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When the velocity structure varies in the lateral direction, the Fourier
transform of velocity is no longer a delta function and the submatrix
*A*_{21} has the form
| |
(15) |

where
Another derivation of submatrix *A*_{21} is possible in space frequency domain
(Kosloff and Kessler, 1987). Whether the derivative in equation (6) is
calculated by the Fourier method as with the ordinary phase-shift method
(Gazdag, 1978), or by finite differences with periodic horizontal boundary
conditions, it can be written as the cyclic convolution

| |
(16) |

where *W*_{i} denotes a convolution operator.
For example, for second-order finite differences,
and , and *W*_{i}=0 for |*i*|>1. For Fourier
second-derivative operator,
and
with *N*_{x}=2*L*+1(Kosloff and Kessler, 1987).
With the spatial discretization and specification of the second-derivative
approximation, the elements of the *N*_{x} by *N*_{x} submatrix *A*_{21} are
given by

| |
(17) |

where *c*_{i}=*c*(*idx*,*z*)

** Next:** Tal-Ezer Method
** Up:** THE GENERALIZED PHASE-SHIFT METHOD
** Previous:** Horizontally Uniform Structures
Stanford Exploration Project

1/13/1998