Vector-valued signals are crucial in science and engineering. The evolving field of hypercomplex signal processing, particularly quaternion algebra, offers a concise and natural approach to handling vectorial data. In multicomponent seismology, for instance, vector-valued signal processing finds a natural fit that has been exploited in several applications. This article provides a concise and practical review of quaternionic methods for handling vector-valued seismic datasets, from historical origins to key concepts and tools in the field of quaternion signal processing, such as the quaternion Fourier transform and quaternion singular value decomposition (SVD). While highlighting existing results, this review also showcases novel developments through source separation applications with quaternions, discussing encountered challenges and outlining potential future trends.

Vector-valued signals are crucial in science and engineering. The evolving field of hypercomplex signal processing, particularly quaternion algebra, offers a concise and natural approach to handling vectorial data. In multicomponent seismology, for instance, vector-valued signal processing finds a natural fit that has been exploited in several applications. This article provides a concise and practical review of quaternionic methods for handling vector-valued seismic datasets, from historical origins to key concepts and tools in the field of quaternion signal processing, such as the quaternion Fourier transform and quaternion singular value decomposition (SVD). While highlighting existing results, this review also showcases novel developments through source separation applications with quaternions, discussing encountered challenges and outlining potential future trends.