Separation of a single mode of Rayleigh waves using quaternion SVD in vertically-heterogeneous media

I propose a wave-separation method that separates a single Rayleigh-wave mode from body waves and from other modes, using quaternions to represent the multi-component seismic data recorded by an array of vector-sensors. This method decomposes the signal into narrow-frequency bands, which are subjected to 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, I refer to this wave packet as “quaternion brick”. It can be proved that this quaternion brick maps into the first quaternion eigenimage of the quaternion SVD (QSVD). I apply this method to vertically-heterogeneous elastic models. The method seems to correctly extract the selected Rayleigh-wave mode and to separate it from body waves. The separation between different modes is more challenging. It results that, when two modes interfere in the near-offset portion, a more accurate separation can be obtained by characterising the mode of interest in the far-offset portion and then extrapolating it in the near-offset portion (where the two modes overlap) using the properties of the right singular vector of the QSVD.