Finsler metric and elastic constants for weak anisotropic media

Differential geometric expressions of elastic constants for a seismic ray path are studied based on Finsler geometry. A Finsler function named mmth root metric is considered to discuss transverse isotropic media in weak anisotropic case. Finsler parameters in the mmth root metric are estimated from phase velocity surfaces. The slight differences from an elliptic wavefront can be expressed by the Finsler parameters. It is found a correlation between the Finsler parameters and the weak anisotropy parameters consisted of elastic constants. Especially, a positivity of weak anisotropy parameter influences on a restriction of Finsler parameter. On the other hand, a geometric condition of Finsler parameter gives a limitation of weak anisotropy parameter. Moreover, the Berwald Gauss curvature of mmth root metric induces a relationship between the spreading ray paths and the weak anisotropy parameter. Therefore, the seismic ray paths in weak isotropic media can be expressed by the Finslerian properties of mmth root metric.