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Given the values of the function , downward continuation consists of finding the
values of . An expression describing this process lays at the end of the
following proof:
Obtain the Helmholtz equation by applying the Fourier transform
defined by (24) to the wave equation (22) while
taking into account the notation (26) and the property
(25) and rearranging:
| |
(29) |
By derivating relation (28) with respect to x and z we
obtain:
| |
(30) |
| |
(31) |
| |
(32) |
| |
(33) |
By plugging into in (29) and eliminating the exponential, we
get:
| |
(34) |
The second derivative with respect to z can be eliminated by
derivating with respect to z, multiplying by , and
adding the result to (34):
| |
(35) |
Note that no approximation has been made between the wave equation
(22) and this point. Eq. 35 is simply the wave
equation in a different coordinate system. Now Qzzz is
approximated by zero:
| |
(36) |
For the case of a homogenous medium, and the equation
turns into the familiar equation:
| |
(37) |
The equation is obtained by neglecting the Qxxz term
also:
| |
(38) |
Downward continuation proceeds by considering
| |
(39) |
then by using one of the equations 36, 37 or
38 to find the values of and by finally finding P by undoing
the variable change:
| |
(40) |
Next: REFERENCES
Up: Appendix
Previous: New notations and definitions
Stanford Exploration Project
6/8/2002