The use of wavenumber normalization in computing spatially averaged coherencies (KRSPAC) of microtremor data from asymmetric arrays

The SPAC method of processing microtremor noise observations for estimation of Vs profiles has a limitation that the array has circular or triangular symmetry in order to allow spatial (azimuthal) averaging of inter-station coherencies over a constant station separation. Common processing methods allow for station separations to vary by typically ±10% in the azimuthal averaging before degradation of the SPAC spectrum is excessive. A limitation on use of high-wavenumbers in inversions of SPAC spectra to Vs profiles has been the requirement for exact array symmetry to avoid loss of information in the azimuthal averaging step. In this paper we develop a new wavenumber-normalised SPAC method (KRSPAC) where instead of performing averaging of sets of coherency versus frequency spectra and then fitting to a model SPAC spectrum, we interpolate each spectrum to coherency versus k.r, where k and r are wavenumber and station separation respectively, and r may be different for each pair of stations. For fundamental mode Rayleigh-wave energy the model SPAC spectrum to be fitted reduces to Jo(kr). The normalization process changes with each iteration since k is a function of frequency and phase velocity and hence is updated each iteration. The method proves robust and is demonstrated on data acquired in the Santa Clara Valley, CA, (Site STGA) where an asymmetric array having station separations varying by a factor of 2 is compared with a conventional triangular array; a 300-mdeep borehole with a downhole Vs log provides nearby ground truth. The method is also demonstrated on data from the Pleasanton array, CA, where station spacings are irregular and vary from 400 to 1200 m. The KRSPAC method allows inversion of data using kr (unitless) values routinely up to 30, and occasionally up to 60. Thus despite the large and irregular station spacings, this array permits resolution of Vs as fine as 15 m for the near-surface sediments, and down to a maximum depth of 2.5 km.