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

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) [1] 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.