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# 2-D TIME EXTRAPOLATION AND ITS CONJUGATE

The acoustic wave equation in two dimensional space is
 (9)
where x and z are the surface and depth axis, respectively. By approximating the derivatives by finite differences, equation (8) becomes

 (10)
where i, j and k represent the time, depth and surface indices, respectively. By rearranging for time-extrapolation, we get the following equation
 (11)

where I assumed .

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 becomes a block tridiagonal matrix as follows :
 (12)
where is a diagonal matrix having the following elements

The matrix represents a tri-diagonal matrix :
 (13)

The conjugate operator can easily be obtained by substitution of the transposed form of the new matrix 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.

Next: Conclusions Up: Ji: Conjugate RTM Previous: Example
Stanford Exploration Project
11/17/1997