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IN THE FREQUENCYWAVENUMBER DOMAIN
Our derivation of the AMO operator starts from
the Fourierdomain formulation of DMO
Hale (1984)
and ``inverse'' DMO
Liner (1990); Ronen (1987).
However, because 3D prestack data is often irregularly sampled,
AMO is most conveniently applied as
an integral operator in the timespace domain.
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
 
(13) 
 
(14) 
The operator is given by the cascades
of and and its impulse response can
can be written as,
 
(15) 
The derivation of the stationaryphase approximation of the
integral in is fairly lengthy and complex.
The following outline
has a similar flavor to the stationaryphase approximation
of the conventional DMO impulse response
presented in Black et al. (1993).
We begin by changing the order of the integrals
and rewriting (15) as
The phase of this integral is,
 
(16) 
where,
 
(17) 
Next we make the following change of variables and let
 
(18) 
Therefore, and become
 
(19) 
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
 

 (20) 
Making one more change of variables, we let
 
(21) 
Setting the derivative of the phase to zero yields the system
of equations:
 
(22) 
which we solve for and (i.e., and )
at the stationary
path . The determinant of the system is given by
 
(23) 
and the solutions for and are
 
(24) 
and
 
(25) 
Now we need to evaluate the phase function along
the stationary path .By respectively multiplying the equations in (23) by
k_{0x} and k_{0y} and summing them together we obtain,
 
(26) 
Substituting this relationship into the expression of the phase
function [equation (17)] we obtain
 
(27) 
The phase function along the stationary path is thus
peaked for
 
(28) 
Substituting equations (25) and (26)
into (29) we obtain (1)
of the main text:
 
(29) 
Next we will derive an expression for the amplitudes of the AMO
impulse response.
The general expression for the stationaryphase approximation
of the integral in equation (15)
is Bleistein and Handelsman (1975),
 
(30) 
Therefore we need to evaluate the determinant and the signature
of the curvature matrix , which is defined as
 
(31) 
Taking the secondorder partial derivatives of with respect to k_{x} and
k_{y} and using the definitions of and yields
the following expressions for
, and :
 
(32) 
 (33) 
 (34) 
With a little algebra, one may verify that the determinant of the curvature
matrix is
 

 (35) 
We notice that the determinant of ,which is the product of the two eigenvalues of ,is always negative,
that is; that the two eigenvalues have opposite signs
and thus the signature of ,which is defined as the number of
positive eigenvalues minus the number of negative eigenvalues,
is always null.
Therefore, the second term
of the phase shift in equation (31) vanishes.
To obtain expressions for the AMO amplitude,
we need to substitute equation (36)
in equation (31),
together with the corresponding expressions
for J_{1} and J_{2}.
For the Jacobian J_{1} of the forward DMO
we can use any of the Jacobians proposed in the literature
by Hale 1984,
Zhang and Black 1988,
and Bleistein 1990.
The Jacobian J_{2} of inverse DMO
can be derived with Beylkin's theory
for the asymptotic inverse of stacking operators
Beylkin (1985); Cohen and Hagin (1985).
The expression for the Jacobian of the
asymptotic inverse for Hale's DMO were derived
by Liner and Cohen 1988.
Chemingui and Biondi 1995 and
Fowler (personal communication) independently derived
the inverse for ZhangBlack's DMO.
As mentioned in the main text, we used ZhangBlack's Jacobians
for the actual application of AMO; that is,
 
(36) 
Finally,
after taking into account he Jacobian of the transformation
from t_{1} to t_{0} ()in the first integral of
equation (16),
we can write the amplitude term for the AMO integral:
 
(37) 
 (38) 
The last substitution, ,enables us to apply the differentiation operator
to the output data;
it is correct because t_{0} and t_{2} are linked
by the linear relationship
.
The expression for the amplitudes presented
in equation (4) of the main text
follows by simple substitution of
the expressions for , ,and , from equations (24), (25) and (26)
into equation (39).
Next: 2D AMO operator
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Stanford Exploration Project
11/11/1997