VANDERMONDE MATRIX ANALYSIS OF LONG-PERIOD SEISMOGRAMS

Abstract

Seismograms recorded by the Global Seismic Network can be processed to obtain receiver strips for each earthquake. The receiver strips for all earthquakes can be analysed simultaneously in the time domain by using the representation in terms of exponentially decaying sinusoids (signal poles). The (time, strip) matrix is the product of two matrices, a Vandermonde matrix of signal poles and a matrix of excitation amplitudes. The shift-invariant property of Vandermonde matrices can be used to determine the signal poles (and hence the complex eigenfrequencies) and, subsequently, the excitation amplitudes. The latter provide the source vectors for each multiplet and earthquake. They can be combined with the eigenfrequencies and amplitudes to form a system of linear equations for the coupling-splitting matrices that constrain 3-D structure. In some cases multiplets whose eigenfunctions are insensible in the upper mantle are quasi-resonantly coupled to well-observed multiplets. The Vandermonde matrix analysis can often be extended to these 'latent' multiplets and their coupling-splitting matrices can be determined. The source vectors for each multiplet and for each earthquake are constraints on the source mechanism of the earthquake. Thus, the investigation of the earthquake mechanism can be performed independently of the investigations of aspherical structure. This separation is one of the strengths of Vandermonde matrix analysis.