Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based Method

The passage of seismic waves through highly heterogeneous media leads to significant scattering of seismic energy and an apparent attenuation of seismic signals emerging from the heterogeneous zone. The size of this scattering attenuation depends on the correlation properties of the medium, the rates of Pand S-wave velocities, and frequency content of the incident waves. An estimate of the effect can be obtained using single scattering theory (first-order Born approximation) for path deviations beyond a minimum scattering angle; smaller deviations require consideration of multiple scattering or a representation in terms of travel-time perturbations. Although an acoustic treatment provides a quantitative reference, full elastic effects need to be taken into consideration to get an accurate attenuation rates. The use of a wavelet-based modeling technique, which is accurate and stable even in highly perturbed media, allows an assessment of the properties of different classes of stochastic media (Gaussian, exponential, von Karman). The minimum scattering angle for these stochastic media is in the range of 60 to 90 . The wavelet-based method provides a good representation of the scattered coda, and it appears that methods such as finite differences may overestimate scattering attenuation when the level of the heterogeneity is high. Introduction One of the most important topics in regional seismic studies is the influence of scattering due to material inhomogeneities and anisotropy in the crust and the upper mantle (Nolet et al., 1994, Wu et al., 1994). Scattering processes modify both the travel times and amplitudes of seismic waves. A full representation of scattering phenomena requires consideration of multiple scattering effects, which are difficult to handle. In consequence, attention has focused on single scattering implemented via a first-order Born approximation for weakly heterogeneous regions (Wu, 1982; Frankel and Clayton, 1986). The single scattering theory is applied mainly to backscattered and side-scattered energy, and the more complex effects in forward scattering are taken care of by including a correction for the induced travel-time shift inside a certain angular range around the propagation direction. The separation between the two different approximation regimes is made at the “minimum (or, cutoff) scattering angle” (Roth and Korn, 1993; Sato and Fehler, 1998; Kawahara, 2002). Estimates of this minimum scattering angle have been made using numerical modeling of stochastic media in an acoustic approximation or with a full elastic treatment (Frankel and Clayton, 1986; Jannaud et al., 1991; Roth and Korn, 1993; Frenje and Juhlin, 2000). Alternatively, estimates of the minimum scattering angle have been made theoretically for random acoustic media (Sato, 1984; Kawahara, 2002). However, there is still some uncertainty as to the appropriate minimum scattering angle for elastic waves because much of the work has been undertaken in the acoustic approximation (Roth and Korn, 1993) or with a scalar wave approach, even for elastic wave studies (Frankel and Clayton, 1986). The scattering pattern of elastic waves is complex and is significantly different from that of scalar waves (Wu and Aki, 1985) due to the inherent characteristics of elastic waves such as wave-type coupling, the radiation patterns in scattering, and complex interferences between the waves. As a result, numerical modeling for elastic waves needs to be compared with theoretical results for a full understanding of the influence of elastic wave scattering. The minimum scattering angle, as one of the key factors in single scattering theory, thus needs to be determined properly and the relation to the acoustic theory explored. Single scattering theory for 3D elastic waves has been developed in several studies. Wu and Aki (1985) compared theoretical scattering coefficients based on the Born approximation with results derived from observations and tried to reveal the characteristics of heterogeneities in the lithosphere. Wu (1989) introduced the “perturbation method” for the scattering of elastic waves in random media, which considers the scattering waves as the response of the perturbations to the incident waves in a sense of a radiation problem. Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based Method 923 Sato and Fehler (1998) followed a similar approach but considered an additional important factor, a travel-time correction applied to the Born approximation, to determine the correct energy loss during scattering. They associated the travel-time shift by the fractional-velocity fluctuation due to the long wavelength component of scattered waves, that is, waves with wavelength more than twice that of the dominant frequency. This approach has been used to determine the minimum scattering angle to be employed in the estimation of scattering attenuation of elastic waves in 3D. It is therefore important to check that the theoretical estimates of the minimum scattering angle match those determined empirically. Although Sato and Fehler’s minimum scattering angle is supported by some numerical studies (Roth and Korn, 1993) for the scalar-wave cases, it has not been fully checked for elastic waves. The numerical studies of elastic waves (Frankel and Clayton, 1986) used the theoretical attenuation curve for scalar waves as the reference curve for determining the minimum scattering angle. However, since numerical modeling for 3D elastic wave propagation still requires considerable computational expense to achieve an adequate domain for the assessment of the scattered energy, we confine our study to 2D elastic waves. For 2D elastic waves, hybrid methods have been used. Fang and Müller (1996) attempted to formulate the governing equation in a rational form by incorporating two formulae for scalar waves with both velocity perturbation (Frankel and Clayton, 1986) and density perturbation (Roth and Korn, 1993). The coefficients of each term in the rational form need to be determined for each stochastic medium by curve fitting to the results from numerical experiments. This approach of Fang and Müller is based on the fundamental assumption that the scattering attenuation pattern of elastic waves is similar to that of scalar waves for the given stochastic medium (e.g., exponential media for Fang and Müller’s study) and that the minimum scattering angle (hmin) would be the same (20 ) for both acoustic and elastic waves. To avoid such assumptions, it is important to develop a fully elastic 2D theory for the variation of scattering attenuation as a function of normalized wavenumber for 2D elastic waves to compare with numerical results, and thereby determine the minimum scattering angle. It is very important that we have not only a correct derivation and implementation of scattering theory for comparisons with numerical results, but also that high-accuracy numerical modeling is available for assessing the value of the minimum scattering angle. The finite difference method (FDM) with fourth-order accuracy in spatial differentiation has been used widely for modeling in random heterogeneous media due to the convenience in treatment of numerical models and simplicity in implementation (Frankel and Clayton, 1986; Jannaud et al., 1991; Roth and Korn, 1993; Fang and Müller, 1996; Frenje and Juhlin, 2000; Fehler et al., 2000). However, Sato and Fehler (1998) have pointed out that derivatives in a FDM scheme are computed in the sense of an average over some grid points in a domain. Therefore, it is still an open question as to whether the fourth-order accuracy in spatial differentiation is sufficient for stable and accurate modeling in random heterogeneous media. High accuracy in spatial differentiation can be achieved with the pseudospectral method, and this approach has been applied in seismic wavefield computation for laterally heterogeneous models on upper mantle and global scales (Furumura et al., 1999). However, it is difficult to achieve a comparable level of accuracy in the representation of the free-surface condition of vanishing traction. Yomogida and Benites (1995) have applied the boundary integral method for modeling media with randomly distributed cavities. Such boundary integral methods can deal well with heterogeneities inside a medium with irregular interfaces (e.g., cavities, cracks). The boundary conditions are satisfied by including effective sources at the boundaries at each time step. For a homogeneous medium, it is possible to get an accurate time response because the necessary Green’s functions can be found analytically. However, it is difficult for the method to be applied to media with heterogeneous backgrounds (including layered media) because the Green’s functions themselves need to be found numerically. Recently, the generalized screen propagators (GSP) method has been developed as a fast computational procedure for modeling of elastic wave propagation in half spaces with small-scale heterogeneities (Wu et al., 2000). However, the approach used in the GSP method ignores the backscattering process and so is not suitable for full representation of scattered waves. In this study, we use a wavelet-based method (WBM: Hong and Kennett, 2002a,b) as an accurate and stable simulator of elastic wave propagation in random media. The accuracy and the stability of the method is addressed by comparisons with the FDM. The WBM is then applied to calculate synthetic seismograms for several styles of stochastic media, from which the scattering attenuation is measured. The nature of the scattering needs to be taken into account to get accurate estimates of the attenuation, since in large-scale heterogeneity, significant deviations in the primary wave field mean that both components of motion need to be considered for a 2D medium. With accurate modeling we are able to place constraints on the minimum scattering angle for 2D elastic waves to the span of 60 –90 . Derivation of from Single Scattering Theory 1 Qs We estimate the scattering attenuation factors ( ) as 1 Qs a function of normalized wavenumber (ka) based on single scattering theory in 2D random heterogeneous media, where k is the wavenumber of incident waves and a is the correlation distance. We represent the wavefield (uj, j x, z) as composed of primary waves ( , j x, z) and scattered waves ( , 0 s u u j j j x, z). The primary waves in 2D elastic media satisfy the relationships 924 T.-K. Hong and B. L. N. Kennett Primary Incident Waves * Receiver