Effect of dense array shapes and directional incidence on seismic surface wave focal spot imaging results

Converging surface wave fields create a large-amplitude feature at the origin referred to as focal spot. Its properties are governed by local medium properties and have long been used in medical imaging approaches such as passive elastography. Modern dense seismic arrays consisting of many hundreds of sensors now allow the application of noise correlation-based focal spot imaging in seismology where they can be obtained from the zero-lag correlation amplitude field. We demonstrated the feasibility of using the Vertical-Vertical and Vertical-Radial components of the focal spot to estimate the Rayleigh wave speed. Azimuthal averaging mitigates anisotropic incidence, which is compatible with related SPAC results in the literature. An important aspect of focal-spot imaging is the emphasis on data collected in the near-field. A clean azimuthal average may be difficult to estimate if sensors are not isotropically distributed around the origin. For this case, an extended description of the seismic interferometry coherence function was developed, that was subsequently extended for mixed components in the SPAC formulation. The objective of the present study is to investigate the resolution power of this new expansion on Rayleigh wave speed estimations in the case of various array shapes and for directional incidence. We perform numerical experiments using an equivalent time-reversal approach to synthesize Rayleigh wave focal spots in an elastic half-space from Green’s functions computed with the AXITRA solver. Simulations are performed using a square 85 x 85 receiver grid separated by 8 m and 72 time-reversal mirror elements that are located at the surface, on a circle, 12 km away from the origin. The regular grid is then adapted to obtain different aspect ratios of the compact, dense array, varying from a 1:1 to a 1:5 ratio. We measure the discrepancy of the imposed Rayleigh wave speed of 2 km/s and the estimates using said 2D parametrization of the amplitude field. We vary systematically the angle of incidence, from 0° (North) to 90° (East), the strength of the anisotropy, the relative position of the origin to the array center, and the frequency between 1 Hz and 10 Hz. Consequently, the ratio of wavelength to array size varies between 0.7 and 33. We illustrate some of our numerical examples with focal spots obtained from USArray data in the 60 s to 120 s period range. The results show that the error on estimations for the Vertical-Vertical components under strong anisotropic incidence is reduced from 12% to less than 1% using the specific expansion. These small values suggest that Rayleigh wave focal spot imaging can robustly be applied for a wider range of array shapes and characteristics of the surface wave field from which the correlation functions are constructed. Further investigations considering biases such as incoherent noise or body wave components are needed to complete the analysis.