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.