nizar@sep.stanford.edu, biondo@sep.stanford.edu

## ABSTRACT
Starting from the definition of Azimuth Moveout ( |

Previously, Biondi and Chemingui introduced a
partial-migration operator named Azimuth-Offset
Moveout (*AMO*) that rotates the data azimuth and
changes the data absolute offset.
The *AMO* operator can be defined as the cascade of an imaging operator
that acts on data with a given offset and azimuth, followed
by a forward modeling operator that reconstructs the data
at a different offset and azimuth.

In the context of amplitude preserving processing, we need to derive
a true amplitude function for *AMO* so that amplitude variations as a
function of offset and azimuth are not distorted by this operation.
Because we derived *AMO* from *DMO*, *AMO* potentially has
amplitude effects similar to those of *DMO*. Starting from the general definition
of *DMO* in the FK domain (, , ) and the
definition of a general inverse *DMO*
(, ), we
derived inverses for Zhang's *DMO* and for Bleistein's Born *DMO*. Our
derivation of true inverses is similar to that
of Liner
for an amplitude-preserved inverse for Hale's *DMO*. The
approach is based on a general formalism for inverting integral solutions
(, ) that we use to derive a solution
for an integral inverse *DMO* that is asymptotically valid.
Our motivation for this approach is to compare our inverses
to an inverse *DMO* formula
of Ronen's and Liner's
. A best amplitude-preserving *DMO* cascaded with
its true amplitude inverse is then selected to define an amplitude
preserving *AMO*. We used this *AMO* operator in an
amplitude-preserved processing sequence consisting of spherical divergence,
normal-moveout (*NMO*), azimuth moveout (*AMO*) and inverse *NMO*.

We conducted numerical experiments by transforming data from a given
azimuth and absolute offset to a new azimuth and offset using different
*AMO* operators defined from kinematically equivalent *DMO*'s and inverse
*DMO*'s. We tested for amplitude preservation by studying the offset-dependent
reflectivity through peak amplitudes along a dipping event
after the *AMO* transformation. According to
most interpreters, ``true-amplitude''
means that each event's peak is proportional to the reflection
coefficient.

In the next section we will briefly review the definition of the *AMO*
operator, describe the general solution for an
asymptotically valid inverse *DMO* () and then derive a
true inverse for Zhang's *DMO* and Bleistein's
Born *DMO* .

AZIMUTH MOVEOUT OPERATOR
We define *AMO* as an operator that transforms 3-D prestack data
with a given offset and azimuth to equivalent data
with different offsets and azimuths ().
Figure shows
a graphical representation of this offset transformation;
the input data with offset
is transformed into data with offset
.*AMO* is not a single-trace to single-trace transformation,
but moves events across midpoints according to their dip.
Therefore, *AMO* is a partial-migration operator.

The *AMO* operator is defined in the the zero-offset frequency and
midpoint wavenumber as

(1) |

Since 3-D prestack data are often irregularly sampled,
it is necessary to define *AMO* as
an integral operator in the time-space domain.
A stationary-phase approximation of amo_freq.eq yields a
time-space representation
of the *AMO* operator where the equation for the kinematics of the
impulse response is ()

(2) |

(3) |

The amplitude behavior of *AMO* is completely controlled by the amplitude
functions of the *DMO* operator and its inverse. An amplitude correct
*AMO* should follow from a true amplitude *DMO* and its amplitude-preserving
inverse. Liner and Cohen argued that an adjoint
*DMO* operator
is a poor representation
of an inverse *DMO*. They showed that for the case of Hale *DMO*, the application
of *DMO* followed by its adjoint inverse can result in a serious amplitude
degradation. They proposed instead a solution for an asymptotically
valid inverse *DMO* and derived a true inverse for Hale's *DMO*. In the next
section we outline their solution and apply it in order
GENERAL FORMALISM FOR INVERSE DMO
*DMO* is a method of transformation of finite-offset data
to zero-offset data. Let the normal moveout corrected input data be
denoted and the zero-offset desired
output denoted . Assume known relationships between
the coordinates of the general form

(4) |

(5) | ||

(6) |

(7) |

(8) |

A detailed derivation of is given by Liner . The method is based on a general formalism (, ) for inverting integral equations such as dmo.eq. The technique mainly involves inserting dmo.eq into dmoinv.eq and expanding the resulting amplitude and phase as a Taylor series and making a change of variables according to Beylkin . The solution provides an asymptotic inverse for dmo.eq, where the weights are given by

(9) |

(10) |

The quantity
is the inverse of the Beylkin determinant, *H*,

(11) |

(12) |

(13) |

(14) |

(15) |

It is very important to recognize at this stage that and in equations omega and wave.numb depend on the coordinate
relationships coord.relat. Therefore, the Beylkin determinant, *H*, is
different for different *DMO* operators. However, as we will demonstrate later
, for kinematically equivalent *DMO* operators the determinant is constant.
We conclude that a general inverse *DMO*
is completely defined given the coordinate
relationships connecting output time and mid point
to their input values. The asymptotic inverse represents an
amplitude-preserving inverse *DMO*.
The methodology was first
applied to Hale's *DMO* by Liner and Cohen ,
and we outline
their results in the next section.

Hale DMO and its inverse Starting from the coordinate relationships between a finite-offset data and its equivalent zero-offset data

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

To compute the Beylkin determinant for Black/Zhang Jacobian we start by
writing the phase phase function in the *DMO* integral kernel as:

(23) | ||

(24) | ||

(25) | ||

(26) |

(27) |

(28) |

Finally, by substituting back in jacob2 and accounting for the factor in the spatial Fourier transform, we obtain an expression for
the weights of an asymptotic inverse for Black/Zhang's *DMO*:

(29) |

(30) |

(31) |

(32) |

(33) |

Summary of true inverse for FK DMO
In this section we analyze the concept of an asymptotic true inverse
and relate it to our application of *AMO*.
Figures spike and comp-idmo compare results of
different inverse *DMO* operators.
Figure spike is similar to the spike test
of Liner .
The left plot is an in-line section from a common offset cube consisting of
five unit-amplitude spikes. The offset is 800m and the CMP spacing is 20 m in
both directions. We compare the output of each true inverse
to the output of Ronen inverse.
The ideal output would be five spikes with unit amplitudes. The table below
summarizes the output of each inverse *DMO* for increasingly deep spikes.

Figure 1

IDEAL Amp. |
Hale |
Black/Zhang |
Bleistein |
Ronen |

1.00 | .76 | .76 | .76 | .35 |

1.00 | .95 | .95 | .95 | .66 |

1.00 | .97 | .97 | .97 | .77 |

1.00 | .97 | .97 | .97 | .82 |

1.00 | .98 | .98 | .98 | .86 |

For both tests the results from the three
different *DMO*'s analyzed were identical. This behavior follows directly
from the kinematic equivalence
of each of these *DMO* operators. On the spike
test we also notice that the results of the inversion are more accurate
from the shallowest to the deepest spike illustrating the asymptotic
nature of the true inverse. Note that the two tests are only conclusive on
the accuracy of the inverse *DMO* solution. Since we are interested
in the cascade of both forward and inverse operators acting at two different
offsets and azimuths, we need to understand what happens in the
intermediate mapping to zero offset prior to the forward modeling step.

Figure 2

TRUE AMPLITUDE AMO FROM TRUE AMPLITUDE DMO
Our goal is to define an amplitude-preserving *AMO* from a true amplitude
*DMO* and its true amplitude inverse. The definition of a true amplitude
inverse follows directly from an amplitude-preserving
forward operator. To select a true amplitude *DMO* we compare the
behavior of various *DMO* algorithms on a dipping bed. Figure comp-dmo
shows the peak amplitudes picked along
the reflection event from the output of several forward *DMO* operators.
The input data
is a constant offset section modeled with a 3-D Kirchhoff
style modeling algorithm. The input was corrected for
spherical-divergence spreading and for *NMO*
effects. On the same plot we also superimpose
the peak amplitudes from a zero-offset section generated with
the Kirchhoff modeling program.
As we notice, the theoretical curve almost coincides with the output amplitudes
of Zhang *DMO*. The amplitudes given by Hale's algorithm fall below the
theoretical curve whereas the peak-amplitude from Bleistein's output
overshoot the correct amplitudes. To understand this behavior we need
to examine the difference between what each *DMO* is trying to accomplish.

The difference between Bleistein's *DMO* and Black/Zhang's *DMO* results from
a philosophical difference about what could be
defined as ``true-amplitude *DMO*''.
While our goal was preserving the peak amplitude of
each reflection event, Bleistein's algorithm is based on preserving the
spectral density of the image wavelet. A second difference results from
the sequence in the processing flow surrounding *DMO*. A divergence
correction must be applied to the input prior to applying Black/Zhang's *DMO*,
whereas both input and output of Bleistein's *DMO* decay
with spherical divergence
factors of and and respectively
().
These two differences account for the *A ^{2}* factor between the two Jacobians
leading to higher weights on Bleistein's

On the other hand, the difference between Black/Zhang's *DMO* and Hale's *DMO*
results from the fact that the former algorithm accounts for the reflection
point smear and, therefore, correctly repositions input events
at their true zero-offset
locations.
The two Jacobians differ by a factor of

(34) |

Consequently, to be consistent with our original definition
of ``amplitude preserved processing'', we chose to define the amplitude function
for the *AMO* operator from the Jacobian of Black/Zhang's *DMO* and the
Jacobian of its corresponding asymptotic true inverse. In the remainder of the
paper we examine the amplitude behavior of *AMO* according to this definition.

Figure 3

AMPLITUDE PRESERVATION BY AMO
The *AMO* operator is defined as the cascade of an imaging operator that
acts on data with a given offset and azimuth, followed by a forward
modeling operator that reconstructs the data at a different offset and
azimuth. *AMO* can also be defined as the cascade of an offset continuation
operator that changes the data absolute offset followed by an azimuth
continuation operator that rotates the data azimuth. These two operations
do commute and in some applications the *AMO* operation may reduce to
simply one or the other.

To examine the amplitude behavior of *AMO*, we
conducted various numerical experiments
and tested for amplitude preservation for a
dipping reflector. In a first experiment we apply *AMO* as an azimuth
continuation operator to change the orientation of the input. In a second
test *AMO* acts a 2-D offset continuation that modifies the offset of the
data. In a final test we apply *AMO* as a vector-offset continuation operator
where both offset and azimuth are modified during the transformation.
For each experiment we compare the peak amplitudes picked on the
*AMO* reconstructed sections to the peak amplitudes extracted from identical
sections generated by synthetic modeling. To better illustrate the difference
in amplitudes, we slightly smooth the curves of amplitude picks
on both sections
Azimuth continuation
Starting from a an input constant-offset section recorded at half offset of
800 m and an angle of 5 degrees measured from the dip direction, we rotate the
azimuth of the data by 40 degrees while keeping the offset constant.
We compare the reconstructed section to a constant offset section modeled
by the 3-D modeling code at an offset of 800m and azimuth of 45 degrees.
Figure azimuth shows the peak amplitudes extracted from
the output of *AMO*
along the dipping event.
On the same graph we also plot the peak amplitudes as picked from
the modeled section. Note that the two curves are very similar with the
reconstructed amplitudes being few percent lower than the predicted
peak amplitudes.

Figure 4

Offset continuation
In the case of no azimuthal change, *AMO* reduces to a 2-dimensional
operator that is equivalent to an offset continuation operator. We apply
*AMO* to the same input constant-offset section recorded at half offset 800m
and 5-dgree azimuth to change
its offset to 400m.
Figure offset shows the peak amplitudes picked along the dipping
event on the reconstructed section together with the theoretical curve
from the modeling program. Again we notice that both plots follow
each other very closely with an error of less than a few percent.

Figure 5

Vector-offset continuation
In a final experiment we apply the *AMO* operator to
transform the input constant-offset section of the first test to a
new section with different
absolute half offset of 400m and azimuth of 45 dgrees.
Figure amo shows the peak amplitudes picked along the dipping event
on the reconstructed section. For comparison, we also plot the peak
amplitudes from a reference section that is modeled with the same vector offset
as the output of *AMO*. The two curves match very closely and the differences
are more contributed to cumulative errors in the processing sequence
surrounding *AMO*, which includes spherical divergence
and *NMO* corrections prior to *AMO* and inverse *NMO* after *AMO*.

Figure 6

CONCLUSION
We presented an amplitude-preserving function for the *AMO* operator.
This amplitude function is based on the FK definition of *DMO*
and the definition of its true inverse. Similar to Liner's formalism
of a true inverse for Hale's *DMO*,
we derived an asymptotically true inverse for Black/Zhang's *DMO*
and Bleinstein's Born *DMO*.
Numerical experiments showed that Black/Zhang *DMO* best presrves
peak amplitudes.
We define amplitude-preserved processing as the preservation of the
offset-dependent reflectivity after *AMO* transformation, where the
reflectivity is considered to be proportional to the peak amplitude
of each event.
We used Black/Zhang's *DMO* cascaded with its true inverse
to define an amplitude function for the AMO operator.
Results showed that we can preserve peak amplitudes for a dipping event
after applying AMO as an azimuth continuation operator, as an offset
continuation operator or a vector-offset transformation.
We conclude that the new AMO amplitude function represents
an amplitude-preserving azimuth moveout.
ACKNOWLEDGMENTS
We thank Norman Bleistein, Paul Fowler, Chris Liner and Mihai Popovici for
useful discussions on DMO. David lumley helped define the problem of
true amplitude processing.
A
CONNECTING FK AMO WITH INTEGRAL AMO
This appendix describes the derivation of the amplitude function for
the AMO impulse response in the time-space domain. The determinant *D*
in Equation amo.amp
of the main text can be written in terms of Biondi and Chemingui
notations as:

(35) |

(36) |

(37) |

(38) |

(39) |

(40) |

(41) |

(42) |

(43) |

[SEP,MISC,EAEG,geophysics,paper]

5/9/2001