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The 2*N*_{x} by 2*N*_{x} matrix A can be partioned according to
| |
(5) |

where *I*_{Nx} denotes the *N*_{x} by *N*_{x} identity matrix and
the *N*_{x} by *N*_{x} submatrix *A*_{21} is given by
| |
(6) |

The horizontal discretization requires an approximation for the derivative
term in equation (6). As with the ordinary phase-shift method (Gazdag, 1978),
this derivative can be calculated by Fourier method
| |
(7) |

Since multiplication in the space domain is equivalent to convolution in the
wavenumber domain, the multiplication of the sloth (squared slowness) becomes
convolution. For horizontally uniform structures, however, the Fourier
transform of the velocity function is a delta function. Therefore, the
submatrix *A*_{21} has the form
| |
(8) |

The diagonal elements of this matrix are the eigenvalues of *A*_{21} and
a matrix in which the columns consist of the eigenvectors form an identity
matrix.
By letting the eigenvalues of submatrix *A*_{21} as
the eigenvalues of matrix A become , for *i*=1,...,*N*_{x}-1.
The corresponding eigenvectors are given by
where *I*_{i} denotes i-th column of identity matrix.
With these eigenvectors we define a 2*N*_{x} by 2*N*_{x} matrix *Q* in which
the columns consist of the eigenvectors of *A*. We then have the relation
| |
(9) |

where is the diagonal matrix
| |
(10) |

and
| |
(11) |

When both sides of equation (9) are multiplied by *Q*^{-1} from the left we
obtain
| |
(12) |

The equation (12) is a matrix representation for a set of equations
| |
(13) |

and
| |
(14) |

The solutions of equations (13) and (14) represent upgoing and downgoing waves
respectively.
When an eigenvalue is real, the component is propagated
by a multiplication by a real exponential, and results in an evanescent
component which should be removed for numerical stability. When an eigenvalue
is imaginary, the corresponding coefficient is
propagated by phase shift.

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** Up:** THE GENERALIZED PHASE-SHIFT METHOD
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Stanford Exploration Project

1/13/1998