Quaternionic rank-reduction methods for vector-field seismic data processing

Abstract

The quaternion signal model can naturally represent vector-field data in a physically consistent approach, plausibly preserving and exploiting its vector relationships. The Singular Spectrum Analysis (SSA) filter can be extended to the case of multicomponent seismic data via the non-commutative quaternion algebra. Adoption of quaternions requires a shift from real or complex signal processing to quaternion-based signal processing. In this paper, the quaternion-defined seismic data is transformed to the frequency–space (f−x) domain via the quaternion Fourier transform (QFT), and rank-reduction can be performed via the quaternion SVD (QSVD) following the SSA methodology. Efficient rank-reduction alternatives, such as Lanczos bidiagonalization via convolutions, are used to allow significant gains in processing time. Simultaneous denoising and reconstruction of synthetic multicomponent seismic records are used to illustrate and compare the quaternionic versions of the SSA against its scalar version.