Finsler metric and elastic constants for weak anisotropic media

Abstract

Differential geometric expressions of elastic constants for a seismic ray path are studied based on Finsler geometry. A Finsler function named m th root metric is considered to discuss transverse isotropic media in weak anisotropic case. Finsler parameters in the m th 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 m th 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 m th root metric.