Seismic reflection and transmission coefficients of a self-similar interface

Details Seismic reflection and transmission coefficients of a self-similar interface Seismic reflection and transmission coefficients of a self-similar interface

Nov 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998geoji.135..585w&link_type=abstract

Geophysical Journal International, Volume 135, Issue 2, pp. 585-594.

Astronomy and Astrophysics

Astronomy

5

Numerical Methods, Seismic Reflection, Transmission

Scientific paper

The derivation of seismic reflection and transmission coefficients is generally based on the assumption that the medium parameters behave as step functions of depth, at least in a finite region around the interface. However, outliers observed in well logs generally behave quite differently from step functions. In this paper we represent an interface by a self-similar singularity, embedded between two homogeneous half-spaces, and we derive its frequency-dependent normal-incidence reflection and transmission coefficients. For ω->0 the expressions for the coefficients reduce to those for a discrete boundary between two homogeneous half-spaces; for ω->∞ they become frequency-independent. These asymptotic expressions have a relatively simple form and depend on the singularity exponent α.
The exact as well as the asymptotic expressions are used to evaluate the time-domain reflection and transmission responses of a self-similar interface. Finally, we use a numerical method to model the response of a smoothed version of a self-similar interface (note that the velocity of a smoothed singularity remains finite). It turns out that smoothing has hardly any effect on the response, provided that the smoothing does not affect the scales corresponding to the seismic frequency range.

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