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## Downward continuation

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