Characterisation and extraction of a Rayleigh-wave mode in vertically heterogeneous media using quaternion SVD

Abstract

We propose a method that identifies a mode of Rayleigh waves and separates it from body waves and from other modes, using quaternions to represent multi-component data. Being well known the abilities of quaternions to handle rotations in space, we use previous results derived from Le Bihan and Mars (2004) to prove that a Rayleigh-wave mode recorded by an array of vector-sensors can be approximated by a sum of trace-by-trace rotating time signals. Our method decomposes the signal into narrow-frequency bands, which undergo both a velocity correction and a polarisation correction. The aim of these corrections is to reduce the mode of interest to a quasi-monochromatic wave packet with infinite apparent velocity and quasi-circular polarisation. Once written in quaternion notation, we refer to this wave packet as quaternion brick. Based on theoretical considerations, we prove that this quaternion brick maps into the first quaternion eigenimage of the quaternion singular value decomposition. We apply this method to synthetic datasets derived from two vertically heterogeneous models to extract the fundamental mode and we prove that it is correctly separated from either a higher mode of propagation or body waves with negligible residual. Results are presented in both time-offset and frequency-phase slowness domains

Quaternions in Signal and Image Processing: A comprehensive and objective overview

“…The Q-SVD plays a crucial role in the definition of any subspace methods over quaternions. In vectorsensor array processing, for example, Q-SVD allows to separate quaternion data into signal and noise subspaces and to take advantage of vector-sensors measurements to perform source localization [25]- [28].…”

Transdimensional and Hamiltonian Monte Carlo inversions of Rayleigh‐wave dispersion curves: a comparison on synthetic datasets

“…In the first case, the observed data (dispersion curves) are semi‐analytically derived from the true model, while in the seismic tests, the dispersion curves are extracted from seismic shot gathers computed on the reference models by means of the reflectivity algorithm (Mallick and Fraser, 1987). In all the following experiments, we limit to consider the fundamental mode as the observed data, although it is known that higher modes are essentially to better constraints the solution in case of shear velocity inversions and/or high stiffness contrasts within the soil column (Feng et al ., 2005; Luo et al ., 2009; Cercato, 2011; Farrugia et al ., 2016; Sajeva and Menanno, 2017; Qiu et al ., 2019). We return to this aspect in more detail in the discussion section.…”

Widely linear denoising of multicomponent seismic data

“…Thanks to the recent developments in quaternion-based signal processing, one can now process vector-valued data following an approach that is not only physically consistent but also offers the possibility of making the most of the additional useful information acquired by multicomponent sensors. In applied seismology, for instance, examples of vector-signal processing through the noncommutative algebra of quaternions include velocity analysis (Grandi, Mazzotti and Stucchi 2007), deconvolution (Menanno and Mazzotti 2012), reconstruction (Stanton and Sacchi 2013) and wavefield separation (Sajeva and Menanno 2017). Other applications of quaternions in the geophysical literature include time-lapse analysis (Witten and Shragge 2006) and anisotropic inversion of cross-dipole sonic logs (Zeng, Yue and Li 2018).…”