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The final step is to take the derivative of the
impulse response of equation (34)
and use the relationships of these derivatives with
and :
 
(35) 
 (36) 
 
Substituting equations (35) and (36)
into the following relationships:
 
(37) 
 (38) 
 
and after some algebraic manipulation, we prove the thesis.
B
This appendix demonstrates equations (19) in the main text:
that for energy dipping at an angle in the (z,x) plane, the wavenumber k_{n} along
the normal to the dip is linked to the wavenumbers k_{z} and k_{x}
by the following relationships:
 
(39) 
For energy dipping at an angle the wavenumbers satisfy the wellknown relationship
 
(40) 
where the positive sign is determined by
by the conventions defined in Figure .
The wavenumber k_{n} is related to k_{x} and k_{z} by the
axes rotation
 
(41) 
Substituting
equation (40) into
equation (41) we obtain
 
(42) 
or,
 
(43) 
C
In this appendix we derive the equations for the
``kinematic migration'' of the reflections from a sphere,
as a function of the ratio between the true constant slowness
S and the migration slowness .For a given we want to find the coordinates of the imaging point
as a function
of the apparent geological dip and
the apparent aperture angle .Central to our derivation is the assumption that
the imaging point lies on the normal
to the apparent reflector dip passing through
,as represented in
Figure .
The first step is to establish the relationships
between the true
and and the apparent
and .This can be done
through the relationships
between the propagation directions of the source/receiver rays
(respectively marked as the angles and in Figure ),
and the event time dips, which
are independent on the migration slowness.
The true and can be thus estimated as follows:
 
(44) 
 (45) 
and then the true and are:
 
(46) 
Next step is to take advantage of the fact that the reflector
is a sphere, an thus that
the coordinates of the
true reflection point are uniquely identified by the
dip angle as follows:
 
(47) 
where (z_{c},x_{c}) are the coordinates of the center
of the sphere and R is its radius.
The midpoint, offset, and traveltime of the event
can be found by applying simple trigonometry
(see Sava and Fomel (2002))
as follows:
 
(48) 
 (49) 
 (50) 
The coordinates of the point ,where the source and the receiver rays cross,
are:
 
(51) 
 
 
 (52) 
and the corresponding traveltime
is:
 
(53) 
Once we have the traveltimes t_{D}
and ,the normal shift
can be easily evaluated by applying
equation (15)
(where the background velocity is and the aperture angle is ),
which yields:
 
(54) 
We use equation (54),
together with
equations (51) and (52),
to compute the lines superimposed onto the images
in
Figure .
D
In this Appendix we derive the expression for the residual moveout (RMO)
function to be applied to ADCIGs computed by wavefield continuation.
The derivation follows
the derivation presented in Appendix C.
The main difference
is that in this appendix we assume the rays to be stationary.
In other words, we assume that the
apparent dip angle and aperture angle are the same as the true angles
and .This assumption also implies that the (unknown) true reflector
position coincides with
the point
where the source and the receiver ray cross.
Given these assumptions, the total traveltime
through the perturbed slowness function
is given by the following expression:
 
(55) 
which is different from the corresponding equation
in Appendix C
[equation (53)].
The difference in traveltimes ,where t_{D} is given by equation equation (50),
is thus a linear function
of the difference in slownesses ;that is,
 
(56) 
As in Appendix C,
the normal shift
can be evaluated by applying
equation (15)
(where the background velocity is and the aperture angle is ),
which yields:
 
(57) 
The RMO function () describes the relative movement
of the image point at any with respect to
the image point for the normalincidence event
.From equation (57),
it follows that the RMO function is:
 

 
 (58) 
The true depth is not known,
but at normal incidence
it can be estimated as a function of the migrated depth
z_{0}
by inverting the following relationship:
 
(59) 
as:
 
(60) 
Substituting relation (60)
in equation (58)
we obtain the result:
 
(61) 
which for flat reflectors simplifies into:
 
(62) 
In Figure ,
the solid lines superimposed into the images
are computed using equation (61),
whereas
the dashed lines
are computed using
equation (62).
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Previous: Evaluation of the impulse
Stanford Exploration Project
7/8/2003