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 perturbations of the SPAC spectrum when sources of seismic noise are confined to a restricted range of azimuths. Such perturbations are especially evident at higher frequencies where wavelengths are on the order of the array radius. The effects are readily modeled and show that the triangular array geometries commonly used for microtremor studies require azimuthal distributions of wave energy on the order of 60° or greater to have a high probability of being free of such perturbations. 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 spectral estimates, and (3) an indication of departures from plane-wave stationarity of the seismic noise wavefield.