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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.

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** Up:** Ji: Conjugate RTM
** Previous:** Example
Stanford Exploration Project

11/17/1997