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%Chapter 2


% Chapter 3
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\lefthead{Biondi}
\righthead{Velocity from beam stacks}
\footer{SEP--61}
\title{Interval velocity estimation from beam-stacked data --- 
field-data results}
\author{Biondo Biondi}
\ABS
{
My velocity estimation from beam-stacked data
successfully reconstructs
a velocity anomaly from a field data set.
Using the estimated velocity model for depth migrating
the data 
corrects the effects of the anomaly on
the positioning and focusing of the reflectors.
The estimated anomaly well explains the perturbations
in the beam-stacked data caused by the actual velocity anomaly.
An analysis of the transformed data 
also shows that the resolution of beam stacks
is sufficient for measuring different effects of the velocity
anomaly on the moveouts of the reflections at different offsets.
}
\INT
Tomographic methods are needed for estimating a velocity model that
varies rapidly along the lateral direction.
In previous reports I presented the theory (Biondi, 1988) and
the synthetic results (Biondi, 1989) of a velocity-estimation
method capable of reconstructing velocity anomalies shorter than the
cable's length.
In this report I present the results of applying the method
to the problem of 
estimating a velocity anomaly from a real data set.
\par
The data set is particularly interesting 
for testing a velocity-estimation method
because the velocity anomaly 
is shorter than the cable's length and it is not located near the surface
but it is centered at a depth of about 1.5 kilometers.
Therefore both the lateral and the vertical resolution
of the estimation method are tested by the problem presented by the data.
Furthermore the pull-down in the reflections caused by the anomaly
can be corrected only
if a good velocity model is used as an input for depth migration.
\mhead{DESCRIPTION OF THE DATA}
The data that I present in this report are part of a
marine survey donated to SEP by AGIP.
The data were recorded with steam-gun sources and a 2.35-km cable. 
The shot spacing is 25 m and the group interval is 50 m;
consequently the midpoint coverage is 48 fold and the midpoint
spacing is 25 m.
I sorted the data in common-midpoint gathers and kept the 341 gathers
that were full fold.
The data were sampled every 2 ms and were wide-band in frequency,
reflected energy was as high as 100 Hz. 
SEP received the data already gained and deconvolved;
I then muted the data to eliminate the first arrivals
and dip-filtered the constant-offset sections to remove some steeply
dipping noise.
\myplottop
{excmpst.fig}
{4.5\figsp}
{\FIGTWO/excmpst}
{CMP gather and beam stacks}
{
A typical CMP gather
of the AGIP data set and
the result of beam stacking the gather with $\ph$ equal to 0.1 s/km.
}
Figure~\ref{excmpst.fig} 
shows a typical common midpoint gather (CMP) of the data
set, together with its beam-stacks.
\par
\myplotpage
{stack.fig}
{7.5\figsp}
{\FIGFOUR/stack}
{Stacked section}
{
The stacked section
of the AGIP data set.
A low velocity anomaly causes the time sag near the midpoint location
of 5.5 km.
}
Figure~\ref{stack.fig} shows the stack of the data.
The velocity anomaly causes the large pull-down
visible in the stacked section near the midpoint location of 5.5 km.
The data set does not need to be migrated before stack,
because the reflections underneath the anomaly stacked coherently,
even if their moveouts were not hyperbolic.
On the other hand,
a poststack depth migration and a good velocity model are needed 
to position correctly the reflectors.
Depth migration is particularly needed for
correctly positioning the top of the anticline,
which has been flattened by the velocity anomaly.
The conventional methods for estimating interval velocity from
stacking velocities cannot be used for this data set
because the anomaly width, which is about 2 kilometers, is 
smaller than length of the cable. 
Therefore this data set is a good test case for a tomographic velocity
estimation method such as the one using beam stacks.
\mhead{DESCRIPTION OF THE VELOCITY ESTIMATION PROCEDURE}
The velocity-estimation method that I described in previous reports
(Biondi, 1989; Biondi, 1988) is based on the
maximization of beam stacks' energy 
at the traveltimes and surface locations predicted by the velocity function
and modeled with raytracing.
Therefore in order to estimate the velocity,
I computed beam stacks from the prestack data for 12 offset ray parameters
$\ph$, from .05 s/km to .336 s/km,
and for three midpoint ray parameters
$\py$, from -.04 s/km to .04 s/km.
I transformed the beam-stacks according to the coordinate transformation
introduced in the previous report (Biondi, 1989), assuming a constant 
velocity of 2.5 km/s.
I then smoothed the transformed data along the midpoint axis 
and the time axis using Gaussian windows. 
The smoothed and transformed data from the
250 midpoint positions located around the anomaly were used
by the velocity estimation.
\par
The starting solution for the iterative estimation algorithm was
a velocity profile function of depth, but constant
in the lateral direction.
This velocity function was derived from 
conventional stacking velocity analysis applied to a few of the CMP gathers.
The estimation procedure started with few conjugate gradient iterations
until the velocity model predicted the gross feature of the beam-stacked
data, and then continued with some Gauss-Newton iterations
for better estimating the finer components of the velocity function.
\myplotpage
{vanom.fig}
{6.5\figsp}
{\FIGFOUR/vanom}
{Estimated velocity model}
{
The velocity model estimated using beam stacks is shown as
the sum of two components:
one background velocity profile
and one anomalous function.
The background velocity is plotted on the right, 
while the anomalous velocity is displayed with a contour plot
superimposed to the result of migrating the stacked data
using as a velocity function the background velocity.
The contour interval is -40 m/s. 
}
\mhead{THE ESTIMATION RESULTS}
The result of the estimation process is a velocity model with
a velocity anomaly located around the midpoint location at 5.5 kilometers
and centered at a depth of 1.6 kilometers. 
Figure~\ref{vanom.fig} shows the resulting velocity
model as the sum of two components: one background velocity
profile, which is a function of depth but is constant in the lateral direction,
and one anomalous model, 
which is a function of both depth and the lateral position.
The anomalous model is shown with a contour plot
superimposed to the result of migrating the stacked section using
the background velocity.  
The contour interval is -40 m/s, and
the maximum amplitude of the anomaly is 200 m/s.
Migrating the stacked section using the background velocity as 
a velocity function 
did not correct the pull-down
in the structure, although it 
approximately focused the data. 
Figure~\ref{vanom.fig} shows that 
the estimated velocity anomaly is correctly positioned
above the sag.
\par
Figure~\ref{migx.fig} shows the result of migrating the stacked section
using as a velocity function the estimated velocity model; 
that is, the background 
plus the anomalous velocity model. 
The inclusion of the anomaly has caused a considerable change in
the positioning of the reflectors, as it can be noticed by comparing
Figure~\ref{migx.fig} with Figure~\ref{migz.fig}.
\myplotpage
{migx.fig}
{7.5\figsp}
{\FIGFOUR/migx}
{Migrated section} 
{
The result of migrating the stacked data using the estimated velocity
as a velocity function.
}
\myplotpage
{migz.fig}
{7.5\figsp}
{\FIGFOUR/migz}
{Migrated section} 
{
The result of migrating the stacked data using the background velocity
as a velocity function.
}
\myplotpage
{compmig.fig}
{7.5\figsp}
{\FIGFOUR/compmig}
{Migrated sections} 
{
The comparison of the migration results using as velocity functions
the background  velocity (top) and the estimated velocity (bottom).
The estimated velocity correctly positions
the top of the anticline and improves the focusing of the reflectors
underneath the anomaly.
}
\par
The two migration results are better compared by a look at the windows
of the data that have been directly affected by the anomaly,
as shown in Figure~\ref{compmig.fig}.
The estimated 
velocity field corrects the mispositioning of the top of the anticline,
both laterally and in depth.
Furthermore, 
the migration with the background velocity did
not perfectly focus the 
deeper reflectors; this misfocusing caused
a decrease in the amplitudes of the migrated reflectors underneath the anomaly.
This focusing effect is corrected in the migration that
used the estimated velocity model as a velocity function.
\shead{Using beam-stacked data to check the estimation results}
When a velocity model to be used for depth migration is built,
it is important to use the prestack data to check the quality of the results.
The beam-stacked data can be readily used for this purpose of quality check:
the actual position of the semblance peaks in the beam-stacked
data can be compared with the positions predicted by the estimated 
velocity model.
\par
I sliced the beam stacks at the traveltimes
corresponding to the reflector at about 4.5 km in the migrated section 
of Figure~\ref{migx.fig}. 
\myplottop
{slicez.fig}
{3.5\figsp}
{\FIGFOUR/slicez}
{Beam-stacks slice}
{
A slice of the beam stacks corresponding to the reflector at about 4.5 km
in the migrated section of Figure~\protect\ref{migx.fig}.
The thinner line superimposed to the semblance plot
corresponds to the transformed offsets predicted by the background
velocity,
and the thicker line corresponds to the transformed offsets predicted 
by the estimated velocity.
}
Figure~\ref{slicez.fig} shows semblance, 
as a function of the transformed offset and midpoint location,
and for a fixed $\ph$ equal to 0.076 s/km and $\py$ equal to zero.
The thinner line superimposed to the semblance plot
corresponds to the transformed offsets predicted by the background
velocity,
and the thicker line corresponds to the transformed offsets predicted 
by the estimated velocity.
The peaks of the beam stacks are well predicted by the final results
of the estimation.
\myplottop
{slicep.fig}
{3.5\figsp}
{\FIGFOUR/slicep}
{Beam-stacks slice}
{
Three slices of 
the beam stacks corresponding to the reflector at about 4.5 km
in the migrated section of Figure~\protect\ref{migx.fig},
and taken at different midpoint locations.
The effects of the velocity anomaly on the beam-stacked data
are different for different values of the offset ray parameter.
}
\par
One of the main
advantages of estimating velocity with local stacking operators,
such as beam stacks,
is the possibility of inverting data that depend locally
on the moveouts of the reflections.
This advantage would be only theoretical if the resolution of
beam stacks were not sufficient to obtain independent
measures for different offset ray parameters.
Figure~\ref{slicep.fig} shows that it is actually possible to measure
the different effects of the velocity anomaly
on the reflections having different offset ray parameters.
The three panels show slices of the beam-stacked data taken
at three midpoint locations, and for the same reflector as in 
Figure~\ref{slicez.fig}, but as a function of $\ph$.
In this figure the thinner line superimposed to the semblance plot
corresponds to the transformed offsets predicted by the background
velocity,
and the thicker line corresponds to the transformed offsets predicted 
by the estimated velocity.
The first and the third panel correspond to midpoints located on
the side of the anomaly, and consequently the
reflections with higher $\ph$ were more strongly
affected by the anomaly than  were
the ones with lower $\ph$.
On the contrary the panel on the center corresponds to a midpoint
located at the center of the anomaly, and consequently the
reflections with lower $\ph$ were more strongly affected by the anomaly than
were the ones with higher $\ph$.
It is useful to notice that
the beam stacks are zero for $\ph$ higher
than 0.206 s/km because the corresponding reflections would have been
recorded off the end of the cable.
\CON
The velocity-estimation method with beam-stacked data was successful
in estimating a low-velocity anomaly from field data.
Depth migrating the data using the estimated velocity model
corrected
the mispositioning of the reflectors caused by the anomaly and improved
the focusing of the result.
\par
The estimated velocity well explains the prestack data and, in particular, the
beam-stacked data. An analysis of the beam-stacked data shows that  
the resolution of beam stacks is sufficient for the measurement of different
effects of the velocity anomaly on reflections stacked at
different values of the offset
ray parameter.
\ACK
I would like to thank AGIP - Hydrocarbon Exploration and
Deutsche Shell for making available the field data I used.
\par
I would like also to thank Bill Harlan for the interesting conversations on the
tomographic problem we had during the summer.
\REF
\reference{Biondi, B., 1988, 
Interval velocity estimation from beam-stacked data --- 2-D gradient operator: 
SEP--{\bf 59}, 103--120.}
\reference{Biondi, B., 1989, 
Interval velocity estimation from beam-stacked data --- 2-D estimation results: 
SEP--{\bf 60}, 1--24.}
%\reference{Kostov, C., and Biondi, B., 1987, Improved resolution of slant stacks using beam stacks:  SEP--{\bf{51}.}, 343--350.}
