** Next:** REFERENCES
** Up:** Appendix
** Previous:** New notations and definitions

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 *Q*_{zzz} 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 *Q*_{xxz} 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