Microtremor surveys based on rotational seismology: theoretical analysis with focus on separation of Rayleigh and Love waves in general wavefield of microtremors

Abstract

SUMMARY The applicability of rotational seismology to the general wavefield of microtremors is theoretically demonstrated based on a random process model of a 2-D wavefield. We show the effectiveness of taking the rotations (i.e. spatial differentiation) of microtremor waveforms in separating the Rayleigh and Love waves in a wavefield where waves are simultaneously arriving from various directions with different intensities. This means that a method based on rotational seismology (a rotational method) is capable of separating Rayleigh and Love waves without adopting a specific array geometry or imposing a specific assumption on the microtremor wavefield. This is an important feature of a rotational method because the spatial autocorrelation (SPAC) method, a conventional approach for determining phase velocities in microtremor array surveys, requires either the use of a circular array or the assumption of an isotropic wavefield (i.e. azimuthal averaging of correlations is required). Derivatives of the SPAC method additionally require the assumption that Rayleigh and Love waves are uncorrelated. We also show that it is possible to apply a rotational method to determine the characteristics of Love waves based on a simple three-point microtremor array that consists of translational (i.e. ordinary) three-component sensors. In later sections, we assume realistic data processing for microtremor arrays with translational sensors to construct a theoretical model to evaluate the effects of approximating spatial differentiation via finite differencing (i.e. array-derived rotation, ADR) and the effects of incoherent noise on analysis results. Using this model, it is shown that in a short-wavelength range compared to the distance for finite differencing (e.g. $\lambda < 3h$, where $\lambda $ and $h$ are the wavelength and distance for finite differencing, respectively), the leakage of unwanted wave components can determine the analysis limit. It is also shown that in a long-wavelength range (e.g. $\lambda > 3h$), the signal intensity gradually decreases, and thus the effects of incoherent noise increase (i.e. the signal-to-noise ratio, SNR decreases) and determine the analysis limit. We derive the relation between the SNR and wavelength. Although the analysis results quantitatively depend on the array geometry used for finite differencing, the qualitative understanding supported by mathematical expressions with a physically clear meaning can serve as a guideline for the treatment of data obtained from ADR.