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## 2-D transverse-isotropic media

The discrete representation for the 2-D case, as well as the derivation of dynamic equations, are presented in Appendix A.

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).

2doper
Figure 4
The spatial operator for a 2-D homogeneous transverse isotropic discrete medium. The line on the bottom, which corresponds to the 36th row, shows that the operator has 13 effective terms, or 17 non-vanishing diagonals.

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.

star
Figure 5
Differencing stars associated with the operator . Each elastic constant contains four stars corresponding to the coupling between the two components (ux and uz) of the wavefield. Only the non-zero values are written.

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.

s2d1
Figure 6
One frame of the horizontal displacement wavefield, for a vertical source using the eigenvalue-decomposition elastic modeling method. The upper 3/4 of the model is isotropic, while the bottom part is anisotropic. Because of the large computation time, the model size is 37 by 37 cells. Press the button to see a movie.

Next: Recursive solution in the Up: Introduction Previous: 1-D discrete systems
Stanford Exploration Project
12/18/1997