Polarity preserving decon in ``N log N'' time

DISCUSSION AND CONCLUSION

This paper introduces the notion that by manipulating the $u_\tau$ we may make improvements on the old mathematical method of blind deconvolution. We were uncommonly successful here in dealing with our most commonly observed wavelet, the Ricker wavelet. This success suggests other improvements might flow from manipulations of the $u_\tau$ for other purposes.

For example, given only a single seismogram, we may wish to limit the number of degrees of freedom for the filter estimation. We have long known this can be done by smoothing the data spectrum. Another method is to limit the range, or taper the range of $u_\tau$ coefficients. Such ideas are untried, so not yet compared.

Likewise, many shot waveforms have been recorded and tabulated. Perhaps it makes sense to map these wavelets to the ``lag-log'' space $u_\tau$ to better understand their statistics.

I see no immediate application, but we might recall that spectral factorization is also applicable for complex-valued signals. Then the spectrum is non-symmetric. This arises when time-dependent signals have been previously Fourier transformed over space.

Shuki asks, ``What about seafloor receivers where there is one ghost, not two?'' I reply, ``Perhaps the same code can be used, but instead of gateing on the range $\pm\tau$ being 3/4 period for the Ricker wavelet, it might be instead 1/4 period for the the primary lobe.

Polarity preserving decon in ``N log N'' time