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\def\figsa{/scr/jos/Figb}
\def\figs{/scr/jos/junk}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\dsp{\displaystyle}
\def\h{h}			% offset 
\def\xm{x_m}			% x coordinate migrated point.
\def\zm{z_m}			% z coordinate migrated point.
\def\of{J}			% objective function
\def\S{{\bf S}}			% semblance trajectory
\def\Spz{\S_{+\Delta z}}	% semblance trajectory
\def\Smz{\S_{-\Delta z}}	% semblance trajectory
\def\DofDzm{\dsp{{\partial \of} \over {\partial \zm}}}
\title{Structural-velocity estimation -- \\
imaging salt structures}
\author{Jos van Trier}
\lefthead{Van Trier}
\righthead{Structural-velocity estimation}
\footer{SEP--61}
\ABS{
Depth migration is often needed for the correct imaging of salt structures.
However, the velocity model necessary for the migration is hard to
determine in salt regions, because of large lateral velocity 
variations and poor data quality.
In previous reports I discussed a gradient optimization method to estimate 
a structural-velocity model in these regions. The method is based on
the inversion of perturbations in prestack migrated events, where the 
events correspond to structural boundaries picked from the migrated image.
The optimization does not require picking of prestack data; 
the data part of the gradient is determined from local semblance calculations.
A preliminary analysis of a dataset recorded above a salt structure
explores the difficulties in salt imaging, and illustrates the gradient
calculation.
}
\INT
Although many migration algorithms are now capable of
accurately imaging seismic data, imaging near salt structures 
remains a hard problem to solve. Complications in the imaging
arise for several reasons. First, because of the high-velocity salt 
intrusion, velocity generally varies strongly, both in depth and laterally.
Reflection events are therefore non-hyperbolic, and time migration does not
produce satisfactory results. Instead, depth migration is needed,
not only for focusing the data, but also for correctly locating salt boundaries
(Larner,~1987).
Second, data quality can often be poor in salt regions. 
The intrusive salt flow breaks up sediments above and alongside the structure,
and these broken-up sediments do not generate clear reflections. 
Third, because of the high reflection coefficient of salt-sediment contrasts,
little seismic energy penetrates the salt structure, and reflections
from sediments below the salt are weak.

In previous reports (Van Trier, 1988; Van Trier, 1989a) I discussed
a method to try to solve the first problem, the depth-migration and
its related velocity-estimation problem -- the second and third problem are
mostly data acquisition problems.
The velocity-estimation method determines structural velocities 
using migrated seismic data. After migration with an initial 
velocity model, structural boundaries 
are picked from the migrated image. The residual perturbations in the
prestack migrated events that correspond to the
picked reflectors are then used to optimize the velocity model.
The optimization method is a gradient method, where the
goal is to maximize semblance in the stacked migrated image. The gradient
calculations in the optimization consist of two parts: a tomographic part that
is calculated using ray tracing, and a data part that calculates
semblance derivatives in the prestack migrated data.

%MENTION STACKING??
In the last report I concentrated on the ray-tracing part; in this report
I discuss the data gradient. 
I first show a data example 
that was recorded over a salt layer.
Then I discuss some of the issues in applying the calculation to
salt data, and, finally, I describe the data-gradient
calculations in detail.

\mhead{FIELD DATA EXAMPLE}
This section describes an exploratory study of a marine
dataset recorded over a salt structure. The dataset is from
Conoco, and was donated to SEP by Western Geophysical Co. that
did most of the preprocessing. It contains 
239 shots, each consisting of 84 geophones. The geophone
spacing equals the shot spacing, and is 30 m. The near-offset distance 
is 75 m. The time sampling
interval is 4 ms, the total time 3 s.

\plot{shot1}{3.75in}{\figs/shots}{Shot gathers located at surface positions
of 1260 m ({\it left}) and 2760 m ({\it right}).}
\plot{cdp1}{3.75in}{\figs/cdps}{CMP gathers located at midpoints 
830 m ({\it left}) and  2710 m ({\it right}).}
Figure~\ref{shot1} shows two shot gathers, Figure~\ref{cdp1} displays two CMP
gathers at about the same location. The reflections at 1.3 s and 1.8 s
are reflections from the top and bottom of a salt layer. The extent of
the salt structure is shown on the near-offset (Figure~\ref{zof})
and stacked section (Figure~\ref{stack}). The velocity function
used for the stack is displayed in Figure~\ref{stvel}. 

\plot{zof}{7in}{\figs/zof}{Near-offset section.}
\plot{stack}{7in}{\figs/stack}{Stacked section. The stacking velocity is 
shown in Figure~\protect\ref{stvel}.}
\plot{stvel}{3.75in}{\figs/stvel}{Stacking velocity used for making the stack
shown in Figure~\protect\ref{stack}.}
The top and bottom of the salt are reasonably well-defined in the left and 
right part of the section, but do not show up clearly in the middle part.
The salt has flowed upward
in this middle part, bending and faulting the sediments above it.
These warped sediments cause lateral velocity variations, which become
even more pronounced below the salt layer because of variable salt thickness.
The moveout of the reflection events is therefore non-hyperbolic as
can be seen in Figure~\ref{cdp1}, and stacking-velocity analysis breaks down. 

Figures~\ref{velan1}~and~\ref{velan2} show the result of such a
velocity analysis at two midpoints. 
The salt top reflection is mostly hyperbolic
in the left part of the section (the semblance peak at 1.2 s in 
Figure~\ref{velan1} is well-defined), whereas the reflection
becomes non-hyperbolic at the dome:
different parts of the moveout curve
stack in at different velocities, broadening the semblance peaks in the 
velocity panels. The non-hyperbolicity is caused by lateral 
velocity variations in
the bended sediments above the dome, and probably also by discontinuities 
in the salt top.
The salt bottom reflection is much less apparent in the velocity
analysis; not only do lateral velocity variations distort the moveout
of this reflection, but it is also much weaker: the high
reflection coefficient at the salt boundary prevents seismic energy
from penetrating the salt structure.
\plot{velan1}{3.75in}{\figs/velan1}{Stacking-velocity analysis for CMP gather
at midpoint position of 500 m.}
\plot{velan2}{3.75in}{\figs/velan2}{Stacking-velocity analysis for CMP gather
at midpoint position of 4580 m.}

\plot{depthmigr}{7in}{\figs/depthmigr}{Depth migration. The migration velocity 
is shown in Figure~\protect\ref{mvel}.}
\plot{mvel}{3.75in}{\figs/mvel}{Migration-velocity profile. The velocity
is laterally invariant.}
\shead{Migration}
Figure~\ref{depthmigr} shows the migrated image after prestack depth migration.
In this example I have concentrated on imaging only the sediments and the top
of the salt, which
already turns out to be a complicated task.

For the migration I have used a 
velocity function that changes linearly with depth, with the exception 
of a constant-velocity layer at the top (Figure~\ref{mvel}). The model
is constant in the lateral direction. In reality, the velocity 
is rapidly changing with surface position, and the migration has done
a poor job in imaging the subsurface. This is apparent from several
features in the stack.
First, coherency in the stack
deteriorates below surface positions ranging from about 2 to 3.5
km, which is caused by a low velocity channel at the top of the section.
Second, the frequencies in the stack are lower than in prestack migrated
data (see Figures~\ref{nof} and~\ref{fof}):
velocity effects cause large residual moveout that smears events when 
they are stacked.
Finally, the top of the salt layer does not stack in into the image, but
does show up in the prestack migrated constant-offset sections
(at a depth of about 1.25 km on the left part of the section, and
about 1.5 km on the right in 
Figures~\ref{nof} and~\ref{fof}). The same applies to the fault plane
reflections.
Although the velocity model is far from correct, the result suffices
for the purpose of analyzing the migrated data.

\plot{nof}{7in}{\figs/nof}{Locally-stacked near-offset migrated section.
The offset of the middle section in the stack is 165 m. }
\plot{fof}{7in}{\figs/fof}{Locally-stacked mid-offset migrated section.
The offset of the middle section in the stack is 825 m. }
The migration method 
is a Kirchhoff method, where I calculate Green's functions using 
finite-difference calculations (Van Trier,~1989b). A Kirchhoff method
allows partial imaging of the subsurface, which is useful if only
major events in the data need to be migrated (see later section).
Although the stacked result stays the same, the prestack migrated
data can be organized in different ways, such as migrated shot profiles
or migrated constant-offset sections. The advantage of 
constant-offset migration is that these sections resemble the
geology, and are individually interpretable. This is an important
feature as will become clear later. 

Examples of 
partial stacks of migrated constant-offset sections are shown
for two different offsets in Figures~\ref{nof}~and~\ref{fof}.
These sections display more detail than the stacked image.
In particular the salt top is apparent in the
section, although its exact shape at the peak of the intrusion
is not distinguishable. The migrated data also reveal the dipping
part of the salt top on the right, which is not visible in the
stacked section (Figure~\ref{stack}). Not enough depths are included 
in the migration
to image the salt bottom on the right. Finally, note the
events below the low velocity channel on the left; these
events have almost completely disappeared in the stacked image.

\plot{csl1}{3.75in}{\figsa/cslot}{Migrated CSL gathers at outer parts of section:
surface locations 1.3 km ({\it left}) and  6.7 km ({\it right}).}
\plot{csl4}{3.75in}{\figsa/cslmd}{Migrated CSL gathers in middle part of section,
above the salt intrusion:
surface locations 4 km ({\it left}) and 4.15 km ({\it right}).}
Figures~\ref{csl1} and \ref{csl4} display slices through the prestack
migrated data at various surface locations. If the correct
migration-velocity would have been used, all the events 
in these panels would have been flat. Most of the events
have been overcorrected, meaning that the migration velocity
is too low. At the outer parts of the section the residual moveout
is reasonably well-behaved, but in the middle part, above the salt dome,
the residual moveout
suggests considerable lateral velocity variations.
%LOW-VELOCITY CHANNEL??

\mhead{VELOCITY INFORMATION FROM MIGRATED DATA}
%The previous section exemplified the complexity of the velocity estimation
%problem in salt regions. 
%Apart from the fact that the data contain few clear
%reflections, large lateral velocity variations occur that result into 
%non-hyperbolic moveout of reflection events.
Strong lateral velocity variations occur in salt regions, which result into 
non-hyperbolic moveout of reflection events.
Velocity-estimation methods that use stacking or other coherency 
measures calculated over the whole cable length may therefore not resolve
local perturbations in the moveout of reflection
events. While having higher resolution, tomographic methods that use
picked traveltimes cannot easily be applied to salt data because
of the mentioned data quality problem.

Migrating the data partially solves the velocity problem: by incorporating
lateral variations in the migration velocity, the velocity-estimation
method only has to backproject residual moveout. However, as was shown
in Figure~\ref{csl4}, the residual moveout pattern can still be
complex in salt data. Therefore, I use a local stacking operator
for the gradient calculations in the optimization, and I constrain
the velocity model by an interactive interpretation of the migrated
image. The next two sections discuss these procedures in detail. 
%CROSSING EVENTS IN UNMIGRATED DATA?

\shead{Gradient calculation}
In SEP-60 I discussed an optimization method that backprojects perturbations
in prestack migrated events onto the velocity model. 
In general, the perturbations are not readily available; picking them
on the constant surface location gathers is cumbersome and not very 
reliable for the data shown here.
However, it may be feasible to pick events from the migrated 
constant-offset sections after they have been locally stacked to 
reduce noise. The sections resemble the geology, at least for the inner 
offsets, as can be seen in Figure~\ref{nof}~and~\ref{fof}.

\plot{sembl}{3.2in}{\figs/sembl}{Schematic explanation of calculation of
semblance derivatives by finite differences. $\Spz$ and $\Smz$ are
stacking trajectories perturbed from the current trajectory $\S$.
Here the current trajectory is a straight line, but it may become
curved during the optimization.} 
A method that avoids picking altogether uses semblance derivatives,
and was described in my previous report. Semblance in the
stacked image is calculated
along stacking trajectories in the prestack migrated data. Instead
of picking perturbations in the stacking trajectory, semblance
derivatives along the trajectory drive the optimization method.
I briefly review the calculations here; for more details I refer the reader to
the SEP-60.

The derivative of semblance $\of$ with respect to $\zm$ for a certain event
is:
\be
\DofDzm (\S, \h) \ =\ {{\of(\Spz(\h))\ -\ \of(\Smz(\h))} \over {2\Delta z}},
\ee
where $h$ is offset, and $(\xm,\zm)$
the position after migration.
$\S$ is the current stacking trajectory, and $\Spz$ and $\Smz$ are 
perturbations thereof. $\Spz(\h)$ is the upward perturbed trajectory; 
it is the same as $\S$, except
for a perturbation by $+\Delta z$ in the $\zm$ direction
at offset $\h$: 
\be
\Spz(\h)\ =\ \left\{ \begin{array}{ll}
		\ (\xm(h), \zm(h)+\Delta z, \h)\ & {\rm at\ offset}\ \h,\\ 
		\ \S\ & {\rm elsewhere.}
			\end{array}
		\right. 
\ee
%$\Spz=(\xm(h), \zm(h)+\Delta z, \h)$, and likewise,
%$\Smz=(\xm(h), \zm(h)-\Delta z, \h)$. 
Likewise, $\Smz(\h)$ is perturbed by $-\Delta z$ at offset $\h$.

The gradient calculation is not very robust if the trajectory is just shifted at
one offset, as described in the above equation.
Therefore, the path is perturbed at several offsets, 
with the perturbations decreasing away from the offset under consideration.
This is schematically illustrated in Figure~\ref{sembl}. 

\shead{Structural analysis}
Since salt has a high-velocity contrast with respect to the surrounding
sediments, and accurate positioning of 
salt boundaries is crucial for oil exploration, it is important to find 
a structural
velocity model in the velocity analysis, rather than one that just
focuses the migrated data.
Also, constraining the velocity model by an (interactive)
 structural interpretation of 
the image reduces the model null space in the inversion, and thus 
speeds up the optimization.
%Finally, the number of events contributing to the velocity
%only few clear reflections are apparent in the data.

Therefore, I construct a structural model from the migrated image
and limit the optimization to the events in the data that correspond
to the major structural boundaries. This significantly reduces the
amount of data in the optimization, and allows me to do prestack
migrations at little cost in different iterations in the migration. 
Kirchhoff migration is well-suited for migrating limited amounts of
data to certain target regions. The targets here are windows around
the boundaries: if the model is updated, a simple event migration determines
the position of the updated boundaries, and Kirchhoff imaging only has
to be done for gates around the updated depth positions of the boundaries.

%I am currently working on an interactive interface to the migration
%programs that allows the user to quickly scan and remigrate the
%prestack migrated events for different velocity models. This facilitates 
%the easy construction of an initial structural model, without which
%the velocity estimation would probably be very difficult.
%The interface runs under the X-window system, and is written in C++ using
%the InterViews toolkit.
%
The constantly changing positions of the windows in the prestack depth-migrated
data can be problematic in the optimization. A possible solution
is to do zero-offset modeling of the migrated constant-offset sections;
the depth axis then gets converted to a pseudo-depth axis, and the
zero-offset section stays constant. The converted data is the same
as DMO-corrected data, where the DMO correction is calculated for
a arbitrary velocity medium (Popovici and Biondi, 1989).

\plot{zofsect}{7in}{\figs/zofmodsect}{Zero-offset modeling of migrated 
constant-offset section shown in Figure~\protect\ref{fof}.}
\plot{zofcsl}{3.75in}{\figs/zofmodcsl}{Constant surface location gathers 
of migrated data, converted to zero-offset time domain.
Surface locations are 4 km ({\it left}) and 6 km ({\it right}).}
This conversion also has some advantages for
the optimization itself: the backprojection operator does not have
to take movements of the zero-offset section into account, 
but instead concentrates on the non-zero-offset behavior of the 
reflections events -- the part that provides the most velocity information.
I discussed this matter in more detail in my previous report (Van Trier,~1989a).
Figure~\ref{zofsect}~and~\ref{zofcsl} show the result of zero-offset 
modeling the migrated constant-offset sections; the plots
show a constant offset and constant surface
location slices through the converted data, respectively. 
A disadvantage of this data representation is that time sections
are harder to interpret for complex structure.
It is hard to
decide which data representation is the most convenient in velocity
analysis without actually applying the velocity inversion;
I will investigate this in the near future.
%Since the velocity model is laterally 
%invariant in this case, the transformation is just a stretch of the
%depth axis.

\mhead{CONCLUSIONS}
In previous reports I have presented a 
method to analyze prestack
migrated data that can be used in a structural-velocity optimization.
The method relies on a structural interpretation of the migrated
image, and does not require picking of prestack migrated data.
In this report I have shown an example of salt data, the type of data that
I intend to apply my method to. 
Several conclusions can be drawn from the results presented here.
First, a structural analysis is necessary to geologically constrain the
velocity model. Salt regions have complex structure, and automated
velocity-estimation methods often fail without the help of interactive
interpretation of the inversion results. Second, even though 
migrating the data with a 2-dimensional velocity model reduces the effects
of lateral velocity variation on the data, the residual moveout in
the data can be complicated.
Therefore, local effects in the residual moveout curves have to 
be taken into account in the velocity estimation to determine correctly
the structural model. 

\ACK
I thank Conoco and Western Geophysical Co., and Oz Yilmaz
in particular, for providing SEP with the data.
%As usual, thanks to my office mates Biondo Biondi and John Etgen
%for interesting discussions on velocity analysis.
%SHELL??

\REF
\reference{Larner, K., 1987, In quest of the flank: Presented at
the 57th Ann. Internat. Mtg., Soc. Explor. Geophys.}
\reference{Popovici, A., and Biondi B., 
Kinematics of Prestack Partial Migration in a variable velocity medium:
SEP--{\bf 61}.}
\reference{Van Trier, J., 1988, Velocity analysis of migrated seismic data
after structural interpretation: SEP--{\bf 59}, 141--150.}
\reference{Van Trier, J., 1989a, Structural velocity analysis using migrated 
seismic data: SEP--{\bf 60}, 41--66.}
\reference{Van Trier, J., 1989b, Finite-difference calculation of traveltimes: 
SEP--{\bf 61}.}