Stationary phase approximation

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 t0 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 .