation.

 

galavl
Figure 9 Result of the LS inverse interpolation plotted in AVS.

 

galavn
Figure 10 Result of the IRLS inverse interpolation plotted in AVS.