Multiwavelet spectral and polarization analyses of seismic records

SUMMARY We present an algorithm, based on the wavelet transform and multiple taper spectral analysis, for providing a low‐variance spectrum estimate of a non‐stationary data process. The‘multiwavelet’algorithm uses, within each frequency band, a number of mutually orthogonal Slepian wavelets, optimally concentrated in frequency. The sum of squared wavelet transforms with the Slepian wavelets results in a spectrum estimate that is both low‐variance and resistant to broad‐band bias. The multiwavelet algorithm is used to estimate the time‐varying spectral density matrix S( f, t ) for two or more time series, in particular for three‐component seismic data. Coherent three‐component motion is described by motion along a single trajectory, with appropriate projections onto the three component axes. This trajectory is found by applying a singular value decomposition (SVD) to a matrix M( f, t ) of wavelet transform values. The normalized first singular value of the SVD determines whether a correlation among the three components of the seismogram is statistically significant. Where significant, coherent particle motion is reconstructed by a linear combination of the wavelets with coefficients specified by the first left‐singular vector. The polarization of this motion with respect to the coordinate axes is given by the first right‐singular vector. Where the wavelets are real‐valued, the usefulness of this method is limited to cases in which the three components of the seismic record oscillate in phase with each other, as is often the case for seismic body waves. Elliptical polarization is handled by pairing even and odd Slepian wavelets into complex‐valued wavelets, capable of detecting phase shifts between components. We demonstrate the mutiwavelet spectrum and polarization estimators on seismic data from a large shallow earthquake in the Solomon Islands, and from the recent deep earthquakes beneath Fiji (1994 March 9) and Bolivia (1994 June 9).