Least-squares migration (LSM) provides subsurface image with higher resolution and more reliable amplitude attribute by minimizing the waveform misfit between synthetic and observed dataset. However, LSM method is highly dependent on the accuracy of background velocity model. Large velocity violation causes cycle skipping problem and makes LSM get trapped into local minimum. To solve this problem, we develop LSM with extended imaging condition under the framework of extended waveform inversion. To mitigate the cycle skipping problems in LSM, we extend the reflectivity model along subsurface offset dimension. Based on Born approximation, we can derive the linearized extended Born modeling operator from acoustic constant-density wave equation. To achieve an optimal extended reflectivity model, we solve a least-squares inverse problem by minimizing the waveform misfit between observed and synthetic data using the derived modeling operator. As the reflectivity model is extended, velocity errors make reflectivity image unfocused and spread over non-zero offset, but still located in the extended image domain, so LSM with extended imaging is less prone to cycle skipping. Moreover, we introduce differential semblance optimization (DSO) operator as regularization operator in solving the inverse problem to impose subsurface image focusing at zero offset, which is reasonable when the background velocity model is not too far from realistic one. It' s also numerically demonstrated that DSO operator greatly improves the inverted image with higher resolution and more balanced amplitude in complex geological settings, especially under the salt bodies. Two models are used to demonstrate our methods. The first one is a simple two-layer model. Compared with RTM (Revers-time migration), LSRTM (Least-squares revers-time migration) not only can remove the amplitude imbalance due to uneven illumination, but also can widen the wavenumber bandwidth and increase the image resolution. After introducing extended imaging condition, LSRTM can avoid cycle skipping and fit the waveform of observed dataset very well no matter how far the background velocity model is from the realistic one. Another model we used in the paper is 2D SEG/EAGE salt model. Compared with RTM results, the reflectivity images obtained from LSRTM show higher resolution and more balanced amplitude, especially under the salt bodies. We also test the dependence of LSRTM on the background velocity model and find that it can be convergent to global minimum after introducting the extended imaging condition even when the velocity model is far from realistic one. Finally, we test the effect of DSO regularization on LSRTM. Numerical results show that it can improve the subsalt image and help focusing the image when the velocity model is unreasonable. Based on model extension and linearized Born approximation, we derive the formulas of linearized Born modeling and RTM with extended imaging condition. With these two adjoint operator, we propose LSRTM with extended imaging condition. Numerical tests on two-layer and SEG/EAGE salt models show that, (1) LSRTM can effectively improve the amplitude reliability and increase the resolution of the reflectivity image; (2) LSRTM with extended imaging condition can mitigate the high dependence of velocity perturbation inversion on the background velocity; (3) Introducing DSO constraints can help focusing of the subsurface image, which not only improve the quality of subsalt image with uneven illumination, but also mitigate the effect of velocity errors on the inverted image.
