Although the passage of singularity information from acoustic impedance to seismic traces is now well understood, it remains unanswered how routine seismic processing, mode conversions, and multiple reflections can affect the singularity analysis of surface seismic data. We make theoretical investigations on the transition of singularity behaviors from acoustic impedances to surface seismic data. We also perform numerical, wavelet-based singularity analysis on an elastic synthetic data set that is processed through routine seismic processing steps (such as stacking and migration) and that contains mode conversions, multiple reflections, and other wave-equation effects. Theoretically, seismic traces can be approximated as proportional to a smoothed version of the (N+1)th derivative of acoustic impedance, where N is the vanishing moment of the seismic wavelet. This theoretical approach forms the basis of linking singularity exponents (Hölder exponents) in acoustic impedance with those computable from seismic data. By using wavelet-based multiscale analysis with complex Morlet wavelets, we can estimate singularity strengths and localities in subsurface impedance directly from surface seismic data. Our results indicate that rich singularity information in acoustic impedance variations can be preserved by surface seismic data despite data-acquisition and processing activities. We also show that high-resolution detection of singularities from real surface seismic data can be achieved with a proper choice of the scale of the mother wavelet in the wavelet transform. Singularity detection from surface seismic data thus can play a key role in stratigraphic analysis and acoustic impedance inversion.

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