A generic formulation for microtremor exploration methods using three-component records from a circular array

We 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.