Least-squares Kirchhoff migration with non-smooth regularisation strategy for subsurface imaging

Abstract

During the past several decades, many types of wave-equation migration methods arise for subsurface structure imaging. The classical Kirchhoff migration, however, is still widely adopted in the petroleum industry owing to its flexibility and computational efficiency. In constant density isotropic acoustic media, a basic assumption of the Kirchhoff migration is that every point of the subsurface model is supposed to be a diffractor which scatters wavefield energy to every direction, and hence collecting the scattered energy of all directions is the basic requirement for focusing the diffractor. Factors influencing the final image quality include incomplete data acquisition, multipathing from the surface to the imaging point, and insufficient illumination under complex overburden. All these factors can be theoretically taken into account in the migration weighting coefficient. However, computation of the weighting coefficient is hard work. In view of this difficulty, a fast regularising least-squares Kirchhoff migration algorithm is presented in this paper. It not only accounts for the irregular and incomplete data sampling (e.g. limited recording aperture, coarse sampling and acquisition gaps), but also compensates for the anomalous ray coverage and multipathing problem except for the shadow zone in the media. For the purpose of attenuating migration artefacts and providing a clear and accurate image of subsurface reflectivity, regularisation strategies are applied. The classical regularisation strategy may easily lead to over-regularisation or insufficient regularisation; we try to balance these two effects in this paper. The method is called the hybrid regularisation which incorporates smoothing and non-smoothing scale operators. The algorithm is implemented using a fast gradient decent solution method based on the Rayleigh quotient being used. Numerical experiments show that this hybrid regularisation method is powerful in handling the sparsity and smoothness of the model parameters.

A fast non-smooth regularisation method is proposed to solve the difficulties involved in the classical least-squares Kirchhoff migration. It not only accounts for the irregular and incomplete data sampling, but also compensates for the anomalous ray coverage and multipathing problem. Numerical experiments show that the method works well.

• fast gradient algorithm
• least-squares migration
• non-smooth regularisation

Acknowledgements

We are grateful to editor M. Tingay and two reviewers for very kind comments and helpful suggestions which greatly improved the quality of our paper. This research is supported by the National Natural Science Foundation of China under grant numbers 41325016 and 91630202. This work is also supported by the Collaboration Project of Seismic Imaging in Elastic Medium of the China National Petroleum Corporation.