Envelope broadening and scattering attenuation of a scalar wavelet in random media having power-law spectra

• Published in: Geophysical journal international Vol. 204; no. 1; pp. 386 - 398
• Main Author: Sato, Haruo
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01-01-2016
• Subjects: Seismic attenuation; Wave scattering and diffraction; Theoretical seismology; Wave propagation
• ISSN: 0956-540X; 1365-246X
• DOI: 10.1093/gji/ggv442

Peak delay and envelope broadening of an S-wavelet with travel distance increasing are seen in short-period seismograms of small earthquakes. Those phenomena are results of scattering by random velocity inhomogeneities in the earth medium. As shown in sonic well-log data we may suppose that random velocity fluctuation has power-law spectra even in the seismic spectral range. As a simple mathematical model, we study how the envelope of a scalar wavelet varies in von Kármán-type random media, which have power-law spectra at large wavenumbers. Since the centre wavenumber of a wavelet is a unique scale in the power-law spectral range, using it as a reference, we divide the random media into the low-wavenumber spectral (long-scale) component and the high-wavenumber spectral (short-scale) component. For the wave propagation through the long-scale component of random media, we may apply the parabolic approximation to the wave equation. Using the Markov approximation, which is a stochastic extension of the phase screen method, we directly synthesize the energy density, which is the mean-square (MS) envelope of a wavelet in a given frequency band. The envelope duration increases according to the second power of travel distance. There is an additional factor, the wandering effect which increases the envelope duration according to the traveltime fluctuation. Wide angle scattering caused by the short-scale component of random media attenuates wave amplitude with travel distance increasing. We use the total scattering coefficient of the short-scale component as a measure of scattering attenuation per distance, which is well described by the Born approximation. Multiplying the exponential scattering attenuation factor by the MS envelope derived by the Markov approximation, we can synthesize the MS envelope reflecting all the spectral components of random media. When the random medium power spectra have a steep role-off at large wavenumbers, the envelope broadening is small and frequency independent, and scattering attenuation is weak. When the random medium power spectra have a small role-off, however, the envelope broadening is large and increases with frequency, and the scattering attenuation is strong and increases with frequency. The proposed synthesis of MS envelopes is fully analytic; therefore, it can be a theoretical basis for the evaluation of random heterogeneity of the earth medium from the analysis of seismogram envelopes.