Isotropy

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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)