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[SSS09-05] Analytical expression of cross-spectrum for ambient-noise surface wave analysis applicable for DAS data using multiple or winding cable
Keywords:Distributed acoustic sensing (DAS), Seismic interferometry
Distributed Acoustic Sensing (DAS), which enables long-range observations over tens of kilometers at intervals of several meters by using optical fibers as sensors, has been increasingly applied in Earth sciences (e.g., Zhan, 2019). The application of ambient-noise surface-wave analysis to dense DAS data is effective for obtaining S-wave velocity (Vs) structures with high lateral resolution both onshore and offshore. Conventionally, DAS data with straight cables or straight sub-sections have been used in ambient-noise surface-wave analysis. In this case, two-dimensional (2-D) Vs structures can be estimated along the cable (e.g., Fukushima et al., 2022). Recently, some DAS observations using winding or multiple cables have already been conducted (e.g., Shinohara et al., 2025). This DAS observation using winding or multiple cables enables us to estimate three-dimensional (3-D) Vs structures by the ambient-noise surface-wave tomography (ANT) method. In ANT, dispersion curves of phase velocity between two stations are required. An analytical expression of the cross-spectrum is often used to estimate dispersion curves between two seismometers (e.g., Ekström et al., 2009; Takagi & Nishida, 2022). Nakahara et al. (2021) derived an analytical expression of cross-spectrum for the case where two DAS stations are located in a line and record axial strain oriented in the same direction at least for one station. This situation roughly corresponds to an analysis using DAS data from a single, straight cable.
In general, however, two DAS observation points in winding or multiple cables are at arbitrary locations, and record axial strain in different directions. Therefore, to estimate 3-D Vs structures using ANT with winding or multiple cables, the analytical expression of cross-spectrum that accounts for axial strain at any position and in any direction is necessary.
In this study, we derived the analytical solution of cross-spectrum for DAS data with winding or multiple cables. In a cylindrical coordinate system, the axial strains at two DAS stations were considered (Fig. 1). The first DAS station is located at the origin and records the axial strain in the azimuthal direction Ψ. The second DAS station is located at a distance r with an azimuth angle η, recording the axial strain with an azimuthal angle inclined for ξ from the η-direction. The resultant cross-spectrum contains contributions from both Love and Rayleigh waves. The relative contributions of these wave types depend on (1) the distance between the two DAS stations, (2) the angle ξ, and (3) the angle Δη, defined as the angle between the azimuthal direction Ψ and η. These findings indicate that the contribution of Rayleigh and Love waves varies depending on the DAS geometry. The newly derived analytical expression of the cross-spectrum enables the estimation of the phase velocity of both Rayleigh and Love waves. This allows for the estimation of high-resolution 3-D Vs structures by using ANT of DAS data with winding or multiple cables.
In general, however, two DAS observation points in winding or multiple cables are at arbitrary locations, and record axial strain in different directions. Therefore, to estimate 3-D Vs structures using ANT with winding or multiple cables, the analytical expression of cross-spectrum that accounts for axial strain at any position and in any direction is necessary.
In this study, we derived the analytical solution of cross-spectrum for DAS data with winding or multiple cables. In a cylindrical coordinate system, the axial strains at two DAS stations were considered (Fig. 1). The first DAS station is located at the origin and records the axial strain in the azimuthal direction Ψ. The second DAS station is located at a distance r with an azimuth angle η, recording the axial strain with an azimuthal angle inclined for ξ from the η-direction. The resultant cross-spectrum contains contributions from both Love and Rayleigh waves. The relative contributions of these wave types depend on (1) the distance between the two DAS stations, (2) the angle ξ, and (3) the angle Δη, defined as the angle between the azimuthal direction Ψ and η. These findings indicate that the contribution of Rayleigh and Love waves varies depending on the DAS geometry. The newly derived analytical expression of the cross-spectrum enables the estimation of the phase velocity of both Rayleigh and Love waves. This allows for the estimation of high-resolution 3-D Vs structures by using ANT of DAS data with winding or multiple cables.