Anisotropic background illumination effects on focal spot imaging configurations with azimuthal asymmetry

Rayleigh surface wave fields can be reconstructed from noise correlations using dense seismic arrays. At zero lag time, the correlation amplitude field forms the so-called focal spot. The equivalence between the time domain focal spot and the frequency domain spatial autocorrelation allows the application of classic SPAC results for Rayleigh wave phase velocity estimates, and it also entails that azimuthal averaging yields accurate velocity estimates for anisotropic background illumination. In this work we explore the effect of directional noise on configurations that do not allow the homogeneous reconstruction of the focal spot at all azimuths. Such configurations are related to stations located towards the edges of quadratic or circular arrays, or to stations in arrays with rectangular or other irregular shapes. We first perform numerical experiments. We systematically test the effects of the ZZ and ZR focal spot position in a rectangular array, the aspect ratio of the array, the strength and direction of the anisotropic incidence, the size of the focal spot, and incoherent noise. Using realistic scenarios and the classic parameterization leads to velocity errors up to 15%, whereas the application of the 2006 Nakahara formulation that parameterizes anisotropic incidence yields errors in the 0.01% range. Second, these two solutions are applied to focal spots that are reconstructed from USArray noise correlations between 114.5° - 102°W and 26° - 50°N. The noise field in the target 60 s to 100 s period range is directional. We obtain phase velocity maps from focal spots using rectangular station geometries with variable aspect ratios similar to the numerical tests. The spatially variable differences between the velocity distributions obtained with the two parameterizations provide empirical evidence of the anisotropic background illumination effects on focal spot imaging configurations with azimuthal asymmetry.