Article,
An analytical treatment of single station triaxial seismic direction finding
Journal of Geophysics and Engineering, 2 (1): 8--15 (March 2005)
DOI: 10.1088/1742-2132/2/1/002
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 x 3 to a 2 x 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.
