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This section discusses the connection between the laws of traveltime
transformation and the laws of the corresponding amplitude
transformation. The change of the wave amplitudes in the OC process
is described by the firstorder partial differential transport
equation (). We can find the general solution of
this equation by applying the method of characteristics. The solution
takes the explicit integral form
 
(179) 
The integral in equation () is defined on a curved
time ray, and A_{n}(t_{n}) stands for the amplitude transported along
this ray. In the case of a plane dipping reflector, the ray amplitude
can be immediately evaluated by substituting the explicit traveltime
and time ray equations from the preceding section
into (). The amplitude expression in this case takes
the simple form
 
(180) 
In order to consider the more general case of a curvilinear reflector,
we need to take into account the connection between the traveltime
derivatives in () and the geometry of the reflector.
As follows directly from the trigonometry of the incident and
reflected rays triangle (Figure ),
 
(181) 
 (182) 
 (183) 
where D is the length of the normal ray. Let be the
zerooffset reflection traveltime. Combining
equations () and () with
(), we can get the following relationship:
 
(184) 
which describes the ``DMO smile'' () found by
Deregowski and Rocca 1981 in geometric terms.
Equation () allows a convenient change of variables
in (). Let the reflection angle be a
parameter monotonically increasing along a time ray. In this case,
each time ray is uniquely determined by the position of the reflection
point, which in turn is defined by the values of D and .According to this change of variables, we can
differentiate () along a time ray to get
 
(185) 
Note also that the quantity in equation () coincides
exactly with the time ray invariant C_{3} found in
equation (). Therefore its value is constant along each
time ray and equals
 
(186) 
Finally, as shown in Appendix ,
 
(187) 
where K is the reflector curvature at the reflection point.
Substituting (), (), and ()
into () transforms the integral to the form
 
(188) 
which we can evaluate analytically. The final formula for the
amplitude transformation takes the form
 

 (189) 
In case of a plane reflector, the curvature K is zero, and ()
coincides with (). Equation () can be rewritten as
 
(190) 
where c is constant along each time ray (it may vary with the reflection point
location on the reflector but not with the offset). We should compare equation
()
with the known expression for the reflection wave amplitude of the leading
ray series term in 2.5D media:
 
(191) 
where C_{R} stands for the angledependent reflection coefficient, G is the
geometric spreading
 
(192) 
and includes other possible factors (such as the source
directivity) that we can either correct or neglect in the preliminary
processing. It is evident that the curvature dependence of the
amplitude transformation () coincides completely with
the true geometric spreading factor () and that the angle
dependence of the reflection coefficient is not provided by the offset
continuation process. If the wavelet shape of the reflected wave on
seismic sections [R_{n} in equation ()] is
described by the delta function, then, as follows from the known
properties of this function,
 
(193) 
which leads to the equality
 
(194) 
Combining equation () with equations ()
and () allows us to evaluate the amplitude after
continuation from some initial offset h_{0} to another offset h_{1},
as follows:
 
(195) 
According to equation (), the OC process described by
equation () is amplitudepreserving in the sense
that corresponds to Born DMO Bleistein (1990); Liner (1991). This means
that the geometric spreading factor from the initial amplitudes is
transformed to the true geometric spreading on the continued section,
while the reflection coefficient stays the same. This remarkable
dynamic property allows AVO (amplitude versus offset) analysis to be
performed by a dynamic comparison between true constantoffset
sections and the sections transformed by OC from different offsets.
With a simple trick, the offset coordinate is transferred to the
reflection angles for the AVO analysis. As follows from ()
and (),
 
(196) 
If we include the factor in the DMO operator
(continuation to zero offset) and divide the result by the DMO section
obtained without this factor, the resultant amplitude of the reflected
events will be directly proportional to , where the
reflection angle corresponds to the initial offset. Of
course, this conclusion is rigorously valid for constantvelocity
2.5D media only.
Black et al. (1993) suggest a definition of trueamplitude DMO different
from that of Born DMO. The difference consists of two important
components:
 1.
 Trueamplitude DMO addresses preserving the peak amplitude
of the image wavelet instead of preserving its spectral density.
In the terms of this chapter, the peak amplitude corresponds to the
preNMO amplitude A from formula () instead of
corresponding to the spectral density amplitude A_{n}. A simple
correction factor would help us take the difference
between the two amplitudes into account. Multiplication by can be easily done at the NMO stage.
 2.
 Seismic sections are multiplied by time to correct for the
geometric spreading factor prior to DMO (or, in our case, offset
continuation) processing.
As follows
from (), multiplication by t is a valid geometric
spreading correction for plane reflectors only. It is the
amplitudepreserving offset continuation based on the OC
equation () that is able to correct for the
curvaturedependent factor in the amplitude. To take into account the
second aspect of Black's definition, we can consider the modified
field such that
 
(197) 
Substituting () into the OC equation () transforms the
latter to the form
 
(198) 
Equations () and () differ
only with respect to the firstorder damping term . This term affects the amplitude behavior but not
the traveltimes, since the eikonaltype equation ()
depends on the secondorder terms only. Offset continuation operators
based on () conform to Black's definition of
trueamplitude processing.
Next: Confirmation of offset continuation
Up: Introducing the offset continuation
Previous: Offset continuation geometry: time
Stanford Exploration Project
12/28/2000