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APPENDIX A
In equation (1) the values of the constant kz,
given by equation (2),
have to be real. Imaginary values of kz do not
satisfy the downward continuation ordinary differential
equation
and have to be excluded. Real values of kz
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 kz, 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 kh, the
integration boundaries for kh become
| |
(20) |
Next: About this document ...
Up: Popovici : PDE for
Previous: Acknowledgments
Stanford Exploration Project
11/17/1997