Incoherent noise-induced distortions of Rayleigh wave ellipticity measurements obtained with three-component beamforming

Abstract

SUMMARY For site characterization, the elliptic particle motion of Rayleigh waves and its frequency dependence is a well-known property that aroused less interest than the frequency dependence of the phase velocity. More than 50 yr ago, ellipticity was already recognized as providing information independent from phase velocity, despite the difficulties inherent to its accurate and precise measurement. Several techniques were developed during the last two decades to extract the ellipticity curve from ambient vibration recordings, with a single three-component (3C) station, with pairs of 3C stations and more recently with 3C arrays. The latter has the advantage over the other approaches that the sign of the ellipticity can be retrieved. Moreover, higher order mode separation is possible under certain conditions. Nevertheless, Rayleigh Three-component BeamForming (RTBF) proposed by Wathelet et al. encounters difficulties in the presence of significant levels of incoherent noise when the true ellipticity is vanishing or when it has a high absolute value. In this work, the analytical expressions of the beam power for a single source wavefield are revised under more realistic assumptions for the incoherent noise azimuthal distribution. The proposed model also includes an asymmetric distribution of the incoherent noise between vertical and horizontal components, which was not the case in the original publication. Switching from ellipticity to angular ellipticity drastically simplifies the formalism. Moreover, it naturally leads to a new steering matrix (all-component ellipticity steering) which solves the limitation around zero and infinity observed for RTBF. Interestingly, the accuracy of the ellipticity is no longer influenced by the absolute level of incoherent noise but by the difference between the incoherent noise ratio on vertical and horizontal components. A method based on the second derivative of the beam power versus the radial wavenumber is finally proposed to experimentally measure the noise ratio difference, which allows experimental values to be corrected. The methodology is compared with classical vertical beamforming and RTBF for a synthetic case and three experimental data sets.