G017
Velocity Model Building by Semblance
Maximization of Modulated-Shot
Migration Gathers
R. Soubaras* (CGG) & B. Gratacos (CGG)
SUMMARY
This paper presents a full wave-equation methodology for velocity model building
based on the non-linear inversion of a semblance criterium with respect to the velocity
field. A new type of migration, called the modulated-shot migration, is used to obtain
the necessary gathers. The semblance of these gathers after spatial averaging is used
as the cost function. This methodology is shown to successfully image the Marmousi
model and the sub-salt part of the Sigsbee model.
EAGE 68th Conference & Exhibition — Vienna, Austria, 12 - 15 June 2006
Introduction
The current methodology of imaging is a velocity model building based on ray theory such as
tomographic inversion and a migration with both Kirchhoff and wave-equation migration.
Although wave-equation migration can improve on Kirchhoff imaging with the same raybased velocity model, the full potential of a more accurate migration can be reached only
when the velocity model is build using the same type of migration as the final one.
Proposed methodology
Our approach is similar to that of Peng et al (2005) although it is implemented differently. At
each iteration, the velocity model is parametrized on a coarse grid. A modulated-shot
migration is computed. This new scheme is a proprietary algorithm producing an image
equivalent to a conventional shot-record migration with the advantage of producing p-indexed
gathers at no extra cost. At each image point, the semblance of the gather is computed. The
cost function is the semblance function averaged over all image points with a user-defined
weight. The gradient of the cost function is then computed and used to feed a non-linear
conjuguate gradient algorithm. More precisely:
Velocity model parametrization
The velocity model is parametrized on a coarse grid. The velocity field is computed from the
coarse grid by a bi-orthogonal wavelet type (rather than a B-spline) interpolation together
with a user-defined mask with a predetermined "frozen" velocity. This mask can be used to
define salt bodies or to perform layer-stripping.
Modulated shot migration
The natural gather for a shot-record migration is the shot-indexed gather. Duquet et al (2001)
and Zhang et al (2003) have proposed to combine shots with linear delays in order to obtain
gathers indexed with the sine of the surface angle. We have used a different approach for
obtaining the same kind of gathers, which is the modulated-shot migration (Soubaras, 2006).
Semblance maximization
We advocate the use of "plain" semblance in favor of any other criteria. The reason behind
this choice is the theoretical justification of semblance as the unknown amplitude best match.
Let's consider the data yn and the model xn(θ) we want to match to the data. If we are unsure
of the amplitude of our model, we solve the following data-fitting problem:
N
min a ,θ C (a,θ ), C (a, θ ) = ∑ y n − ax n (θ )
2
(1)
n =1
which is, in the case where yn=1, equivalent to maximizing in θ the semblance of xn(θ).
Cost function
The cost function is the mean semblance weighted by a user-defined function. This weight
can be used to limit edge effects, to concentrate on some reflectors, or to perform layerstripping.
Synthetic data examples
The Marmousi model
We have tested our methodology on the Marmousi velocity model. We have reconstructed the
shot gathers using a one-way modeling while keeping the original geometry (240 shots of 96
receivers) and a frequency bandwidth of 5-40 Hz. The initial velocity model is a mere
constant vertical velocity gradient ranging from 1500m/s at the surface to 3000m/s at the
bottom. Figure 1 shows this initial velocity model with the associated migration. This can be
compared with the final velocity and associated migration represented in Figure 2 and the true
EAGE 68th Conference & Exhibition — Vienna, Austria, 12 - 15 June 2006
velocity and associated migration represented in Figure 3. Down to a depth of 2 km the
reconstructed image is almost perfect, with a very good definition of the faults, despite the
crude initial model, and from 2 to 3 km the focusing is as good as the original image although
the depthing is not perfect. Looking at the final velocity model of Figure 2, fine details of the
true model can be recognized. Considering that the data has a maximum offset limited to 2.5
km, the results of Figure 2 are very satisfactory. The corresponding gathers for the locations
3.84 km and 5.76 km are shown in Figure 4. We can see how far from optimal the initial
gathers are, and can check that the focusing for the final velocity is as good as for the true
velocity.
The Sigsbee model
Our methodology has also been tested on a difficult sub-salt velocity estimation problem: to
image the fault located between 40 and 45 kft and 20 and 30 kft in depth. This fault is
underneath a complex top salt. We have taken as a starting model the true velocity for the
sediments above the salt and the true salt geometry. The sub-salt velocity was again a constant
vertical gradient. The migration with the initial velocity model, estimated velocity model and
true velocity model are shown in Figure 5. The focusing is again as good with the estimated
velocity model as with the true model. The fault is very clearly defined as is the diffraction
point near the fault. Although depthing errors remain, the bottom reflector has moved much
closer to its true position. Figure 6 shows the gathers at location 42.5kft confirming that the
focusing for the estimated model is as good as the final model.
Conclusion
We have exposed a methodology for velocity model building based on a modulated-shot
wave-equation migration and mean semblance maximization of the corresponding p-indexed
gathers. This methodology gives very satisfactory results in automatically estimating the
Marmousi and Sigsbee velocity models. This technique can be used in a semi-automatic way
by making use of the frozen velocity and the weight of the semblance.
References
Shen, P., Symes, W. W., Morton, S. and Calandra, H. [2005] Differential semblance velocity
analysis via shot-profile migration. SEG 75th Annual International Meeting, Expanded
Abstracts 2249-2252.
Soubaras, R. [2006] Modulated shot migration. EAGE Annual Meeting, Vienna, 2006
(submitted)
Figure 1: Initial velocity and migration
Figure 2: Final velocity and migration
Figure 3: True velocity and migration
(a)
(b)
(c)
Figure 4: Gathers for initial velocity (a), final velocity (b), true velocity (c)
EAGE 68th Conference & Exhibition — Vienna, Austria, 12 - 15 June 2006
Figure 5: Migration with initial velocity (a), final velocity (b), true velocity (c)
Figure 6: Gathers at x=42.5kft with initial velocity(a), final velocity(b), true velocity(c)