Seismic ray theory for structural medium based on Kawaguchi and Finsler geometry

In seismological studies, ray theory for high frequency wave has been discussed and extended by many researchers (e.g. Achenbach et al. 1982, ervený 2002). The seismic ray can be obtained either by solving the elasto-dynamic equations, (i.e. Euler-Lagrange equation or Hamilton equation) or by using a generalization of Fermat's variational principle (Bóna and Slawiski 2003). These geometrical ray theories are similar to the previous seismic ray theory (Teisseyre 1955, Babich 1961, 1994) and the other ray theories in acoustic ray (Ugincius 1972, Meyer and Schroeter 1981) and light (Babich 1987, Joets and Ribotta 1994). Since a micromorphic continuum (Eringen and Suhubi 1964) can express a continuum with microstructure such as earthquake structure (Teisseyre 1973, Nagahama and Teisseyre 2000) and an anisotropic texture and seismic anisotropy in polycrystals (Mainprice and Nicolas 1989, Siegesmund et al. 1989, Muto et al. 2005), a new theory of the micromorphic continuum is needed for the propagation theory of seismic wave. The seismic wave propagating through the anisotropic medium can be regarded as a velocity vector field on each material point. The geometry of the medium consists of the crustal material point and the direction of the velocity attached on each point, and seismic rays are geodesics in Finsler space (Bernstein and Gerver 1978, Hanyga 1984). Moreover, Antonelli et al. (2003) have introduced the seismic Finsler metric for anisotropic and inhomogeneous medium. From standpoints of higher-order geometry, the intrinsic behaviour of the ray velocity attached toeach point can be represented by the base connection in Kawaguchi space (Kawaguchi A 1931, 1966, Kawaguchi M 1962). Here, we discuss a new seismic ray theory from the view point of higher-order geometry. Firstly, we introduce the Finsler geometry for seismic ray, and point out that the intrinsic behaviour of seismic ray velocity can be given by two covariant derivatives in higher-order space. Then, we proposed a relation between a metric of Kawaguchi space and a seismic Finsler metric. From the view point of the differential geometry, we discuss how to estimate the anisotropy of crustal materials (structural medium) from seismic ray path. This section is an extensional version of our previous study (Yajima and Nagahama 2004).