Modal Identification and Extraction of Rayleigh Waves by Means of Quaternion SVD and Group Velocity Curves

We present a method which employs quaternion algebra to extract Rayleigh Waves (RWs). Using quaternions we compactly represent multi-component data, and, at the same time, we are able to handle the vectorial nature of RWs. This method extracts a single RW mode per time. The mode is identified by its group velocity curve, which must be previously measured. Quaternion SVD (QSVD), which is an extension of SVD to quaternions, is applied to the quaternion data matrix on windows with slopes equal to the group velocities of the mode. In order to honor the frequency dependence of group velocities, a frequency dependent extraction is performed. After QSVD analysis, only the first q-autoimage is taken. Dispersive phenomena, which are present in RWs, can be represented as trace by trace phase shifts in the quaternion domain. Furthermore, properly manipulating the amplitudes of the spatial components, it is possible to force the polarization of the RWs mode from elliptical to circular and then to use QSVD to compactly describe it. In this paper, after a very brief introduction on the basics of quaternion algebra, we describe the quaternion extraction method and we discuss the results of its application to synthetic and real datasets.