Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

Abstract

Observations of seismic waves from earthquakes at depths between 100 and 200 km beneath Japan show that the initial portion of the S-wave arrival has greater duration than can be accounted for by earthquake source-time duration. The observed long duration of S waves has been explained as being caused by multiple forward scattering around the ray path between source and receiver. Array observations of Lg waveforms have also shown that multiple forward scattering along the path between the source and receiver are important influences on the character of Lg waveforms and that the scattering cannot be explained only by vertical variations in velocity. Multiple forward scattering in 3D has been modeled using the Markov approximation for the parabolic-wave equation, which allows the calculation of seismogram envelopes in statistically characterized random media. To test the range of validity of the Markov approximation-derived solutions, we made 2D numerical calculations of wavefields in random media using approximations to the parabolic wave equation, which only models forward scattering, and by finite-difference solution of the scalar-wave equation, which gives complete wavefields. Media of background velocity 4 km/sec are characterized using a Gaussian autocorrelation function with a 5 km correlation distance and 5% rms fractional fluctuation. To compare with envelopes obtained from the Markov approximation, we calculated wavefields for source-receiver distances ranging from 50 to 200 km for several statistically identical realizations of random media. We obtain ensemble average envelopes by averaging envelopes from the realizations. We find a good agreement between ensembleaverage envelopes obtained from the numerical solution of the parabolic-wave equation and envelopes obtained from the Markov approximation. The later portion of the ensemble-average envelopes calculated using finite difference have larger amplitudes than those from the Markov approximation, which is probably due to the late-arriving energy that has been scattered at wide angles from the global propagation direction between the source and the receiver. We observe that the variations among the numerically calculated envelopes of individual realizations of random media are well fit by a Rayleigh distribution, which describes the distribution of envelope amplitude when signals of one frequency but random phase are summed. Our results show that the Markov approximation provides reliable information about envelope shapes for forward-scattered wavefields but that the influence of wide-angle scattering and backscattering have some influences on envelope shapes and should be considered when analyzing data using random media models.

Envelope syntheses of cylindrical vector-waves in 2-D random elastic media based on the Markov approximation

“…theoretically solved the envelope broadening of spherically outgoing waves in von Kármán-type 3-D random media that have a realistic power-law spectrum at large wavenumbers. Fehler et al (2000) confirmed the validity of the Markov approximation for scalar waves from a comparison with finite difference simulations in 2-D random media. Wegler et al (2006) numerically examined the applicable range of the Markov approximation, and the radiative transfer solution with Born scattering amplitudes and that with isotropic scattering for scalar waves taking the finite difference simulation envelope as a reference in 2-D random media.…”

“…According to Fehler et al (2000), we first define the TFMCF as the correlation of field U between two locations r θ and r θ within a small distance from the global ray direction θ =0 on the transverse line at distance r at different angular frequencies ω and ω , 2 (r, θ , θ , ω , ω where…”

“…which is the ISD of a cylindrical scalar wave as derived in Fehler et al (2000). For the derivation of the last line in Eq.…”

High Resolution/High Fidelity Seismic Imaging and Parameter Estimation for Geological Structure and Material Characterization

“…Sato, Fehler, and Wu, 2002;Pride et al, 2003). We have used numerical modeling of wave propagation through random media along with various analytical approaches to better understand what characteristics of the medium can be inferred from analysis of seismic traces (Fehler, Sato, and Huang, 2000;Saito et al 2003;Sat0 et al, 2003). We have recently been investigating an approach for using AVO analysis to characterize the spatial heterogeneity in a layer.…”

Advanced Wave-Equation Migration

“…We have also used some of these methods for modeling primary forward wave propagation in random media Huang and Fehler, 1998b;Fehler et al, 2000). In this paper, we present an overview of these dual-domain migration methods.…”