DISCUSSION AND CONCLUSION

Results were disappointing. I thought of switching to synthetic data but instead switched to a high quality data set (Yilmaz and Cumro number 27). The quality of the results remained poor. The data extensions found in figure 1 do not look as expected. The stepouts are only roughly appropriate. Further, they are numerically too small to be easily visible, so the amplitudes in the display were replaced by the signed square roots.

Is there a program bug or a conceptual bug? I don't know. (This is a progress report.) The theory is given above and the program listing in an appendix. Various pitfalls and nagging doubts are summarized below.

• The zero-offset truncations look more severe than the wide offset and maybe they should have been handled first.
• The simplifications arising from taking recorded data to be noiseless could be reexamined.
• The presumption that we could ignore the lead term in (2) should be reexamined.
• Notice that the trace closest to the known data is smaller than the others. This can be understood as follows: If a trace can extinguish its neighbor by being equal to it but of opposite polarity, then in a power minimization this will happen. Theoretically, however it should not happen because two nearby traces of opposite polarity imply signal in velocity space at slownesses beyond practical interest which unfortunately are likely to be omitted from the computation. (This is an argument for parameterizing with slowness instead of slowness squared.)
• The weighting function should be different.
• The theoretical approach could follow that of Bill Harlan.
• The theoretical approach could follow some entropy theory. For example extremalizing some quadratic form like the ratio of the terms in 2 instead of the sum would not have solutions biased towards zero because the ratio is scale invariant.
• Possible program bug.