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To derive a time-space representation of the AMO impulse response
from its frequency-wavenumber representation,
we evaluate the stationary-phase approximation of
the inverse Fourier transform along the midpoint coordinates.
The operator and its inverse, , can be
defined in the zero-offset frequency
and the midpoint wavenumber as
| |
(63) |
| |
(64) |
The operator is given by the cascades
of and and its impulse response can
be written as,
| |
(65) |
The derivation of the stationary-phase approximation of the
integral in is similar to the one presented in
Black et al. (1993b) for deriving a time-space formulation
for the conventional DMO impulse response.
We begin by changing the order of the integrals
and rewriting amo_freq.eq as
| |
|
| (66) |
The phase of this integral is,
| |
(67) |
where,
| |
(68) |
Next we make the following change of variables and let
| |
(69) |
Therefore, and become
| |
(70) |
The derivatives of and with respect to the
in-line component of the wavenumber kx
and the cross-line component ky can be written as
| |
|
| (71) |
Making one more change of variables, we let
| |
(72) |
Setting the derivative of the phase to zero yields the system
of equations:
| |
(73) |
which we solve for and (i.e., and )
at the stationary
path . The determinant of the system is given by
| |
(74) |
and the solutions for and are
| |
(75) |
and
| |
(76) |
Now we need to evaluate the phase function along
the stationary path .By respectively multiplying the equations in eq36 by
k0x and k0y and summing them together we obtain,
| |
(77) |
Substituting this relationship into the expression of the phase
function [equation phase_app.eq] we obtain
| |
(78) |
The phase function along the stationary path is thus
peaked for
| |
(79) |
Substituting equations eq44 and eq48
into ratio.eq we obtain amo_surf.eq
of the main text:
| |
(80) |
Next: AMO amplitudes
Up: Derivation of integral AMO
Previous: Derivation of integral AMO
Stanford Exploration Project
1/18/2001