Short Note Surface-Wave Green’s Tensors in the Near Field

We demonstrate the connection between theoretical expressions for the correlation of ambient noise Rayleigh and Love waves and the exact surface-wave Green’s tensors for a point force. The surface-wave Green’s tensors are well known in the far-field limit. On the other hand, the imaginary part of the exact Green’s tensors, including near-field effects, arises in correlation techniques such as the spatial autocorrelation (SPAC) method. Using the imaginary part of the exact Green’s tensors from the SPAC method, we find the associated real part using the Kramers–Kronig relations. The application of the Kramers–Kronig relations is not straightforward, however, because the causality properties of the different tensor components vary. In addition to the Green’s tensors for a point force, we also derive expressions for a general point moment tensor source. Introduction Recently, Haney et al. (2012) extended the spatial autocorrelation (SPAC) method of Aki (1957) to the full tensor of possible three-component correlations, including correlations of mixed components. The complete RayleighφR and LoveφL wave correlation coefficient tensors are φR r;ω φZZ φ R ZR φ R ZT φRZ φ R RR φ R RT φTZ φ R TR φ R TT 2 64 3 75 PR ω × J0 ωr=cR −RJ1 jωjr=cR 0 RJ1 jωjr=cR R J0 ωr=cR − J2 ωr=cR =2 0 0 0 R J0 ωr=cR J2 ωr=cR =2 2 64 3 75 1