[S-SS09] Seismic wave propagation: Theory and Application

[SSS09-09] Connection between the correlation of strain and rotation seismograms and Green’s function: Surface-wave case

*Hisashi Nakahara¹ (1.Solid Earth Physics Laboratory, Department of Geophysics, Graduate School of Science, Tohoku University)

Keywords:cross correlation, Green's function, seismic interferometry, strain, rotation

Dynamic strain (or strain rate) is recently measured by Distributed Acoustic Sensing (DAS) techniques along a fiber-optic cable with high spatial resolutions of 10m or less. Rotational ground motion is measured by rotational seismometers that are sometimes used in combination with translational seismometers (called the six-component (6C) observation) (e.g., Schmelzbach et al., 2018). Therefore, we need to accustom ourselves to using strain and rotation seismograms by adapting the methods used for traditional translational seismograms to these new kinds of seismograms. Nakahara et al. (2020, SSJ) formulated the SPAC method for DAS axial-strain seismograms. Nakahara et al. (2021) further extended the formulation to the other components of strain, rotation and tilt records. According to seismic interferometry, it is possible to connect SPAC to Green's function (e.g., Nakahara, 2006; Sanchez-Sesma and Campillo, 2006). In this study, we clarify the connection between cross-correlation functions of strain and rotation seismograms and Green's functions for the moment tensor under an assumption that surface waves only exist. Since the formulation is conducted in the frequency domain, the connection between the cross-spectral matrix of strain and rotation at two receivers and the Green's tensor is concerned. A general framework of seismic interferometry for strain and rotation was already provided by Paitz et al. (2019). However, the physical meanings of cross-correlation functions of strain and rotation are not yet fully understood.

We start from the strict expressions of surface-wave Green’s tensors from moment-tensor sources in the Cartesian coordinate derived by Haney and Nakahara (2016). First, we convert the expression to the cylindrical coordinate and calculate the displacement gradients of Green's tensor. Then, we derive analytical expressions for strain and rotation Green's functions from moment-tensor sources. Finally, we compare these expressions with the SPAC expressions for strain and rotation derived by Nakahara et al. (2021). We obtain the following results for the isotropic incidence of random waves with a specific ratio between Rayleigh and Love wave energies. The (i, j)-component cross-spectral matrix of strain is found to be proportional to the strain Green's tensor at one receiver for the sum of (i,j)- and (j, i)-component moment tensor source at the other receiver. Regarding the rotational component, the (i, j)-component cross-spectral matrix is found to be proportional to the rotation of Green's tensor at one receiver for the difference of (i,j)- and (j, i)-component moment tensor source at the other receiver.

So far, we proved these relations for surface waves by the concrete calculation using the analytical expressions. Now we speculate that these relations hold for body waves as well. An extension of the proof to body waves will be the next target.