Stratigraphic filtering of seismic waves: The influence of mode conversion multiples onP-wave andS-wave phase and attenuation

Abstract

We demonstrate how multiples, generated at the interfaces of plane parallel beds, modify the propagation characteristics of an originally coherent seismic wave. For waves propagating at an angle to the bedding plane we find that theSV andP-waves couple so that neither is a pure mode. TheSH-wave, while modified in its propagation characteristics by multiples, remains a pure mode. The coupling ofSV-multiples into the quasi-P-mode appears weaker than the coupling ofP-wave multiples into the quasi-SV mode; at least this is so for the two simple cases of (a) density fluctuations only and (b) correlatedV _p andV _s fluctuations which conserve Poisson's ratio.

We also find that the coupling is sensitive to both the angle of propagation and frequency. In addition there is a cut-off angle forP-wave multiples influencing the quasi-SV mode. Propagation angles larger than the cut-off permit theP-multiples to modify the phase of the quasi-SV mode, but not its effective attenuation. No such cut-off effect is found for SV-multiples influencing the quasi-P mode, whose angle-dependent and frequency-dependent phase distortion and effective attenuation are influenced both byP-wave multiples andSV-multiples.

In view of the mathematical complexity of the expressions describing the phase, and effective attenuation of modes when allowance is made forP-andS-wave multiples, we strongly advocate numerical coding of the major mathematical formulae. By so doing a systematic study can be undertaken of the frequency and offset dependence of seismic waves as a function of seismic source input and power spectral behavior of the fluctuations in density and elastic constants of beds. It is our opinion that the full mathematical expressions are too involved to permit an analytic, systematic investigation to be given of the phase and attenuation of seismic waves with any degree of sophistication or generality.