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 ${\bf u}$ 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 ${\bf u}$, the one-dimensional representation of ${\bf u}$ will be a $2 \times N \times M$ vector given by

$${\bf u} = (u^{1,1}_x, u^{2,1}_x, \ldots, u^{N,1}_x, u^{1,2}_... ...}, \ldots, u^{N,M}_x, u^{1,1}_z, u^{2,1}_z, \ldots, u^{N,M}_z),$$

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 ${\bf A}$ of equation (3) takes the form of a $2NM \times 2NM$ 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 ${\bf A}$ for the case of a homogeneous medium. It's worthwhile to notice that the u_x - u_x star for C₅₅ represents an x-smoothed version of $\partial^2/\partial z^2$. 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 ${\bf A}$. Each elastic constant contains four stars corresponding to the coupling between the two components (u_x and u_z) 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 $8 (N \times M)^3$. 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 ${\bf A}$, 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 ($\sim$ 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.