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Downward continuation using the DSR equation for a separate
constant-offset section requires the computation of the integral
in equation (14). This can be done for a single offset by finding a stationary
phase approximation to the integral
| |
(20) |

Integrals of the form
are approximated asymptotically (Zauderer, 1989) when
by
| |
(21) |

where *t*_{0} is the ``stationary point'' in which the derivative of
the phase is zero. The approximation described here assumes the
second derivative is non-zero, which is the case here, as Popovici (1993) has demonstrated.
The phase of the exponential for homogeneous isotropic media is

| |
(22) |

In order to evaluate the stationary point, we need to find the
roots of the equation
| |
(23) |

To find the root we need to solve a sixth-order polynomial.
Approximate solutions are presented in Appendix B.
The second derivative of the phase is non-zero, because it consists
of the sum of four positive terms (Popovici, 1993):

| |
(24) |

This ensures that there is no change in curvature, and that the phase
always has a maximum or a minimum and, therefore, a stationary point.
Indeed, for a fixed pair of values , Figure A-1
shows the phase function for several depth levels.

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Stanford Exploration Project

11/11/1997