1:45 PM - 2:00 PM
[SSS07-06] Energy partitioning among strain components in diffuse wave fields (1)
Keywords:Energy partitioning, strain, diffuse wave field
Equi-partition is an equilibrated state in which energy is equally distributed in all modes. This state is expected for diffuse wave fields that can be realized after multiple conversion scatterings. For the equipartition state in infinite media, the ratio between P-wave and S-wave energies can be analytically expressed with the ratio between P-wave velocity and S-wave velocity (Vp/Vs ratio) (e.g., Weaver, 1982; Snieder, 2002). Energy partitioning among different displacement components is also calculated based on seismic interferometry (e.g. Sanchez-Sesma and Campillo, 2006). According to Sanchez-Sesma et al. (2008), energy is found to be equally partitioned to all the displacement components in infinite homogeneous media. Thanks to distributed acoustic sensing (DAS), recently we can measure strain time series at very dense spatial points. However, only a single component of strain tensors, axial strains along optical fibers, can be recorded. Therefore, we need to know how seismic energy is distributed in this single component. We are also curious about the coda of DAS records. To address these questions, we need to study energy partitioning among different strain components. But there are no previous studies about that as far as we know. Here, we report our results on the strain energy partitioning for diffuse fields in two-dimensional infinite media where only body waves exist.
We calculate cross spectra of strain components at two stations in two-dimensional diffuse wave fields where isotropic and equi-partitioned P and S waves are incident. The formulation can be made by slightly modifying the formulation for surface waves by Nakahara et al. (2021). When the two stations are the same, and the same strain component is considered, cross spectra become proportional to energy densities. Then, we can calculate how energy is partitioned among different strain components at a station. We can express the partitioning among different strain components with the Vp/Vs ratio. We recognize that energy is not equally partitioned into respective strain components. Actually, energy is slightly more partitioned into axial strains than shear strains and areal strains. For Poisson solids with the Vp/Vs ratio of the square root of 3, the contribution of P waves is one-third of S waves in axial strains, the same as the partitioning between P-wave and S-wave energies in the entire medium. On the other hand, P-wave contribution is only one-ninth of S waves in shear strains, which confirms the predominance of S waves in shear strains. Areal strains are composed of only P waves, and its energy partition is slightly smaller than those of axial and shear strains. We also confirm that cross terms between different components of strains contribute to the strain energy.
We can validate these results by a different approach based on the fact that the imaginary part of collocated Green's function is proportional to energy density (e.g. Sanchez-Sesma et al., 2008). Strain Green's tensors for moment-tensor sources (Nakahara and Haney, 2022) are needed to calculate energy densities for strains. By doing these calculations, we confirm that our calculations are correct.
This study clarifies for the first time that energy is not equally distributed among different strain components in diffuse wave fields for infinite two-dimensional media. This helps us understand how P-wave and S-wave energies are partitioned into axial strains. So far, the theory is limited to two-dimensional cases. It is necessary to extend the theory to three-dimensional cases, which will help understand the energy partitioning into axial strains that can be measured with DAS. This is our future study.
We calculate cross spectra of strain components at two stations in two-dimensional diffuse wave fields where isotropic and equi-partitioned P and S waves are incident. The formulation can be made by slightly modifying the formulation for surface waves by Nakahara et al. (2021). When the two stations are the same, and the same strain component is considered, cross spectra become proportional to energy densities. Then, we can calculate how energy is partitioned among different strain components at a station. We can express the partitioning among different strain components with the Vp/Vs ratio. We recognize that energy is not equally partitioned into respective strain components. Actually, energy is slightly more partitioned into axial strains than shear strains and areal strains. For Poisson solids with the Vp/Vs ratio of the square root of 3, the contribution of P waves is one-third of S waves in axial strains, the same as the partitioning between P-wave and S-wave energies in the entire medium. On the other hand, P-wave contribution is only one-ninth of S waves in shear strains, which confirms the predominance of S waves in shear strains. Areal strains are composed of only P waves, and its energy partition is slightly smaller than those of axial and shear strains. We also confirm that cross terms between different components of strains contribute to the strain energy.
We can validate these results by a different approach based on the fact that the imaginary part of collocated Green's function is proportional to energy density (e.g. Sanchez-Sesma et al., 2008). Strain Green's tensors for moment-tensor sources (Nakahara and Haney, 2022) are needed to calculate energy densities for strains. By doing these calculations, we confirm that our calculations are correct.
This study clarifies for the first time that energy is not equally distributed among different strain components in diffuse wave fields for infinite two-dimensional media. This helps us understand how P-wave and S-wave energies are partitioned into axial strains. So far, the theory is limited to two-dimensional cases. It is necessary to extend the theory to three-dimensional cases, which will help understand the energy partitioning into axial strains that can be measured with DAS. This is our future study.