Practical Issues

The splitting of model and wave field does not address the problem of interpolation of quantities to the desired grid points. That has to be handled separately. In practical implementations one problem can arise when calculating derivatives of the quantities. The derivatives have to be interpolated onto collocation points. This can result in a potential loss in accuracy and diminish improvements achieved by the operator adaption. Examining the partitioned constitutive equation, one can see how a commonly used staggered grid is chosen.

$$\pmatrix{\pmatrix{\sigma_1 \cr \sigma_2 \cr \sigma_3 \cr } \... ...on_3} \cr \pmatrix{\epsilon_4 \cr \epsilon_5 \cr \epsilon_6}}$$ (7)

Stiffness values in the first diagonal quadrant and the first part of the $\sigma$ and $\epsilon$ vectors are chosen to lie on the primary grid. The second diagonal quadrant and the second part of $\sigma$ and $\epsilon$are then necessarily chosen to lie on the secondary grid. That scheme works fine as long as the derivative computation evaluates the quantities at the correct collocation points. In anisotropic elastic media up to orthorhombic symmetry, the previous notion works fine, since the off-diagonal elements of the stiffness matrix are zero and thus do not have to be evaluated. In more anisotropic systems (beyond orthorhombic) those elements are non zero, such that even in conventional schemes interpolation to collocation points has to be included for staggered grid methods. When splitting the wave operator, one has to take care of proper collocation interpolation. Thus the interpolation issue can complicate the primary goal of adapting derivative operators to the quantities.