  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.
