Next: IMPLEMENTATION
Up: THEORY
Previous: The basis for discretization
For the case of a 2D elastic medium, the relations for finding
the equivalent 1D Young modulus (equation (2)) needs
to be extended for all the involved 2D elastic moduli. This extension
is provided by the group theory of Shoenberg and Muir (1989),
which defines the equations for finding the equivalent elastic parameters.
The general elastic wave equation is given by
 
(4) 
where indices i and j refers to the spatial components (x,y,z), f_{i}
is the i component of the internal forces, and is the ij
component of the stress field (Auld, 1990).
In the 2D transverse isotropic case without internal sources,
the x component of equation (4) becomes
where
Combining these equations we find for the x component
 

 (5) 
and for the z component
 

 (6) 
The equivalent stiffness constants are obtained with the
ShoenbergMuir algebra. For displacements in the x direction,
 

 (7) 
 
and for displacements in the z direction,
 

 (8) 
 
Next: IMPLEMENTATION
Up: THEORY
Previous: The basis for discretization
Stanford Exploration Project
12/18/1997