2-D TIME EXTRAPOLATION AND ITS CONJUGATE

The acoustic wave equation in two dimensional space is

$${\partial^2 P \over {\partial t^2}} = v^2 ({\partial^2 P \over {\partial x^2}} + {\partial^2 P \over {\partial z^2}})$$ (9)

where x and z are the surface and depth axis, respectively. By approximating the derivatives by finite differences, equation (8) becomes

$${1 \over {\Delta t^2}}( P_{i-1,j,k} - 2 P_{i,j,k} + P_{i+1,j,k})$$

$$= {v_{j,k}^2 \over {\Delta z^2}}( P_{i,j-1,k} - 2 P_{i,j,k} ... ...2 \over {\Delta x^2}}( P_{i,j,k-1} - 2 P_{i,j,k} + P_{i,j,k+1})$$ (10)

where i, j and k represent the time, depth and surface indices, respectively. By rearranging for time-extrapolation, we get the following equation

$$P_{i+1,j,k} = 2(1-2\alpha_{j,k})P_{i,j,k} + \alpha_{j,k}(P_{i,j-1,k} + P_{i,j+1,k} + P_{i,j,k-1} + P_{i,j,k+1}) - P_{i-1,j,k}$$ (11)

$$\alpha_{j,k} = {v_{j,k}^2 \Delta t^2 \over {\Delta z^2}}$$

where I assumed $\Delta x = \Delta z$.

The matrix representation of this operator is the same as equation(4) except that the size of each matrix is now larger than the corresponding matrix in equation(4) because I need to express the two-dimensional wavefield with an abstract vector form. For a space of (z,x)=(5,6), the matrix ${\bf T}$ becomes a block tridiagonal matrix as follows :

$${\bf T}= \left[ \begin{array} {cccccc} {\bf R_1}&\bf A&\bf 0... ...\bf 0&\bf 0&\bf 0&\bf 0&{\bf A}&{\bf R_6}\\ \end{array}\right],$$ (12)

where ${\bf A}$ is a diagonal matrix having the following elements

$$\alpha_{j,k} = {v_{j,k}^2 \Delta t^2 \over {\Delta z^2}}.$$

The matrix ${\bf R_k}$ represents a tri-diagonal matrix :

$${\bf R}_k= \left[ \begin{array} {ccccc} 2(1-2\alpha_{1,k})&\... ...k}\\ 0&0&0&\alpha_{5,k}&2(1-2\alpha_{5,k})\\ \end{array}\right]$$ (13)

The conjugate operator can easily be obtained by substitution of the transposed form of the new matrix ${\bf T}$into the equation(7). The code for 2-D algorithm, which has passed dot-product test, can be found in the CD-rom version of this report.