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Downward continuation using the DSR equation for a separate
constant-offset sections requires the computation of the integral
in equation (A-1). This can be done for all offsets by
taking the inverse transform of the downward continuation
exponential, or for a single offset by finding a stationary
phase approximation to the integral
| |
(10) |

Integrals of the form
are approximated asymptotically (Zauderer, 1989) when
by
where *t*_{0} is the ``stationary point'' where the derivative of
the phase is zero. The approximation described here assumes the
second derivative is non-zero, which I will prove is the situation in
this case.
The phase of the exponential in this case is

| |
(11) |

In order to evaluate the stationary point we need to find the
roots of the equation
| |
(12) |

I was not able to find these roots, as it involves an equation of
sixth degree. However, I will point out some properties of the
phase, which should make the numerical computation simpler than
the full integration.
The second derivative of the phase is non-zero, as it consists
of the sum of four positive terms
multiplied by :

| |
(13) |

This ensures that there is no change in curvature, and 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.
The sign of determines if the phase is positive
or negative.
**App1
**

Figure 8 Plot of the phase function for a constant pair of values .The different plots correspond to increasing depth levels.

a. Phase corresponding to positive .

b. Phase corresponding to negative .

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Stanford Exploration Project

11/16/1997