Angle-dependent image-domain least-squares migration using the amplitude-preserving migration operator, Part II: Inverse problem

Least-squares migration is a powerful technique for building high-resolution and high-fidelity seismic images in complex geological structures relative to seismic migration alone. Least-squares migration can be effectively and efficiently performed in the image domain with the help of an explicit Hessian matrix. This research introduces an angle-dependent image-domain least-squares migration method using the amplitude-preserving migration operator. This work is split into two parts. In this paper, Part II of a two-part series, I will focus on the solution of the inverse problem in the angle-dependent least-squares migration method, explain the benefits of amplitude-preserving migration operator to the inverse problem, and showcase the numerical experiments. Specifically, I have derived the alternating direction method of multipliers for the regularized least-squares migration method and compared the least-squares migration methods in terms of adjoint and amplitude-preserving migration operators. Through numerical experiments with synthetic and field data, I test the effectiveness of the proposed least-squares migration method and highlight three key benefits. First, the proposed least-squares migration method formulated in terms of the amplitude-preserving migration operator can provide a faster convergence rate and invert angle-domain common-image gathers with higher spatial resolution and better amplitude fidelity than that formulated in terms of the adjoint operator, thanks to a small condition number of the Hessian operator. Second, the proposed least-squares migration method can efficiently and effectively recover the high-resolution and high-fidelity angle-domain common-image gathers in the case of inhomogeneous migration velocity. Furthermore, the amplitude variation with the reflection angle from the proposed image-domain inversion can match the reference value in the presence of the complex overburden. Third, the angle-dependent least-squares migration method requires the simultaneous application of at least two regularization terms to effectively retrieve a high-resolution image while suppressing migration artifacts in the inverted angle-domain common-image gathers.