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
where N and M are, respectively, the horizontal and vertical dimensions of the model, as measured in terms of the number of cells.
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 shows the ``equivalent differencing stars" associated with for the case of a homogeneous medium. It's worthwhile to notice that the ux - ux star for C55 represents an x-smoothed version of . This form is similar to the McClellan transform used by Hale (1990) to construct a numerically isotropic 3-D migration operator. Here however, the suitable form of the star comes naturally, from the direct application of physical laws.
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 n3, 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.