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) $$