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IN THE TIMESPACE DOMAIN
In this appendix, we present an alternative derivation of the AMO
operator. The entire derivation is carried out in the timespace
domain. It applies the idea of cascading DMO and inverse DMO,
developed in appendix A, but uses the integral
formulation of DMO
Deregowski and Rocca (1981); Deregowski (1986); Hale (1991)
in place of the frequencydomain DMO.
Let be the input of an AMO
operator (commonazimuth and commonoffset seismic
reflection data after normal moveout correction) and
be the output. Then the
threedimensional AMO operator takes the following general form:
 
(39) 
where is the differentiation operator
(equivalent to multiplication by in the frequency domain),
is the difference vector between the input
and the output midpoints, is the summation path,
and w_{12} is the weighting function.
To derive (40) in the timespace domain we cascade
an integral DMO operator of the form
 
(40) 
with an inverse DMO of the form
 
(41) 
Where and are the
summation paths of the DMO and inverse DMO operators
Deregowski and Rocca (1981):
 
(42) 
w_{10} and w_{02} are the corresponding weighting functions
(amplitudes of impulse responses);
is the component of
along the azimuth; is the component
of along the azimuth;
and , . stands for the operator of halforder
differentiation (equivalent to the multiplication in
Fourier domain).
Both DMO and inverse DMO operate as 2D operators on 3D seismic data,
because their apertures are defined on a line.
This implies that for a
given input midpoint , the corresponding location of must belong to the line going through , with the azimuth
defined by the input offset . Similarly, must be on the line going through with the azimuth of .These geometrical considerations lead us to the following
conclusion: For a given pair of input and output midpoints
and of the AMO operator, the corresponding midpoint
on the intermediate zerooffset gather is determined by the
intersection of two lines drawn through and in the
offset directions.
Applying the geometric connection among the three
midpoints, we can find the cascade of the DMO and inverse DMO
operators in one step. For this purpose, it is sufficient to notice
that the angles in the triangle, formed by the midpoints ,, and , satisfy the law of sines:
 
(43) 
Substituting equation (41) into (42), taking
into account (44),
and neglecting the loworder asymptotic terms,
produces the 3D integral AMO operator (40), where
 
(44) 
 
(45) 
Equation (45) is the reciprocal of,
and thus equivalent to
equation (1) in the main text.
The factor
in the denominator of the
equation (46) appears as the result
of the midpointcoordinate transformation
.
The timeandspace analogue of amplitudepreserving DMO
Black et al. (1993) has the weighting function
 
(46) 
while its asymptotic inverse
has the weighting function
 
(47) 
Inserting (47) and (48) into (46),
and using the equality ,similarly to appendix A, yields
 

 (48) 
which is equivalent to equation (4) in the main text.
Next: 2D AMO operator
Up: Biondi, Fomel & Chemingui:
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Stanford Exploration Project
11/11/1997