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``Focusing'' eikonal equation and global tomography

Biondo Biondi, Sergey Fomel, and Tariq Alkhalifah

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


The transformation of the eikonal equation from depth coordinates (z,x) into vertical-traveltime coordinates ($\tau,\xi$)enables the computation of reflections traveltimes independent of depth-mapping. This separation allows the focusing and mapping steps to be performed sequentially even in the presence of complex velocity functions, that otherwise would ``require'' depth migration.

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.

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