Next: Conclusion Up: Shan and Biondi: Residual Previous: Amplitude preserving one-way wave

# WKBJ and its first order approximation

In this section, we demonstrate that the first order approximation of WKBJ is the same as the amplitude preserving one-way wave equation for v=v(z).

The one-way wave equation with the WKBJ amplitude correction is
 (7)
The WKBJ amplitude correction term can be rewritten as
 (8)
Then in equation (8) can be linearized to , so we have
 (9)
From

and

we have
 (10)
Because
 (11)
and from the dispersion relation , we have
 (12)
and
 (13)
From equation (8) to equation (13), we have
 (14)
So equation (7) can be rewritten as
 (15) (16)
Comparing the amplitude preserving one-way wave equation (equation (4)) with first order approximation of WKBJ (equation (16)), we find they are same. So we demonstrate theoretically that the amplitude preserving one-way wave equation is equivalent to the first order approximation of WKBJ.

Next: Conclusion Up: Shan and Biondi: Residual Previous: Amplitude preserving one-way wave
Stanford Exploration Project
10/14/2003