Synthetic seismograms in attenuating media

Abstract

Synthetic seismograms are computed in a two‐dimensional medium that contains contrasts both in the elastic and in the attenuation (Q) properties with depth. The case of a liquid‐solid interface, with and without a Q contrast between the two media, is examined in detail. We show the results when the S velocity in the solid is greater than the P velocity in the liquid. The synthetic seismograms show that the effect of the attenuation contrast is not noticeable on precritical reflections when the change in elastic properties is large even in the case of large Q contrast. When this situation occurs, there is a relative increase in the amplitude of the postcritical reflections and a decrease in amplitude of the head waves. It is interesting to notice that when two head waves are generated, the P head wave always has a very low amplitude with respect to the S head wave. This result agrees with what has been observed in borehole data. We also show the results obtained at a solid‐solid interface with and without Q contrast. The synthetic seismogram shows the same type of results as in the previous case but less pronounced due to the smaller portion of the total path occurring in attenuating media. It also shows that the converted waves (S — P and P — S) are of opposite polarities and so nearly cancel each other. The synthetic seismograms are calculated in the horizontal wavenumber‐frequency [Formula: see text] domain, and Fourier transformed back into offset‐time (x, t) domain. The choice of the [Formula: see text] domain as a working space is motivated by the fact that it allows us to include easily the effects of attenuation by specifying complex frequency‐dependent elastic moduli. The reflection and transmission coefficients in the ([Formula: see text], ω) domain are derived from a plane wave theory. Accurate geometric divergence is represented by using the exact Green’s functions in a two‐dimensional medium. The constant Q model for attenuation, derived by Kjartansson, is used.