Next: AMO amplitudes
Up: Derivation of integral AMO
Previous: Derivation of integral AMO
To derive a timespace representation of the AMO impulse response
from its frequencywavenumber representation,
we evaluate the stationaryphase approximation of
the inverse Fourier transform along the midpoint coordinates.
The operator and its inverse, , can be
defined in the zerooffset 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 stationaryphase approximation of the
integral in is similar to the one presented in
Black et al. (1993b) for deriving a timespace 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
inline component of the wavenumber k_{x}
and the crossline component k_{y} 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
k_{0x} and k_{0y} 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