Connection between the cross correlation and the Green's function: strain and rotation of surface waves

Recent developments of distributed acoustic sensing (DAS) techniques and rotational seismometers enable us to record strain and rotational seismograms. This prompts us to adapt the methods used for traditional translational seismograms to strain and rotational seismograms. Recently, we extended the formulation of the spatial autocorrelation (SPAC) method to strain, rotation and tilt records. According to seismic interferometry for translation seismograms, cross correlations have clear connections to Green's functions under the isotropy and equipartition of noise wavefields. In this study, we clarify similar connections for strain and rotation seismograms. Because we extend the formulation in the frequency domain, we actually study the connection between the cross-spectral matrix of strain and rotation at two receivers and the Green's tensor. First, we provide a proof under a general framework by simply extending the proof for translation to that for strain and rotation. The proof shows the following results: (1) The cross-spectral matrix of (i, j)-component strains at two receivers is found to be proportional to the strain Green's tensor at one receiver for the sum of (i, j)- and (j, i)-components of the moment tensor source at the other receiver. (2) The cross-spectral matrix of (i, j)-component rotations at two receivers is found to be proportional to the rotation Green's tensor at one receiver for the difference of (i, j)- and (j, i)-components of the moment tensor source at the other receiver. Necessary assumptions for the proof are the isotropy and equipartition of the wavefield. We then check this proof with specific calculations for surface waves observed on the free surface. The proof is confirmed for the isotropic incidence of random-phase surface waves with a specific ratio between Rayleigh and Love wave energies. Seismic interferometry for strain and rotation was already studied. However, the connection of cross correlations to the Green's function for strain and rotation seismograms is discovered by this study for the first time. A specific proof for body waves that is assured by the general proof will be our next study.