An analytical treatment of single station triaxial seismic direction finding

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 a 3 × 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 terms of the triaxial signal amplitudes and phases.