


 Theory of differential offset continuation  

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Appendix
A
Secondorder reflection traveltime derivatives
This appendix contains a derivation of equations connecting
secondorder partial derivatives of the reflection traveltime with the
geometric properties of the reflector in a constant velocity medium.
These equations are used in the main text of this paper to describe
the amplitude behavior of offset continuation. Let be the
reflection traveltime from the source to the receiver .
Consider a formal equality

(114) 
where is the reflection point parameter, corresponds to the
incident ray, and corresponds to the reflected ray.
Differentiating (A1) with respect to and yields
According to Fermat's principle, the twopoint reflection ray path must
correspond to the traveltime stationary point. Therefore

(117) 
for any and . Taking into account (A4) while
differentiating (A2) and (A3), we get
where
Differentiating equation (A4) gives us the additional
pair of equations
where
Solving the system (A8)  (A9) for
and
and substituting
the result into (A5)  (A7) produces the
following set of expressions:
In the case of a constant velocity medium, expressions (A10) to
(A12) can be applied directly to the explicit
equation for the twopoint eikonal

(126) 
Differentiating (A13) and taking into account the trigonometric
relationships for the incident and reflected rays (Figure
1), one can
evaluate all the quantities in (A10) to (A12) explicitly.
After some heavy algebra, the resultant expressions for the traveltime
derivatives take the form

(131) 

(132) 

(133) 
Here is the length of the normal (central) ray, is its dip angle
(
,
),
is the reflection angle
, is the reflector
curvature at the reflection point
, and
is the dimensionless function of and defined in (47).
The equations derived in this appendix were used to obtain the equation

(134) 
which coincides with (50) in the main text.
Appendix
B
The kinematics of offset continuation
This Appendix presents an alternative method to derive equation
(70), which describes the summation path of the
integral offset continuation operator. The method is based on the
following considerations.
The summation path of an integral (stacking) operator coincides with
the phase function of the impulse response of the inverse operator.
Impulse response is by definition the operator reaction to an impulse
in the input data. For the case of offset continuation, the input is a
reflection commonoffset gather. From the physical point of view, an
impulse in this type of data corresponds to the special focusing
reflector (elliptical isochrone) at the depth. Therefore, reflection
from this reflector at a different constant offset corresponds to the
impulse response of the OC operator. In other words, we can view
offset continuation as the result of cascading prestack commonoffset
migration, which produces the elliptic surface, and commonoffset
modeling (inverse migration) for different offsets. This approach
resemble that of Deregowski and Rocca (1981). It was also applied to a
more general case of azimuth moveout (AMO) by
Fomel and Biondi (1995) and fully generalized by
Bleistein and Jaramillo (2000). The geometric approach implies that
in order to find the summation pass of the OC operator, one should
solve the kinematic problem of reflection from an elliptic reflector
whose focuses are in the shot and receiver locations of the output
seismic gather.
In order to solve this problem , let us consider an elliptic surface of
the general form

(135) 
where . In a constant velocity medium, the reflection
ray path for a given sourcereceiver pair on the surface is controlled
by the position of the reflection point . Fermat's principle
provides a required constraint for finding this position. According to
Fermat's principle, the reflection ray path corresponds to a
stationary value of the traveltime. Therefore, in the neighborhood of
this path,

(136) 
where and stand for the source and receiver locations on the
surface, and is the reflection traveltime

(137) 
Substituting equations (B3) and (B1) into
(B2) leads to a quadratic algebraic equation on the
reflection point parameter . This equation has the explicit
solution

(138) 
where , , , and
. Replacing in equation
(B3) with its expression (B4) solves
the kinematic part of the problem, producing the explicit traveltime
expression

(139) 
where
The two branches of equation (B5) correspond to the
difference in the geometry of the reflected rays in two different
situations. When a sourceandreceiver pair is inside the focuses of
the elliptic reflector, the midpoint and the reflection point
are on the same side of the ellipse with respect to its small
semiaxis. They are on different sides in the opposite case (Figure
B1).
Reflections from an ellipse. The three pairs of reflected rays
correspond to a common midpoint (at 0.1) and different offsets. The
focuses of the ellipse are at 1 and 1.
If we apply the NMO correction, equation (B5) is transformed to

(140) 
Then, recalling the relationships between the parameters of the
focusing ellipse , and and the parameters of the
output seismic gather (Deregowski and Rocca, 1981)

(141) 
and substituting expressions (B7) into equation (B6) yields the
expression

(142) 
where
It is easy to verify algebraically the mathematical equivalence of
equation (B8) and equation (70) in the
main text. The kinematic approach described in this appendix applies
equally well to different acquisition configurations of the input and
output data. The sourcereceiver parameterization used in
(B8) is the actual definition for the summation path of
the integral shot continuation operator
(Bagaini and Spagnolini, 1996,1993). A family of these summation
curves is shown in Figure B2.
Summation paths of the integral shot continuation. The output source
is at 0.5 km. The output receiver is at 0.5 km. The indexes of the
curves correspond to the input source location.



 Theory of differential offset continuation  

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