## Abstract

Triaxial seismic direction finding can be performed by eigenanalysis of the complex coherency matrix (or cross power matrix). By splitting the symmetric Hermitian coherency matrix C to D + E (where det(E) = 0 and D is diagonal), we shift unpolarized (or inter-channel uncorrelated) data into D and then E becomes 'random noise free'. Without placing any restrictions on the signal set - P, S, Rayleigh - matrix E has only one non-zero eigenvalue (at least for the case of a single mode arriving from a single direction). But for real data (polychromatic transients with correlated noise), it will have two non-zero eigenvalues. By rotating one axis of the triaxial geophone recorded signals to lie normal to the principal eigenvector, it is possible to reduce the coherency matrix from a3 × 3 to a 2 × 2 matrix. For the case of a perfectly polarized monochromatic signal, we interpret this to mean that the particle trajectory can only be elliptical. It seems as though particles can only move in a plane: they cannot move in three dimensions. In practice, the signal is made up of a band of frequencies, there are multiple arrivals in the time window of interest, and noise is invariably present, which causes the ellipse to wobble in a 3D orbit. Explicit analytical expressions are derived in this paper to yield the eigenvalues and eigenvectors of the coherency matrix in tenus of the triaxial signal amplitudes and phases.

Original language | British English |
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Pages (from-to) | 8-15 |

Number of pages | 8 |

Journal | Journal of Geophysics and Engineering |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2005 |

## Keywords

- Coherency matrix
- Direction finding
- Eigenanalysis
- Multi-component seismology