Next: Clapp: REFERENCESVelocity uncertainty
Up: Appendix
Previous: New notations and definitions
Given the values of the function 472#472, downward continuation consists of finding the
values of 473#473. An expression describing this process lays at the end of the
following proof:
Obtain the Helmholtz equation by applying the Fourier transform
defined by () to the wave equation () while
taking into account the notation () and the property
() and rearranging:
By derivating relation () with respect to x and z we
obtain:
By plugging into in () and eliminating the exponential, we
get:
The second derivative with respect to z can be eliminated by
derivating with respect to z, multiplying by 480#480, and
adding the result to ():
Note that no approximation has been made between the wave equation
() and this point. Eq. is simply the wave
equation in a different coordinate system. Now Qzzz is
approximated by zero:
For the case of a homogenous medium, 483#483 and the equation
turns into the familiar 429#429 equation:
The 444#444 equation is obtained by neglecting the Qxxz term
also:
Downward continuation proceeds by considering
then by using one of the equations , or
to find the values of 487#487 and by finally finding P by undoing
the variable change:
Next: Clapp: REFERENCESVelocity uncertainty
Up: Appendix
Previous: New notations and definitions
Stanford Exploration Project
6/7/2002