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## APPENDIX A

In equation (1) the values of the constant *k*_{z},
given by equation (2),
have to be real. Imaginary values of *k*_{z} do not
satisfy the downward continuation ordinary differential
equation

and have to be excluded. Real values of *k*_{z}
from equation (2) require the conditions:
which can be reduced to the condition
| |
(16) |

After the change of variable from to
in equation (5)
we want to express function of and
determine the integration boundaries for .We start
with the expression for :
| |
(17) |

and after reducing
and grouping
we have
| |
(18) |

The discriminant is
From the conditions on *k*_{z}, is always positive
and therefore is always real within
the limits.
The integration limits for are found by
starting with the limits for :

and after we square both sides
and replacing in the equation for we have
| |
(19) |

The integration in is done from
and from
. After changing
the order of integration from to *k*_{h}, the
integration boundaries for *k*_{h} become
| |
(20) |

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Stanford Exploration Project

11/17/1997