In the 2-D case equation (3) keeps the same form but
is now a one-dimensional representation of the bi-dimensional
displacement field. If we use indices *i* and *j* to specify a given
cell (or a position in the plane), and indices *x* and *z* to denote the
horizontal and vertical components of , the one-dimensional
representation of will be a vector given by

Appendix A shows that for a 2-D system, the operator of equation (3) takes the form of a sparse matrix with diagonal structure. This operator has only 13 effective terms corresponding to 17 non-vanishing diagonals as illustrated in Figure , which was computed for an isotropic model of size 8 by 8 (64 cells).

Figure 4

Figure shows the ``equivalent differencing stars" associated
with for the case of a homogeneous medium. It's worthwhile to
notice that the *u*_{x} - *u*_{x} star for *C _{55}* represents an

Figure 5

The method is implemented here by first reducing the operator to
tridiagonal form using an algorithm based in Givens rotations
and then using a tridiagonal eigenvalue decomposition routine.
For an *n* by *n* matrix, the cost in both stages (reduction and decomposition)
is proportional to *n ^{3}*, which in this case represents .
Whereas the cost in the reduction stage can be substantially reduced
with the use of a more specific algorithm that takes advantage of
the sparseness of , the price of the decomposition stage
is still extremely high.

Figure shows a time frame of the horizontal component of the displacement wavefield evaluated with the eigenvalue-decomposition modeling for a two-layer model. The upper part ( 3/4) is isotropic, while the bottom is transverse isotropic, with a total size of 37 by 37 cells, and a vertical impulsive displacement source at the center.

Figure 6

12/18/1997