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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.
| |
(7) |

Stiffness values in the first diagonal
quadrant and the first part of the and vectors
are chosen to lie on the primary grid.
The second diagonal quadrant and the second part of and 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.

** Next:** SYNTHETIC EXAMPLES
** Up:** ADAPTION OF THE DERIVATIVE
** Previous:** ADAPTION OF THE DERIVATIVE
Stanford Exploration Project

11/17/1997