On bias and noise in passive seismic data from finite circular array data processed using SPAC methods

The finite nature of typical small seismic arrays used in conjunction with spatial autocorrelation SPAC processing for observing the microtremor wavefield causes predictable perturbationsoftheSPACspectrumwhensourcesofseismic noiseareconfinedtoarestrictedrangeofazimuths.Suchperturbations are especially evident at higher frequencies where wavelengths are on the order of the array radius. The effects arereadilymodeledandshowthatthetriangulararraygeometries commonly used for microtremor studies require azimuthal distributions of wave energy on the order of 60° or greatertohaveahighprobabilityofbeingfreeofsuchperturbations. The imaginary component of the SPAC spectrum, which is ideally zero for a sufficiently dense circular array and/or a sufficiently isotropic wavefield, is in practice often nonzero and provides three quality-control indicators: 1 an indication of insufficient spatial averaging, 2 an empirical measure of the level of statistical uncertainty in SPAC spectralestimates,and3anindicationofdeparturesfromplanewavestationarityoftheseismicnoisewavefield.