Correction to “Envelope broadening of spherically outgoing waves in three‐dimensional random media having power law spectra” by Tatsuhiko Saito, Haruo Sato, and Masakazu Ohtake

Abstract

High-frequency S wave seismogram envelopes are broadened with increasing travel distance due to diffraction and scattering. The basic mechanism of the broadening has been studied on the basis of the scattering theory with the parabolic approximation for the scalar wave equation in random media. However, conventional models are not realistic enough since the plane wave modeling is too simple and the Gaussian autocorrelation function (ACF) is far from the reality to represent the inhomogeneity in the Earth. Focusing on the early part of envelopes, we formulated the envelope broadening of spherically outgoing scalar waves in three-dimensional von Kármán-type random media, of which the spectra decay according to a power law at large wave numbers. Random media are characterized by three parameters: RMS fractional velocity fluctuation ε, correlation distance a, and order κ that controls the gradient of the power law spectra. This model predicts that the envelope duration increases with both travel distance and frequency when short-wavelength components are rich in random media, while the duration is independent of frequency when short-wavelength components are poor. Introducing phenomenological attenuation Q, we developed a method for estimating the parameters of inhomogeneity and attenuation from the envelope duration. Applying this method to S wave seismogram envelopes for the frequency range from 2 to 32 Hz in northeastern Honshu Japan, we estimated the random inhomogeneity parameters as κ = 0.6, ε2.2a−1 ≈ 10−3.6 [km−1] and f/Q = 0.0095 [s−1], where f is frequency. The power law portion of the estimated power spectral density function is P(m) ≈ 0.01 m−4.2 [km3], where m is wave number.