SEISMIC-WAVE ATTENUATION OPERATORS FOR ARBITRARY-Q

Under the minimum-phase condition attenuation operators are derived from arbitrary intrinsic or scattering Q as a function of frequency, and a simple method is described to calculate these operators with the aid of Fourier transformation. In the case of scattering attenuation, the medium may contain both 1-D, 2-D or 3-D heterogeneities. For illustration, the method is applied to a 1-D model of statistically layered media with 12 per cent standard deviation of relative velocity fluctuation. The theoretical scattering Q, deduced from single-scattering theory, is determined for a numerical autocorrelation function and an exponential autocorrelation function, as well as for a modified exponential autocorrelation function, which is introduced here. The corresponding attenuation operators are calculated. The theoretical Q and simulated seismograms, calculated with the attenuation operators, agree very well with the exact results of synthetic-seismogram calculations.