Assessing the applicability of the spatial autocorrelation method: A theoretical approach

Abstract

[1] We present a rigorous theoretical framework that allows one to assess the range of applicability of the spatial autocorrelation (SPAC) method, a technique of microtremor exploration that is widely used to infer phase velocities of Rayleigh waves using vertical-motion records from a circular array of seismic sensors. The magnitude of systematic errors (biases) that depend on the number of seismic sensors deployed around the circle, and the magnitude of systematic errors that arise from the presence of incoherent noise, are both evaluated analytically, and their general properties are discussed. The relationship between the magnitude of stochastic errors, inherent in the analysis results, and the duration of measurement (or to put it more accurately, the data's degree of freedom) is also elucidated. The validity of our theory is corroborated by checks against the results of both real data analysis and numerical experiments, and an example is given of how the theory can be adapted to account for practical situations encountered in the field. Discussions on the range of applicability of the SPAC method, which have heretofore often fallen back on empirical observations, have now obtained a theoretical ground on which to stand, providing a basis for strategies to make maximal use of the SPAC method's capabilities.Citation: Cho, I., T. Tada, and Y. Shinozaki (2008), Assessing the applicability of the spatial autocorrelation method: A theoretical approach,

Miniature array analysis of microtremors

“…Although the FK method may appear to be more flexible, owing to the arbitrary array shape, in reality this method requires a considerable number of seismometers (e.g., Capon, 1969;Asten and Henstridge, 1984). The "circular array" used in the SPAC method, on the other hand, can be satisfactorily constructed by arranging a single seismometer at the center of a circle and three seismometers on the circumference, to form a triangular shape (Figure 1a; this array is referred to as a "four-point array" in this paper) (Okada, 2003;Asten, 2006;Cho et al, 2008a). Furthermore, the SPAC method can also be applied for wavelength ranges much wider than those that can be handled using the FK method (e.g., Okada, 2003).…”

Statistical Validity Control on SPAC Microtremor Observations Recorded with a Restricted Number of Sensors

“…Asten (2003Asten ( , 2006a numerically analyzed the expected behavior of the SPAC spectrum with different array configurations and azimuth samplings. Cho et al (2004Cho et al ( , 2006Cho et al ( , 2008 further evaluated the maximum expected errors on SPAC spectra with respect to frequency and number of sensors using different wave-field scenarios. Okada (2006) completed a similar study, evaluating the variations of an error term with frequency and number of sensors.…”

The Effect of Lateral Heterogeneity on Horizontal-to-Vertical Spectral Ratio of Microtremors Inferred from Observation and Synthetics

“…As shown in this paper, the isotropic amplitude distribution of ground motion would not be realized for the case of a medium with distinctive geometrical features, even under the strong scattering situation or random wave incidences. There are also circumstantial evidences of the wavefield being diffuse such as the studies of spatial autocorrelation techniques to retrieve apparent velocity of surface waves from array measurements of microtremors (e.g., Aki, 1957;Cho et al, 2008). The recent great success of the Green's function retrieval from cross correlation of ambient seismic noise also supports the reality of the diffuse field concept (e.g., Shapiro and Campillo, 2004;Campillo, 2006;Stehly et al, 2008;Prieto and Beroza, 2008).…”