Isotropy

A medium is seismically isotropic when all directions of wave propagation are equivalent.

The stiffness tensor is defined by two independent elastic constants: the Lamé parameters:

$C_{ijkl}^{\mathrm {iso} }=\lambda \delta _{ij}\delta _{kl}+\mu (\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})$

or,

$C^{\mathrm {iso} }=\left({\begin{matrix}\lambda +2\mu &\mu &\mu &0&0&0\\\mu &\lambda +2\mu &\mu &0&0&0\\\mu &\mu &\lambda +2\mu &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{matrix}}\right)$

then

$C^{\mathrm {VTI} }=\left({\begin{matrix}C_{11}&C_{11}-2C_{66}&C_{13}&0&0&0\\C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{55}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{matrix}}\right)$