Testing Equipartition for S-Wave Coda Using Borehole Records of Local Earthquakes

Equipartition is a state in which energy of all possible wave modes becomes equal due to multiple scattering among the modes. This is important to identify the scattering regime and constitutes the basic principle underlying seismic interferometry. In this study, we calculate partitioning of total kinetic energy into the horizontal and vertical components, similar to the horizontal-to-vertical ratio, for S-wave coda of 60 local earthquake records at three borehole stations in Japan. S-wave coda in lapse times of 40-80 s and in the 2-16 Hz frequency band is used. At the surface receivers of two rock sites, the horizontal components are dominant and their contribution gradually increases with frequency. At a soft site, the energy partitioning shows stronger variations with frequency, reflecting the presence of low-velocity layers. Subsurface receivers at about 100 m depths show larger contribution of the horizontal components, irrespective of site conditions. We quantitatively explain the observations with random wave fields at equipartition in horizontally layered structures derived from well logging. Through the modeling, we test three formulations of equipartition: a mixture of (1) only body waves, (2) only surface waves, and (3) both body and surface waves. Subsurface receivers are critically important to distinguish among the three formulations. For S-wave coda at frequencies lower than about 5 Hz, equipartition holds among both body and surface waves. For higher frequencies, equipartition holds predominantly among body waves. Equipartition of coda waves enables a simple modeling of the horizontal-to-vertical ratio, which contrasts with ambient noise.