The SPAC Method for a Finite M-Station Circular Array Using Horizontally Polarized Ambient Noise

Abstract

The spatial autocorrelation (SPAC) method is a useful tool for interpreting ambient noise, and several theoretical tools based on this method are available for data collection, processing, and analysis. The method uses the three components of seismic microtremors recorded simultaneously with arrays of different configurations and yields dispersion curves of Rayleigh and Love waves. Existing theories of the SPAC method are based on an idealized circular array with an infinite number of stations (the continuous model). Errors due to the use of a finite number of stations have been derived theoretically for vertically polarized ambient noise. This paper presents an extension of that error theory to horizontally polarized ambient noise so that it is valid for a circular array with M uniformly spaced stations. The quantities that are obtained as the output of the SPAC method, however, depend on the ratio of the power spectral densities (PSDs) of Rayleigh and Love waves, which are difficult to identify beforehand. Instead we introduce a quantity, κ , that does not depend on the PSD ratio and therefore is more convenient for separating the properties of Rayleigh and Love waves. We derive an expression of κ for a finite, M-station circular array ( κ M) and compare it with what it should be in the continuous model ( κ T ). We illustrate κ M and κ T using field data from a site in Tehran and discuss their discrepancies. Finally, we demonstrate how we have estimated Love-wave phase velocities using the calculated κ M.