We have developed a new method to determine phase velocities from the vertical component of microseisms recorded with an array of seismic sensors spaced around the circumference of a circle. We calculate two different time histories by taking the average of the seismograms with differing sets of weights for the sensor stations. The spectral ratio of these two time histories contains no information on the arrival directions or on the amplitudes of the incoming waves but depends solely on the phase velocities of the arriving modes. Theoretical considerations indicate that the effects of directional aliasing caused by the use of a finite number of sensors in the field implementation of our method are small in most situations except for short wavelengths. The presence of incoherent noise limits the efficacy of our method for long wavelengths. In field tests using arrays of three seismic sensors, we obtained appropriate estimates of phase velocities in the wavelength range from 5r to 30r where r , the array radius, was on the order of a few meters.

S U M M A R YWe present a generic formulation for analysis methods that estimate phase velocities of Rayleigh and Love waves, by way of intermediary quantities called 'spectral ratios', using three-component records of microtremors from a circular array of sensors. At each time instant, the set of records are expanded in a Fourier series with respect to azimuth, so that we obtain a set of Fourier coefficients that are represented in the form of complex time histories. We then estimate power-and cross-spectral densities of those Fourier coefficients. The spectral densities, thus obtained, generally contain information on the phase velocities, powers and arrival directions of individual modes of Rayleigh and Love waves. By taking the quotient of two different sorts of such spectral densities, we can cancel out information on their powers and arrival directions, and extract information on their phase velocities alone. The spectral ratios have to be estimated, in practice, on the basis of records from a finite number of seismic sensors that are either evenly or unevenly spaced around a circumference. We describe a general procedure for their estimation, and discuss the effects of directional aliasing that the finite number of sensors and their configuration have on the estimates of spectral ratios. We also discuss biases in the estimates of spectral ratios caused by the presence of incoherent noise. By using our method, it is also possible to estimate the central arrival direction of the microtremors, the ellipticity of the Rayleigh waves, and whether the Rayleigh waves are prograde or retrograde.

[1] The centerless circular array (CCA) method, proposed by ourselves in an earlier work, is an algorithm of microtremor exploration which can be used to estimate phase velocities of Rayleigh waves by analyzing vertical component records of microtremors that are obtained with an array of three or five seismic sensors placed around a circumference. We have confirmed, through field tests, the applicability of our CCA method to arrays on the order of several to several hundred meters in radii and have revealed its remarkably high performance in long-wavelength ranges, the upper resolution limit extending as far as several 10 times the array radius. We have also invented a mathematical model that enables to evaluate signal-to-noise ratios in a given microtremor field. Scrutiny of field data has borne out our hypothesis that noise is the principal factor that biases the analysis results of the CCA method in long-wavelength ranges and that its longest resolvable wavelength is determined by the signal-to-noise ratio. Combined use of the CCA method and our new method of signal-to-noise ratio analysis provides a powerful methodological tool that allows one to extract maximal information from microtremor records obtained with a simple seismic array.Citation: Cho, I., T. Tada, and Y. Shinozaki (2006), Centerless circular array method: Inferring phase velocities of Rayleigh waves in broad wavelength ranges using microtremor records,