Apparent and intrinsic Q: the one-dimensional case
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The propagation of plane waves through statistically layered media is investigated both numerically and with single-scattering theory in the one-dimensional case. Exact apparent or stratigraphic Q, Qs, is determined from synthetic seismograms with the spectral-ratio method. Maximum velocity (impedance) fluctuations up to 30% (~40%) are studied; the fluctuations are uniformly distributed with zero mean. In all cases the trend of Qs as a function of frequency is well described by the analytical Qs, as determined from single-scattering theory under the assumption of an exponential autocorrelation function of the impedance fluctuations. The frequency dependence of the analytical Qs-1 follows a Debye-peak function, its maximum is γ2/2 and corresponds to the wavelength 4πa (γ2 = variance of relative impedance fluctuation, a = correlation distance). In further numerical calculations intrinsic or anelastic Q, Qa, is introduced, and it is shown that total attenuation Q -1 agrees very well with the sum of apparent and anelastic attenuation, Qs-1+Qa-1. Finally, a simple, minimum-phase stratigraphic attenuation operator is derived which describes the amplitude decay and the dispersion in a one-dimensional random medium with good accuracy. Stratigraphic attenuation is similar to the anelastic attenuation of a standard linear solid.
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