The effect of scattering in surface wave tomography

Abstract

Summary We present a new technique in surface wave tomography that takes the finite frequency of surface waves into account using first‐order scattering theory in a SNREI Earth. Physically, propagating surface waves with a finite frequency are diffracted by heterogeneity distributed on a sphere and then interfere at the receiver position. Paradoxically, surface waves have the largest sensitivity to velocity anomalies off the path of the geometrical ray. The non‐ray geometrical effect is increasingly important for increasing period and distance. Therefore, it is expected that the violation of ray theory in surface wave tomography is most significant for the longest periods. We applied scattering theory to phaseshift measurements of Love waves between periods of 40 and 150 s to obtain global phase velocity maps expanded in spherical harmonics to angular degree and order 40. These models obtained with scattering theory were compared with those constructed with ray theory. We observed that ray theory and scattering theory predict the same structure in the phase velocity maps to degree and order 25–30 for Love waves at 40 s and to degree and order 12–15 for Love waves at 150 s. For reasons of spectral leakage, a smoothness condition was included in the phaseshift inversions to construct the phase velocity maps, so we could not access the small length‐scale structure in the obtained Earth models. We carried out a synthetic experiment for phase velocity measurements to investigate the limits of classical ray theory in surface wave tomography. In the synthetic experiment, we computed, using the source–receiver paths of our surface wave data set, the discrepancy between ray theoretical and scattering theoretical phase velocity measurements for an input model with slowness heterogeneity for increasing angular degree. We found that classical ray theory in global surface wave tomography is only applicable for structures with angular degrees smaller than 25 (equivalent to 1600 km) and 15 (equivalent to 2700 km) for Love waves at 40 and 150 s, respectively. The synthetic experiment suggests that the ray theoretical great circle approximation is appropriate to use in present‐day global surface wave tomography. On the other hand, in order to obtain reliable models with a higher resolution we must take the non‐ray geometrical effect of surface waves into account.

Geometrical approach to surface wave finite frequency effects

“…In the case of surface waves, this occurs mainly because a kernel has a p/4 phase shift [Spetzler et al, 2002]. Our results actually agree with it as equation (6) shows explicitly.…”

“…[4] Most derivations of the Fresnel zone for surface waves were based on scattering theory [e.g., Spetzler et al, 2002;Snieder and Nolet, 1987] or on extended ray theory [Yoshizawa and Kennett, 2002]. Instead, we start from a formula in the normal mode theory, derived by Woodhouse and Girnius [1982]:…”

Development of Surface Wave Dispersion and Attenuation Maps and Improved Methods for Measuring Surface Waves

“…Yoshizawa and Kennett (2004) and Kennett and Yoshizawa (2002) use a similar technique with a narrower kernel that they believe to be more representative of realistic surface waves. Spetzler et al (2001Spetzler et al ( , 2002 discuss the implications of scattering and diffraction for surface wave tomography. Friedrich et al (1993), Friedrich (1999 and Maupin (2001) extend the Born approximation to incorporate multiple scattering.…”

The state of the upper mantle beneath southern Africa

“…We choose L corr = 400 km, thus favoring a smooth model considering our ray density and shortest wavelength used (about 200 km at 50 s period). However, although using shorter paths allows better resolution compared to global tomography, theoretical limitations resulting from the great circle approximation suggest that finescale structures should be interpreted cautiously (Spetzler et al, 2002).…”