Angle-dependent image-domain least-squares migration using the amplitude-preserving migration operator — Part I: Hessian operator and imaging resolution function

This research presents an angle-dependent image-domain least-squares migration method using the amplitude-preserving migration operator. This work is split into two parts. In Part I, I derive the explicit formulas of the angle-dependent Hessian operator and imaging resolution function using the amplitude-preserving migration operator. The benefit of the amplitude-preserving migration operator is that it can improve spatial resolution and provide partial illumination compensation, compared to the adjoint migration operator with the cross-correlation imaging condition. Hence, the angle-dependent Hessian operator through the amplitude-preserving migration operator will be closer to unity, and its condition number is explicitly decreased. In addition, I clarify the relation between the angle-dependent Hessian operator and the imaging resolution function. The angle-dependent imaging resolution function can generally be assumed to be a blurring kernel localized at its spatial position. Therefore, the angle-dependent Hessian operator can be approximately reconstructed through the localized version of imaging resolution functions, which will contribute to the forward Hessian operator to efficiently simulate the angle-domain common-image gathers. Through some numerical experiments, I test the effectiveness of the Hessian operator through the imaging resolution functions and obtain two insights. Firstly, the forward Hessian operator through the imaging resolution functions can efficiently and effectively simulate the angle-domain common-image gathers and capture the accurate effects of spatial resolution, wavelet stretching, illumination, and amplitude variation observed in the migrated angle-domain common-image gathers. Thanks to the localization property of imaging resolution functions, the application of the Hessian operator to an angle-dependent reflectivity image achieves computational efficiency three orders of magnitude greater than that of forward modeling and migration operators. Secondly, the amplitude-preserving migration operator will be more suitable for constructing the angle-dependent Hessian operator and imaging resolution function, due to its higher spatial resolution and better fidelity in amplitude.