biondo@sep.stanford.edu, sergey@sep.stanford.edu, tariq@sep.stanford.edu

## ABSTRACT
The transformation of the eikonal equation from depth
coordinates ( The traveltimes satisfying the transformed eikonal equation can be numerically evaluated by solving the associated ray tracing equations. The application of Fermat's principle leads to the expression of linear relationships between perturbations in traveltimes and perturbations in focusing velocity. This linearization, in conjunction with ray tracing, can be used for a tomographic estimation of focusing velocity. |

Velocity has a dual role in reflection-seismic imaging. It is needed to focus the data through migration, and to map the reflectors in depth by converting arrival times into depths. These two imaging goals are often conflicting. The velocity function that best focuses the data is not necessarily the velocity that performs the correct depth mapping.

The *focusing* velocity is the velocity that best
predicts the relative delays between
reflections originated at the same point in the subsurface
and recorded at different offsets and midpoints.
We can measure these relative delays and
try to estimate the focusing velocity by
solving an inverse problem.
On the contrary, the *mapping* velocity
mostly affects the absolute delays of the reflections.
If we do not know the depth of the reflectors,
we cannot estimate the mapping velocity from reflection data.
To estimate mapping velocity we need other source of information,
such as well data and a priori geological information.

This distinction between focusing and mapping velocity
is routinely used when the data it time imaged,
and is one of the source of robustness of the time-imaging procedure.
In time imaging, the data are first focused by
determining stacking and/or RMS velocities,
then map-migrated to depth along
the image rays using an appropriate mapping velocity
Hubral (1977); Larner et al. (1981).
Unfortunately this useful separation is lost when
the data are imaged using depth migration.
In this case the same velocity field is used
to focus the data and to map the reflectors.
The main reason for this shortcoming of depth migration
is not conceptual, but it is imposed by limitations
in current depth-migration algorithms.
In this paper we present a method for computing
migration operators for Kirchhoff-like migrations
as a function of the focusing velocity.
The method is based on
a coordinate transformation
from depth to two-way vertical traveltime
applied to the eikonal equation.
We demonstrate that under fairly mild assumptions
on the relation between focusing and mapping velocities,
the traveltimes computed using the transformed eikonal
are only functions of the focusing velocity.
As a result, we call the transformed eikonal equation
the *focusing eikonal*.

The focusing eikonal has potential applications to the estimation of focusing velocities. It enables us to apply to depth-migration problems the same sequential estimation of the two interval velocities that is possible in time imaging. One problem with the joint estimation of the two velocities is that a velocity perturbation not only causes changes in the focusing of the data but it also causes a vertical shift of both the reflectors and velocity that are below the perturbation. These vertical shifts present a challenge to reflection tomography, and are one of the reasons why layer-stripping procedures are considered to be more robust than global tomographic procedures. However, because global tomography has the potential to be more accurate than layer stripping, there are many incentives to stabilizing global tomography. We propose to perform the whole velocity estimation in the vertical traveltime domain. When the velocity function and the reflector geometry are expressed in the vertical-traveltime coordinates, their position is only weakly dependent on velocity. Therefore, we may ameliorate the problems associated with inaccuracies in the mapping velocity.

- Focusing eikonal equation
- Reflection tomography in
- CONCLUSIONS
- REFERENCES
- Reflections in
- About this document ...

10/10/1997