Stationary phase approximation

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  

$$I= \int dk_h \; e^{i \omega [p_{\tau}(p_x,p_h)\tau+ 2 p_h h]}.$$ (20)

Integrals of the form

$$I(k)=\int_{-\infty}^{\infty} e^{ik\phi(t)}f(t)\; dt$$

are approximated asymptotically (Zauderer, 1989) when $k \rightarrow \infty$ by  

$$I(k) \approx e^{ik\phi (t_0)} f(t_0) e^{{\rm sign} (\phi''(... ...t[{{2\pi} \over {k \mid \phi''(t_0) \mid }} \right]^{1 \over 2}$$ (21)

where t₀ 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

$$\phi(p_h)= 0.5 \tau \left[ \; \sqrt{1.0 - v^2 (p_x+p_h)^2}+ \sqrt{1.0 - v^2 (p_x-p_h)^2} \; \right] + 2 p_h h.$$ (22)

In order to evaluate the stationary point, we need to find the roots of the equation

$$\phi ' (p_h)= X + 0.5 \tau \left[ {{p_h+p_x} \over {\sqrt{1... ...)^2}}}+ {{p_h-p_x} \over {\sqrt{1.0-v^2(p_x-p_h)^2}}} \right].$$ (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):

$$\begin{array} {lcl} \phi '' (p_h) & = & \left. {0.5 \over {\... ....0- v^2(p_x+p_h)^2} \right ]^{3 \over 2}}} \right. .\end{array}$$ (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 $\omega,p_x$, Figure A-1 shows the phase function for several depth levels.