The problem of subsurface imaging in poorly illuminated areas is an increasingly important one. Our search for hydrocarbons often concentrates on the potential reservoirs near salt bodies in the subsurface, often under deep water. The strong velocity contrast between the salt and the surrounding rock causes seismic energy to be redirected or become evanescent. This energy will not be recorded during the seismic survey, resulting in poor illumination. In such areas, the high cost of drilling wells makes accurate imaging essential, and justifies algorithms that have a high computational cost.
Subsurface imaging is typically done with a migration algorithm. In complex areas such as those around salt bodies, it is common to use some type of wave-equation depth migration. The 2-D migration algorithm used in this thesis is a downward-continuation migration and the 3-D migration algorithm is a common azimuth migration. Both generate angle-domain (offset ray parameter) common image gathers. When using these migration algorithms, poor illumination will be manifested in the common image gathers as gaps in the events. It will appear as shadow zones in the Common Reflection Point (CRP)-depth plane. It is not possible for a migration algorithm to compensate for seismic energy that has been directed outside of the survey bounds. Fortunately, it is possible to use a migration operator in a least-squares inversion to obtain a better seismic image than that obtained from migration alone.
The inversion algorithm I described in this thesis, Regularized Inversion with model Preconditioning (RIP), can be designed to improve imaging in poorly illuminated areas. RIP is a regularized least-squares inversion that uses not only the migration operator but also a regularization operator to compensate for the sudden amplitude decrease that occurs when a seismic event passes into a shadow zone. Geophysical regularization acts along the offset ray parameter (equivalent to reflection angle) axis. It can be thought of as a derivative operator that minimizes large changes in amplitude. This geophysical regularization can be combined with geological regularization, which acts along specified dips in the CRP-depth plane. The geological regularization operator is a steering filter composed of dip annihilator filters at every point in the model. The geological regularization operator is created by an interpreter from a migration result or a geological model.
The geophysical regularization acts horizontally along the offset ray parameter axis, which means that the velocity model used for the migration should be correct to get the best results. However, as long as the velocities are reasonably close so that the moveout is fairly small, geophysical RIP will still provide a better image than migration alone.
Geological RIP has its own difficulties. The geological regularization operator is dependent on an interpreter's idea of how the seismic events should behave in the CRP-depth plane. If the geological regularization conflicts with recorded data, the inversion process will reject the attempted regularization.
The regularization, by filling in the gaps caused by poor illumination in the model, can be thought of as recovering seismic energy that was directed outside of the survey bounds. To keep this energy, we must pad the dataspace so it has somewhere to go. When calculating the residual in the conjugate gradient step, this energy must be masked out or the inversion will try to suppress it. In addition to this energy, the regularization also introduces frequencies into the model that are not included in the original data. Therefore we must pad the frequencies as well. These must also be masked out during the calculation of the residual.
Regularized inversion with model preconditioning is a framework that can be used for many different problems with many different regularization schemes. Concentrating on the problem of poor illumination, it is possible that RIP could be used to jointly invert two or more 3-D datasets shot in different directions over the same subsurface volume. In this case, the illumination patterns of the different datasets would be different, allowing us to construct a regularization operator that would combine the information to produce a model with the best illumination from each dataset. RIP is very flexible and is really only limited by the creativity of the researcher.
The single greatest drawback in the use of RIP is that it is very computationally expensive when used for seismic imaging problems. Fortunately, parallel processing and improved processors and memory are making it possible to handle the enormous amounts of data now seen as commonplace in the oil exploration industry. It will soon be reasonable to use regularized inversion with model preconditioning to image complex areas where hydrocarbons are suspected.